Study Materials for Stabilization Policy I | ECON 611, Exams of Economics

Download Study Materials for Stabilization Policy I | ECON 611 and more Exams Economics in PDF only on Docsity! Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 1 Conditions Implied by Dynamic Optimization 1.1 Introduction Two of the hallmarks of both macroeconomics and finance are the concern with time and with uncertainty. In the first half of this book, the focus is on time. The key to much of economics is an understanding of choices made with an eye to the future. The consequences of such choices unfold over time, but it is the view of the future at the time of decision that governs such choices. Throughout the first half of this book, we will make the certainty- equivalence approximation of looking at the decisions agents would make if all uncertainty vanished and they were certain to face the expected values of future variables. Using the certainty-equivalence approximation, one proceeds as if all the agents in a model had perfect foresight.1 As we proceed to study “perfect-foresight” models, it is important to keep sight of their purpose of providing a certainty-equivalence approximation to stochastic models. 1In the second half of the book, we will see how far optimal decisions under uncertainty depart from the certainty equivalence approximation. One simple generalization is that when only aggregate, economy-wide uncertainty is at issue, the certainty-equivalence approxima- tion is typically quite a good approximation. The certainty-equivalence approximation is often not a very good approximation when the idiosyncratic risk faced by heterogeneous households and firms is at issue. (The law of averages helps to make aggregate uncertainties much smaller as a percentage of mean values than the risks faced by individual households and firms.) In any case, the certainty-equivalence approximation is the foundation on which higher-order approximations will be built. 2 CHAPTER 1. NECESSARY CONDITIONS In order to develop the intuition behind dynamic optimization, this chapter moves from a two-period model, to several related T-period models, and finally to a continuous-time model. For now, the focus is on deriving and interpreting the basic conditions that are required in order for a set of dynamic choices to be optimal. 1.2 A Two-Period Model of Production, Consump- tion, and Capital Accumulation Many of the most important principles of dynamic optimization are apparent in a two-period model of production, consumption and capital accumulation, with a fixed amount of inelastically supplied labor in the background. Consider an agent facing the optimization problem max c0, c1, k0, k1, k2 u0(c0) + u1(c1) subject to k0 = k̄ (1.1) k1 − k0 = f(k0)− c0 (1.2) k2 − k1 = f(k1)− c1 (1.3) k2 ≥ 0. (1.4) The subscripts are for the two active time periods 0 and 1 and any later periods used for accounting purposes. The letters are the traditional labels: u : single-period utility function f : production function (net of depreciation) c : consumption k : capital stock. The utility function is additively time-separable with no explicit time discount- ing (though the possibility of time discounting is implicit in the distinction between u0 and u1). Equation (1.1), which says that the initial capital stock k0 is fixed at k̄ is the initial condition. Equation (1.4), which says that the 1.3. T-PERIOD MODEL OF PRODUCTION AND SAVING 5 The Lagrangian for the T-period problem is L = u0(c0) + u1(c1) + u2(c2) + . . . + uT−1(cT−1) +Λ0[k̄ − k0] +Λ1[f(k0)− c0 + k0 − k1] +Λ2[f(k1)− c1 + k1 − k2] (1.16) + ... +ΛT [f(kT−1)− cT−1 + kT−1 − kT ] +ΛT+1kT . The first-order conditions obtained by setting the derivative of L with respect to ct equal to zero for t = 0, . . . , T − 1, are u′t(ct) = Λt+1 (t = 0, . . . , T − 1) (1.17) The Euler equations obtained by setting the derivative of L with respect to kt equal to zero for t = 0, . . . , T − 1 are Λt = [1 + f ′(kt)] Λt+1 (t = 0, . . . , T − 1). (1.18) In addition, setting the derivative with respect to kT equal to zero, ΛT = ΛT+1. (1.19) Finally, the transversality condition and associated nonnegativity condition for ΛT+1 obtained from the Kuhn-Tucker conditions for (1.15) are ΛT+1kT = 0 (1.20) ΛT+1 ≥ 0. (1.21) As in the two-period model, the first-order conditions say that the marginal utility of consumption in each period is equal to the marginal value of capital anticipated in the next period. Thinking of the net marginal product of capital as the real interest rate, the Euler equations say that the ratio between the marginal value of capital now and in the next period is equal to one plus the real interest rate. The transversality condition says that the value of all capital left over, evaluated at its marginal value, must be zero. 6 CHAPTER 1. NECESSARY CONDITIONS 1.4 The T-Period Model Viewed from Another An- gle What happens if we write the T-period model of production, consumption, and capital accumulation in a different, but equivalent form? Viewing the same model from various different angles can often give additional insights and strengthen one’s intuition. Checking the consistency of the various solutions to the model stated in different ways can suggest important questions where there is a gap in one’s understanding as well as helping one to detect simple, but potentially troublesome mistakes. Here, among other things, the alternative formulation allows us to see what happens when the capital stock is in the objective function. Consider the same T-period economic problem with (net) investment (i) taking center stage instead of consumption: max i,k T−1∑ t=0 ut(f(kt)− it) subject to k0 = k̄ (1.22) kt+1 − kt = it (t = 0, . . . , T − 1) (1.23) kT ≥ 0. (1.24) The Lagrangian is L = u0(f(k0)− i0) + u1(f(k1)− i1) + . . . + uT−1(f(kT−1 − iT−1) +Λ0[k̄ − k0] +Λ1[i0 + k0 − k1] +Λ2[i1 + k1 − k2] (1.25) + ... +ΛT [iT−1 + kT−1 − kT ] +ΛT+1kT . The first-order conditions obtained by setting the derivative of L with respect to it equal to zero for t = 0, . . . , T − 1, are u′t(f(kt)− it) = Λt+1 (t = 0, . . . , T − 1) (1.26) 1.5. GENERAL T-PERIOD MODEL 7 Since f(kt) − it = ct, this is equivalent to the first-order conditions (1.17) above. The Euler equations obtained by setting the derivative of L with respect to kt equal to zero for t = 0, . . . , T − 1 are Λt = Λt+1 + f ′(kt)u′(f(kt)− it) (t = 0, . . . , T − 1). (1.27) Using (1.26) to substitute in Λt+1 for u′(f(kt)− it) yields Λt = [1 + f ′(kt]Λt+1, which is identical to (1.18). The remaining equations are identical to (1.19), (1.20), and (1.21). 1.5 A General T-Period Model of Dynamic Opti- mization With the preceding models as background, it is time to look at a general T- period model that will yield general results that apply to a wide variety of more specific models. Consider the general problem max x,k T−1∑ t=0 U(kt, xt, t) subject to k0 = k̄ (1.28) kt+1 − kt = A(kt, xt, t) (t = 0, . . . , T − 1) (1.29) kT ≥ 0. (1.30) U is a general objective function. A is a general accumulation function. The dependence of U and A on time is shown within the parentheses rather than as a subscript in order to save subscripts on U and A for partial derivatives. The correspondence with the preceding sections is as follows: Section: (1.3) (1.4) U(kt, xt, t) : ut(xt) ut(ft(kt)− xt) A(kt, xt, t) : ft(kt)− xt xt xt : ct it 10 CHAPTER 1. NECESSARY CONDITIONS As h → 0, the Euler equation (1.43) guarantees that Λt will be continuously differentiable with respect to time. Thus, Λt+h → Λt as h → 0. Dividing (1.42) and (1.43) by h and taking the limit as h → 0, the continuous-time first-order condition is Ux(kt, xt, t) + ΛtAx(kt, xt, t) = 0 (t ∈ [0, T ) ). (1.46) Using a dot for time-derivatives (e.g., Λ̇t = dΛtdt ), the continuous-time Euler equation is Λ̇t = −[Uk(kt, xt, t) + ΛtAk(kt, xt, t)] (t ∈ [0, T ) ). (1.47) 1.7 Summary of the Continuous-Time Results As h → 0, the continuous-time model can be written (omitting t-subscripts) as max x ∫ T 0 U(k, x, t) dt subject to k0 = k̄ (1.48) k̇ = A(k, x, t) (1.49) kT ≥ 0. (1.50) As is traditional, the maximization is here represented as being over only x alone because the accumulation equation (1.49) implies that the path of kt is determined by the path of xt. Following this line of thinking, in the dynamic optimization literature, variables like k that have rates of change constrained by differential equations like (1.49) are called state variables; while variables like x that the agent could vary discontinuously from one moment to the next at will (though it may not be optimal to do so) are called control variables. Multipliers like Λ are called costate variables. This is useful terminology.5 Omitting arguments as well as t subscripts for clarity, the four key results above that are implied by optimization are the first-order condition 5Even though k is not called a control variable, it is at least indirectly controllable. It is good to remember that the Euler equations above arise from considering changes in the path of k in conformance with the accumulation equation. 