Intermediate Growth Theory.pdf

Harald Wiese

Intermediate Growth Theory

10. Juli 2006

Springer-VerlagBerlin Heidelberg NewYorkLondon Paris TokyoHongKong BarcelonaBudapest

Inhaltsverzeichnis

A. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

B. Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B.2 Growth in discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3B.3 Growth in continuous time . . . . . . . . . . . . . . . . . . . . . . . . . 6

B.3.1 From discrete to continuous time . . . . . . . . . . . . . . 6B.3.2 Defining continuous-time growth rates . . . . . . . . . . 9B.3.3 Using the natural logarithm to express growth. . . 10

B.4 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13B.5 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

C. The simple Solow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17C.1 Cobb-Douglas production functions . . . . . . . . . . . . . . . . . . 17C.2 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20C.3 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22C.4 Comparative statics and the golden rule . . . . . . . . . . . . . . 26C.5 Heterogeneous population groups . . . . . . . . . . . . . . . . . . . . 28

C.5.1 Marginal product payment . . . . . . . . . . . . . . . . . . . . 28C.5.2 Dynamics and steady state . . . . . . . . . . . . . . . . . . . 30C.5.3 Labour and capital immigration . . . . . . . . . . . . . . . 31

C.6 Technological change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32C.7 Exogenous growth? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35C.8 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

D. Solow and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41D.1 Neoclassical production functions . . . . . . . . . . . . . . . . . . . 42

D.1.1 Constant returns to scale . . . . . . . . . . . . . . . . . . . . . 42

X Inhaltsverzeichnis

D.1.2 Decreasing marginal productivities and Inada con-ditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

D.2 Dynamics and steady state for neoclassical productionfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

D.3 Comparative statics and the golden rule . . . . . . . . . . . . . . 52D.4 Dynamics and steady state for AK production functions 52D.5 Conditional convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . 55D.6 Dynamics and steady state for Harrod-Domar produc-

tion functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58D.7 CES production functions . . . . . . . . . . . . . . . . . . . . . . . . . . 64

D.7.1 Elasticity of substitution . . . . . . . . . . . . . . . . . . . . . 64D.7.2 Special cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66D.7.3 Inada conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69D.7.4 Dynamics and steady state for CES production

functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72D.8 Poverty trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73D.9 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

E. Ramsey model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87E.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87E.2 Static optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87E.3 Dynamische Optimierung . . . . . . . . . . . . . . . . . . . . . . . . . . 91

E.3.1 Das Maximierungsproblem . . . . . . . . . . . . . . . . . . . . 91E.3.2 Die Hamilton-Funktion (Gegenwartswert) . . . . . . . 92E.3.3 Die Hamilton-Funktion (aktueller Wert) . . . . . . . . 94

E.4 Ein einfaches Modell - nur ein Agent . . . . . . . . . . . . . . . . . 98E.4.1 Modellbeschreibung . . . . . . . . . . . . . . . . . . . . . . . . . . 98E.4.2 Anwendung des Lösungsalgorithmus’ . . . . . . . . . . . 100E.4.3 Konkavität von u und ein Elastizitätsmaß . . . . . . 102E.4.4 Interpretation der Optimalitätsbedingung für in-

tertemporalen Konsum . . . . . . . . . . . . . . . . . . . . . . . 103E.4.5 Dynamik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

E.5 Literaturhinweise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107E.6 Lösungen zu den Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . 107

Inhaltsverzeichnis XI

F. Schumpetersche Modelle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111F.1 Kreative Zerstörung bei Schumpeter . . . . . . . . . . . . . . . . . 111F.2 Der Erfolg von Forschungsbemühungen als Poissonprozess113F.3 Intertemporale Nutzenfunktion. . . . . . . . . . . . . . . . . . . . . . 116F.4 Produktionsfunktion: Endprodukt, Zwischenprodukt und

Prozessinnovation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117F.5 Die Produktion des Zwischenproduktes . . . . . . . . . . . . . . 117F.6 Zwei Gleichgewichtsbedingungen . . . . . . . . . . . . . . . . . . . . 118F.7 Gewinnmaximierung des Prozessinnovators . . . . . . . . . . . 124F.8 Wachstumsgleichgewicht und Stationarität . . . . . . . . . . . 127F.9 Komparative Statik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

F.9.1 Gleichgewichtige Forschungsanstrengungen . . . . . . 129F.9.2 Gleichgewichtige Wachstumsrate . . . . . . . . . . . . . . . 130

F.10Die wohlfahrtsoptimale Wachstumsrate . . . . . . . . . . . . . . 131F.11Lösungen zu den Aufgaben . . . . . . . . . . . . . . . . . . . . . . . . . 135

G. Overlapping generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137G.1 Overview of the model and overlapping generations . . . . 137G.2 Saving, investment and capital . . . . . . . . . . . . . . . . . . . . . . 139G.3 Utility and optimal saving . . . . . . . . . . . . . . . . . . . . . . . . . . 139

G.3.1 Solving the intertemporal household problem . . . . 139G.3.2 Comparative statics (the savings function) . . . . . . 141

G.4 The (representative) firm . . . . . . . . . . . . . . . . . . . . . . . . . . . 143G.5 Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

G.5.1 Temporary equilibrium . . . . . . . . . . . . . . . . . . . . . . . 145G.5.2 Resource constraint and net production . . . . . . . . 147G.5.3 Intertemporal equilibrium . . . . . . . . . . . . . . . . . . . . 148

G.6 Steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151G.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

G.7.1 Logarithmic utility and Cobb-Douglas production 152G.7.2 Logarithmic utility and CES production . . . . . . . . 153

G.8 Golden rule, efficiency, and equilibrium outcome . . . . . . 155G.8.1 Definition of efficiency and Pareto optimality . . . . 155G.8.2 Golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156G.8.3 Is the golden rule equivalent to efficiency? . . . . . . 158G.8.4 Does the first welfare theorem hold? . . . . . . . . . . . 159

XII Inhaltsverzeichnis

G.9 Transfers from young to old and pension systems . . . . . . 161G.9.1 Fully funded systems . . . . . . . . . . . . . . . . . . . . . . . . . 161G.9.2 Pay-as-you-go systems . . . . . . . . . . . . . . . . . . . . . . . . 161

G.10Ricardian equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164G.11Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

H. Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169H.1 Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169H.2 Logarithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169H.3 Rule of de l’Hospital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170H.4 Rules of differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

H.4.1 Product rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170H.4.2 Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

H.5 Rules of integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171H.5.1 Partial differentiation . . . . . . . . . . . . . . . . . . . . . . . . 171

Literaturverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Für Corinna, Ben, Jasper und Samuel

Preface and Acknowledgments

This is a manual on growth theory. It is certainly not better than manyof the very good textbooks around. Indeed, I learnt a lot from Aghi-on/Howitt (1998), Barro/Sala-i-Martin (1999), Romer (1996),and de la Croix/Michel (2002). As in these books, I will to de-al with the so-called endogenous growth theory, with Schumpeteriangrowth theory and with overlapping generations.In contrast to these and other books, I will try to aim at an in-

termediate level, hence the title Intermediate Growth Theory. Sincemany students will not be able to cope with the rather sophisticatedlevel of mathematics normally found in books on growth theory, I triedmy best to guide the student into the difficult matter, to offer manyexercises (with solutions to be found at the end of each chapter) and,especially in the beginning, to state the obvious. Also, an appendixsummarizes important formulae.This book has profited from many readers who freely gave their

criticism and suggestions. In particular, I like to thank MarkusWagner,Lothar Tröger,

A. Introduction

Growth theory examines relationships between central macroeconomicvariables, such as output, capital, consumption, and investment. Moreadvanced models also treat innovation, unemployment, learning bydoing, education, foreign trade, business cycles, pensions, or publicdebts. Obviously, growth theory is concerned with most topics thatshould be of interest to students of economics and to politicians alike.Growth theory helps to address the following questions:

— How do different growth rate determine output over time?— Which growth rate do we need to double output in 10 years?— What is the relationship between the growth rate of output per headand growth rate of the population?

— Can a pay-as-you-go pension scheme be transformed into a fullyfunded system?

— Under what circumstances does the Ricardian equivalence hold?— What effect do learning by doing and formal education have on grow-th?

B. Growth

B.1 Introduction

In this chapter, we deal with growth rates, in discrete as well as incontinuous time.

B.2 Growth in discrete time

We take some economic (or other) variable y whose evolution we wantto consider. By yt we denote the value of y at time t. Normally, we havet = 0, 1, ... Our definition of a growth rate in discrete time presupposessome given time interval, for example a year or a month.

Definition B.2.1. The discrete-time growth rate of y is defined by

γh1iy :=yt+1 − yt

yt.

Growth rates are often denoted by γ (”gamma”), the Greek letterfor g. Superscript h1i refers to the full time interval, a year, say. ”:=”indicates that γh1iy (on the :-side) is defined to be equal to yt+1−yt

yt(on

the =-side). Note that γh1iy does not carry the time index. Sometimes,this will mean that γh1iy is constant over time, but at other times, theauthor is just to lazy to write down the time index or does not wantto bother the reader with too much notational garbage.

Exercise B.2.1. What are the growth rates of xt, yt, and zt, givenby

xt : = t,

yt : = t+ 4 and

zt : = 100t?

4 B. Growth

Multiplyingyt

by the growth factor

1 + γh1iy = 1 +yt+1 − yt

yt

yields, at the end of a year,

yt

µ1 +

yt+1 − ytyt

¶= yt+1.

t years later, a given y0 (y at time 0) has become

yt = y0

³1 + γh1iy

´t. (B.1)

For example, if you take Euro 100,- to the bank to earn an interest ofr = 5

100 = 5%, at the end of five years, you collect

100

µ1 +

5

100

¶5≈ 100 · 1.276 = 127.6.

ApproximationsIn growth theory, yt often denotes income per head at time t, i.e.,

yt =YtLt,

where Yt is the income and Lt the labor force, both at time t. Onewould, of course, think that the growth rates of y, Y and L are closelyconnected. Indeed, we obtain

yt+1 − ytyt

=

Yt+1Lt+1

− YtLt

YtLt

=

Yt+1Lt+1

− YtLt

YtLt

Yt+1−YtYt

− Lt+1−LtLt

µYt+1 − Yt

Yt− Lt+1 − Lt

Lt

¶

=

Yt+1Lt−YtLt+1Lt+1Lt

YtLt

Yt+1Lt−YtLt−(Lt+1Yt−LtYt)YtLt

µYt+1 − Yt

Yt− Lt+1 − Lt

Lt

¶

B.2 Growth in discrete time 5

=

1Lt+1Lt

YtLt

1YtLt

µYt+1 − Yt

Yt− Lt+1 − Lt

Lt

¶

=

1Lt+1Yt1

YtLt

µYt+1 − Yt

Yt− Lt+1 − Lt

Lt

¶=

LtLt+1

µYt+1 − Yt

Yt− Lt+1 − Lt

Lt

¶.

Thus, the growth rate of

y =Y

Lis close to the growth rate of Y minus the growth rate of L if Lt isclose to Lt+1,

yt+1 − ytyt

≈ Yt+1 − YtYt

− Lt+1 − LtLt

.

Very much the same holds for the product of two variables. Let usconsider the production function

Yt = LtKt,

which supposes that income Yt is the product (in mathematical terms)of labor Lt and capital Kt. We have

Lt+1 − LtLt

+Kt+1 −Kt

Kt

=(Lt+1 − Lt)Kt + (Kt+1 −Kt)Lt

LtKt

=Lt+1Kt+1 − Lt+1Kt+1 + (Lt+1 − Lt)Kt + (Kt+1 −Kt)Lt

LtKt

=Lt+1Kt+1 − LtKt − Lt+1Kt+1 + Lt+1Kt − LtKt +Kt+1Lt

LtKt

=Yt+1 − Yt

Yt+Lt (Kt+1 −Kt)− Lt+1 (Kt+1 −Kt)

LtKt

≈ Yt+1 − YtYt

.

The growth rate of the product of two variables is approximately equalto the sum of the growth rates of the two factors. Now, if the timeintervals are ”very small”, both approximations are very good. Indeed,if we define growth in continuous time, they will turn out to be exact.

6 B. Growth

B.3 Growth in continuous time

B.3.1 From discrete to continuous time

Let us consider half-yearly instead of yearly growth rates. For example,the bank may pay out interest every six months. To make up for thehalf-yearly interest payment, the interest rate is halfed. Instead of thegrowth factor ³

1 + γh1i´t

for the yearly growth rate γh1iy , we have the growth factor⎛⎝Ã1 + γh1i

2

!2⎞⎠t = Ã1 + γh1i

2

!2t.

for the half-yearly growth rate γh1i

2 .

In this textbook, we invite the reader to pause and think under theheading of ”food for though”. Trying an answer to these questions willhelp the reader to stay awake and prepare him for the next steps.

Food for thought. Would you prefer an interest payment of γh1i

2 ,

two times a year, to an interest rate of γh1i, paid out only once a year?

Since we earn interest on the interest, these two factors are notequal: Ã

1 +γh1i

2

!2t>³1 + γh1i

´t.

We now look for a growth rate that makes the investor indifferentbetween half-yearly payments and yearly payments. That is, we defineγh 12i implicitly by Ã

1 +γh

12i2

!2t=³1 + γh1i

´t.

Food for thought. Would you expect γh 12i > γh1i or γh 12i < γh1i?

B.3 Growth in continuous time 7

From Ã1 +

γh 12i2

!2t=³1 + γh1i

´twe obtain

1 +γh

12i2

=

⎛⎝Ã1 + γh12i2

!2t⎞⎠ 12t

=

µ³1 + γh1i

´t¶ 12t

=³1 + γh1i

´ 12

and thenγh 12i = −2 + 2

q1 + γh1i.

We can now conclude

γh1i > 0

⇒³γh1i

´2> 0

⇒³γh1i

´2+ 4 (1 + γ) > 4 (1 + γ)

⇒³γh1i + 2

´2> 4 (1 + γ)

⇒ γh1i + 2 > 2p1 + γ

⇒ γh1i > −2 + 2p1 + γ = γh

12i

We now decrease the time interval even further. Generally, we con-sider an interest payment n times a year with interest rate γh1i/n.

Then, at the end of t years, we obtainÃÃ1 +

γh1i

n

!n!t

=

Ã1 +

γh1i

n

!nt.

It can be shown (but we will not do that here) that this growth fac-

tor is an increasing function of n. The sequenceµ³1 + γh1i

n

ńt¶n∈N

converges (gets closer and closer to some value) and we have

limn→∞

Ã1 +

γh1i

n

!nt= eγ

h1it.

8 B. Growth

81

6

4

2

x

1−x

xln

Abbildung B.1. The natural logarithm

Again, because of the interest on the interest, one prefers to obtaincontinuous interest payments. Analogous to γh 12i, we are now lookingfor γh0i, which is the rate at which indifference to a yearly interest rateobtains:

eγh0it =

³1 + γh1i

´t.

Applying the natural logarithm on both sides and deviding by t yields

γh0i = ln³1 + γh1i

´(B.2)

We would like to confirm γh0i < γh1i. Indeed, it is well-known that

lnx < x− 1 for x > 0, x 6= 1

holds. In fig. B.1, the reader can see the logarithm which cuts theabscisse at x = 1. So does x− 1. Replacing x by 1 + y, we obtain

ln (1 + y) < y for y > −1, y 6= 0

from where we find the desired inequality:

γh0i = ln³1 + γh1i

´< γh1i for γh1i > −1, γh1i 6= 0

The growth rates γh1i and γh0i are close for small rates, as can beseen from the following table:


γh1i γh0i (approximation)0, 001 (one-tenth of a percent) 0, 0009995

0, 01 (one percent) 0, 0099503

0, 1 (10 percent) 0, 09531

0, 2 (20 percent) 0, 18232

0, 3 (30 percent) 0, 26236

B.3.2 Defining continuous-time growth rates

In discrete time, the growth rate of y is defined by

γh∆tiy :=

yt+∆t−yt(t+∆t)−tyt

.

Taking the limit with respect to ∆t yields

lim∆t→0

γh∆tiy = lim∆t→0

yt+∆t−yt(t+∆t)−tyt

= lim∆t→0

∆yt∆t

yt

=dytdt

yt.

Definition B.3.1. The continuous-time growth rate of y is defined by

γy := γy,t :=dytdt

yt

where the time index is often suppressed.

It is important for the reader to understand that yt is just anotherway to write y (t) , i.e., we have a function y which takes one argument,t. Therefore, we could as well have written dy(t)

dt or dydt instead ofdytdt .

Writing time as an index rather than a functional argument is theusual procedure in growth theory.Assuming a constant growth rate g, we obtain

g =dytdt

yt,

10 B. Growth

which does not define γy,t (as in the definition above) but claims thatthe growth rate of y, i.e., dytdt /yt, is equal to the constant g. This isa differential equation, i.e., an equation that contains a function y(with argument t) together with the first (or higher order) derivativeof y. ”Solving a differential equation” means to state the function yexplicitly. In this case, exponential growth given by

yt = y0egt (B.3)

does the trick.

Exercise B.3.1. Calculate dytdt /yt for yt = y0e

gt. Hint: the derivativeof ex is ex, but do not forget the chain rule.

The upshot of this exercise is γy = g so that we can (and will) write

yt = y0eγyt.

This is the first of many exercises in this textbook. The reader willfirst solve the exercises for himself and then (and only then!) refer tothe solutions to check his results.If y stands for capital, it is also possible to interpret γy as the rate

of return on capital, i.e., the gain in capital over the capital used toproduce that gain.

B.3.3 Using the natural logarithm to express growth

In analyzing the growth of some x, it is sometimes expedient not toconsider and draw

yt

directly, but rather take recourse to

byt = ln yt.The reason is this: The derivative of by with respect to t is equal to thegrowth rate of y. To see this, note d lnx

dx = 1x . We obtain


dbydt

=d ln ytdt

=1

yt

dy

dt(chain rule!)

=

·ytyt

Therefore, if by is plotted against t, the growth rate of y can be seendirectly from the slope of the by-graph.Lemma B.3.1. The growth rate of y is given by

·ytyt=d ln ytdt

.

Exercise B.3.2. Try to find the relationship between the (continuous-time) growth rates of Y, K and L for Yt = LtKt. Hint: apply theproduct rule of differentiation and use ln (LK) = lnL+ lnK.

Of course, the solution to our exercise cannot surprise the readerwho has gone through section B.2 on pp. 4. Also, for

y =Y

L

we findγy = γY − γL.

To sum up, in continuous time we obtain:

— The growth rate of a product is equal to the sum of the growth ratesof its factors.

— The growth rate of a ratio is equal to the difference of the growthrates of nominator and denominator.

An application concerns the relationship between the monetary in-terest rate and the real interest rate. By r we denote the monetaryinterest rate which is the growth rate for an asset K, a deposit in abank or a government bond. If π denotes the rate of inflation, the realinterest rate is given by r − π. Let us explain, why.

12 B. Growth

Now, if we denote the price level by P and the real capital (in abank, say) by Kr, we have

Kr :=K

P

and, applying the above rule, the real interest rate is given by

γKr= γK − γP = r − π.

Exercise B.3.3. Assuming a constant growth rate, apply the naturallogarithm to the exponential-growth formula

yt = y0eγyt

in order to confirm

γy =ln yt − ln y0t− 0 =

1

tlnyty0.

Hint: lnx is the inverse of ex, i.e., ln ex = x.

Finally, we can use the natural logarithm to justify a handy rule ofthumb. According to this rule the number of years needed to doublesome variable y is approximately equal to

70

γy · 100.

For example, if you take some money to the bank and you get aninterest rate of 2%, you need 35 years to double your capital.Inversely, in order to achieve a doubling in t years, a growth rate

(in percentage points) of70

t

is needed. If you hope to double your capital within 10 years, you haveto ask for an interest rate of 7%.The confirmation of this rule is not difficult. We are looking for the

growth rate γy and/or the time span needed to double y, i.e., we needto solve

y0eγyt = 2y0.

B.4 Convergence 13

Deviding by y0 and taking the logarithm leads to

γyt = ln¡eγyt

¢= ln 2 ≈ 0, 69315.

Solving for t or γy, we obtain

t ≈ 70

γy · 100

andγy · 100 ≈

70

t,

respectively. This approximation formula yields the following table:

Growth ratein percentage points

Years neededfor doubling(approximation)

Years neededfor doubling(correct,continuous time)

Years neededfor doubling(correct,yearly interest)

0.1 (one-tenth of a percent) 700 ≈ 693.15 ≈ 693.491 (one percent) 70 ≈ 69.31 ≈ 69.6610 (ten percent) 7 ≈ 6.93 ≈ 7.2720 (twenty percent) 312 ≈ 3.46 ≈ 3.8030 (thirty percent) 213 ≈ 2.31 ≈ 2.64

The reader will note that we used two approximations for this formula.First, we have ln 2 instead of 0.7 (no problem), second, we use thecontinuous-time growth rate instead of the more usual yearly one.

B.4 Convergence

One of the central questions of growth theory is whether or not diffe-rent economies converge over time. In this section, we will stress theneed to distinguish between weak and strong convergence. Considertwo variables xt and yt with 0 < x0 < y0. Weak convergence meansthat x grows faster than y. Alternatively, the ratio of y over x decreasesin time. Put formally:

Definition B.4.1. Weak convergence between xt and yt is said to holdif, whenever 0 < x0 < y0, the growth rates obey γx > γy for all t ≥ 0.

14 B. Growth

Lemma B.4.1. Weak convergence between xt and yt hold iff, whene-

ver 0 < x0 < y0,dytxtdt < 0.

”Iff” is short for ”if and only if”. Differently put, the lemma provi-des a criterium for weak convergence. The proof is easy:

d yxdt< 0

⇔dydt x−

dxdt y

x2< 0

⇔dydt

x−

dxdt

x

y

x< 0

⇔dydt

y−

dxdt

x< 0 (multiply by

x

y)

⇔ γy < γx

Weak convergence may hold even if x and y never get close. Forexample, weak convergence exists between

xt = t and

yt = 2t+ 2.

Exercise B.4.1. Show that weak convergence holds between xt andyt.

Strong convergence requests that the two variables do indeed getcloser and closer.

Definition B.4.2. Strong convergence between xt and yt is said tohold if weak convergence between xt and yt holds and if

limt→∞

ytxt= 1.

Exercise B.4.2. Show that strong convergence does not hold betweenxt = t and yt = 2t+ 2.

B.5 Solutions

B.2.1. We obtain

B.5 Solutions 15

xt+1 − xtxt

=t+ 1− t

t=1

t

yt+1 − ytyt

=t+ 5− (t+ 4)

t+ 4=

1

t+ 4<1

tand

zt+1 − ztyt

=100 (t+ 1)− 100t

100t=1

t

B.3.1. You have found

dytdt

yt=

d(y0egt)dt

y0egt

=y0e

gtγyy0egt

= g.

If another result were correct, our solution to the differential equationwould have been false.

B.3.2. Using the original definition, we obtain

γY =

·YtYt=

d(KtLt)dt

KtLt

=dKtdt Lt +

dLtdt Kt

KtLt

=dKtdt

Kt+

dLtdt

Lt= γK + γL.

Using the logarithm, we have

γY =d lnYtdt

=d ln (KtLt)

dt

=d (lnKt + lnLt)

dt

=d lnKtdt

+d lnLtdt

= γK + γL.

16 B. Growth

B.3.3. Applying the natural logarithm on both sides of the equationyields

ln yt = ln y0 + ln eγyt

= ln y0 + γyt

and hence

γy =ln yt − ln y0t− 0 =

ln yty0t− 0 .

B.4.1. Obviously, y0 > x0. Now,

γy =2

2t+ 2

=1

t+ 1(multiply by

1/2

1/2)

<1

t= γx

B.4.2. While x and y converge in a weak sense, they do not in a strongsense:

limt→∞

2t+ 2

t

= limt→∞

µ2 +

2

t

¶= 2 + lim

t→∞2

t= 2 > 1

ytxtdecreases (by weak convergence), but yt > 2xt for all t.

C. The simple Solow model

The standard Solow (1956) model uses a constant-returns productionfunction in order to trace the capital-per-head trajectory in terms ofthe rate of saving (s), the depreciation rate (δ), the growth rate ofthe (working) population (n) and the initial capital per head (k0). Inthis chapter we will present the Solow model on the basis of a Cobb-Douglas production function.This chapter will guide the reader

— through the dynamics of the Solow model for Cobb-Douglas produc-tion functions,

— to the equilibrium concept employed by growth theorists,— to an elementary discussion of technological progress, and— to an understanding of the terms ”exogenous” versus ”endogenous”growth.

C.1 Cobb-Douglas production functions

The Cobb-Douglas production function F is given by

Y = F (K,L) = AKαL1−α, A > 0, 0 < α < 1.

Y is total output, K and L denote the amount of capital and laborthat enter into production, and A is a technological coefficient. TheCobb-Douglas (for short, CD) production function exhibits a numberof interesting properties. First of all, it shows constant returns to scale.

Definition C.1.1. A production function F exhibits constant returnsto scale, if we have

18 C. The simple Solow model

F (λK,λL) = λF (K,L) ,K ≥ 0, L ≥ 0

for any λ ≥ 0.

Exercise C.1.1. Can you prove that the CD production function isconstant-returns? Hint: you will use (a1a2)

b = ab1ab2 and a

bac = ab+c.

Second, the marginal productivity of each factor is positive anddecreasing. The more capital (or labor) we employ, the higher theoutput, but the additional output of additional input of capital getssmaller and smaller. Indeed, for L > 0, we obtain

∂F

∂K= AαKα−1L1−α

= AαL1−α

K1−α

= Aα

µL

K

¶1−α> 0

and it is easy to see that the marginal product of capital decreases withincreasing K. In other words: F is a concave function of K (and of L,too). In expressing the marginal product of capital, we have written∂F∂K instead of dFdK , because F carries two arguments, K and L, and weneed to apply the partial derivative with respect to K, while holdingL constant.CD production functions feature positive and decreasing marginal

productivities in an extreme fashion. On one hand, if we keep on in-creasing K, the marginal product finally becomes zero,

limK→∞

∂F

∂K= 0.

It cannot decrease any further. On the other hand, if we let K vanish,the marginal product becomes very high,

limK→0

∂F

∂K=∞.

This two properties are called Inada conditions.Third, the production elasticity of capital is constant and equal to

α while the production elasticity of labor is equal to 1 − α. Thus, a

C.1 Cobb-Douglas production functions 19

one percent change in the quantity of capital (labor) results in an α

percent (1 − α percent) change in output. Formally, the productionelasticity of capital is given by

εY,K =∂YY∂KK

=∂Y

∂K

K

Y.

Exercise C.1.2. Can you confirm that the production elasticity ofcapital is equal to α?

We will later (section C.5) make use of a fourth interesting propertyof CD production functions. Assume that factors of production arepaid their marginal product. Then, factor payments equal output. Thisis the exhaustion theorem, due to Euler.

Exercise C.1.3. Prove Euler’s theorem for CD production functions:

∂F

∂K·K +

∂F

∂L· L = F (K,L) .

For now, we will assume that Y, K, and L may change over time,while A and α are constant. However, in section C.6 below, we willconsider technological change by making A dependent on t.In the last chapter, we learnt about growth rates. For a function

yt, they are defined by·ytytord ln ytdt

.

Exercise C.1.4. Assuming a CD production function, show how thegrowth rate of output depends on the growth rates of capital andlabor. Hint: you will use the product and chain rule of differentiation(first growth-rate definition) or the rules for manipulating the naturallogarithm (second growth-rate definition).

In growth theory, we are often concerned with per-head variables.In particular, per-head consumption might be taken as a measure ofwelfare. CD production functions are very suitable in this context be-cause they allow to express per-head output

y :=Y

L


as a function of capital per head,

k :=K

L.

Indeed, we find

y =AKαL1−α

L= A

Kα

Lα= Akα =: f (k) .

f is called the production function in intensive form. Its (one) argu-ment is capital per head.

C.2 Dynamics

We are now ready to introduce the Solow growth model for CD pro-duction functions. At every point in time, output is devided betweenconsumption and investment. We assume that output can be used forboth purposes. For example, animals such as cows (output) can beslaughtered and eaten (consumption) or used to produce additionalanimals.In the standard Solow model, one works with the plausible con-

sumption functionC := (1− s)Y

where C is overall consumption and s ≥ 0 the constant saving rate.This is the behaviorist tradition. Alternatively, and closer to micro-economics, one assumes a representative agent who chooses his con-sumption path over his whole life time. We take up this optimizingtradition in chapter ??.Pursuing the behaviorist tradition, per-head consumption is given

by

c :=C

L= (1− s) Y

L= (1− s) y.

