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Macroeconomic Theory

(some basics)

Table of contents

1 Macroeonomic schools and methodological views 2

2 Some analytical tools 6

2.1 Optimization in continuous time . . . . . . . . . . . . . . . . . . . . . 6

2.2 Discrete time optimization . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Log-linear approximation of a system of non-linear equations . . . . . 10

3 Examples of postulated models 11

3.1 IS-LM-AD-AS and expectations: from the Keynesian to the neoclassi-

cal approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Economic growth: the Solow-Swan model . . . . . . . . . . . . . . . . 14

4 Models with micro foundations 18

4.1 The Kydland-Prescott model of real business cycles . . . . . . . . . . 18

4.1.1 Basic RBC model . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1.2 Construction and analysis of the RBC models . . . . . . . . . 19

4.2 The Ramsey-Cass-Koopmans model of economic growth . . . . . . . 22

4.2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1

4.2.2 Firms and the market structure . . . . . . . . . . . . . . . . . 23

4.2.3 Optimization results and the general equilibrium . . . . . . . . 24

1 Macroeonomic schools and methodological views

Basic literature:

1. Arnold L.G. (2002) Business Cylce Theory. OUP. (Ch.1)

2. Wickens, M. (2012) Macroeconomic Theory: A Dynamic General Equilibrium Approach.

PUP. (Ch.1)

3. Phelps E.S. (1998) Seven Schools of Macroeconomic Thought. OUP.

4. Lucas R.E. (1976). Econometric policy evaluation: A critique. In K. Brunner

and A.H. Meltzer, eds., The Phillips curve and labour markets. Amsterdam:

North Holland.

5. Mankiw, N.G. (2006) The macroeconomist as scientist and engineer,

Journal of Economic Perspectives 20(4), 29-46.

Main approaches

• Keynesian economics

• Monetarism

• Supply side proponents

• New Classical economics

• Real business cycles theory

• New Keynesian economics

2

Concepts and tasks

Group and discuss the role of:

• the Great Depression

• fiscal and monetary interventions

• general macroeconomic equilibrium (of flows, markets, subjects)

• existence and uniqueness of equilibrium

• aggregation problem

• uncertainty

• availability of data

• oil price shocks and stagflation

• precision of forecasts

• Lucas critique

• formation of expectations (naive, adaptive, model consistent, forward-looking)

• Ricardian equivalence

• time inconsistency of decision makers

• flexibility of wages and prices

• market structure

• nominal and real rigidities

• the Great Contraction

3

Exercise. Characterize the methodological views of different schools in terms of:

• general equilibrium

• representative agents

• objective functions and optimizing behaviour

• budget, technical/production, and market (flows) constraints

• postulation/derivation of structural relationships

• "deep parameters"

• linearization of equations

• market perfection

• rigidity of prices

• expectations

• efficiency of government interventions

• efficiency of demand-side and supply-side policies

• efficiency of fiscal and monetary policies

• welfare analysis and benevolent social planner

An example of postulated relations with equilibrium of markets. In the Keynesian

type IS-LM model, the IS equation characterizes the relationship between the pro-

duction level/income Y and real interest rate r with exogenously given public ex-

penditure G, which satisfies the equilibrium condition in the goods market (Y =

�− �r+ �G, �, �, � > 0). The LM equation defines the relationship between Y and

the interest rates (Y = − + �r + �MP, , �, � > 0), which gives equilibrium in the

4

money market given exogenous supply of money M and fixed price level P . Then the

’general equilibrium’ consistent with equilbria in both the goods and money markets

is obtained as a solution of the system consisting of these two equations for Y and

r given the exogenous levels of G (fiscal policy), M (monetary policy), and P (price

level).

Exercise. Find the ’general equilibrium’ solution for Y and r as functions of ex-

ogenous variables in the example provided above. Does it always exist? If public

expenditure increased twice, what effect it would have on income and (real) inter-

est rates? Nominal interest rates? Why? If a central bank would strive to keep

the interest rates at the same level as before the fiscal intervention, what monetary

change were required? After both interventions, does the income level stay at the

pre-interventional level? Why?

A framework for optimization-based models.