1.7. SUMMARY 11 Ux + ΛAx = 0, (1.51) the Euler equation Λ̇ = −[Uk + ΛAk], (1.52) the transversality condition ΛT kT = 0 (1.53) and the nonnegativity condition for ΛT ΛT ≥ 0. (1.54) 1.7.1 The Hamiltonian A good way to remember the two most important necessary conditions for dynamic optimization—the first order condition and the Euler equation—is by the mnemonic device of the Hamiltonian. The Hamiltonian is analogous to, but not identical to the Lagrangian. The Hamiltonian H is defined as H = U(k, x, t) + ΛA(k, x, t). (1.55) In terms of the Hamiltonian H, the first-order condition (1.51) can be recast as Hx = 0. (1.56) The Euler equation can be recast as Λ̇ = −Hk. (1.57) Interestingly enough, if one is willing to take a partial derivative with respect to Λ, even the accumulation equation (1.49) can be recast in terms of the Hamiltonian as k̇ = HΛ. (1.58) The most important fact about the Hamiltonian is that a number of nec- essary conditions for dynamic optimization can be summarized in the single statement that for each k and t on an optimal path, the Hamiltonian must be maximized over x. In addition to the first order condition (1.56) implied by 12 CHAPTER 1. NECESSARY CONDITIONS H being maximized over x at each point in time, optimization requires, for example, that Hxx ≤ 0. (1.59) We leave the proof of (1.59) to an exercise. A general proof that the Hamil- tonian must be maximized at each point in time (the Pontryagin maximum principle) can be found in [?]. 1.8 The Maximum Principle The Pontryagin Maximum Principle states that given the value of the costate variable Λ at each point in time, the Hamiltonian must necessarily be max- imized globally over all possible values of the control variables. It applies in the strict sense only in continuous time, though it will be approximately true in discrete time if the time intervals are short. We will not prove it here, but it is not hard to explain the essence of the proof of the Maximum Principle. As far as the overall goal is concerned, only two things matter about what is being done this instant: (1) the value of the objective function this instant and (2) the accumulation of the state variable that occurs during this instant: H = U + ΛA. Since the amount of accumulation that can occur in an instant is infinitesimal, there is no problem in valuing that accumulation at a fixed marginal value measured by Λ even if following an alternative policy for an instant involves a large change in the rate of accumulation for that instant. In particular, suppose the Hamiltonian was not at a global maximum at some point when following a proposed policy. Modifying that policy by switching to the global maximum for an instant at that point would improve the sum of the value of the objective function at that instant and the value of the accumulation at that instant. In discrete time, switching to the global maximum of the Hamiltonian from some other local maximum has a noticeable effect on the amount of accumu- lated in the period, so the costate variable Λ that appears in the definition of the Hamiltonian may be noticeably altered, so that the global maximum given the original value of Λ might not be the global maximum given the new value of Λ. However, in discrete time the Hamiltonian still must be lo- cally optimized, since any local change in the control variables causes only an infinitesimal change in the amount of accumulation. As the length of the period gets shorter, it becomes impossible for the local maximum achieved Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 2 Conditions Sufficient for an Optimum 2.1 Introduction: Sufficient Conditions for Static Optimization In the straightforward calculus problem max x f(x) with differentiable f , maximization implies as a minimal requirement the nec- essary condition f ′(x) = 0. All of the necessary conditions for dynamic opti- mization discussed in the previous chapter are akin to this “flat top” condition. But finding where the derivative equals zero is not sufficient to guarantee that one has found a global maximum. Even if the second-order condition is sat- isfied strictly (f ′′(x) < 0) it guarantees only a local maximum—that one has found a hilltop, but not necessarily the highest hilltop. The simplest addi- tional condition sufficient to guarantee that if f ′(x) = 0, one has found a global maximum is for f to be globally concave. A function f is globally concave if and only if its graph lies beneath all of its tangent lines (or all of its tangent hyperplanes).1 Symbolically, if f is 1if: If the graph of f lies beneath all of its tangent lines, then the tangent line through any point on the curve—and therefore the point itself—must be above the secant line segment connecting any two points on either side of the first point. Points on the curve being above all relevant secant line segments is the primary definition of concavity. only if: If f is concave, the separating hyperplane theorem guarantees that the graph of f lies below its tangent lines. 16 CHAPTER 2. SUFFICIENT CONDITIONS concave, f(x) ≤ f(x∗) + f ′(x∗)[x− x∗], (2.1) where there is a tangency at x∗ and x represents any other point on the curve. If x∗ satisfies the first-order condition f ′(x∗) = 0 with f concave, the inequality (2.1) simplifies to f(x) ≤ f(x∗) —guaranteeing that (x∗, f(x∗)) is a global maximum. 2.2 Relating Dynamic Optimization to Static Opti- mization Sufficient conditions for dynamic optimization are closely related to sufficient conditions for static optimization. Indeed, one of the most instructive sufficient conditions for dynamic optimization is static optimization at each instant of the classic Lagrangian L(k, x, t) defined by L(k, x, t) = H(k, x, t) + Λ̇tk = U(k, x, t) + ΛtA(k, x, t) + Λ̇tk (2.2) over k and x. More precisely, consider the problem max x,k ∫ T 0 U(k, x, t) dt subject to k0 = k̄ (2.3) k̇ = A(k, x, t) (2.4) kT ≥ 0. (2.5) A program is feasible only if it satisfies (2.3), (2.4) and (2.5). If (k∗t , x∗t ) is feasible, and there is any function of time Λt for which the maximum of L(k, x, t) = U(k, x, t) + ΛtA(k, x, t) + Λ̇tk over k and x at each t is attained by (k∗t , x∗t ), and for which 2.2. RELATING DYNAMIC OPTIMIZATION TO STATIC OPTIMIZATION 17 ΛT k∗T = 0 (2.6) ΛT ≥ 0, (2.7) then (k∗t , x∗t ) achieves a global maximum for the dynamic optimization problem over all feasible programs. Moreover, this result holds even if one modifies the dynamic optimization problem by adding the constraint x ∈ Xt at each time t where Xt is a time- varying constraint set. 2.2.1 Proof Integrating by parts, ∫ T 0 Λ̇tkt dt = ΛT kT − Λ0k0 − ∫ T 0 Λtk̇t dt. (2.8) Therefore, for any function of time Λt, (2.4) and integration by parts together yield the identity (2.9) ∫ T 0 U(kt, xt, t) dt = ∫ T 0 {U(kt, xt, t) + Λt[A(kt, xt, t)− k̇t]} dt = ∫ T 0 [U(kt, xt, t) + ΛtA(kt, xt, t) + Λ̇tkt] dt −ΛT kT + Λ0k0 = ∫ T 0 L(kt, xt, t) dt− ΛT kT + Λ0k0. If the program (k∗t , x∗t ) maximizes L(k, x, t) for each t, then for any other feasible program (kt, xt), ∫ T 0 L(kt, xt, t) dt ≤ ∫ T 0 L(k∗t , x ∗ t , t) dt. (2.10) Both programs being feasible implies that Λ0k0 = Λ0k̄ = Λ0k∗0, (2.11) Finally, by (2.7), (2.5) and the transversality condition (2.6) for the pro- gram (k∗t , x∗t ), 20 CHAPTER 2. SUFFICIENT CONDITIONS 4. Arrow’s Theorem: Define the function L?(k, t) = max x L(k, x, t). • Show that if L? is concave in k, then the necessary conditions for dynamic optimization are also sufficient. • Show that if U and A are each jointly concave in k and x and Λt is positive, then L? will be concave in x. 5. Show that if L(k, x, t) = g(Q(k, x, t)) where Q is concave jointly in k and x and g is monotonically increasing, then the necessary conditions are sufficient. Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 3 Discounting and an Infinite Horizon 3.1 Introduction Assuming a preference for earlier rather than later consumption (“impa- tience”) has a long history going back at least to Irving Fisher. At the in- dividual level, the evidence for such a preference is equivocal at best,1 but in a dynastic setting, it corresponds to the not-implausible assumption that people care about the average welfare of all of their children more than they care about their own welfare.2 Beyond a belief in impatience or imperfect altruism, there are important technical reasons for the interest economists have had in models with utility discounting. Dynamic control theory predicts that, with a finite horizon, as the end-time T approaches, an agent’s behavior will be strongly influenced by that approaching end. In a saving and investment problem, this is reflected in increases in consumption toward the end as the end when the capital stock will become worthless approaches. It is often desirable to keep one’s analysis from being dominated by such “endgame” considerations by letting the end-time T go to infinity. An infinite horizon does not typically present serious technical 1When asked to imagine themselves in a hypothetical situation, slightly more than half of a sample of older people in the Health and Retirement Survey and slightly less than half of a sample of undergraduates choose an upward-sloping consumption profile even in the face of a zero real interest rate. Other evidence also calls into question whether people in general prefer earlier to later consumption. See [?]. 2How such a preference for one’s own welfare over the average welfare of all one’s children can persist in the fact of sociobiological pressures is unclear. 22 CHAPTER 3. DISCOUNTING AND AN INFINITE HORIZON problems as long as the integral of the objective function ∫∞ 0 U(kt, xt, t) dt converges. Convergence of this integral requires U to eventually decrease as t gets large. Ordinarily, the long-run decline in U results from a negative direct dependence on t in the long run—utility discounting. 