Now, we can put down the changes in the stock of capital:

·K= sY − δK (C.1)

where sY is the income’s share not consumed and hence invested, andδ is the constant depreciation rate. Eq. C.1 is based on the assumption

C.2 Dynamics 21

that savings and investments are equal so that s can be addressed asthe rate of savings or the rate of gross investments.Since per-head output y depends on per-head capital endowment

k, we are interested in knowing the dynamics of capital per head. Incase of zero investments, capital endowments per head decrease for tworeasons. First, depreciation reduces the amount of capital available inour economy. Second, if the population grows, per-head endowment ofcapital is reduced even if overall capital stays constant. Our calcula-tions will show the increase of capital needed in order to make up forthese two effects.We proceed in two steps. First, we apply the quotient rule:

·k =

·µK

L

¶=

·K L−

·L K

L2

=

·K

L−

·L

L

K

L

=

·K

L− nk

and find how the change in per-head capital depends on the change of

capital·K, labor supply L, the growth rate of labor n := γL and the

capital per head k.Second, we insert eq. C.1 and find

·k =

sY − δK

L− nk

= sY

L− δ

K

L− nk

= sAkα − (δ + n) k.

By deviding with k > 0, we obtain the growth rate of per-head capital

γk =

·k

k=

sA

k1−α− (δ + n) . (C.2)

Capital per head increases if the actual investment per head

sY

L


lies above the break-even investment per head

(δ + n) k.

Note that the break-even investment per head reflects the two effectsdiscussed above.Let us consider two economies that differ only in capital endowment

per head but have equal parameters s, δ, n and A. Then, the capital-poor economy (measured in capital per head) will witness a greatergrowth of capital per head. This can easily be seen from eq. C.2. Thisis the property of β-convergence that we will look at in greater detailin chapter D.

C.3 Steady state

Apart from the dynamics, we are interested in wether our variableswill settle down in the long run. If they do, a so-called steady statehas been achieved.

Definition C.3.1. A steady state is a tuple of relevant economic va-riables that grow at constant rates.

The reader will note that this equilibrium concept is very differentfrom microeconomic equilibrium concepts. Indeed, the definition doesnot refer to any economic actors that have preferences, endowmentsand actions or strategies. So far, growth theory is devoid of preferences,optimization, and other ingredients typical of economic theory. We willturn to a more actor-based growth theory in later chapters and, in avery restrictive manner, in the next section.For the Solow model, one might consider the tuples (Y,K,L) oder

(Y, y, k, L) . Since the per-head capital endowment is a central variable,

a steady state implies that·kk and hence (see eq. C.2)

sA

k1−α− (δ + n)

is constant. However, since s, δ and n are constant, this term, can beconstant only if k does not change. Formally, this can be seen fromd

µsA

k1−αt

−(δ+n)¶

dt = 0.

C.3 Steady state 23

Indeed, we obtain

d¡sAk1−α − (δ + n)

¢dt

=d¡sAk−1+α

¢dt

= sA (−1 + α) k−2+αdk

dt

= sA (−1 + α)1

k2−αdk

dt

which is equal to zero

— for s = 0, in which case the (constant!) growth rate of per-headcapital is equal to − (δ + n) and per-head capital approaches zero,or

— for dkdt = 0, so that the growth rate of per-head capital is zero and

the steady state is characterized by

sA

(k∗)1−α= δ + n, (C.3)

sA (k∗)α = (δ + n) k∗, or (C.4)

k∗ =

µsA

δ + n

¶ 11−α

. (C.5)

In the steady state, k∗ is constant and so are

y∗ = f (k∗) = A (k∗)α = A

µsA

δ + n

¶ α1−α

and

c∗ = (1− s)AµsA

δ + n

¶ α1−α

.

On the other hand, K, Y, and C grow at rate n.

Exercise C.3.1. Show this. Hint: remember K = kL, Y = yL, andC = cL.

The dynamics and the steady state can be visualized as in fig. C.1and C.2. In both figures, k is the abscisse variable. The first depictsthe change in per-head capital, the second the growth rate.Both figures suggest that the per-head endowment of capital incre-

ases as long as it is smaller than the steady-state value. This can also

be shown algebraically: 0 < k < k∗ =³sAδ+n

´ 11−α

implies


kn)( +δ

( ) αsAkksf =

*k k

Break-even investment

Actual investment

Inve

stm

ent p

er h

ead

Abbildung C.1. Break-even versus actual investment

k0

n+δ

( )α−= 1k

sAkkfs

0<kγ

0>kγ

*k

Abbildung C.2. Positive and negative growth rates

C.3 Steady state 25

γk =sA

k1−α− (δ + n)

>sA

(k∗)1−α− (δ + n)

=sAµ³

sAδ+n

´ 11−α¶1−α − (δ + n)

= 0.

Inversely, capital per head goes down if it is larger than the steady-state value. In this sense, the steady state at k∗ is stable.Another steady state exists at k = 0. Here, output is zero and

investment, too. However, for a small k > 0 capital per head increases(if s > 0 holds) and converges towards k∗.Both figures show how changes in s, δ, n, and α influence the steady-

state capital per head. This is the topic of the next section.An alternative way to obtain the steady state is to solve the diffe-

rential equation.k= sAk

α − (n+ δ)k.

The solution is

kt =

µsA

n+ δ+

µk1−α0 − sA

n+ δ

¶e−(1−α)(n+δ)t

¶ 11−α

.

Exercise C.3.2. Can you confirm that the solution is correct? Youneed to form the time derivative of kt and find the above differentialequation. Hint: this is not easy, so this time you will be pardoned fornot trying for yourself.

Now, by letting the time index go towards infinity, we find

limt→∞

e−(1−α)(n+δ)t = limt→∞

1

e(1−α)(n+δ)t= 0

and see that kt converges towards its steady state:

limt→∞

kt

=

µsA

n+ δ+

µk1−α0 − sA

n+ δ

¶limt→∞

e−(1−α)(n+δ)t¶ 1

1−α


=

µsA

n+ δ+

µk1−α0 − sA

n+ δ

¶· 0¶ 1

1−α

=

µsA

n+ δ

¶ 11−α

= k∗.

C.4 Comparative statics and the golden rule

Comparative statics means: How do the (exogenous) parameters of ourmodel influence the (endogenous) variables? In the Solow model, theparameters are s, δ, n, and α. The central variable is capital per head,

k∗ =

µsA

δ + n

¶ 11−α

.

It is a positive function of

— the production elasticity of capital α,— the saving rate s, and— the technological parameter A

but depends negatively on

— the rate of depreciation δ and— the growth rate of the population n.

It is important to note that a change in these parameters doesindeed change capital per head, but does not change the growth ratesof the most important variables (which are 0 and n, respectively).With respect to developing countries, Robert Solow posed the fa-

mous question: ”Why are we so rich and they so poor?” His modeldoes provide a partial answer. Consumption per head may be taken asan indicator of richness. Indeed, looking at

c∗ = (1− s)AµsA

δ + n

¶ α1−α

,

we find that a high rate of depreciation (no repair of public and priva-te capital) and a high rate of population growth depress consumption

C.4 Comparative statics and the golden rule 27

per head. Also, a high production elasticity of capital increases con-sumption because it increases capital per head and output per head.However, while saving increases capital per head, it does not unam-biguously increase consumption per head. On one hand, increasing sdecreases consumption immediately, on the other hand, increasing sleads to a higher per-head capital and income.This brings us to the question of which saving rate is optimal, where

optimality is defined in terms of steady-state consumption per headc∗. Setting the derivative of

c∗ = (1− s) y∗

= A (k∗ (s))α − sA (k∗ (s))α

= A (k∗ (s))α − (δ + n) k∗ (eq. C.4)

with respect to s equal to zero yields

Aα (k∗ (s))α−1dk∗

ds− (δ + n) dk

∗

ds= 0,

hence

kgold!=

µαA

δ + n

¶ 11−α

.

kgold is the capital per head in the steady state that maximizes steady-state consumption per head. The optimal capital per head is positi-ve function of α (the production elasticity of capital) and a negativefunction of both δ (depreciation rate) and n (rate of growth of thepopulation). Comparing,

kgold!=

µαA

δ + n

¶ 11−α

and k∗ (s) =µsA

δ + n

¶ 11−α

yields

sgold!= α.

Here, the exclamation marks ! expresses the fact that kgold and sgoldare the result of an optimization. A capital endowment per head abovekgold is ”dynamically inefficient”, i.e., it is possible to consume morein every period by saving less.


goldk k0

kn)( +δ

goldc

( ) αAkkf =

( ) αAkskfs goldgold =

n+= δslope

Abbildung C.3. The golden rule of capital accumulation

Fig. C.3 helps to visualize the golden rule. sgold leads to a steady-state capital per head kgold and a steady-state consumption per headcgold. Note that steady-state consumption c∗ is the difference of

A (k∗ (s))α and (δ + n) k∗.

Hence, in order to maximize this difference, the slope of A (k∗ (s))α

has to equal the slope of (δ + n) k∗ which is δ + n.

C.5 Heterogeneous population groups

C.5.1 Marginal product payment

We will present a Solow-type model where the population is dividedin m groups which differ in s, δ, n, and k0. We assume that factors arepaid their marginal products. We then obtain interdependent dynamicsfor the m groups.This model will help to answer the following question: Under what

circumstances will (capital and/or labor immigration) be welcomed

C.5 Heterogeneous population groups 29

by the incumbent groups? We have two results. First, immigration ofboth labor and capital (capital imports) will be welcomed only for in-creasing returns to scale. Second, the relative capital richness of theincumbent groups are decisive for their attitude towards immigration.In particular, capital-rich groups are more welcoming towards laborimmigration and capital-poor groups are more welcoming towards ca-pital immigration.The size of the population, or labor force, of group i, i = 1, ...,m is

denoted by Lit ≥ 0. Them groups grow at rates n1, ..., nm, respectively.Therefore, at time t, population sizes are

Lit = Li0enit, i = 1, ...,m

for the m groups and

Lt :=mXi=1

Lit

for the whole population.The population groups are differentiated with respect to the capital

they own. Let kit be the amount of capital owned by a worker of groupi at time t. The amount of capital at time t is equal to

Kit : = kitLit for group i, and

Kt : =mXi=1

kitLit for all groups together.

Paying the factors according to their marginal product yields

yit =∂F (Kt, Lt)

∂Kt· kit +

∂F (Kt, Lt)

∂Lt· 1

= A

"α

µLtKt

¶1−αkit + (1− α)

µKtLt

¶α#

for each worker in group i at time t and

Yit = Lityit = A

"α

µLtKt

¶1−αKit + (1− α)

µKtLt

¶α

Lit

#

for the whole group.


C.5.2 Dynamics and steady state

Taking depreciation into account, the capital stock of group i developsin accordance with

d (Litkit)

dt=dKitdt

= siYit − δiKit.

Now, because of

d (Litkit)

dt=dLitdtkit +

dkitdtLit

we obtaindkitdt

=d(Litkit)

dt

Lit−

dLitdt

Litkit

and then, inserting the first equation,

dkitdt

=siYit − δiKit

Lit−

dLitdt

Litkit

= siyit − kit (δi + ni)

= siA

"α

µLtKt

¶1−αkit + (1− α)

µKtLt

¶α#− kit (δi + ni)

and hence·kitkit= siA

"α

µLtKt

¶1−α+ (1− α)

µKtLt

¶α

· 1kit

#− (δi + ni) .

Note that m = 1 (dropping the i index) yields the Solow equation

·ktkt=

sA

k1−αt

− (δ + n) .

Exercise C.5.1. Show this!

For two groups i and j with kit > kjt, we obtain the difference ingrowth rates of capital per head

·kitkit−

·kjtkjt

= (si − sj)AαµLtKt

¶1−α+

µsikit− sjkjt

¶A (1− α)

µKtLt

¶α

− (δi + ni) + (δj + nj)

C.5 Heterogeneous population groups 31

We now turn to the special case of equal rates of growth for both sectorsof the population, n := ni = nj , equal depreciation rates, δ := δi = δj ,

and equal rates of saving, s := si = sj . Then, we have convergence ofgrowth rates in capital per head in the following sense: If one grouphas a higher endowment of capital per head, this group’s capital perhead will grow at a slower rate. To see this, note that kit > kjt impliesskit< s

kjt.

C.5.3 Labour and capital immigration

Let us now examine the question of whether incumbent agents (ofgroup i) welcome additional groups, via immigration. To fix ideas, weassume that a ”small” group m+ 1 joins the economy.

Definition C.5.1. We have labor immigration into an economy attime t, if Lm+1,t > 0 and km+1,t = 0 hold. We have capital immigration(capital imports) into an economy at time t, if Lm+1,t = 0, km+1,t > 0hold.

Note that capital imports imply that Ym+1,t is paid to agents out-side our economy. For practical purposes, however, there is no harmin assuming some very small population Lm+1,t > 0 earning Ym+1,t.

Definition C.5.2. Group i is said to be welcoming towards labor (ca-pital) immigration (at time t), if dYitdLt

> 0 (dYitdKt> 0) holds.

Since consumption is defined by Cit := (1− si)Yit, ”welcoming”could equivalently be defined with respect to consumption.Above, we found,

Yit = Lityit = A£αL1−αt Kα−1

t Kit + (1− α)L−αt Kαt Lit

¤.

Therefore,

dYitdKt

= A£α (α− 1)L1−αt Kα−2

t Kit + α (1− α)L−αt Kα−1t Lit

¤and


dYitdKt

> 0

⇔ α (1− α)L−αt Kα−1t Lit > α (1− α)L1−αt Kα−2

t Kit

⇔ Lit > L1tK

−1t Kit

⇔ KtLt> kit

Exercise C.5.2. Can you confirm

dYitdLt

> 0⇔ KtLt< kit?

Proposition C.5.1. Considering immigration at time t into an eco-nomy, we find:

— Group i benefits from labor immigration if KtLt

< kit holds. Themore-than-average capital-rich groups will welcome labor immigrati-on, while the less-than-average capital-rich groups will oppose laborimmigration. If group i is as capital-rich as the economy as a whole(kit = Kt

Lt), group i will be indifferent towards labor immigration.

— Group i benefits from capital immigration if KtLt> kit holds. The less-

than-average capital-rich groups will welcome capital immigration,while the more-than-average capital-rich groups will oppose capitalimmigration. If group i is as capital-rich as the economy as a whole(kit = Kt

Lt), group i will be indifferent towards capital immigration.

— There is no group that welcomes both capital and labor immigration.

It may be interesting to speculate about whether some group ibenefitting from immigration will later come to regret it. Consider, forexample, a capital-rich group that welcomes labor immigration butopposes capital immigration. If an initially capital-poor group has ahigh rate of saving, group i may finally be harmed.

C.6 Technological change

So far, parameter A in our CD production function

Yt = F (Kt, Lt) = AKαt L

1−αt

C.6 Technological change 33

has been assumed constant. We will now turn to the case of variable A.An increase of A represents technological progress. We suppose thattechnological progress ”just happens”. In chapter ??, we will endoge-nize A by explicitly considering research and development.Parameter A in any production function (not necessarily of the

CD-type)Y = F (At,Kt, Lt) = AtF (Kt, Lt)

represents ”neutral” technological progress which has been introducedby Hicks (1932). Hicks neutrality refers to the fact that the ratio ofthe marginal products,

∂F∂K∂F∂L

,

is not influenced by whether A is low or high. Note that we abusenotation by employing F both for a three-argument and for a two-argument function.The other two classes of technological progress usually discussed

in the literature are called labor-augmenting and capital-augmentingtechnological progress. All three classes are summarized in the follo-wing table

Type of technological progressY = F (K,AL) labor augmenting Harrod neutralY = F (AK,L) capital augmenting Solow neutralY = AF (K,L) output augmenting Hicks neutral

While the defining characteristic of Hicks neutrality is easy to see,Harrod and Solow neutrality are more involved. Harrod neutrality ispresent if the ratio of factor payments,

∂F∂KK∂F∂LL

,

is unaffected by A, while Solow neutrality amounts to a constant ratioof factor payments with the proviso that the labor/output ratio isconstant. We will not delve into these differences.Our model works particularly well with a labor augmenting pro-

gress. So, we start from


Yt = F (Kt, AtLt) = Kαt (AtLt)

1−α .

Effective labor, denoted by

L = AL,

will increase whenever A or L increases. We now go on to define k := KL

and y := YL. We obtain

k =k

A,

y =Y

L=KαL1−α

L=

µK

L

¶α

= kα =

µk

A

¶α

Now, transferring the results from eq. C.2 (p. 21) and C.5 (p. 23)(where A := 1), we find

γk =dkdt

k=

s

k1−α− (δ + n) ,

k∗ =

µs

δ + n

¶ 11−α

,

γk∗ = γy∗ = γc∗ = 0

and hence, by k = Ak,

γk∗ = γA + γk∗ = γA.

By y = YL = Ay and c =

(1−s)YL = (1− s)Ay, we obtain

γy∗ = γy∗ + γA = γA and

γc∗ = 0 + γy∗ + γA = γA.

Furthermore,γK = γY ∗ = γC∗ = γA + n.

It would also be possible to go through the same procedure for Solow-neutral and Hicks-neutral technological progress. Can you do it? Howe-ver, the formulae would be somewhat different because the parameterA enters differently. For CD production functions these are spurious

C.7 Exogenous growth? 35

differences. For example, labor augmenting technological progress isoutput augmenting by

F (K,L) = Kα (AL)1−α

= A1−α¡KαL1−α

¢and also capital augmenting.

Exercise C.6.1. Show that labor-augmenting technological progressimplies capital-augmenting technological progress for CD productionfunctions. Hint: the trick is to manipulate the exponents for A.

C.7 Exogenous growth?

The upshot of the preceding discussion is this: our model generatessteady-state growth rates

— γA for the per-head variables k, y, and c (in the labor-augmentingcase) and

— γA + n for the total variables K, Y and C.

This growth is often called exogenous because it is driven directlyby the exogenously given growth rates of technology and population.We will see that other production functions lead to a growth processwhere the variables grow in the absence of technological progress andpopulation growth. We will then talk about endogenous growth.In a sense, ”endogenous” growth is a misnomer. In all these models,

growth is an endogenous variable resting on parameters, in our case s,δ, n, and γA. In the model just discussed, γC happens to be the sum ofγA and n. This does not make γC an exogenous parameter. It is stillan endogenous variable driven by the equilibrium concept employed(steady state).

C.8 Solutions

C.1.1. Constant returns to scale are easy to show:


F (λK,λL) = A (λK)α (λL)1−α

= AλαKαλ1−αL1−α

= λαλ1−αAKαL1−α

= λF (K,L) .

C.1.2. ∂Y∂K is just another expression of ∂F

∂K , therefore

εY,K =∂F

∂K

K

Y

= Aα

µL

K

¶1−α K

AKαL1−α

= α.

C.1.3. You have found (haven’t you?)

∂F

∂K·K +

∂F

∂L· L = Aα

µL

K

¶1−α·K +A (1− α)

µK

L

¶α

· L

= A

∙αL1−α

K1−α ·K + (1− α)Kα

Lα· L¸

= A£αKαL1−α + (1− α)KαL1−α

¤= F (K,L) .

C.1.4. For Yt = F (Kt, Lt) = AKαt L

1−αt , we find

γY =dYtdt

Yt

=

d(AKαt L

1−αt )

dt

AKαt L

1−αt

=A£αKα−1

tdKdt L

1−αt +Kα

t (1− α)L−αtdLdt

¤AKα

t L1−αt

(product rule and chain rule)

= αdKdt

Kt+ (1− α)

dLdt

Lt= αγK + (1− α) γL.

Alternatively, we have

C.8 Solutions 37

γY =d lnYtdt

=d ln

¡AKα

t L1−αt

¢dt

=d (lnA+ α lnKt + (1− α) lnLt)

dt

=d lnA

dt+d (α lnKt)

dt+d ((1− α) lnLt)

dt

= 0 + αd lnKtdt

+ (1− α)d lnLtdt

= αγK + (1− α) γL.

C.3.1. Since K is the product of k and L, we have

γK = γk + γL = 0 + n = n.

Analogously, we obtain γY = γC = n.

C.3.2.

— We need to show that

kt =

∙sA

n+ δ+ e(1−α)(−n−δ)t

µk1−α0 − sA

n+ δ

¶¸ 11−α

(C.6)

solves the differential equation

.k= sAk

α − (n+ δ)k. (C.7)

We define

E (t) :=sA

n+ δ+ e(1−α)(−n−δ)t

µk1−α0 − sA

n+ δ

¶.

Then, we have

kt = (E (t))1

1−α and (C.8)

E (t) = (kt)1−α (C.9)

Before forming the derivative of eq. C.6 with respect to t, we takenote of the following:


1. Eq. C.6 can be rewritten:

k1−αt = e(1−α)(−n−δ)tµk1−α0 − sA

n+ δ

¶+

sA

n+ δ(C.10)

and

e(1−α)(−n−δ)t · k1−α0 = k1−αt +sA

n+ δ

³e(1−α)(−n−δ)t − 1

´. (C.11)

2. It is easy to see

1

1− α− 1 = 1

1− α− 1− α

1− α=

α

1− α(C.12)

3. Finally, forming the derivative of the term in square brackets ofC.6 yields

dE (t)

dt

= (1− α)(−n− δ)e(1−α)(−n−δ)tµk1−α0 − sA

n+ δ

¶= e(1−α)(−n−δ)tk1−α0 (1− α)(−n− δ)

+(1− α)e(1−α)(−n−δ)tsA (C.13)

Now, we form the derivative of kt with respect to t and obtain thedesired result:

dktdt

=1

1− α(E (t))

11−α−1 · dE (t)

dt(chain rule, applied to C.8)

=1

1− α(E (t))

α1−α

dE (t)

dt(C.12)

= kαt ·he(1−α)(−n−δ)tk(0)1−α(−n− δ) + e(1−α)(−n−δ)tsA

i(C.9,C.13)

= kαt ·∙µk1−αt +

sA

n+ δ

³e(1−α)(−n−δ)t − 1

´¶(−n− δ) + e(1−α)(−n−δ)tsA

¸(C.11)

= kαt ·hk1−αt (−n− δ)− sA

³e(1−α)(−n−δ)t − 1

´+ e(1−α)(−n−δ)tsA

i= kαt ·

hk1−αt (−n− δ)− sAe(1−α)(sA−n−δ)t + sA+ e(1−α)(−n−δ)tsA

i= kαt ·

£k1−αt (−n− δ) + sA

¤= kt(−n− δ) + sAk(t)α

C.8 Solutions 39

C.5.1. Dropping the group index and using kt = Kt/Lt, we obtain

·ktkt

= sA

"α

µLtKt

¶1−α+ (1− α)

µKtLt

¶α

· 1KtLt

#− (δ + n)

= sA

"α

µLtKt

¶1−α+ (1− α)

µKtLt

¶α−1#− (δ + n)

= sA

"α

µLtKt

¶1−α+ (1− α)

µLtKt

¶1−α#− (δ + n)

= sA

µLtKt

¶1−α− (δ + n)

=sA

k1−α− (δ + n) .

C.5.2. You have obtained (haven’t you?)

dYitdLt

> 0

⇔ α (1− α)L−αt Kα−1t Kit > α (1− α)L−α−1t Kα

t Lit

⇔ K−1t Kit > L−1t Lit

⇔ KtLt< kit.

C.6.1. Starting from labor-augmenting technological progress you ar-rive at capital-augmenting progress by

F (K,L) = Kα (AL)1−α

= A1−αKαL1−α

=³A

1−αα

´αKαL1−α

=³³A

1−αα

´K´αL1−α.

D. Solow and beyond

In the previous chapter, we looked at the Solow model where the pro-duction function is of the Cobb-Douglas variety. This is unduly re-strictive, motivated by didactic considerations, only. In general terms,the Solow model presupposes a neoclassical production function (themost prominent example being Cobb-Douglas). A production functionY = F (K,L) is called neoclassical if F has two properties:

1. constant returns to scale and2. decreasing marginal productivities obeying the Inada conditions.

We will expound the implications of these conditions in section D.1.This chapter has two aims:

— First, we will go through the Solow model a second time and usea general neoclassical production function. We will explain why themodel works for this type of production function. Since the Solowmodel is at the heart of modern growth theory, some repetition willdo no harm.

— Second, using the same basic differential equations, we will exami-ne the growth implications of CES, AK, and Leontief productionfunctions. We will see that all production functions considered inthis book (CD, AK, Leontief) are special cases of CES productionfunctions.

42 D. Solow and beyond

D.1 Neoclassical production functions

D.1.1 Constant returns to scale

Neoclassical production functions are of constant returns which willturn out to be a very powerful property. Constant returns to scale isa special sort of homogeneity:

Definition D.1.1. A production function F is homogeneous of degreed, if we have

F (λK,λL) = λdF (K,L) ,K ≥ 0, L ≥ 0.

A production function F exhibits constant returns to scale if it is ho-mogeneous of degree 1.

Exercise D.1.1. Prove that the production function given by

F (K,L) = A£αK−ρ + (1− α)L−ρ

¤−1/ρ, 0 < α < 1, ρ > −1, ρ 6= 0

exhibits constant returns to scale.

Exercise D.1.2. Can you show that the Leontief production function,given by

Y = F (K,L) = min (AK,BL) ,

is also constant-returns?

The production function of the above exercise (which is not neoclas-sical!) is called CES production function because it shows a ”constantelasticity of substitution” which we will demonstrate below on in sec-tion D.7.Constant returns to scale are not a very plausible assumption. Inde-

ed, for very low endowments of capital and labor, constant returns willnot hold because gains from specialization need a certain size of theeconomy. Also, the reader might think of public goods such as software(production) which can be used by everyone in a small economy and ahuge economy alike. On the other hand, if all gains from specializationhave been exhausted and no public goods exist, constant returns toscale might hold for λ ≥ 1.

D.1 Neoclassical production functions 43

Exercise D.1.3. Can you prove F (0, 0) = 0 for any constant-returnsproduction function F?

The expedience of constant returns lies in the possibility of expres-sing the output per head as a function of capital per head. Indeed, forλ := 1

L , we obtain

F

µK

L, 1

¶= F

µ1

LK,

1

LL

¶=1

LF (K,L) .

By defining

k : =K

L,

y : =Y

L, and

f (k) : = F (k, 1)

we find

y =F (K,L)

L= F

µK

L, 1

¶= f (k) . (D.1)

Hence, f (k) is the output per head for a per-head endowment of capitalk. Romer (1996, S. 9) suggests the following interpretation of thisequation. Imagine that the economy is devided in L small economies,each of which endowed with 1 unit of labor and k = K

L units of capital.Because of constant returns, each of these small economies producesan Lth part of the total economy. f is called the production functionin intensive form.

Exercise D.1.4. Determine the intensive form of the CES productionfunction.

Constant returns is a powerful restriction that turns out to havemany interesting consequences concerning the marginal productivities

∂F

∂Kand

∂F

∂L.

We assume that our production functions are ”twice continuously dif-ferentiable”. Note that every differentiable function is continuous. The-refore, a twice differentiable function is continuous and its first deriva-tive is continuous, too. Inserting ”continuously” in ”twice continuously


differentiable” ensures that the second-order derivatives are also con-tinuous. Our production function has four second-order derivates, twoof them mixed:

∂2F

(∂K)2: =

∂ ∂F∂K

∂K,

∂2F

(∂L)2: =

∂ ∂F∂L

∂L,

∂2F

∂K∂L: =

∂ ∂F∂L

∂K, and

∂2F

∂L∂K: =

∂ ∂F∂K

∂L

The two mixed derivates are equal. The effect of a marginal increaseof capital on the marginal productivity of labor is equal to the effectof a marginal increase of labor on the marginal productivity of capital.Sometimes, it is necessary to explicitly state where (for which ca-

pital and labor values) a derivative is calculated. If no such values aregiven, (K,L) is assumed, for example,

∂F

∂K=

∂F

∂K

¯(K,L)

.

We can now prove some identities that follow from the fact that F isconstant-returns.

