Week 3 35

Week 3

Government Spending in the Ramsey Model We examine here a severely limited model of how government spending affects the

Ramsey model: government spending uses resources but doesn’t affect utility or

productivity.

Government budget constraint and Ricardian equivalence

It is easiest to start with two-period model again

o Suppose that government must balance its budget over two periods

o 2 21 11 1

T GT G

r r

corresponds to usual intertemporal budget constraint of

household, with sources of funds on the left and uses on the right (assuming

no initial debt or assets)

o If households must pay these taxes (assume lump-sum), then

2 2 21 1 1

2 2 21 1 1

1 1

1 1 1

Y T CY T C

r rY T C

Y T Cr r r

o Substituting in from government budget constraint:

2 2 21 1 11 1 1

Y G CY G C

r r r

This is demonstration of the “Ricardian equivalence” principle (re) discovered by

Barro in 1974: If the government is on its budget constraint, then the timing of lump-

sum taxes does not matter, and deficits are irrelevant.

The corresponding condition for the Ramsey model is

0 0

,n g t n g trt rt

t te T t e dt e G t e dt

where T(t) is taxes per effective labor unit

and G(t) is government spending per effective labor unit.

This means that the new Ramsey budget constraint is

0 0

0n g t n g trt rt

t te c t e dt k e w t G t e dt

o Note that for the household: K Y C B T , and for the government

B G T so K Y C G and government borrowing vs. taxation does

not affect the household’s budget constraint.

Again, the timing of taxation is irrelevant in this model due to

o Lump-sum taxation

o Infinite lifetimes so you can’t die before the tax bill comes due

o No Ponzi schemes (including by government)

36 Government Spending in the Ramsey Model

o Perfect capital markets in which people and governments borrow and lend at

a single interest rate

Increase in G shifts the 0k curve parallel downward by the amount of the change

in G

o Government spending takes some output that could have been used for

capital investment and uses it instead for government.

o k f k c G n g k

o 0k c f k G n g k

Effects of government spending

Consider an increase in G: There will be no “substitution effect” here because the

lump-sum taxes assure that the taxes do not (directly) change the rate of return on

capital

o Contrast this with the homework problem you will do in which there is a tax

on capital income, creating a substitution effect, but in which the money

collected is returned (lump-sum) to households, negating the income effect.

How much the increase in G affects households’ lifetime income depends on how

long it lasts.

o We consider two cases: permanent G and temporary G.

o In both cases we assume that the change is totally unexpected prior to its

occurrence.

o It is also possible to examine the effect of anticipated changes, which will

cause changes in consumption behavior as soon as they are known (ever

before the change actually happens)

Permanent increase in G

o G suddenly (and unexpectedly) goes up and is expected to stay up forever

o 0k curve shifts permanently downward

o k* is unchanged.

o New e1 is directly below e0.

o Economy jumps vertically to new saddle path, which means it moves

instantly to e1

o No transition dynamics: convergence is instantaneous.

o Question: Why would “consumption smoothers” choose a discrete change in

their consumption path?

Consumption smoothing applies only to the planned consumption

path, not to changes in response to new information.

For households will no formal “adjustment costs” (such as mortgage

contracts), they will adjust their future lifetime consumption plan to

any changes in their expected lifetime income path. This planned

consumption path will be smooth, but may be discretely higher or

lower than the current value on the previous path.

Government Spending in the Ramsey Model 37

Temporary increase in G: unexpected but duration is known with certainty

o If the increase in G is temporary from T0 to T1 (such as a war), two things

happen differently

The change in lifetime wealth is smaller

There is an expected change in disposable income and the 0k

curve in the future

o One feasible option that households won’t choose is to drop consumption to

c1* during the war and return to c0* after the war.

They could do this, but would not want to

They would rather smooth their consumption around the instant that

the war ends (T1) rather than having a discrete jump at that date

They can increase utility if they take some consumption from just

after T1 when it is high and transfer it back to just before T1 when

consumption is low by borrowing before T1 and repaying after T1

The same argument extends backward to all dates during the war vs.

all dates after

o Households will reduce consumption at T0 but not all the way to e1

o Consider dynamics of point like e2 when G is high (during the war)

Dynamics during war are governed by system centered on e1

At a point like a, the dynamics move the economy initially due west (

0k because consumption has increase and saving has fallen), then

northwest as it moves off of the 0c line (as k starts to fall, MPK

rises raising r and providing encouragement to postpone

consumption, so 0c ).

k

c

k*

c0* e0

c1* e

1

38 Government Spending in the Ramsey Model

Households will determine the point a so that the movement along

the unstable northwesterly path brings them to b exactly as the war

ends at T1

At time T1, government spending falls again, which means that the

dynamics of the system return to those centered on e0.

