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CAS EC 102
Graham Wilson

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Notes on Macroeconomic Theory Chapter 1 Simple Representative Agent Models This chapter deals with the most simple kind of macroeconomic model, which abstracts from all issues of heterogeneity and distribution among economic agents. Here, we study an economy consisting of a representative ¯rm and a representative consumer. As we will show, this is equivalent, under some circumstances, to studying an economy with many identical ¯rms and many identical consumers. Here, as in all the models we will study, economic agents optimize, i.e. they maximize some objective subject to the constraints they face. The preferences of consumers, the technology available to ¯rms, and the endowments of resources available to consumers and ¯rms, combined with optimizing behavior and some notion of equilibrium, allow us to use the model to make predictions. Here, the equilibrium concept we will use is competitive equilibrium, i.e. all economic agents are assumed to be price-takers. 1.1 A Static Model 1.1.1 Preferences, endowments, and technology There is one period and N consumers, who each have preferences given by the utility function u(c; `); where c is consumption and ` is leisure. Here, u(¢; ¢) is strictly increasing in each argument, strictly concave, and 1 2 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS twice di®erentiable. Also, assume that lim c!0u1(c; `) = 1; ` > 0; and lim`!u (c; `) = 1; c > 0: Here, (c; `) is the partial derivative with respect to argument i of u(c; `): Each consumer is endowed with one unit of time, which can be allocated between work and leisure. Each consumer also owns k0 N units of capital, which can be rented to ¯rms. There are M ¯rms, which each have a technology for producing consumption goods according to y = zf(k; n); where y is output, k is the capital input, n is the labor input, and z is a parameter representing total factor productivity. Here, the function f(¢; ¢) is strictly increasing in both arguments, strictly quasiconcave, twice di®erentiable, and homogeneous of degree one. That is, production is constant returns to scale, so that ¸y = zf(¸k; ¸n); (1.1) for ¸ > 0: Also, assume that limk!f1(k; n) = 1; lk!11(k; n) = 0; limn!02(k; n) = 1; and lin!1f(k; n) = 0: 1.1.2 Optimization In a competitive equilibrium, we can at most determine all relative prices, so the price of one good can arbitrarily be set to 1 with no loss of generality. We call this good the numeraire. We will follow convention here by treating the consumption good as the numeraire. There are markets in three objects, consumption, leisure, and the rental services of capital. The price of leisure in units of consumption is w; and the rental rate on capital (again, in units of consumption) is r: Consumer's Problem Each consumer treats w as being ¯xed, and maximizes utility subject to his/her constraints. That is, each solves max c;`sk u(c; `) 1.1. A STATIC MODEL 3 subject to c · w(1 ¡ `) + s(1.2) 0 · ks· k0 N (1.3) 0 · ` · 1 (1.4) c ¸ 0 (1.5) Here, ks is the quantity of capital that the consumer rents to ¯rms, (1.2) is the budget constraint, (1.3) states that the quantity of capital rented must be positive and cannot exceed what the consumer is endowed with, (1.4) is a similar condition for leisure, and (1.5) is a nonnegativity constraint on consumption. Now, given that utility is increasing in consumption (more is preferred to less), we must have k s= k0 N ; and (1.2) will hold with equality. Our restrictions on the utility function assure that the nonnegativity constraints on consumption and leisure will not be binding, and in equilibrium we will never have ` = 1; as then nothing would be produced, so we can safely ignore this case. The optimization problem for the consumer is therefore much simpli¯ed, and we can write down the following Lagrangian for the problem. L = u(c; `) + ¹(w + r k0 N ¡ w` ¡ c); where ¹ is a Lagrange multiplier. Our restrictions on the utility function assure that there is a unique optimum which is characterized by the following ¯rst-order conditions. @L @c = u1 ¡ ¹ = 0 @L @` = u2 ¡ ¹w = 0 @L @¹ = w + r k0 N ¡ w` ¡ c = 0 Here, u iis the partial derivative of u(¢; ¢) with respect to argument i: The above ¯rst-order conditions can be used to solve out for ¹ and c to obtain wu 1(w + r k0 N ¡ w`; `) ¡ 2(w + r k0 N ¡ w`; `) = 0; (1.6) 4 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS which solves for the desired quantity of leisure, `; in terms of w; r; and k0 N : Equation (1.6) can be rewritten as u2 u1 = w; i.e. the marginal rate of substitution of leisure for consumption equals the wage rate. Diagrammatically, in Figure 1.1, the consumer's budget constraint is ABD, and he/she maximizes utility at E, where the budget constraint, which has slope ¡w; is tangent to the highest indi®erence u1rve, where an indi®erence curve has slope ¡ 2 : Firm's Problem Each ¯rm chooses inputs of labor and capital to maximize pro¯ts, treating w and r as being ¯xed. That is, a ¯rm solves max k;n [zf(k; n) ¡ rk ¡ wn]; and the ¯rst-order conditions for an optimum are the marginal product conditions zf1= r; (1.7) zf2= w; (1.8) where f idenotes the partial derivative of f(¢; ¢) with respect to argument i: Now, given that the function f(¢; ¢) is homogeneous of degree one, Euler's law holds. That is, di®erentiating (1.1) with respect to ¸; and setting ¸ = 1; we get zf(k; n) = z1k + zf2n: (1.9) Equations (1.7), (1.8), and (1.9) then imply that maximized pro¯ts equal zero. This has two important consequences. The ¯rst is that we do not need to be concerned with how the ¯rm's pro¯ts are distributed (through shares owned by consumers, for example). Secondly, suppose k¤ and n ¤are optimal choices for the factor inputs, then we must have zf(k; n) ¡ rk ¡ wn = 0 (1.10) 1.1. A STATIC MODEL 5 Figure 1.1: 6 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS for k = k¤and n = n ¤: But, since (1.10) also holds for k = ¸k ¤and n = ¸n¤ for any ¸ > 0; due to the constant returns to scale assumption, the optimal scale of operation of the ¯rm is indeterminate. It therefore makes no di®erence for our analysis to simply consider the case M = 1 (a single, representative ¯rm), as the number of ¯rms will be irrelevant for determining the competitive equilibrium. 1.1.3 Competitive Equilibrium A competitive equilibrium is a set of quantities, c; `; n; k; and prices w and r; which satisfy the following properties. 1. Each consumer chooses c and ` optimally given w and r: 2. The representative ¯rm chooses n and k optimally given w and r: 3. Markets clear. Here, there are three markets: the labor market, the market for consumption goods, and the market for rental services of capital. In a competitive equilibrium, given (3), the following conditions then hold. N(1 ¡ `) = n (1.11) y = Nc (1.12) k0 = k (1.13) That is, supply equals demand in each market given prices. Now, the total value of excess demand across markets is Nc ¡ y + w[n ¡ N(1 ¡ `)] + r(k ¡ k0); but from the consumer's budget constraint, and the fact that pro¯t maximization implies zero pro¯ts, we have Nc ¡ y + w[n ¡ N(1 ¡ `)] + r(k ¡ k0) = 0: (1.14) Note that (1.14) would hold even if pro¯ts were not zero, and were distributed lump-sum to consumers. But now, if any 2 of (1.11), (1.12), 1.1. A STATIC MODEL 7 and (1.13) hold, then (1.14) implies that the third market-clearing condition holds. Equation (1.14) is simply Walras' law for this model. Walras' law states that the value of excess demand across markets is always zero, and this then implies that, if there are M markets and M ¡ 1 of those markets are in equilibrium, then the additional market is also in equilibrium. We can therefore drop one market-clearing condition in determining competitive equilibrium prices and quantities. Here, we eliminate (1.12). The competitive equilibrium is then the solution to (1.6), (1.7), (1.8), (1.11), and (1.13). These are ¯ve equations in the ¯ve unknowns `; n, k; w; and r; and we can solve for c using the consumer's budget constraint. It should be apparent here that the number of consumers, N; is virtually irrelevant to the equilibrium solution, so for convenience we can set N = 1, and simply analyze an economy with a single representative consumer. Competitive equilibrium might seem inappropriate when there is one consumer and one ¯rm, but as we have shown, in this context our results would not be any di®erent if there were many ¯rms and many consumers. We can substitute in equation (1.6) to obtain an equation which solves for equilibrium `: zf2(k0; 1 ¡ `)1(zf(k0; 1 ¡ `); `) 2(zf(k0; 1 ¡ `); `) = 0 (1.15) Given the solution for `; we then substitute in the following equations to obtain solutions for r; w; n; k, and c: zf1(k0; 1 ¡ `) = r (1.16) zf2(k0; 1 ¡ `) = w (1.17) n = 1 ¡ ` k = k0 c = zf(k0; 1 ¡ `) (1.18) It is not immediately apparent that the competitive equilibrium exists and is unique, but we will show this later. 8 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS 1.1.4 Pareto Optimality A Pareto optimum, generally, is de¯ned to be some allocation (an allocation being a production plan and a distribution of goods across economic agents) such that there is no other allocation which some agents strictly prefer which does not make any agents worse o®. Here, since we have a single agent, we do not have to worry about the allocation of goods across agents. It helps to think in terms of a ¯ctitious social planner who can dictate inputs to production by the representative ¯rm, can force the consumer to supply the appropriate quantity of labor, and then distributes consumption goods to the consumer, all in a way that makes the consumer as well o® as possible. The social planner determines a Pareto optimum by solving the following problem. max c;` u(c; `) subject to c = zf(k0; 1 ¡ `) (1.19) Given the restrictions on the utility function, we can simply substitute using the constraint in the objective function, and di®erentiate with respect to ` to obtain the following ¯rst-order condition for an optimum. zf2(k0; 1 ¡ `)1[zf(k; 1 ¡ `); `] ¡2[zf(0; 1 ¡ `); `] = 0 (1.20) Note that (1.15) and (1.20) are identical, and the solution we get for c from the social planner's problem by substituting in the constraint will yield the same solution as from (1.18). That is, the competitive equilibrium and the Pareto optimum are identical here. Further, since u(¢; ¢) is strictly concave and f(¢; ¢) is strictly quasiconcave, there is a unique Pareto optimum, and the competitive equilibrium is also unique. Note that we can rewrite (1.20) as zf2= u2 u1 ; where the left side of the equation is the marginal rate of transformation, and the right side is the marginal rate of substitution of consumption for leisure. In Figure 1.2, AB is equation (1.19) and the Pareto 1.1. A STATIC MODEL 9 optimum is at D, where the highest indi®erence curve is tangent to the production possibilities frontier. In a competitive equilibrium, the representative consumer faces budget constraint AFG and maximizes at point D where the slope of the budget line, ¡w; is equal to ¡ u2 u1 : In more general settings, it is true under some restrictions that the following hold. 1. A competitive equilibrium is Pareto optimal (First Welfare Theorem). 2. Any Pareto optimum can be supported as a competitive equilibrium with an appropriate choice of endowments. (Second Welfare Theorem). The non-technical assumptions required for (1) and (2) to go through include the absence of externalities, completeness of markets, and absence of distorting taxes (e.g. income taxes and sales taxes). The First Welfare Theorem is quite powerful, and the general idea goes back as far as Adam Smith's Wealth of Nations. In macroeconomics, if we can successfully explain particular phenomena (e.g. business cycles) using a competitive equilibrium model in which the First Welfare Theorem holds, we can then argue that the existence of such phenomena is not grounds for government intervention. In addition to policy implications, the equivalence of competitive equilibria and Pareto optima in representative agent models is useful for computational purposes. That is, it can be much easier to obtain competitive equilibria by ¯rst solving the social planner's problem to obtain competitive equilibrium quantities, and then solving for prices, rather than solving simultaneously for prices and quantities using marketclearing conditions. For example, in the above example, a competitive equilibrium could be obtained by ¯rst solving for c and ` from the social planner's problem, and then ¯nding w and r from the appropriate marginal conditions, (1.16) and (1.17). Using this approach does not make much di®erence here, but in computing numerical solutions in dynamic models it can make a huge di®erence in the computational burden. 