• market subjects (housholds/consumers, firms/producers, public sector, foreign

sector, etc.);

• objective functions of subjects (utility function(s) of consumers, profit func-

tion(s) of producers, government function of political dividends, Taylor’s func-

tion of central bank, etc.);

• technological an bugetary constraints (production technology, inter-temporal

budget constraint(s), etc.);

• market structure (perfect competition, monopolistic competition, etc.);

• (general) equilibrium analysis;

• analysis of dynamics.

5

2 Some analytical tools

Basic literature:

1. Barrow R.J. ir Sala-i-Martin X. (2004) Economic Growth. The MIT Press,

Cambridge (2nd edition). Appendix A.3.


PUP. (Ch.16.2,17)

3. Heer B. ir A. Maussner (2005) Dynamic General Equilibrium Modelling: Computational

Methods and Applications. Springer, Belrin. Chapter 2.

4. Ferguson B.S. ir G.C. Lim (2003). Dynamic Economic Models in Discrete Time.

Routledge, London. Chapters 5-6.

2.1 Optimization in continuous time

A typical task A decision maker can control values of control variables (c(t)) and

aims at maximizing the objective function under certain dynamic constraints of state

variables (k(t)):

max{c(t)}U(0) =

∫ T

0

v[k(t), c(t), t]dt,

s.t.:

a)dk(t)

dt= g[k(t), c(t), t],

b) k(0) > 0,

c) k(T )e−r(T )T ≥ 0.

Here U(0) is an objective function—e.g. utility function of consumers—at an initial

period (0), r(t) =∫ t

0r(�)d� , and T - the last period which also could be T =∞.

6

2.1 Optimization in continuous time

The solution of such a task is found leaning on the Hamiltonian

H[k(t), c(t), t, �(t)] = v[k(t), c(t), t] + �(t)g[k(t), c(t), t],

where �(t) denotes a dynamic Lagrange multiplier. The optimal path of c(t) which

maximizes the above-defined constraint is obtained by solving the system of differen-

tial equations:∂ℋ∂k(t)

= −�(t),

∂ℋ∂c(t)

= 0,

with the cut-off condition �(T )k(T ) = 0, provided T is finite, and limt→∞ℋ(⋅, t) = 0

in the infinite case. Here for any variable z, z(t) ≡ dz(t)dt

. In most of the cases, it can

be simplified to limt→∞�(t)k(t) = 0.

Example. Let us find a solution of the continuous time version of consumer

maximization problem where they strive to maximize the utility function: U(0) =∫∞0e−�tlog(c(t))dt with a capital accumulation constraint dk(t)

dt= k(t)�− c(t)− �k(t),

k(0) = 1, limt→∞k(t)e−r(t)t ≥ 0.

The Hamiltonian of such a task is

H[k(t), c(t), t, �(t)] = e−�tlog(c(t)) + �(t)(k(t)� − c(t)− �k(t)),

and first order derivatives:

∂H∂k(t)

= ��(t)k(t)�−1 − ��(t) = −d�(t)d(t)

⇒ d�(t)d(t)

= �k(t)�−1 − �,∂H∂c(t)

= e−�t

c(t)− �(t)0 ⇒ d�(t)/dt

mu(t)= −�− dc(t)/d(t)

c(t).

Hence, it is easy to derive the optimal path for consumption:

dc(t)

d(t)= �k�−1 − � − �.

7

2.2 Discrete time optimization


Suppose that the representative houshold aims at maximizing the utility of (infinite)

life

Ulife = Et

∞∑t=t

��−tU(C� , 1− L� ) (2.1)

given the capital accumulation constraint1

Kt+1 = (1− �)Kt + AtF (Kt, Lt)− Ct, (2.2)

by choosing the optimal amounts of consumption {C� , � = t, . . . ,∞} and labour

{L� , � = t, . . . ,∞}. Let U� := U(C� , L� ), ∂U�∂C�

> 0, ∂U�∂L�

< 0, which is discounted

to the present value by �. � denotes the norm of depreciation of capital, while

AtF (Kt, Lt) = Yt is a production function satisfying the standard homogeneity, Inada,

etc. conditions. For notational simplicity let Ft := F (Kt, Lt)).