Lemma D.1.1. Let F be homogeneous of degree 1. Then,

1. the marginal productivities are homogeneous of degree 0 :

∂F

∂K

¯(λK,λL)

=∂F

∂K

¯(K,L)

and (D.2)

∂F

∂L

¯(λK,λL)

=∂F

∂L

¯(K,L)

(D.3)

2. the second-order derivatives are homogenous of degree −1 :

λ∂2F

(∂K)2

¯(λK,λL)

=∂2F

(∂K)2

¯(K,L)

and (D.4)

λ∂2F

(∂L)2

¯(λK,λL)

=∂2F

(∂L)2

¯(K,L)

(D.5)


3. the marginal productivities can be expressed as functions of capitalper head, k :

∂F

∂K=

df

dkand (D.6)

∂F

∂L= f (k)− k df

dk=: ω (k) (D.7)

4. Euler’s theorem holds:

F (K,L) =∂F

∂KK +

∂F

∂LL (D.8)

and, finally,5. the second-order derivatives relate to each other in a simple man-ner:

∂2F

∂K∂L= −k ∂2F

(∂K)2, (D.9)

∂2F

∂K∂L= −1

k

∂2F

(∂L)2, and (D.10)

∂2F

(∂K)2∂2F

(∂L)2=

µ∂2F

∂K∂L

¶2. (D.11)

The first item of the above lemma can be solved by the reader.

Exercise D.1.5. Prove

∂F

∂K

¯(λK,λL)

=∂F

∂K

¯(K,L)

and

∂F

∂L

¯(λK,λL)

=∂F

∂L

¯(K,L)

for a constant-returns production function F, i.e., show that the mar-ginal productivities are homogeneous of degree 0 (note λ0 = 1). Hint:form the derivative of F (λK,λL) = λF (K,L) with respect to K andL and use the chain rule of differentiation.

The proof of the second item is quite analogous to the proof of thefirst. The trick is to form the derivative of ∂F

∂K

¯(λK,λL)

= ∂F∂K

¯(K,L)

withrespect to K and to proceed analogously with labor.


Turning to the third item, we find

∂F (K,L)

∂K=

∂£Lf¡KL

¢¤∂K

= Ldf

d¡KL

¢ ∂ ¡KL ¢∂K

= Ldf

dk

1

L=df

dk.

In case of constant returns, the marginal product of labor is designatedby ω (k) . Here, ω is reminiscent of w as in wage rate. Indeed, if factorsare paid their marginal products, the wage rate is equal to ω (k) .

Exercise D.1.6. Show ∂F∂L = f (k)−k

dfdk . Hint: Beginn with

∂F (K,L)∂L =

∂(Lf(KL−1))∂L and apply the product rule of differentiation.

Exercise D.1.7. Prove Euler’s theorem (item 4). Hint: You need theresults from item 3.

Euler’s theorem claims that factor payments according to margi-nal products exhaust the total product. Also useful: Euler’s theoremimplies item 5. Forming the derivative with respect to K and L yield

∂F

∂K=

µ∂2F

(∂K)2K +

∂F

∂K

¶+

∂2F

∂K∂LL and

∂F

∂L=

∂2F

∂K∂LK +

µ∂2F

(∂L)2L+

∂F

∂L

¶and hence the desired equalities

∂2F

∂K∂L= −k ∂2F

(∂K)2and

∂2F

∂K∂L= −1

k

∂2F

(∂L)2

which imply the third one,

∂2F

(∂K)2∂2F

(∂L)2=

µ∂2F

∂K∂L

¶2.

This concludes the proof of lemma D.1.1.


D.1.2 Decreasing marginal productivities and Inadaconditions

The second property of neoclassical production functions concern themarginal productivities of the factors. For capital (and analogously,for labor), neoclassical production functions require positive and de-creasing marginal productivity:

∂F

∂K> 0 for L > 0, (D.12)

∂2F

(∂K)2< 0. (D.13)

That is, F is a concave function of K (and of L, too). Neoclassicalproduction functions are also defined by the Inada conditions. The-se require that the marginal product (being positive and decreasing)finally vanishes:

limK→∞

∂F

∂K= 0 (D.14)

If, on the other hand, K tends to 0, the marginal product (havingincreased all the way with decreasing K) gets infinite:

limK→0

∂F

∂K=∞ (D.15)

The Inada conditions also require the corresponding properties for la-bor.We now need to know whether f inherits these properties from F .

Indeed, we find that

— the marginal product per head of capital per head is positive byD.12:

df

dk=

∂F (k, 1)

∂k> 0, (D.16)

— the marginal product per head of capital per head decreases by D.13:

d2f

(dk)2=

∂2F (k, 1)

(∂k)2< 0, (D.17)

and also


— the Inada conditions hold by D.14 and D.15:

limk→∞

df

dk= lim

k→∞

∂F (k, 1)

∂k= 0, (D.18)

limk→0

df

dk= lim

k→0

∂F (k, 1)

∂k=∞. (D.19)

We have shown above that constant returns to scale imply F (0, 0) =0. Furthermore, using the Inada conditions, Barro/Sala-i-Martin(1999, p. 52) show that F (0, L) = F (K, 0) = 0 hold for any L and K.Hence, we also have

f (0) = F (0, 1) = 0. (D.20)

D.2 Dynamics and steady state for neoclassicalproduction functions

Very similar to the procedure in chapter C, we derive the dynamicsfor capital per head. First of all, we have

·Kt= sYt − δKt (D.21)

where

— Kt is the economy’s stock of capital,

—·Kt is the change of this stock

— due to sYt, the share s (=saving rate) of income Yt invested and— due to δKt, the depreciated capital at a depreciation rate δ.

Since output per head y can be expressed as a function of capitalper head k, we consider the dynamics of k :

·k =

·µK

L

¶=

·K L−

·L K

L2

=

·K

L− nk

=sY − δK

L− nk

= sf (k)− (n+ δ) k (D.22)

D.2 Dynamics and steady state for neoclassical production functions 49

where n is the (working) population’s growth rate. In order to grow(in terms of capital per head), the actual investment per head

sf (k)

has to make up for both per-head depreciation of capital δk and thedilution effect through population growth nk.The growth rate of per-capita endowment of capital is given by

γk =

·k

k= s

f (k)

k− (δ + n) , k > 0. (D.23)

In a steady state, growth rates of all relevant economic variables haveto be constant so that

sf (k)

k− (δ + n)

needs to be constant, too. Therefore, if k is the steady-state capitalper head, sf(k)k − (δ + n) may not change. Therefore according to thequotient rule and the chain rule,

0 =dhsf(k)k − (δ + n)

idt

= sdf(k)kdt

= sdfdkdkdt k −

dkdt f (k)

k2

= sdfdkk − f (k)

k

dkdt

k

= −sf (k)− df

dkk

kγk

= −s∂F∂L

kγk (eq. D.7).

Since the marginal product of labor is positive, a steady state occursat s = 0 or γk = 0. Leaving aside zero savings for the moment, eq.D.23 implies

sf (k∗) = (δ + n) k∗ (D.24)

for the steady-state variable k∗.


kn )( +δ

( )ksf

*k k


Actual investment

Inve

stm

ent p

er h

ead

Abbildung D.1. Break-even versus actual investment

Output and consumption per head are also constant,

y∗ = f (k∗) and

c∗ = (1− s) y∗,

while K = kL, Y = yL, and C = cL grow at rate n.We depict the dynamics and the steady state in figures D.1 and

D.2. The first depicts the change in per-head capital. When the actualinvestment outpasses the break-even investment, capital per head in-creases by eq. D.22. However, a second steady state exists at k = 0,

with output and investment equal to zero. This steady state is not sta-ble. Indeed, for a small k > 0 capital per head increases (if s > 0 holds)and converges towards k∗. Fig. D.2 shows the development towards k∗

by depicting eq. D.23.Both figures assume s > 0. If we have s = 0, according to eq.

D.23, the growth rate of capital per head is negative and constant at− (δ + n) .

Exercise D.2.1. Draw the equivalents of figures D.1 and D.2 for s =0.

D.2 Dynamics and steady state for neoclassical production functions 51

k0

n+δ( )kksf

0<kγ

0>kγ

*k

Abbildung D.2. Positive and negative growth rates

Returning to s > 0, we now want to show more formally that k = 0and k∗ > 0 are the two values of capital per head where the actualinvestment equals the break-even investment.

— For sufficiently small endowments of capital per head k > 0, actualinvestment per head sf (k) is greater than the break-even investment

(δ + n) k by the Inada condition D.19. Hence,·k= sf (k)− (n+ δ) k

is positive and capital per head increases.— For sufficiently large k, actual investment per head is smaller thanbreak-even investment, by the other Inada condition D.18. Therefo-re, capital per head decreases.

— Summarizing, sf (k) − (n+ δ) k is positive for small k and ne-gative for large ones. Therefore, we should find a k∗ in betweenwhere sf (k∗) − (n+ δ) k∗ is zero. This follows from the so-calledintermediate-value theorem which holds for continuous functions.(sf (k)− (n+ δ) k is continuous for k > 0.)

— k = 0 is a steady state by f (0) = 0 (see p. 48).— Finally, f and hence sf (k) − (n+ δ) k is concave by D.17 so thatfurther nulls are excluded.


D.3 Comparative statics and the golden rule

In general, the comparative-statics results found in chapter C do notchange by considering a general neoclassical production function in-stead of a CD production function. In particular, k∗ is a positive func-tion of s (dk∗/ds > 0) which is clear from fig. D.1.Starting from

c∗ (s) = (1− s) f (k∗ (s))= f (k∗ (s))− (δ + n) k∗ (s) (eq. D.24)

we maximize consumption per head by

f 0 (k∗ (s))dk∗

ds− (δ + n) dk

∗

ds!= 0

⇔ f 0 (k∗ (s))!= (δ + n) (note

dk∗

ds> 0).

We callf 0 (kgold)

!= δ + n

the golden rule of capital accumulation, depicted in fig. D.3. Indeed,the slope of f (k∗ (s)) has to be equal to the slope of (δ + n) k∗ (s)because steady-state consumption c∗ is the difference of output perhead f (k∗ (s)) and (break-even) investment (δ + n) k∗.

D.4 Dynamics and steady state for AK productionfunctions

We now turn to the AK production function. It is given by

Y = F (K) = AK, A > 0.

In this production function, K can be thought to hold both physicaland human capital. Obviously, the AK production function is the limitof a CD production function

Y = F (K,L) = AKαL1−α, A > 0, 0 < α < 1,

obtained for α = 1. The AK production function is not neoclassical.

D.4 Dynamics and steady state for AK production functions 53

goldk k0

kn)( +δ

goldc

( )kf

( )kfsgold

n+= δslope

Abbildung D.3. The golden rule of capital accumulation

Exercise D.4.1. In section D.1, the defining properties of a neoclas-sical production function are put forward, constant returns to scale,positive and decreasing marginal productivity of capital (and labor),and the Inada conditions

limK→∞

∂F

∂K= 0, lim

K→0

∂F

∂K=∞.

Which of these hold for AK production functions?

Exercise D.4.2. Can you express y = YL as a function of k = K

L ?

Determine dfdk and

f(k)k .

The dynamics of the AK model is simple. Indeed, as in the Solowmodel, we obtain

·k= sy − (δ + n) k.

Now, usingy = f (k) = Ak

from the preceding exercise, we find


k0

n+δ

( )kkfssA

0>kγ

Abbildung D.4. Endogenous growth in the AK model

γk =

·k

k

=sy − (δ + n) k

k

= sf (k)

k− (δ + n)

= sA− (δ + n) .

This growth rate does not depend on k and is constant. Hence, we havealready found the steady-state growth rate of k! In case of sA > δ+n,

the stock of capital grows exponentially. This is depicted in fig. D.4.In case of sA < δ + n, the stock of capital shrinks exponentially,converging to 0. If, by chance, sA = δ + n, the growth rate of capitalper head is 0.The steady-state value of capital per head therefore given by

k∗t = k0e(δ+n)t.

Also, we find steady-state output per head

y∗ = Ak∗

D.5 Conditional convergence 55

and steady-state consumption per head

c∗ = (1− s) y∗ = (1− s)Ak∗.

All of these per-head variables grow at rate γk = γy = γc = sA −(δ + n) . The corresponding overall values K, Y and C grow at

sA− (δ + n) + n = sA− δ.

The reader will note that the AK production functions make growthpossible in the absence of technological progress or population growth.Therefore, this is our first model of so-called ”endogenous growth” (seep. 35).

D.5 Conditional convergence

An important question of growth theory (and development theory)concerns the convergence of economies. Do we have theoretical and/orempirical evidence that (poor) countries whose capital per head is lowwill grow faster than rich countries? Growth theorists distinguish bet-ween absolute convergence and conditional convergence. Absolute con-vergence means that convergence occurs irrespective of whether basicparameters are identical. Conditional convergence claims convergenceonly for those economies who are sufficiently similar. The Solow mo-del supports conditional convergence: All countries per-head variableswill finally converge towards the steady state. The Solow model alsorefutes absolute convergence. Countries with different saving rates ordepreciation rates might permanently differ in economic well-being (interms of consumption per head).While absolute convergence does not show up in the data, the em-

pirical support for conditional convergence (where the parameters ofthe steady state are similar) is quite strong. In chapter B (see pp. 13)we present two different definitions of convergence, weak and strongconvergence, that apply to both conditional and absolute convergence.While weak convergence allows that the rich economy to stay foreverahead of the poor economy, this is ruled out by strong convergence.


Exercise D.5.1. Consider the production function

Y = F (K,L) = AK +BKαL1−α, A > 0, B > 0, 0 < α < 1,

taken from ( Barro/Sala-i-Martin 1999, p. 41-42).

— Show that F is constant-returns.— Show that F is not neoclassical. Hint: consider limK→∞ ∂F

∂K .

— Find the intensive form.— Calculate the average productivity f(k)

k and find the limit as k ap-proaches infinity.

— Using the Solow-Swan differential equation

·k

k= s

f (k)

k− (δ + n) ,

do you find weak convergence? Hint: weak convergence is present ifthe higher variable grows more slowly (see pp. 13).

— Determine the steady-state growth of capital per head for δ+n > sA.

We now pursue the above exercise. In fig. D.5 and D.6, the readerwill note that f(k)k is decreasing and converging towards A. The diffe-rence between the two figures is that the first obeys δ+n > sA, whilethe opposite holds in the second. The first admits a steady state at k∗

as calculated in the exercise.For sA > δ+n (fig. D.6), we have endogenous growth which declines

for every k > 0 but converges towards sA− (δ + n). Strictly speaking,we cannot identify a steady-state growth since k is changing forever.However, these changes get smaller and smaller so that we might bejustified in taking sA− (δ + n) as the growth rate of capital per headin the steady state.In fig. D.6, we obviously have weak convergence: the higher k, the

lower γk. However, strong convergence does not hold. To see this, note(you do not need to check) that the solution to the differential equation

.k

k= s

µA+

B

k1−α

¶− (δ + n)

found in the above exercise is given by

D.5 Conditional convergence 57

*k k0

kkfs /)(⋅

sA0<kγ

0>kγ

n+δ

Abbildung D.5. A steady state exists for a sufficiently small saving rate

k0

n+δ

kkfs /)(⋅

sA0>kγ

Abbildung D.6. A steady state exists in the limit for a sufficiently large savingrate


kt =

µe(1−α)(sA−n−δ)t

µk1−α0 +

sB

sA− n− δ

¶− sB

sA− n− δ

¶1−α.

We now consider two economies, a poor one, p, and a rich one, r.At time 0, the capital per person in these economies is kp0 and kr0,respectively. To check on strong convergence, we calculate the limit ofthe ratio of capital-per-head endowments krtkpt :

limt→∞

kr(t)

kp(t)= lim

t→∞

³e(1−α)(sA−n−δ)t

³k1−αr0 + sB

sA−n−δ

´− sB

sA−n−δ

´1−α³e(1−α)(sA−n−δ)t

³k1−αp0 + sB

sA−n−δ

´− sB

sA−n−δ

´1−α=

⎛⎝ limt→∞

e(1−α)(sA−n−δ)t³k1−αr0 + sB

sA−n−δ

´− sB

sA−n−δ

e(1−α)(sA−n−δ)t³k1−αp0 + sB

sA−n−δ

´− sB

sA−n−δ

⎞⎠1−α

=

⎛⎝ limt→∞

e(1−α)(sA−n−δ)t³k1−αr0 + sB

sA−n−δ

é(1−α)(sA−n−δ)t

³k1−αp0 + sB

sA−n−δ

´⎞⎠1−α

=

Ãk1−αr0 + sB

sA−n−δk1−αp0 + sB

sA−n−δ

!1−α> 1.

Therefore, strong conditional convergence does not hold.

D.6 Dynamics and steady state for Harrod-Domarproduction functions

The Harrod-Domar model rests on a Leontief production function:

Y = F (K,L) = min (AK,BL) , A,B > 0,

where the factors are perfect complements and the isoquants are L-shaped. Unless AK happens to be equal to BL, we have unemployedworkers or idle capital. In case of

AK < BL,

capital is the restricting factor and

D.6 Dynamics and steady state for Harrod-Domar production functions 59

Lempl :=A

BK < L, Lunempl := L−

A

BK > 0 (D.25)

are the workers actually employed and the unemployed workers, re-spectively. In case of

AK > BL,

labor is fully employed while

Kempl :=B

AL < K, Kunempl := K −

B

AL > 0

is the capital used in the actual production process and the idle capital,respectively.Since the Leontief production function is of constant returns (exer-

cise p. 42), we obtain

y =F (K,L)

L=min (AK,BL)

L= min (Ak,B) =: f (k) .

We now transfer the above two cases to per-head values. For

AK < BL⇔ Ak < B ⇔ k <B

A=: k

part of the labor force is unemployed. See fig. D.7, taken fromBarro/Sala-i-Martin (1999, S. 47). In this case, f (k) = Ak and

df

dk=f (k)

k= A.

For k > BA , part of the capital remains idle. Output does not increase

in k, but stays constant at

f (B/A) = min (A (B/A) , B) = B.

Hence,df

dk= 0 and

f (k)

k=B

k.

Exercise D.6.1. Are the Inada conditions fulfilled?

Using the case distinction, we have


B

AB k

( )kf

idlecapital

unem-ployment

full employment of both factors

Abbildung D.7. The Leontief production function in intensive form

γk =

·k

k= s

f (k)

k− (δ + n)

=

(sA− (δ + n) , k ≤ B

A

sBk − (δ + n) , k > BA

We now distinguish eight cases which can be found in the followingtable:

Parameters Growth rate γk signCase 1 sA = δ + n k0 ≤ B

A sA− (δ + n) = 0

Case 2 k0 >BA sBk − (δ + n) < 0

Case 3 sA < δ + n k0 ≤ BA sA− (δ + n) < 0

Case 4 k0 >BA sBk − (δ + n) < 0

Case 5 sA > δ + n k0 ≤ BA sA− (δ + n) > 0

Case 6a k0 >BA sBk = δ + n sBk − (δ + n) = 0

Case 6b k0 >BA sBk < δ + n sBk − (δ + n) < 0

Case 6c k0 >BA sBk > δ + n sBk − (δ + n) > 0

In cases 1 through 4, the growth rate is zero or negative. In case 5,the growth rate is positive, while case 6 necessitates further distincti-ons.


AB k0

n+δ

( )kkfs

case 1 case 2

Abbildung D.8. Cases 1 and 2

Exercise D.6.2. Show that the growth rate is negative in case 2! Whyis the growth rate positive in case 6c?

We now work through the eight cases. Cases 1 and 2 are depictedin fig. D.8. Growth is zero for k ≤ B

A (case 1) and negative for k >BA

(case 2). In case 1, any capital endowment per head k∗ obeying

0 ≤ k∗ ≤ BA

is in a steady state. The growth rate of capital per head is zero whilethe population grows at rate n. So does K = kL. For k∗ < B

A , we haveunemployment which also grows at rate n.

Exercise D.6.3. Using eq. D.25, show that Lunempl grows at rate n.Hint: you can use γL = γK = n.

Since both the labor force and the unemployed workers grow at raten, the unemployment rate remains constant at

L0 − ABK0

L0=

ent

entL0 − A

BK0

L0


k0

n+δ

( )kkfs

case 3 case 4

sA0<kγ

AB


=L0e

nt − ABK0e

nt

L0ent

=Lt − A

BKt

Lt.

In the second case, the capital endowment per head shrinks until k = BA

is reached. Both factors are then fully employed.However, these two cases are of no real interest because the equality

of sA and δ + n is only fulfilled by chance.Fig. D.9 depicts cases 3 and 4. Case 4 will finally turn into case 3

which seems to admit a steady state. Here, the (negative) growth rateof capital per head is constant, at sA− (δ + n).

Exercise D.6.4. What is the growth rate of K and of Lunempl in case3?

Although the growth rate of k is constant, the growth rate of theunemployed population is not constant. Hence, case 3 does not admita steady state.


k0

n+δ

( )kkfs

case 5 case 6c

sA

case 6b

case 6a

*kAB


We now turn to cases 5 and 6 which are shown in fig. D.10. Incase 5, the growth rate of per-head capital is constant at a positiverate until k finally approaches case 6c, where again capital per headincreases, albeit with a decreasing growth rate. In contrast, in case 6b,capital per head decreases. In case 6a, bordering 6c and 6b, we cansolve for k and obtain k∗ = sB

δ+n . Indeed, k∗ is the steady-state growth

rate of capital per head for cases 5 and 6.By k∗ > k = B

A , a part of capital remains idle. Nevertheless, Kgrows (together with L) at rate n. Of course, this is a curious resultgiven a constant saving rate s. Endogenizing s is the topic of chapterE.Summarizing, we find two steady states. The first (a continuum of

steady states, see case 1) is linked to unemployment that grows at raten. If, by chance, not only sA = δ + n, but also k0 = B

A , there is nounemployment. This is the famous knife-edge growth by Harrod andDomar. The second steady state (case 6a) witnesses idle capital, againgrowing at rate n.


D.7 CES production functions

D.7.1 Elasticity of substitution

We now turn to the CES production function which is given by

F (K,L) = A£αK−ρ + (1− α)L−ρ

¤−1/ρ, 0 < α < 1, ρ > −1, ρ 6= 0.

Parameter ρ is most important in specifying this production function.While ρ = −1 and ρ = 0 are excluded in order to simplify somecalculations, both values provide important examples of productionfunctions.

Exercise D.7.1. What kind of production function do you find forρ = −1? Sketch the isoquants.

CES stands for constant elasticity of substitution. We will nowintroduce the (somewhat difficult) concept of elasticity of substitutionwhich bears on the isoquants of production functions. We begin byreminding the reader of the marginal rate of technical substitutionbetween labor and capital, MRTSK,L, which is given by

MRTSK,L =

¯dK

dL

¯.

The interpretation of MRTSK,L is this: Imagine you want to holdoutput constant, i.e., you want to stay on a given isoquant. If youincrease labor by one unit you can decrease capital byMRTSK,L units.Normally, the easiest way to obtain the marginal rate of technicalsubstitution is to calculate the marginal productivities:

MRTSK,L =dFdLdFdK

.

Exercise D.7.2. Calculate the marginal rate of technical substitutionfor the CES production function.

Before proceeding, we remind the reader of some trigonometrics,the relationships holding in right triangles. In particular, we now needthe tangent tan γ of an angle γ, which is defined by

tan γ =side oppositeside adjacent

.

D.7 CES production functions 65

Exercise D.7.3. Draw a diagram with K and L on the axes (K onthe x-coordinate or on the y-coordinate?) and mark a point in thatdiagram so that the tangent of the angle leading to this point is equalto k = K

L .

Now, the elasticity of substitution captures the following aspect ofisoquants. If we change the slope of the isoquant, how will the ratio ofcapital over labor (which is our variable k) change? Consider fig. D.11with the two points C and D on an isoquant. At these points, we havecapital per head of

k = tan γ (point C),

k = tan δ (point D).

Going from C to D, both capital per head k and marginal rate oftechnical substitution MRTSK,L increase. The elasticity of substitu-tion measures this while

— considering small changes in k and— looking at relative changes, i.e., dkk =

d(K/L)K/L and dMRTSK,L

MRTSK,L.

Thus, the elasticity of substitution is defined by

εKL,MRTSK,L

=

dKLKL

dMRTSK,LMRTSK,L

=dKL

dMRTSK,L

MRTSK,LKL

.

It is often convenient to consider

1

εKL,MRTSK,L

=

dMRTSK,LMRTSK,L

dKLKL

=dMRTSK,L

dKL

KL

MRTSK,L.

Then, we change the capital-labor ratio to look for the correspondingchange in the marginal rate of technical substitution, both in percen-tage terms.For CES production functions, we find MRTSK,L = 1−α

α kρ+1 sothat we obtain the constant (!) elasticity of substitution

1

1 + ρ.

Exercise D.7.4. Yes! Your turn! Hint: Calculate 1εKL,MRTSK,L

.


L

K

C

D

γδ

Abbildung D.11. Elasticity of substitution

D.7.2 Special cases

We now take a look back to the case of perfect substitutability(ρ = −1). Here, isoquants are linear and marginal rates of techni-cal substitution constant along any isoquant. Therefore, the MRTSdoes not change while allowing k to change dramatically (compare fig.D.11). Hence, the elasticity of substitution should be∞ which is equalto 1

1+(−1) .

On the other end of the ρ-spectrum, we have ρ → ∞ and obtain(without proof) the Leontief production function

Y = F (K,L) = Amin (K,L) .

By the way, if we were to reproduce the Leontief production functionused in section D.6,

Y = F (K,L) = min (AK,BL) , A,B > 0,

we would need to start from the CES production function

F (K,L) = A

"αK−ρ + (1− α)

µB

AL

¶−ρ#−1/ρ, 0 < α < 1, ρ > −1, ρ 6= 0, A,B > 0.


L

K

δ

Abbildung D.12. Elasticity of substitution for perfect complements

Turning to the elasticity of substitution for L-shaped isoquants, wenote that we can change the marginal rate of substitution withoutaffecting the capital-labor ratio (see fig. D.12). Therefore, the elasticityof substitution is 0 = limρ→∞

11+ρ . Strictly speaking, however, the

marginal rate of technical substitution is not defined at K = L (orAK = BL).

Exercise D.7.5. Calculate the elasticity of substitution for the CDproduction function given by

Y = F (K,L) = AKαL1−α, A > 0, 0 < α < 1.

Hint: Remember ∂F∂K = Aα

¡LK

¢1−αand ∂F

∂L = A (1− α)¡KL

¢αand

calculate the MRTSK,L as a function of k.

It can be shown (see Barro/Sala-i-Martin 1999, p. 310) thatwe obtain the CD production function when ρ converges to 0. Hence,the constant elasticity of substitution of CD production functions is11+0 = 1. If we increase the marginal rate of technical substitution byone percent, capital per head increases by one percent.


1−=ρ

3−=ρ

0=ρ

∞→ρ

L

K

Abbildung D.13. Isoquants

It has become obvious that ρ, or 11+ρ , determine shape of isoquants.

Fig. D.13 depicts four isoquants.

— ρ = −1 represents perfect substitutes,— ρ =∞ stands for perfect complements, and— ρ = 0 yields Cobb-Douglas production function with intermediatesubstitutability.

These three production technologies are convex, i.e., the marginalrate of technical substitution decreases in L. On the other hand, ρ =−3 (in fact, any ρ < −1) leads to a concave production technologywhich we do not want to consider in detail.Another way to summarize the discussion so far is presented in fig.

D.14. Note that the ρ-axis goes from left to right while the elasticityof substitution 1

1+ρ points to the left. A high elasticity of substitution11+ρ > 1 (ρ < 0) indicates high substitutability, while a low elasticityof substitution 1

1+ρ < 1 (ρ > 0) indicates that the factors exhibitcomplementarity. CD production functions are moderate in featuringneither high substitutes nor high complements.


1−ρ

0

perfectsubstitutes

CobbDouglas

ρ+11

1∞

∞

0

perfectcomplements

highsubstitutes

h i g h c o m p l e m e n t s

convex production technologyconcave production technology

Abbildung D.14. Summary of three special cases of CES production functions

D.7.3 Inada conditions

In order to prepare for the analysis of the dynamics resulting from CESproduction functions, we need to know whether they are neoclassical.In section D.1.1 above, we found that a CES production function is ofconstant returns and that its intensive form is given by

f (k) = A£αk−ρ + (1− α)

¤−1/ρ.