10 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS Figure 1.2: 1.1. A STATIC MODEL 11 1.1.5 Example Consider the following speci¯c functional forms. For the utility function, we use u(c; `) = c1¡°¡ 1 1 ¡ ° + `; where ° > 0 measures the degree of curvature in the utility function with respect to consumption (this is a \constant relative risk aversion" utility function). Note that lim °!1 c1¡°¡ 1 1 ¡ ° = lim d!1 d°[e(1¡°) lo¡ 1] d d°(1 ¡ °) = log c; using L'Hospital's Rule. For the production technology, use f(k; n) = ® n1¡®; where 0 < ® < 1: That is, the production function is Cobb-Douglas. The social planner's problem here is then max ` ( [zk® 0(1 ¡ `)1¡®1¡°¡ 1 1 ¡ ° + ` ) ; and the solution to this problem is ` = 1 ¡ [(1 ¡ ®)(zk® 0)1¡°] 1 ®+(1¡®(1.21) As in the general case above, this is also the competitive equilibrium solution. Solving for c; from (1.19), we get c = [(1 ¡ ®) 1¡(zk ® 0)] 1 ®+(1¡®; (1.22) and from (1.17), we have w = [(1 ¡ ®) 1¡®(zk ® 0)] ° ®+(1¡®(1.23) From (1.22) and (1.23) clearly c and w are increasing in z and k 0: That is, increases in productivity and in the capital stock increase aggregate consumption and real wages. However, from equation (1.21) the e®ects 12 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS on the quantity of leisure (and therefore on employment) are ambiguous. Which way the e®ect goes depends on whether ° < 1 or ° > 1: With ° < 1; an increase in z or in k 0 will result in a decrease in leisure, and an increase in employment, but the e®ects are just the opposite if ° > 1: If we want to treat this as a simple model of the business cycle, where °uctuations are driven by technology shocks (changes in z); these results are troubling. In the data, aggregate output, aggregate consumption, and aggregate employment are mutually positively correlated. However, this model can deliver the result that employment and output move in opposite directions. Note however, that the real wage will be procyclical (it goes up when output goes up), as is the case in the data. 1.1.6 Linear Technology - Comparative Statics This section illustrates the use of comparative statics, and shows, in a somewhat more general sense than the above example, why a productivity shock might give a decrease or an increase in employment. To make things clearer, we consider a simpli¯ed technology, y = zn; i.e. we eliminate capital, but still consider a constant returns to scale technology with labor being the only input. The social planner's problem for this economy is then max ` u[z(1 ¡ `); `]; and the ¯rst-order condition for a maximum is ¡zu 1[z(1 ¡ `); `] + 2[z(1 ¡ `); `] = 0: (1.24) Here, in contrast to the example, we cannot solve explicitly for `; but note that the equilibrium real wage is w = @y @n = z; so that an increase in productivity, z, corresponds to an increase in the real wage faced by the consumer. To determine the e®ect of an increase 1.1. A STATIC MODEL 13 in z on `; apply the implicit function theorem and totally di®erentiate (1.24) to get [¡u1¡ z(1 ¡ `)11+ u 21(1 ¡ `)]dz +(z2u 11¡ 2zu12 + u22)d` = 0: We then have d` dz = u1 + z(1 ¡ `)11 ¡ 21(1 ¡ `) z2u11 ¡ 2zu12+ u 22 : (1.25) Now, concavity of the utility function implies that the denominator in (1.25) is negative, but we cannot sign the numerator. In fact, it is easy to construct examples where d` dz> 0; and where d` dz< 0: The ambiguity here arises from opposing income and substitution e®ects. In Figure 1.3, AB denotes the resource constraint faced by the social planner, c = z 1(1¡`); and BD is the resource constraint with a higher level of productivity, 2 > z1: As shown, the social optimum (also the competitive equilibrium) is at E initially, and at F after the increase in productivity, with no change in ` but higher c: E®ectively, the representative consumer faces a higher real wage, and his/her response can be decomposed into a substitution e®ect (E to G) and an income e®ect (G to F). Algebraically, we can determine the substitution e®ect on leisure by changing prices and compensating the consumer to hold utility constant, i.e. u(c; `) = h; (1.26) where h is a constant, and ¡zu1(c; `) + 2(c; `) = 0 (1.27) Totally di®erentiating (1.26) and (1.27) with respect to c and `; and using (1.27) to simplify, we can solve for the substitution e®ect d` dz(subst:) as follows. d` dz (subst:) = u1 z2u11 ¡ 2zu12+ u 22 < 0: From (1.25) then, the income e®ect d` dz(inc:) is just the remainder, d` dz (inc:) = z(1 ¡ `)11¡ u21(1 ¡ `) z2u11 ¡ 2zu12+ u22 > 0; 14 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS Figure 1.3: 1.2. GOVERNMENT 15 provided ` is a normal good. Therefore, in order for a model like this one to be consistent with observation, we require a substitution e®ect that is large relative to the income e®ect. That is, a productivity shock, which increases the real wage and output, must result in a decrease in leisure in order for employment to be procyclical, as it is in the data. In general, preferences and substitution e®ects are very important in equilibrium theories of the business cycle, as we will see later. 1.2 Government So that we can analyze some simple ¯scal policy issues, we introduce a government sector into our simple static model in the following manner. The government makes purchases of consumption goods, and ¯- nances these purchases through lump-sum taxes on the representative consumer. Let g be the quantity of government purchases, which is treated as being exogenous, and let ¿ be total taxes. The government budget must balance, i.e. g = ¿: (1.28) We assume here that the government destroys the goods it purchases. This is clearly unrealistic (in most cases), but it simpli¯es matters, and does not make much di®erence for the analysis, unless we wish to consider the optimal determination of government purchases. For example, we could allow government spending to enter the consumer's utility function in the following way. w(c; `; g) = u(c; `) + v(g) Given that utility is separable in this fashion, and g is exogenous, this would make no di®erence for the analysis. Given this, we can assume v(g) = 0: As in the previous section, labor is the only factor of production, i.e. assume a technology of the form y = zn: Here, the consumer's optimization problem is c;` u(c; `) 16 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS subject to c = w(1 ¡ `) ¡ ¿; and the ¯rst-order condition for an optimum is ¡wu 1+ u2 = 0: The representative ¯rm's pro¯t maximization problem is max n (z ¡ w)n: Therefore, the ¯rm's demand for labor is in¯nitely elastic at w = z: A competitive equilibrium consists of quantities, c; `; n; and ¿; and a price, w; which satisfy the following conditions: 1. The representative consumer chooses c and ` to maximize utility, given w and ¿: 2. The representative ¯rm chooses n to maximize pro¯ts, given w: 3. Markets for consumption goods and labor clear. 4. The government budget constraint, (1.28), is satis¯ed. The competitive equilibrium and the Pareto optimum are equivalent here, as in the version of the model without government. The social planner's problem is max c;` u(c; `) subject to c + g = z(1 ¡ `) Substituting for c in the objective function, and maximizing with respect to `; the ¯rst-order condition for this problem yields an equation which solves for ` : ¡zu1[z(1 ¡ `) ¡ g; `] 2[z(1 ¡ `) ¡ g; `] = 0: (1.29) In Figure 1.4, the economy's resource constraint is AB, and the Pareto optimum (competitive equilibrium) is D. Note that the slope of the resource constraint is ¡z = ¡w: 1.2. GOVERNMENT 17 Figure 1.4: 18 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS We can now ask what the e®ect of a change in government expenditures would be on consumption and employment. In Figure 1.5, g increases from g 1to g2; shifting in the resource constraint. Given the government budget constraint, there is an increase in taxes, which represents a pure income e®ect for the consumer. Given that leisure and consumption are normal goods, quantities of both goods will decrease. Thus, there is crowding out of private consumption, but note that the decrease in consumption is smaller than the increase in government purchases, so that output increases. Algebraically, totally di®erentiate (1.29) and the equation c = z(1 ¡ `) ¡ g and solve to obtain d` dg = ¡zu11 + u12 z2u11 ¡ 2zu12+ u 22 < 0 dc dg = zu 12¡ u22 z2u11 ¡ 2zu12+ u 22 < 0 (1.30) Here, the inequalities hold provided that ¡zu 11 + u12> 0 and zu 12 ¡ u22 > 0; i.e. if leisure and consumption are, respectively, normal goods. Note that (1.30) also implies that dy dg < 1; i.e. the \balanced budget multiplier" is less than 1. 1.3 A \Dynamic" Economy We will introduce some simple dynamics to our model in this section. The dynamics are restricted to the government's ¯nancing decisions; there are really no dynamic elements in terms of real resource allocation, i.e. the social planner's problem will break down into a series of static optimization problems. This model will be useful for studying the e®ects of changes in the timing of taxes. Here, we deal with an in¯nite horizon economy, where the representative consumer maximizes time-separable utility, 1X t=0 ¯ u(c t; t); where ¯ is the discount factor, 0 < ¯ < 1: Letting ± denote the discount rate, we have ¯ = 1 1+± ; where ± > 0: Each period, the consumer is endowed with one unit of time. There is a representative ¯rm 1.3. A \DYNAMIC" ECONOMY 19 Figure 1.5: 20 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS which produces output according to the production function y t= z n : The government purchases g tunits of consumption goods in period t; t = 0; 1; 2; :::; and these purchases are destroyed. Government purchases are ¯nanced through lump-sum taxation and by issuing one-period government bonds. The government budget constraint is g t+ (1 + r )b t= ¿ t+ b t+1; (1.31) t = 0; 1; 2; :::; where b tis the number of one-period bonds issued by the government in period t¡1: A bond issued in period t is a claim to 1+r t+1units of consumption in period t+1; where r t+1is the one-period interest rate. Equation (1.31) states that government purchases plus principal and interest on the government debt is equal to tax revenues plus new bond issues. Here, b 0= 0: The optimization problem solved by the representative consumer is max ft+;c;tg1t =0; 1X t=0 ¯ u(c t; t) subject to ct= w (1 ¡ `t) ¡ ¿t¡ st+1+ (1 + r )st; (1.32) t = 0; 1; 2; :::; s0= 0; where s t+1is the quantity of bonds purchased by the consumer in period t, which come due in period t + 1: Here, we permit the representative consumer to issue private bonds which are perfect substitutes for government bonds. We will assume that lim n!1 sn Q n¡1 i=1(1 + r ) = 0; (1.33) which states that the quantity of debt, discounted to t = 0; must equal zero in the limit. This condition rules out in¯nite borrowing or \Ponzi schemes," and implies that we can write the sequence of budget constraints, (1.32) as a single intertemporal budget constraint. Repeated substitution using (1.32) gives c0 + 1X t=1 ctQ ti =1(1 + r ) = w 0 (1 ¡ 0 ) ¡ ¿0+ 1X t=1 w t(1 ¡ t) ¡ ¿tQ ti =1(1 + r ) : (1.34) 1.3. A \DYNAMIC" ECONOMY 21 Now, maximizing utility subject to the above intertemporal budget constraint, we obtain the following ¯rst-order conditions. ¯ u 1(ct; t) ¡ ¸ Q ti =1(1 + r ) = 0; t = 1; 2; 3; ::: ¯ u 2(ct; t) ¡ ¸w tQ ti =1(1 + r ) = 0; t = 1; 2; 3; ::: u 1(c0; `0) ¡ ¸ = 0 u 2(c0; `0) ¡ ¸w0 = 0 Here, ¸ is the Lagrange multiplier associated with the consumer's intertemporal budget constraint. We then obtain u 2(ct; t) u 1(ct; t) = w t; (1.35) i.e. the marginal rate of substitution of leisure for consumption in any period equals the wage rate, and ¯u 1(c t+; `t+) u 1(ct; t) = 1 1 + r t+1 ; (1.36) i.e. the intertemporal marginal rate of substitution of consumption equals the inverse of one plus the interest rate. The representative ¯rm simply maximizes pro¯ts in each period, i.e. it solves max nt (zt¡ w t)nt; and labor demand, n ; is perfectly elastic at w t= z t: A competitive equilibrium consists of quantities, fc ; `; nt; st+; bt+1; ¿tg1t=0; and prices fw t; t+1g1t=0 satisfying the following conditions. 