The economy is on the saddle path back to e0 and converges over time

back to the original equilibrium

Interest rate rises slowly through the war and then declines slowly

after war ends

o Long vs. short war: The longer the war, the closer a is to e1 and the longer it

takes to get from a to b for two reasons:

a is further from b

a is very close to the 0k locus, so the initial shortfall of saving is

very small and the movement in k to the left is very slow.

o A very short war would not move consumption down very much in

comparison.

o If the end of the war were to be delayed unexpectedly, the economy would

jump vertically downward from the a → b path onto a lower path on which it

moves to a point lower than b on the peacetime saddle path at the delayed

ending date

Barro’s JME paper (fifth paper of week) is discussed on Romer’s pp. 75–77 and looks

at behavior of interest rates during English wars from 1729 to 1918.

k

c

k*

c0* e0

c1* e

1

a b

The Diamond Model: Overlapping Generations 39

The Diamond Model: Overlapping Generations Skip this, but just discuss the differences in modeling framework

Differences from the Ramsey model

Discrete time

Finite (two-period) lifetimes

Distinct phases of life (work and retirement)

Similarities to Ramsey model

Same basic idea of consumption/saving

Same production and growth dynamics

Dynamic assumptions

Lifetime assumptions

o Individual lives two periods

o No links to earlier or later generations

o Works first period and lives off saving (with interest) second period

o Old gradually sell off capital to the young throughout the old-age period

o Young save by buying capital from old, then diverting output from

consumption to create additional new capital as desired

Size of population

o Lt = number of young in year t

They are the only workers

o 1 1t tL n L

Labor force and population grow at annually compounded rate n

Consumption notation

o 1,tC consumption per person by young in period t

o 2,tC consumption per person by old in period t

Utility

Period felicity function is again CRRA

Consumption choices by person who is young in t

o Consumes 1,tC when young and 2, 1tC when old

Utility is 1 11, 2, 11

,1 1 1

t tC CU

with > 0 and > 0.

o Utility is discounted at annually compounded rate

40 The Diamond Model: Overlapping Generations

Production and dynamics

,t t t tY F K A L with constant returns to scale and usual marginal product

conditions

o t ty f k as before

o t tr f k (assume no depreciation)

o t t t tw f k k f k = wage of each effective labor unit

t tA w = wage per worker

t t tA w L total wages earned in economy (= income of young)

All capital is owned by the old at the beginning of each period

1 1,t t t t tK L w A C

o Expression in parentheses is saving by each young person, which consists of

buying up capital from oldies plus perhaps diverting some new production to

further capital.

o Multiply by number of young people to get total capital put away for next

period

Budget constraint and utility maximization

Budget constraint over lifetime is 2, 1

1,11

tt t t

t

CC A w

r

o Or 2, 1 1 1,1t t t t tC r w A C

Individual maximization problem:

1, 2, 1

1 11, 2, 1 2, 1

1,,

1

1max , subject to

1 1 1 1t t

t t tt t t

C Ct

C C CC A w

r

Can do this as a Lagrangean, but just as easy to solve and substitute for 2, 1tC :

o

1,

11

1 1,1,11

max1 1 1t

t t t tt

C

r w A CC

First-order conditions for this maximization come from 1,

0t

dUdC

o

1

2, 1 1

1,

1

1t t

t

C rC

o Budget constraint: 2, 1 1 1,1t t t t tC r w A C

o Note similarity of first condition to Euler equation in Ramsey model:

C increases or decreases over time as r > or r <

Sensitivity of consumption path to r depends on 1/

The Diamond Model: Overlapping Generations 41

Plugging the budget constraint back into the other first-order condition yields

1

1, 11 1

1

11

1 1t t t t t t

t

C A w s r A wr

o Note that numerator and denominator terms are always positive and that the

denominator is always larger

o If ≈ r then we consume about ½ of income in first period and ½ in second,

which is consistent with basic consumption smoothing

We can show that

1

11 1 1

1

10

1 1

tt

t

rs r

r

iff < 1

o Change in r has income and substitution effects

Reward to saving is higher if r↑

Don’t need to save as much for retirement if r↑

o Remember that 1/ is elasticity of intertemporal substitution

If 1/ is large, then substitution effect is strong and s > 0

If 1/ is small, then income effect dominates and s < 0

o Intermediate case = 1 has s = 0 and saving rate does not depend on interest

rate

Recall that the CRRA utility function approaches u = ln(c) as → 1.

