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

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 sustained growth is driven by exogenous growth in labor productivity. Then we turn our 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 economy matches what a benevolent planner would choose. Where the market outcome doesn’t match the planner’s outcome, there can be room for welfare-improving government interventions. The objective of the chapter is to shed light on the mechanisms at work in different models. We try to facilitate comparison by using the same production function for most of our discussion while changing the meaning of one of its arguments. Paul Romer’s work has been an impetus to the revived interest in the the- ory of economic growth. In the spirit of Arrow’s (1962) model of learning by doing, Romer (1986) presents an endogenous growth model where the accumu- lation of capital (or knowledge) is associated with a positive externality on the available technology. The aggregate of all agents’ holdings of capital is positively related to the level of technology, which in turn interacts with individual agents’ savings decisions and thereby determines the economy’s growth rate. Thus, 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 falls short of the socially optimal one. Another approach to generating endogenous growth is to assume that all production factors are reproducible. Following Uzawa (1965), Lucas (1988) formulates a model with accumulation of both physical and human capital. The joint accumulation of all inputs ensures that growth will not come to a halt even though each individual factor in the final-good production function is subject – 443 – 444 Economic Growth to diminishing returns. In the absence of externalities, the growth rate in the competitive equilibrium coincides in this model with the social optimum. Romer (1987) constructs a model where agents can choose to engage in research that produces technological improvements. Each invention represents a technology for producing a new type of intermediate input that can be used in the production of final goods without affecting the marginal product of existing intermediate 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 agents will only expend resources on research if they are granted property rights over their inventions. Under the assumption of infinitely lived patents, Romer solves for a monopolistically competitive equilibrium that exhibits the classic tension between static and dynamic efficiency. Patents and the associated market power are necessary for there to be research and new inventions in a decentralized equilibrium, while the efficient production of existing intermediate inputs would require marginal-cost pricing, that is, the abolition of granted patents. The monopolistically competitive equilibrium is characterized by a smaller supply of each intermediate input and a lower growth rate than would be socially optimal. Finally, we revisit the question of when nonreproducible factors may not pose an obstacle to growth. Rebelo (1991) shows that even if there are non- reproducible factors in fixed supply in a neoclassical growth model, sustained growth 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 the ever-increasing relative scarcity of a nonreproducible factor, Rebelo finds that its 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 his earlier model (1987). Now, if labor is in fixed supply and workers’ innate pro- ductivity is constant, it follows immediately that growth must asymptotically come to an halt. To make sustained growth feasible, we can take a cue from our earlier discussion. One modeling strategy would be to introduce an externality that enhances researchers’ productivity, and an alternative approach would be to assume that researchers can accumulate human capital. Romer adopts the first type of assumption, and we find it instructive to focus on its role in overcoming a barrier to growth that nonreproducible labor would otherwise pose. The economy 445 14.2. The economy The economy has a constant population of a large number of identical agents who order consumption streams {c t } ∞ t=0 according to ∞  t=0 β t u (c t ) , with β ∈ (0, 1) and u (c)= c 1−σ − 1 1 −σ for σ ∈ [0, ∞) , (14.2.1) 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 upper case letters stand for aggregate quantities. For most part of our discussion of economic growth, the production function takes the form F (K t ,X t )=X t f  ˆ K t  , where ˆ K t ≡ K t X t . (14.2.2) 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 