1. Consumers choose fc ; `; st+1; g1t=0optimally given f¿ g and fw t; t+1g 1t=: 2. Firms choose fn g 1t=0optimally given fw g 1t=: 3. Given fg tg1t=0; fbt+1; ¿g 1t=0satis¯es the sequence of government budget constraints (1.31). 22 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS 4. Markets for consumption goods, labor, and bonds clear. Walras' law permits us to drop the consumption goods market from consideration, giving us two market-clearing conditions: st+1 = bt+1; t = 0; 1; 2; :::; (1.37) and 1 ¡ `t= n ; t = 0; 1; 2; ::: Now, (1.33) and (1.37) imply that we can write the sequence of government budget constraints as a single intertemporal government budget constraint (through repeated substitution): g 0+ 1X t=1 g tQti =1(1 + r ) = ¿ 0 + 1X t=1 ¿ tQ ti =1(1 + r ) ; (1.38) i.e. the present discounted value of government purchases equals the present discounted value of tax revenues. Now, since the government budget constraint must hold in equilibrium, we can use (1.38) to substitute in (1.34) to obtain c0 + 1X t=1 ctQ ti =1(1 + r ) = w 0(1 ¡ `0) ¡ g0 + 1X t=1 w (1 ¡ `t) ¡ gtQ ti =1(1 + r ) : (1.39) Now, suppose that fw t; t+1g 1t=0are competitive equilibrium prices. Then, (1.39) implies that the optimizing choices given those prices remain optimal given any sequence f¿ tg 1t=0satisfying (1.38). Also, the representative ¯rm's choices are invariant. That is, all that is relevant for the determination of consumption, leisure, and prices, is the present discounted value of government purchases, and the timing of taxes is irrelevant. This is a version of the Ricardian Equivalence Theorem. For example, holding the path of government purchases constant, if the representative consumer receives a tax cut today, he/she knows that the government will have to make this up with higher future taxes. The government issues more debt today to ¯nance an increase in the government de¯cit, and private saving increases by an equal amount, since the representative consumer saves more to pay the higher taxes in the future. 1.3. A \DYNAMIC" ECONOMY 23 Another way to show the Ricardian equivalence result here comes from computing the competitive equilibrium as the solution to a social planner's problem, i.e. max fg1t =0 1X t=0 ¯tu[z (1 ¡ `) ¡ gt; t] This breaks down into a series of static problems, and the ¯rst-order conditions for an optimum are ¡ztu1[zt(1 ¡ `) ¡ g; `] + u 2[z(1 ¡ `) ¡ gt; t] = 0; (1.40) t = 0; 1; 2; ::: . Here, (1.40) solves for ` ; t = 0; 1; 2; :::; and we can solve for ctfrom c t= zt(1 ¡ `): Then, (1.35) and (1.36) determine prices. Here, it is clear that the timing of taxes is irrelevant to determining the competitive equilibrium, though Ricardian equivalence holds in much more general settings where competitive equilibria are not Pareto optimal, and where the dynamics are more complicated. Some assumptions which are critical to the Ricardian equivalence result are: 1. Taxes are lump sum 2. Consumers are in¯nite-lived. 3. Capital markets are perfect, i.e. the interest rate at which private agents can borrow and lend is the same as the interest rate at which the government borrows and lends. 4. There are no distributional e®ects of taxation. That is, the present discounted value of each individual's tax burden is una®ected by changes in the timing of aggregate taxation. 24 CHAPTER 1. SIMPLE REPRESENTATIVE AGENT MODELS Chapter 2 Growth With Overlapping Generations This chapter will serve as an introduction to neoclassical growth theory and to the overlapping generations model. The particular model introduced in this chapter was developed by Diamond (1965), building on the overlapping generations construct introduced by Samuelson (1956). Samuelson's paper was a semi-serious (meaning that Samuelson did not take it too seriously) attempt to model money, but it has also proved to be a useful vehicle for studying public ¯nance issues such as government debt policy and the e®ects of social security systems. There was a resurgence in interest in the overlapping generations model as a monetary paradigm in the late seventies and early eighties, particularly at the University of Minnesota (see for example Kareken and Wallace 1980). A key feature of the overlapping generations model is that markets are incomplete, in a sense, in that economic agents are ¯nite-lived, and agents currently alive cannot trade with the unborn. As a result, competitive equilibria need not be Pareto optimal, and Ricardian equivalence does not hold. Thus, the timing of taxes and the size of the government debt matters. Without government intervention, resources may not be allocated optimally among generations, and capital accumulation may be suboptimal. However, government debt policy can be used as a vehicle for redistributing wealth among generations and inducing optimal savings behavior. 25 26CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS 2.1 The Model This is an in¯nite horizon model where time is indexed by t = 0; 1; 2; :::;1. Each period, L ttwo-period-lived consumers are born, and each is endowed with one unit of labor in the ¯rst period of life, and zero units in the second period. The population evolves according to Lt= L 0(1 + n) ; (2.1) where L 0 is given and n > 0 is the population growth rate. In period 0 there are some old consumers alive who live for one period and are collectively endowed with K 0 units of capital. Preferences for a consumer born in period t; t = 0; 1; 2; :::; are given by u(c y t; co t+); where c y tdenotes the consumption of a young consumer in period t and cot is the consumption of an old consumer. Assume that u(¢; ¢) is strictly increasing in both arguments, strictly concave, and de¯ning v(cy; co) ´ @u @c y @u @c o ; assume that lim cy!ov(cy; co) = 1 for co > 0 and lim co!ov(cy; co) = 0 for cy> 0: These last two conditions on the marginal rate of substitution will imply that each consumer will always wish to consume positive amounts when young and when old. The initial old seek to maximize consumption in period 0: The investment technology works as follows. Consumption goods can be converted one-for-one into capital, and vice-versa. Capital constructed in period t does not become productive until period t+1; and there is no depreciation. Young agents sell their labor to ¯rms and save in the form of capital accumulation, and old agents rent capital to ¯rms and then convert the capital into consumption goods which they consume. The representative ¯rm maximizes pro¯ts by producing consumption goods, and renting capital and hiring labor as inputs. The technology is given by Y t= F(K t; L); 2.2. OPTIMAL ALLOCATIONS 27 where Y tis output and K tand L tare the capital and labor inputs, respectively. Assume that the production function F(¢; ¢) is strictly increasing, strictly quasi-concave, twice di®erentiable, and homogeneous of degree one. 2.2 Optimal Allocations As a benchmark, we will ¯rst consider the allocations that can be achieved by a social planner who has control over production, capital accumulation, and the distribution of consumption goods between the young and the old. We will con¯ne attention to allocations where all young agents in a given period are treated identically, and all old agents in a given period receive the same consumption. The resource constraint faced by the social planner in period t is F(K ; Lt) + K t= K t+1+ c y tLt+ c ot L t¡; (2.2) where the left hand side of (2.2) is the quantity of goods available in period t; i.e. consumption goods produced plus the capital that is left after production takes place. The right hand side is the capital which will become productive in period t + 1 plus the consumption of the young, plus consumption of the old. In the long run, this model will have the property that per-capita quantities converge to constants. Thus, it proves to be convenient to express everything here in per-capita terms using lower case letters. De¯ne k t´K t Lt (the capital/labor ratio or per-capita capital stock) and f(kt) ´ F(kt; 1): We can then use (2.1) to rewrite (2.2) as f(kt) + kt= (1 + n)k t+1 + cy t+ cot 1 + n (2.3) De¯nition 1 A Pareto optimal allocation is a sequence fc y t; cot ; kt+1g1t=0 satisfying (2.3) and the property that there exists no other allocation f^cy t; ^cot ; ^kt+1g 1t=0which satis¯es (2.3) and ^c o 1 ¸ co 1 u(^c y t; ^co t+1) ¸ u(cy t; co t+) for all t = 0; 1; 2; 3; :::; with strict inequality in at least one instance. 28CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS That is, a Pareto optimal allocation is a feasible allocation such that there is no other feasible allocation for which all consumers are at least as well o® and some consumer is better o®. While Pareto optimality is the appropriate notion of social optimality for this model, it is somewhat complicated (for our purposes) to derive Pareto optimal allocations here. We will take a shortcut by focusing attention on steady states, where k t= k; cy t= cy; and c ot = co; where k; c y; and c oare constants. We need to be aware of two potential problems here. First, there may not be a feasible path which leads from k 0to a particular steady state. Second, one steady state may dominate another in terms of the welfare of consumers once the steady state is achieved, but the two allocations may be Pareto non-comparable along the path to the steady state. The problem for the social planner is to maximize the utility of each consumer in the steady state, given the feasibility condition, (2.2). That is, the planner chooses c y; o; and k to solve max u(c y; o) subject to f(k) ¡ nk = cy+ co 1 + n : (2.4) Substituting for c oin the objective function using (2.4), we then solve the following max c;k u(c ; [1 + n][f(k) ¡ nk ¡ ]) The ¯rst-order conditions for an optimum are then u1 ¡ (1 + n)u2 = 0; or u1 u2 = 1 + n (2.5) (intertemporal marginal rate of substitution equal to 1 + n) and f0(k) = n (2.6) (marginal product of capital equal to n): Note that the planner's problem splits into two separate components. First, the planner ¯nds the 2.3. COMPETITIVE EQUILIBRIUM 29 capital-labor ratio which maximizes the steady state quantity of resources, from (2.6), and then allocates consumption between the young and the old according to (2.5). In Figure 2.1, k is chosen to maximize the size of the budget set for the consumer in the steady state, and then consumption is allocated between the young and the old to achieve the tangency between the aggregate resource constraint and an indi®erence curve at point A. 2.3 Competitive Equilibrium In this section, we wish to determine the properties of a competitive equilibrium, and to ask whether a competitive equilibrium achieves the steady state social optimum characterized in the previous section. 2.3.1 Young Consumer's Problem A consumer born in period t solves the following problem. max t;ot +1;t u(c y t; co t+) subject to cy t= w t¡ st(2.7) co t+1= s t(1 + r t+) (2.8) Here, w tis the wage rate, r tis the capital rental rate, and s t is saving when young. Note that the capital rental rate plays the role of an interest rate here. The consumer chooses savings and consumption when young and old treating prices, w tand r t+1; as being ¯xed. At time t the consumer is assumed to know r t+1: Equivalently, we can think of this as a rational expectations or perfect foresight equilibrium, where each consumer forecasts future prices, and optimizes based on those forecasts. In equilibrium, forecasts are correct, i.e. no one makes systematic forecasting errors. Since there is no uncertainty here, forecasts cannot be incorrect in equilibrium if agents have rational expectations. 30CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS Figure 2.1: 2.3. COMPETITIVE EQUILIBRIUM 31 Substituting for c y tand c o t+1in the above objective function using (2.7) and (2.8) to obtain a maximization problem with one choice variable, st; the ¯rst-order condition for an optimum is then ¡u 1(w ¡s ; st(1+r t+1))+u 2(w t¡s; st(1+r t+1))(1+r t+1) = 0 (2.9) which determines s ; i.e. we can determine optimal savings as a function of prices st = s(w t; rt+): (2.10) Note that (2.9) can also be rewritten as u1 u2 = 1 + r t+1; i.e. the intertemporal marginal rate of substitution equals one plus the interest rate. Given that consumption when young and consumption when old are both normal goods, we have @s @w t > 0; however the sign of @s @r t+1 is indeterminate due to opposing income and substitution e®ects. 2.3.2 Representative Firm's Problem The ¯rm solves a static pro¯t maximization problem max Kt;t [F(K t; Lt) ¡ wtLt ¡ tK ]: The ¯rst-order conditions for a maximum are the usual marginal conditions F 1(K ;L ) ¡ rt= 0; F 2(K ;L ) ¡ w t= 0: Since F(¢; ¢) is homogeneous of degree 1, we can rewrite these marginal conditions as f0(kt) ¡ t= 0; (2.11) f(kt) ¡ tf0(k) ¡ w t= 0: (2.12) 2.3.3 Competitive Equilibrium De¯nition 2 A competitive equilibrium is a sequence of quantities, fkt+1; sg 1t=0and a sequence of prices fw ; rg 1t=0; which satisfy (i) consumer optimization; (ii) ¯rm optimization; (iii) market clearing; in each period t = 0; 1; 2; :::; given the initial capital-labor ratio k 0: 32CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS Here, we have three markets, for labor, capital rental, and consumption goods, and Walras' law tells us that we can drop one marketclearing condition. It will be convenient here to drop the consumption goods market from consideration. Consumer optimization is summarized by equation (2.10), which essentially determines the supply of capital, as period t savings is equal to the capital that will be rented in period t+1: The supply of labor by consumers is inelastic. The demands for capital and labor are determined implicitly by equations (2.11) and (2.12). The equilibrium condition for the capital rental market is then kt+1(1 + n) = s(w t; t+1); (2.13) and we can substitute in (2.13) for w tand r t+1from (2.11) and (2.12) to get kt+1(1 + n) = s(f(k t) ¡ k0(k ); 0(k t+)): (2.14) Here, (2.14) is a nonlinear ¯rst-order di®erence equation which, given k0; solves for fk g 1t=. Once we have the equilibrium sequence of capitallabor ratios, we can solve for prices from (2.11) and (2.12). We can then solve for fs tg1t=0from (2.10), and in turn for consumption allocations. 2.4 An Example Let u(c y; co) = ln c y+ ¯ ln c o; and F(K; L) = °K ®L 1¡®; where ¯ > 0; ° > 0; and 0 < ® < 1: Here, a young agent solves max st [ln(w t¡ s) + ¯ ln[(1 + r t+1)s)]; and solving this problem we obtain the optimal savings function st= ¯ 1 + ¯ w : (2.15) Given the Cobb-Douglass production function, we have f(k) = °k ® and f 0(k) = °®k ®¡1: Therefore, from (2.11) and (2.12), the ¯rst-order conditions from the ¯rm's optimization problem give rt= °®k ®¡1 t; (2.16) 2.4. AN EXAMPLE 33 w t= °(1 ¡ ®)k ® t: (2.17) Then, using (2.14), (2.15), and (2.17), we get kt+1(1 + n) = ¯ (1 + ¯) °(1 ¡ ®)k ® t: (2.18) Now, equation (2.18) determines a unique sequence fk tg1t=1given k0 (see Figure 2m) which converges in the limit to k ¤; the unique steady state capital-labor ratio, which we can determine from (2.18) by setting kt+1= kt= k ¤and solving to get k¤ = " ¯°(1 ¡ ®) (1 + n)(1 + ¯) # 1 1¡® : (2.19) Now, given the steady state capital-labor ratio from (2.19), we can solve for steady state prices from (2.16) and (2.17), that is r¤= ®(1 + n)(1 + ¯) ¯(1 ¡ ®) ; w ¤= °(1 ¡ ®) " ¯°(1 ¡ ®) (1 + n)(1 + ¯) # ® 1¡® : We can then solve for steady state consumption allocations, cy= w ¤ ¡ ¯ 1 + ¯ w ¤= w ¤ 1 + ¯ ; co = ¯ 1 + ¯ w ¤(1 + ¤): In the long run, this economy converges to a steady state where the capital-labor ratio, consumption allocations, the wage rate, and the rental rate on capital are constant. Since the capital-labor ratio is constant in the steady state and the labor input is growing at the rate n; the growth rate of the aggregate capital stock is also n in the steady state. In turn, aggregate output also grows at the rate n: Now, note that the socially optimal steady state capital stock, ^k; is determined by (2.6), that is °®^k ®¡1= n; 34CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS or ^k = μ ®° n 1¡® : (2.20) Note that, in general, from (2.19) and (2.20), k ¤ 6= ^k; i.e. the competitive equilibrium steady state is in general not socially optimal, so this economy su®ers from a dynamic ine±ciency. There may be too little or too much capital in the steady state, depending on parameter values. That is, suppose ¯ = 1 and n = :3: Then, if ® < :103; k ¤> ^k; and if ® > :103; then k ¤< ^k: 2.5 Discussion The above example illustrates the dynamic ine±ciency that can result in this economy in a competitive equilibrium.. There are essentially two problems here. The ¯rst is that there is either too little or too much capital in the steady state, so that the quantity of resources available to allocate between the young and the old is not optimal. Second, the steady state interest rate is not equal to n; i.e. consumers face the \wrong" interest rate and therefore misallocate consumption goods over time; there is either too much or too little saving in a competitive equilibrium. The root of the dynamic ine±ciency is a form of market incompleteness, in that agents currently alive cannot trade with the unborn. To correct this ine±ciency, it is necessary to have some mechanism which permits transfers between the old and the young. 2.6 Government Debt One means to introduce intergenerational transfers into this economy is through government debt. Here, the government acts as a kind of ¯nancial intermediary which issues debt to young agents, transfers the proceeds to young agents, and then taxes the young of the next generation in order to pay the interest and principal on the debt. Let B t+1denote the quantity of one-period bonds issued by the government in period t: Each of these bonds is a promise to pay 1+r t+1 2.6. GOVERNMENT DEBT 35 units of consumption goods in period t +1: Note that the interest rate on government bonds is the same as the rental rate on capital, as must be the case in equilibrium for agents to be willing to hold both capital and government bonds. We will assume that B t+1= bL ; (2.21) where b is a constant. That is, the quantity of government debt is ¯xed in per-capita terms. The government's budget constraint is B t+1+ Tt= (1 + rt)Bt; (2.22) i.e. the revenues from new bond issues and taxes in period t, T t; equals the payments of interest and principal on government bonds issued in period t ¡ 1: Taxes are levied lump-sum on young agents, and we will let ¿ tdenote the tax per young agent. We then have T = ¿ tLt: (2.23) A young agent solves max st u(w t¡ st¡ ¿t; (1 + rt+1)st); where s tis savings, taking the form of acquisitions of capital and government bonds, which are perfect substitutes as assets. Optimal savings for a young agent is now given by st= s(w t¡ ¿ ; rt+): (2.24) As before, pro¯t maximization by the ¯rm implies (2.11) and (2.12). A competitive equilibrium is de¯ned as above, adding to the de¯nition that there be a sequence of taxes f¿ tg1t=0 satisfying the government budget constraint. From (2.21), (2.22), and (2.23), we get ¿ t= μ rt¡ n 1 + n ¶ b (2.25) The asset market equilibrium condition is now kt+1(1 + n) + b = s(w t¡ ¿t; t+1); (2.26) 36CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS that is, per capita asset supplies equals savings per capita. Substituting in (2.26) for w t; ¿t; and r t+; from (2.11), we get kt+1(1+n)+b = s à f(kt) ¡ tf0(kt) ¡ à f0(kt) ¡ n 1 + n ! b; f0(kt+1) ! (2.27) We can then determine the steady state capital-labor ratio k ¤(b) by setting k ¤(b) = k t= k t+1in (2.27), to get k¤(b)(1+n)+b = s à f(k¤(b)) ¡ k¤ (b)f0(k¤(b)) ¡ à f0(k¤(b)) ¡ n 1 + n ! b; f0(k¤(b)) ! (2.28) Now, suppose that we wish to ¯nd the debt policy, determined by b; which yields a competitive equilibrium steady state which is socially optimal, i.e. we want to ¯nd ^b such that k ¤(^b) = ^k: Now, given that f0(^k) = n; from (2.28) we can solve for ^b as follows: ^b = ¡^k(1 + n) + s ³ f(^k) ¡ ^kn; n ´ (2.29) In (2.29), note that ^b may be positive or negative. If ^b < 0; then debt is negative, i.e. the government makes loans to young agents which are ¯nanced by taxation. Note that, from (2.25), ¿ t= 0 in the steady state with b = ^ b; so that the size of the government debt increases at a rate just su±cient to pay the interest and principal on previouslyissued debt. That is, the debt increases at the rate n, which is equal to the interest rate. Here, at the optimum government debt policy simply transfers wealth from the young to the old (if the debt is positive), or from the old to the young (if the debt is negative). 2.6.1 Example Consider the same example as above, but adding government debt. That is, u(cy; co) = ln cy+ ¯ ln co; and F(K; L) = °K ®L1¡®; where ¯ > 0; ° > 0; and 0 < ® < 1: Optimal savings for a young agent is st= à ¯ 1 + ¯ ! (w t¡ ¿): (2.30) 2.7. REFERENCES 37 Then, from (2.16), (2.17), (2.27) and (2.30), the equilibrium sequence fkg 1t=0is determined by kt+1(1 + n) + b = à ¯ 1 + ¯ !" (1 ¡ ®)°k® t¡ (®°k ®¡1 t¡ n)b 1 + n # ; and the steady state capital-labor ratio, k ¤(b); is the solution to k¤(b)(1+n)+b = à ¯ 1 + ¯ !" (1 ¡ ®)° (k¤(b))® ¡ (®° (k¤(b))®¡1 ¡ n)b 1 + n # Then, from (2.29), the optimal quantity of per-capita debt is ^b = à ¯ 1 + ¯ ! (1 ¡ ®)° μ ®° n ¶® 1¡® ¡ μ ®° n 1¡® (1 + n) = ° μ ®° n ¶® 1¡® " ¯(1 ¡ ®) 1 + ¯ ¡ ® n # : Here note that, given °; n; and ¯; ^b < 0 for ® su±ciently large, and ^b > 0 for ® su±ciently small. 2.6.2 Discussion The competitive equilibrium here is in general suboptimal for reasons discussed above. But for those same reasons, government debt matters. That is, Ricardian equivalence does not hold here, in general, because the taxes required to pay o® the currently-issued debt are not levied on the agents who receive the current tax bene¯ts from a higher level of debt today. Government debt policy is a means for executing the intergenerational transfers that are required to achieve optimality. However, note that there are other intergenerational transfer mechanisms, like social security, which can accomplish the same thing in this model. 2.7 References Diamond, P. 1965. \National Debt in a Neoclassical Growth Model," American Economic Review 55, 1126-1150. 38CHAPTER 2. GROWTHWITH OVERLAPPING GENERATIONS Blanchard, O. and Fischer, S. 1989. Lectures on Macroeconomics, Chapter 3. Kareken, J. and Wallace, N. 1980. Models of Monetary Economies, Federal Reserve Bank of Minneapolis, Minneapolis, MN. Chapter 3 Neoclassical Growth and Dynamic Programming Early work on growth theory, particularly that of Solow (1956), was carried out using models with essentially no intertemporal optimizing behavior. That is, these were theories of growth and capital accumulation in which consumers were assumed to simply save a constant fraction of their income. Later, Cass (1965) and Koopmans (1965) developed the ¯rst optimizing models of economic growth, often called \optimal growth" models, as they are usually solved as an optimal growth path chosen by a social planner. Optimal growth models have much the same long run implications as Solow's growth model, with the added bene¯t that optimizing behavior permits us to use these models to draw normative conclusions (i.e. make statements about welfare). This class of optimal growth models led to the development of stochastic growth models (Brock and Mirman 1972) which in turn were the basis for real business cycle models. Here, we will present a simple growth model which illustrates some of the important characteristics of this class of models. \Growth model" will be something of a misnomer in this case, as the model will not exhibit long-run growth. One objective of this chapter will be to introduce and illustrate the use of discrete-time dynamic programming methods, which are useful in solving many dynamic models. 