If = 1, then 1

2s

= constant

We shall use log utility as a special case because it is simple

What does this mean in terms of indifference curves?

C2,t + 1

C1,t

Atwt

slope = – (1 + rt + 1)

Income and substitution effects cancel

out as C1,t is the same after increase in

rt + 1 steepens the budget constraint.

42 Analysis of the Diamond Model

Analysis of the Diamond Model

Dynamics

The basic equation of motion of this model is 1 1t t t t tK s r L A w

We want to translate this into kt + 1 :

11 1

1 1 1 1

1 11 1 1 1

t t t tt t

t t t t

t t ttt t

t t

t t

K A w Lk s r

A L A L

f k k f kws r s f k

L A n gL A

o This equation gives kt + 1 implicitly as a function of kt, but it can’t be solved in

the general case.

Steady-state condition

o What would correspond to 0k ?

o 1 1 0t t tk k k would be the equivalent in discrete time

o Setting 1 0tk (or 1 *t tk k k ) gives the steady-state condition:

* * * ** ,

1 1

s f k f k k f kk

n g

which implicitly defines k* the steady-

state value of k.

o This equation is difficult to work with because we don’t know the form of f

and we don’t even know the sign of s. o Depending on the forms of s and f, the function on the right can have a

variety of shapes.

Special case: = 1 (log utility) and Cobb-Douglas production function y = k

o In this case, 1

2s r

, 1 ,f k k 1 .w k

o

1

1

2 1 1t

t t

kk Dk

n g

with positive constant D

o In this case, we can graph 1tk as a function of tk and know its basic shape:

Analysis of the Diamond Model 43

o In this case, if we start at k0, we will converge to 1

1*k D by the “cobweb”

path shown

o Note effects of parameters:

↑ D↓

n↑ D↓

g↑ D↓

Effects are similar to Ramsey (and Solow) model

Properties of Diamond Steady State

In steady-state equilibrium

o k and y are stable

o Y/L grows at g

o Y, K grow at n + g

Speed of convergence: o 1 *t t tk k k k

o ≈ 1/3, so economy moves 1/3 of way to equilibrium in each “period”

Note that “period” is half a lifetime, so this is not so different from

Solow/Ramsey result

Diamond Model General Case?

If we abandon the comfortable home of log utility and Cobb-Douglas we admit to

strange possibilities:

o Can have multiple equilibria

kt + 1

kt

kt + 1

= kt

e

k*

k*

k0 k1

44 Analysis of the Diamond Model

o Can have no non-zero equilibrium at all

Dynamic Inefficiency in Diamond Model

Equilibrium in Ramsey model was Pareto efficient

Diamond model admits the possibility of inefficiency

o It is possible that k* is greater than Golden Rule k

o Why? Because survival in old age depends on saving lots of income

regardless of rate of return.

o It is possible that the saving rate that is optimal for an individual might be

higher than the saving rate that leads to Golden Rule level of k*

Suppose that k* > kGR

kt + 1

kt

kt + 1

= kt

e

k*

k*

kt + 1

= g(kt)

kt + 1

kt

kt + 1

= kt

e

k*

k*

kt + 1

= g(kt)

Analysis of the Diamond Model 45

o Suppose that there was another (than capital) way of transferring money from

youth to old age (Social Security) so that saving could go down

o Young would be better off because they could consume more

o Future generations would be better off because k*↓ means higher steady-state

c*

o Everyone is made better off and no one worse off, so original equilibrium must

not have been Pareto optimal

How can model by inefficient? Where is the market failure?

o No externalities, but an absent market: no way for current generation to trade

effectively with future generations

o Only way to provide for retirement is through saving, even if rate of return is zero

or negative

o From social standpoint, it is desirable to avoid low- or negative-return

investment, but for individual facing retirement this is only choice

o If benevolent government were to establish transfer scheme from young to old

(like Social Security), then they would not need to accumulate useless capital in

order to eat in retirement

o You will do a problem this week looking at the effect of alternative Social

Security regimes in the Diamond model

Is this empirically relevant?

o k > kGR means that f k g n

o Are interest rates lower than the GDP growth rate?