For neoclassical production functions, we have f (0) = 0 (p. 48). Thisdoes not, in general, hold true for CES production functions. Indeed,we have to distinguish negative from positive ρ. We obtain

limk→0

f (k) =

(limk→0A [αk−ρ + (1− α)]

−1/ρ, −1 < ρ < 0

limk→0A [αk−ρ + (1− α)]−1/ρ

, ρ > 0

=

⎧⎨⎩ limk→0A [αk−ρ + (1− α)]−1/ρ

, −1 < ρ < 0

limk→0A

[α 1kρ+(1−α)]1/ρ

, ρ > 0

=

(limk→0A (1− α)−1/ρ , −1 < ρ < 0

0, ρ > 0


If the elasticity of substitution is high ( 11+ρ > 1 ⇔ −1 < ρ < 0)

and, hence, the factors are high substitutes, labor alone can achive apositive output. If, on the other hand, the elasticity of substitution isrelatively low ( 1

1+ρ < 1⇔ ρ > 0), the factors are high complements sothat both factors are needed for a positive output.Since f (0) = 0 follows from constant returns plus Inada conditions,

we already know that the Inada conditions must be violated. However,we want to look closer. Turning to the marginal productivities, we find

∂F∂K

A=

dfdk

A(eq. D.6, p. 45)

= −1/ρ£αk−ρ + (1− α)

¤− 1ρ−1

α (−ρ) k−ρ−1 (chain rule)

= α£αk−ρ + (1− α)

¤− 1ρ−1k−ρ−1

= α£αk−ρ + (1− α)

¤− 1+ρρ (kρ)−

1+ρρ (

³abć= abc)

= α£¡αk−ρ + (1− α)

¢(kρ)

¤− 1+ρρ ( (a1a2)

b = ab1ab2)

= α£αk−ρkρ + (1− α) kρ

¤− 1+ρρ

= α [α+ (1− α) kρ]−1+ρρ (aba−b = 1) (D.26)

> 0

and

∂2F

(∂K)2=

∂ dfdk∂K

=

∂

µAα

hα+ (1− α)

¡KL

¢ρi−1+ρρ

¶∂K

= Aα

µ−1 + ρ

ρ

¶ ∙α+ (1− α)

µK

L

¶ρ¸− 1+ρρ−1

· (1− α) ρ

µK

L

¶ρ−1 1

L(chain rule, twice)

= −Aα (1 + ρ)

∙α+ (1− α)

µK

L

¶ρ¸− 1+ρρ−1

· (1− α)

µK

L

¶ρ−1 1

L


< 0 (ρ > −1).

Hence, the marginal product of capital is a concave function (as is themarginal product of labor).These calculations hold for any ρ > −1, ρ 6= 0. In order to examine

the Inada conditions, we again need to distinguish negative parameters

ρ from positive ones. Using ∂F∂K = df

dk = Aα [α+ (1− α) kρ]−1+ρρ , we

have

limK→∞

∂F

∂K= lim

K→∞Aα

∙α+ (1− α)

µK

L

¶ρ¸− 1+ρρ

=

⎧⎪⎪⎨⎪⎪⎩limK→∞Aα

∙α+ 1−α

(KL )−ρ

¸ 1+ρ−ρ, −1 < ρ < 0

limK→∞Aα

[α+(1−α)(KL )ρ]1+ρρ, ρ > 0

=

⎧⎪⎪⎨⎪⎪⎩Aα limK→∞

∙α+ 1−α

(KL )−ρ

¸ 1+ρ−ρ, −1 < ρ < 0

Aα

limK→∞[α+(1−α)(KL )ρ]1+ρρ, ρ > 0

=

(Aαα

1+ρ−ρ , −1 < ρ < 0

0, ρ > 0

=

(Aα

1−ρ , −1 < ρ < 0

0, ρ > 0

and

limK→0

∂F

∂K=

⎧⎪⎪⎨⎪⎪⎩Aα limK→0

∙α+ 1−α

(KL )−ρ

¸ 1+ρ−ρ, −1 < ρ < 0

Aα

limK→0[α+(1−α)(KL )ρ]1+ρρ, ρ > 0

=

(∞, −1 < ρ < 0Aα

α1+ρρ, ρ > 0

=

(∞, −1 < ρ < 0A

α1/ρ, ρ > 0

For ρ > 0, limK→∞∂F∂K = 0 and limk→0 f (k) = 0 hold, while

limK→0∂F∂K is not inifinity. Therefore, the Inada conditions are not


fulfilled and CES production functions are not neoclassical for anyρ 6= 0 (Cobb-Douglas). This does not, of course, preclude to analyzethe growth dynamics for CES production functions.

D.7.4 Dynamics and steady state for CES productionfunctions

In order to apply the Solow-Swan differential equation

γk =

·k

k= s

f (k)

k− (δ + n) , k > 0,

we need to calculate f(k)k which is equal to

f (k)

k=

A [αk−ρ + (1− α)]−1/ρ

k

=A [αk−ρ + (1− α)]

−1/ρ

(k−ρ)−1/ρ

= A

µαk−ρ + (1− α)

k−ρ

¶−1/ρ= A [α+ (1− α) kρ]−1/ρ .

Now, the Solow-Swan differential equation for CES production functi-on is given by

γk = sf (k)

k− (δ + n)

= sA [α+ (1− α) kρ]−1/ρ − (δ + n) . (D.27)

To specify the dynamics, we need to know the limiting behaviour off(k)k :

limk→∞

f (k)

k= lim

k→∞A [α+ (1− α) kρ]−1/ρ

=

(limk→∞A

£α+ 1−α

k−ρ¤1/(−ρ)

, −1 < ρ < 0

limk→∞A

(α+(1−α)kρ)1/ρ, ρ > 0

=

(Aα1/(−ρ), −1 < ρ < 0

0, ρ > 0

D.8 Poverty trap 73

and

limk→0

f (k)

k= lim

k→0A [α+ (1− α) kρ]−1/ρ

=

(limk→0A

£α+ 1−α

k−ρ¤1/(−ρ)

, −1 < ρ < 0

limk→0A

(α+(1−α)kρ)1/ρ, ρ > 0

=

(∞, −1 < ρ < 0A

α1/ρ, ρ > 0

We now distinguish four cases:

— ρ < 0, sAα1/(−ρ) < δ + n,

— ρ < 0, sAα1/(−ρ) > δ + n,

— ρ > 0, sAα1/ρ

< δ + n,

— ρ > 0, sAα1/ρ

> δ + n.

Case 1 : ρ < 0, sAα1/(−ρ) < δ + n: We have high substitutes andlimk→∞

f(k)k = Aα1/(−ρ). Saving is insufficient for endogenous growth.

Fig. D.15 shows that the economy approaches a steady state at k∗

where all variables per head remain constant.

Case 2 : ρ < 0, sAα1/(−ρ) > δ + n: We have high substitutes andlimk→∞

f(k)k = Aα1/(−ρ). Here, saving is larger (or δ and n smaller)

so that endogenous growth occurs. Fig. D.16 shows that the economykeeps on growing forever. The growth rate approaches

sAα1/(−ρ) − (δ + n) > 0.

Cases 3 and 4 : ρ > 0: In cases 3 and 4, f(k)k approaches 0 as k

approaches infinity. Hence, growth will finally come to an end. Indeed,in case 3, the economy will vanish (fig. D.17), while case 4 features asteady state at positive, but constant per-head values.

Exercise D.7.6. Draw the picture for case 4!

D.8 Poverty trap

In the AK model f(k)k is constant. We will now assume that marginalproductivity declines, increases, and finally declines again (fig. D.18).


*k k0

kkfs /)(⋅

ρα −1

sA0<kγ

0>kγ

n+δ

Abbildung D.15. Case 1: high substitutes, insufficient saving

k0

kkfs /)(⋅

ρα −1

sA

0>kγ

n+δ

Abbildung D.16. Case 2: high substitutes, sufficient saving

D.9 Solutions 75

k0

kkfs /)(⋅ρα1

sA0<kγ

n+δ

Abbildung D.17. Case 3: high complements, insufficient saving

We then may obtain a situation with two equilibria, one with a lowper-head endowment of capital and one with a higher per-head capital.If such a f(k)

k curve were to exist it might help to explain why somecountries are poor while others are rich. The poor countries are simplythose who happened to have a low capital endowment per head, belowkm. The political consequences are obvious. Foreign capital should flowinto these countries to lift k above km. From then on, self-sustaininggrowth would occur. Alternatively, an increase of s or a decrease of nor δ might also do the trick.

D.9 Solutions

D.1.1. For any λ ≥ 0, we obtain

F (λK,λL) = A£α (λK)−ρ + (1− α) (λL)−ρ

¤−1/ρ= A

£αλ−ρK−ρ + (1− α)λ−ρL−ρ

¤−1/ρ= A

¡λ−ρ

£¡αK−ρ + (1− α)L−ρ

¢¤¢−1/ρ=

¡λ−ρ

¢−1/ρA£¡αK−ρ + (1− α)L−ρ

¢¤−1/ρ


k0

n+δ

( )kkfs

)stabil(

*kleink

)instabil(

*mittelk

)stabil(

*hochk

Abbildung D.18. Poverty trap

D.9 Solutions 77

= λ−ρ·(−1/ρ)F (K,L)

= λF (K,L)

and confirm that F is constant-returns.

D.1.2. First, we note F (0 ·K, 0 · L) = 0·F (K,L), so that the equalityholds for λ = 0. For λ > 0, we have

AK ≤ BL⇔ λ (AK) ≤ λ (BL)

and hence

F (λK,λL) = min (A (λK) , B (λL))

= min (λ (AK) ,λ (BL))

= λmin (AK,BL) .

D.1.3. For λ := 0, the desired equations follow easily:

F (0, 0) = F (0 ·K, 0 · L) = 0 · F (K,L) = 0

D.1.4. The intensive form of the CES production function is given by

f (k) = F

µK

L, 1

¶

= A

"α

µK

L

¶−ρ+ (1− α) · 1−ρ

#−1/ρ= A

£αk−ρ + (1− α)

¤−1/ρD.1.5. The derivative of F (λK,λL) = λF (K,L) with respect to Kyields

∂F (λK,λL)

∂K=

∂ [λF (K,L)]

∂K

⇔ ∂F (λK,λL)

∂ (λK)

d (λK)

dK= λ

∂ [F (K,L)]

∂K

⇔ ∂F (λK,λL)

∂ (λK)=

∂ [F (K,L)]

∂K

⇔ ∂F

∂K

¯(λK,λL)

=∂F

∂K

¯(K,L)

.


Analogously, forming the derivative with respect to L leads to

∂F (λK,λL)

∂L=

∂ [λF (K,L)]

∂L

⇔ ∂F

∂L

¯(λK,λL)

=∂F

∂L

¯(K,L)

.

D.1.6. You have found

∂F (K,L)

∂L=

∂¡Lf¡KL−1

¢¢∂L

= f¡KL−1

¢+ L

∂f

∂ (KL−1)

d¡KL−1

¢dL

= f (k) + L∂f

∂k(−1)KL−2

= f (k)− dfdkk.

D.1.7. Euler’s theorem:

∂F

∂KK +

∂F

∂LL =

df

dkK +

µf (k)− k df

dk

¶L

=df

dkK + f (k)L− K

L

df

dkL

= Lf (k)

= F (K,L) .

D.2.1. Compare fig. D.19 and D.20.

D.4.1. The AK production function features constant returns to sca-le. Indeed,

F (λK) = A (λK) = λ (AK) = λF (K) für alle λ ≥ 0

holds. The marginal productivity of capital is given by dFdK = A > 0

and is positive and constant (not decreasing). Obviously, the Inadaconditions for K are not fulfilled.

D.4.2. Per-head income is

y =Y

L=AK

L= A

K

L= Ak =: f (k) .

Hence,df

dk= A =

df (k)

k.

D.9 Solutions 79

kn)( +δ

( )kf⋅0

0* =k k


Actual investment

Inve

stm

ent p

er h

ead

Abbildung D.19. Break-even versus actual investment for zero savings

k

n+δ

( )k

kf⋅0

0<kγ

0* =k

Abbildung D.20. Positive and negative growth rates


D.5.1.

— Constant returns: For λ > 0, we obtain

F (λK,λL) = A (λK) +B (λK)α (λL)1−α

= λAK + λα+1−αBKαL1−α

= λF (K,L) .

— Marginal product and Inada conditions: The marginal product ofcapital is

∂F

∂K= A+ αB

µL

K

¶1−α,

which is positive and decreasing (as neoclassical production functi-ons). However, we have

limK→∞

∂F

∂K

= limK→∞

ÃA+ αB

µL

K

¶1−α!

= A+ limK→∞

αB

µL

K

¶1−α= A+ 0

> 0.

which violates an Inada condition.— Intensive form: The intensive form of the production function is thesum of the intensive forms for AK and CD production function:

f (k) = Ak +Bkα.

— Average productivity: We have

f (k)

k=Ak +Bkα

k= A+

B

k1−α.

The limit is

limk→∞

f (k)

k= limk→∞

µA+

B

k1−α

¶= A+ lim

k→∞

B

k1−α

= A.

D.9 Solutions 81

— Growth rate of capital per head and convergence: We obtain.k

k= s

f(k)

k− (δ + n)

= s

µA+

B

k1−α

¶− (δ + n) . (D.28)

and find weak convergence: The higher k, the lower the growth rateof k.

— Steady state: Referring again to eq. D.28, since s, A, B, δ, and n areconstant, so is k in the steady state. Therefore γk∗ = 0 and

k∗ =

µsB

δ + n− sA

¶ 11−α

.

D.6.1. While limk→∞dfdk = limk→∞

Bk = 0, we have limk→0

dfdk =

limk→0A <∞.

D.6.2. In case 2, we have k > BA and hence growth rate

γk = sB

k− (δ + n)

< sBBA

− (δ + n)

= sA− (δ + n)= 0.

In case 6c, by k > BA , we find the growth rate of capital per head

γk = sB

k− (δ + n)

which is positive by sBk > δ + n.

D.6.3. It is straightforward to calculate the growth rate of the unem-ployed population:

γLunempl =

d(Lt−ABKt)

dt

Lt − ABKt

=dLtdt

Lt − ABKt

−ABdKtdt

Lt − ABKt


=Lt

dLtdtLt

Lt − ABKt

−ABKt

dKtdtKt

Lt − ABKt

=Ltn

Lt − ABKt

−ABKtn

Lt − ABKt

= n.

D.6.4. For K, we have

γK = γkL = γk + n = sA− δ.

Similar to the above exercise, we find

γLunempl =

d(Lt−ABKt)

dt

Lt − ABKt

=Lt

dLtdtLt

Lt − ABKt

−ABKt

dKtdtKt

Lt − ABKt

=Ltn

Lt − ABKt

−ABKt (sA− δ)

Lt − ABKt

+ 0

=Ltn

Lt − ABKt

−ABKt (sA− δ)

Lt − ABKt

−ABKtn

Lt − ABKt

+ABKtn

Lt − ABKt

=Ltn

Lt − ABKt

−ABKtn

Lt − ABKt

−ABKt (sA− δ)

Lt − ABKt

+ABKtn

Lt − ABKt

= n+ABKt (δ + n− sA)

Lt − ABKt

= n+ABK0e

(sA−δ)t (δ + n− sA)L0ent − A

BK0e(sA−δ)t

= n+ABK0 (δ + n− sA)L0e(δ+n−sA)t − A

BK0.

D.7.1. ρ = −1 leads to the CES production function

F (K,L) = A [αK + (1− α)L] , 0 < α < 1.

Capital and labor are perfectly substitutable and the isoquants arelinear.

D.9 Solutions 83

L

K

C

γ

K

L

γtan==LKk

Abbildung D.21. Capital per head and the tangent

D.7.2. We find

MRTSK,L =

¯dK

dL

¯=

dFdLdFdK

=A (−1/ρ) [αK−ρ + (1− α)L−ρ]−1/ρ−1 (1− α) (−ρ)L−ρ−1

A (−1/ρ) [αK−ρ + (1− α)L−ρ]−1/ρ α (−ρ)K−ρ−1

=(1− α)L−ρ−1

αK−ρ−1

=1− α

α

µK

L

¶ρ+1

=1− α

αkρ+1.

D.7.3. Fig. D.21 provides the answer.

D.7.4. We find

1

εKL,MRTSK,L

=d¡1−αα kρ+1

¢dk

k1−αα kρ+1


= (ρ+ 1)1− α

αkρ

k1−αα kρ+1

= ρ+ 1

and henceεKL,MRTSK,L

=1

ρ+ 1.

D.7.5. First of all, we note

MRTSK,L =dFdLdFdK

=A (1− α)

¡KL

¢αAα

¡LK

¢1−α=

1− α

α

K

L

=1− α

αk.

Then, we obtain

1

εKL,MRTSK,L

=d¡1−αα k

¢dk

k1−αα k

=1− α

α

k1−αα k

= 1.

D.7.6. Fig. D.22 is the solution.

*k k0

kkfs /)(⋅

0<kγ0>kγn+δ

ρα1

sA

Abbildung D.22. Case 4: high complements, sufficient saving

E. Ramsey model

E.1 Introduction

So far, the saving rate s was constant. With the exception of commentspertaining to the golden rule, s was also exogenous. In this chapter, wewill allow for varying saving rates which will be determined from anoptimization program. A representative agent decides on a path of s(a function t 7→ st) with the objective to maximize his utility. Ramseydeveloped such a model as early as 1928 which was refined by Cass(1965) and Koopmans (1965). Therefore, the model is addressed asRamsey model or Ramsey-Cass-Koopmans model.We will see that a dynamically inefficient rate of saving (which

leads to a capital per head above kgold) cannot obtain. For reasonableparameter values, we obtain a constant rate of saving, as in the Solowmodel.

E.2 Static optimization

In this chapter, we need some knowledge on dynamic optimization.In preparation, we will present the static case. For optimization pro-blems with equality constraints, the Lagrange approach is useful. Afamous application concerns household theory where consumers con-sume goods 1 and 2 in quantities x1 and x2 and prices are denotedby p1 and p2, respectively. We assume that a household has a utilityfunction u which reflects monotonic and convex preferences.The households problem is to maximize

u (x1, x2)

subject to the budget constraint (m denotes income)

88 E. Ramsey model

m− (p1x1 + p2x2) = 0.

If you have to deal with an inequality constraint (in our case, m −p1x1 − p2x2 ≥ 0), a more complex optimization procedure has to beapplied, the so-called Kuhn-Tucker -Verfahren.Bei monotonen Präferenzen möchte der Haushalt möglichst viele

Einheiten von Gut 1 (beispielsweise) konsumieren. Allerdings sorgtdie Nebenbedingung dafür, dass er den Mehrkonsum von Gut 1 mitdem Minderkonsum von Gut 2 zu bezahlen hat. Die LagrangefunktionL ist nun so aufgebaut, dass sie einerseits den Nutzen maximiert, aberandererseits dabei die Nebenbedingung beachtet:

L (x1, x2,λ) = u (x1, x2) + λ (m− p1x1 − p2x2) .

Die Idee des Lagrange-Multiplikators ist diese: Wenn man x1 erhöht,hat dies einen direkten (positiven) Einfluss auf den Nutzen und einenindirekten (negativen) über die Restriktion. Bei der Budgetrestriktionaus der Haushaltstheorie wird der Mehrkonsum von Gut 1 durch denMinderkonsum von Gut 2 ”bezahlt”. In der Lagrange-Gleichung wirddies so berücksichtigt, dass nach Maßgabe von λ der Mehrkonsumvon Gut 1, der zur Ausgabenerhöhung p1 führt, in Nutzeneinheitenumgerechnet wird. Der optimale λ-Wert wird dabei simultan mit denoptimalen Konsummengen x∗1 und x

∗2 bestimmt.

Wie arbeitet man nun mit der Lagrangefunktion?

1. Zunächst wird die Lagrangefunktion nach allen Variablen abgelei-tet und die Ableitung gleich Null gesetzt. Wir erhalten also dreiBedingungen erster Ordnung:

∂L (x1, x2,λ)

∂x1=

∂u (x1, x2)

∂x1− λp1

!= 0, (E.1)

∂L (x1, x2,λ)

∂x2=

∂u (x1, x2)

∂x2− λp2

!= 0, (E.2)

∂L (x1, x2,λ)

∂λ= m− p1x1 − p2x2 !

= 0.

Die dritte Optimalbedingung ist die Nebenbedingung.2. Anschließend gewinnt man aus den ersten beiden Optimierungsbe-dingungen

E.2 Static optimization 89

∂u(x1,x2)∂x1

p1

!= λ

!=

∂u(x1,x2)∂x2

p2

und somit das zweite Gossen’sche Gesetz. Man kann auch so um-formen:

MRS =MU1MU2

=

∂u(x1,x2)∂x1

∂u(x1,x2)∂x2

!=p1p2.

Allgemein gibt der Lagrange-Multiplikator λ den zusätzlichen Nut-zen an, der durch die Lockerung der Nebenbedingung um eine Einheitentsteht. In der Haushaltstheorie ist

λ =du

dm

also der Grenznutzen des Einkommens. Mit dieser Interpretation kannman nun die Optimierungsbedingung Gl. E.1 so schreiben:

∂u (x1, x2)

∂x1

!=du

dmp1.

Im Optimum konsumiert der Haushalt von jedem Gut so, dass derGrenznutzen des Konsums (linke Seite) gleich dem Nutzen ist, den eraus der Lockerung der Budgetbeschränkung bei Nichtkonsum erhält(rechte Seite). Denn bei Nichtkonsum wird die Budgetbeschränkunggelockert (um p1 bei einer Einheit von Gut 1) und diese Lockerungerhöht den Nutzen. Hinter dem Nutzen der Lockerung steht natürlichder Konsum anderer Güter, der nun möglich wird.Nur die ganz Verwegenen, die sich für einen formalen Beweis von

λ = dudm interessieren, sollten jetzt noch weiterlesen. Streng genommen

macht dudm insofern keinen Sinn, als u als Argumente die konsumierten

Gütermengen hat und nicht das Einkommen. Man kann jedoch die sogenannte indirekte Nutzenfunktion V durch

V (p1, p2,m) := u (x1 (p1, p2,m) , x2 (p1, p2,m)) (E.3)

definieren, wobei x1 (p1, p2,m) die bei den Preisen p1 und p2 und beimEinkommen m nutzenmaximal nachgefragte Menge von Gut 1 meint.Die indirekte Nutzenfunktion ordnet also den Preisen und dem Ein-kommen den maximal erreichbaren Nutzen zu. Abb. E.1 stellt die Nut-zenfunktion der indirekten Nutzenfunktion gegenüber.

90 E. Ramsey model

Funktion Argumente optimale Gütermengen

Nutzenfunktion Gütermengen x1 (p1, p2,m) , x2 (p1, p2,m)

Indirekte Nutzenfunktion Einkommen, Preise x1 (p1, p2,m), x2 (p1, p2,m)

Abbildung E.1. Direkte und indirekte Nutzenfunktion

Wir wollen nun als Übung die indirekte Nutzenfunktion für dieCobb-Douglas-Nutzenfunktion u (x1, x2) = xa1x

1−a2 (0 < a < 1) be-

stimmen. Wir erhalten

V (p1, p2,m) = u (x∗1, x∗2)

=

µam

p1

¶aµ(1− a) m

p2

¶1−a=

µa

p1

¶aµ1− ap2

¶1−am.

Durch Differenziation der obigen Definitionsgleichung E.3 nach merhalten wir

∂V

∂m=

∂u

∂x1

∂x1∂m

+∂u

∂x2

∂x2∂m

.

Die Summe kommt dadurch zustande, dass der Nutzen

u (x1 (p1, p2,m) , x2 (p1, p2,m))

indirekt durch das Einkommen m beeinflusst wird, indem sich sowohlder Konsum von Gut 1 als auch der Konsum von Gut 2 ändern. DurchEinsetzen der Gl. E.1 und E.2 erhalten wir

∂V

∂m= λp1

∂x1∂m

+ λp2∂x2∂m

= λ

µp1∂x1∂m

+ p2∂x2∂m

¶. (E.4)

Um den Klammerausdruck zu bestimmen, leiten wir die Budgetglei-chung im Optimum,

p1x1 (p1, p2,m) + p2x2 (p1, p2,m) = m,

nach m ab. Dadurch ergibt sich

E.3 Dynamische Optimierung 91

p1∂x1∂m

+ p2∂x2∂m

= 1,

sodass wir aus Gl. E.4 schließlich

∂V

∂m= λ (E.5)

erhalten: Der Lagrange-Multiplikator ist gleich der Ableitung der in-direkten Nutzenfunktion nach dem Einkommen m.

E.3 Dynamische Optimierung

E.3.1 Das Maximierungsproblem

In diesem und in späteren Kapiteln benötigen wir Kenntnisse der dyna-mischen Optimierung, die kochbuchmäßig bereitgestellt werden. Aus-gangslage ist im endlichen Fall (bis zum Zeitpunkt T ) das Optimie-rungsproblem

maxc(t)

U = maxc(t)

Z T

0v [c (t) , t] dt

unter den Nebenbedingungen

·k (t) = g [k (t) , c (t) , t] (Bewegungsgleichung),

k (0) = k0 > 0 (Startwert)

k (T ) ≥ 0 (Ausschluss von Schulden).

Unter v werden wir den Barwert des Nutzens verstehen, unter c denKonsum und unter k den Kapitalstock pro Kopf. U ist dann der Ge-samtnutzen in Barwertformulierung. Die Bewegungsgleichung gibt an,wie der Kapitalstock in Abhängigkeit vom Kapitalstock und vom Kon-sum wächst. Die zweite Gleichung legt den Startwert für den Kapital-stock fest.Man nennt k die Zustandsvariable und c die Kontrollvariable. Die

Kontrollvariable hat über die Bewegungsgleichung Einfluss auf dieÄnderung der Zustandsvariable. Das Problem besteht darin, einenc−Pfad (bzw. s−Pfad) so zu finden, dass U unter Beachtung der Ne-benbedingungen maximiert wird. Kurzfristig erhöht sich der Nutzenmit c, während sich der Zuwachs des Kapitals verringert.

92 E. Ramsey model

Schließlich ist die dritte Ungleichung zu kommentieren: Am Endedes Planungszeitraumes darf der Kapitalstock nicht negativ sein undder repräsentative Agent nicht mit Schulden aus dem Leben scheiden.Falls nun der Planungshorizont unendlich ist, erhalten wir folgende,

ganz ähnliche Formulierung des Optimierungsproblems:

maxc(t)

U = maxc(t)

Z ∞

0v [c (t) , t] dt

unter den Nebenbedingungen

·k (t) = g [k (t) , c (t) , t] (Bewegungsgleichung),


limT→∞

k (T )

er(T )T≥ 0 (Ausschluss von Schneeballsystemen).

Im Wesentlichen ist nur die dritte Ungleichung verändert, wobei r (t)die durchschnittliche Diskontierungsrate zwischen 0 und t darstellt. DieBedingung sagt nun, dass k (T ) negativ sein darf und sogar anwachsendarf; die Wachstumsrate der Schulden darf jedoch nicht größer als r (t)sein. Ansonsten könnte sich das Individuum immer größere Geldbeträ-ge leihen, mit denen es dann seine wachsenden Schulden befriedigt undsich zusätzlich Konsummöglichkeiten verschafft.

Exercise E.3.1. Show that k (T ) := −e−r`T and r` < rh implylimT→∞

k(T )

erhT≥ 0.

Es gibt zwei verschiedene, aber äquivalente Formulierungen der Op-timalbedingungen. Wir beginnen mit der üblichen Hamilton-Funktionmit Barwert. Im nächsten Abschnitt wird die Hamilton-Funktion mitaktuellem Wert dargestellt. Dabei ändert sich recht wenig. Diese zwei-te Formulierung ist jedoch diejenige, die für die Optimierungsproblemein dieser Vorlesung häufiger benutzt wird.

E.3.2 Die Hamilton-Funktion (Gegenwartswert)

Bei der Lösung des Optimierungsproblems kann man in fünf Schrittenso vorgehen:


— Zunächst wird die Hamilton-Funktion

H = v (c, t) + μ (t) g (k, c, t)

konstruiert. μ (t) ist ein dynamischer Lagrange-Multiplikator, derSchattenpreis einer zusätzlichen Einheit des Kapitalstocks in t, aus-gedrückt in Einheiten des Nutzens zum Zeitpunkt 0. Dieser Schat-tenpreis ”übersetzt” somit die Kapitaländerungen in Nutzenände-rungen. Damit drückt die Hamilton-Funktion aus, wie der Konsumden Nutzen direkt über v und indirekt über eine Änderung von kbeeinflusst.

