Download ECON 5010 Class Notes Endogenous Growth Theory 1 Introduction

ECON 5010 Class Notes
Endogenous Growth Theory
1
Introduction
One drawback of the Solow model is that long-run growth in per capita income is entirely exogenous. In the
absence of exogenous technological growth, income per capita would be static in the long run. This is an
implication of diminishing marginal returns to capital. To generate long-run endogenous economic growth,
we need to relax the assumption of diminishing marginal returns to capital.
To introduce endogenous growth, it is necessary to have increasing (or at least non-decreasing) returns
to capital.
To incorporate increasing returns, we need to modify the Neoclassical growth model.
Two
possibilities:
1. Positive Production Externality.
Models with increasing returns can be compatible with perfect
competition if private returns to capital are diminishing (e.g., Arrow’s model).
2. Imperfect Competition.
If …rms have some market power, then the model can support increasing
private marginal returns to capital (e.g., Romer’s model).
2
A Simple Endogenous Growth Model
Consider a simple growth model where consumers choose ct to maximize
U0 =
1
X
t
log(ct ):
(1)
ct
(2)
t=0
The law of motion for capital is
kt+1 = yt
with 100% depreciation of capital. The production function is
yt = kt Kt
where 0 <
< 1,
+
=
1, and 0 <
< 1.
Lower case kt refers to the individual …rm.
(3)
Upper
case Kt is the economy-wide average of the capital stock. This is the positive production externality. If all
…rms are identical, then kt = Kt . This implies that the social production function is
yt = kt :
1
(4)
The individual …rm will treat Kt as given so the private production function takes the standard form:
yt = At kt
(5)
where At = Kt .
2.1
Competitive Equilibrium
There exists a production externality – individual …rms act as if their investment has no impact on other
…rms’productivity.
Consumers choose fkt g1
t=1 to maximize
U0 =
1
X
t
log(At kt
kt+1 )
(6)
t=0
where (2) and (5) have been substituted directly into the objective function, changing the choice variable to
kt . Individual …rms treat At as given. The …rst-order condition is
@U0
=
@kt+1
1
t
kt Kt
+
kt+1
t+1
kt+11 Kt+1
= 0;
kt+1 Kt+1 kt+2
(7)
for all t. This simpli…es to
1
kt Kt
kt+1
=
kt+11 Kt+1
;
kt+1 Kt+1 kt+2
(8)
which is a non-linear second-order di¤erence equation.
We can solve (8) using the method of undetermined coe¢ cients (see Blanchard and Fischer’s toolkit on
p. 261). Start by guessing the form of the solution
kt+1 = kt Kt :
(9)
Using the guess, we get
1
kt Kt
kt Kt
=
kt+11 Kt+1
;
kt+1 Kt+1
kt+1 Kt+1
(10)
which can be simpli…ed to
(1
1
=
)kt Kt
(1
)kt+1
:
This is a non-linear …rst-order di¤erence equation. Equation (11) implies that
kt+1 =
kt Kt :
2
(11)
=
and the solution is
(12)
Imposing the equilibrium condition kt = Kt , we get
kt+1 =
kt :
(13)
) + log(kt ):
(14)
Next, we can log linearize:
log(kt+1 ) = log(
Repeated substitutions in (14) gives
log(kt+1 ) = (1 + +
2
+
t
+
) log(
)+
t+1
log(k0 ):
(15)
Equation (15) and its lagged version can be combined to get
CE
k
= log
kt+1
kt
=
CE
k )
Therefore, the growth in the capital stock (
t
[log(
)+(
1) log(k0 )] :
(16)
may be decreasing, zero, or increasing depending on the
initial capital stock. If the initial capital stock is
)1=(1
k0 = (
then kt will settle down to a steady state.
)
(17)
Higher initial values lead to increasing growth in kt ; lower
initial values lead to decreasing growth in kt . The model displays multiple equilibria and dependence on k0 .
Standard Neoclassical models (e.g., Solow or Ramsey) display a unique steady state and no dependence on
k0 .
2.2
Social Planner’s Problem
The social planner’s problem is similar but she will not treat At as given. The social planner chooses fkt g1
t=1
to maximize
U0 =
1
X
t
log(kt
kt+1 ):
(18)
t=0
The …rst-order condition is
@U0
=
@kt+1
t
kt
1
+
kt+1
t+1
kt+11
= 0;
kt+1 kt+2
(19)
which simpli…es to
kt
kt+11
1
=
:
kt+1
kt+1 kt+2
3
(20)
Guessing the form of the solution as
kt+1 = kt ;
and substituting into (20) gives
=
(21)
. Thus, the optimal path is
kt+1 =
kt :
(22)
Log linearizing and following similar steps, we get
SP
k
2.3
= log
kt+1
kt
t
=
[log(
)+(
1) log(k0 )] :
(23)
Summarizing
To simplify, consider the case of
= 1 or the so-called "AK Growth Model". In this case, the growth rates
are
SP
k
= log( ) >
CE
k
= log(
).