39 40CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 3.1 Preferences, Endowments, and Technology There is a representative in¯nitely-lived consumer with preferences given by 1X t=0 ¯tu(ct); where 0 < ¯ < 1; and c tis consumption. The period utility function u(¢) is continuously di®erentiable, strictly increasing, strictly concave, and bounded. Assume that lim c!0u0(c) = 1: Each period, the consumer is endowed with one unit of time, which can be supplied as labor. The production technology is given by yt= F(k t; t); (3.1) where y tis output, ktis the capital input, and n tis the labor input. The production function F(¢; ¢) is continuously di®erentiable, strictly increasing in both arguments, homogeneous of degree one, and strictly quasiconcave. Assume that F(0; n) = 0; lim k!0F (k; 1) = 1; and limk!1F (k; 1) = 0: The capital stock obeys the law of motion kt+1= (1 ¡ ±)kt+ it; (3.2) where i tis investment and ± is the depreciation rate, with 0 · ± · 1 and k0 is the initial capital stock, which is given. The resource constraints for the economy are ct+ it· y; (3.3) and nt· 1: (3.4) 3.2 Social Planner's Problem There are several ways to specify the organization of markets and production in this economy, all of which will give the same competitive equilibrium allocation. One speci¯cation is to endow consumers with 3.2. SOCIAL PLANNER'S PROBLEM 41 the initial capital stock, and have them accumulate capital and rent it to ¯rms each period. Firms then purchase capital inputs (labor and capital services) from consumers in competitive markets each period and maximize per-period pro¯ts. Given this, it is a standard result that the competitive equilibrium is unique and that the ¯rst and second welfare theorems hold here. That is, the competitive equilibrium allocation is the Pareto optimum. We can then solve for the competitive equilibrium quantities by solving the social planner's problem, which is max =0;n;;kt+g1t 1X t=0 ¯ u(c t) subject to ct + it· F(k ; n ); (3.5) kt+1 = (1 ¡ ±)k t + i; (3.6) n t· 1; (3.7) t = 0; 1; 2; :::; and k 0 given. Here, we have used (3.1) and (3.2) to substitute for y tto get (3.5). Now, since u(c) is strictly increasing in c; (3.5) will be satis¯ed with equality. As there is no disutility from labor, if (3.7) does not hold with equality, then n t and c tcould be increased, holding constant the path of the capital stock, and increasing utility. Therefore, (3.7) will hold with equality at the optimum. Now, substitute for i tin (3.5) using (3.6), and de¯ne f(k) ´ F(k; 1); as n t= 1 for all t: Then, the problem can be reformulated as max fc;t+1g1t =0 1X t=0 ¯ u(c t) subject to ct + k t+1= f(k ) + (1 ¡ ±)k ; t = 0; 1; 2; :::; k0 given. This problem appears formidable, particularly as the choice set is in¯nite-dimensional. However, suppose that we solve the optimization problem sequentially, as follows. At the beginning of any period t; the utility that the social planner can deliver to the consumer depends only on k t; the quantity of capital available at the beginning of the period. Therefore, it is natural to think of k tas a \state 42CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING variable" for the problem. Within the period, the choice variables, or \control" variables, are c tand k t+1: In period 0, if we know the maximum utility that the social planner can deliver to the consumer as a function of k 1; beginning in period 1, say v(k 1); it is straightforward to solve the problem for the ¯rst period. That is, in period 0 the social planner solves max c0;1 [u(c 0) + ¯v(k 1)] subject to c0 + k 1= f(k 0) + (1 ¡ ±)k 0: This is a simple constrained optimization problem which in principle can be solved for decision rules k 1 = g(k 0); where g(¢) is some function, and c 0= f(k 0) + (1 ¡ ±)k 0¡ g(k 0): Since the maximization problem is identical for the social planner in every period, we can write v(k 0) = max c0;1 [u(c 0) + ¯v(k 1)] subject to c0 + k 1= f(k 0) + (1 ¡ ±)k 0; or more generally v(k ) = max c;kt+1 [u(c ) + ¯v(k t+1)] (3.8) subject to ct+ k t+1= f(k ) + (1 ¡ ±)k t: (3.9) Equation (3.8) is a functional equation or Bellman equation. Our primary aim here is to solve for, or at least to characterize, the optimal decision rules k t+1= g(k ) and c t= f(kt)+(1¡±)k t¡g(k ): Of course, we cannot solve the above problem unless we know the value function v(¢). In general, v(¢) is unknown, but the Bellman equation can be used to ¯nd it. In most of the cases we will deal with, the Bellman equation satis¯es a contraction mapping theorem, which implies that 1. There is a unique function v(¢) which satis¯es the Bellman equation. 3.2. SOCIAL PLANNER'S PROBLEM 43 2. If we begin with any initial function v 0(k) and de¯ne v i+(k) by vi+1(k) = max c;0 [u(c) + ¯v (k0)] subject to c + k 0 = f(k) + (1 ¡ ±)k; for i = 0; 1; 2; :::; then, limi!1vi+1(k) = v(k): The above two implications give us two alternative means of uncovering the value function. First, given implication 1 above, if we are fortunate enough to correctly guess the value function v(¢); then we can simply plug v(k t+1) into the right side of (3.8), and then verify that v(k t) solves the Bellman equation. This procedure only works in a few cases, in particular those which are amenable to judicious guessing. Second, implication 2 above is useful for doing numerical work. One approach is to ¯nd an approximation to the value function in the following manner. First, allow the capital stock to take on only a ¯nite number of values, i.e. form a grid for the capital stock, k 2 fk 1; k2; :::m g = S; where m is ¯nite and k i< k i+1: Next, guess an initial value function, that is m values v i0 = v 0(ki); i = 1; 2; :::;m: Then, iterate on these values, determining the value function at the j thiteration from the Bellman equation, that is vij = max `;c [u(c) + ¯v ` j¡] subject to c + k `= f(k ) + (1 ¡ ±)k i: Iteration occurs until the value function converges. Here, the accuracy of the approximation depends on how ¯ne the grid is. That is, if ki¡ ki¡= °; i = 2; :::m; then the approximation gets better the smaller is ° and the larger is m: This procedure is not too computationally burdensome in this case, where we have only one state variable. However, the computational burden increases exponentially as we add state variables. For example, if we choose a grid with m values for each state variable, then if there are n state variables, the search for a maximum on the right side of the Bellman equation occurs over m n grid points. This problem of computational burden as n gets large is sometimes referred to as the curse of dimensionality. 44CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 3.2.1 Example of \Guess and Verify" Suppose that F(k ; nt) = k® tn 1¡® t; 0 < ® < 1; u(c ) = ln c t; and ± = 1 (i.e. 100% depreciation). Then, substituting for the constraint, (3.9), in the objective function on the right side of (3.8), we can write the Bellman equation as v(k ) = max kt+1 [ln(k® t¡ kt+1) + ¯v(k t+1)] (3.10) Now, guess that the value function takes the form v(k ) = A + B ln k ; (3.11) where A and B are undetermined constants. Next, substitute using (3.11) on the left and right sides of (3.10) to get A + B ln k t= max kt+1 [ln(k® t¡ kt+1) + ¯(A + B ln k t+1)]: (3.12) Now, solve the optimization problem on the right side of (3.12), which gives kt+1 = ¯Bk ® t 1 + ¯B ; (3.13) and substituting for the optimal k t+1in (3.12) using (3.13), and collecting terms yields A + B ln k t= ¯B ln ¯B ¡ (1 + ¯B) ln(1 + ¯B) + ¯A +(1 + ¯B)® ln k t: (3.14) We can now equate coe±cients on either side of (3.14) to get two equations determining A and B: A = ¯B ln ¯B ¡ (1 + ¯B) ln(1 + ¯B) + ¯A (3.15) B = (1 + ¯B)® (3.16) Here, we can solve (3.16) for B to get B = ® 1 ¡ ®¯ : (3.17) 3.2. SOCIAL PLANNER'S PROBLEM 45 Then, we can use (3.15) to solve for A; though we only need B to determine the optimal decision rules. At this point, we have veri¯ed that our guess concerning the form of the value function is correct. Next, substitute for B in (3.13) using (3.17) to get the optimal decision rule for kt+1; kt+1= ®¯k ® t: (3.18) Since c t= k® t¡ kt+; we have ct= (1 ¡ ®¯)k ® t: That is, consumption and investment (which is equal to k t+1given 100% depreciation) are each constant fractions of output. Equation (3.18) gives a law of motion for the capital stock, i.e. a ¯rst-order nonlinear di®erence equation in k t; shown in Figure 3.1. The steady state for the capital stock, k ¤; is determined by substituting k t= kt+1= k ¤in (3.18) and solving for k ¤ to get k¤ = (®¯) 1 1¡®: Given (3.18), we can show algebraically and in Figure 1, that k tconverges monotonically to k ¤; with k tincreasing if k 0< k ¤; and k tdecreasing if 0 > k : Figure 3.1 shows a dynamic path for k twhere the initial capital stock is lower than the steady state. This economy does not exhibit long-run growth, but settles down to a steady state where the capital stock, consumption, and output are constant. Steady state consumption is c¤= (1 ¡ ®¯)(k ¤)®; and steady state output is y ¤ = (k¤)®: 3.2.2 Characterization of SolutionsWhen the Value Function is Di®erentiable Benveniste and Scheinkman (1979) establish conditions under which the value function is di®erentiable in dynamic programming problems. Supposing that the value function is di®erentiable and concave in (3.8), we can characterize the solution to the social planner's problem using ¯rst-order conditions. Substituting in the objective function for c tusing in the constraint, we have v(k ) = max kt+fu[f(kt) + (1 ¡ ±)kt¡ kt+] + ¯v(k t+1)g (3.19) 46CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING Figure 3.1: 3.2. SOCIAL PLANNER'S PROBLEM 47 Then, the ¯rst-order condition for the optimization problem on the right side of (3.8), after substituting using the constraint in the objective function, is ¡u0 [f(t) + (1 ¡ ±)k t¡ kt+1] + ¯v 0(kt+1) = 0: (3.20) The problem here is that, without knowing v(¢); we do not know v 0(¢): However, from (3.19) we can di®erentiate on both sides of the Bellman equation with respect to k tand apply the envelope theorem to obtain v0(k ) = u 0[f(kt) + (1 ¡ ±)k t¡ kt+][f0(kt) + 1 ¡ ±]; or, updating one period, v0(k t+) = u 0[f(kt+1) + (1 ¡ ±)k t+1¡ kt+2][0(k t+) + 1 ¡ ±]: (3.21) Now, substitute in (3.20) for v 0(kt+1) using (3.21) to get ¡u0 [f(t) + (1 ¡ ±)k t¡ kt+1] +¯u 0[f(kt+) + (1 ¡ ±)k t+1 ¡ kt+][f0(kt+1) + 1 ¡ ±] = 0; (3.22) or ¡u0 (c) + ¯u 0(ct+1)[0(k t+) + 1 ¡ ±] = 0; The ¯rst term is the bene¯t, at the margin, to the consumer of consuming one unit less of the consumption good in period t; and the second term is the bene¯t obtained in period t + 1; discounted to period t; from investing the foregone consumption in capital. At the optimum, the net bene¯t must be zero. We can use (3.22) to solve for the steady state capital stock by setting k t= kt+1 = kt+2 = k¤ to get f0(k¤) = 1 ¯ ¡ 1 + ±; (3.23) i.e. one plus the net marginal product of capital is equal to the inverse of the discount factor. Therefore, the steady state capital stock depends only on the discount factor and the depreciation rate. 48CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING 3.2.3 Competitive Equilibrium Here, I will simply assert that the there is a unique Pareto optimum that is also the competitive equilibrium in this model. While the most straightforward way to determine competitive equilibrium quantities in this dynamic model is to solve the social planner's problem to ¯nd the Pareto optimum, to determine equilibrium prices we need some information from the solutions to the consumer's and ¯rm's optimization problems. Consumer's Problem Consumers store capital and invest (i.e. their wealth takes the form of capital), and each period they rent capital to ¯rms and sell labor. Labor supply will be 1 no matter what the wage rate, as consumers receive no disutility from labor. The consumer then solves the following intertemporal optimization problem. max =0;t+g1t 1X t=0 ¯ u(c t) subject to ct+ k t+1 = w t+ rtkt+ (1 ¡ ±)k ; (3.24) t = 0; 1; 2; :::; k0given, where w tis the wage rate and r tis the rental rate on capital. If we simply substitute