Universidad de Montevideo Macroeconomia II. The RamseyCassKoopmans Model


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1 Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The RamseyCassKoopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous and constant. By not allowing consumers to behave optimally, it does not allow us to discuss how incentives affect the behavior of the economy. In order to have a more complete framework, we need to allow for the path of consumption and, hence, the saving rate to be determined by optimizing agents (households and firms) that interact on competitive markets. We deal here with infinitelylived households that choose consumption and saving to maximize their dynastic utility, subject to an intertemporal budget constraint. This is the RamseyCass Koopmans model (from now on, the Ramsey model). Let us start now with a simplified version of the Ramsey model, with a single infinitelylived agent or, what is the same, the Robinson Crusoe model. Indeed, the infinite horizon assumption turns out to have strong implications: together with the assumptions of competitive markets, constant returns to scale in production, and homogeneous agents, it typically implies that the allocation of resources achieved by a decentralized economy will be the same as that of a single individual or that chosen by a central planner who maximizes the utility of the representative agent. 2 The Model The agent s problem is to maximize its lifetime utility defined as U = 0 u(c(t))e ρt, (1) where u( ) is the instantaneous utility function, assumed to be nonnegative and concave, c(t) is per capita consumption, and ρ is a constant, strictly positive rate of time preferene, subject to dk f(k(t)) c(t) δk(t) (2) 1
2 where k is the per capital stock of capital, f(k) is the production function in per capita terms (derived from an original CRS function) and δ is the percentage rate of depreciation. In this case, k is the state variable, which is determined by past history. A state variable cannot change discretely in the context of the working of the model, although it could as a result of an exogenous shock the usual experiment of an earthquake, which at a point in time destroys part of the capital stock. The variable c, instead, is a control variable, which can take any value at any point in time sometimes called jumpers. What follows is the sketch of the optimization procedure by optimal control. 1 This procedure implies setting the Hamiltonian 2 H = u(c)e ρt + λ[f(k) c δk], where λ is the costate variable associated with the state variable k, or the multiplier. The value of λ is is the marginal value as of time 0 of an additional unit of capital at time t. Now, solve for the derivatives in order to find the first order conditions: H c = e ρt λ = 0 (3) λ t λ. = H k = λ[f (k) δ] (4) H λ = dk = f(k) c δk (5) Notice that this method is the analogue of the alternative method of deriving optimality conditions in the static case. Differentiating the first order condition (3) with respect to time, we obtain c ċe ρt ρe ρt =. λ (6) Now, divide this expression by the same condition to obtain, c ċe ρt ρe ρt e ρt =. λ λ. (7) Substituting (4) into (7), and operating on it, we have the key equation of the Ramsey model, the Euler condition: 1 For a detailed discussion regarding optimization in continuous time, I encourage you to read Appendix A.3 in Barro and SalaiMartin. And, for a very quick cookbook procedure, I suggest you to read in particular Section A For simplicity, let me remove the timet arguments accompanying the variables of the model. 2
3 ρ [ cucc ] ċ c = f (k) δ. (8) The expression cc / reflects the curvature of the utility function. It is equal to the elasticity of marginal utility with respect to consumption. Finally, we have the transversality condition (TVC): or equivalently, lim λk = 0, t lim ke ρt = 0. (9) t The TVC, for the case in which the time horizon is finite (T instead of infinite) says that a rational individual will end its life with a zero stock of assets. The expression above is simply the horizon version of this condition. Notice the relationship between the TVC and the NoPonzigame condition, which requires, for the finite horizon case, that the individual cannot die with negative assets. We will turn back later to this condition when we introduce the decentralized economy. The Economic Interpretation of the Euler Condition Notice that (8) has been derived from the equivalent form ( duc ) 1 = [f (k) δ ρ]. (10) The economic interpretation of this condition (KeynesRamsey rule) can be discussed as follows: Take two points in time (dates), t and t + 1, and consider the effects on welfare of moving some consumption from t to t + 1 or the other way around. The change in utility at date t is (c t ) c t (11) The change in the capital stock between two dates, and the change in consumption at t + 1, is equal to c t+1 = c t [ 1 + f (k t ) δ ], which will result in a change in utility, at date t + 1, equal to 3