Probably not in U.S. steady state

46 Introduction to Endogenous Growth Models

Introduction to Endogenous Growth Models Paul Romer’s 1986 model and Robert Lucas’s (1988) human capital model.

These models get around the diminishing marginal returns to “capital” assumption

by broadening the definition of capital to include knowledge or human capital, both

of which may have positive externalities.

We need some kind of external effects in order to have a model in which

o individual firms do not have increasing overall returns to scale, so they do not

expand infinitely and become economy-wide monopolies

o the economy as a whole has increasing returns to scale, so that returns to

“capital” or “produced inputs” can be constant

Endogenous growth model have several features that economists have found

attractive

o They endogenize key parameters of the model such as g

o They can explain lack of convergence

o They allow s and related policy variables to affect the growth rate of GDP,

not just the level of the growth path

Text book begins with a simplified model of knowledge production via research and

development in Chapter 3.

o Uses constant saving assumption as in Solow model

o Incorporating Ramsey saving model does not change basic dynamics

Key characteristic leading to endogenous growth: constant returns to scale in

produced inputs.

o In Solow and Ramsey models, capital was only produced input and had

diminishing returns

David Romer’s R&D model

Dynamics and behavioral assumptions

Economy has two sectors: goods-producing sector and R&D (knowledge-producing)

sector

Each sector uses labor and capital

o aL and aK are the shares of labor and capital allocated to the knowledge sector

o These should be determined by choices of owners of labor and capital

allocating them to their highest return

o Romer simplifies the model by taking these to be exogenous

o Econ 454 studies models in which the rewards to capital and labor in the two

sectors are explicitly modeled and these decisions are allowed to be

endogenous

David Romer’s R&D model 47

We assume a Cobb-Douglas CRTS production function for goods:

11 1K LY t a K t A t a L t

Knowledge is produced according to a Cobb-Douglas that may or may not have

CTRS:

K LA t B a K t a L t A t

o (Different than in Ramsey model) Note that = 1 K L

AB a K a L

A

which means growth rate of A (our old g) depends on the amounts of K and L

devoted to research and is constant if those amounts are constant.

o Replication argument cannot be used to justify CRTS here

Same knowledge produced by two people is not twice as valuable

Positive spillovers could yield increasing returns to scale

Are other discoveries substitutes or complements for the next

discovery?

No depreciation and constant saving rate mean K t sY t

Exogenous growth of labor force: L t nL t

Analysis of R&D Model

Romer begins with a model in which there is no physical capital ( = = 0)

o We won’t analyze this model in detail, but note the equations of the model if

= 1 and n = 0 (so L is constant)

1 L

L

Y t A t a L

A tB a L

A t

Capital (K) Knowledge (A) Labor (L)

Goods-producing

sector Knowledge-

producing sector

Output

(Y)

Consumption

(C)

Investment

New knowledge

1 – aK a

K 1 – a

L

aL

s 1 – s

48 David Romer’s R&D model

Output is proportional to A and the growth rate of A is a constant, so

this model has a constant growth rate of output that is determined by

B, aL, and L (and ). Higher R&D productivity, more labor being used in the labs, and a

bigger population all lead to a higher growth rate (not just to a higher,

parallel growth path)

For the full model (with K), we have two state variables, A and K

o We denote the growth rates of A and K by gA and gK

Dynamics of K

1 1 11 1k LK t sY t s a a K t A t L t

o The term in brackets is a constant (over time) that we shall call cK

o The growth rate of K at every moment t is

1

K K

K t A t L tg t c

K t K t

o We seek a steady state in which K grows at a constant rate *Kg , so we want to

analyze the change in or growth rate of the growth rate

o 1K

A KK

g tg n g

g t

using our rules for growth rates

* *

0 if

0 if 0 :

0 if

K K A

K K A K K A

K K A

g t g t g t n

g t g t g t n g g g n

g t g t g t n

The 0Kg curve is a line with slope of one and intercept on the gK

axis at n ≥ 0.

Below the line, 0Kg and above the line 0Kg , so the

arrows point vertically toward the line

Dynamics of A

1

A K L

A tg t Ba a K t L t A t

A t

o The term in brackets is constant over time and called cA

o As above, the growth rate of the growth rate at every moment t is

1A

K AA

g tg t n g t

g t

* *

10 if

1 10 if 0 :

10 if

A K A

A K A A K A

A K A

ng t g t g t

n ng t g t g t g g g t

ng t g t g t

David Romer’s R&D model 49

The 0Ag curve is a line with slope 1

and intercept on the

vertical (gK) axis at –n/ ≤ 0.