in the objective function using (3.24), then we can reformulate the consumer's problem as max ft+g 1t =0 1X t=0 ¯ u(w t+ rtkt+ (1 ¡ ±)k t¡ k t+) subject to k t¸ 0 for all t and k 0given. Ignoring the nonnegativity constraints on capital (in equilibrium, prices will be such that the consumer will always choose k t+1 > 0), the ¯rst-order conditions for an optimum are ¡¯tu 0(w t+ rtkt+ (1 ¡ ±)k t¡ k t+) +¯ t+1u0(w t+1 + rt+1kt+1+ (1 ¡ ±)k t+1¡ kt+2)(t+1 + 1 ¡ ±) = 0 (3.25) 3.2. SOCIAL PLANNER'S PROBLEM 49 Using (3.24) to substitute in (3.25), and simplifying, we get ¯u 0(c t+) u 0(ct) = 1 1 + r t+1¡ ± ; (3.26) that is, the intertemporal marginal rate of substitution is equal to the inverse of one plus the net rate of return on capital (i.e. one plus the interest rate). Firm's Problem The ¯rm simply maximizes pro¯ts each period, i.e. it solves max k;nt [F(k ; n ) ¡ w n t¡ tkt]; and the ¯rst-order conditions for a maximum are F 1(kt; n) = rt; (3.27) F 2(kt; n) = w t: (3.28) Competitive Equilibrium Prices The optimal decision rule, k t+1 = g(k ); which is determined from the dynamic programming problem (3.8) allows a solution for the competitive equilibrium sequence of capital stocks fk g 1t=1given k 0: We can then solve for fc g 1t=0using (3.9). Now, it is straightforward to solve for competitive equilibrium prices from the ¯rst-order conditions for the ¯rm's and consumer's optimization problems. The prices we need to solve for are fw t; tg1t=0; the sequence of factor prices. To solve for the real wage, plug equilibrium quantities into (3.28) to get F 2(kt; 1) = w : To obtain the capital rental rate, either (3.26) or (3.27) can be used. Note that r t¡ ± = f 0(k) ¡ ± is the real interest rate and that, in the steady state [from (3.26) or (3.23)], we have 1+r ¡± = 1 ¯ ; or, if we let ¯ = 1 1+´; where ´ is the rate of time preference, then r¡± = ´; i.e. the real interest rate is equal to the rate of time preference. 50CHAPTER 3. NEOCLASSICAL GROWTH AND DYNAMIC PROGRAMMING Note that, when the consumer solves her optimization problem, she knows the whole sequence of prices fw t; g1t=: That is, this a \rational expectations" or \perfect foresight" equilibrium where each period the consumer makes forecasts of future prices and optimizes based on those forecasts, and in equilibrium the forecasts are correct. In an economy with uncertainty, a rational expectations equilibrium has the property that consumers and ¯rms may make errors, but those errors are not systematic. 3.3 References Benveniste, L. and Scheinkman, J. 1979. \On the Di®erentiability of the Value Function in Dynamic Models of Economics," Econometrica 47, 727-732. Brock, W. and Mirman, L. 1972. \Optimal Economic Growth and Uncertainty: The Discounted Case," Journal of Economic Theory 4, 479-513. Cass, D. 1965. \Optimum Growth in an Aggregative Model of Capital Accumulation," Review of Economic Studies 32, 233-240. Koopmans, T. 1965. \On the Concept of Optimal Growth," in The Econometric Approach to Development Planning, Chicago, Rand- McNally. Chapter 4 Endogenous Growth This chapter considers a class of endogenous growth models closely related to the ones in Lucas (1988). Here, we use discrete-time models, so that the dynamic programming methods introduced in Chapter 2 can be applied (Lucas's models are in continuous time). Macroeconomists are ultimately interested in economic growth because the welfare consequences of government policies a®ecting growth rates of GDP are potentially very large. In fact, one might argue, as in Lucas (1987), that the welfare gains from government policies which smooth out business cycle °uctuations are very small compared to the gains from growth-enhancing policies. Before we can hope to evaluate the e±cacy of government policy in a growth context, we need to have growth models which can successfully confront the data. Some basic facts of economic growth (as much as we can tell from the short history in available data) are the following: 1. There exist persistent di®erences in per capita income across countries. 2. There are persistent di®erences in growth rates of per capita income across countries. 3. The correlation between the growth rate of income and the level of income across countries is low. 4. Among rich countries, there is stability over time in growth rates 51 52 CHAPTER 4. ENDOGENOUS GROWTH of per capita income, and there is little diversity across countries in growth rates. 5. Among poor countries, growth is unstable, and there is a wide diversity in growth experience. Here, we ¯rst construct a version of the optimal growth model in Chapter 2 with exogenous growth in population and in technology, and we ask whether this model can successfully explain the above growth facts. This neoclassical growth model can successfully account for growth experience in the United States, and it o®ers some insights with regard to the growth process, but it does very poorly in accounting for the pattern of growth among countries. Next, we consider a class of endogenous growth models, and show that these models can potentially do a better job of explaining the facts of economic growth. 4.1 A Neoclassical GrowthModel (Exogenous Growth) The representative household has preferences given by 1X t=0 ¯tN t c° t ° ; (4.1) where 0 < ¯ < 1; ° < 1; c tis per capita consumption, and N tis population, where N t= (1 + n) N 0; (4.2) n constant and N 0given. That is, there is a dynastic household which gives equal weight to the discounted utility of each member of the household at each date. Each household member has one unit of time in each period when they are alive, which is supplied inelastically as labor. The production technology is given by Y t= K ® t(N A t1¡®; (4.3) where Y tis aggregate output, K tis the aggregate capital stock, and A t is a labor-augmenting technology factor, where A t= (1 + a) A 0; (4.4) 4.1. A NEOCLASSICAL GROWTHMODEL (EXOGENOUS GROWTH)53 with a constant and A 0 given. We have 0 < ® < 1; and the initial capital stock, K 0; is given. The resource constraint for this economy is N c t+ K t+1= Y : (4.5) Note here that there is 100% depreciation of the capital stock each period, for simplicity. To determine a competitive equilibrium for this economy, we can solve the social planner's problem, as the competitive equilibrium and the Pareto optimum are identical. The social planner's problem is to maximize (4.1) subject to (4.2)-(4.5). So that we can use dynamic programming methods, and so that we can easily characterize longrun growth paths, it is convenient to set up this optimization problem with a change of variables. That is, use lower case variables to de¯ne quantities normalized by e±ciency units of labor, for example y t´Y t AtNt : Also, let xt´ ct At : With substitution in (4.1) and (4.5) using (4.2)-(4.4), the social planner's problem is then max fx;t+1g1t =0 1X t=0 [¯(1 + n)(1 + a) ]t à x° t ° ! subject to xt + (1 + n)(1 + a)k t+1= k ® t; t = 0; 1; 2; ::: (4.6) This optimization problem can then be formulated as a dynamic program with state variable k tand choice variables x tand k t+1: That is, given the value function v(k ); the Bellman equation is v(k ) = max x;kt+1 " x° t ° + ¯(1 + n)(1 + a) °v(k t+1) # subject to (4.6). Note here that we require the discount factor for the problem to be less than one, that is ¯(1 +n)(1 +a) °< 1: Substituting in the objective function for x tusing (4.6), we have v(k ) = max kt+1 " [k® t¡ kt+1(1 + n)(1 + a)] ° ° + ¯(1 + n)(1 + a) °v(k t+1) # (4.7) 54 CHAPTER 4. ENDOGENOUS GROWTH The ¯rst-order condition for the optimization problem on the right side of (4.7) is ¡(1 + n)(1 + a)x °¡1 t+ ¯(1 + n)(1 + a) °v 0(kt+1) = 0; (4.8) and we have the following envelope condition v0 (kt) = ®k ®¡1 tx °¡1 t: (4.9) Using (4.9) in (4.8) and simplifying, we get ¡(1 + a) 1¡x °¡1 t+ ¯®k ®¡1 t+1x °¡1 t+1= 0: (4.10) Now, we will characterize \balanced growth paths," that is steady states where x t= x¤ and k t= k ¤; where x ¤ and k ¤are constants. Since (4.10) must hold on a balanced growth path, we can use this to solve for k ¤; that is k¤ = " ¯® (1 + a) 1¡° # 1 1¡® (4.11) Then, (4.6) can be used to solve for x ¤to get x¤ = " ¯® (1 + a) 1¡° 1¡® " (1 + a) 1¡° ¯® ¡ (1 + n)(1 + a) # : (4.12) Also, since y t= k ® t; then on the balanced growth path the level of output per e±ciency unit of labor is y¤ = (k¤)® = " ¯® (1 + a) 1¡° # ® 1¡® : (4.13) In addition, the savings rate is st= K t+1 Y t = kt+1(1 + n)(1 + a) k® t ; so that, on the balanced growth path, the savings rate is s¤ = (k¤)1¡®(1 + n)(1 + a): 4.1. A NEOCLASSICAL GROWTHMODEL (EXOGENOUS GROWTH)55 Therefore, using (4.11) we get s¤ = ¯®(1 + n)(1 + a) °: (4.14) Here, we focus on the balanced growth path since it is known that this economy will converge to this path given any initial capital stock K 0> 0: Since k ¤; ¤; and y ¤ are all constant on the balanced growth path, it then follows that the aggregate capital stock, K , aggregate consumption, N c ; and aggregate output, Y t; all grow (approximately) at the common rate a + n; and that per capita consumption and output grow at the rate a: Thus, long-run growth rates in aggregate variables are determined entirely by exogenous growth in the labor force and exogenous technological change, and growth in per capita income and consumption is determined solely by the rate of technical change. Changes in any of the parameters ¯; ®; or ° have no e®ect on long-run growth. Note in particular that an increase in any one of ®; ¯; or ° results in an increase in the long-run savings rate, from (4.14). But even though the savings rate is higher in each of these cases, growth rates remain una®ected. This is a counterintuitive result, as one might anticipate that a country with a high savings rate would tend to grow faster. Changes in any of ®; ¯; or ° do, however, produce level e®ects. For example, an increase in ¯; which causes the representative household to discount the future at a lower rate, results in an increase in the savings rate [from (4.14)], and increases in k ¤and y ¤; from (4.11) and (4.13). We can also show that ¯(1+n)(1+a) ° < 1 implies that an increase in steady state k ¤will result in an increase in steady state x¤: Therefore, an increase in ¯ leads to an increase in x ¤: Therefore, the increase in ¯ yields increases in the level of output, consumption, and capital in the long run. Suppose that we consider a number of closed economies, which all look like the one modelled here. Then, the model tells us that, given the same technology (and it is hard to argue that, in terms of the logic of the model, all countries would not have access to A t); all countries will converge to a balanced growth path where per capita output and consumption grow at the same rate. From (4.13), the di®erences in the level of per capita income across countries would have to be explained by di®erences in ®; ¯; or °: But if all countries have access 56 CHAPTER 4. ENDOGENOUS GROWTH to the same technology, then ® cannot vary across countries, and this leaves an explanation of di®erences in income levels due to di®erences in preferences. This seems like no explanation at all. While neoclassical growth models were used successfully to account for long run growth patterns in the United States, the above analysis indicates that they are not useful for accounting for growth experience across countries. The evidence we have seems to indicate that growth rates and levels of output across countries are not converging, in contrast to what the model predicts. 