There are several ways to solve the problem, but let us consider the one which is

based on the value function approach. Let the value function be defined as a function

maximizing the utility for a given level of a state variable Kt in terms of control

variables {C� , L�}∞�=t:

V (Kt) ≡ max{C� ,L�}Et

∞∑t=t

��−tU(C� , L� ) = max{Ct,Lt}U(Ct, Lt) + Et�V (Kt+1)

(2.3)

s.t.

Kt+1 = (1− �)Kt + AtF (Kt, Lt)− Ct.

Such a formulation in terms of the so called Bellman equation re-structures the infinite1Actually, this constraint is a result of three equations: 1) physical capital accumulation constraint

Kt+1 = (1− �)Kt+ It, 2) technical production constraint Yt = AtF (Kt, Lt), and 3) product/incomedistribution constraint in a closed economy (without public sector) Yt = Ct + It.

8


horizon problem into the current period optimization and the functional equation in

terms of the value function.

The necessary conditions of maximization of the value function in terms of Ct and

Lt are:∂V (Kt)

∂Ct= 0 ⇒ ∂Ut

∂Ct= �EtV

′(Kt+1), (2.4)

∂V (Kt)

∂Lt= 0 ⇒ ∂Ut

∂(1− Lt)= �At

∂Ft∂Lt

EtV′(Kt+1). (2.5)

If the functional form of V (⋅) was known, these two equations augmented with the

capital accumulation constraint would be sufficient to solve the solution of the three

unknowns (C, K ir L) that satisfy the optimality condition. However, the form of

V (⋅) is not known in the general case. It is possible to guess its form and check

whether the Bellman equation holds i.e. equality is satisfied. However, this problem

can be overcome in a different way.

Let us first differentiate the value function definition:

V ′(Kt) = �EtV′(Kt+1)

(1− � + At

∂Ft∂Kt

). (2.6)

Using it, and from (2.4) solving for the expected value differential:

(2.4)+(2.6): ∂Ut∂Ct

= V ′(Kt)(

1−�+At∂Ft∂Kt

)−1

⇒ EtV′(Kt+1) = Et

∂Ut∂Ct

(1−�+At

∂Ft∂Kt

).

(2.7)

considering EtV (Kt+1) as an unknown variable and using equations (2.3)-(2.7), it is

simple to get three equations already without the unknown value function expression:

(2.4)+(2.5): ∂Ut∂(1− Lt)

=∂Ut∂Ct

At∂Ft∂Lt

, (2.8)

(2.4)+(2.7): ∂Ut∂Ct

= Et∂Ut+1

∂Ct+1

(1− � + At+1

∂Ft+1

∂Kt+1

), (2.9)

9

2.3 Log-linear approximation of a system of non-linear equations

apribojimas iš (2.3): Kt+1 = (1− �)Kt + AtF (Kt, Lt)− Ct. (2.10)

Hence, we have a system of three equations in three endogenous variables (Ct, Lt, Kt).

In the general case, this is a non-linear system (with rational expectations).

2.3 Log-linear approximation of a system of non-linear equa-

tions

The analysis of a non-linear system of equations is, in the general case, quite cum-

bersome and difficult to analyse analytically. Hence, instead of defining the global

properties of the system, it is usual to look at its local properties by using a kind of

approximation around the point of interest, for instance, some points of equilibrium.

For any variable Xt, xt := ln(Xt/X), where Xt denotes the actual, and X the

steady-state value. Then, using the first-order Taylor approximation of any func-

tion f f(Xt) ≃ f(X) + f ′(Xt)∣Xt=X(Xt − X), it is easy to derive these (first-order)

approximation rules:

1. �Xt + �Zt ≈ �X(1 + xt) + �Z(1 + zt)

2. XtZ�t ≈ XZ�(1 + xt + �zt)

3. f(Xt) ≈ f(X)(1 + �xt), � = f ′(Xt)f(Xt)

Xt

∣∣∣Xt=X

.

Exercise. Derive these (approximate) rules from the first-order Taylor expansion

considering the higher-order terms as ’insignificant’. Provide any case where such

an approximation-based analysis would clearly fail (e.g. think of several points of

equilibrium).