(24)
The economy experiences continual growth because the marginal product of capital does not diminish. Also,
the growth rate is higher for the social planner because the social marginal product of capital is higher than
in the private case.
Question. How do you reach the Pareto optimal path in a competitive economy?
Answer. Have Barack Obama subsidize …rms at rate s.
Question. What is s?
Answer. The subsidy rate s should equate the social and private marginal products of capital. Solve
the following equation for s:
1
M PkP = (1 + s) kt
= kt
1
= M PkS ,
(25)
which gives
s = = :
(26)
Question. How do you pay for s?
Answer. Lump-sum tax,
t.
Society’s budget constraint is
kt+1 = (1 + s)kt
ct
t;
(27)
which implies that the optimal tax is
t
= skt .
4
(28)
3
Learning-by-Doing
Arrow (1962) was a key early paper in the movement toward endogenous growth theory. One way of gaining
knowledge is to learn through experience. Since knowledge is embedded in At in the standard Neoclassical
model, learning is treated as exogenous.
Arrow endogenizes the learning process and incorporates it in the Neoclassical growth model. His famous
example is that the labor hours required to build airplane frames declines with the third root of the number
of frames built.
3.1
3.1.1
Model
Technology
The production function is
Yi = Ki (Ai Li )1
(29)
Ai = BK
(30)
where
is knowledge that results from producing new capital goods.
Knowledge spills over to all …rms and is a
function of K, the economy-wide average capital stock. Substituting for Ai , the production function is
Yi = B 1
K
(1
)
Ki L1i
:
(31)
There are decreasing returns to Ki at the private level. At the social level, we have non-decreasing returns
to capital if
3.1.2
1. If
= 1, we have the "AK" growth model where A = B 1
and Li = 1.
Consumers
Identical and in…nitely lived consumers choose c(t) to maximize
U (t) =
Z
1
e
(
n)t
u (c(t)) dt
(32)
0
subject to the ‡ow budget constraint
a(t)
_
= w(t) + r(t)a(t)
5
c(t)
na(t)
(33)
where a(t) is the asset, a(0) given,
> n, and consumers take input prices as given. The consumption path
must also satisfy the "No Ponzi Game" constraint:
Rt
lim e
0
t!1
(r(s) n)ds
a(t)
0:
(34)
Labor grows at constant exponential rate, L(t) = L(0)ent , where L(0) is normalized to one. The CES utility
function is
u (c(t)) =
c(t)(1
1
)
1
(35)
with derivatives
u0 (c(t))
= c(t)
u00 (c(t))
=
>0
c(t)
(36)
(1+ )
< 0:
(37)
The intertemporal elasticity of substitution ( ) is given by
=
1
u0
= :
00
cu
(38)
To solve the consumer’s maximization problem, we form the Hamiltonian function:
J =e
(
n)t
u (c(t)) + (t) [w + ra(t)
c(t)
na(t)]
(39)
where c(t) is the control variable, a(t) is the state variable, and (t) is the costate variable. The conditions
for a maximum are
@J
@c(t)
=
_ (t)
=
lim (t)a(t)
=
t!1
e
n)t 0
(
u (c(t))
@J
=
@a(t)
(t) = 0
(t)(r
(40)
n)
(41)
0:
(42)
Di¤erentiate (40) with respect to t gives
_ (t) = e
(
n)t 00
u (c(t)) c(t)
_
(
n)e
(
(
n)e
n)t 0
u (c(t)) :
(43)
n)t 0
(44)
Substituting from (41) and rearranging, we get
(t)(r
n) = e
(
n)t 00
u (c(t)) c(t)
_
6
(
u (c(t)) :
Substituting from (40), we get
e
(
Dividing through by
n)t 0
u (c(t)) (r
(
e
n) = e
n)t 00
(
u (c(t)) c(t)
_
(
n)e
(
n)t 0
u (c(t)) :
(45)
n)t 0
u (c(t)), we get
(r
u00 (c(t))
c(t)
_ +(
u0 (c(t))
n) =
n):
(46)
Rearranging and substituting in for , we get
(r
c(t)
_
=
c(t)
)
:
(47)
This is the Euler equation or optimal rule for consumption.
