Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 14 pptx

Themonopolistically competitive equilibrium is characterized by a smaller supply ofeach intermediate input and a lower growth rate than would be socially optimal.. Alongsuch a steady-sta

Chapter 14

Economic Growth

14.1 Introduction

This chapter describes basic nonstochastic models of sustained economic growth

We begin by describing a benchmark exogenous growth model, where sustainedgrowth is driven by exogenous growth in labor productivity Then we turnour attention to several endogenous growth models, where sustained growth of

labor productivity is somehow chosen by the households in the economy We

describe several models that differ in whether the equilibrium market economymatches what a benevolent planner would choose Where the market outcomedoesn’t match the planner’s outcome, there can be room for welfare-improvinggovernment interventions The objective of the chapter is to shed light on themechanisms at work in different models We try to facilitate comparison byusing the same production function for most of our discussion while changingthe meaning of one of its arguments

Paul Romer’s work has been an impetus to the revived interest in the ory of economic growth In the spirit of Arrow’s (1962) model of learning bydoing, Romer (1986) presents an endogenous growth model where the accumu-lation of capital (or knowledge) is associated with a positive externality on theavailable technology The aggregate of all agents’ holdings of capital is positivelyrelated to the level of technology, which in turn interacts with individual agents’savings decisions and thereby determines the economy’s growth rate Thus, the

the-households in this economy are choosing how fast the economy is growing but

do so in an unintentional way The competitive equilibrium growth rate fallsshort of the socially optimal one

Another approach to generating endogenous growth is to assume that allproduction factors are reproducible Following Uzawa (1965), Lucas (1988)formulates a model with accumulation of both physical and human capital Thejoint accumulation of all inputs ensures that growth will not come to a halt eventhough each individual factor in the final-good production function is subject

– 443 –

to diminishing returns In the absence of externalities, the growth rate in thecompetitive equilibrium coincides in this model with the social optimum.

Romer (1987) constructs a model where agents can choose to engage inresearch that produces technological improvements Each invention represents

a technology for producing a new type of intermediate input that can be used inthe production of final goods without affecting the marginal product of existingintermediate inputs The introduction of new inputs enables the economy to ex-perience sustained growth even though each intermediate input taken separately

is subject to diminishing returns In a decentralized equilibrium, private agentswill only expend resources on research if they are granted property rights overtheir inventions Under the assumption of infinitely lived patents, Romer solvesfor a monopolistically competitive equilibrium that exhibits the classic tensionbetween static and dynamic efficiency Patents and the associated market powerare necessary for there to be research and new inventions in a decentralizedequilibrium, while the efficient production of existing intermediate inputs wouldrequire marginal-cost pricing, that is, the abolition of granted patents Themonopolistically competitive equilibrium is characterized by a smaller supply ofeach intermediate input and a lower growth rate than would be socially optimal

Finally, we revisit the question of when nonreproducible factors may notpose an obstacle to growth Rebelo (1991) shows that even if there are non-reproducible factors in fixed supply in a neoclassical growth model, sustainedgrowth is possible if there is a “core” of capital goods that is produced with-out the direct or indirect use of the nonreproducible factors Because of theever-increasing relative scarcity of a nonreproducible factor, Rebelo finds thatits price increases over time relative to a reproducible factor Romer (1990)assumes that research requires the input of labor and not only goods as in hisearlier model (1987) Now, if labor is in fixed supply and workers’ innate pro-ductivity is constant, it follows immediately that growth must asymptoticallycome to an halt To make sustained growth feasible, we can take a cue from ourearlier discussion One modeling strategy would be to introduce an externalitythat enhances researchers’ productivity, and an alternative approach would be toassume that researchers can accumulate human capital Romer adopts the firsttype of assumption, and we find it instructive to focus on its role in overcoming

a barrier to growth that nonreproducible labor would otherwise pose

and σ = 1 is taken to be logarithmic utility.1 Lowercase letters for quantities,

such as c t for consumption, are used to denote individual variables, and uppercase letters stand for aggregate quantities

For most part of our discussion of economic growth, the production functiontakes the form

F (K t , X t ) = X t f

ˆ

That is, the production function F (K, X) exhibits constant returns to scale in

its two arguments, which via Euler’s theorem on linearly homogeneous functionsimplies