Exercise. Find the log-linear approximations to these equations:

a) �It + �Ct = �Yt,

10

b) Yt = At

(Kt

Lt

)�,

c)Ct+1

Ct= �Et(1− � + At+1�

Yt+1

Kt+1

),

3 Examples of postulated models

3.1 IS-LM-AD-AS and expectations: from the Keynesian to

the neoclassical approach

Basic literature:

1. Arnold L.G. (2002) Business Cylce Theory. OUP. (Ch.2-4)


PUP. (Ch.13)

3. Muth J.F. (1961) Rational expectations and the theory of price movements,

Econometrica, 29.

4. Hoover, K.D. (1992) The Rational Expectations Revolution: An Assessment,

The Cato Journal 12(1), 81-106.

5. Romer, D. (2000) Keynesian Macroeconomics without the LM Curve,

Journal of Economic Perspectives 14(2), 149-169.

• Aggregate price level

• Production and short-term aggregate supply functions

• Models of aggregate expenditure

• Money demand and supply

11

3.1 IS-LM-AD-AS and expectations: from the Keynesian to the neoclassicalapproach

• IS: goods market

• LM: money market

• Labour demand and supply

• Wage setting environment

• Expected inflation, nominal and real interest rates

• Price expectations

• Lucas’ production function

A stylized (log-linearized) IS-LM-AD-AS model:

IS: yt = �− �rt + �gt,

LM: mt − pt = + �yt − �it,

Fisher: it = rt + �et , �et = pet − pt−1,

AS: yt − y∗ = (pt − wt),

W: wt = pet ,

Expectations: pet = Et−1pt;

where:

yt ir y∗ - actual level and exogenously given potential production levels, respectively,

rt - ir it - real and nominal interest rates, correspondingly,

gt - public expenditure (exogenous),

mt - money supply (exogenous),

pt, pet , and �et - actual and expected price levels, and the expected inflation, corre-

spondingly

wt - wages.

In such a model, a change only of expectations can lead from the Keynesian to the

New Classical type of economy behaviour.

12

3.1 IS-LM-AD-AS and expectations: from the Keynesian to the neoclassicalapproach

• Naive expectations (Keynesian): pet+1 = pt;

• Adaptive expectations (Monetarist): pet+1 = pt + �t

or more generally pet+1 =∑k

i=0 �ipt−i,∑k

i=0 �i = 1, k ∈ N.

• Rational (New Classical): pet+1 = Etpt+1, where teh expectations Et are based

on some information set available in period t.

If the information set is not defined in the rational expectations, one understands

that all available information belongs to the conditioning set. In practice, this is

hard to implement. To operationalize the information set, Muth(1961) proposed the

model-consistent expectations. It should be pointed out that Etpt+1 can be simply

considered as an additional variable in the system.

Example Taking the expectations of the above defined system of equations pro-

duces:

IS: y∗ = �− �Et−1rt + �gt,

LM: mt − Et−1pt = + �y∗ − �Et−1it,

Fisher: Et−1it = Et−1rt + Et−1pt − pt−1,

AS: Et−1yt = y∗

W: Et−1wt = Et−1pt.

Then, from the IS, LM, and Fisher equations of this system of (expected) variables,

the (rational and model consistent) expected price is obtained as a function of exoge-

nous variables, historical (known) values and model parameters:

Et−1pt =mt − 1− �

− � + �

1− �y∗ +

�

1− �[� + �gt − pt−1]. (3.1)

Exercize. Given this solution of the expected price, find the solution of all endoge-

nous variables as functions of exogenous, historical values, and parameters i.e. find

the reduced form of the model. Show that the AS and (3.1) equations in the reduced

13

3.2 Economic growth: the Solow-Swan model

form model imply that the expected change of public expenditure and/or money will

affect only prices, but not the level of production.

Exercize. What effects would be observed on the real and nominal interst rates a)

an increase in public expenditure, b) a decrease in money supply?

Open economy formulations of the IS-LM models

• Foreign trade, foreign exchange, relative prices and the nominal interst rates.

• The purchasing power and interest rates parities.

• The Mundell-Fleming model with fixed and flexible exchange rates.

• The Dornbush ’overshooting’ model.