— Dann wird die Hamilton-Funktion nach der Kontrollvariablen abge-leitet und diese Ableitung gleich null gesetzt:

∂H

∂c=

∂v

∂c+ μ

∂g

∂c!= 0.

Hier hat also der Grenznutzen des Konsums gleich den Grenzkostendes Konsums zu sein. Der erhöhte Konsum erhöht den Nutzen ei-nerseits (∂v/∂c), führt aber andererseits zu einer verringerten Kapi-talakkumulation, die zukünftige Nutzenminderungen nach sich zieht(μ∂g/∂c). Der Leser beachte, dass wir bei unserer Problemstellung∂g/∂c < 0 zu erwarten haben. Im Optimum müssen diese beidenEffekte sich ausbalancieren.

— Die vorherige Optimierungsbedingung kann natürlich nur richtigeErgebnisse hervorbringen, falls die Schattenpreise μ (t) in richtigerWeise den Beitrag von Kapitalerhöhungen zum Gesamtnutzen Uwiedergeben. Man kann zeigen (mehr dazu im nächsten Abschnitt),dass die Ableitung der Hamilton-Funktion nach der Zustandsvaria-blen gleich der negativen Zeitableitung des Lagrange-Multiplikatorszu setzen ist:

∂H

∂k= μ

∂g

∂k!= −dμ

dt.

— Sodann ist die Transversalitätsbedingung aufzustellen. Hier sind dreiFälle zu unterscheiden:

— Endlicher PlanungshorizontDas Produkt aus Schattenpreis und Kapitalstock am Endes desPlanungshorizontes ist Null:

94 E. Ramsey model

μ (T ) k (T )!= 0.

Denn ”am Ende” muss das Kapital ganz aufgebraucht (k (T ) = 0)oder wertlos (μ (T ) = 0) sein.

— Unendlicher Planungshorizont bei DiskontierungHier gilt die obige Bedingung im Grenzübergang:

limt→∞

μ (t) k (t)!= 0.

Falls also der Kapitalstock durchgängig zunimmt, hat der Schat-tenpreis abzunehmen, und zwar schneller, als der Kapitalstock zu-nimmt.

— Die vorstehenden Optimierungsbedingungen zusammen liefern dieLösung unseres Optimierungsproblems.

E.3.3 Die Hamilton-Funktion (aktueller Wert)

Bei der intertemporalen Optimierung des Haushalts gehen wir häufigvon einer Zielfunktion der FormZ T

0v [c (t) , t] dt =

Z T

0u [c (t)] e−ρtdt

aus. Man beachte, das u nicht direkt von t abhängt. ρ is the time pre-ference rate. The higher ρ, the higher the consumer’s wish to consumeearly. ρ can also be interpreted as the rate of return on consumption.This rate can be defined as

u (c (t)) e−ρt − u (c (t+∆t)) e−ρ(t+∆t)

∆tu (c (t)) e−ρt

.

It is the increase in discounted utility for consuming now (at time t)rather than later (at time t +∆t). The advantage of consuming nowand not later comes out most clearly, if we evaluate this expression byassuming that consumption does not change over time, i.e., c (t) = c.Letting ∆t go towards zero, we obtain


lim∆t→0

u (c (t)) e−ρt − u (c (t+∆t)) e−ρ(t+∆t)

∆tu (c (t)) e−ρt

= − lim∆t→0

u (c (t+∆t)) e−ρ(t+∆t) − u (c (t)) e−ρt∆t

u (c (t)) e−ρt

= −du (c) e−ρt

dtu (c) e−ρt

= −de−ρt

dte−ρt

= ρ

Man könnte nun wie im vorangegangenen Abschnitt die Hamilton-Funktion

H = u [c (t)] e−ρt + μ (t) g (k, c, t)

verwenden. Hierbei bedeutet μ (t) den Schattenpreis einer zusätzlichenEinheit des Kapitalstocks in t, ausgedrückt in Einheiten des Nutzenszum Zeitpunkt 0. Wir definieren nun

q (t) := μ (t) · eρt

und schreiben

H = e−ρt (u [c (t)] + q (t) g (k, c, t)) .

Während der Schattenpreis μ den Nutzen einer zusätzlichen Einheitdes Kapitalstocks in t in Einheiten des Nutzens zum Zeitpunkt 0 aus-drückt, bezieht sich q auf den Nutzen der aktuellen Periode t.Jetzt können wir die Hamilton-Funktion mit aktuellem Wert defi-

nieren: bH := eρtH = u [c (t)] + q (t) g (k, c, t) .

Wir variieren die im vorigen Absatz unternommenen Schritte leicht.Vor allem der erste Schritt ist dabei wichtig.

— Zuerst stellt man die Hamilton-Funktion mit aktuellem Wert

bH = u [c (t)] + q (t) g (k, c, t)

96 E. Ramsey model

auf. q (t) ist ein Lagrange-Multiplikator, der Schattenpreis einer zu-sätzlichen Einheit des Kapitalstocks in t, ausgedrückt in Einheitendes Nutzens zum Zeitpunkt t.

— Differenzieren nach der Kontrollvariablen c und Nullsetzen lieferthier:

∂ bH∂c

=∂u

∂c+ q

∂g

∂c!= 0.

Alternativ und äquivalent ist weiterhin ∂H∂c

!= 0 zu verwenden.

— Die Frage, wie die richtigen dynamischen Lagrange-Multiplikatorengestaltet sein müssen, wollen wir hier nun etwas ausführlicher be-leuchten. Der Leser erinnere sich daran, dass im statischen Fall derkorrekte Lagrange-Multiplikator λ = ∂V

∂m beträgt, also als Grenznut-zen des Einkommens wiederzugeben ist. In analoger Weise hat manhier

q (t)!=

∂Umax

∂k (t)

zu setzen, wobei Umax den maximal erreichbaren Nutzen meint.∂Umax/∂k (t) fragt also danach, wie sich der maximal erreichbareNutzen erhöht, falls in t exogen der Kapitalstock um eine kleineEinheit erhöht wird. Man kann nun zeigen (siehe de la Fuente(2000, 568 ff.)), dass diese Bedingung gerade

∂ bH∂k

!= −

·q +ρq (t)

impliziert. Eine alternative Herleitung aus der Optimalbedingung∂H∂k = μ∂g

∂k!= −dμdt im Barwert-Fall soll ebenfalls erwähnt werden:

∂ bH∂k

= eρt∂H

∂k

= eρtµμ∂g

∂k

¶!= eρt

µ−dμdt

¶= eρt

Ã−d¡q (t) e−ρt

¢dt

!= −eρt

³ ·q e−ρt + q (t) e−ρt (−ρ)

´= −

·q +ρq (t) .

Wir könnten übrigens alternativ weiterhin ∂H∂k

!= −dμdt benutzen.

Barro/Sala-i-Martin (1999, S. 510) weisen darauf hin, dass


∂ bH∂k

!= −

·q +ρq (t) bzw.

∂ bH∂k +

·q

q!= ρ

auf folgende Weise interpretiert werden kann:

— ∂ bH∂k ist der zusätzliche Nutzen eines höheren Kapitaleinsatzes beiKonstanz des (Schatten-)Preises (Dividende in aktuellen Nutzen-einheiten).

—·q ist die Preisänderung des Kapitals. Damit gibt ∂ bH

∂k +·q an, wie der

Agent von einem höheren Kaitaleinsatz direkt (Dividende) und in-direkt (Änderung des Kapitalpreises) profitiert. Die Division durchden Kapitalpreis lässt den Ausdruck links des Gleichheitszeichendann als Rendite erscheinen, wobei hier im Zähler und im Nennerauf eine Einheit des (mit dem Schattenpreis) bewerteten Kapital-einsatzes Bezug genommen wird. (Beispielsweise ist die Gewinn-rendite durch den Quotienten von Gewinn und Kapitaleinsatz de-finiert.)

— ρ ist die Ertragsrate für die alternative Verwendung des Geldes,für Konsum.

— Sodann ist die Transversalitätsbedingung aufzustellen. Hier sind wie-derum drei Fälle zu unterscheiden:

— Endlicher Planungshorizont:

q (T ) e−ρTk (T )!= 0.

— Unendlicher Planungshorizont bei Diskontierung:

limt→∞

q (t) e−ρtk (t)!= 0.

— Schließlich kombiniert man die Bedingungen aus dem 2. und 3.Schritt mit der Bewegungsgleichung und erhält so ein System vonGleichungen, das sich auf zwei Differenzialgleichungen reduzierenlässt. Um eindeutige Lösungen zu bekommen, benötigt man denStartwert und die geeignete Transversalitätsbedingung.

98 E. Ramsey model

E.4 Ein einfaches Modell - nur ein Agent

E.4.1 Modellbeschreibung

Wir betrachten ein Modell mit einem repräsentativen Agenten. Dieserlebt unendlich lange. Man kann dies damit rechtfertigen, dass endlichlebende Haushalte in altruistischer Weise mit den ihnen folgenden Ge-nerationen verknüpft sind. Hierbei ist jedoch Heirat ausgeschlossen,weil diese Beziehungen zwischen Familien herstellt. Wir nehmen hieran, dass die Größe des Haushalts sich im Zeitablauf nicht ändert undauf L = 1 normiert ist. Hierdurch entfällt die Notwendigkeit, Pro-Kopf-Größen einzuführen. Vielleicht kann man an Robinson Crusoedenken, der ein sehr langes Leben vor sich sieht.Der Haushalt habe das Ziel, seinen Nutzen

U =

Z ∞

0u [c (t)] e−ρtdt

unter Nebenbedingungen (siehe unten) zu maximieren. Dabei meint

— c (t) den Konsum pro Kopf in Periode t,— u den Nutzen des Periodenkonsums, der

u0 > 0, u00 < 0

und die Inada-Bedingungen,

limc→∞

u0 = 0, limc→0

u0 =∞,

erfüllt,— ρ > 0 die Zeitpräferenzrate und— U den Nutzen des Haushaltes über den gesamten Zeitraum.

Die an u gestellten Bedingungen bewirken, dass der Haushalt einInteresse daran hat, den Konsum über die Zeit gleichmäßig zu gestal-ten (intertemporaler Ausgleich des Konsums). ρ > 0 bedeutet, dasseine Präferenz dafür besteht, den Konsum lieber früh als spät zu tä-tigen. Dies impliziert, dass Personen während ihres Lebens spätereneigenen Konsum genau so abdiskontieren wie den Konsum spätererGenerationen.

E.4 Ein einfaches Modell - nur ein Agent 99

Um die Modellelemente besser zu verstehen, lösen Sie, bitte, diefolgende Aufgabe für den Zwei-Perioden-Fall. In dieser Aufgabe mussgenau zwischen der Ertragsrate des Konsums, ρ, und der Ertragsratedes Kapitals, r, unterschieden werden. Wenn Ihnen nicht geläufig ist,wie diese beiden zu berechnen und zu interpretieren sind, schauen Sienochmals auf den Seiten 94 und 10 nach.

Exercise E.4.1. Setzen Sie die zweiperiodige Nutzenfunktion

U = u (c1) + u (c2) ·1

1 + ρ

voraus, wobei u wiederum die oben genannten Eigenschaften aufweiseund ci den Konsum der i-ten Periode darstelle. Gehen Sie zudem, wie inWiese (2005, S. 130 ff.), von der intertemporalen Budgetbeschränkung

(1 + r) c1 + c2 = (1 + r)m1 +m2

aus, wobei m1 und m2 die realen Einkommen in den Perioden 1 und2 darstellen. Bestimmen Sie die Bedingung erster Ordnung für dasOptimum. Unter welchen Bedingungen für r und ρ konsumiert derHaushalt in der ersten Periode mehr als in der zweiten?Hinweis: Stellen Sie die Optimierungsbedingung so um, dass 1+r

1+ρ aufeiner Seite der Gleichung auftaucht.

Der Haushalt produziert, bei vollem Einsatz seiner Arbeitskraft,entsprechend der Produktionsfunktion f (k) , die sich aus den neoklas-sischen Bedingungen oder aus dem AK-Modell ergeben könnte. DerKapitalstock wird mit der Abschreibungsrate δ in jeder Periode ver-mindert. Wegen L = 1 gilt die Bewegungsgleichung

·k = g [k (t) , c (t) , t] = f (k)− δk − c oder

f (k)− δk =.k +c

Der um den abgeschriebenen Teil korrigierte Output kann entwederinvestiert oder konsumiert werden. Die Bewegungsgleichung ist eineNebenbedingung der obigen Maximierungsaufgabe. Die anderen zweisind auch alte Bekannte:


limT→∞

k (T ) · e−r(T )T ≥ 0 (Ausschluss von Schneeballsystemen).

100 E. Ramsey model

E.4.2 Anwendung des Lösungsalgorithmus’

Wir wenden jetzt das fünf-Schritte-Verfahren an. k ist die Zustandsva-riable und c die Kontrollvariable. Wir nehmen die Hamilton-Funktionmit aktuellem Wert, bH.— Die Hamilton-Funktion mit aktuellemWert lautet für unser Problem

bH = u [c (t)] + q (t) (f (k)− δk − c) .

q (t) ist ein Lagrange-Multiplikator, der Schattenpreis einer zusätz-lichen Einheit des Kapitalstocks in t, ausgedrückt in Einheiten desNutzens zum Zeitpunkt t.

— Durch Ableiten der Hamilton-Funktion bH nach der Kontrollvaria-blen erhält man die Optimalbedingung:

∂ bH∂c

=du

dc− q !

= 0. (E.6)

Im Optimum bringt dem Haushalt Konsum genauso viel wie Inve-stition. Eine zusätzliche Konsumeinheit bringt dem Haushalt du/dc,eine zusätzliche Investitionseinheit bringt dem Haushalt q (t), das istgleich dem aktuellen Wert (auf t bezogen) des Stroms der zusätz-lichen (zukünftigen) Nutzen, der durch die zusätzliche Einheit desKapitals erzeugt wird.

— Durch Ableiten der Hamilton-Funktion bH nach der Zustandsvaria-blen erhält man die Optimalbedingung (Euler-Gleichung)

∂ bH∂k

= q (t)¡f 0 (k)− δ

¢ != −

·q +ρq (t)

bzw.

ρ!= f 0 (k)− δ +

·q

q (t)=q (t) (f 0 (k)− δ)+

·q

q (t). (E.7)

Die zweite Formulierung kann man so lesen. Auf der rechten Seitesteht im Zähler der Nutzen einer zusätzlichen Kapitaleinheit: Einezusätzliche Einheit Kapital führt zur Erhöhung des Kapitalstocksum f 0 (k) − δ, der Schattenpreis sagt, was diese Erhöhung des Ka-

pitalstocks wert ist. Zusätzlich wird im Ausmaß·q das Kapital und


damit auch die letzte zusätzliche Einheit wertvoller. Teilt man diesdurch den Schattenpreis, erhält man eine Art Rendite dafür, eine zu-sätzliche Kapitaleinheit zu halten. Diese Rendite muss mindestensso groß sein wie die Rendite der alternativen Verwendung, des Kon-sums, also ρ.

— Die Transversalitätsbedingung lautet:

limt→∞

q (t) e−ρtk (t)!= 0. (E.8)

Sie soll verhindern, dass zuviel Kapitalaufbau betrieben wird; Kon-sum ist der Zweck der Produktion.

— Man erhält das Gleichungssystem

·k = f (k)− δk − c, (Bewegungsgleichung)du

dc= q, (Ableitung nach der Kontrollvariablen)

q (t)¡f 0 (k)− δ

¢= −

·q +ρq (t) . (Euler-Gleichung)

Setzt man nun die zweite Gleichung in die dritte ein, erhält man

du

dc

¡f 0 (k)− δ

¢= −

d¡dudc

¢dt

+ ρdu

dc,

woraus sich durch Auflösen nach ρ

ρ = f 0 (k)− δ +d¡dudc

¢dt

· 1dudc

= f 0 (k)− δ +d2udcdtdcdt

c1dudc

··c

c

= f 0 (k)− δ +d2u

(dc)2c1dudc

··c

c(siehe unten)

= f 0 (k)− δ +

d2u(dc)2

dudc

c ··c

c(E.9)

= f 0 (k)− δ +u00

u0c · γc (E.10)

ergibt, wobei wir

102 E. Ramsey model

d2udcdtdcdt

=d2u

(dc)2

verwendet haben (wird wohl richtig sein, ergibt sich durch hem-mungsloses Kürzen).

E.4.3 Konkavität von u und ein Elastizitätsmaß

In der Herleitung des letzten Abschnitts kommt (ohne Minuszeichen)der Term

u00 (c) · cu0 (c)

vor, der an das Maß für die relative Risikoaversion erinnert, das aus derEntscheidungstheorie bekannt ist. (Dort ist u dann die von-Neumann-Morgenstern-Nutzenfunktion und c das Endvermögen.) Grob gilt: Jegrößer dieser (positive) Wert, desto gekrümmter oder ”konkaver” istdie Perioden-Nutzenfunktion. Man kann diesen Wert auch so schrei-ben:

εu0,c =

du0(c)u0(c)dcc

,

also als Elastizität der ersten Ableitung u0 in Bezug auf c. Er sagt aus:Um wie viel Prozent ändert sich der Grenznutzen, wenn c um ein Pro-zent erhöht wird. εu0,c wird auch als Grenznutzen-Elastizität bezeich-net. Der Leser beachte, dass εu0,c bei konkaven Perioden-Nutzenfunktionennegativ ist.Nehmen wir als Perioden-Nutzenfunktion die durch

u (c) =c1−θ

1− θ, θ > 0, θ 6= 1

bestimmte Funktion.

Exercise E.4.2. Bestimmen Sie die Perioden-Nutzenfunktion für θ =0 und θ → 1.

Für die obige Perioden-Nutzenfunktion gilt

u0 (c) =(1− θ) c−θ

1− θ= c−θ > 0,

u00 (c) = −θc−θ−1 < 0,


und daher

εu0,c =u00 (c) · cu0 (c)

=

¡−θc−θ−1

¢· c

c−θ= −θ.

Die Nutzenfunktion weist also eine konstante Elastizität des Grenz-nutzens auf.

Exercise E.4.3. Bestimmen Sie für die Perioden-Nutzenfunktion u (c) =ln (c) die Grenznutzen-Elastizität.

E.4.4 Interpretation der Optimalitätsbedingung fürintertemporalen Konsum

Die auf S. 101 (Gl. E.10) hergeleitete Optimalitätsbedingung könnenwir nun auch so:

ρ!= f 0 (k)− δ + εu0,c · γc (E.11)

oder so:f 0 (k)− δ

!= ρ− εu0,c · γc (E.12)

schreiben. Wir wollen nun eine Interpretation dieser Optimierungsbe-dingung versuchen. Im Solow-Modell haben wir die goldene Regel derKapitalakkumulation hergeleitet. Sie lautet:

f 0 (kgold)!= δ + n.

Sie ergibt sich aus der Maximierung des Pro-Kopf-Konsums im Gleich-gewicht. Wir sehen nun in diesem Modell von Bevölkerungswachstumab und haben also unsere Optimalitätsbedingung mit

f 0 (kgold)− δ!= 0

zu vergleichen. Bei ρ = 0 (späterer Konsum genauso gut wie jetzi-ger) und εu0,c = 0 (kein Glättungsmotiv) würden wir auf die goldeneRegel der Kapitalakkumulation geführt. Im vorliegenden Modell sindzwei zusätzliche Effekte zu berücksichtigen, zum einen das Konsumaus-gleichsmotiv aufgrund der konkaven Nutzenfunktion und zum anderendie Gegenwartspräferenz.

Exercise E.4.4. Unter welchen Bedingungen entfällt das Konsum-ausgleichsmotiv? Welche Konsequenz hat dies für die Optimalitätsbe-dingung?

104 E. Ramsey model

Ein positiver ρ-Wert führt im Optimum zu einer höheren Grenzpro-duktivität des Kapitals und somit zu einem geringeren Kapitaleinsatz:Der Haushalt konsumiert verstärkt auf Kosten des Kapitalaufbaus.Formal sieht man dies daran, dass die Pro-Kopf-Grenzproduktivitätmit der Pro-Kopf-Kapitalausstattung sinkt, die f 0 (k)-Kurve also kon-kav verläuft (siehe Kap. D, S. 47).Ist die Zeitpräferenzrate ρ gleich Null, erhält man die Optimalitäts-

bedingung

f 0 (k)− δ!=¡−εu0,c

¢· γc.

Wir nehmen nun an, der optimale Konsumpfad steige an (γc > 0).Dann ist f 0 (k)−δ positiv und der Haushalt konsumiert ”heute” mehr,als er dies bei einer linearen Nutzenfunktion täte. Unter dieser Bedin-gung erfolgt der Konsum verstärkt auf Kosten des Kapitalaufbaus,falls das Konsumausgleichsmotiv wichtig ist, falls also die Elastizitätvom Betrag her relativ hoch ist. Betrachten wir nun die umgekehrteAnnahme: Der optimale Konsumpfad sei fallend (γc < 0). Dann führtdas Konsumausgleichsmotiv (ein hoher Wert für den Betrag der Ela-stizität) zu folgendem Ergebnis: Je konkaver die Nutzenfunktion, destoeher verzichtet der Haushalt auf den Konsum ”heute”.Um die Optimierungsbedingung noch weiter interpretieren zu kön-

nen, betrachten wir das Unternehmen, das dem Haushalt gehört. Eshat die Produktionsfunktion f (k) und den Gewinn

G (k) = f (k)− (r + δ) k − w,

wobei r den Zinssatz für Kapital darstelle und w den Lohnsatz fürdie eine Arbeitseinheit. Zu den Faktorkosten rk zählen zusätzlich dieAbschreibungskosten δk. Setzt man die Ableitung der Gewinnfunktiongleich Null erhält man die ”Grenz(wert)produkt = Faktorpreis”-Regel,die hier formal als

f 0 (k)!= r + δ

bzw.f 0 (k)− δ

!= r

zu schreiben ist.


Die Optimierungsbedingung E.12 lässt sich dann auch so schreiben:

r!= ρ− εu0,c · γc.

Bei r = ρ entspricht der Zinssatz genau der Zeitpräferenzrate. DerHaushalt wird für seinen Konsumverzicht genau entschädigt und wählteinen konstanten Konsumpfad (γc = 0) oder hegt nicht den Wunschnach Konsumausgleich (εu0,c = 0). Ist der Zinssatz höher als die Zeit-präferenzrate, so wählt der Haushalt einen ansteigenden Konsumpfad(γc > 0), der allerdings bei stark ausgeprägtem Wunsch nach Konsu-mausgleich (hohes −εu0,c) nicht sehr stark ansteigt. Der Leser erinneresich an den Zwei-Perioden-Fall der Aufgabe E.4.1. Dort konnten wirein sehr ähnliches Phänomen feststellen.Wir versuchen es nochmals etwas anders und gehen von der Un-

gleichungf 0 (k)− δ > ρ

aus. Sie besagt, dass eine Einheit Kapital mehr bringt als eine zusätz-lich Einheit Konsum, wenn man nur die Zeitpräferenz bedenkt. Vondaher ist sie als Aufforderung zu verstehen, den heutigen Konsum zu-gunsten der Investition einzuschränken. Bei einem (ohnehin) steigen-den Konsumpfad könnte dies jedoch mit dem Konsumausgleichsmotivkonfligieren. Tatsächlich kann die obige Ungleichung ein Gleichgewichtanzeigen, wenn

εu0,c · γcgenau die Differenz ausmacht.Abschließend betrachten wir den Spezialfall der Perioden-Nutzen-

funktion

u (c) =c1−θ

1− θ, θ > 0, θ 6= 1

die Gegenstand von Übung E.4.2 auf S. 102 ist. Setzt man zusätzlichfür f 0 (k)− δ den Zinssatz r, kann man die Optimalitätsbedingung als

r − ρ

θ!= γc

schreiben. Für große θ−Werte (d.h. eine hohe Elastizität) haben wirselbst bei relativ großen Differenzen zwischen Zinssatz und Zeitprä-ferenzrate einen recht egalitären Konsumpfad zu erwarten. Für kleine

106 E. Ramsey model

θ−Werte genügt dagegen schon ein geringfügiges Übersteigen von r ge-genüber ρ, um den Konsum deutlich in spätere Perioden zu verlagern.Für kleine θ−Werte und ρ > r würde der Konsum im Wesentlichenin den ersten Perioden stattfinden. Bei θ = 0 und r = ρ ist jederKonsumpfad, den er sich leisten kann, dem Haushalt gleich lieb.

E.4.5 Dynamik

Wir nehmen nun an, die Inada-Bedingungen gelten sowohl für die Pro-duktionsfunktion als auch für die Periodennutzenfunktion. Dann, sobehaupten wir ohne Beweis, wird die Wachstumsrate des Kapitalstocks(pro Kopf) im Wachstumsgleichgewicht weder positiv noch negativ

sein. Die Bewegungsgleichung führt dann wegen·k= 0 zu

c = f (k)− δk, (E.13)

sodass die Wachstumsrate des Konsums ebenfalls gleich Null ist. Dergleichgewichtige Kapitalstock ist dann wegen Gl. E.11 implizit durch

f 0 (k∗) = ρ+ δ

definiert. Aus Gl. E.7 folgt·q= 0 bei k∗. Der gleichgewichtige Schat-

tenpreis ergibt sich wegen Gl. E.6 durch

q∗ := u0 (c∗) = u0 (f (k∗)− δk∗) .

Der Leser betrachte die Abb. E.2. Rechts von der·q= 0-Linie steigt

der Schattenpreis an. Dies ergibt sich aus Gl. E.7 und folgender Über-legung: Der Haushalt ist nur dann bereit Kapital k > k∗ zu halten,dessen Ertrag f 0 (k) − δ (f 0 (k) sinkt mit steigendem Kapitaleinsatz!)unter seiner Zeitpräferenzrate ρ liegt, wenn er durch einen Anstieg

des Wertes des Kapitals (·q > 0) belohnt wird. Daher zeigen die Pfeile

rechts der·q= 0-Linie nach oben und links davon nach unten.

Wegen Gl. E.6 geht ein hoher Schattenpreis q mit niedrigem Kon-sum einher (der Grenznutzen fällt mit c). Wegen Gl. E.13 geht ein ho-her Schattenpreis daher auch mit einem niedrigen Wert von f (k)−δk,d.h. mit einem niedrigen Kapitalstock einher. Dies erklärt die Steigung

E.5 Literaturhinweise 107

Aufbrauchen desKapitalstocks

*k k

q 0=q&

0=k&

0

*q

stabiler PfadVerletzung derTransversalität

Abbildung E.2. Der stabile Pfad

der·k= 0-Linie in der Abbildung. Man kann es auch ökonomischer aus-

drücken: Ein höherer Kapitaleinsatz impliziert einen höheren Output,der einen höheren Konsum ermöglicht. Der höhere Konsum ist jedochaufgrund des abnehmenden Grenznutzens nicht so viel wert, sodassder Schattenpreis des Kapitals ebenfalls abnehmen muss. Oberhalb

der·k= 0-Linie ist q so hoch und c so niedrig, dass der Kapitalstock

anwächst. Unterhalb dieser Linie nimmt er ab. Dies erkärt die nachlinks bzw. nach rechts gerichteten Pfeile.Pfade nach Nordosten verletzen die Transversalitätsbedingung: Es

wird dauerhaft zu wenig konsumiert. Pfade nach Südwesten brauchenden gesamten Kapitalstock auf. Dies ist auch nicht optimal. Der Haus-halt hat sich zum Kapitalstock k (0) denjenigen q-Wert zu suchen, derihn schließlich auf (k∗, q∗) hinführt.

E.5 Literaturhinweise

E.6 Lösungen zu den Aufgaben

E.3.1. Indeed, we find

108 E. Ramsey model

limT→∞

k (T )

e−rh(T )T= limT→∞

−e−r`Te−rhT

= − limT→∞

1

e(rh−r`)T= 0.