3.1.1 Additive Time-Separability, Stationarity and Time Con- sistency Economists in general, and macroeconomists in particular, have begun to take their utility functions for granted. When a particular form has become tradi- tional, it is useful to step back and ask which aspects of the traditional form are solidly based and which aspects are arbitrary and dispensable. The famil- iar additively time-separable, exponentially discounted utility—utility of the form ∫ ∞ 0 e−ρtu(ct) dt —is actually less arbitrary than it might appear. Let us examine some of the bases for choosing this form of utility function. To begin with, additive time-separability is an implication of ordinary sepa- rability holding between every two subsets of time periods whenever there are at least three time periods.3 Given additive time-separability, exponential discounting is the only form of discounting that is both stationary and allows time consistency.4 Stationar- ity means that the preferences for consumption after some date in relation to that starting date does not depend on the starting date. In other words, a util- ity function is stationary if preferences between earlier and later consumption are the same when viewing the year 2011 from the year 2010 as when viewing the year 2001 from the year 2000. Time-consistency means that preferences between earlier and later consumption look the same in advance as they do when one gets there. In other words, a utility function is time-consistent if preferences between earlier and later consumption are the same when viewing the comparison of 2010 and 2011 from the vantage point of the year 2000 as when viewing the comparison of 2010 and 2011 from the nearer vantage point of 2010 itself. Much more can be said about these issues, but the only point here is to motivate an interest in the traditional exponential discounting.5 3See the first exercise for an idea why. 4See [?]. 5Time-consistency without stationarity is possible if there is a time-varying utility dis- 3.3. CONTINUOUS TIME 25 for practical purposes, this almost always suffices as a transversality condition. More accurately, the transversality condition should be stated in terms of the size of a feasible differential reduction in k, −∆k. For any feasible differential reduction in k, dynamic optimization requires lim t→∞β tλt[−∆kt] = 0. (3.13) If, as is typically the case, the size of a feasible differential reduction in k grows in proportion to k itself as t → ∞, then (3.12) is exactly right. See [?] for more details on the transversality condition in an infinite horizon setting. 3.3 Continuous Time 3.3.1 Discounting One could readily derive results for the continuous-time model with discount- ing by taking a limit of the discrete-time model as the length of a time period goes to zero,6 but it is easier to start with the continuous-time necessary con- ditions and add discounting. In the continuous-time model, let U(k, x, t) = e−ρtU(k, x, t), (3.14) so that the overall objective is given by ∫ T 0 e−ρtU(kt, xt, t) dt. As in the discrete-time case, U(k, x, t) is the objective function expressed in a present value as viewed from time zero, while U(k, x, t) is the objective function expressed as a current value as of time t. (Much, perhaps all, of the dependence of U(k, x, t) on time is in the e−ρt term multiplying U(k, x, t), but we have allowed for the possibility of some other dependence on time that is better modeled separately from this term.) In order to be consistent and have a form of the costate variable that expresses the marginal value of capital as a current value as of time t (in current utils) rather than as a present value viewed from time zero (in time- zero utils), let us also define the current value marginal value of capital, λt, by 6See exercise 2. 26 CHAPTER 3. DISCOUNTING AND AN INFINITE HORIZON Λt = e−ρtλt, (3.15) or equivalently λt = eρtΛt. (3.16) Substituting the definitions (3.14) and (3.15) into (??) yields the first-order condition e−ρtUx(kt, xt, t) + e−ρtλtAx(kt, xt, t) = 0 (3.17) or equivalently, Ux(kt, xt, t) + λtAx(kt, xt, t) = 0. (3.18) Substituting the above definitions into (??) yields the Euler equation e−ρtλ̇− ρe−ρtλ = −[e−ρtUk(kt, xt, t) + e−ρtλtAk(kt, xt, t)] (3.19) or equivalently, λ̇t = ρλt − [Uk(kt, xt, t) + λtAk(kt, xt, t)]. (3.20) Alternatively, λ̇t = (ρ−Ak(kt, xt, t))λt − Uk(kt, xt, t). (3.21) The transversality condition becomes e−ρT λT kT = 0 (3.22) which, when T is finite, is equivalent to λT kT = 0. (3.23) Finally, the associated nonnegativity condition comes to λT ≥ 0. (3.24) 3.4. SUMMARY OF RESULTS 27 3.3.2 An Infinite Horizon Again, the only equation affected in form by letting T go to infinity is the transversality condition. The straightforward infinite counterpart to (3.22) is lim t→∞ e −ρtλtkt = 0. (3.25) for practical purposes, this almost always suffices as a transversality condition. More accurately, the transversality condition should be stated in terms of the size of a feasible differential reduction in k, −∆k. For any feasible differential reduction in k, dynamic optimization requires lim t→∞ e −ρtλt[−∆kt] = 0. (3.26) If, as is typically the case, the size of a feasible differential reduction in k grows in proportion to k itself as t →∞, then (3.25) is exactly right.7 3.4 Summary of Results 3.4.1 Discrete Time With U = βtU and Λt = βtλt the first order condition is Ux + βλt+1Ax = 0. (3.27) (Here and below, all of the arguments of U and A are at time t.) The Euler equation is λt = Uk + β[1 + Ak]λt+1. (3.28) The transversality condition, in a form that remains essentially valid if T →∞, is βT λT kT = 0. (3.29) 3.4.2 Continuous Time Omitting the time t subscripts, with U = e−ρtU and Λ = e−ρtλ the first order condition is 7See [?] for more details on the transversality condition in an infinite horizon setting. 30 CHAPTER 3. DISCOUNTING AND AN INFINITE HORIZON Where δ is rate at which capital depreciates. (From here on, we will list the maximization as being over the control variables rather than also over the state variables implied by the policies for the control variables. For brevity, we will also typically omit the initial condition k0 = k̄ in the statement of a problem since this condition seldom varies in form.) The current-value Hamiltonian for (3.41) is H = u(c) + λ[f(k)− δk − c]. The first order condition is Hc = u′(c)− λ = 0, or u′(c) = λ. Thus, the marginal utility of consumption equals the marginal value of capital. The Euler equation is λ̇ = ρλ−Hk = [ρ + δ − f ′(k)]λ, or λ̇ λ = ρ + δ − f ′(k). Thus, the proportional growth rate of the marginal value of capital is equal to the difference between the utility discount rate and the net marginal product of capital f ′(k)−δ. (In comparing with earlier results, pay particular attention to the special case when the depreciation rate δ is zero.) 3.6 Extension: A Variable Discount Rate Especially in dealing with the behavior of profit-maximizing firms, for whom the relevant discount rate is the interest rate r, it is important to know how to deal with variable discount rates. Consider a general problem with a variable discount rate r: max x ∫ T 0 e− ∫ t 0 rt′ dt ′ U(kt, xt, t) dt (3.43) 3.6. EXTENSION: A VARIABLE DISCOUNT RATE 31 subject to k̇ = A(kt, xt, t), (3.44) where t′ is just a dummy variable. For this problem, where we want to dis- count by the variable interest rate r, define the current-value marginal value of capital λ by λt = e ∫ t 0 rt′ dt ′ Λt. (3.45) In order to differentiate λ, we need the fundamental theorem of calculus, which says that8 d dt ∫ t 0 r dt′ = r, so that by the chain rule, d dt e ∫ t 0 r dt′ = re ∫ t 0 r dt′ and the product rule give us λ̇ = re ∫ t 0 r dt′Λ + e ∫ t 0 r dt′Λ̇ = rλ + e ∫ t 0 r dt′Λ̇. Together with the general Euler equation (??), this implies λ̇ = rλ− [Uk(k, x, t) + λAk(k, x, t)] (3.46) = rλ−Hk, where the current value Hamiltonian H is defined as H = U + λA. Thus, the formula for the Euler equation has essentially the same form even when the discount rate varies with time! It is also easy to show that the first-order condition Hx = 0 and the more general principle of maximization of the current-value Hamil- tonian is still valid in terms of the current-value Hamiltonian with a variable discount rate. 8For clarity, we will drop the time and time-dummy subscripts from here on in this section. 32 CHAPTER 3. DISCOUNTING AND AN INFINITE HORIZON Exercises Additive Time-Separability. This exercise is meant to clarify why separability every which way implies additive separability as long as there are at least three periods. To see the logic of this result, consider the case of exactly three time periods and objective function Ω(c1, c2, c3). Separability every which way implies that each ratio of marginal utilities is a function of only the two immediately relevant variables; that is, ∂Ω(c1, c2, c3) ∂c1 ÷ ∂Ω(c1, c2, c3) ∂c2 = φ(c1, c2) ∂Ω(c1, c2, c3) ∂c2 ÷ ∂Ω(c1, c2, c3) ∂c3 = χ(c2, c3) ∂Ω(c1, c2, c3) ∂c1 ÷ ∂Ω(c1, c2, c3) ∂c3 = ψ(c1, c3) Note that this implies ψ(c1, c3) = χ(c2, c3)φ(c1, c2) for any c2. Starting from an arbitrary reference point (c̄1, c̄2, c̄3), define u′1(c1) = φ(c1, c̄2) φ(c̄1, c̄2) u′2(c2) = 1 φ(c̄1, c2) u′3(c3) = 1 ψ(c̄1, c3) . Exercise: Show that, given these definitions, u′1(c1) u′2(c2) = ∂Ω(c1, c2, c3) ∂c1 ÷ ∂Ω(c1, c2, c3) ∂c2 , u′2(c1) u′3(c2) = ∂Ω(c1, c2, c3) ∂c2 ÷ ∂Ω(c1, c2, c3) ∂c3 , u′1(c1) u′3(c2) = ∂Ω(c1, c2, c3) ∂c1 ÷ ∂Ω(c1, c2, c3) ∂c3 , 3.6. EXTENSION: A VARIABLE DISCOUNT RATE 35 H = pF (k, n)− wn− pii + q[kφ(i/k)]. The first-order conditions for n and i are Hn = pFn − w = 0 Hi = −pi + qφ′(i/k) = 0, or pFn = w (3.49) q = pi φ′(i/k) . (3.50) The Euler equation is q̇ = rq −Hk = [r + (i/k)φ′(i/k)− φ(i/k)]q − pFk. 36 CHAPTER 3. DISCOUNTING AND AN INFINITE HORIZON Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 