To the left of the line 0Ag and to the right of the line 0Ag , so the

arrows point horizontally toward the line

Equilibrium dynamics

o The nature of the equilibrium depends crucially on two properties of the

parameters:

n > 0 vs. n = 0

This determines whether there is any exogenous source of

growth in the model

If n > 0 as in the Solow and Ramsey models, sustained

growth in total GDP is possible through exogenous growth in

L

+ = 1 vs. + < 1 (or + > 1)

This determines “returns to scale in produced inputs”

Note that K and A are “produced” in the model

The production function for goods always has constant

returns in produced inputs because K has exponent and A

has exponent 1 –

The production function for knowledge has returns to scale in

the two produced inputs equal to the sum of their exponents:

+

If + = 1, then the model can sustain ongoing

“endogenous” growth even if n = 0 because increases in both

K and A together are not subject to diminishing returns

Dynamics with n > 0 and n = 0

o With n > 0, the 0Kg line intercept is positive and the 0Ag line intercept

is negative

o If n = 0, both lines pass through the origin

o Case I: + < 1 (diminishing returns in produced inputs)

Slope of 0Ag line is 1

1

, so it is steeper than the 0Kg line

50 David Romer’s R&D model

Economy converges to unique equilibrium from all points in space

Solving algebraically, we can show

*

*

1

11

A

K

g n

g n

Growth here is exogenous in the sense that if n = 0, both K and A

stop growing. (Note that both lines intercept at the origin if n = 0.)

This case replicates the dynamics of the Solow model with g

determined endogenously as a function of n

o Case II: + = 1

In this case, the slope of the 0Ag line 1

1

and the two lines are

parallel (or coincident)

If n > 0, then they are parallel

gK

gA

gK*

n

0 g

A*

David Romer’s R&D model 51

The economy will move into the channel between the lines

and then growth in both K and A will accelerate forever.

Intuitively, a bigger economy means more scientists means

more discoveries means faster growth. As long as n > 0, the

exogenous growth in the labor force leads to accelerating

growth.

If n = 0, then the two lines coincide

Economy converged to the line and on the line, both growth

rates are constant and equal (because the line has slope of

one)

gK

gA

n

0

gK

gA

0

52 Microeconomics of R&D

Because * *K Ag g , K/A is constant in the steady state

o There is a unique K/A* that will sustain equal growth

in K and A and a unique common growth rate g* that

is consistent with that K/A*

o You will work out the algebra in Problem 3.5.

Example of this case: Suppose that B↑ so that cA increases.

o This raises gA and moves the economy to a point to the

right of the original equilibrium.

o Economy converges back up and to the left to a new

equilibrium that is higher than original.

Endogenous growth occurs in this case: economy sustains

positive growth even when there is no exogenous source (n =

0)

Growth rate depends (positively) on s, B, aK, aL, and L

o Case III: + > 1

In this case, the 0Ag line is flatter than the 0Kg because

11

This case looks like Case II, but the lines are not parallel.

In this case, we get explosive growth even when n = 0.

Microeconomics of R&D The key question that we have dodged in Romer’s R&D model: What determines aK?

o Capital owners must decide whether to build factories or labs

o Economists would assume that the choose the use of their capital that

provides the higher rate of return

So in equilibrium the amount of capital in the two sectors would have

to balance the marginal rates of return

o Rate of return on factories is straightforward: They produce output that is

sold to earn revenue

How do labs earn money?

o In the real world, there are lots of funding sources for R&D

Corporate funding

Government grants

Tuition from university students

Since we don’t model government or university research, we are

interested mostly in corporate-funded research and development

o In our model, knowledge is purely non-rival and non-excludable

Any discovery is immediately useful to all producers

There is no “appropriability” of knowledge for private benefit

New knowledge cannot be sold or used profitably

Model of Learning by Doing 53

Why would capital owners put money into labs that earn nothing?

They wouldn’t, so we would need to build a model of how lab

owners can earn money from R&D in order to pay for the

capital and labor that is used.