functions implies F (K, X)=F 1 (K, X) K + F 2 (K, X) X, (14.2.3) where F i (K, X) is the derivative with respect to the ith 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 cap- ital. Past investments are reversible. It follows that the relative price of capital 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 differ among the various setups that we will examine. We assume that the production function satisfies standard assumptions of positive but diminishing marginal products, F i (K, X) > 0,F ii (K, X) < 0, for i =1, 2; and the Inada conditions, lim K→0 F 1 (K, X) = lim X→0 F 2 (K, X)=∞, lim K→∞ F 1 (K, X) = lim X→∞ F 2 (K, X)=0, 1 By virtue of L’Hˆopital’s rule, the limit of (c 1−σ −1)/(1 −σ)islog(c)asσ goes to one. 446 Economic Growth which imply lim ˆ K→0 f   ˆ K  = ∞, lim ˆ K→∞ f   ˆ K  =0. (14.2.4) 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 ˆ K .Inparticular,wehave F 1 (K, X)= ∂Xf(K/X) ∂K = f   ˆ K  , (14.2.5a) F 2 (K, X)= ∂Xf(K/X) ∂X = f  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 .The literature uses the term “balanced growth path” to denote a situation where all endogenous variables grow at constant (but possibly different) rates. Along such a steady-state growth path (and during any transition toward the steady state), 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 = F 1 (K t ,X t )=f   ˆ K t  . (14.2.6) Households maximize utility given by equation (14.2.1) subject to the sequence of budget constraints c t + k t+1 = r t k t +(1− δ) k t + χ t , (14.2.7) where χ t stands for labor-related budget terms. The first-order condition with 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 t  σ = β  f   ˆ 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 sus- tained in an equilibrium by a constant rate of return on the right-hand side. It was also for this reason that we chose the class of utility functions in equation (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 sus- tain steady-state consumption growth when the labor input X t is constant over 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 constant consumption level and a capital-labor ratio ˆ K  given by f   ˆ K   = β −1 − (1 − δ) . (14.2.10) In chapter 5 we derived a closed-form solution for the transition dynamics toward such a steady state in the case of logarithmic utility, a Cobb-Douglas production function, and δ =1. 14.3. Exogenous growth As in Solow’s (1956) classic article, the simplest way to ensure steady-state consumption growth is to postulate exogenous labor-augmenting technological change at the constant rate 1 + µ ≥ 1, X t = A t L, with A t =(1+µ) A t−1 , where L is a fixed stock of labor. Our conjecture is then that both consump- tion and physical capital will grow at that same rate 1 + µ along a balanced growth path. The same growth rate of K t and A t implies that the ratio ˆ K and, therefore, the marginal product of capital remain constant in the steady state. 2 To ensure well-defined maximization problems, a maintained assumption throughout 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 c 0 } ∞ t=0 into equation (14.2.1). 448 Economic Growth A time-invariant rate of return is in turn consistent with households choosing a constant 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  is given 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 ˆ K  is such that the implied rate of return on capital induces the agents to choose a consumption growth rate of 1+µ.Ascan be seen, a higher degree of patience (a larger β ), a higher willingness intertem- porally to substitute (a lower σ ) and a more durable capital stock (a lower δ) each yield a higher ratio ˆ K  , and therefore more output (and consumption) at a point in time; but the growth rate remains fixed at the rate of exogenous labor-augmenting technological change. It is straightforward to verify that the competitive equilibrium outcome is Pareto optimal, since the private return to capital 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, w t = F 2 (K t ,X t ) d X t d L = F 2 (K t ,X t ) A t . (14.3.2) So, by equation (14.2.3), we have r t K t + w t L = F (K t ,A t L) . Factor payments are equal to total production, which is the standard result of a competitive equilibrium with constant-returns-to-scale technologies. However, it is interesting to note that if A t were a separate production factor, there could not exist a competitive equilibrium, since factor payments based on marginal products would exceed total production. In other words, the dilemma would 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 ,andL, 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) suggests that economic growth can be endogenized by assuming that technology grows because of aggregate spillovers coming from firms’ production activities. The problem 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 tech- nological advancementbeexternaltofirms. 