4 Multiple Control Variables and Multiple State Variables 4.1 Introduction Nothing in the derivation of the general first-order conditions and Euler equa- tions in the previous chapters prevents the control x and the state k from being vectors. The costate λ will be a vector of the same size as the state k. Looking at the components of these vectors, (1) to each control variable there is a first-order condition, (2) each state variable comes endowed with an accumulation equation and (3) each state-variable/costate-variable pair has an Euler equation. Thus, accumulation equations and Euler equations come paired according to the state variable they relate to. All of the equations have the same form for each variable as if that variable were the only control variable or the only state variable. Although state and costate variables are linked together in pairs so that there are exactly as many costate variables as there are state variables, the number of control variables can be very different from the number of state variables. Indeed, it is surprisingly easy to work with models with one state variable and many control variables—allowing one to address a wide range of phenomena. In later chapters we will find that additional state variables add much more to the difficulty of a complete analysis of a model than do additional control variables, but in this chapter we are only trying to get the basic equations of each model. Additional state variables do not cause any trouble in getting the basic equations of a model, so we will look at both models with one state 40 CHAPTER 4. MULTI-VARIABLE CONTROL 4.3 Constraints on the Control Variables In this section, we will focus on dealing with constraints on the control vari- ables. The Pontryagin Maximum Principle states that the Hamiltonian must be maximized at each point in time (globally in the case of continuous time, at least locally in discrete time). The easiest way to maximize the Hamiltonian subject to constraints is to augment the Hamiltonian with these constraints and their associated multipliers and then get first-order conditions and some- times Kuhn-Tucker conditions in the familiar way. The multipliers associated with these constraints must be carefully distinguished from the costate vari- ables. In particular, there are no Euler equations attached to the multipliers for the constraints on the control variables. 4.3.1 The Cass Model Revisited We can make both consumption and investment explicit in the Cass model if we also make an explicit constraint that consumption plus investment equals output. That is, max c,i ∫ ∞ 0 e−ρtu(c) dt (4.3) subject to k̇ = i− δk (4.4) f(k) = c + i. (4.5) The augmented current-value Hamiltonian is H = u(c) + λ[i− δk] + µ[f(k)− c− i]. The costate variable is λ, since λ is the multiplier for the accumulation equa- tion (4.4). By constrast, µ is an ordinary Lagrange multiplier associated with the maximization of the stripped-down Hamiltonian u(c) + λ(i− δk). (With- out the constraint, maximization of this stripped down Hamiltonian would be easy: infinitely high consumption c and infinitely high investment i.) Using the augmented Hamiltonian, the first-order conditions are Hc = u′(c)− µ = 0 Hi = λ− µ = 0. (4.6) 4.3. CONSTRAINTS ON THE CONTROL VARIABLES 41 Not surprisingly, µ can be eliminated, yielding the familiar equation u′(c) = λ. There is no Euler equation for µ, since it is not a costate variable. The Euler equation for λ is still λ̇ = ρλ−Hk = (ρ + δ)λ− f ′(k)µ = [ρ + δ − f ′(k)]λ, where the last step requires the fact that µ = λ due to (4.6). It does not matter whether one calculates the Hk in the Euler equation from the stripped-down Hamiltonian or from the augmented Hamiltonian as here. For one thing, the augmented and stripped-down Hamiltonians are iden- tically equal since, for each constraint, either the constraint or the associated multiplier is always equal to zero making the product identically zero. (In this particular case, f(k) − c − i is identically equal to zero.) Furthermore, by the Envelope Theorem, the change in the maximized Hamiltonian is equal to the partial derivative of either version of the Hamiltonian with respect to k. (See Section (??) for more on the Envelope Theorem. The Envelope Theorem states that the derivative of a maximized objective with respect to a parameter—that is, with respect to any variable not maximized over—is equal to the partial derivative with respect to that parameter.) 4.3.2 A Model with Consumer Durables Consumer Durables as Another State Variable Consider a model with both nondurable consumption c and a stock of a durable consumer good d. The quantity of the durable good is another state variable, in addition to k. Let us put these two types of consumption in the context of a growth model. Thus, the social planner’s problem is max c,ik,id ∫ ∞ 0 e−ρtu(c, d) dt (4.7) subject to k̇ = ik − δkk ḋ = id − δdd f(k) = c + ik + id. (4.8) The augmented current-value Hamiltonian is 42 CHAPTER 4. MULTI-VARIABLE CONTROL H = u(c, d) + λd[id − δdd] + λk[ik − δkk] + µ[f(k)− c− id − ik]. The first order conditions are Hc = uc(c, d)− µ = 0 (4.9) Hid = λd − µ = 0 Hik = λk − µ = 0. (4.10) (Subscript represent partial derivatives on the utility function u but not oth- erwise.) There are two Euler equations, for λd and λk: λ̇d = ρλd −Hd = (ρ + δd)λd − ud(c, d) (4.11) λ̇k = ρλk −Hk = (ρ + δk)λk − f ′(k)µ. (4.12) The first-order conditions imply uc(c, d) = µ = λd = λk. Substituting λk for every occurence of λd and µ in the Euler equations, this means that [ρ + δk − f ′(k)]λk = λ̇k = [ρ + δd]λk − ud(c, d) (4.13) Equation (??) implies, in turn, that ud(c, d) = [f ′(k)− δk + δd]λk, (4.14) and in view of (4.9) that ud(c, d) uc(c, d) = f ′(k)− δk + δd. (4.15) In words, the ratio of the marginal utility of the durable to the marginal utility of nondurable consumption is proportional to the effective price ratio: the net marginal product of capital plus the depreciation rate for consumer durables. 4.3. CONSTRAINTS ON THE CONTROL VARIABLES 45 when ik = 0 but id > 0. When both levels of gross investment are zero, both λd and λk can be less than µ and uc and it is unclear which of λd or λk is smaller. Intuitively, if the marginal value of consumer durables is especially low, it can shut off investment in consumer durables entirely. Because investment must be nonnegative, investment in consumer durables may not be able to go low enough to allow equalization of this low marginal value of consumer durables with the marginal value of capital or the marginal utility of consump- tion. If the marginal value of investment in capital is especially low, it can shut off investment in capital entirely. 46 CHAPTER 4. MULTI-VARIABLE CONTROL Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 5 Characterizing Phase Diagrams in Fully Optimizing Concave Models with One State Variable 5.1 Introduction There is more structure to the phase diagrams that arise from dynamic op- timization than many economists realize. This chapter derives general char- acteristics of the phase diagram for fully optimizing models with a concave objective function and a concave accumulation function. In this chapter, we discuss models with one state variable, for which the phase diagram can be graphed in two dimensions. In the next chapter we extend certain key princi- ples laid out in this chapter to the case of many state variables, and point out which principles cannot be generalized to the many-state-variable case. For the sake of concreteness, we will continue to call the state variable k “capital” and the costate variable λ the “marginal value of capital.” Keep in mind that the following results, because they are purely mathematical, apply to any fully optimizing dynamic optimization model with one state variable and concave objective and accumulation functions. We will standardize by always putting the state variable k on the horizontal axis and the costate variable λ (rather than one of the control variables) on the vertical axis. Formally, consider the following general dynamic optimization problem: 50 CHAPTER 5. CHARACTERIZING PHASE DIAGRAMS: I see the λ̇ = 0 locus as separating the right from left and to see the k̇ = 0 separating above from below. Then the general rules can be stated simply as (1) to the right, λ is rising; (2) to the left, λ is falling; (3) up high, k is accumulating; (4) down low, k is decumulating. Schematically, Figure 1 gives the right idea.1 [Figure 1 has a shaded area in the shape of a Maltese cross in the middle to indicate a vaguely vertical λ̇ = 0 locus of indefinite slope, and a vaguely horizontal k̇ locus of indefinite slope. The four quadrants have arrows showing the dynamics in each quadrant.] 5.2.1 Proof that H?k is decreasing in k. H?k must be decreasing in k (when it is well-defined) if H ? is concave in k, as it will be if H is jointly concave in k and x. To see this consider that if x?0 maximizes H at (k0, λ) while x ? 1 maximizes H at (k1, λ), and H is jointly concave in k and x, then for any a ∈ [0, 1], H?(ak0 + (1− a)k1, λ) = max x H(ak0 + (1− a)k1, x, λ) (5.4) ≥ H(ak0 + (1− a)k1, ax?0 + (1− a)x?1, λ) ≥ aH(k0, x?0, λ) + (1− a)H(k1, x?1, λ) = aH?(k0, λ) + (1− a)H?