E.4.1. Die betragsmäßige Steigung der Budgetgeraden ist

1 + r,

die Grenzrate der Substitution beträgt

MRS =

¯dc2dc1

¯=MU1MU2

=u0 (c1)

u0 (c2) · 11+ρ

,

sodass die Optimalbedingung

u0 (c1)

u0 (c2)(1 + ρ)

!= 1 + r

oderu0 (c1)

u0 (c2)!=1 + r

1 + ρ

lautet.Sind r und ρ gleich, erhält man die Gleichheit der Grenznutzen und

damit die Gleichheit des Konsums in beiden Perioden. Dieser Ausgleichdes Konsums rührt von der konkaven Nutzenfunktion u her.Ist der Zinssatz kleiner als die Zeitpräferenzrate, ist u0(c1)

u0(c2)kleiner

als 1 und u0 (c1) kleiner als u0 (c2). Aufgrund der Konkavität der Nut-zenfunktion können wir dann auf c1 > c2 schließen; der Haushalt kon-sumiert also in der ersten mehr als in der zweiten. Die Ertragsratedes Konsums, ρ, ist ja größer als die Ertragsrate des Kapitals, r. Istumgekehrt der Zinssatz größer als die Zeitpräferenzrate, wird in derzweiten Periode mehr als in der ersten Periode konsumiert.

E.4.2. Für θ = 0 ist die Perioden-Nutzenfunktion durch u (c) = c

bestimmt. Für θ → 1 geht der Zähler gegen 1 und der Nenner gegennull, sodass der Limes unendlich beträgt.

E.4.3. Man erhält

u0 (c) =1

c= c−1 > 0,

u00 (c) = −1c−2 = − 1c2< 0,

und daher

εu0,c =u00 (c) · cu0 (c)

=− 1c2· c

1c

= −1.

E.6 Lösungen zu den Aufgaben 109

E.4.4. Das Konsumausgleichsmotiv entfällt bei u00 = 0, wenn also diePeriodennutzenfunktion nicht streng konkav ist. Dann erhält man alsOptimierungsbedingung

ρ!= f 0 (k)− δ.

F. Schumpetersche Modelle

F.1 Kreative Zerstörung bei Schumpeter

Joseph A. Schumpeter hat 1942 in den USA ein Buch mit dem Titel ”-Capitalism, Socialism and Democracy” veröffentlicht. In diesem Buchversucht er zu begründen, warum der Kapitalismus nicht überlebenkann und warum Sozialismus überlebensfähig und dem Kapitalismusin mancherlei Hinsicht überlegen sei. Teil II seines Buches (Can Ca-pitalism Survive) enthält ein Kapital über ”The Process of CreativeDestruction”.Dieser Prozess der kretiven Zerstörung ist dem Kapitalismus we-

senseigen. Schumpeter (1976, S. 82f) schreibt:”Capitalism ... is by nature a form or method of economic change

and not only never is but never can be stationary. And this evolutio-nary character of the capitalist process is not merely due to the factthat economic life goes on in a social and natural environment whichchange and by its change alters the data of economic action; this factis important and these changes (wars, revolutions and so on) oftencondition industrial change, but they are not its prime movers. Nor isthis evolutionary character due to a quasi-automatic increase in po-pulation and capital or to the vagaries of monetary systems of whichexactly the same thing holds true. The fundamental impulse that setsand keeps the capitalist engine in motion comes from the new con-sumers’ goods, the new methods of production or transportation, thenew markets, the new forms of industrial organization that capitalistenterprise creates.”Schumpeter (1976, S. 83) beschreibt dann einige organisatorische,

produktionstechnische Revolutionen mit folgenden Worten:

112 F. Schumpetersche Modelle

”... the history of the productive apparatus of a typical farm, fromthe beginnings of the rationalization of crop rotation, plowing andfattening to the machanized thing of today - linking up with elevatorsand railroads - is a history of revolutions. So is the history of theproductive apparatus of the iron and steel industry from the charcoalfurnace to our own type of furnace, or the history of the apparatusof power production from the overshot water wheel to the modernpower plant, or the history of transportation from the mailcoach tothe airplane.The opening up of new markets, foreign or domestic, andthe organizational development from the craft shop and factory to suchconcerns as U.S. Steel illustrate the same process of industrial mutation- if I may use that biological term - that incessantly revolutionizes theeconomic structure from within, incessantly destroying the old one,incessantly creating a new one. This process of Creative Destructionis the essential fact about capitalism. It is what capitalism consists inand what every capitalist concern has got to live in.”Schließlich kritisiert er gängige ökonomische Analysen (S. 84f.): ”E-

conomists are at long last emerging from the stage in which price com-petition was all they saw. As soon as quality competition and sales ef-fort are admitted into the sacred precincts of theory, the price variableis ousted from its dominant position. However, it is still competitionwithin a rigid pattern of invariant conditions, methods of productionand forms of industrial organization in particular, that practically mo-nopolizes attention. But in capitalist reality as distinguished from itstextbook picture, it is not that kind of competition which counts butthe competition from the new commodity, the new technology, the newsource of supply, the new type of organization ... competition whichcommands a decisive cost or quality advantage and which strikes notat the margins of the profits and the outputs of the existing firms butat their foundations and their very lives. This kind of competition is asmuch more effective than the other as a bombardement is in compari-sion with forcing a door, and so much more important that it becomesa matter of comparative indifference whether competition in the or-dinary sense functions more or less promptly: the powerful lever thatin the long run expands output and brings down prices is in any casemade of other stuff.”

F.2 Der Erfolg von Forschungsbemühungen als Poissonprozess 113

Schumpeter weist in diesem Zusammenhang auf die Rolle des po-tentiellen Wettbewerbs hin (S. 85): ”It disciplines before it attacks.”

F.2 Der Erfolg von Forschungsbemühungen alsPoissonprozess

Forschungsanstrengungen müssen nicht notwendigerweise zum Erfolgführen. Aghion and Howitt modellieren den Erfolg der Forschungsan-strenungen mithilfe eines so genannten Poisson-Prozesses.Ein Poisson-Prozess ist durch eine bestimmte Verteilungsfunktion

F gekennzeichnet, die angibt, mit welcher Wahrscheinlichkeit ein Er-eignis bis zu einem vorgegebenen Zeitpunkt T nach dem Zeitpunkt 0erstmalig eintritt. Diese Funktion ist durch

F (T ) = 1− e−μT ,μ > 0

gegeben.

Exercise F.2.1. Bestimmen Sie

F (0) und lim T→∞F (T ) .

Die Dichte dieser Verteilungsfunktion lautet

f (T ) = F 0 (T ) = −e−μT (−μ) = μe−μT > 0.

Dies bedeutet, dass die Wahrscheinlichkeit des Eintritts zwischen Tund T + dt annäherungsweise

μe−μTdt

beträgt. Und die Wahrscheinlichkeit des Eintritts zwischen 0 und 0+dt(d.h. die Wahrscheinlichkeit des Eintritts innerhalb von dt) ist

μdt.

Man nennt μ auch die Flusswahrscheinlichkeit des Ereignisses oder dieAnkunftsrate.


Exercise F.2.2. Bestimmen Sie die Wahrscheinlichkeit des Eintrittszwischen T und T + dt, falls bis T das Ereignis nicht eingetreten ist.Hinweis: Gefragt ist hier nach der bedingten Wahrscheinlichkeit. All-gemein ergibt sich für Ereignisse A und B die Wahrscheinlichkeit vonA unter der Voraussetzung, dass B sich ereignet, als

w (A |B ) = w (A ∧B)w (B)

.

Für unsere Aufgabe haben wir

A : Eintritt zwischen T und T + dt

B : bisheriges Nichteintreten

zu setzen.

1− e−13T , 1− e−1

5T

T 420-2-4

0

-1

-2

-3

-4

13e− 13T , 15e

− 15T

T 420-2-4

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

F.2 Der Erfolg von Forschungsbemühungen als Poissonprozess 115

Je größer die Ankunftsrate ist, desto höher ist die Wahrscheinlich-keit, bis zu einem bestimmten vorgegebenen Zeitpunkt eine Innovationzu erreichen.Bei unabhängigen, gleichzeitig ablaufenden Poisson-Prozessen kann

man die Ankunftsraten addieren. Sind μ1 und μ2 die Ankunftsratenzweier Poisson-Prozesse bezüglich desselben Ereignisses, so ist

μ1 + μ2

die Ankunftsrate des Gesamtprozesses. Zur Begründung: Beim Ge-samtprozess stellt sich das Ereignis ein, wenn es sich im ersten oder imzweiten oder auch in beiden Prozessen ergibt. Das simultane Auftretenin beiden Prozessen kann man jedoch vernachlässigen.Man kann nun zwei unterschiedliche Fragen stellen:

— Wie lange hat man durchschnittlich auf das erste Eintreten des Er-eignisses zu warten? Diese Zeitdauer ist durch den ErwartungswertZ ∞

0Tf (T ) dT

=

Z ∞

0Tμe−μTdT

=1

μ(F.1)

gegeben, wobei der letzte Rechenschritt in Kürze erläutert werdenwird.

— Alternativ kann man fragen, wie viele Ereignisse man innerhalb einerZeiteinheit zu erwarten hat. Wir haben soeben festgestellt, dass in-nerhalb der Zeitspanne 1μ ein Ereignis zu erwarten ist. Daher kommtman innerhalb einer Zeiteinheit auf μ Ereignisse. Und innerhalb derZeitspanne ∆ hat man μ∆ Ereignisse zu erwarten.

— Schließlich fragen wir nach der Wahrscheinlichkeit für m Ereignis-se innerhalb einer vorgegebenen Zeitspanne ∆. Wir geben sie ohneBeweis durch

g (m) =(μ∆)m e−μ∆

m!

wieder.


Die durchschnittliche Wartezeit (siehe Gl. F.1) kann man mithil-fe der partiellen Integration (siehe Appendix) berechnen. Man erhältzunächst für ein τ > 0Z τ

0Tf (T ) dT = TF (T )|τ0 −

Z τ

0F (T ) dT

= T¡1− e−μT

¢¯τ0−Z τ

0

¡1− e−μT

¢dT

= τ¡1− e−μτ

¢− 0

¡1− e−μ0

¢−Z τ

01dT +

Z τ

0e−μTdT

=¡τ − τe−μτ

¢− 0− τ +

µ− 1μe−μT

¯τ0

¶= −τe−μτ − 1

μe−μτ +

1

μe−μ·0

= − τ

eμτ− 1

μeμτ+1

μ

und dann durch Grenzübergang und Anwendung der Regel von del’Hospital (Appendix)

limτ→∞

Z τ

0Tf (T ) dT = lim

τ→∞

∙− τ

eμτ− 1

μeμτ+1

μ

¸= − lim

τ→∞τ

eμτ− lim

τ→∞1

μeμτ+ lim

τ→∞1

μ

= − limτ→∞

τ

eμτ− 0 + 1

μ

= − limτ→∞

1

μeμτ− 0 + 1

μ(Regel von de l’Hospital)

=1

μ.

F.3 Intertemporale Nutzenfunktion

Die Ökonomie besteht aus einer kontinuierlichen Masse aus L Indivi-duen, deren intertemporale Präferenzen durch die Nutzenfunktion

U (c) =

Z ∞

0c (τ) e−rτdτ

F.4 Produktionsfunktion: Endprodukt, Zwischenprodukt und Prozessinnovation 117

wiedergegeben werden, wobei r den Zinssatz und die gleich hohe Zeit-präferenzrate darstellt. Jedes Individuum verfügt über eine Einheit desFaktors Arbeit, so dass L das Arbeitsangebot darstellt.

F.4 Produktionsfunktion: Endprodukt,Zwischenprodukt und Prozessinnovation

Das in einer Periode Produzierte wird in derselben Periode konsumiert(keine Ersparnis), wobei die Produktionsfunktion

(c (t) =) y = Axα, 0 < α < 1

gilt. x ist ein Zwischenprodukt, das wesentlich (die anderen Faktorenwerden vernachlässigt) in die Herstellung von y eingeht. A steht fürdie erfolgreiche Prozessinnovation. Ein höherer A-Wert schlägt sichin einer größeren Menge des Endproduktes bei gegebener Menge desZwischenproduktes nieder.

Exercise F.4.1. Bestimmen Sie

εy,x =

∂yy

∂xx

=∂y

∂x

x

y!

Modelliert wird die erfolgreiche Prozessinnovation durch eine Er-höhung des Technologieparametes A. Anstelle von At bei bisher terfolgten Innovationen hat man aufgrund einer weiteren InnovationAt+1 = Atβ mit β > 1. Jede weitere Innovation führt somit zu ei-ner weiteren Erhöhung des Technologieparameters. Während wir mitt also die die Anzahl der bisher erfolgten Innovationen bezeichnen,reservieren wir τ für den Zeitindex. Aus Gründen der Schreib- undLeseökonomie lassen wir t häufig weg.

F.5 Die Produktion des Zwischenproduktes

Eine Einheit des Zwischenprodukts benötigt eine Einheit Arbeit. BeiL Arbeitskräften verbleibt ein Rest von L−x, der der Forschung- undEntwicklung dient:


a = L− x.

Die Ankunftsrate der Innovation beträgt annahmegemäß

λa,

wobei a die Anzahl der Arbeitskräfte wiedergibt, die in der Forschungtätig sind, und damit die Forschungsanstrengungen. λ ist ein Parame-ter ist, der die Produktivität der Forschungstechnologie misst.Entsprechend den Schumpeter’schen Gedanken wird ein erfolgreich

innovierendes Unternehmen zum Monopolisten, bis es seinerseits auf-grund einer Innovation durch ein anderes Unternehmen vom Marktverschwindet. Man hat sich dabei vorzustellen, dass eine Innovationdurch Patent geschützt ist, das Konkurrenten jedoch nicht davon ab-halten kann, von dem neuen Stand der Technologie Kenntnis zu erlan-gen.

F.6 Zwei Gleichgewichtsbedingungen

Das Modell ist durch zwei Gleichgewichtsbedingungen gekennzeichnet.Die erste lautet

L = xt + at. (F.2)

Hier meint t wiederum die Phase nach der t-ten Innovation. Die zweitebetrifft die Anreize, im Forschungs- oder im Produktionssektor tätigzu werden. Im Modell wird a endogen bestimmt durch die Arbitrage-Bedingung

wt|{z}Lohnsatz im

Produktionssektor

!= λ|{z}Ankunftsrateder t+ 1-tenInnovation

Vt+1|{z}Barwert

der Gewinneaus dert+ 1-tenInnovation

. (F.3)

wt ist der Lohnsatz in der Produktion des Zwischenprodukts nach dert-ten Innovation und Vt+1 der Barwert der Gewinne aus der t+1−ten

F.6 Zwei Gleichgewichtsbedingungen 119

Innovation. λVt+1 ist dann der erwartete Barwert aufgrund der An-kunftsrate λ, die sich bei einem Arbeiter ergibt. Dabei ist daran zu er-innern, dass bei der Flusswahrscheinlichkeit μ die Wahrscheinlichkeitfür das Eintreten eines Ereignisses zwischen Zeitpunkt 0 und Zeitpunktdt gleich μdt ist. Die gleiche Wahrscheinlichkeit gilt für das Eintretenzwischen T und T+dt ist, wenn bisher ein solcher Eintritt nicht erfolgtist. Nimmt man nun dt = 1 (eine ganz kleine Zeitspanne), so setzt dieArbitrage-Bedingung den Lohnsatz für diese kleine Zeitspanne im Pro-duktionssektor gleich den erwarteten Gewinnen im Forschungssektor.Irritieren mag hierbei, dass die Anzahl der forschenden Wettbewer-

ber offenbar keinen Einfluss auf die Wahrscheinlichkeit dafür hat, dasein gegebener Forscher der erste ist. Man hat sich dies damit zu er-klären, dass die Zeitspanne, in der ein Forscher durch einen anderenüberholt werden kann, als sehr klein zu nehmen ist. Formal gilt ja dieAdditivität der Ankunftsraten. Für einen einzelnen Forscher hat manalso

λi =Xj

λj −Xj 6=i

λj .

Um den Barwert Vt+1 genauer betrachten zu können, müssen wirzunächst wiederholen, was der Barwert der so genannten ewigen Renteist. Man erhält ihn aufgrund einer Formel für die unendliche geome-trische Reihe. Eine unendliche geometrische Reihe ist eine unendlicheSumme, wobei ein Summand aus dem ihm vorangehenden Summan-den durch Multiplikation mit einem festen Faktor hervorgeht. BeimAnfangswert A und beim Faktor k ergibt sich die unendliche geome-trische Reihe

s = A+Ak +Ak2 + ... .

Durch Betrachten von s−sk kann man zeigen, dass s für einen Faktork mit −1 < k < 1 gleich

A

1− kist.Mit diesen Vorüberlegungen können wir die ewige Rente bestim-

men. Ein Individuum erhalte am Ende jeder Periode τ = 1, 2, 3, ... dieAuszahlung π. Beim Zinssatz r (0 < r < 1) bzw. beim Diskontfaktor11+r < 1 beträgt der Barwert somit


π

1 + r+

π

(1 + r)2+

π

(1 + r)3+ ...

=π1+r

1− 11+r

=π1+r

1+r1+r −

11+r

=π

r.

Exercise F.6.1. Alternativ könnte man den Barwert der ewigen Ren-te im kontinuierlichen Fall durchZ ∞

0πe−rτdτ

bestimmen. Was erhält man?

Wir bezeichnen den Gewinn nach der t+1-ten Innovation mit πt+1.Somit ist πt+1

r der Barwert der Gewinne, wie er sich ergäbe, wenn abder t + 1-ten Innovation keine weitere mehr erfolgt. Jedoch wird miteiner gewissen Wahrscheinlichkeit eine neue Innovation erfolgen, diesomit den Gewinnstrom abbrechen lässt. Hierbei wird angenommen,dass der etablierte Monopolist (der Innovator der t + 1-Innovation)selbst nicht innoviert. Tatsä chlich hat der Monopolist selbst insofernden geringsten Anreiz zur Innovation, weil er sich im Erfolgsfall nurselbst ersetzen würde (Arrow-Effekt). Zu den Innovationsanreizen kannman Pfähler/Wiese (2006, Kap. G) konsultieren.Den Wert Vt+1 kann man sich nun so bestimmen:

rVt+1| {z }Zinseinkommenaus der t+ 1-tenInnovation

= πt+1|{z}Gewinn ausder t+ 1-tenInnovation

− λat+1| {z }Ankunftsrateder t+ 2-tenInnovation beiat+1 Forschern

Vt+1|{z}Barwert

der Gewinneaus dert+ 1-tenInnovation

.

Hierbei ist rVt+1 das aus einer Lizenz generierte erwartete Einkommenaus der t + 1−ten Innovation während einer Zeiteinheit. at+1 ist die


für F&E aufgewandte Arbeit nach der t + 1−ten Innovation. Das Li-zenzeinkommen ist gleich dem Gewinnstrom πt+1, den der Monopolistdes Zwischengutes erhält, abzüglich des erwarteten KapitalverlustesVt+1, das mit einer Flusswahrscheinlichkeit λat+1 erfolgt. λat+1 nenntman auch Obsoleszenzrate. Schreibt man nach Division mit r die obigeGleichung als

Vt+1 =πt+1r− λat+1Vt+1

r

so liegt folgende Interpretation nahe: Der Wert der t + 1−ten Inno-vation ist gleich dem Barwert πt+1

r der zukünftigen Gewinne aus derMonopolposition abzüglich des Erwartungswertes des Barwertes derzukünftigen Verluste (aus dem Risiko, dass ein anderes Unternehmeninnoviert).Durch Auflösen nach Vt+1 erhält man nun

Vt+1 =πt+1

r + λat+1. (F.4)

Der Wert der Innovation kann also als ewige Rente aufgefasst werden,wobei r + λat+1 der um die Obsoleszenzrate erweiterte Zinssatz ist.Nun mag man die obige (elegante) Herleitung mit Skepsis betrach-

ten. Wir wollen daher zwei direkte Wege gehen, die zum gleichen Er-gebnis führen. Mit τ := τ (t+ 1) bezeichnen wir im Folgenden denZeitpunkt, ab dem die t + 1 Innovation erfolgt. Wir erhalten dannwunschgemäß


Vt+1 =

Z ∞

τ(t+1)

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

πt+1|{z}Monopol-

gewinn in dT

e−r(T−τ)| {z }Diskont-

faktor

ab τ

h1−

³1− e−λat+1(T−τ)

í| {z }

Wahrscheinlichkeit,

dass bis T keine

weitere Innovation

erfolgt,

bei Flussrate λat+1ab τ

+ 0|{z}Zerstörung

in dT

e−r(T−τ)| {z }Diskont-

faktor

ab τ

h1−

³1− e−λat+1(T−τ)

í| {z }

Wahrscheinlichkeit,

dass bis T

(mindestens) eine

weitere Innovation

erfolgt,

bei Flussrate λat+1ab τ

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

dT

= πt+1

Z ∞

τ(t+1)e−(r+λat+1)(T−τ)dT

= πt+1

Ãe−(r+λat+1)(T−τ)

− (r + λat+1)

¯¯∞

τ

!


= πt+1

ÃlimT→∞

e−(r+λat+1)(T−τ)

− (r + λat+1)− e

−(r+λat+1)(τ−τ)

− (r + λat+1)

!

= πt+1

Ãe−(r+λat+1)(τ−τ)

r + λat+1

!=

πt+1r + λat+1

Vielleicht noch schöner einsichtig ist die folgende Berechnung. Dazuermitteln wir zunächst den Barwert PV (τ , T ) der Monopolgewinnevon τ bis T. Dieser ist

PV (τ , T )| {z }Barwert

der Monopolgewinne

von τ bis T

=

Z T

τπt+1|{z}

Monopol-

gewinn in dT

e−r(z−τ)| {z }Diskont-

faktor

ab τ

dz

= πt+1

Z T

τ(t+1)e−r(z−τ)dz

= πt+1e−r(z−τ)

−r

¯¯T

τ

= πt+1e−r(T−τ)

−r − e−r(τ−τ)

−r

= πt+11− e−r(T−τ)

r.

Damit erhalten wir den Erwartungswert

Vt+1 =

Z ∞

τPV (τ , T )| {z }Barwert

der Monopolgewinne

von τ bis T

hλat+1e

−λat+1(T−τ)idT| {z }

Wahrscheinlichkeit

der Innovation

zwischen T und dT,

ab τ

=

Z ∞

τπt+1

1− e−r(T−τ)r

hλat+1e

−λat+1(T−τ)idT


=πt+1r

Z ∞

τ

hλat+1e

−λat+1(T−τ)idT − πt+1

r

Z ∞

τe−r(T−τ)

hλat+1e

−λat+1(T−τ)idT

=πt+1r

λat+1

Z ∞

τ

he−λat+1(T−τ)

idT − πt+1

rλat+1

Z ∞

τ

he−λat+1(T−τ)−r(T−τ)

idT

=πt+1r

λat+1

Z ∞

τ

he−λat+1(T−τ)

idT − πt+1

rλat+1

Z ∞

τ

he−(λat+1+r)(T−τ)

idT

=πt+1r

λat+1e−λat+1(T−τ)

−λat+1

¯¯∞

τ

− πt+1r

λat+1e−(λat+1+r)(T−τ)

− (λat+1 + r)

¯¯∞

τ

=πt+1r

λat+1

ÃlimT→∞

e−λat+1(T−τ)

−λat+1− e

−λat+1(τ−τ)

−λat+1

!

−πt+1r

λat+1

ÃlimT→∞

e−(λat+1+r)(T−τ)

− (λat+1 + r)− e

−(λat+1+r)(τ−τ)

− (λat+1 + r)

!

=πt+1r

λat+1

µ1

λat+1

¶− πt+1

rλat+1

µ1

λat+1 + r

¶=

πt+1r− πt+1

r

λat+1λat+1 + r

=πt+1r

µ1− λat+1

λat+1 + r

¶=

πt+1λat+1 + r

(juchu!)

F.7 Gewinnmaximierung des Prozessinnovators

Wir haben nun noch πt und xt zu bestimmen. Der Monopolist produ-ziert das Zwischenprodukt und hat dabei (unter Weglassen von t)

maxx[p (x)x− wx]

zu lösen.Zunächst haben wir zu klären, welcher Preis p sich für die Menge

x des Zwischenproduktes einstellen wird. Aghion und Howitt nehmenan, dass der Endprodukt-Markt wettbewerblich ist. Dann nimmt der(repräsentative) Produzent des Endproduktes den Preis pt als Datumund maximiert

y − px= Axα − px,

F.7 Gewinnmaximierung des Prozessinnovators 125

indem er x aufgrund der ”Faktorpreis = Grenzwertprodukt”-Regel be-stimmt:

p!= Aαxα−1.

(Dies ist eine etwas waghalsige Konstruktion. Entweder gibt es nureinen einzigen Produzenten; dann ist unklar, warum er den Preis alsDatum nehmen sollte. Oder aber es gibt viele Produzenten, sagen wira an der Zahl; dann würde beim Preis p die Nachfrage nach dem Zwi-schenprodukt a ·x betragen.) Schließlich steht hinter dieser Konstruk-tion das zweistufige Spiel, nach dem der Zwischenprodukt-Monopolistzunächst einen Preis p festlegt und der (bzw. die) Endprodukt-Herstellerdie Faktornachfrage (das ist die Nachfrage nach dem Zwischenprodukt)festlegt.Die Gewinnfunktion des Monopolisten lautet damit:

π = p (x)x− wx= Aαxα−1x− wx= Aαxα − wx.

Er bestimmt den Preis, indem er diesen Gewinn maximiert. DurchDifferenzieren nach x erhält man

Aα21

x1−α−w,

woraus sich durch Nullsetzen und Auflösen nach x

x∗ =

µAα2

w

¶ 11−α

(F.5)

ergibt. Der vom Monopolisten festgesetzte Preis ist demnach

p∗ = p (x∗) = Aα (x∗)α−1

= Aα

µAα2

w

¶α−11−α

=w

α

Wir setzen nun ω := wA und haben dann


x∗ =

µα2

ω

¶ 11−α

und

p∗ =ωA

α

Man kann ω als den produktivitätsbereinigten Lohnsatz bezeichnen.Bei der gewinnmaximalen Menge des Zwischenproduktes x∗ ergibt sichder Gewinn

π (ω) = p (x∗)x∗ − wx∗

= [p (x∗)− w]x∗

=hwα−w

ix∗

=

∙1

α− 1¸wx∗ (F.6)

= A

∙1

α− 1¸ωx∗ (F.7)

bzw. der produktivitätsbereinigte Gewinn

eπ : =1

Atπ

=

∙1

α− 1¸ωtxt. (F.8)

Nach Einsetzen von x erhält man

eπ =

∙1

α− 1¸ω

µα2

ω

¶ 11−α

=

∙1

α− 1¸ωα

21−α

ω1

1−α

=

∙1

α− 1¸

α2

1−α

ω1

1−α−1

=

∙1

α− 1¸α

21−α

ωα

1−α(F.9)

mit∂eπ∂ω

< 0 wegen α < 1.

F.8 Wachstumsgleichgewicht und Stationarität 127

F.8 Wachstumsgleichgewicht und Stationarität

Zur Erinnerung: Wir haben neben der Gleichung

L = xt + at, (F.10)

die auf die Räumung des Arbeitsmarktes zielt, die Arbitrage-Bedingung

wt!= λVt+1 = λ

πt+1r + λat+1

bzw.