4 (c t+1 ) c t+1 = (c t+1 ) c t [ 1 + f (k t ) δ ]. (12) Equate now the change (loss) in utility ad date t, expression (11), to the present value at date t of the change (gain) in utility at t + 1, the RHS of (12), to obtain (c t+1 ) (c t )(1 + ρ) = f (k t ) δ, where the LHS is the marginal rate of substitution between consumption at times t and t+1, and the RHS is the marginal rate of transformation, from production, between consumption at times t and t + 1. This is the direct analogue of the usual condition for the consumer optimization condition in the static case. This expression can be written as (c t+1 ) (c t ) = (1 + ρ) 1 + f (k t ) δ, which, for the interval between the two dates becoming sufficiently small, results in expression (10), i.e., ( duc ) 1 = [f (k) δ ρ]. Rewrite equation (8) in order to have the following alternative formulation. c c = 1 [ cucc ][f (k) δ ρ] Notice that the Euler condition implies that consumption increases, remains constant, or decreases depending on whether the marginal product of capital (net of depreciation) exceeds, is equal to, or is less than the rate of time preference. 3 It roughly means that if the subjective discount rate is low relative to the net return on capital, then the agent will prefer to move current consumption toward future consumption, so consumption will be rising over time on the optimal path. 3 Steady State and Dynamics The optimal path is characterized by equations (8), (9), and the constraint (2). Setting aside for the moment the TVC, the model solution can be summarized by the following two expressions: 3 Recall that the utility function is nonnegative and concave in consumption (U c 0 and U cc < 0). 4
5 . c c = 1 [ cucc ][f (k) δ ρ] (13). k f(k) c δk (14) Let us start with the steady state. In steady state both k and c are constant. As in the Solow model, denote the steady state of these variables by k and c. 4 From (13), with ċ = 0, we have the modified golden rule relationship: f (k ) δ = ρ (15) The marginal product of capital net of depreciation, in steady state, is equal to the rate of time preference. Corresponding to k is the steady state level of consumption implied by (14) with k. = 0: c = f(k ) δk (16) Notice that steady state consumption is maximized when f (k ) δ = 0 it comes from equalizing to zero the result of differentiating equation (16) with respect to k. This is, in fact, the golden rule itself. The modification in (15) is that the capital stock is reduced below the GR level by an amount that depends on the rate of time preference. To study the dynamics, we use the phase diagram, drawn in (k, c) space (see the figure below). The locus k. = 0 starts from the origin, reaches a maximum at the GR capital stock, k g, at which f (k) = δ and cross the horizontal axis where f(k) = δ +ρ (call it k max ). Notice from (13) that the locus ċ = 0 will be vertical at the modified GR capital, k. Anywhere above the k. = 0 locus, c > f(k) δk, then k. < 0. Similarly, k is increasing at points below that curve. Anywhere to the left of the ċ = 0 locus, k will be less than k. Given that f(k) is concave, then f (k) > f (k ) = δ + ρ. As a consequence, consumption will be increasing to the left of the ċ = 0 line, and will be falling to the right. There are three equilibria: (k = 0, c = 0), (k = k, c = c ), and (k = k c=0, c = 0). However, for each initial capital stock only one trajectory, the saddle point path, that converges to the equilibrium (k = k, c = c ) satisfies the three necesary conditions (8), (9), and (2). The equilibrium (k = 0, c = 0) does not satisfy the Ramsey condition, while the equilibrium (k = k max, c = 0) does not satilsfy the TVC. 4 Notice that now we do not have either population growth or technological progress. Then, both aggregate and per capita variables will be constant at this state. 5
6 c c = 0 k = 0 k* k 4 The Decentralized Economy Suppose now that instead of having a single agent in the model, we assume that there is a constant population N; it can be thought of as many identical households. The labor force is equal to the population, with labor supplied inelastically. Ouput, as before, is produced using capital and labor. There is no productivity growth. Finally, suppose that the economy is decentralized. There are two factor markets, one for labor and one for capital services. We denote the rental price of labor, the wage, as w, and the rental price of capital as R. There is also a debt market in which households can borrow and lend at the interest rate r. Each household decides, at any point in time, how much labor and capital to rent to firms and how much to save or consume. They can save by either accumulating capital or lending to other households. Since the capital stock depreciates at the rate δ, the net rate of return to a household that owns a unit of capital is R δ. Since capital and loans are perfect substitutes as stores of value, we must have r = R δ or, equivalently, R = r + δ. There are many identical firms, each with the same technology as described above, f(k). Competitive, pricetaker firms rent the services of capital and labor to produce output. Both households and firms have perfect foresight; that is they know both current and future values of w and r and take them as given same as rational expectations for a certainty context. 6