(k1, λ). The first and last equalities in (5.4) follow from the definition of H?. The first inequality follows from the nature of maximization, with the convexity of the k−x region formed by X(k, z) guaranteeing that the indicated point is in the domain of H. The second inequality follows from the joint concavity of H in k and x. Intuitively, joint concavity of U and A in k and x guarantees that the marginal product of capital (the combined value of its contribution to utility and to accumulation) must fall as k increases, even with x adjusting optimally along the way. (The adjustment in the vector of control variables x makes the resulting fall in the marginal product of capital as small as possible, but cannot stop the decline.) In many applications, constant returns to scale in production will mean that A is only weakly concave, but strict concavity of U will typically induce strict concavity of H? in k. Any one of various types of increasing returns to scale in production may overturn the concavity of 1Figure 1 is appropriate for all three of the stable cases discussed below. But it is not appropriate for the unstable case discussed below, since Figure 1 assumes that the λ̇ = 0 locus is more nearly vertical than the k̇ = 0 locus. 5.3. DYNAMICS IN THE NEIGHBORHOOD OF THE STEADY STATE51 H?, leading to a qualitatively different phase diagram. (See for example Paul Romer’s first paper on endogenous growth, 19??.) 5.2.2 Proof that H?λ is increasing in λ. If x?0 maximizes the Hamiltonian H at (k, λ0), while x ? 1 maximizes the Hamil- tonian H at (k, λ1), then by definition, U(k, x?0) + λ0A(k, x ? 0) ≥ U(k, x?1) + λ0A(k, x?1) (5.5) and U(k, x?1) + λ1A(k, x ? 1) ≥ U(k, x?0) + λ1A(k, x?0). (5.6) Therefore, (λ1 − λ0)[H?λ(k, λ1)−H?λ(k, λ0)] = (λ1 − λ0)[A(k, x?1)−A(k, x?0)] = {U(k, x?1) + λ1A(k, x?1)− U(k, x?0)− λ1A(k, x?0)} +{U(k, x?0) + λ0A(k, x?0)− U(k, x?1)− λ0A(k, x?1)} ≥ 0. (The same result can be obtained by subtracting (5.5) from (5.6) and rear- ranging.) In words, H?λ moves in the same direction as λ. Intuitively, a higher marginal value of capital λ makes capital accumu- lation more valuable and therefore leads to more accumulation by means of adjustments in the control variables. Paul Milgrom and John Roberts, in “Comparing Equilibria” AER June 1994, 441–449, indicate the generality of this type of result. In particular, H?λ being increasing in λ in no way depends on concavity. 5.3 Dynamics in the Neighborhood of the Steady State To go further in our analysis, we need to zoom in on the neighborhood of the steady state (or of a steady state). Making a first order Taylor approximation of (5.2) and (5.3) around the steady state (k∗, λ∗) on the assumption that H? is twice differentiable, 2 2Note the distincition between a five-pointed star ? for maximization and a regular asterisk ∗ for a steady-state value. 52 CHAPTER 5. CHARACTERIZING PHASE DIAGRAMS: I k̇ ≈ 0 + H?λk(k∗, λ∗)[k − k∗] + H?λλ(k∗, λ∗)[λ− λ∗] λ̇ ≈ 0−H?kk(k∗, λ∗)[k − k∗] + (ρ−H?kλ(k∗, λ∗))[λ− λ∗]. Using a tilde (̃) to represent a small deviation from a steady-state value that can be treated as a differential for calculus, these equations can be rewritten in matrix form as   ˙̃ k ˙̃ λ   =   H?λk H ? λλ −H?kk ρ−H?kλ     k̃ λ̃   , (5.7) where all of the derivatives of H? are evaluated at (k∗, λ∗) and ˙̃ k = k̇ − ˙(k∗) = k̇ ˙̃ λ = λ̇− ˙(λ∗) = λ̇, both regarded as small. The key characteristics of the 2x2 dynamic matrix in (5.7) are (1) by Lemma 1 above, both off-diagonal elements (H?λλ and −H?kk) are positive; and (2) by the symmetric equality of mixed partial derivatives, H?λk = H ? kλ, making the trace (the sum of the elements on the main diagonal) equal to ρ. Using ϑ to denote an eigenvalue, the characteristic equation of a 2x2 matrix is ϑ2 − trace ϑ + det = 0, where det is the determinant. The determinant of the dynamic matrix is det = ρH?kλ − (H?λk)2 + H?kkH?λλ. If H?kλ is negative or ρ is small, the determinant will be negative, since both −(H?kλ)2 and H?kkH?λλ are negative. (See Lemma 1.) But if H?kλ is positive and ρ is large, the determinant may be positive. The sign of the determinant mat- ters because, as a consequence of the characteristic equation, the determinant is equal to the product of the two eigenvalues. The only way there can be one positive and one negative root for saddlepoint stability is if the determinant is negative. 5.4. A TYPOLOGY 55 necessary to take these induced movements into account. Using ddk and d dλ to denote full derivatives including this induced variation in x, we can calculate [ H?kk H ? kλ H?λk H ? λλ ] = [ Hkk Hkλ Hλk Hλλ ] (5.10) + [ Hkx Hλx ] [ dxdk dx dλ ] = [ Ukk + λAkk Ak Ak 0 ] + [ Ukx + λAkx (Ax)T ] [ dxdk dx dλ ] = [ Ukk + λAkk + (Ukx + λAkx)dxdk Ak + (Ukx + λAkx) dx dλ Ak + (Ax)T dxdk (Ax) T dx dλ ] , where (Ax)T is the transpose of Ax. Because the off-diagonal elements of the dynamic matrix are always positive, the qualitative distinctions all have to do with the sign and magnitude of H?kλ = H ? λk at the steady state. There are two equivalent ways to express this cross derivative, depending on the order of differentiation. In order to make the equivalence of the two expressions Ak + Ax dx dk = H?λk = H ? kλ = Ak + [Ukx + λAkx] dx dλ (5.11) less mysterious, consider the case in which the control variables are all at an interior solution, so that dxdk and dx dλ can be obtained by differentiating Hx = 0: Hxk + Hxx dx dk = 0, so that dx dk = −[Hxx]−1Hxk = −[Uxx + λAxx]−1[Uxk + λAxk], and Hxλ + Hxx dx dλ = 0, so that dx dλ = −[Hxx]−1Hxλ = −[Uxx + λAxx]−1Ax. is inconvenient for a general discussion. 56 CHAPTER 5. CHARACTERIZING PHASE DIAGRAMS: I Thus, using either expression for H?kλ, (plus the fact that a scalar is its own transpose), H?kλ = Ak − [Ukx + λAkx][Uxx + λAxx]−1Ax. (5.12) However, the symmetry equation (5.11) remains true even when one of the control variables is at a corner solution, as long as the constraint set X does not depend on k. 5.4.2 Thrift To aid in interpreting the sign and magnitude of H?kλ we will call the change in behavior indicated by dxdλ an increase in thrift. That is, increased thrift is the set of actions an agent uses in order to accumulate faster when λ, the marginal value of the stock to be accumulated, increases. Since by (5.10), (Ax)T dx dλ = H?λλ ≥ 0, an increase in thrift does in fact raise the rate of accumulation. In a particular model, the increased thrift induced by an increase in λ will be a particular pattern of change in behavior. Identifying the behavior associated with thrift and getting an intuitive grasp on the nature of thrift in any particular model is one of the keys to giving an intuitive interpretation of the dynamics of a model. Identifying thrift clearly may require rewriting the optimization problem to eliminate k from all but the accumulation function A and the objective function U . For example, in the Basic Real Business Cycle Model described in the Introduction (Chapter ??), greater thrift is a combination of more labor and less consumption. Eliminating k from all but the accumulation equation and the objective eliminates the control variables i and y. In the Cass model, greater thrift is just a matter of less consumption. In Mankiw’s model, greater thrift is a combination of reduced consumption of nondurables and reduced consumption of the services of durables. 5.4.3 Divide and Conquer Already, by recasting a model so that the constraint set does not depend on k, we have concentrated all dependence on k into the objective and accumulation functions. Now, in order to further the economic interpretation of the different types of phase diagram dynamics, we will further concentrate the dependence 5.4. A TYPOLOGY 57 on k into either the objective function alone or the accumulation function alone. Looking at cases in which either U or A is not a function of k is more general than it seems. The first-order condition Ux + λAx = 0 is valid in all of the directions in which the control variables can be freely varied, or more formally, if the differential dx is consistent with the constraint set, then [Ux + λAx]T dx = 0. Unless both terms are individually zero, [Ux]T dx = 0 and [Ax]T dx = 0, in all of the directions in which the control variables can be freely varied, it is always possible (at least in the neighborhood of the steady state) to eliminate k from either the objective function U or the accumulation function A, whichever is chosen.4 When k is only in the Objective Function (Ak ≡ 0). When Ak ≡ 0, by (5.11), H?kλ = Ak + (Ukx + λAkx) dx dλ = Ukx dx dλ . In words, when the dependence on k is concentrated into the objective function U , H?kλ is positive when k and thrift are complements in the objec- tive function, but negative when k and thrift are substitutes in the objective function. Since the dynamic matrix for this case looks like [ Ukx dx dλ + + ρ− Ukx dxdλ ] , the slope of the k̇ = 0 locus depends on whether k and thrift are complements or substitutes in U . The k̇ = 0 locus is downward-sloping if k and thrift are complements, and upward-sloping if k and thrift are substitutes. 4Since in the neighborhood of the steady state, x can be freely varied in this direction, any dependence of the constraint set on k introduced by this recasting of the model will be irrelevant for dynamics in the neighborhood of the steady state. If both [Ax] T dx and [Ux] T dx are zero in all of the directions dx in which the control variables can be freely varied, then by concavity, U and A are each separately maximized in those dimensions of x. Except when a constraint on the vector x in some other dimension is exactly at the critical point between binding and not binding (where the balance could be tipped by a change in λ) this means that a change in λ will lead to no change in the value of x needed to maximize H = U +λA, so that dx dλ = 0. (In other words, there is no appropriate way for the agent to alter actions for increased thrift.) That makes this a very special case. (See Exercise ?? for more details.) 