ωt =wtAt

!=1

Atλπ (ωt+1)

r + λat+1=

βAtAtβAt

λπ (ωt+1)

r + λat+1=

β

At+1λπ (ωt+1)

r + λat+1(F.11)

zu beachten.EinWachstumsgleichgewicht kann nun nicht durch konstante Wachs-

tumsraten definiert sein. Denn das Wachstum erfolgt in Sprüngen,die zufallsabhängig sind. Wir definieren daher ein Wachstumsgleich-gewicht im Schumpeter-Modell dadurch, dass die Gl. F.10 und F.11erfüllt sind und alle Größen bei einer neuen Erfindung mit konstantemFaktor (z.B. β oder 1) zu multiplizieren sind.Aufgrund von Gl. F.10 hat a stationär zu sein. Denn wenn a bei

jeder Innovation mit einem Faktor größer als 1 wüchse, wäre schließlicha größer als L, also mehr als L Beschäftigte in der Forschung tätig. Undwenn a bei jeder Innovation mit einem Faktor kleiner als 1 schrümpfe,müsste x mit einem konstanten Faktor größer als 1 wachsen. Und dannwären bald mehr als L Beschäftigte im Produktionssektor tätig. DieAufteilung der Beschäftigten in Forschungs- und Produktionsarbeiterist also nicht variabel.Die Menge des Endproduktes, yt = Atx

αt , steigt somit bei jeder

Innovation mit dem Faktor β an. Aufgrund von Gl. F.5 folgt aus derStationarität von x auch die Stationarität von ω = w/A. Aus dieserStationarität folgt wegen Gl. F.9 auch diejenige des produktivitätsbe-reinigten Gewinns eπ, während der Gewinn π = eπA, der Lohn w = ωA

und der Preis des Zwischenproduktes p = wα mit jeder Innovation um

den Faktor β steigen.Aufgrund der Stationarität von x, a,ω und eπ kann man bei diesen

Größen den Index t weglassen und erhält die beiden Gleichgewichts-bedingungen


a

ω

0a L

A

L

Abbildung F.1. Das Gleichgewicht des Aghion-Howitt-Modells

L!= x (ω) + a (bL)

und, unter Beachtung der Definition von eπ,ω

!= λ

βeπ (ω)r + λa

. ( bA)Beide Bedingungen definieren ω implizit als Funktion von a. Zur

Gleichung bA: Wenn a ansteigt, sinktωeπ (ω) = λ

β

r + λa,

was ein Sinken von ω bedeutet (man beachte deπ(ω)dω < 0 wegen Gl. F.9).Die Kurve für bA ist also fallend. Bei Gleichung bL bedeutet ein Anstiegvon a bei Konstanz von L ein Fallen von x (ω) und somit wegen Gl.F.5 ein Anstieg von ω. Die bL-Kurve steigt also an. Es wird sich somitein Gleichgewicht einstellen, wie wir Abb. F.1 entnehmen können.Wir können beide Gleichgewichtsbedingungen zusammenführen, in-

dem wir zunächst aus den Gleichungen F.8 und bLeπ (ω) = ∙ 1

α− 1¸ω (L− a)

entnehmen und hiermit Gleichung bA so:

F.9 Komparative Statik 129

ω = λβeπ (ω)r + λa

= λβ£1α − 1

¤ω (L− a)

r + λa

bzw. so:

H (ba,λ,β, L, r) = 1 = λβ£1α − 1

¤(L− ba)

r + λba (F.12)

schreiben. Hierbei deutet ba auf die gleichgewichtige Anzahl der in FuEBeschäftigten hin.

F.9 Komparative Statik

F.9.1 Gleichgewichtige Forschungsanstrengungen

Im Gleichgewicht hängt der Anteil der Beschäftigten, die im FuE-Sektor tätig sind, von den Parametern r (Zinssatz/Zeitpräferenzrate),L (Größe der Bevölkerung), λ (Produktivität der FuE-Anstrengungen),β (Innovationsschrittweite) und α (Produktionselastizität) ab. Wir un-tersuchen die Wirkung dieser Parameter auf ba der Reihe nach.Wir beginnen mit dem Zinssatz. Eine Erhöhung des Zinssatzes

führt, so lehrt uns Gl. bA??, zu einer Verschiebung der bA-Kurve nachunten. Anhand von Abb. F.1 sehen wir, dass der gleichgewichtige An-teil der im FuE-Sektor Beschäftigten abzunehmen hat. Ein höhererZinssatz reduziert nämlich den Wert der Innovation V (als ewige Ren-te) und damit die Anreize zur Innovation. Formal sieht man dies mit-hilfe von Gl. F.12 und durch Anwendung des ”MRS=Verhältnis derGrenznutzen”-Kalküls:

dbadr

= −∂H∂r∂H∂ba

= −∂

µλβ 1−αα (L−ba)

r+λba¶

∂r

∂


r+λba¶

∂ba= − L− ba

r + λL< 0.


Wegen Gl. bL führt eine Erhöhung der Arbeitskräfte L insgesamt zueiner Verschiebung der bL-Kurve nach rechts und damit zu einem niedri-geren Lohn und einer Erhöhung der FuE-Beschäftigten. Der niedrigereLohn erhöht den Wert der Innovation (über den Gewinn π). Zusätz-lich werden die Kosten der Innovation aufgrund der niedrigeren Löhneauch geringer.Mit einer größeren Innovationsschrittweite steigt der GewinnA [αx (ω)α − ωx (ω)]

und damit der Wert der Innovation.Auch λ lässt ba ansteigen, wie man berechnen kann.

Exercise F.9.1. Bestimmen Sie dbadλ = −

∂H∂λ∂H∂ba .

Dies ist jedoch ein eher zufälliges Ergebnis des Modells. Denn zumeinen führt ein hohes λ zu einer hohen Obsoleszenzrate und lässt vondaher Innovationen nicht attraktiv scheinen. Zum anderen, und dieserEffekt überwiegt offenbar, werden mit λ die Grenzkosten der Forschungreduziert insoweit, als eine gegebene Anzahl von Arbeitsstunden mitgrößerer Wahrscheinlichkeit zum Erfolg führt.Schließlich interessieren wir uns für den Parameter α. Nach Gl. F.12

ist ba eine fallende Funktion von α.

Exercise F.9.2. Berechnen Sie die Preiselastizität der Nachfrage nachdem Zwischenprodukt x aus der inversen Nachfragefunktion

p (x) = αAxα−1.

Je elastischer die Nachfrage nach dem Zwischenprodukt ist (d.h.je weniger der Innovator-Monopolist über Marktmacht verfügt), destogeringer sind demnach die Forschungsanstrengungen.

F.9.2 Gleichgewichtige Wachstumsrate

Wir haben bisher bestimmt, wie sich der Output oder der Konsumändert, wenn wieder ein neuer Forschungsschritt gelingt. Wie verän-dert sich der Output jedoch in der Zeit? Zwischen der t-ten und der(t+ 1)-ten Innovation ist der Outputfluss gleich

yt = Atxα = At (L− ba)α

F.10 Die wohlfahrtsoptimale Wachstumsrate 131

und es giltyt+1 = βyt.

Zwischen einem Zeitpunkt τ und dem Zeitpunkt τ + 1 gebe es ε (τ)Innovationen. Dann gilt

y (τ + 1) = βε(τ)y (τ) .

Hieraus folgt für die diskrete Wachstumsrate

γ1y =y (τ + 1)− y (τ)

y (τ)= βε(τ) − 1.

Aus Gl. B.2 in Kap. B wissen wir, welche kontinuierliche Wachstums-rate γy der diskreten Wachstumsrate γ

1y entspricht:

γy = ln£1 + γ1y

¤.

Zusammen folgt somit

γy = ln£1 + γ1y

¤= ln

hβε(τ)

i= ε (τ) ln [β] .

Aus Abschnitt F.2 wissen wir, dass es zwischen τ und τ +1 durch-schnittlich λba Innovationen gibt. Die erwartete Wachstumsrate wollenwir auch γy nennen. Es gilt also

γy = λba ln [β] .Die erwartete Wachstumsrate hängt nun positiv von L und negativ

von r und α ab, denn diese beeinflussen die gleichgewichtige FuE-Beschäftigung. Die Parameter λ und β wirken dagegen sowohl direkt(über die obige Gleichung) als auch indirekt über ba auf die Wachs-tumsrate positiv ein.

F.10 Die wohlfahrtsoptimale Wachstumsrate

Wir gehen jetzt der Frage nach, welche Wachstumsrate ein benevolen-ter Diktator wählen würde. Die Wohlfahrtsfunktion soll den erwartetenNutzen (gleich erwarteter Konsum) maximieren.Der Konsum im Zeitpunkt τ ist stochastisch gleich


y (τ) =∞Xt=0

prob (t, τ)Atxα,

wobei prob (t, τ) die Wahrscheinlichkeit ist, dass es bis zum Zeitpunktτ genau t Innovationen gibt. In Abschnitt F.2 haben wir festgehalten,dass die Wahrscheinlichkeit fürm Ereignisse innerhalb eines Zeitraums∆ durch

g (m) =(μ∆)m e−μ∆

m!

gegeben ist, wobei hier μ die Ankunftsrate darstellt. Damit ist

prob (t, τ) =(λaτ)t e−λaτ

t!.

Der Dikator maximiert dann

U =

Z ∞

0e−rτy (τ) dτ

=

Z ∞

0e−rτ

Ã ∞Xt=0

prob (t, τ)Atxα

!dτ

=

Z ∞

0e−rτ

Ã ∞Xt=0

(λaτ)t e−λaτ

t!Atx

α

!dτ

=

Z ∞

0e−rτ

Ã ∞Xt=0

(λaτ)t e−λaτ

t!A0β

txα

!dτ

=

Z ∞

0e−rτ

Ã ∞Xt=0

(λaτ)t e−λaτ

t!A0β

t (L− a)α!dτ

= A0 (L− a)αZ ∞

0e−rτ

Ã ∞Xt=0

(βλaτ)t e−λaτ

t!

!dτ

= A0 (L− a)αZ ∞

0e−rτ

Ã ∞Xt=0

(βλaτ)t e−βλaτ e−λaτ

e−βλaτ

t!

!dτ

= A0 (L− a)αZ ∞

0e−rτ

Ã ∞Xt=0

(βλaτ)t e−βλaτ

t!eλaτ(β−1)

!dτ

= A0 (L− a)αZ ∞

0e−rτeλaτ(β−1)

Ã ∞Xt=0

(βλaτ)t e−βλaτ

t!

!dτ

F.10 Die wohlfahrtsoptimale Wachstumsrate 133

= A0 (L− a)αZ ∞

0e−rτeλaτ(β−1) · 1dτ (Summe der Wahrs.)

= A0 (L− a)αZ ∞

0e[−r+λa(β−1)]τdτ

= A0 (L− a)α ·1

− (r − λa (β − 1))e−(r−λa(β−1))τ

¯∞0

= A0 (L− a)α ·1

(r − λa (β − 1))

=A0 (L− a)α

(r − λa (β − 1)) .

Er wählt also a so, dass

0!=d A0(L−a)α(r−λa(β−1))da

=A0α (L− a)α−1 (−1) (r − λa (β − 1))−A0 (L− a)α (−λ (β − 1))

(r − λa (β − 1))2

bzw.

1 =λ (β − 1) 1α (L− a∗)r − λa∗ (β − 1) . (F.13)

Analog zu den Überlegungen im vorigen Abschnitt produzierendie wohlfahrtsoptimalen Forschungsanstrengungen a∗ eine optimaleWachstumsrate

γ∗y = λa∗ ln [β] .

Um den Vergleich zwischen a∗ und ba bzw. zwischen γ∗y und bγyherzustellen, erinnern wir an die Gleichgewichtsbedingung F.12:

1 = λβ 1−αα (L− ba)r + λba .

Die beiden Bedingungen unterscheiden sich in dreifacher Hinsicht:

1. Die private Diskontierungsrate berücksichtigt neben dem Zinssatzr die Obsoleszenz. Dies tut auch die soziale Diskontierungsrate,die ihrerseits jedoch um den Summanden λa∗β geringer ist alsdie private Diskontierungsrate. Der wohlwollende Diktator berück-sichtigt, dass der Nutzen der nächsten Innovation aufgrund dermultiplikativen Struktur (A, βA, β2A, ...) fortwährt, während die


private Unternehmung dies nicht honoriert. Dies nennt man denFuE-Effekt.

2. Der zweite Unterschied besteht darin, dass nur bei den privatenAnreizen der Faktor (1− α) zusätzlich auftaucht. Aghion und Ho-witt (S. 62) sehen hier den Aneignungseffekt. Allerdings ist dieserdoch eher in α repräsentiert: Der Preis p (x) ist gleich dem Grenz-produkt

αAxα−1

und nicht gleich dem Durchschnittsprodukt

Axα

x= Axα−1.

Oder anders: Der Gesamtoutput des Endprodukts beträgt Axα,während der Zwischenproduzent nur p (x)x = αAxα−1x = αAxα

erhält. Alternativ könnte man folgende Interpretation versuchen.Man schreibt die Gleichgewichtsbedingung so auf:

1 = λβ 1α (L− ba)(r + λba) · 1

1−α

= λβ 1α (L− ba)

(r + λba) · |εx,p| .Je größer die Preiselastizität der Nachfrage ist, desto geringer derAnreiz Innovation zu betreiben, d.h. desto geringer ist ba.

3. Der dritte Unterschied betrifft β−1 in der Optimalbedingung ver-sus β in der Gleichgewichtsbedingung. Der Monopolist berücksich-tigt nicht, dass er dem alten Monopolisten seinen Gewinn nimmt(business stealing).

Zuviel FuE-Tätigkeit und daher zu hohes Wachstum ist möglichund insbesondere dann zu erwarten, wenn die Monopolmacht groß ist(α nahe bei Null) und somit der Konsumentenrenten-Effekt niedrigund wenn die Innovationssprünge nicht so groß sind (β eher klein) undsomit der FuE-Effekt niedrig. Dann kann der Business-Stealing-Effektüberwiegen. Es gibt dann zu viel Wachstum und anfänglich zu wenigKonsum.

F.11 Lösungen zu den Aufgaben 135

F.11 Lösungen zu den Aufgaben

F.2.1. Man erhält

F (0) = 0 und limT→∞F (T ) = 1.

F.2.2. Wenn das Ereignis zwischen T und T +dt eintritt, ist es bishernicht eingetreten. Daher ist die Wahrscheinlichkeit für das Eintretenzwischen T und T + dt (A) und das bisherige Nichteintreten (B) ein-fach gleich der Wahrscheinlichkeit für das Eintreten zwischen T undT + dt (A ∩ B = A). Man erhält daher die gesuchte bedingte Wahr-scheinlichkeit als

w (A)

w (B)=

μe−μTdt

1− F (T ) =μe−μTdt

1− (1− e−μT ) = μdt.

F.4.1. Die Produktionselastizität (des einzigen Faktors ”Zwischenpro-dukt”) ist

εy,x =∂y

∂x

x

y

= αAxα−1x

Axα= α.

F.6.1. Man errechnet:

Z ∞

0πe−rτdτ

= π

Z ∞

0e−rτdτ

= π limT→∞

Ã−1re−rτ

¯T0

!

= π limT→∞

µ−1re−rT −

µ−1re−r·0

¶¶= π

µ−1rlimT→∞

e−rT +1

rlimT→∞

e−r·0¶

=π

r.


F.9.1. Man errechnet

dbadλ

= −∂H∂λ∂H∂ba

= −∂


r+λba¶

∂λ

∂


r+λba¶

∂ba= r

L− ba(r + λL)λ

> 0.

F.9.2. Die Preiselastizität der Nachfrage nach dem Zwischenproduktx lautet

dx

dp· px

=1dpdx

· px

=1

(α− 1)αAxα−2αAxα−1

x

=1

(α− 1)

= − 1

1− α.

G. Overlapping generations

G.1 Overview of the model and overlapping generations

In this chapter, we will present the basic overlapping-generations mo-del. The chapter is based on the textbook by de la Croix/Michel(2002). In this section, we will give a short overview of the model whichwill be expounded in more detail in the sections to come.Overlapping-generations models are of interest for a number of re-

asons. They can show

— that debt does not need to be neutral (as claimed by the Ricardoequivalence),

— how asset bubbles can develop,— that competitive equilibria may be inefficient,— how pension schemes can be analyzed.

The most simple overlapping-generations model assumes that agentslive for two periods. In each period t = 0, 1, 2, ..., Nt persons are born.In order to get the model started, we assume a retired population attime 0. The birth of these N−1 agents belongs to pre-history. There-fore, in every period t = 0, 1, 2, ..., a new generation of Nt agents isborn while Nt−1 persons are retired who will be dead in t+1. Fig. G.1(adapted from the above-mentioned textbook) depicts this setup.Assuming a constant discrete growth rate of the population, the

number of historic agents is given recursively by

Nt = (1 + n)Nt−1, t ≥ 0.

This describes the population dynamics. We now provide a broad-brush view of the consumption, saving, and capital dynamics. Every

138 G. Overlapping generations

0

N1 retired agentsof generation 1

1 2 3 4

generations

1−

0

1

2

3

N1 youngstersof generation 1

N2 retired agentsof generation 2 live at the same timeas N3 youngsters of generation 3

time

Abbildung G.1. Youngsters and retired agents live at the same time

agent lives two periods. In the first period of their life (active period),the homogeneous Nt agents born in period t ≥ 0 work and receivewage earnings wt. The wage is determined competitively on the labormarket. The agents decide how much of this wage to consume in time t,ct, and how much to invest, st. The capital buildup takes one period oftime so that savings in time t are transformed into productive capital intime t+1. For simplicity, it is assumed that capital is productive for oneperiod only. Therefore,Kt+1, the capital available at time t+1, is equalto Ntst. The agents’ saving decisions depend on the return on theircapital. This return is an expected value in time t and the uncertaintyresolves in time t + 1 when the owners of the (representative) firm(the investors from a period before) receive all the profit. The retiredagents eat up the total share of the profit.We consider two different equilibrium concepts. The temporary

equilibrium deals with one period of time, t, only. It depends on thenumber of young and old people alive, the capital available at timet, and the expectations about the firm’s profits in which the youngpeople invest.

G.2 Saving, investment and capital 139

G.2 Saving, investment and capital

The young people supply their one unit of labor and earn a wage wt.They choose how devide their income wt between consumption ct andsaving st:

wt = ct + st, t ≥ 0. (G.1)

Savings are invested in capital. Building capital takes one unit of timeand the capital is productive for one unit of time, only. Therefore, thecapital Kt productive in time t, is equal to

Kt = Nt−1st−1, t ≥ 0.

The idea is that the capital is owned by the retired agents who sharethe profits. Capital at time 0 (which is build up before we look at theeconomy) is equal to K0. Therefore, we let

s−1 :=K0N−1

be the capital share owned by each pre-historical agent.The saving allow these agents to consume dt+1 during their retire-

ment. Retirement consumption depends on the rate of interest rt+1 :

dt+1 = (1 + rt+1) st, t ≥ 0.

This interest rate has to bear a specific relationship to the productivityof capital.

G.3 Utility and optimal saving

G.3.1 Solving the intertemporal household problem

We assume the two-period utility function

U¡ct, d

et+1

¢= u (ct) + u

¡det+1

¢· 1

1 + ρ

where u is period utility function, c the consumption in the first period(young age) and d the expected consumption in the second period (old


age). ρ is the time preference rate (or the rate of return to consump-tion). We assume a concave period utility u (u0 > 0, u00 < 0) with theInada condition

limc→0

u0 (c) =∞

which ensures a positive consumption in each period if a positive in-come can be distributed over time. An example of a suchlike utilityfunction is given by

u (c) =c1−θ

1− θ, θ > 0, θ 6= 1

introduced on p. 102. We observe

u0 (c) =(1− θ) c−θ

1− θ=1

cθ> 0,

u00 (c) = −θc−θ−1 < 0,

and also note the elasticity of marginal utility

εu0,c =u00 (c) · cu0 (c)

=

¡−θc−θ−1

¢· c

c−θ= −θ,¯

εu0,c¯= θ.

As can be seen from the previous section, the intertemporal budgetconstraint is given by

ct +1

1 + ret+1det+1 = wt. (G.2)

From household theory (with ct being the first, and det+1 the secondgood), we know the optimization condition¯

ddet+1dct

¯=MRS =

u0 (ct)

u0¡det+1

¢· 11+ρ

!=

11

1+ret+1

= 1 + ret+1. (G.3)

The above example yieldsµdet+1ct

¶θ!=1 + ret+11 + ρ

. (G.4)

Assume an expected interest rate above the time preference rate,ret+1 > ρ. Then, for any θ > 0, retirement consumption will be above

G.3 Utility and optimal saving 141

the consumption of the active phase. If the elasticity of marginal uti-lity is high (θ > 1), the consumer tries to smooth consumption anddet+1ctwill insignificantly be above 1. If, on the other hand, the elasticity

of marginal utility is low (θ < 1), consumption smoothing is of a les-ser importance so that the ratio of retirement consumption to workerconsumption can be relatively high.

G.3.2 Comparative statics (the savings function)

The optimization program of the preceding section yields the optimalconsumption in both periods and hence the savings function

s (w, 1 + r) = argmaxs∈R

u (w − s) + u ((1 + r) s) · 1

1 + ρ.

It is characterized by

ϕ (s, w, 1 + r) : =du

dc

dc

ds+du

dd

dd

ds

1

1 + ρ

= −dudc+du

dd

1 + r

1 + ρ= 0.

By the implicit function theorem (the ”MRS = marginal utility overmarginal utility” - rule) we find

ds

dw= −

dϕdwdϕds

= −− d2u(dc)2

d2u(dc)2

+ d2u(dd)2

(1+r)2

1+ρ

=− d2u(dc)2

− d2u(dc)2

− d2u(dd)2

(1+r)2

1+ρ

so that

0 <ds

dw< 1.

On the other hand, we have

ds

d (1 + r)= −

dϕd(1+r)

dϕds


= −dudd

11+ρ +

d2u(dd)2

s1+r1+ρ

d2u(dc)2

+ d2u(dd)2

(1+r)2

1+ρ

= −dudd

11+ρ + εu0,d

duddd s

1+r1+ρ

d2u(dc)2

+ d2u(dd)2

(1+r)2

1+ρ

(note εu0,c =u00 (c) · cu0 (c)

)

=

dudd

11+ρ

¡1 + εu0,d

¢− d2u(dc)2

− d2u(dd)2

(1+r)2

1+ρ

(note d = s (1 + r) )

=

⎧⎪⎨⎪⎩> 0, εu0,d > −10, εu0,d = −1< 0, εu0,d < −1

=

⎧⎪⎨⎪⎩> 0,

¯εu0,d

¯< 1

0,¯εu0,d

¯= 1

< 0,¯εu0,d

¯> 1

A rise in the rate of interest gives the household an incentive to increasehis savings because consumption today has become more expensive(in terms of forgone consumption tomorrow, see eq. G.2). This is thesubstitution effect. It will work out particularly strong, if the householddoes not aspire to consumption smoothing. On the other hand, theincome effect means that the household can afford more in both periodsand may therefore be tempted to decrease savings. The income effectwill be particularly strong if supported by a consumption smoothingmotive which is hidden behind a relatively great elasticity of marginalutility

¯εu0,d

¯> 1. By

dudd

11+ρ

¡1 + εu0,d

¢− d2u(dc)2

− d2u(dd)2

(1+r)2

1+ρ

=dudd

¡1 + εu0,d

¢− d2u(dc)2

(1 + ρ)− d2u(dd)2

(1 + r)2,

if the rate of time preference is high, dsd(1+r) is low. The household

wants to consume a lot in the first period and needs strong interest-rate incentives to change his saving behavior.The period utility function

u (c) =c1−θ

1− θ, θ > 0, θ 6= 1

G.4 The (representative) firm 143

yields eq. G.4 which can be written asµ(1 + r) s

w − s

¶θ!=1 + r

1 + ρ

or(1 + r) s

w − s!=

µ1 + r

1 + ρ

¶ 1θ

s (w, 1 + r) =1³

1 + (1 + r)1−1θ (1 + ρ)

1θ

´wwhence we obtain the savings function

s (w, 1 + r) =1

1 + 1+r³1+r1+ρ

´−θ w =1³

1 + (1 + r)1−1θ (1 + ρ)

1θ

´| {z }

propensity to save

w.

According to this function, savings can be written as the product ofthe wage and the propensity to save. If the rate of time preferenceincreases, the household will consume more and save less. If the rateof interest increases, the household will save less if 1− 1

θ > 0⇔ θ > 1,

i.e., if the consumption-smoothing motive is strong.In the logarithmic case, we have

s (w, 1 + r) =1

2 + ρ| {z }propensity to save

w =

11+ρ

1 + 11+ρ| {z }

propensity to save

w.

G.4 The (representative) firm

The representative firm can use a stock of capital K. At the prevailingwage w the firm maximizes profits

π = maxLF (K,L)− wL

where the output price is normalized to 1. We obtain the ”marginalproduct equals factor price”-rule of optimization:


∂F

∂L= w.

We assume constant returns and can make use of eq. D.7 (p. 45):

∂F

∂L= f (k)− k df

dk=: ω (k)

where k is capital per head and f the production function in intensiveform. The reader will note the use ω (k) as a shorthand for f (k)−k dfdk .ω is the Greek letter for a long o whose form is similar to the w whichstands for wage. For the Cobb-Douglas production function

Y = F (K,L) = AKαL1−α, A > 0, 0 < α < 1

we have

y =Y

L=AKαL1−α

L= A

Kα

Lα= Akα =: f (k)

and hence

ω (k) = f (k)− k dfdk= Akα − kAαkα−1

= A (1− α) kα.

Therefore, the income share of labor is

wL

Y=w

y=A (1− α) kα

Akα= 1− α

From a previous section, we have investment It equal to

It = Ntst = Kt+1, t ≥ 0.

Returning to the general case of constant-returns production func-tions, the maximum profit is given by

π = F (K,L)− wL

=∂F

∂KK +

∂F

∂LL− wL (Euler’s theorem, p. 45)

=∂F

∂KK (marginal product of labor equals wage).

G.5 Equilibria 145

This profit is to be devided among the capital owners of the previousperiod. We thereby define the interest rate accruing to the savers:

∂F

∂K

¯(Kt,Lt)

Kt = (1 + rt)Nt−1st−1 (dividends, no undistributed profits)

= (1 + rt)Kt (capital equals investment).

Therefore, we have

1 + rt =∂F

∂K

¯(Kt,Lt)

=∂f

∂k

¯KtLt

. (G.5)

G.5 Equilibria

G.5.1 Temporary equilibrium

A temporary equilibrium at a given point in time t ≥ 0, is based onthe number of young people Nt, the number of old people Nt−1, thecapital available at time t, and the expectations of future returns, ret+1.We have the following definition:

Definition G.5.1. Given the parameters Kt, Nt, Nt−1 and ret+1, atemporary equilibrium at time t is defined

1. by intertemporal optimization, i.e., by the saving st = s¡wt, 1 + r

et+1

¢,

2. by profit maximization, i.e., by the wage wt = ω (kt) with kt = KtLt,

3. by the distribution of total profits to the investors born at time t−1,i.e., by retirement consumption dt = f 0 (kt) Kt

Nt−1,

4. by one of the following two market equilibrium conditions:

— an equilibrium on the labor market, i.e., by Lt = Nt (inelasticsupply) and by the wage rate wt = ω (kt) , in sum by

ω

µKtNt

¶∂F

∂L

¯(Kt,Nt)

= ω

µKtNt

¶= f

µKtNt

¶− KtNt

df

dk

¯KtNt

= wt,

or— an equilibrium on the goods market, i.e.,

Yt (Kt, Nt) = Nt−1dt +Nt (ct + st) .


Note that we have N instead of L in the labor-market condition.The labor-market equilibrium assumes that all young people inelasti-cally supply one unit of labor. The wage rate assures that this inelasticsupply meets sufficient demand.The goods-market equilibrium makes sure that the output in time

t is used up by the old people and by the young people. Note that theyoung people’s demand on output consists of both consumption ct andinvestment st.While, in general, the temporary equilibrium depends on the ex-

pectation on future profits (embodied in ret+1), in specific cases, thisneed not to be the case as the following exercise shows.

Exercise G.5.1. Assume the logarithmic period utility function u (c) =ln c and the Cobb-Douglas production function in intensive formf (k) = Akα. Find the equilibrium values for saving, wage, and con-sumption for young and old, all at time t.

From general equilibrium theory, we know Walras law. It says thatit cannot happen that all markets but one are cleared. Differently put,if we have n markets and n− 1 of these markets are cleared, the nthmarket is also cleared. In our model, this implies that we do not needto check the clearing of both the labor and goods markets. Let usassume that the labor market is cleared. Then,

Nt−1dt = Nt−1∂f

∂k

¯KtLt

KtNt−1

(distribution of total profits)

=∂f

∂k

¯KtNt

Kt (labor market clearing)

and

Nt (ct + st) = Ntwt (eq. G.1, p. 139)

= Ntω

µKtLt

¶(wage equals marginal product of labor)

= Ntω

µKtNt

¶(labor market clearing)

= Ntf

µKtNt

¶−Kt

df

dk

¯KtNt

G.5 Equilibria 147

= Yt −Ktdf

dk

¯KtNt

.

Indeed, Walras law also holds in our model: The sum of Nt−1dt andNt (ct + st) is Yt.