F (K, X) = F1(K, X) K + F2(K, X) X, (14.2.3) where F i (K, X) is the derivative with respect to the i th argument [and F ii (K, X) will be used to denote the second derivative with respect to the i th argument] The input K t is physical capital with a rate of depreciation equal to δ New

capital can be created by transforming one unit of output into one unit of ital Past investments are reversible It follows that the relative price of capital

cap-in terms of the consumption good must always be equal to one The second

argument X t captures the contribution of labor Its precise meaning will differamong the various setups that we will examine

We assume that the production function satisfies standard assumptions ofpositive but diminishing marginal products,

We will also make use of the mathematical fact that a linearly homogeneous

function F (K, X) has first derivatives F i (K, X) homogeneous of degree 0; thus,

the first derivatives are only functions of the ratio ˆK In particular, we have

F1(K, X) = ∂ Xf (K/X)

ˆ

K

− f ˆ

K

ˆ

K (14.2.5b)

14.2.1 Balanced growth path

We seek additional technological assumptions to generate market outcomes with

steady-state growth of consumption at a constant rate 1 + µ = c t+1 /c t Theliterature uses the term “balanced growth path” to denote a situation whereall endogenous variables grow at constant (but possibly different) rates Alongsuch a steady-state growth path (and during any transition toward the steadystate), the return to physical capital must be such that households are willing

to hold the economy’s capital stock

In a competitive equilibrium where firms rent capital from the agents, the

rental payment r t is equal to the marginal product of capital,

r t = F1(K t , X t ) = f 

ˆ

respect to k t+1 is

u  (c t ) = βu  (c t+1 ) (r t+1+ 1− δ) (14.2.8) After using equations ( 14.2.1 ) and ( 14.2.6 ) in equation ( 14.2.8 ), we arrive at

the following equilibrium condition:



c t+1 c

K t+1

+ 1− δ (14.2.9)

Exogenous growth 447

We see that a constant consumption growth rate on the left-hand side is tained in an equilibrium by a constant rate of return on the right-hand side Itwas also for this reason that we chose the class of utility functions in equation

sus-( 14.2.1 ) that exhibits a constant intertemporal elasticity of substitution These

preferences allow for balanced growth paths.2

Equation ( 14.2.9 ) makes clear that capital accumulation alone cannot tain steady-state consumption growth when the labor input X t is constant over

sus-time, X t = L Given the second Inada condition in equations ( 14.2.4 ), the limit

of the right-hand side of equation ( 14.2.9 ) is β(1 −δ) when ˆ K approaches

infin-ity The steady state with a constant labor input must therefore be a constantconsumption level and a capital-labor ratio ˆK  given by

f 

ˆ

consump-therefore, the marginal product of capital remain constant in the steady state

2 To ensure well-defined maximization problems, a maintained assumptionthroughout the chapter is that parameters are such that any derived consump-

tion growth rate 1+µ yields finite lifetime utility; i.e., the implicit restriction on parameter values is that β(1 + µ) 1−σ < 1 To see that this condition is needed,

substitute the consumption sequence {c t } ∞

t=0 ={(1 + µ) t c0} ∞

t=0 into equation

( 14.2.1 ).

A time-invariant rate of return is in turn consistent with households choosing aconstant growth rate of consumption, given the assumption of isoelastic prefer-ences.

Evaluating equation ( 14.2.9 ) at a steady state, the optimal ratio ˆ K  isgiven by

(1 + µ) σ = β



f 

ˆ

K 

+ 1− δ (14.3.1) While the steady-state consumption growth rate is exogenously given by 1 + µ ,

the endogenous steady-state ratio ˆK  is such that the implied rate of return on

capital induces the agents to choose a consumption growth rate of 1 + µ As can

be seen, a higher degree of patience (a larger β ), a higher willingness porally to substitute (a lower σ ) and a more durable capital stock (a lower δ )

intertem-each yield a higher ratio ˆK , and therefore more output (and consumption)

at a point in time; but the growth rate remains fixed at the rate of exogenouslabor-augmenting technological change It is straightforward to verify that thecompetitive equilibrium outcome is Pareto optimal, since the private return tocapital coincides with the social return

Physical capital is compensated according to equation ( 14.2.6 ), and labor

is also paid its marginal product in a competitive equilibrium,

then be that the production function F (K t , A t L) exhibits increasing returns to

scale in the three “inputs” K t , A t , and L , which is not compatible with the

existence of a competitive equilibrium This problem is to be kept in mind as

we now turn to one way to endogenize economic growth

Externality from spillovers 449

14.4 Externality from spillovers

Inspired by Arrow’s (1962) paper on learning by doing, Romer (1986) suggeststhat economic growth can be endogenized by assuming that technology growsbecause of aggregate spillovers coming from firms’ production activities Theproblem alluded to in the previous section that a competitive equilibrium fails

to exist in the presence of increasing returns to scale is avoided by letting nological advancement be external to firms.3 As an illustration, we assumethat firms face a fixed labor productivity that is proportional to the currenteconomy-wide average of physical capital per worker.4 In particular,

tech-X t= ¯K t L, where ¯K t= K t

L .