60 CHAPTER 5. CHARACTERIZING PHASE DIAGRAMS: I 5.5.2 A Model with Unstable Dynamics A good example of a model with unstable dynamics (IIB) is a modified version of Mankiw’s Model. First, for simplicity, let the agent consume only durables. Second, allow for the possibility that k, instead of being a stock of machines that depreciates, is a biological stock with a positive natural rate of increase. The rate of increase can be enhanced by procedures that require labor (which is in inelastic supply) and which interfere with use as a consumer durable. Use of the stock by the government for its purposes actually uses up the stock. Formally, the model is max x ∫ ∞ 0 e−ρtu(k − x) dt s.t. k̇ = µk + f(x)− g. Both u and f are increasing and concave and µ > 0. (Note that if µ were negative, this would be a special case of Mankiw’s model, since we could set −δ = µ.) u′(0) = +∞ insures that we need not worry about the implicit constraint x ≤ k, but let us make the constraint x ≥ 0 explicit in case f ′(0) is finite. The current-value Hamiltonian is H = u(k − x) + λ[µk + f(x)− g]. Rewriting the model in terms of the part of the stock used for consumer durables d = k − x would allow one to eliminate x and concentrate all of the dependence on k into the accumulation function, demonstrating the thrift-k substitutability in the accumulation function that guarantees that the λ̇ = 0 locus is downward-sloping. (The recast Hamiltonian would be H = u(d) + λ[µk + f(k − d) − g].) But for the rest of the analysis, it is best to focus on the behavior of the part of the stock involved in enhanced production x. The first order condition for x can be written u′(k − x) = λf ′(x) when k > 0, with u′(k − x) ≥ λf ′(x) when x = 0. It is clear from the first order condition that an increase in λ leads to an increase in x: dxdλ ≥ 0. The Euler equation is λ̇ = (ρ− µ)λ− u′(k − x) = λ[ρ− µ− f ′(x)], 5.5. EXAMPLES OF THE DYNAMIC TYPES 61 where the last equality holds as long as x > 0. Exercise 9 gives an example showing that this model can have a steady state for at least some functional forms and parameter values. The cross-derivative of H? is easily calculated as H?kλ = Ak + (Ukx + λAkx) dx dλ = µ− u′′(k − x)dx dλ > 0. Thus, the k̇ = 0 locus is also downward-sloping, putting the dynamics squarely in type II. The remaining issue is whether the dynamics are stable or unstable. The dynamics will be unstable if and only if the k̇ = 0 locus is steeper than the λ̇ = 0 locus. In the stable case, with the λ̇ = 0 locus steeper than the k̇ = 0 locus, points on the k̇ = 0 locus to the right of the steady state will have λ̇ > 0. However, in the unstable case, with both loci downward sloping, the k̇ = 0 locus to the right of the steady state is to the left of the λ̇ = 0 locus and so has λ̇ < 0. Which case are we dealing with here? In order to hold k̇ = µk + f(x)− g constant as k increases, x must fall and λ̇ λ = ρ− µ− f ′(x) must also fall. Since λ̇ = 0 at the steady state, in that neighborhood a fall in λ̇ λ requires a fall in λ̇ itself. Thus, the dynamics are unstable. (See Figure 3.) Interpreting Unstable Dynamics With unstable dynamics, it is clear that one will not end up at the steady state if one does not have the steady state capital stock to begin with. To make sense of the model’s dynamics, it is best to think of the corresponding finite horizon problem with end-time T and then take the limit as T → ∞. With a finite end-time T , the transversality condition λ(T )k(T ) = 0 must be satisfied. Looking at the dynamic paths shown in Figure 3, if initially, k > k∗ the way to satisfy this condition when T is large is have λ start very slightly below the downward-sloping explosive path, follow along very close to the downward-sloping explosive path for a long time, then veer off down to the λ = 0 axis near T . Thus, as T → ∞, the appropriate dynamics are to 62 CHAPTER 5. CHARACTERIZING PHASE DIAGRAMS: I follow the downward-sloping explosive path (which is also the slower explosive path) more and more closely. In some ways this downward-sloping explosive path acts like a saddle-path even though it is not a saddle path. If, initially, k < k∗, as T → ∞ it may be impossible to stay away from both k = 0 and λ = 0 for more than a finite length of time. If the axes are far away, it will still be appropriate to follow the downward-sloping saddle-path for some time before veering down toward λ = 0.5 Exercises 1. Using the proof of Lemma 1 as a guide, prove that maximization of the Hamiltonian U(k, x) + λA(k, x) implies that U(k, x) falls with λ (given k). 2. If you had a model in which more of the state variable was always unde- sirable, how could you recast the model in order to fit it into the frame- work for characterizing phase diagrams in this chapter? What does this mean for the original way of looking at the model? 3. Draw the three borderline cases for dynamics: (I/II), (II/III) and the borderline stable case. What are the dynamics like in the borderline stable case? 4. The slope of the saddle-path and of the explosive path. (a) Verify that the explosive path is upward sloping for each case by graphing the intersection of each “quadrant” divided up by the k̇ = 0 and λ̇ = 0 loci with the right-angled area made by the arrow- angle for that quadrant with the vertex put at the steady state. Explain why the explosive path must lie in this region. Show that in each of the stable cases, every point in this region is above and to the right or below and to the left of the steady state, implying that the explosive path is upward sloping. (b) Verify that the saddle-path must be downward sloping for each stable case by graphing the intersection of each quadrant with the right-angled area made by the opposite of the arrow angle for that 5One can conjecture that when T is large enough, the optimal thing will be to have λ jump to the path that leads to the origin and follow that path, as the way to spend the most possible time before reaching λ = 0. Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 6 More on Characterizing Phase Diagrams 6.1 N State Variables Consider the general dynamic optimization problem discussed in the previous chapter, max x ∫ ∞ 0 e−ρtU(k, x; z)dt, s.t. k̇ = A(k, x; z) (6.1) and x ∈ X(k, z), but with k and A interpreted as vectors. Again, assume that the objective function U and the accumulation function A are (twice-differentiable and) concave and that each component of k has nonnegative value. The current- value Hamiltonian is H(k, λ, x) = U(k, x) + λT A(k, x), where λ is the vector of costate variables, the marginal values for each compo- nent of k, and λT is its tranpose. The term λT A(k, x) can be thought of also 66 CHAPTER 6. CHARACTERIZING PHASE DIAGRAMS: II as the vector dot product λ ·A(k, x). Defining the maximized Hamiltonian H? as before by H?(k, λ) = max x H(k, x, λ), the dynamics of the system are described by the pair of vector equations k̇ = Hλ(k, x, λ) = H?λ(k, λ) (6.2) λ̇ = ρλ−Hk(k, x, λ) = ρλ−H?k(k, λ). (6.3) 6.1.1 The Saddle Shape of H? The dependence of H? on k and λ can be characterized as follows: Lemma 2 H?(k, λ) is concave in k as long as H(k, x, λ) is jointly concave in k and x; H?(k, λ) is convex in λ even without this assumption. This result is akin to the concavity of a firm’s profit function in the vec- tor of quasi-fixed factors and convexity in factor and output prices. In this comparison the vector k corresponds to the quasi-fixed factors, the vector λ corresponds to the prices and the vector x corresponds to the variable factors. Proof. The proofs of the one-state variable lemma (1) can be followed line by line to prove the multi-state-variable version of the lemma (as you are asked to do in Exercise ??). This approach directly establishes the concavity of H? in k. For λ one obtains the result [λ1 − λ0]T [H?λ(k, λ1)−H?λ(k, λ0)] = [λ1 − λ0]T [A(k, λ1)−A(k, λ0)] ≥ 0, for any pair of costate vectors λ0 and λ1, which is equivalent to convexity of H? in λ. 6.1. N STATE VARIABLES 67 Interpretation By the same token, the concavity of H? in k is equivalent to [k1 − k0]T [H?k(k1, λ)−H?k(k0, λ)] ≤ 0. Thus, one can say that a change in the costate vector λ changes k̇ in a direction with a positive dot product to the change in λ, while a change in the vector k changes λ̇ in a direction with a positive dot product to the change in k. In particular, this means that an increase in one scalar component of λ by itself will raise the rate of increase of the corresponding state variable, while an increase in any scalar component of k by itself will raise the rate of increase in the corresponding costate variable. In a two-dimensional slice of the 2N-dimensional phase diagram with the component state variable ki on the horizontal axis and the corresponding com- ponent costate variable λi on the vertical axis, rules like those stated in the previous chapter apply: above the k̇i = 0 locus, k̇i > 0 while k̇i < 0 below; to the right of the λ̇i = 0 locus, λ̇i > 0 while λ̇i < 0 to the left. Moreover, the same can be said for any linear combination of component state variables and the same linear combination of component costate variables. 6.1.2 The Dynamic Matrix The dynamics in the neighborhood of the steady state for the N-state-variable case are given by [ ˙̃ k ˙̃ λ ] = [ H?λk H ? λλ −H?kk ρI −H?kλ ] [ k̃ λ̃ ] . (6.4) Hkλ = (Hλk)T . The key characteristics of the 2Nx2N dynamic matrix in (6.4) are (1) by Lemma 2 above, both off-diagonal elements (H?