Models of aK

o Corporate R&D is profitable if there is an effective way for the company to

appropriate the knowledge

o This usually occurs by preventing other firms from using the knowledge

created through some kind of “appropriability mechanism”

May also involve licensing

Note that either is inefficient, because once created the knowledge is

nonrival and “should” be universally used for free

o Two common appropriability mechanisms are intellectual property rights

(patents) and secrecy

Both are flawed

Some kinds of intellectual property are better protected by patents,

some by secrecy, and others are virtually unprotectable

o Effective patent protection or secrecy gives an effective (but usually

temporary) monopoly on the use of the knowledge to the firm doing the

R&D

o Two common models for aK are based on this:

A model of product innovation in which R&D can produce new

varieties of (intermediate) goods on which the innovating firm holds a

monopoly

A model of process innovation in which R&D can advance

productive efficiency of one (intermediate) good (of many) and have a

cost advantage in production until another firm leap-frogs it

Both models add complexity to Romer’s R&D model because both

require multiple goods in order to have more than one firm

We study both models in Econ 454

Model of Learning by Doing Romer’s short section on learning by doing develops the essence of Paul Romer’s first

(1986) endogenous growth model.

o Kenneth Arrow developed a model in the 1960s based on the idea that a

firm’s A would be increased as it produced output, so A Y

o Paul Romer’s version of this was slightly different

Firms’ learning is related to capital accumulation rather than output

Knowledge is non-appropriable

New knowledge occurs as a by-product of capital investment

54 Model of Learning by Doing

Firms have (some) incentive to invest, so knowledge creation

happens despite pure nonrivalry

Learning by doing with a constant saving rate

o

1Y t K t A t L t

A t BK t

o Solving out A yields

1 11

1 11

Y t K t B K t L t

K t sY t sB L t K t

o This model converges, has endogenous growth, or explodes as < 1, = 1,

> 1

o Case of = 1 is the endogenous-growth case

Let n = 0 so there is no exogenous growth

1Y t BL K t bK t

K t sY t sbK t

K tsb

K t

Thus, growth in the capital stock and output is constant at rate sb

Any increase in saving, in the productivity of learning, or in the labor

force would increase growth

Ramsey consumers in the learning-by-doing model (not done this way in 4th edition)

o Assume = 1 and n = 0, so we have the endogenous-growth case

o Aggregate knowledge is proportional to aggregate capital stock (but this is not

the case at the firm level)

Firms take aggregate knowledge as given and do not consider how

their own investment will add to it because they are small

o

1

1 11

i i i

i i i

Y t K t A t L t

A t BK t

Y t B K t K t L t

o The private marginal product of capital is

1

11i i

i i

Y t K tB K t r t

K t L t

(there is no depreciation)

Each firm sets its i

i

K

L so that the private marginal product equals the

economy-wide interest rate r

This means that all firms have the same i

i

K

L that is equal to the

aggregate K/L

Model of Learning by Doing 55

Setting Ki/Li = K/L,

1

11

1 1 11

1 1

K tr t B K t

L t

B K t K t L t

BL t BL r

This rate of return r is constant over time (with n = 0) and depends

on the rate of knowledge accumulation through investment B, , and

the size of the labor force

Note that the marginal social product of capital (varying K as well as

Ki) is larger than the marginal private product

1i

iK K

i K K

YMSP BL MPP

K

This means that individual firms will underinvest in capital

They do not take into account the positive social externality

that their investment conveys on all firms through increased

knowledge

This means that the privately generated growth rate will be

lower than the socially optimal growth rate

o Ramsey consumers, as usual, choose a consumption path that satisfies the

Euler equation

C t r t

C t

.

In this case, r t r and

1C t BLrg

C t

o To satisfy the economy’s budget constraint, Y must grow at the same rate as

C, so the economy grows at g at every instant.

o There are no convergence dynamics: wherever an economy is, it just grows at

g from there. (Poor countries with same parameters do not catch up.)

o Growth rate g depends on parameters of economy: , B, L, , .

o Once again, we have “scale effects” because a larger L means a faster growth

rate.

If we allow n > 0, then we have both endogenous and exogenous

growth and the growth rate accelerates over time.

Are scale effects realistic? Some argue no, but Kremer’s argument for

Eurasia, Australia, and Tasmania seems to provide some support.

In addition, there is much evidence that growth has accelerated over

the centuries (as population has grown).

56 (Paul) Romer model (not worth doing the details)

o Non-optimality: social planner would internalize the knowledge externality

and use 1 1*r BL r BL

leading to faster growth at

1*

BLg g

(Paul) Romer model (not worth doing the

details) What’s different about this model?

o We model the incentives for production of knowledge explicitly

o We introduce the “Ethier production function” and the now-ubiquitous

model of a continuum of “intermediate goods”