The competitive rental rate of capital is still given by equation ( 14.2.6 ) but we

now trivially have ˆK t = 1 , so equilibrium condition ( 14.2.9 ) becomes

Note first that this economy has no transition dynamics toward a steady state

Regardless of the initial capital stock, equation ( 14.4.1 ) determines a

time-invariant growth rate To ensure a positive growth rate, we require the

param-eter restriction β[f (1) + 1− δ] ≥ 1 A second critical property of the model is

that the economy’s growth rate is now a function of preference and technologyparameters

The competitive equilibrium is no longer Pareto optimal, since the privatereturn on capital falls short of the social rate of return, with the latter returngiven by

4 This specific formulation of spillovers is analyzed in a rarely cited paper byFrankel (1962)

where the last equality follows from equations ( 14.2.5 ) This higher social rate

of return enters a planner’s first-order condition, which then also implies a higheroptimal consumption growth rate,

equilib-t=0 , the production function F (k t , ¯ K t l t) exhibits

constant returns to scale in k t and l t So, once again, factor payments in acompetitive equilibrium will be equal to total output, and optimal firm size

is indeterminate Therefore, we can consider a representative agent with oneunit of labor endowment who runs his own production technology, taking thespillover effect as given His resource constraint becomes

c t + k t+1 = F k t , ¯ K t

+ (1− δ) k t= ¯K t f

and the private gross rate of return on capital is equal to f  (k t / ¯ K t) + 1− δ

After invoking the equilibrium condition k t= ¯K t, we arrive at the competitive

equilibrium return on capital f (1) + 1−δ that appears in equation (14.4.1) In

contrast, the planner maximizes utility subject to a resource constraint wherethe spillover effect is internalized,

All factors reproducible 451

14.5 All factors reproducible

14.5.1 One-sector model

An alternative approach to generating endogenous growth is to assume that allfactors of production are producible Remaining within a one-sector economy,

we now assume that human capital X t can be produced in the same way as

physical capital but rates of depreciation might differ Let δ X and δ K be therates of depreciation of human capital and physical capital, respectively.The competitive equilibrium wage is equal to the marginal product of hu-man capital

w t = F2(K t , X t ) (14.5.1) Households maximize utility subject to budget constraint ( 14.2.7 ) where the term χ t is now given by

χ t = w t x t+ (1− δ X ) x t − x t+1

The first-order condition with respect to human capital becomes

u  (c t ) = βu  (c t+1 ) (w t+1+ 1− δ X ) (14.5.2) Since both equations ( 14.2.8 ) and ( 14.5.2 ) must hold, the rates of return on the

two assets have to obey

F1(K t+1 , X t+1)− δ K = F2(K t+1 , X t+1)− δ X ,

and after invoking equations ( 14.2.5 ),

f

ˆ

K t+1

= δ X − δ K , (14.5.3)

which uniquely determines a time-invariant competitive equilibrium ratio ˆK ,

as a function solely of depreciation rates and parameters of the productionfunction.5

5 The left side of equation (14.5.3) is strictly increasing, since the derivative

with respect to ˆK is −(1 + ˆ K)f ( ˆK) > 0 Thus, there can only be one

solu-tion to equasolu-tion ( 14.5.3 ) and existence is guaranteed because the left-hand side

After solving for f ( ˆK  ) from equation ( 14.5.3 ) and substituting into tion ( 14.2.9 ), we arrive at an expression for the equilibrium growth rate

no longer any discrepancy between private and social rates of return.6

The problem of optimal taxation with commitment (see chapter 15) is ied for this model of endogenous growth by Jones, Manuelli, and Rossi (1993),who adopt the assumption of irreversible investments

stud-ranges from minus infinity to plus infinity The limit of the left-hand side whenˆ

K approaches zero is f (0) − lim Kˆ→0 f ( ˆK) , which is equal to minus infinity by

equations ( 14.2.4 ) and the fact that f (0) = 0 [Barro and Sala-i-Martin (1995)

show that the Inada conditions and constant returns to scale imply that all

production factors are essential, i.e., f (0) = 0 ] To establish that the left side of equation ( 14.5.3 ) approaches plus infinity when ˆ K goes to infinity, we can define

the function g as F (K, X) = Kg( ˆ X) where ˆ X ≡ X/K and derive an

alterna-tive expression for the left-hand side of equation ( 14.5.3 ), (1 + ˆ X)g ( ˆX) −g( ˆ X) ,

for which we take the limit when ˆX goes to zero.