λλ and −H?kk) are positive semidefinite; and (2) the following lemma: Lemma 3 All of the eigenvalues of the dynamic matrix come in pairs adding up to ρ. Lemma 3 has at least four important consequences. 70 CHAPTER 6. CHARACTERIZING PHASE DIAGRAMS: II of the objective function does not converge, and the dynamics in this case are identical to the limit as ρ → 0 from above. To understand the role of the discount rate ρ in the model, it is important to see that ρ affects only the dynamics. Given k and λ, the solution to H?(k, λ) = max x U(k, x) + λA(k, x) is the same regardless of the value of ρ. The optimizing value of x, and the values of H?, U , and A are all fixed by a knowledge of k and λ alone, without any reference to ρ. Since k̇ = A, this means that the k̇ = 0 locus is unaffected by changes in ρ. On the other hand, since λ̇ = ρλ−H?k(k, λ), an increase in ρ raises λ̇ at a given point (k, λ) on the phase diagram. Points that were on the old λ̇ = 0 locus will have λ̇ > 0; the new λ̇ = 0 locus will be to the left of the old one, since a reduction in k lowers λ̇. As long as the dynamics are stable, the leftward shift in the λ̇ = 0 accom- panying an increase in ρ always reduces the steady state capital stock k∗. The effect of the increase in ρ on λ∗ depends on the slope of the unmoving k̇ = 0 locus. If the dynamics are unstable, the leftward shift of the λ̇ = 0 locus accom- panying an increase in ρ raises the steady-state capital stock k∗. This is more intuitive than it may seem at first. In the unstable case, greater impatience as measured by ρ raises the critical level of capital at which the economy can “take off” on a rightward path of increasing k—making such a take-off more difficult and making a gradual collapse of the economy on a leftward path of shrinking k more likely. 6.2.2 The Isograms of the Hamiltonian “Isogram” is the general word for any curve on which some quantity is con- stant. Isoquants, indifference curves, and the k̇ = 0 and λ̇ = 0 loci are all examples of isograms. There is a remarkable connection between the dynam- ics of a dynamic control model with ρ = 0 and the isograms of the Hamiltonian, or iso-Hamiltonian curves. In general, k̇ = H?λ(k, λ) and 6.2. MORE ON THE ONE-STATE-VARIABLE CASE 71 λ̇ = ρλ−H?k(k, λ) imply that Ḣ = Ḣ? (6.5) = H?k(k, λ)k̇ + H ? λ(k, λ)λ̇ = H?k(k, λ)H ? λ(k, λ) + H ? λ(k, λ)[ρλ−H?k(k, λ)] = ρλH?λ(k, λ) = ρλk̇. When ρ = 0, Ḣ = 0—the Hamiltonian is constant. In other words, the dynamic paths (“flow lines”) followed by the model when ρ = 0 are the iso- grams for the Hamiltonian (the “iso-Hamiltonian curves”). As iso-Hamiltonian curves, these curves are unchanged as curves in k − λ space if ρ is increased to a positive number, although the dynamics change with ρ so that the iso- Hamiltonian curves are then no longer flow lines. When ρ > 0, the value of the Hamiltonian rises and falls with k when following a flow line. On the saddle path this means that H is rising when going down (and right) along the saddle path. H is falling when going up (and left) along the saddle path. Other flow lines either have H continually increasing or continually decreasing along with k or have a maximum or minimum value of H where the flow line crosses the k̇ = 0 locus. There is one more intriguing fact about H?. Since H?kk ≤ 0 and H?k = ρλ− λ̇, the λ̇ = 0 locus for ρ = 0 is the solution to max k H?(k, λ) for any given λ. Similarly, since H?λλ ≥ 0 and H?λ = k̇, the k̇ = 0 locus is the solution to min λ H?(k, λ) for any given k. Thus, the steady state when ρ = 0 can be viewed as the equilibrium in a zero-sum game with payoff H? with one player controlling k and the other player controlling λ. 72 CHAPTER 6. CHARACTERIZING PHASE DIAGRAMS: II 6.2.3 The Isograms of the Objective Function U It is often useful to know the behavior of U . We can state several quite general results. The Saddle Path and U First, going down along the saddle path (to the right), the Hamiltonian U +λA is increasing, since with ρ > 0, the Hamiltonian moves in the same direction as k along any flow line, including the saddle path. But going down the saddle path (to the right), λA is decreasing both to the left of the steady state and for at least some distance to the right of the steady state (since λA must decline from 0 to something negative on the saddle path immediately to the right of the steady state). Therefore, in the neighborhood of the steady state, the objective function U must be increasing when going down (to the right) along the saddle path to allow the Hamiltonian to be increasing, and the objective function U is decreasing when going up (to the left) along the saddle path in the neighborhood of the steady state. λ and U Second, if we define the stripped Hamiltonian G and its maximized counterpart G? by G(k, x, `) = G(k, x, 1 λ ) = U(k, x) λ + A(k, x) = `U(k, x) + A(k, x) (` = 1λ) and G?(k, `) = max x `U(k, x) + A(k, x), then the same value of x maximizes the stripped Hamiltonian G as maxi- mizes the regular Hamiltonian H. Writing the value of the objective function associated with maximizing H and G as U?(k, λ), that is U?(k, λ) = U(k, x?(k, λ)), the envelope theorem tells us that U?(k, λ) = G?`(k, `) = G ? ` ( k, 1 λ ) . Thus, G?` tell us the value of the objective function at each point on the phase diagram. The set of curves on which U is constant (“iso-U curves”) is identical to the set of curves on which G?`(k, 1 λ) is constant. 6.2. MORE ON THE ONE-STATE-VARIABLE CASE 75 The Road to Oblivion City and U In dynamic type I, in which the λ̇ = 0 locus is upward sloping, the iso-U curve must be downward-sloping by the third principle, but slope down less than the k̇ = 0 locus by the fourth principle. This sandwiching of the iso-U curve through the steady state between a horizontal line and the downward-sloping k̇ = 0 locus pins things down pretty well. What of those cases (II and III) in which the λ̇ = 0 locus is downward- sloping? The iso-U curve through the steady state must be upward-sloping and steeper than the k̇ = 0 curve (a fact that bites only in dynamic type III in which the k̇ = 0 locus slopes upward). But is there any limit to how steep the upward slope of the iso-U curve through the steady state can be? Indeed there is. The fifth principle about iso-U curves is that, in the stable case, starting from the steady state, the iso-U curve must be flatter or slope up less than the explosive eigenvector. This sandwiches the iso-U curve through the steady state either between a horizontal line and the explosive path or an upward- sloping k̇ = 0 locus and the explosive path. To begin with, the explosive path is a flow line. By (6.5), along a flow line, Ḣ = U̇ + λ̇A + λȦ = ρλk̇. Therefore, using the fact that k̇ = A, U̇ = (ρλ− λ̇)A− λȦ. Considering the stable case first, and writing −κ for the negative eigenvalue, the positive eigenvalue must be ρ + κ since the sum of the eigenvalues equals the trace ρ. Therefore, on the explosive path in the neighborhood of the steady state, Ȧ = (ρ + κ)A, so that U̇ = −(κλ + λ̇)A. Since λ̇ is small in the neighborhood of the steady state, U̇ has the same sign as −κA. In the stable case, this means that U declines when the explosive path is followed to the right, while U rises when the explosive path is followed to the left. In either direction, U is negatively related to k when going along the explosive path. The iso-U curve must be flatter so that the fall in U one would get by going along the explosive path to the right is counteracted by a reduction in λ and corresponding shot in the arm for U . 76 CHAPTER 6. CHARACTERIZING PHASE DIAGRAMS: II In the unstable case, we can use the smaller positive eigenvalue −κ and the equations above will still be valid. But in this case, U rises when following the explosive path to the right. Therefore, in the unstable case, the iso-U curve through the steady state must be steeper than the explosive path. Exercises 1. Spell out in detail the proof of Lemma 2, following the pattern of the proof of Lemma 1. 2. Show that if [ υk υλ ] is a right eigenvector of the dynamic matrix with eigenvalue ϑ, then [ υTλ −υTk ] is a left eigenvector of the dynamic matrix with eigenvalue ρ− ϑ. Real Business Cycle Theory: A Semiparametric Approach, by Miles S. Kimball Copyright c© 2002 Please do not quote or reproduce without authorization September 7, 2004 at 16:22 Chapter 7 Why Steady State Growth Requires the King-Plosser-Rebelo Form of the Utility Function What we have done so far makes it clear that the King-Plosser-Rebelo utility function works well with steady-state growth. In this section, we would like to give a rough argument for why the King-Plosser-Rebelo form of the util- ity function is necessary—why there isn’t a more general form of the utility function that would do the trick.1 7.1 Labor, Consumption and the Real Wage Over the course of the last century or two, real wages and per capita consump- tion have risen by at least an order of magnitude (that is, by a factor of 10) [some data would be good here]. Yet the average workweek N has changed little in comparison with the dramatic increases in real wages and in con- sumption. Also, the ratio of labor income to consumption WNC has remained relatively constant despite those dramatic changes. We need to ask what kind of utility function can reproduce these stylized facts.2 1A more formal proof can be found in King, Plosser and Rebelo’s separate appendix. 