G.5.2 Resource constraint and net production

The goods-market equilibrium

Yt (Kt, Nt) = Nt−1dt +Nt (ct + st)

is also called the resource constraint. At any given time t, the output isto be divided between saving (Ntst), consumption of the young (Ntct)and consumption of the old (Nt−1dt). This identity is reminiscent ofthe so-called Mackenroth theorem 1952 (Gerhard Mackenroth, 1903-1955, German sociologist and statistician):”Nun gilt der einfache und klare Satz, dass aller Sozialaufwand

immer aus dem Volkseinkommen der laufenden Periode gedeckt wer-den muss. Es gibt gar keine andere Quelle und hat nie eine andereQuelle gegeben, aus der Sozialaufwand fließen könnte, es gibt keineAnsammlung von Periode zu Periode, kein ”Sparen” im privatwirt-schaftlichen Sinne, es gibt einfach gar nichts anderes als das laufendeVolkseinkommen als Quelle für den Sozialaufwand. Das ist auch nichteine besondere Tücke oder Ungunst unserer Zeit, die von der Hand inden Mund lebt, sondern das ist immer so gewesen und kann nie anderssein.”Even in our simple model, the Mackenroth theorem is false. Agents

save at time t and thereby increase production at time t+1, when theyare old. Of course, capital buildup is somewhat strange in our model.If capital would not depreciate totally at the end of each period, aneconomy could save by building up capital (roads, machines, humancapital) and desaving by being somewhat slow on replacement. TheMackenroth theorem is wrong for a second reason: If we allow for for-eign trade, consumption can be shifted between periods in one countryby having the opposite shift in other countries.The resource constraint can also be written in per-capita values.

We then get


f (kt) =Yt (Kt, Nt)

Nt=Nt−1Nt

dt + ct +Kt+1Nt+1

Nt+1Nt

= ct +dt1 + n

+ kt+1 (1 + n) . (G.6)

Definition G.5.2. Per-head net production is given by

φ (kt, kt+1) := f (kt)− kt+1 (1 + n) .

By the goods-market equilibrium, the net production is equal toct+

dt1+n . Therefore, it is to be understood as the consumption available

in present-value terms if the capital stock is kt at time t and kt+1 attime t+1.Now, we will rule out the possibility of negative consumption.

Definition G.5.3. Per-head net production is feasible if

φ (kt, kt+1) ≥ 0 (G.7)

holds.Let (kt)t≥0 be a sequence of capital stocks. If the above inequality

holds for all t ≥ 0, (kt)t≥0 is called a feasible path of capital.

G.5.3 Intertemporal equilibrium

While the temporary equilibrium takes a snapshot of the economy, theintertemporal equilibrium ensures that the links to the past and intothe future are consistent. In particular, we need to specify how agentsform expectations about future profits. In our model, the agents saveat time t on the basis of 1+ret+1, i.e., on the basis of their expectationsfor period-t+1 profit (see fig. G.2). Aggregating for all Nt agents yieldsthe total savings which equal total investment by the capital-marketequilibrium which is already the capital at time t + 1 after the one-period buildup. When investors are paid their marginal product ofcapital, they receive df

dk

¯Kt+1Nt+1

per unit invested.

We will shortly deal with rational expectations but will present aprominant alternative first, myopic or adaptive expectations. There,agents expect the world to be as in the past or an extrapolation of thepast. In our model, adaptive expectations would be expressed by

G.5 Equilibria 149

equality in case ofrational expectations

etr+1 ts tK

t

tNKdk

df

savings function

aggregationand capital-marketequilibrium

payment toinvestorsaccording tomarginalproduct of capital

Abbildung G.2. Expectation of future profits determine future profits

1 + ret+1 =df

dk

¯KtNt

.

Here, agents expect to obtain the payoff for their capital in the futureas it is in the presence. Then, the capital available at time t+ 1 is anexplicit function of the capital available at time t :

Kt+1 = s

⎛⎝ω (kt) ,df

dk

¯KtNt

⎞⎠Nt, t ≥ 0.The drawback of adaptive expectations is that agents are wrong. Sin-ce there are infinitely many ways to err, we like to deal with rationalexpectations instead. Rational expectations or perfect forsight meansthat agents foresee the economy’s development as perfectly as the mo-deler can. In case of rational expectations, these interlinkages depictedin fig. G.2 are perfectly understood by the agents so that

1 + ret+1 =df

dk

¯Kt+1Nt+1

results.

Definition G.5.4. Given the parameters K0, and N−1, an intertem-poral equilibrium with perfect foresight (or with rational expectations)is defined


1. by a sequence of temporary equilibria at times t = 0, 1, ...,2. by a consistent population growth, i.e., by

Nt+1 = (1 + n)Nt, t ≥ −1,

3. by equilibrium on the capital market (savings equal investment),i.e., by

Kt+1 = s¡ω (kt) , 1 + r

et+1

¢Nt, t ≥ 0,

4. by rational expectations, i.e., by

1 + ret+1 =df

dk

¯Kt+1Nt+1

, t ≥ 0.

Thus, apart from rational expectations, the definition of an inter-temporal equilibrium ensures that the temporary equilibria are fulfil-led, that the population trajectory is consistent, and that the savingsin period t properly feed into investments in period t and capital inperiod t+ 1.A priori, it is unclear, whether an equilibrium (any equilibrium)

exists. Rational expectations make it particularly hard to prove exi-stence because the t + 1-capital is implicitly (not explicitly) definedby

Kt+1 = s

⎛⎝ω (kt) ,df

dk

¯Kt+1Nt+1

⎞⎠Nt, t ≥ 0.Indeed, de la Croix/Michel (2002, pp. 20) have a lot to say on both

— existence (does any equilibrium exist?) and— uniqueness (given several equilibria, are they identical?).

As in the Solow model, the trajectory of capital per head is ofupmost importance. By dividing the above equation through Nt+1, weobtain

kt+1 =Kt+1Nt+1

= s

⎛⎝ω (kt) ,df

dk

¯Kt+1Nt+1

⎞⎠ NtNt+1

= s

Ãω (kt) ,

df

dk

¯kt+1

!1

1 + n, t ≥ 0.

G.6 Steady state 151

Here, kt+1 is implicitly defined. If, for all t ≥ 0, this equation hasone and only one solution, the existence and uniquenss problems aresolved.The above equation can be rewritten as follows:

(1 + n) kt+1|{z}=Kt+1

Nt+1,

labor-market

equilibrium

=|{z}capital-market

equilibrium,

constant growth

rate of the population

st =|{z}life-cycle utility

maximizing

saving

s

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ω (kt)| {z }

marginal product

of labor

,df

dk

¯kt+1| {z }

rational

expectations

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.

G.6 Steady state

Sometimes, we are even lucky and can get an explicit solution to theimplicit definition of kt+1. This is a function g that links time-t capitalper head to time-t+ 1 capital:

kt+1 = g (kt) .

Proposition G.6.1. Let an intertemporal equilibrium with rationalexpectations and a g-function be given. For every k0, the sequence g (kt)is monotonic.

Definition G.6.1. Let an intertemporal equilibrium with rational ex-pectations and a g-function be given. Then, k∗ > 0 is called a steadystate if

k∗ = g (k∗) .

k∗ = 0 is called a corner steady state if 0 = g (0) holds.


Proposition G.6.2. Let an intertemporal equilibrium with rationalexpectations and a g-function be given. k∗ = 0 is a corner steady stateiff ω (0) = 0. For every k0, the sequence g (kt) converges towards asteady state, towards 0 or tends to infinity.

G.7 Examples

G.7.1 Logarithmic utility and Cobb-Douglas production

First of all, we will revisit the example of the logarithmic period utilityfunction and the Cobb-Douglas production function. In the exerciseabove, we found that expectations do not have any interesting role toplay in the temporary equilibrium. We will easily find the g-function.From

st =

11+ρ

1 + 11+ρ

A (1− α) kαt

we get

kt+1 = g (kt)

=

11+ρ

1 + 11+ρ

A (1− α)1

1 + nkαt .

Obviously, we have a corner steady state. Furthermore, forming thederivative,

dkt+1dkt

= α

11+ρ

1 + 11+ρ

A (1− α)1

1 + n

1

k1−αt

= α (1− α)A1

2 + ρ

1

1 + n

1

k1−αt

we see that the greater k, the smaller the increase in k. By

limkt→0

dkt+1dkt

=∞

we see that this corner steady state is not stable. However, exactly onesteady state exists at

G.7 Examples 153

k∗ =11+ρ

1 + 11+ρ

A (1− α)1

1 + n(k∗)α

⇔ k∗

(k∗)α=

11+ρ

1 + 11+ρ

A (1− α)1

1 + n

⇔ k∗ =

Ã11+ρ

1 + 11+ρ

A (1− α)1

1 + n

! 11−α

At k∗, we have

dkt+1dkt

¯k∗

= α

11+ρ

1 + 11+ρ

A (1− α)1

1 + n

1

k1−α

= α

11+ρ

1 + 11+ρ

A (1− α)1

1 + n

1Ãµ1

1+ρ

1+ 11+ρ

A (1− α) 11+n

¶ 11−α!1−α

= α

11+ρ

1 + 11+ρ

A (1− α)1

1 + n

11

1+ρ

1+ 11+ρ

A (1− α) 11+n

= α.

With the help of fig. G.3, it is easy to confirm that their is only onestable steady state which is at k∗.At

k∗ =

Ã11+ρ

1 + 11+ρ

A (1− α)1

1 + n

! 11−α

we have

s∗ =

11+ρ

1 + 11+ρ

A (1− α)

Ã11+ρ

1 + 11+ρ

A (1− α)1

1 + n

! α1−α

=

Ã11+ρ

1 + 11+ρ

A (1− α)

! 11−α µ 1

1 + n

¶ α1−α

.

G.7.2 Logarithmic utility and CES production

If we have CES production functions, different things can happen(please refer to chapter D, pp. 64):


tk

( )tkg

*k

slope α

Abbildung G.3. A unique stable steady state

— If the elasticity of substitution is in between, we have the Cobb-Douglas case analyzed in the previous section.

— If the elasticity of substitution is high, a zero capital per head en-dowment still leads to a positive production so that capital can beginto grow. The picture is very much like fig. G.3, but g (0) > 0. Thereis no corner steady state and one stable steady state.

— If the elasticity of substitution is low, the factors are high comple-ments so that both are needed for a positive output. Then, we haveg (0) = 0 and the corner steady state is stable. We typically have oneof the two g-functions depicted in fig. G.4. In the lower one, thereis no other steady state. In the upper one, there are two, but onlyone of them is stable. Obviously, the upper g-function lends itself topoverty-trap discussion as in chapter D, on p. 73.

Exercise G.7.1. In fig. G.4, looking at the upper g-function, is k∗a ork∗b stable?

G.8 Golden rule, efficiency, and equilibrium outcome 155

tk

( )tkg

*ak *

bk

Abbildung G.4. None or two steady states

G.8 Golden rule, efficiency, and equilibrium outcome

G.8.1 Definition of efficiency and Pareto optimality

We will now present two different definitions, one defining efficiency,the second Pareto optimality. Efficiency is concerned with the con-sumption possible at any point in time. Pareto optimality points at-tention to the life-cycle utility.

Definition G.8.1. Let (kt)t≥0 be a feasible path of capital (fulfillinginequality G.7, p. 148). This path is called inefficient if another path³bkt´

t≥0exists such that per-head net production is never smaller at³bkt´

t≥0, but sometimes greater. If no such other path exists, (kt)t≥0

is called efficient.

Definition G.8.2. Let k0 ≥ 0 the initial capital per head and let(ct, dt, kt+1)t≥0 be a feasible allocation, i.e., the resource constraints

f (kt) = ct +dt1 + n

+ kt+1 (1 + n) , t ≥ 0


hold. (ct, dt, kt+1)t≥0 is Pareto optimal if no other feasible allocationcan increase the utility of one household without decreasing the utilityof another household.

Under our usual requirements associated with the households’ uti-lity, an inefficient path of capital allows to increase utility of at leastone household without diminishing the utility of another household.However, it might well be possible to decrease per-head net productionat one point in time and make up for it (in life-cycle utility) in theperiod immediately before or after.

Lemma G.8.1. Let (kt)t≥0 be a feasible path of capital. If (kt)t≥0 isinefficient, it violates Pareto optimality.

We will now introduce the golden rule and examine its relationshipwith both efficiency and the outcome in the steady-state intertemporalequilibrium.

G.8.2 Golden rule

The golden rule comes in two parts, a golden-rule per-head capitalendowment and a golden-rule division of consumption between youngand old. If k does not change over time (as is the case in a steadystate), the resource constraint (eq. G.6, p. 148) can be written as

f (k)− (1 + n) k = c+ d

1 + n.

It seems obvious that per-head net production

φ (k, k) = f (k)− (1 + n) k

should be maximal in order to allow a maximal consumption of first-and second-period agents.Maximizing f (k)−(1 + n) k yields the golden-rule capital per head

kgold, defined byf 0 (kgold) = 1 + n. (G.8)

It is depicted in fig. G.5. In case of a Cobb-Douglas production func-tion, we obtain


goldk k0

kn)1( +

goldgold dn

c+

+1

1

( )kf

n+= 1slope

over-accumu-lation

under-accumu-lation

Abbildung G.5. The golden rule for overlapping generations

kgold =

µαA

1 + n

¶ 11−α

.

This amounts to savings

sgold = (1 + n) kgold = (1 + n)

µαA

1 + n

¶ 11−α

. (G.9)

For Cobb-Douglas production technology, we note

ω (kgold) = f (kgold)− kgolddf

dk

¯kgold

= αA

µαA

1 + n

¶ α1−α−µ

αA

1 + n

¶ 11−α

(1 + n)

Definition G.8.3. Assume a steady-state intertemporal equilibriumat a capital per head k∗. If we have

f 0 (k∗) > 1 + n,

a state of under-accumulation of capital is said to exist. Conversely,


f 0 (k∗) < 1 + n,

is denoted by over-accumulation of capital.

Under-accumulation means that the net production

φ (k∗, k∗) = f (k∗)− (1 + n) k∗

can be increased by aiming for some k > k∗. By the concavity of f,f 0 (k∗) > f 0 (kgold) implies k∗ < kgold as depicted in fig. G.5.Since agents live for two periods, we do not only need to consider

how to maximize production and consumption but also how to divideconsumption between the old and the young. When the old todayget the same as the old tomorrow, we are back to our intertemporalhousehold problem and need to maximize

u (c) +1

1 + ρu (d)

= u (c) +1

1 + ρu³(1 + n) f (k)− (1 + n)2 k − c (1 + n)

´.

Forming the derivative with respect to c yields

du

dc+

1

1 + ρ

du

dd(− (1 + n))

so that the golden-rule for dividing consumption is

du

dc!=1 + n

1 + ρ

du

dd. (G.10)

G.8.3 Is the golden rule equivalent to efficiency?

It may come as a surprise that the golden rules are not equivalent toefficiency. While the golden rules imply efficiency, the reverse is nottrue. We will present an informal argument and refer the reader to dela Croix/Michel (2002, pp. 83) for a more mathematical treatment.We consider, first, the case of over-accumulation. Here, the golden

rule is violated and so is efficiency. The idea is to save less in eachperiod (and lower capital per head to the golden-rule level) and to


increase consumption (of young and/or old) instead. This is possiblefrom each period onward.It is important to understand that this argument works only be-

cause time never ends. If, instead, time were to end, the last-periodagents would have no reason to save. They would just eat up all of

Nsecond to lastdlast +Nlastclast + 0 ·Nlast.

However, since production is lower under the golden rule than underover-accumulation, these last-period agents are hurt.Thus, over-accumulation does not contradict the equivalence of the

golden rule with efficiency.However, under-accumulation violates the golden rule but is consi-

stent with efficiency. Looking at the resource constraint

f (kt)− kt+1 (1 + n) = ct +dt1 + n

we see that we cannot increase k1 in period 0, but at the cost of loweringnet production in period 0. Similarly, we proceed for the followingperiods. Thus, starting from a situation of under-accumulation, wecan make no efficiency case for increasing capital per head. Of course,efficiency does not speak against the golden-rule capital per head.

G.8.4 Does the first welfare theorem hold?

Now, we want to consider the question of whether Pareto-optimal con-sumption is attained in a steady-state intertemporal equilibrium cha-racterized by k∗ (or in any intertemporal equilibrium where the equi-librium capital-per-head values converge towards k∗). From GeneralEquilibrium Theory, we know that the Walras equilibrium is Paretooptimal. While we have price takership in our overlapping-generationsmodel, we will see

— that k∗ may imply under- as well as over-accumulation of capital perhead,

— that over-accumulation is not Pareto efficient so that the first welfaretheorem does not hold while

— under-accumulation is Pareto efficient.


To prove the first claim, we consider logarithmic utility and Cobb-Douglas production. From section G.7.1, we know

k∗ =

Ã11+ρ

1 + 11+ρ

1

1 + nA (1− α)

! 11−α

and

s∗ =

Ã11+ρ

1 + 11+ρ

A (1− α)

! 11−α µ 1

1 + n

¶ α1−α

.

Therefore,

k∗ Q kgold

⇔Ã

11+ρ

1 + 11+ρ

1

1 + nA (1− α)

! 11−α

Qµ

αA

1 + n

¶ 11−α

⇔11+ρ

1 + 11+ρ

Q α

1− α.

Hence, if the rate of time preference ρ and/or the elasticity of pro-ductivity α are relatively small, we have more capital in our steady-state intertemporary equilibrium k∗ than prescribed by the golden rule.Under-accumulation is also possible.This was the first claim and we now show the second. We know

from the preceeding section that over-accumulation is inefficient whichimplies a violation of Pareto optimality by lemma G.8.1.From the preceeding section, we know that under-accumulation

does not violate inefficiency. One might think that matters are dif-ferent with respect to Pareto optimality. Should the young in one pe-riod not have sufficient reason to save if they, as firm owners, receivethe benefit a period later? Without going into details (consult, again,the textbook), we report that this is not the case. The basic reason isthat, in the intertemporal equilibrium, the young generation chooses tounder-accumulate.A second avenue might be to let the old help the young to build

capital. However, the old of the first period have no interest to do so.

G.9 Transfers from young to old and pension systems 161

G.9 Transfers from young to old and pension systems

We will now look at pension systems.

G.9.1 Fully funded systems

Fully funded systems are those where the young are forced to save someamount at which they receive in their old age with the appropriateinterest. The capital build-up then has two sources, private saving stand pension payments at. At t+ 1, capital is equal to

Kt+1 = Ntst|{z}private

savings

+ Ntat|{z}pension

contributions

.

The pension fund invests also in the firm and receives the same interest.Therefore,

dt+1 =¡1 + ret+1

¢st +

¡1 + ret+1

¢at

is the consumption agents when old.Obviously, st + at plays the same role as st before. Therefore, any

contributions to pension funds are offset by private savings if the pen-sion contributions do not exceed the optimal saving.

G.9.2 Pay-as-you-go systems

A balanced system of transfers. We now introduce transfers fromyoung to old agents. If transfers are denoted by at, the youngs’ budgetconstraint becomes

wt − at = ct + st, t ≥ 0

and these agents will expect to consume

dt+1 =¡1 + ret+1

¢st + z

et+1

in their old age. Note that at is not invested into a firm but givendirectly to the old of the present period.


A balanced system of transfers makes sure not to run any deficitsor surpluses, i.e.,

Ntat|{z}paymentscollectedfrom agentsyoung in t

= Nt−1zt| {z }paymentsreceivedby agentsold in t

.

By Nt = (1 + n)Nt−1, the balancedness requires

(1 + n) at = zt (G.11)

If the population grows, n > 0, the old people can receive more thanthe young people give.

The savings function. We now consider the question of how thesetransfers influence savings. Denoting

ω1 : = w − a,ω2 : = z,

we like to find the optimal saving

argmaxsu (ω1 − s) +

1

1 + ρu (ω2 + (1 + r) s)

= : s (ω1,ω2, 1 + r) .

We now reformulate our decision problem in order to make the link tosaving functions without transfers. First of all, we rewrite

u (ω1 − s) +1

1 + ρu (ω2 + (1 + r) s)

= u

⎛⎜⎜⎝ ω1 +ω21 + r| {z }

life-cycle income

−µs+

ω21 + r

¶⎞⎟⎟⎠+ 1

1 + ρu

µ(1 + r)

µs+

ω21 + r

¶¶.

The idea is to push ω2 (discounted!) into the first period and save thisdiscounted value. Now, the situation is as if

G.9 Transfers from young to old and pension systems 163

— ω1 +ω21+r were the first-period income,

— there were no second-period income, and— s+ ω2

1+r were the saving.

Comparing with

s (w, 1 + r) = argmaxs∈R

u (w − s) + u ((1 + r) s) · 1

1 + ρ

we see that

s (ω1,ω2, 1 + r) +ω21 + r

= s

µω1 +

ω21 + r

, 1 + r

¶,

or

s (ω1,ω2, 1 + r) = s

µω1 +

ω21 + r

, 1 + r

¶− ω21 + r

.

The reader will note∂s

∂ω1=

∂s

∂w

and∂s

∂ω2=

∂s

∂w

1

1 + r− 1

1 + r.

Comparative statics. We can now do comparative statics. By

0 <∂s

∂w< 1

we find

0 <∂s

∂ω1< 1.

It is more complicated to confirm

∂s

∂a< 0.

First of all, we assume

dzet+1 = (1 + n) dat, (G.12)

i.e., a simultaneous increase of expected future pension and contribu-tion. This means that people who pay more today expect to get morewhen they are old. Note, that this does not follow from requirementG.11. We will now form the total derivative of


st¡at, z

et+1

¢= s

¡wt − at, zet+1, 1 + ret+1

¢and obtain

dst =∂s

∂adat +

∂s

∂zet+1dzet+1

=ds

dω1(−1) dat +

∂s

∂ω2(1 + n) dat (eq. G.12)

= − ∂s

∂wdat +

µ∂s

∂w

1

1 + r− 1

1 + r

¶(1 + n) dat

=

∙− ∂s

∂w−µ1− ∂s

∂w

¶1 + n

1 + r

¸dat

and hencedstdat

= − ∂s

∂w−µ1− ∂s

∂w

¶1 + n

1 + r< 0.

This is an important result and confirmed by empirical work: If theextent of a pay-as-you-go system is enlarged, savings go down.Note also

dstdat

< −1⇔ 1− ∂s

∂w<

µ1− ∂s

∂w

¶1 + n

1 + r⇔ 1 + r < 1 + n.

This means that the transfers crowd out saving by more than 100% ifthe rate of population growth exceeds the rate of interest.

G.10 Ricardian equivalence

Ricardo claimed that debt incurred by a government is neutral in thefollowing sense: If agents know that their government is indebted theyprepare for a future increase in taxation (to pay back) by saving. Theywould not have saved if the government had financed its programs bytaxation. In real terms, debt financing or taxation is equivalent.It can be shown that this result breaks down in an overlapping-

generations setup. However, altruism between generations can restorethe equivalence. Then, government policies with respect to taxes anddebt are compensated by private transfers (bequests). We will not godeeply into this matter but will only define the basic notions of debtand balanced budgets.

G.10 Ricardian equivalence 165

The government budget constraint is given by

Bt|{z}bonds =

government

debt

+ Ntτyt| {z }

lump-sum

tax on young

generation

+ Nt−1τot| {z }

lump-sum

tax on old

generation

=(1 + rt)Bt−1| {z }repayment

of bonds

+ Gt|{z}government

expenditure

.

Definition G.10.1. The budget is balanced if government expenditureis financed totally by taxes,

Ntτyt| {z }

lump-sum

tax on young

generation

+ Nt−1τot| {z }

lump-sum

tax on old

generation

= Gt|{z}government

expenditure

.

Exercise G.10.1. Imagine that the government issued a debt B0 intime t, but runs a balanced budget from time t = 1 onward. What isthe debt issued in time t = 3?

The government is said to run a surplus, if taxes exceed expendi-ture. It is clear from the previous exercise that the fact of running asurplus does not necessarily prevent government debt to increase.

Exercise G.10.2. How would you define ”keine Neuverschuldung”from time t onward? Hint: The obvious answer is wrong.

Definition G.10.2. A Ponzi debt (Bt)t≥0 is a sequence of debt thatsatisfies, for all t,

Bt ≥ (1 + rt)Bt−1 and Bt−1 > 0.

Exercise G.10.3. What is the relationship between a balanced bud-get and a Ponzi debt?

...


G.11 Solutions

G.5.1. We first solve the intertemporal consumption problem and find

u0 (ct)

u0¡det+1

¢· 11+ρ

=1ct

1det+1

· 11+ρ

!= 1 + ret+1

whence we havedet+1ct

=1 + ret+11 + ρ

.

By the profit maximization, the wage (for Cobb Douglas) is equal to

wt = A (1− α) kαt ,

where kt, by labour market clearing, is

kt =KtNt.

Therefore,

wt = ct +1

1 + ret+1det+1 = ct +

1

1 + ret+1

1 + ret+11 + ρ

ct

=

µ1 +

1

1 + ρ

¶ct

and hence

ct =wt

1 + 11+ρ

=A (1− α) kαt1 + 1

1+ρ

and

st = wt − ct =µ1 +

1

1 + ρ

¶ct − ct

=ct1 + ρ

=

A(1−α)kαt1+ 1

1+ρ

1 + ρ

=

11+ρ

1 + 11+ρ

A (1− α) kαt .

Finally, we have the consumption of the old people,

G.11 Solutions 167

dt = (1 + rt)KtNt−1

= f 0 (kt)KtNt−1

= αA1

k1−αt

KtNt−1

.

G.7.1. If we have stable and unstable equilibria, often every otherequilibrium is stable, here k∗b .

G.10.1. The bonds debt are issued to repay the previous bonds. The-refore, we have

B1 = (1 + r1)B0,

B2 = (1 + r2)B1 = (1 + r2) (1 + r1)B0, and

B3 = (1 + r3)B2 = (1 + r3) (1 + r2) (1 + r1)B0.

G.10.2. It may be tempting to define ”keine Neuverschuldung” byBt = 0 for t ≥ t. In that case however, the debt would be eliminatedin t. ”Keine Neuverschuldung” means Bt = Bt−1 for t ≥ t. This implies

Ntτyt +Nt−1τ

ot = rtBt−1 +Gt

so that government expenditure is lower than actual taxes by the in-terest payment.

G.10.3. For positive debts, a balanced budget implies a Ponzi debt.

H. Appendix

In this appendix, we provide some formulae that will often be used inour textbook. Most of these formulae should be well-known to moststudents. If nothing to the contrary is said, let a, a1, a2 > 0, be positivereal numbers and let b, c be any real numbers.

H.1 Powers

You will find useful the following identities:

ab =1

a−b,

abac = ab+c,

(a1a2)b = ab1a

b2,µ

a1a2

¶b=

ab1ab2and³

abć

= abc

Also, note

a0 = ab−b = aba−b =ab

ab= 1.

H.2 Logarithms

The natural logarithm ln has three remarkable properties. First, wehave

170 H. Appendix

ln (a1 · a2) = ln a1 + ln a2,

ln³ab´= b ln a and

lna1a2

= ln a1 − ln a2

Food for thought. Can you confirm ln 1 = 0 from the above proper-ties?

Second, the derivative is the inverse,

d lnx

dx=1

x.

Third, the natural logarithm is closely related to the magical numbere = 2. 7183... and we have

lnx = y ⇔ ey = x,

ln ex = x, and

elnx = x

Food for thought. Do you see that

xα = eα lnx

follows from elnx = x?

H.3 Rule of de l’Hospital

For f (x) = g (x) = 0 (or f (x) = g (x) = ∞), if f and g are differen-tiable with g0 (x) 6= 0 (or f 0 (x) 6=∞), we have

limx→x

f (x)

g (x)= limx→x

f 0 (x)

g0 (x).

H.4 Rules of differentiation

H.4.1 Product rule

For two functions F and G, we have

(FG)0 = F 0G+G0F.

H.5 Rules of integration 171

H.4.2 Chain rule

For two functions F and G, we have

(F ◦G)0 = F 0G0

H.5 Rules of integration

H.5.1 Partial differentiation

Let f and g be the derivatives of F and G, respectively. Then, by theproduct rule of differentiation, we have

FG =

Z(FG)0 dx =

Z(fG+ Fg) dx =

ZfGdx+

ZFgdx

and hence ZFgdx = FG−

ZfGdx.

Introducing the boundaries of integration, we haveZ b

aF (x) g (x) dx = F (x)G (x)|ba −

Z b

af (x)G (x) dx.

Hint: Take F to be function whose derivative is simpler than F itself.

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de la Croix, David/Michel, Philippe (2002). A Theory of Economic Growth.Dynamics and Policy in Overlapping Generations, Cambridge University Press,Cambridge (UK) et al.

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Schumpeter, Joseph A. (1976). Capitalism, Socialism and Democracy, 5. Aufl.,Routledge, London/New York.

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