6 It is instructive to compare the present model with two producible factors,

F (K, X) , to the previous setup with one producible factor and an externality,

˜

F (K, X) with X = ¯ KL Suppose the present technology is such that ˆ K = 1

and δ K = δ X, and the two different setups are equally productive; i.e., we

assume that F (K, X) = ˜ F (2K, 2X) , which implies f ( ˆ K) = 2 ˜ f ( ˆ K) We can

then verify that the present competitive equilibrium growth rate in equation

( 14.5.4 ) is the same as the planner’s solution for the previous setup in equation ( 14.4.3 ).

All factors reproducible 453

14.5.2 Two-sector model

Following Uzawa (1965), Lucas (1988) explores endogenous growth in a sector model with all factors being producible The resource constraint in thegoods sector is

two-C t + K t+1 = K t α (φ t X t)1−α+ (1− δ) K t , (14.5.5a)

and the linear technology for accumulating additional human capital is

X t+1 − X t = A (1 − φ t ) X t , (14.5.5b) where φ t ∈ [0, 1] is the fraction of human capital employed in the goods sector,

and (1− φ t) is devoted to human capital accumulation (Lucas provides analternative interpretation that we will discuss later.)

We seek a balanced growth path where consumption, physical capital, andhuman capital grow at constant rates (but not necessarily the same ones) and

the fraction φ stays constant over time Let 1 + µ be the growth rate of consumption, and equilibrium condition ( 14.2.9 ) becomes

capi-equation ( 14.5.5a ) through by K t and applying equation ( 14.5.6 ) we obtain

By definition of a balanced growth path, K t+1 /K t is constant, so equation

( 14.5.7 ) implies that C t /K t is constant; that is, the capital stock must grow atthe same rate as consumption

Substituting K t = (1 + µ)K t −1 into equation ( 14.5.6 ),

which directly implies that human capital must also grow at the rate 1+ µ along

a balanced growth path Moreover, by equation ( 14.5.5b ), the growth rate is

1 + µ = 1 + A (1 − φ) , (14.5.8)

so it remains to determine the steady-state value of φ

The equilibrium value of φ has to be such that a unit of human capital

receives the same factor payment in both sectors; that is, the marginal products

of human capital must be the same,

p t A = (1 − α) K α

t [φX t]−α ,

where p t is the relative price of human capital in terms of the composite

con-sumption/capital good Since the ratio K t /X t is constant along a balanced

growth path, it follows that the price p t must also be constant over time nally, the remaining equilibrium condition is that the rates of return on humanand physical capital are equal,

Fi-p t (1 + A)

p t −1 = αK

α −1

t [φX t]1−α+ 1− δ,

and after invoking a constant steady-state price of human capital and

equilib-rium condition ( 14.5.6 ), we obtain

1 + µ = [β (1 + A)] 1/σ (14.5.9) Thus, the growth rate is positive as long as β(1 + A) ≥ 1, but feasibility requires

also that solution ( 14.5.9 ) falls below 1+A which is the maximum growth rate of human capital in equation ( 14.5.5b ) This parameter restriction, [β(1+A)] 1/σ <

(1 + A) , also ensures that the growth rate in equation ( 14.5.9 ) yields finite

lifetime utility

As in the one-sector model, there is no discrepancy between private andsocial rates of return, so the competitive equilibrium is Pareto optimal Lucas(1988) does allow for an externality (in the spirit of our earlier section) wherethe economy-wide average of human capital per worker enters the productionfunction in the goods sector, but as he notes, the externality is not needed togenerate endogenous growth