Basic literature:

1. Romer D. (2001) Advanced Macroeconomics. McGraw-Hill, Boston (2-as leidi-

mas). 1 skyrius.


PUP. (Ch.3.3)

3. Barrow, R.J., and Sala-i-Martin, X. (2004) Economic Growth. The MIT Press,

(2nd ed.). (Ch.1)

Let us denote the production function by F : ℝ2+ → ℝ+, K stand for the stock

of capital, AL for the efficient level of labour; A to represent the labour augmenting

technological development, and L to denote the labour force. F is also assumed to

satisfy:

14


• ∂F∂Z

> 0, ∂2F∂Z2 < 0 for any factor Z;

• �F (K,AL) = F (�K, �AL), ∀� > 0;

• the Inada conditions: limk→∞∂F∂Z

= 0, limk→0∂F∂Z

=∞.

The postulated equations of the model

- The production level (Y ) is determined by the technological constraint:

Y = F (K,AL). (3.2)

- The capital accumulation constraint is given by:

K = I − �K, kur K ≡ dK/dt, (3.3)

where investments I increase and the depretiation at the rate � decrease the stock of

capital.

- The closed economy equilibrium of flows:

Y = C + I. (3.4)

- The postulated behavioural equations:

a fixed proportion (0 < s < 1) of income goes to savings (S), whereas the rest is

spent for consumption (C):

S = �Y, (3.5)

C = (1− �)Y. (3.6)

In addition, it is assumed that the exogenous growth of labour (L) and technology

(A) is given by:

L = nL, (3.7)

15


A = gA. (3.8)

Consequently, the growth rate of the efficient unit of labourAL is given by ˙(AL)/(AL) =

L/L+ A/A = n+ g.

The reduced form of the model

From the structural mode, it is easy to get

K = sF (K,AL)− �K, (3.9)

or, introducing the efficient level of stock of capital by k ≡ KAL

,

k =K

AL= �f(k)− (n+ g + �)k, (3.10)

where f(k) ≡ F (k, 1).

Exercise. Use the structural model and the definition of k to derive the differential

equation (3.10).

Equilibrium

The dynamic equilibrium k = 0 holds for a k = k∗ such that

�f(k∗) = (n+ g + �)k∗. (3.11)

Exercise. From the Inada conditions for F derive that the following holds: limk→∞ f′(k) =

0 ir limk→0 f′(k) =∞.

Exercise. Explain, how the Inada conditions limk→∞ f′(k) = 0, limk→0 f

′(k) = ∞

and smoothness of the function ensures the existence of an equilibrium. For k > 0, is

the equilibrium unique? Why?

16


Exercise. Suppose that the production function is of the Cobb-Douglas form i.e.

F (K,AL) = K0.3(AL)0.7. Find k∗, when s = 0.1, � = 0.05, and n = g = 0.01. What

happens when the savings rate increases?

Exercise. From the general model show that Y /Y = n+ g in equilibrium. What

are the growth rates of K, Y/L, y, C? Which level of savings maximizes the con-

sumption per unit of labour?

Exercise. Define the effect of a change in � for the non-equilibrium rate of growth

of income.

Exercise. Let us now introduce two otherwise similar economies but with �1>�2.

Find k∗1, k∗2 and Y1/L, Y2/L in each of the closed economies. What happens when the

economies opens borders for flows of capital? Find the values of the same variables

for each country.

17

4 Models with micro foundations

4.1 The Kydland-Prescott model of real business cycles

Basic literature:


PUP. (Ch.2,16)

2. Lutz G.A. (2002) Business Cycle Theory. OUP. (Ch.5)

4.1.1 Basic RBC model

- Representative firm

The representative firm maximizes in perfect competition real profits �(t) for each

period, given the exogenous prices of capital and labour—i.e. the required rate of

return r(t) and wages W (t)—and the production technology F which satisfies the

standard conditions.

max(K(t),L(t)) �(t), �(t)/P (t) = Y (t)− r(t)K(t)−W (t)L(t), (4.1)

with the technological constraint

Y (t) = F(K(t), A(t)L(t)

).