3.1.3
Firms
Each …rm’s problem is essentially static. The …rm chooses Ki to maximize
i
= Yi
RKi
wLi
(48)
subject to equation (31)
Yi = B 1
K
(1
)
Ki L1i
;
where R is the rental price of capital and w is the wage rate. Written in per worker form, the …rm chooses
ki to maximize
i
= yi
Rki
w
(49)
subject to
(50)
yi = Aki
where A = B 1
K
(1
)
. If capital and loans are perfect substitutes, then
r=R
:
(51)
The …rst-order condition is
@ i
= Aki
@ki
1
7
(r + ) = 0;
(52)
which states that the net marginal product of capital, M Pkprivate
, must equal r. Imposing the equilibrium
condition ki = k = Ki =Li = K=L, the private marginal product of capital is
M Pkprivate = B 1
K
(1
)
1
K
L1
= L(1
)
h
B1
k(
1)(1
)
i
:
(53)
Because M Pkprivate depends on the size of the labor force, economic growth is subject to "scale e¤ects".
3.1.4
Equilibrium
For simplicity, assume n = 0. Substituting (51) into the Euler equation for consumption (47) gives
c(t)
_
=
c(t)
c
=
1
(M Pkprivate
):
(54)
Also, setting a = k in the budget constraint (33) gives
_
k(t)
=
k(t)
=
k
1
(w(t)
k(t)
c(t)) + r(t):
(55)
Because the production is homogenous of degree one in Ki and Li , we can write
yi = Rki + w = M Pkprivate ki + w.
(56)
Substituting w and r(t) into (55) gives
k
=
1
y(t)
k(t)
M Pkprivate k(t)
c(t) + M Pkprivate
=
1
(y(t)
k(t)
c(t))
.
(57)
Because the transversality condition implies c(t) = ' k(t), we get
k
Also, because y(t) = Ak(t)
+ (1
)
=
c:
(58)
, we get
y
= [ + (1
)]
k.
(59)
Therefore, we have the following cases...
If
< 1, then (
c
=
k
>
y)
and the model exhibits convergence.
When
= 0, we get the
Neoclassical model.
If
= 1, then (
c
=
k
=
y)
and the model exhibits no transition dynamics. This is the AK growth
8
model.
If
3.1.5
> 1, then (
c
=
k
<
y)
and the model exhibits multiple equilibria.
Externality and Pareto Optimality
Continue to let n = 0. There exists an externality because individual …rms do not account for the positive
spillover e¤ects from investment. The social planner chooses c(t) to maximize
Z
1
e
t
u (c(t)) dt
(60)
0
subject to the law of motion for capital. The solution for the growth in consumption is
c
=
1
(M Pksocial
);
h
) + )k (
where
M Pksocial = L(1
)
B1
( (1
(61)
1)(1
)
i
:
(62)
The solution for the competitive market is
c
=
1
(M Pkprivate
):
The social optimum can be achieved by subsidizing output at rate s = (1
4
(63)
)= .
Romer’s Model of R&D Driven Growth
Here, I outline Paul Romer’s (1990) seminal article "Endogenous Technological Change" published in the
Journal of Political Economy.
4.1
Introduction
Romer’s model has the following distinguishing features:
As in the Solow model, technological change fuels growth.
Technological change arises from research and development (R&D).
R&D is the result of pro…t-maximizing private …rms.
The production of ideas from R&D can be used over and over again with no cost.
Newly discovered ideas are non-rivalrous but partially excludable.
9
Partial excludability implies that …rms have market power and can earn positive rents.
Romer’s model incorporates market power through imperfect competition.
4.2
4.2.1
The Model
Final-Goods Production
There are four inputs into the production of …nal output:
1. Overall capital stock: K =
discovered inputs, and
RA
i=0
x(i)di, where x(i) are durable inputs, A measures the number of
are the units of forgone consumption necessary to produce x(i).
2. Labor: L is the constant skills available from a healthy physical body.
3. Human capital: H = HY + HA is rivalrous and can be used to produce output or new ideas.
4. Knowledge: A is non-rivalrous and measures the number of discovered ideas.
The production function is a modi…ed Cobb-Douglas:
Y = Y (HY ; L; x; A) = HY L
RA
i=0
x(i)1
di:
(64)
The production function (64) is homogeneous of degree one in the inputs so it can support competitive,
price-taking …rms.
4.2.2
Intermediate-Goods Production
The production of x(i) has the following features:
Each distinct …rm i produces durable good i.
Once a …rm designs i (or purchase the rights to produce i), it can convert
units of …nal output into
one unit of durable good i.
A …rm can obtain an in…nitely lived patent on durable good i.
The intermediate …rm can rent x(i) to the …nal-good producing …rm for p(i).
Intermediate …rm i faces a downward-sloping demand for x(i).