Lucas provides an alternative interpretation of the technologies in equations

( 14.5.5 ) Each worker is assumed to be endowed with one unit of time The

Research and monopolistic competition 455

time spent in the goods sector is denoted φ t, which is multiplied by the agent’s

human capital x t to arrive at the efficiency units of labor supplied The ing time is spent in the education sector with a constant marginal productivity

remain-of Ax t additional units of human capital acquired Even though Lucas’s pretation does introduce a nonreproducible factor in form of a time endowment,the multiplicative specification makes the model identical to an economy withonly two factors that are both reproducible One section ahead we will study asetup with a nonreproducible factor that has some nontrivial implications

inter-14.6 Research and monopolistic competition

Building on Dixit and Stiglitz’s (1977) formulation of the demand for tiated goods and the extension to differentiated inputs in production by Ethier(1982), Romer (1987) studies an economy with an aggregate resource constraint

differen-of the following type:

where one unit of the intermediate input Z t+1 (i) can be produced from one unit

of output at time t , and Z t+1 (i) is used in production in the following period

t + 1 The continuous range of inputs at time t , i ∈ [0, A t] , can be augmented

for next period’s production function at the constant marginal cost κ

In the allocations that we are about to study, the quantity of an

interme-diate input will be the same across all existing types, Z t (i) = Z t for i ∈ [0, A t]

The resource constraint ( 14.6.1 ) can then be written as

C t + A t+1 Z t+1 + (A t+1 − A t ) κ = L 1−α A t Z t α (14.6.2)

If A t were constant over time, say, let A t = 1 for all t , we would just have

a parametric example of an economy yielding a no-growth steady state given

by equation ( 14.2.10 ) with δ = 1 Hence, growth can only be sustained by

allocating resources to a continuous expansion of the range of inputs But thisapproach poses a barrier to the existence of a competitive equilibrium, since the

production relationship L 1−α A t Z α

t exhibits increasing returns to scale in itsthree “inputs.” Following Judd’s (1985a) treatment of patents in a dynamic

setting of Dixit and Stiglitz’s (1977) model of monopolistic competition, Romer(1987) assumes that an inventor of a new intermediate input obtains an infinitelylived patent on that design As the sole supplier of an input, the inventor can

recoup the investment cost κ by setting a price of the input above its marginal

cost

14.6.1 Monopolistic competition outcome

The final-goods sector is still assumed to be characterized by perfect competition

because it exhibits constant returns to scale in the labor input L and the existing continuous range of intermediate inputs Z t (i) Thus, a competitive outcome

prescribes that each input is paid its marginal product,

w t= (1− α) L −α  A t

0

Z t (i) α di, (14.6.3)

p t (i) = αL 1−α Z t (i) α −1 , (14.6.4) where p t (i) is the price of intermediate input i at time t in terms of the final

good

Let 1 + R m be the steady-state interest rate along the balanced growthpath that we are seeking In order to find the equilibrium invention rate of newinputs, we first compute the profits from producing and selling an existing input

i The profit at time t is equal to

π t (i) = [p t (i) − (1 + R m )] Z t (i) , (14.6.5) where the cost of supplying one unit of the input i is one unit of the final

good acquired in the previous period; that is, the cost is the intertemporal

price 1 + R m The first-order condition of maximizing the profit in equation

( 14.6.5 ) is the familiar expression that the monopoly price p t (i) should be set

as a markup above marginal cost, 1 + R m, and the markup is inversely related

to the absolute value of the demand elasticity of input i , | t (i) |;

Research and monopolistic competition 457

The constant marginal cost, 1 + R m, and the constant-elasticity demand curve

( 14.6.4 ),  t (i) = −(1−α) −1, yield a time-invariant monopoly price which

substi-tuted into demand curve ( 14.6.4 ) results in a time-invariant equilibrium quantity

By substituting equation ( 14.6.7 ) into equation ( 14.6.5 ), we obtain an input

producer’s steady-state profit flow,

In an equilibrium with free entry, the cost κ of inventing a new input must

be equal to the discounted stream of future profits associated with being thesole supplier of that input,

The profit function Ωm (R) is positive, strictly decreasing in R , and convex,

as depicted in Figure 11.1 It follows that there exists a unique intersection

between Ω(R) and Rκ that determines R m Using the corresponding version

of equilibrium condition ( 14.2.9 ), the computed interest rate R m characterizes

a balanced growth path with

as long as 1 + R m ≥ β −1; that is, the technology must be sufficiently productive

relative to the agents’ degree of impatience.7 It is straightforward to verify that

7 If the computed value 1 + R m falls short of β −1, the technology does notpresent sufficient private incentives for new inventions, so the range of interme-diate inputs stays constant over time, and the equilibrium interest rate equals

β −1

substi-tuted into demand curve ( 14. 6.4 ) results in a time-invariant equilibrium quantity

By substituting equation ( 14. 6.7...

solu-tion to equasolu-tion ( 14. 5.3 ) and existence is guaranteed because the left-hand side