4.2 A Simple Endogenous Growth Model In attempting to build a model which can account for the principal facts concerning growth experience across countries, it would seem necessary to incorporate an endogenous growth mechanism, to permit economic factors to determine long-run growth rates. One way to do this is to introduce human capital accumulation. We will construct a model which abstracts from physical capital accumulation, to focus on the essential mechanism at work, and introduce physical capital in the next section. Here, preferences are as in (4.1), and each agent has one unit of time which can be allocated between time in producing consumption goods and time spent in human capital accumulation. The production technology is given by Y t= ®h tutN ; where ® > 0; Y tis output, h tis the human capital possessed by each agent at time t, and u tis time devoted by each agent to production. That is, the production function is linear in quality-adjusted labor input. Human capital is produced using the technology h t+1= ±h t(1 ¡ u ); (4.15) where ± > 0, 1¡u tis the time devoted by each agent to human capital accumulation (i.e. education and acquisition of skills), and h 0 is given. Here, we will use lower case letters to denote variables in per capita terms, for example y t´ Yt Nt : The social planner's problem can then 4.2. A SIMPLE ENDOGENOUS GROWTH MODEL 57 be formulated as a dynamic programming problem, where the state variable is h tand the choice variables are c t, ht+1; and u : That is, the Bellman equation for the social planner's problem is v(h ) = max c;u;ht+1 " t° ° + ¯(1 + n)v(h t+1) # subject to ct = ®h u t(4.16) and (4.15). Then, the Lagrangian for the optimization problem on the right side of the Bellman equation is L = c° t ° + ¯(1 + n)v(h t+1) + ¸t(®h tu t¡ c) + ¹t[±h t(1 ¡ ut) ¡ ht+1]; where ¸ tand ¹ tare Lagrange multipliers. Two ¯rst-order conditions for an optimum are then @L @c t = c °¡1 t¡ ¸t= 0; (4.17) @L @h t+1 = ¯(1 + n)v 0(ht+1) ¡ ¹t= 0; (4.18) (4.15) and (4.16). In addition, the ¯rst derivative of the Lagrangian with respect to u tis @L @u t = ¸t®h t¡ t±h t Now, if @L @u t > 0; then u t= 1: But then, from (4.15) and (4.16), we have h s= c s = 0 for s = t + 1; t + 2; ::: . But, since the marginal utility of consumption goes to in¯nity as consumption goes to zero, this could not be an optimal path. Therefore @L @u · 0: If @L @u t < 0; then u t= 0; and ct = 0 from (4.16). Again, this could not be optimal, so we must have @L @u t = ¸t®h t¡ t±h t= 0 (4.19) at the optimum. 58 CHAPTER 4. ENDOGENOUS GROWTH We have the following envelope condition: v0(h t) = ®u t¸t+ ¸t®(1 ¡ u ); or, using (4.17), v0(h t) = ®c °¡1 t(4.20) From (4.17)-(4.20), we then get ¯(1 + n)±c °¡1 t+1¡ c°¡1 t= 0: (4.21) Therefore, we can rewrite (4.21) as an equation determining the equilibrium growth rate of consumption: ct+1 ct = [¯(1 + n)±] 1 1¡: (4.22) Then, using (4.15), (4.16), and (4.22), we obtain: 1¯(1 + n)±] 1¡= ±(1 ¡ u t)ut+1 u t ; or u t+1= [¯(1 + n)± °] 1 1¡u t 1 ¡ u t (4.23) Now, (4.23) is a ¯rst-order di®erence equation in u tdepicted in Figure 4.1 for the case where [¯(1+n)] 1¡± ¡°< 1, a condition we will assume holds. Any path fu g 1t=0satisfying (4.23) which is not stationary (a stationary path is u t= u; a constant, for all t) has the property that lim t!u t= 0; which cannot be an optimum, as the representative consumer would be spending all available time accumulating human capital which is never used to produce in the future. Thus the only solution, from (4.23), is 1 t= u = 1 ¡ [¯(1 + n)± °] 1¡° for all t: Therefore, substituting in (4.15), we get h t+1 h t = [¯(1 + n)±] 1 1¡; 4.3. ENDOGENOUS GROWTHWITH PHYSICAL CAPITAL AND HUMAN CAPITAL59 and human capital grows at the same rate as consumption per capita. If [¯(1 + n)±] 1 1¡> 1 (which will hold for ± su±ciently large), then growth rates are positive. There are two important results here. The ¯rst is that equilibrium growth rates depend on more than the growth rates of exogenous factors. Here, even if there is no growth in population (n = 0) and given no technological change, this economy can exhibit unbounded growth. Growth rates depend in particular on the discount factor (growth increases if the future is discounted at a lower rate) and ±; which is a technology parameter in the human capital accumulation function (if more human capital is produced for given inputs, the economy grows at a higher rate). Second, the level of per capita income (equal to per capita consumption here) is dependent on initial conditions. That is, since growth rates are constant from for all t; the level of income is determined by h 0; the initial stock of human capital. Therefore, countries which are initially relatively rich (poor) will tend to stay relatively rich (poor). The lack of convergence of levels of income across countries which this model predicts is consistent with the data. The fact that other factors besides exogenous technological change can a®ect growth rates in this type of model opens up the possibility that di®erences in growth across countries could be explained (in more complicated models) by factors including tax policy, educational policy, and savings behavior. 4.3 Endogenous GrowthWith Physical Capital and Human Capital The approach here follows closely the model in Lucas (1988), except that we omit his treatment of human capital externalities. The model is identical to the one in the previous section, except that the production technology is given by Y t= K ® t(N h u )1¡®; where K tis physical capital and 0 < ® < 1; and the economy's resource constraint is N c t+ K t+1= K ® t(N h u )1¡® 60 CHAPTER 4. ENDOGENOUS GROWTH Figure 4.1: 4.3. ENDOGENOUS GROWTHWITH PHYSICAL CAPITAL AND HUMAN CAPITAL61 As previously, we use lower case letters to denote per capita quantities. In the dynamic program associated with the social planner's optimization problem, there are two state variables, k tand h t; and four choice variables, u ; ct; ht+; and k t+1: The Bellman equation for this dynamic program is v(k ; ht) = max c;u;kt;ht+1 " c° t ° + ¯(1 + n)v(k t+1; ht+) # subject to ct+ (1 + n)k t+1= k ® t(h u )1¡® (4.24) h t+1= ±h t(1 ¡ ut) (4.25) The Lagrangian for the constrained optimization problem on the right side of the Bellman equation is then L = c° t ° +¯(1+n)v(k t+; ht+1)+¸ [k® t(h u )1¡®¡c ¡(1+n)k t+1]+¹t[±h (1¡u t)¡ht+1] The ¯rst-order conditions for an optimum are then @L @c t = c °¡1 t¡ ¸t= 0; (4.26) @L @u t = ¸t(1 ¡ ®)k ® th 1¡® tu ¡® t¡ ¹±h t= 0; (4.27) @L @h t+1 = ¯(1 + n)v 2(kt+1; ht+1) ¡ t= 0; (4.28) @L @k t+1 = ¡¸t(1 + n) + ¯(1 + n)v 1 (kt+1; ht+) = 0; (4.29) (4.24) and (4.25). We also have the following envelope conditions: v1(k ; ht) = ¸t®k ®¡1 t(h u )1¡® (4.30) v2(k ; ht) = ¸t(1 ¡ ®)k ® th ¡® tu 1¡® t+ ¹t±(1 ¡ u t) (4.31) Next, use (4.30) and (4.31) to substitute in (4.29) and (4.28) respectively, then use (4.26) and (4.27) to substitute for ¸ t and ¹ tin (4.28) and (4.29). After simplifying, we obtain the following two equations: ¡c°¡1 t+ ¯c °¡1 t+1®k ®¡1 t+1(h t+u t+1)1¡®= 0; (4.32) 62 CHAPTER 4. ENDOGENOUS GROWTH ¡c°¡1 tk® th ¡® tu ¡® t+ ±¯(1 + n)c °¡1 t+1k ® t+1h¡® t+1u¡® t+1= 0: (4.33) Now, we wish to use (4.24), (4.25), (4.32), and (4.33) to characterize a balanced growth path, along which physical capital, human capital, and consumption grow at constant rates. Let ¹ k; ¹h; and ¹ cdenote the growth rates of physical capital, human capital, and consumption, respectively, on the balanced growth path. From (4.25), we then have 1 + ¹ h= ±(1 ¡ u ); which implies that u t= 1 ¡ 1 + ¹ h ± ; a constant, along the balanced growth path. Therefore, substituting for u , ut+1; and growth rates in (4.33), and simplifying, we get (1 + ¹ c)1¡(1 + ¹ k)¡®(1 + ¹h)® = ±¯(1 + n): (4.34) Next, dividing (4.24) through by k t, we have ct kt + (1 + n) kt+1 kt = k ®¡1 t(h u )1¡®: (4.35) Then, rearranging (4.32) and backdating by one period, we get (1 + ¹ c)1¡° ¯® = k ®¡1 t(h u )1¡® (4.36) Equations (4.35) and (4.36) then imply that ct kt + (1 + n)(1 + ¹ k) = (1 + ¹ c)1¡° ¯® : But then ct kt is a constant on the balanced growth path, which implies that ¹c = ¹k: Also, from (4.36), since u tis a constant, it must be the case that ¹ k = ¹h: Thus per capita physical capital, human capital, and per capita consumption all grow at the same rate along the balanced growth path, and we can determine this common rate from (4.34), i.e. 1 + ¹ c= 1 + ¹ k = 1 + ¹ h = 1 + ¹ = [±¯(1 + n)] 1¡: (4.37) 4.4. REFERENCES 63 Note that the growth rate on the balanced growth path in this model is identical to what it was in the model of the previous section. The savings rate in this model is st= K t+1 Y t = kt+1(1 + n) ktk®¡1 t(htu)1¡® Using (4.36) and (4.37), on the balanced growth path we then get 1t= ® [± ¯(1 + n)] 1¡(4.38) In general then, from (4.37) and (4.38), factors which cause the savings rate to increase (increases in ¯; n; or ±) also cause the growth rate of per capita consumption and income to increase. 4.4 References Lucas, R.E. 1987. Models of Business Cycles, Basil Blackwell, New York. Lucas, R.E. 1988. \On the Mechanics of Economic Development," Journal of Monetary Economics 22, 3-42. 64 CHAPTER 4. ENDOGENOUS GROWTH Chapter 5 Choice Under Uncertainty In this chapter we will introduce the most commonly used approach to the study of choice under uncertainty, expected utility theory. Expected utility maximization by economic agents permits the use of stochastic dynamic programming methods in solving for competitive equilibria. We will ¯rst provide an outline of expected utility theory, and then illustrate the use of stochastic dynamic programming in a neoclassical growth model with random disturbances to technology. This stochastic growth model is the basis for real business cycle theory. 5.1 Expected Utility Theory In a deterministic world, we describe consumer preferences in terms of the ranking of consumption bundles. However, if there is uncertainty, then preferences are de¯ned in terms of how consumers rank lotteries over consumption bundles. The axioms of expected utility theory imply a ranking of lotteries in terms of the expected value of utility they yield for the consumer. For example, suppose a world with a single consumption good, where a consumer's preferences over certain quantities of consumption goods are described by the function u(c); where c is consumption. Now suppose two lotteries over consumption. Lottery i gives the consumer c 1i units of consumption with probability p , and c 2i units of consumption with probability 1¡p ; where 0 < p i< 1; i = 1; 2: 65 66 CHAPTER 5. CHOICE UNDER UNCERTAINTY Then, the expected utility the consumer receives from lottery i is piu(c 1i ) + (1 ¡ p)u(c 2i ); and the consumer would strictly prefer lottery 1 to lottery 2 if p1u(c 11 ) + (1 ¡ p1)u(c21 ) > p2u(c 12 ) + (1 ¡ p2)u(c22 ); would strictly prefer lottery 2 to lottery 1 if p1u(c 11 ) + (1 ¡ p1)u(c21 ) < p2u(c 12 ) + (1 ¡ p2)u(c22 ); and would be indi®erent if p1u(c 11 ) + (1 ¡ p1)u(c21 ) = p2u(c 12 ) + (1 ¡ p2)u(c22): Many aspects of observed behavior toward risk (for example, the observation that consumers buy insurance) is consistent with risk aversion. An expected utility maximizing consumer will be risk averse with respect to all consumption lotteries if the utility function is strictly concave. If u(c) is strictly concave, this implies Jensen's inequality, that is E[u(c)] · u (E[c]) ; (5.1) where E is the expectation operator. This states that the consumer prefers the expected value of the lottery with certainty to the lottery itself. That is, a risk averse consumer would pay to avoid risk. If the consumer receives constant consumption, ¹c; with certainty, then clearly (5.1) holds with equality. In the case where consumption is random, we can show that (5.1) holds as a strict inequality. That is, take a tangent to the function u(c) at the point (E[c]; u(E[c])) (see Figure 1). This tangent is described by the function g(c) = ® + ¯c; (5.2) where ® and ¯ are constants, and we have ® + ¯E[c] = u(E[c]): (5.3) Now, since u(c) is strictly concave, we have, as in Figure 1, ® + ¯c ¸ u(c); (5.4) 5.1. EXPECTED UTILITY THEORY 67 for c ¸ 0; with strict inequality if c 6= E[c]: Since the expectation operator is a linear operator, we can take expectations through (5.4), and given that c is random we have ® + ¯E[c] > E[u(c)]; or, using (5.3), u(E[c]) > E[u(c)]: As an example, consider a consumption lottery which yields c 1units of consumption with probability p and c 2units with probability 1 ¡ p; where 0 < p < 1 and c 2> c1: In this case, (5.1) takes the form pu(c 1) + (1 ¡ p)u(c2) < u (pc1 + (1 ¡ p)c2) : In Figure 2, the di®erence u (pc 1+ (1 ¡ p)c2) ¡ [pu(c1) + (1 ¡ p)u(c2)] is given by DE. The line AB is given by the function f(c) = c2u(c 1) ¡ 1u(c2) c2 ¡ 1 + " u(c ) ¡ u(c1) c2 ¡ 1 # c: A point on the line AB denotes the expected utility the agent receives for a particular value of p; for example p = 0 yields expected utility u(c ) or point A, and B implies p = 1: Jensen's inequality is re°ected in the fact that AB lies below the function u(c): Note that the distance DE is the disutility associated with risk, and that this distance will increase as we introduce more curvature in the utility function, i.e. as the consumer becomes more risk averse. 