The aggregate stock of durables, K, evolves according to
_
K(t)
= Y (t)
10
C(t):
(65)
4.2.3
Production of Ideas/Designs: R&D Sector
The production function for new designs by researcher j is given by
H j A;
(66)
where
is a productivity parameter;
H j is the human capital available to researcher j; and
A is the total stock of knowledge or ideas.
The total stock of ideas therefore evolves according to
A_ = HA A;
where HA =
R
j
(67)
H j dj. Equation (67) has three important properties:
1. The growth rate of ideas is increasing in HA . Devoting more human capital to R&D leads to more
ideas.
2. A larger stock of ideas increases the productivity in the R&D sector.
3. Intermediate producer j does not have property rights over j in the R&D sector (see Romer’s widget/wodget discussion).
4.2.4
Pro…t Maximization
The representative …nal-goods producer solves:
max
x
Z
A
HY L x(i)1
p(i)x(i) di:
(68)
p(i) = 0:
(69)
0
The …rst-order condition for each i is
(1
)HY L x(i)
( + )
Therefore, the demand curve for each intermediate good is downward-sloping and given by:
p(i) = (1
)HY L x(i)
11
( + )
:
(70)
The intermediate-goods producer solves:
max
x
= p(x)x
)HY L x1
r x = (1
r x;
(71)
where r is the interest rate applied to x units of output needed to produce x durables.
The …rst-order
condition is
@
= (1
@x
)2 HY L x
( + )
r = 0:
(72)
Substituting (69) into (72) gives
(1
)p(i)
r =0
which implies that the monopoly price is a simple markup over marginal cost:
1
p=
r :
1
(73)
The decision to undertake R&D depends on whether the discounted stream of net revenue:
Z
1
e
rt
( )d
PA (t)
(74)
t
is positive, where PA (t) is the price of designs. Competition will drive (74) to zero, which implies that
(t) = r(t)PA :
(75)
In other words, the instantaneous pro…ts must cover the interest costs of the initial design expense.
4.2.5
Consumers
Ramsey consumers will maximize
Z
0
1
e
t
C(t)1
1
1
dt
(76)
subject to the standard intertemporal budget constraint. The familiar optimal consumption rule is
_
C(t)
r
=
C(t)
:
Consumers also inelastically supply L units of labor and H units of human capital.
intermediate-goods-producing …rms and have pro…ts paid back as dividends.
4.2.6
Equilibrium
An equilibrium is a path of prices and quantities where...
12
(77)
Consumers own the
consumers maximize lifetime utility;
consumers decide optimally where to allocate their human capital;
…nal goods producers maximize pro…ts by choosing labor, human capital and durable inputs;
each intermediate …rm i choose x(i) to maximize pro…ts given its downward-sloping demand;
…rms take PA (t) as given when deciding whether to invest in R&D to discover a new design; and
the supply of each good equals its demand.
4.3
Solution
Assume for the moment that A is …xed. All durable goods will be supplied at the same level, x, and the
aggregate capital stock is given by K = Ax. The production function is therefore given by
Y = HY L Ax1
= (HY A) (AL) K 1
+
1
:
(78)
This is the standard Solow production function with diminishing marginal returns to capital and convergence.
The solution has a balanced growth path where C, A, K, and Y grow at (endogenous) constant exponential rate g. The equilibrium level of human capital used in the output sector is
HY =
1
(1
)( + )
r.
(79)
From equation (67), we know that
g = HA = (H
HY ) = H
r;
(80)
where
=
(1
)( + )
.
(81)
We also know that
C_
r
=g=
C
(82)
so that the growth rate of the economy can be written as
g=
4.4
H
+1
Interpretation of the Solution
There are several notable implications of the model:
13
:
(83)
1. An increase in r will lower growth.
The bene…ts from R&D come in the future while the cost is
immediate. Higher r imply a lower present discounted value for the stream of revenue from R&D and
less R&D.
2. There are no scale e¤ects with L.
A larger population increases the return to human capital in
manufacturing and in research. The e¤ects cancel so there is no change in the allocation of human
capital to manufacturing (HY ) or to research (HA ).
3. The appropriate scale factor is H.
4. An increase in
will decrease growth. The bene…ts from R&D come in the future. Less patience will
result in less R&D.
5. An increase in the intertemporal elasticity of substitution (1= ) will increase growth.
6. A direct subsidy to physical capital (e.g., Arrow’s learning-by-doing model) is not best.
A better
policy would reduce r and encourage more human capital in research.
7. The private growth rate, g, in equation (83) is too low because (i) discoveries from R&D are not
compensated for their use in the research sector and (ii) the inventors of designs charge too much
because of their market power. The socially optimal growth rate is
H
+ (1
gsocial =
where
= =( + ).
14
)
>g