7 Households As before, each household maximizes its lifetime utility given by (1), subject to the budget constraint dω(t) where ω(t) k(t) + a(t). w(t) + r(t)ω(t) c(t) for all t, k 0 given, (17) Houshold wealth is given by ω, which is equal to holdings of capital, k, plus households net assets, a. Given that capital is given at any point in time, and labor is supplied inelastically, the household decision at each t will be how much to consume or save. Firms Firms maximize profits at each point in time. Given the the features of technology and prices described above, first order conditions for profit maximization imply that 5 f (k) = r + δ, f(k) kf (k) = w. (18) Given a sequence of wages and rental rates, households will choose a path of consumption and wealth accumulation. Given that private assets must always be equal to zero in the aggregate, wealth accumulation will determine capital accumulation. The path of capital will in turn imply a path of wages and rental rates. The equilibrium paths of these prices are in fact defined as those paths that reproduce themselves given optimal decisions by firms and households. The NoPonzi Game Condition We need an additional condition on the maximization problem in order to prevent households to choose succesive high levels of borrowing (ω < 0) to meet interest payments on the existing debt (the socalled Ponzi game). At the same time, notice that we do not want to impose a condition that rules out temporary indebtedness. A natural condition, then, is to require that household debt not increase asymptotically faster than the interest rate: 5 Once again, for simplicity of notation I remove timet arguments. t lim t ωe 0 r(v)dv 0. (19) 7
8 This condition is known as a noponzigame condition. Although (19) is stated as an inequality, it is clear that as long as marginal utility is positive, households will not want to have increasing wealth forever at rate r, and that the condition will hold as an equality. Thus, in what follows we use this condition directly as an equality. The Equilibrium Maximization of (1) subject to (17) and (19), carried up by setting up a Hamiltonian as the one studied before, implies the following necessary and sufficient conditions ( duc ) 1 = [r ρ] (20) lim ωe ρt = 0. (21) t In equilibrium, aggregate private assets, a(t), must always be equal to zero (since all households are identical, in equilibrium there is neither lending nor borrowing). ω = k. Using this and equations (18) for w and r, and replacing in (17) and (20) gives ( duc dk Thus, f(k) c δk (22) ) 1 = [f (k) δ ρ]. (23) Equations (21), (22), and (23) characterize the behavior of the decentralized economy. Note that they are identical to equations (9), (2), and (10) which characterize the behavior of the RobinsonCrusoe economy. Thus, the dynamic behavior of the decentralized economy will be the same as that of Robinson Crusoe and, consequently, as that of the centrally planned economy. 5 The Government Let us introduce the government into the model. We assume that the government spending is fixed exogenously, and we examine the effects on the economy s equilibrium of a balancedbudget change in the level of government spending. Suppose the government s per capita demand, g, is exogenous and it does not directly affect the marginal utility of consumption of the representative household. Also, assume the government levies per capita lumpsum taxes τ = g. The household s flow budget constraint now becomes 8
9 c + dω which, using the noponzigame condition, integrates to 6 0 or equivalently, ce t 0 r(v)dv = ω = w + rω τ, (24) we t 0 r(v)dv 0 τe t 0 r(v)dv, (25) 0 ce t 0 r(v)dv = ω 0 + h 0 G 0, (26) where G 0 is the present discounted value of government spending, which is equal by assumption to the present discounted value of lumpsum taxes. Suppose that the government demands a constant small g. The phase diagram is the same as before, except that now the output available for private consumption is reduced by the uniform amount g (see the figure below). The dynamics toward the steady state will have the same features, but now the steadystate consumption will be smaller than in the previous steady state. In steady state, government completely crowds out private consumption but has no effect on the capital stock. Now, does an unanticipated change in g have dynamic effects on capital accumulation? Departing from steady state, a permanent change only affects consumption by the same amount (it instantaneously jumps onto the new steady state). However, a transitory change in g will indeed affect capital accumulation. Suppose g rises during the period t = t 0 through t = t 1. At impact, the increase in g leads to a fall in consumption that, optimally chosen by the agent, is less in absolute value than the change in g. Consumption will be higher than that needed to be on the new. k = 0 locus. Then, optimality conditions governed by the new system will lead capital to fall, at the same time that consumption would start rising. When government goes back to its old spending at t 1, the agent reaches the old saddle path governed by the original. k = 0 locus. Both capital and consumption will be rising in direction toward the original steady state, c = c and k = k. Notice that during the period where g is higher, capital is falling, then the interest rate will rise (f (k) will be above relative to that of the steady state). Of course, it will go back later to the original interest rate. Finally, notice that allowing for forwardlooking behavior yields richer results than those of the traditional view, where consumption depended only on current disposable income. 6 This equation is the result of applying the following procedure. First, integrate the flow budget constraint from time 0 to time T, and discount to time zero. Then, using the NPG condition, together with letting T go to, finally gives the intertemporal budget constraint. This constraint implies that the present value of consumption is equal to nonhuman wealth, ω 0, plus human wealth, h 0 the present value of labor income. 9
10 c c = 0 g k = 0 k* k 10
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