- Representative consumer

The representative consumer maximizes the utility which depends on the level of

consumption (positively) and the supplied work effort (negatively):

max{C(�),L(�)}∞�=t Et

∞∑�=t

��−tU(C� , 1− L� ) (4.2)

18


under the capital accumulation constraint

Kt+1 = (1− �)Kt + It

and the macroeconomic flows constraint

It = Yt − Ct.

- Technology

The technological process At = At ⋅ At. Here At is the long run growth process

given by At = A0e�, � > 0, and At is a stationary stochastic component defined by

At = �At−1e� with � being white noise. In the general case, any stationary process

can be considered.

4.1.2 Construction and analysis of the RBC models

• Model formulation with specific forms of functions F and U

• Setting up a (non-linear) system of equations defining the optimal choice from

the necessary conditions of (maximization of) objective functions and techno-

logical, budgetary and other flows constraints.

• Finding the (dynamic) equilibrium solution

• Calibration of parameters of the model

• An approximation (e.g. log-linear) of the nonlinear system of equations

• Findin the rational expectation solution

• Application of the model for simulations, etc.

19


An example of specific functional choices.

Technological constraint: Yt = AtK�t L

1−�t , 0 < � < 1.

Capital accumulation: Kt+1 = (1− �)Kt + It, 0 < � < 1.

Macroeconomic flows: Yt = Ct + It.

Utility of consumers: Et∑∞

�=t ��−t[log(C� ) + log(1− L� )].

Using these it is easy to reduce the model to:

Utility: Et∑∞

�=t ��−t[log(C� ) + log(1− L� )].

Capital accumulation: Kt+1 = (1− �)Kt + AtK�t L

1−�t − Ct, 0 < � < 1.

The non-linear system of equations defining the optimal choice.

Using the above-defined utility and production functions:∂Ut

∂(1−Lt) = 1−Lt ,

∂Ut∂Ct

= 1Ct

, ∂Ut+1

∂Ct+1= 1

Ct+1, ∂Ft∂Lt

= (1− �)At

(KtLt

)�ir ∂Ft

∂Kt= �At

(KtLt

)�−1

.

Using these in (2.8)-(2.10), the optimal choice is governed by:

(1− Lt)=

1

CtAt

(Kt

Lt

)�, (4.3)

1

Ct= Et

1

Ct+1

[1− � + At+1

(Kt+1

Lt+1

)�−1], (4.4)

Kt+1 = (1− �)Kt + AtK�t L

1−�t − Ct. (4.5)

The solution of dynamic equilibrium. In the (constant) dynamic equilibrium, ∀ t, At =

A, Ct = C, Kt = K, Lt = L. Hence:

(1− L)=

1

CA(KL

)�, (4.6)

1

C=

1

C

[1− � + A

(KL

)�−1], (4.7)

K = (1− �)K + AK�L1−� − C. (4.8)

20


It should be pointed out that the expectation operator here is absent, because the

equilibrium values are fixed.

Exercise. Find C, K, L.

Calibration of the model. The number of unknown parameters in the RBC models

is usually larger than the number of equations. Hence, it is usual to use some extra-

information to pin-down the values of some parameters, whereas the remaining ones

can be solved from the given equations. However, various more formal approaches

to model estimation are available using various moments of variables as well as the

Bayesian estimation of parameters.

Exercise. In the case of the model provided above, find the number of parame-

ters that would be needed to calibrate "freely" using some extraneous information.

Which parameter values are ’widely’ accepted and what parameters can be reasonably

estimated?

Log-linear approximation. Using the rules defined in section 2.2, we get the follow-

ing log-linear approximation of the non-linear system of equations defined above:

1

CA(KL

)�ct −

(1− L)

L

1− Llt =

1

CA(KL

)�(at + �(kt − lt)), (4.9)

1

C(1− ct) = Et

1

C(1− ct+1)

[1− � + A

(KL

)�−1

(1 + at + �(kt − lt))], (4.10)

K(1+kt+1) = (1−�)K(1+kt)+AK�L1−�(1+at+�kt+(1−�)lt)−C(1+ct). (4.11)

Exercise. Simplify this system by using the equilibrium conditions and dropping

the second order terms.

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4.2 The Ramsey-Cass-Koopmans model of economic growth


The Ramsey model is a direct analogue of the Solow model, but having the micro

foundations of the main market participants.