3 As an illustration, we assume that firms face a fixed labor productivity that is proportional to the current economy-wide average of physical capital per worker. 4 In particular, 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 ˆ K t = 1, so equilibrium condition (14.2.9) becomes  c t+1 c t  σ = β [f  (1) + 1 −δ] . (14.4.1) 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 technology parameters. The competitive equilibrium is no longer Pareto optimal, since the private return on capital falls short of the social rate of return, with the latter return given by d F  K t , K t L L  d K t = F 1 (K t ,K t )+F 2 (K t ,K t )=f (1) , (14.4.2) 3 Arrow (1962) focuses on learning from experience that is assumed to get embodied in capital goods, while Romer (1986) postulates spillover effects of firms’ investments in knowledge. In both analyses, the productivity of a given firm is a function of an aggregate state variable, either the economy’s stock of physical capital or stock of knowledge. 4 This specific formulation of spillovers is analyzed in a rarely cited paper by Frankel (1962). 450 Economic Growth 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 higher optimal consumption growth rate,  c t+1 c t  σ = β [f (1) + 1 − δ] . (14.4.3) Let us reconsider the suboptimality of the decentralized competitive equilib- rium. Since the agents and the planner share the same objective of maximizing utility, we are left with exploring differences in their constraints. For a given sequence of the spillover { ¯ K t } ∞ 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 a competitive equilibrium will be equal to total output, and optimal firm size is indeterminate. Therefore, we can consider a representative agent with one unit of labor endowment who runs his own production technology, taking the spillover effect as given. His resource constraint becomes c t + k t+1 = F  k t , ¯ K t  +(1−δ) k t = ¯ K t f  k t ¯ K t  +(1−δ) k t , 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 where the spillover effect is internalized, C t + K t+1 = F  K t , K t L L  +(1−δ) K t =[f (1) + 1 − δ] K t . 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 all factors 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 the rates 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 = F 2 (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 F 1 (K t+1 ,X t+1 ) −δ K = F 2 (K t+1 ,X t+1 ) −δ X , and after invoking equations (14.2.5), f  ˆ K t+1  −  1+ ˆ K t+1  f   ˆ K t+1  = δ X − δ K , (14.5.3) which uniquely determines a time-invariant competitive equilibrium ratio ˆ K  , as a function solely of depreciation rates and parameters of the production function. 5 5 The left side of equation (14.5.3) is strictly increasing, since the derivative with respect to ˆ K is −(1 + ˆ K)f  ( ˆ K) > 0. Thus, there can only be one solu- tion to equation (14.5.3) and existence is guaranteed because the left-hand side 452 Economic Growth After solving for f  ( ˆ K  ) from equation (14.5.3 ) and substituting into equa- tion (14.2.9), we arrive at an expression for the equilibrium growth rate  c t+1 c t  σ = β   f  ˆ K   1+ ˆ K  +1− δ X + ˆ K  δ K 1+ ˆ K    . (14.5.4) As in the previous model with an externality, the economy here is void of any transition dynamics toward a steady state. But this implication hinges now critically upon investments being reversible so that the initial stocks of physical capital and human capital are inconsequential. In contrast to the previous model, the present competitive equilibrium is Pareto optimal because there is no longer any discrepancy between private and social rates of return. 6 The problem of optimal taxation with commitment (see chapter 15) is stud- ied for this model of endogenous growth by Jones, Manuelli, and Rossi (1993), who adopt the assumption of irreversible investments. ranges from minus infinity to plus infinity. The limit of the left-hand side when ˆ K approaches zero is f(0) − lim ˆ K→0 f  ( ˆ K), 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  ( ˆ X) −g( ˆ X), for which we take the limit when ˆ X 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)withX = ¯ 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). [...]... −1 , (14. 