5.1.1 Anomalies in Observed Behavior Towards Risk While expected utility maximization and a strictly concave utility function are consistent with the observation that people buy insurance, some observed behavior is clearly inconsistent with this. For example, many individuals engage in lotteries with small stakes where the expected payo® is negative. 68 CHAPTER 5. CHOICE UNDER UNCERTAINTY Another anomaly is the \Allais Paradox." Here, suppose that there are four lotteries, which a person can enter at zero cost. Lottery 1 involves a payo® of $1 million with certainty; lottery 2 yields a payo® of $5 million with probability .1, $1 million with probability .89, and 0 with probability .01; lottery 3 yields $1 million with probability .11 and 0 with probability .89; lottery 4 yields $5 million with probability .1 and 0 with probability .9. Experiments show that most people prefer lottery 1 to lottery 2, and lottery 4 to lottery 3. But this is inconsistent with expected utility theory (whether the person is risk averse or not is irrelevant). That is, if u(¢) is an agent's utility function, and they maximize expected utility, then a preference for lottery 1 over lottery 2 gives u(1) > :1u(5) + :89u(1) + :01u(0); or :11u(1) > :1u(5) + :01u(0): (5.5) Similarly, a preference for lottery 4 over lottery 3 gives :11u(1) + :89u(0) < :1u(5) + :9u(0); or :11u(1) < :1u(5) + :9u(0); (5.6) and clearly (5.5) is inconsistent with (5.6). Though there appear to be some obvious violations of expected utility theory, this is still the standard approach used in most economic problems which involve choice under uncertainty. Expected utility theory has proved extremely useful in the study of insurance markets, the pricing of risky assets, and in modern macroeconomics, as we will show. 5.1.2 Measures of Risk Aversion With expected utility maximization, choices made under uncertainty are invariant with respect to a±ne transformations of the utility function. That is, suppose a utility function v(c) = ® + ¯u(c); 5.1. EXPECTED UTILITY THEORY 69 where ® and ¯ are constants with ¯ > 0: Then, we have E[v(c)] = ® + ¯E[u(c)]; since the expectation operator is a linear operator. Thus, lotteries are ranked in the same manner with v(c) or u(c) as the utility function. Any measure of risk aversion should clearly involve u 00(c); since risk aversion increases as curvature in the utility function increases. However, note that for the function v(c); that we have v 00(c) = ¯u00(c); i.e. the second derivative is not invariant to a±ne transformations, which have no e®ect on behavior. A measure of risk aversion which is invariant to a±ne transformations is the measure of absolute risk aversion, ARA(c) = ¡ u00(c) u0(c) : A utility function which has the property that ARA(c) is constant for all c is u(c) = ¡¡®; ® > 0: For this function, we have ARA(c) = ¡¡® 2e¡®c ®e ¡®c= ®: It can be shown, through Taylor series expansion arguments, that the measure of absolute risk aversion is twice the maximum amount that the consumer would be willing to pay to avoid one unit of variance for small risks. An alternative is the relative risk aversion measure, RRA(c) = ¡c u00(c) u0(c) : A utility function for which RRA(c) is constant for all c is u(c) = c1¡°¡ 1 1 ¡ ° ; where ° ¸ 0: Here, RRA(c) = ¡c¡°c ¡(1+°) c¡°= ° 70 CHAPTER 5. CHOICE UNDER UNCERTAINTY Note that the utility function u(c) = ln(c) has RRA(c) = 1: The measure of relative risk aversion can be shown to be twice the maximum amount per unit of variance that the consumer would be willing to pay to avoid a lottery if both this maximum amount and the lottery are expressed as proportions of an initial certain level of consumption. A consumer is risk neutral if they have a utility function which is linear in consumption, that is u(c) = ¯c; where ¯ > 0: We then have E[u(c)] = ¯E[c]; so that the consumer cares only about the expected value of consumption. Since u 00(c) = 0 and u 0(c) = ¯; we have ARA(c) = RRA(c) = 0: 5.2 Stochastic Dynamic Programming We will introduce stochastic dynamic programming here by way of an example, which is essentially the stochastic optimal growth model studied by Brock and Mirman (1972). The representative consumer has preferences given by E 0 1X t=0 ¯tu(c ); where 0 < ¯ < 1; c tis consumption, u(¢) is strictly increasing, strictly concave, and twice di®erentiable, and E 0 is the expectation operator conditional on information at t = 0: Note here that, in general, c twill be random. The representative consumer has 1 unit of labor available in each period, which is supplied inelastically. The production technology is given by yt= z tF(kt; n); where F(¢; ¢) is strictly quasiconcave, homogeneous of degree one, and increasing in both argument. Here, k tis the capital input, n tis the labor input, and z tis a random technology disturbance. That is, fz g 1t=0is a sequence of independent and identically distributed (i.i.d.) random variables (each period z tis an independent draw from a ¯xed probability distribution G(z)): In each period, the current realization, z , is learned 5.2. STOCHASTIC DYNAMIC PROGRAMMING 71 at the beginning of the period, before decisions are made. The law of motion for the capital stock is kt+1= it+ (1 ¡ ±)k t; where i tis investment and ± is the depreciation rate, with 0 < ± < 1: The resource constraint for this economy is ct+ it= y : 5.2.1 Competitive Equilibrium In this stochastic economy, there are two very di®erent ways in which markets could be organized, both of which yield the same unique Pareto optimal allocation. The ¯rst is to follow the approach of Arrow and Debreu (see Arrow 1983 or Debreu 1983). The representative consumer accumulates capital over time by saving, and in each period he/she rents capital and sells labor to the representative ¯rm. However, the contracts which specify how much labor and capital services are to be delivered at each date are written at date t = 0: At t = 0; the representative ¯rm and the representative consumer get together and trade contingent claims on competitive markets. A contingent claim is a promise to deliver a speci¯ed number of units of a particular object (in this case labor or capital services) at a particular date (say, date T) conditional on a particular realization of the sequence of technology shocks, fz 0; z1; z2; :::; Tg: In a competitive equilibrium, all contingent claims markets (and there are potentially very many of these) clear at t = 0; and as information is revealed over time, contracts are executed according to the promises made at t = 0: Showing that the competitive equilibrium is Pareto optimal here is a straightforward extension of general equilibrium theory, with many state-contingent commodities. The second approach is to have spot market trading with rational expectations. That is, in period t labor is sold at the wage rate w t and capital is rented at the rate r : At each date, the consumer rents capital and sells labor at market prices, and makes an optimal savings decision given his/her beliefs about the probability distribution of future prices. In equilibrium, markets clear at every date t for every 72 CHAPTER 5. CHOICE UNDER UNCERTAINTY possible realization of the random shocks fz 0; z1; z2; :::; g: In equilibrium expectations are rational, in the sense that agents' beliefs about the probability distributions of future prices are the same as the actual probability distributions. In equilibrium, agents can be surprised in that realizations of z tmay occur which may have seemed, ex ante, to be small probability events. However, agents are not systematically fooled, since they make e±cient use of available information. In this representative agent environment, a rational expectations equilibrium is equivalent to the Arrow Debreu equilibrium, but this will not be true in models with heterogeneous agents. In those models, complete markets in contingent claims are necessary to support Pareto optima as competitive equilibria, as complete markets are required for e±cient risk sharing. 5.2.2 Social Planner's Problem Since the unique competitive equilibrium is the Pareto optimum for this economy, we can simply solve the social planner's problem to determine competitive equilibrium quantities. The social planner's problem is max ft;t+g1t=0 E 0 1X t=0 ¯ u(c t) subject to ct+ k t+1 = ztf(kt) + (1 ¡ ±)k ; where f(k) ´ F(k; 1): Setting up the above problem as a dynamic program is a fairly straightforward generalization of discrete dynamic programming with certainty. In the problem, given the nature of uncertainty, the relevant state variables are k tand z ; where k tis determined by past decisions, and z tis given by nature and known when decisions are made concerning the choice variables c tand k t+1: The Bellman equation is written as v(k ; zt) = max c;kt+1 [u(c ) + ¯E v(k t+1; zt+)] subject to ct + k t+1= z f(k ) + (1 ¡ ±)k t: 5.2. STOCHASTIC DYNAMIC PROGRAMMING 73 Here, v (¢; ¢) is the value function and E tis the expectation operator conditional on information in period t: Note that, in period t; c t is known but c t+ii = 1; 2; 3; :::; is unknown. That is, the value of the problem at the beginning of period t + 1 (the expected utility of the representative agent at the beginning of period t + 1) is uncertain as of the beginning of period t: What we wish to determine in the above problem are the value function, v(¢; ¢); and optimal decision rules for the choice variables, i.e. k t+1 = g(k t; z) and c t= z tf(kt) + (1 ¡ ±)k t¡ g(k ; zt): 5.2.3 Example Let F(k t; n) = k ® tn 1¡® t; with 0 < ® < 1; u(c t) = ln c ; ± = 1; and E[ln z t] = ¹: Guess that the value function takes the form v(k ; zt) = A + B ln k t+ Dln z t The Bellman equation for the social planner's problem, after substituting for the resource constraint and given that n t= 1 for all t, is then A+B ln k +Dln z t= max kt+fln[z k ® t¡ kt+1] + ¯E [A + B ln k t+1+ Dln z t+1]g ; or A+B ln k +Dln z t= max kt+fln[z k ® t¡ kt+1] + ¯A + ¯B ln k t+1+ ¯D¹g : (5.7) Solving the optimization problem on the right-hand side of the above equation gives kt+1 = ¯B 1 + ¯B ztk® t: (5.8) Then, substituting for the optimal k t+1 in (5.7), we get A + B ln k t+ Dln z t= ln à ztk® t 1 + ¯B ! + ¯A + ¯B ln à ¯Bz tk® t 1 + ¯B ! + ¯D¹ (5.9) 74 CHAPTER 5. CHOICE UNDER UNCERTAINTY Our guess concerning the value function is veri¯ed if there exists a solution for A;B; and D: Equating coe±cients on either side of equation (5.9) gives A = ln à 1 1 + ¯B ! + ¯A + ¯B ln à ¯B 1 + ¯B ! + ¯D¹ (5.10) B = ® + ®¯B (5.11) D = 1 + ¯B (5.12) Then, solving (5.10)-(5.12) for A; B; and D gives B = ® 1 ¡ ®¯ D = 1 1 ¡ ®¯ A = 1 1 ¡ ¯ " ln(1 ¡ ®¯) + ®¯ 1 ¡ ®¯ ln(®¯) + ¯¹ 1 ¡ ®¯ # We have now shown that our conjecture concerning the value function is correct. Substituting for B in (5.8) gives the optimal decision rule kt+1= ®¯z k® t; (5.13) and since c t= zk ® t¡ kt+; the optimal decision rule for c tis ct= (1 ¡ ®¯)z tk® t: (5.14) Here, (5.13) and (5.14) determine the behavior of time series for c tand kt(where k t+1is investment in period t): Note that the economy will not converge to a steady state here, as technology disturbances will cause persistent °uctuations in output, consumption, and investment. However, there will be convergence to a stochastic steady state, i.e. some joint probability distribution for output, consumption, and investment. This model is easy to simulate on the computer. To do this, simply assume some initial k 0, determine a sequence fz gT t=0using a random number generator and ¯xing T; and then use (5.13) and (5.14) to determine time series for consumption and investment. These time series 5.3. REFERENCES 75 will have properties that look something like the properties of post-war de
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