Basic literature:

1. Romer D. (2001) Advanced Macroeconomics. McGraw-Hill, Boston (2nd ed.).

(Ch.2 part A).

2. Barrow R.J. ir Sala-i-Martin X. (2004) Economic Growth. MIT (2nd ed.).

(Ch.2).

4.2.1 Households

The number of identical households H is assumed to be constant. A household at a

moment t consists of H/L(t) individs, where the total number of inhabitants/workers

L(t) = ent is growing exogenously at a rate n.

The assets a(t) of a households are accumulating in accordance with the constraint:

a(0)

H+

∫ ∞t=0

e−R(t)W (t)

L(t)

L(t)

Hdt ≥

∫ ∞t=0

e−R(t)C(t)

L(t)

L(t)

Hdt, (4.12)

where R(t) =∫∞�=0

r(�)d� stands for the "average" interst rate.

Given such a constraint, households maximise:

UH({C(t), t ≥ 0}) =

∫ ∞t=0

e−�tU[C(t)

L(t)

]L(t)

Hdt, (4.13)

where � > 0 reveals the inter-temporal preference and U [∙] stands for the instanta-

neous utility function. Assume that it has the following form:

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U(z) =z1−�

1− �, � > 0. (4.14)

The L’Hopital rulle implies that under the � = 0, this utility is logarithmic.

Exercise. Derive this claim.

Then the optimal choice results from the solution of the constrained maximization

problem:

max{C(t),t≥0}LH = UH({C(t), t ≥ 0})+�HH

[a(0)+

∫ ∞t=0

e−R(t)(W (t)−C(t))dt]. (4.15)

Exercise. Find the necessary conditions and derive the Euler equation.

Exercise. Write the necessary conditions in terms of the ’efficient’ or normalized

units: a) z(t) ≡ Z(t)A(t)L(t)

for any for any Z(t); b) z(t) ≡ Z(t)L(t)

for any for any Z(t).

4.2.2 Firms and the market structure

Firms produces product Y (t) from the capital K(t) and labour inputs L(t) under the

exogenous technological progress A(t) = egt, g > 0. The production function

Y (t) = F(K(t), A(t)L(t)

), (4.16)

satisfies the standard conditions.

In each moment firms maximize profits

�(t) = P (t)Y (t)− r(t)K(t)− W (t)L(t), (4.17)

given exogenous product and factor prices P (t), r(t) and W (t) in perfect competition.

Then the optimal choice of firms is define by the necessary conditions of the con-

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strained maximization:

max(K(t),L(t))LF = �(t) + �F

[Y (t)− F

(K(t), A(t)L(t)

)]. (4.18)

Exercise. Find the necessary conditions of firm maximization problem. Re-write

them by changing the interest rate and wages into their real analogues: r(t) = r(t)P (t)

,

W (t) = W (t)P (t)

.

Exercise. Re-write the necessary conditions in terms of the efficient units.

4.2.3 Optimization results and the general equilibrium

Households

Differentiation of the Lagrangian of households yields

Be−[�−n−(1−�)g]tc(t)−�(1− �)− �He−R(t)a(0)L(0)e(n+g)t = 0,

where B is a function of constants (parameters). Taking a logarithm and time differ-

ential of it produces the optimal path for consumption growth:

c′

c=r(t)− �

�− g. (4.19)

Apart the parameters of the model, only r(t) is not known.

Firms

Differentiation of the objective function of firms with respect to labour and capital

yields the following necessary conditions:

F ′LA = w(t), wℎere w(t) =w(t)

p(t),

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ir

F ′K = r(t), wℎere r(t) =r(t)

p(t).

Leaning on the constant rate of return assumption, f(k) = F (k, 1), where k = KAL

.

Then the conditions provided above have the following analogues:

f(k)− f ′(k) =w(t)

A(t), (4.20)

and

f ′(k) = r(t), (4.21)

Making the assumption about the general equilibrium and using these conditions

together with the necessary conditions of households allows finding the solution of

unknown values of variables in terms of the exogenous processes and parameters of

the model.

Exercise. Derive the reduced form equations of the model for all the endogenous

variables.

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