6.6) (i) ∂ pt (i) Zt (i) ∂ Zt (i) pt (i) −1 < 0 Research and monopolistic competition 457 The constant marginal cost, 1 + Rm , and the constant-elasticity demand curve (14. 6.4 ), t (i) = −(1−α)−1 , yield a time-invariant monopoly price which substituted into demand curve (14. 6.4 ) results in a time-invariant equilibrium quantity of input i : pt (i) = 1 + Rm , α Zt (i) = α2 1 + Rm (14. 6.7a)... Ωs (R) 460 Economic Growth We can also show that the laissez-faire supply of an input falls short of the socially optimal one, Zm < Zs ⇐⇒ α 1 + Rs < 1 1 + Rm (14. 6.16) To establish condition (14. 6.16 ), divide equation (14. 6.7b ) by equation (14. 6.12 ) Thus, the laissez-faire equilibrium is characterized by a smaller supply of each intermediate input and a lower growth rate than would be socially optimal... kt+1 = rt kt + (1 − δ) pt kt + χt (14. 7.5) The first-order condition with respect to capital is ct+1 ct σ =β (1 − δ) pt+1 + rt+1 pt (14. 7.6) After substituting rt+1 = pt+1 A from equation (14. 7.2 ) and steady-state rates of change from equation (14. 7.4 ) into equation (14. 7.6 ), we arrive at the following equilibrium condition: 1−α(1−σ) [1 + ρ (φ)] = β (1 − δ + A) (14. 7.7) 462 Economic Growth Thus,... long as β (1 − δ + A) ≥ 1 (14. 7.8a) Moreover, the maintained assumption of this chapter that parameters are such that derived growth rates yield finite lifetime utility, β(ct+1 /ct )1−σ < 1 , imposes here the parameter restriction β[β(1 − δ + A)]α(1−σ)/[1−α(1−σ)] < 1 which can be simpli ed to read α(1−σ) β (1 − δ + A) < 1 (14. 7.8b) Given that conditions (14. 7.8 ) are satis ed, there is a unique equilibrium... time zero, capital is redistributed across households (i.e., some people must surrender capital and others get their capital) ii) Half of the households are required to pay a lump sum tax The proceeds of the tax are used to finance a transfer program to the other half of the population iii) Two thirds of the households are required to pay a lump sum tax The proceeds of the tax are used to finance the purchase... equilibrium condition (14. 2.9 ) becomes σ 1−α α−1 (1 + µ) = β αKt [φXt ] +1−δ (14. 5.6) That is, along the balanced growth path, the marginal product of physical capital must be constant With the assumed Cobb-Douglas technology, the marginal product of capital is proportional to the average product, so that by dividing equation (14. 5.5a) through by Kt and applying equation (14. 5.6 ) we obtain Ct Kt+1... = α2 1 + Rm (14. 6.7a) 1/(1−α) L ≡ Zm (14. 6.7b) By substituting equation (14. 6.7 ) into equation (14. 6.5 ), we obtain an input producer’s steady-state profit flow, πt (i) = (1 − α) α1/(1−α) α 1 + Rm α/(1−α) L ≡ Ωm (Rm ) (14. 6.8) 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 the sole supplier of that... quantities of intermediate inputs that maximize At L1−α At Zt (i)α di − (1 + Rs ) 0 Zt (i) di, 0 with the following first-order condition with respect to Zt (i), Zt (i) = α 1 + Rs 1/(1−α) L ≡ Zs (14. 6.12) Thus, the quantity of an intermediate input is the same across all inputs and constant over time Hence, the planner’s problem is simpli ed to one where utility function (14. 2.1 ) is maximized subject to... same rate along a balanced growth path It then remains 464 Economic Growth to determine which consumption growth rate given by equation (14. 7.12 ), is supported by Euler equation (14. 6.11 ); 1 + ηL − Rm = [β (1 + Rm )]1/σ α (14. 7.13) The left side of equation (14. 7.13 ) is monotonically decreasing in Rm , and the right side is increasing It is also trivially true that the left-hand side is strictly... shows how this result can be overturned if the production function for final goods on the right side of equation (14. 6.1 ) is multiplied by the input range raised to some power ν , Aν It then becomes possible that the laissez-faire growth rate exceeds t the socially optimal rate because the new production function disentangles input producers’ market power, determined by the parameter α , and the economy’s . is simpli ed to one where utility function (14. 2.1) is maximized subject to resource constraint (14. 6.2) with quantities of intermediate inputs given by equation (14. 6.12). The first- order condition. by f   ˆ K   = β −1 − (1 − δ) . (14. 2.10) In chapter 5 we derived a closed-form solution for the transition dynamics toward such a steady state in the case of logarithmic utility, a Cobb-Douglas production function,. steady state. 2 To ensure well-defined maximization problems, a maintained assumption throughout the chapter is that parameters are such that any derived consump- tion growth rate 1+µ yields finite