2If the per capita quantity of labor N did have a secular trend, we could not use steady- state analysis at all. N is bounded below by zero and bounded above by the amount of time physically available, and so cannot have an unending trend. In a sense, any drift in N would 80 CHAPTER 7. APPENDIX: WHY KING-PLOSSER-REBELO UTILITY and the proportional growth rate of consumption ċc can both be constant. Given u(c, n) = U(ln(c)− v(n)), the marginal utility of consumption λ is λ = 1 c U ′(ln(c)− v(n)). (7.3) In words, this equation says among other things that, moving along an in- difference curve—which means holding ln(c) − v(n) constant—the marginal utility of consumption is inversely proportional to consumption c. Graphi- cally, this is clear from the fact that the vertical distance ∆c between any two indifference curves is proportional to c, so that with ∆u between the two indifference curves is constant (by definition), ∆u∆c is proportional to 1 c . Since ṅ = 0 along a steady state growth path, taking logarithms of both sides of (7.3) and differentiating with respect to time yields the following relationship: λ̇ λ = d dt ln(λ) = d dt [ln(U ′(ln(c)− v(n)))− ln(c)] = [ U ′′(ln(c)− v(n)) U ′(ln(c)− v(n)) − 1 ] ċ c . Therefore, r = ρ + [ 1− U ′′(ln(c)− v(n)) U ′(ln(c)− v(n)) ] ċ c (7.4) along a steady state growth path on which ṅ = 0. Given (7.4), the only way both the proportional growth rate of consump- tion ċc and the real interest rate r can be constant along this path is if U ′′(x) U ′(x) is a constant. Let us call that constant 1 − β, so that the coefficient of ċc in (7.4) is equal to β. Then solving the implied differential equation, U(·) must be of the form U(x) = e(1−β)x 1− β , 7.2. CONSUMPTION GROWTH AND THE REAL INTEREST RATE 81 or U(x) = x if β = 1 so that U ′′(x) U ′(x) = 0. (The two additional arbitrary constants that arise in solving the differential equation transform the utility function linearly and so do not affect the implied preferences.) Finally, substituting in x = ln(c)− v(n), we must have u(c, n) = U(ln(c)− v(n)) = c 1−β 1− β e (β−1)v(n) or u(c, n) = ln(c)− v(n) if β = 1. This is the King-Plosser-Rebelo utility function. 82 CHAPTER 7. APPENDIX: WHY KING-PLOSSER-REBELO UTILITY 8.2. CONSTRUCTING THE PHASE DIAGRAM 85 to construct a two-dimensional phase diagram, it is necessary to choose which two variables to represent on the axes. For the Ramsey model itself, there are two equally good choices—putting k and c or putting k and λ on the axes. (In the Ramsey model, the qualitative effect of switching between k−λ space and k− c space is just to flip everything in the graph upside down.) The great advantage of the choice of k and λ is that it generalizes to other models in a way that the choice of k and c does not. State variables and the associated costate variables such as k and λ are intrinsic to the structure of any dynamic optimization problem, while the centrality of consumption as a preeminent control variable is peculiar to the Ramsey model. Being able to compare and contrast different models is crucial to developing the economic intuition. Thus, whenever possible, we will follow a standard convention of putting the state variable (k) on the horizontal axis and the costate variable (λ) on the vertical axis of phase diagrams throughout this book. The key to the dynamics of the phase diagram are the locus of points at which k̇ = 0 and the locus of points at which λ̇ = 0. These are sometimes called the k-isocline and the λ-isocline. The intersection of these two curves is the steady state. 8.2.1 The λ̇ = 0 Locus Setting λ̇ to zero in (8.4) yields the equation for the λ̇ = 0 locus (the λ-isocline): f ′(k) = ρ + δ. (8.6) Since ρ and δ are constants, and f ′(k) is decreasing in k, this equation determines a particular value of k. In particular, if (for use throughout the book) we define K = f ′−1(ρ + δ), (8.7) then the λ̇ = 0 locus is given by k = K. This is a vertical line on the phase diagram, as shown in Figure 8.2.1. Because f ′(·) is decreasing, (8.4) implies that to the right of this isocline (k > K), f ′(k) < f ′(K) = ρ + δ, so that λ̇ > 0. To the left of this isocline (k < K), λ̇ < 0. 86 CHAPTER 8. THE RAMSEY MODEL VIA PHASE DIAGRAMS Indeed, Chapter 5 demonstrates the general result that in any optimal control problem with one endogenous state variable and concave objective and accumulation functions (thereby satisfying the key sufficiency conditions of Chapter 2), λ̇ > 0 to the right of the λ̇ = 0 locus and λ̇ < 0 to the left of the λ̇ = 0 locus. Being able to state general principles like this is an important advantage of standardizing the phase diagram by putting the costate variable on the vertical axis and the state variable on the horizontal axis. 8.2.2 The k̇ = 0 Locus Setting k̇ to zero in (8.2) yields the equation f(k)− δk − g = c. By (8.3), λ is a monotonically decreasing function of c, which makes c a monotonically decreasing function of λ: c = u′−1(λ). The k̇ = 0 locus can be written as f(k)− δk − g = u′−1(λ) (8.8) and is shaped as shown in Figure 8.2.2. For comparison, Figure cass c k isocline shows what the k̇ = 0 isocline would look like in k − c space. As k increases, the left-hand-side of (8.2.2)—which gives consumption when k̇ = 0— increases as long as f ′(k) > δ—which is always true to the left of the λ̇ = 0 locus where f ′(k) = ρ + δ and for some distance to the right of the λ̇ = 0 locus. Beyond the point where f ′(k) = ρ, a higher level of capital would reduce the level of consumption when k̇ = 0 in a reflection of dynamic inefficiency. Consumption and the marginal utility of consumption λ are inversely related. Therefore, in (8.2.2), the k-isocline is downward sloping to the left of the λ- isocline and for some distance to the right of it, eventually becoming upward sloping. As indicated by (8.5), at higher values of λ and lower values of c than the k̇ = 0 locus, net investment will be positive, making k̇ > 0. At lower values of λ and higher values of c, net investment will be negative, making k̇ < 0. Thus, in the standardized phase diagram with λ on the vertical axis and k on 8.2. CONSTRUCTING THE PHASE DIAGRAM 87 the horizontal axis, k̇ > 0 above the k̇ = 0 locus and k̇ < 0 below the k̇ = 0 locus. Indeed, Chapter 5 demonstrates the general result that in any optimal control problem with one endogenous state variable and concave objective and accumulation functions that k̇ > 0 above the k̇ = 0 locus and k̇ < 0 below the k̇ = 0 locus. Standardizing the phase diagram by putting the costate variable on the vertical axis and the state variable on the horizontal axis makes this common thread stand out. 8.2.3 The Saddle Path Combining the dynamics for λ shown by (8.2.1) and the dynamics for k shown by (8.2.2) yields Figure 8.2.3. Note that the dynamic paths are precisely horizontal when crossing the λ̇ = 0 locus, and are precisely vertical when crossing the k̇ = 0 locus. The intersection of the k̇ = 0 locus and the λ̇ = 0 locus is the steady state. We will denote steady-state values of all variables by asterisks. Thus, on the phase diagram, the steady state is the point (k∗, λ∗). At the steady state there is no tendency to move toward anywhere else. Most of the dynamic paths shown lead off to infinity or off the edges of the graph. The exception is the pair of dynamic paths shown in bold, which lead in to the steady state. These are the saddle paths. Often, both together are referred to in the singular as “the saddle path.” The importance of the saddle path arises from the fact that while the initial value of k is given by history, the initial value of λ is not. While k is pointed toward the past, λ is pointed toward the future. The initial value of λ is determined by the need to have a future that is (1) consistent with rational expectations, (2) consistent with the transversality condition and (3) otherwise sensible. In practice, the paths that go off to infinity or off the edges of the graph are all eliminated by one of these three criteria.1 Therefore, the focus is on paths that lead eventually to a steady state. When no further movements of exogenous shifters can be foreseen, that leads to a focus on the saddle path. Given an initial value of k by history, the initial value of λ is given by the saddle path. Based on the saddle-path, the prediction of the model is that if the capital stock k begins below its long-run steady-state value, the marginal value of 1See exercises .... 90 CHAPTER 8. THE RAMSEY MODEL VIA PHASE DIAGRAMS capital λ falls. In view of the results above, the gradual increase in the capital stock leads to a gradual rise in output y and the real wage w and a gradual fall in the real rental rate R and the real interest rate r. The gradual fall in the marginal value of capital leads to a gradual rise in consumption c. By the nature of the saddle path, k̇ gradually falls from its positive value toward zero, while λ̇ gradually rises from its negative value toward zero (at least in the neighborhood of the steady state). Finally, since investment i is positively related to λ, investment will grad- ually fall if the saddle path cuts the investment isograms from above, but investment will gradually rise if the saddle path cuts the investment isograms from below. Either case is possible. If δ = 0, then gross and net investment are the same, and the investment isograms parallel the k̇ = 0 locus. Since the saddle path is steeper than the k̇ = 0 locus, in this case the saddle path cuts the investment isograms from above (at least in the neighborhood of the steady state). If, on the other hand, δ becomes large enough, the invest- ment isograms will become close to vertical and the saddle path will cut the investment isograms from below.3 If the initial capital stock is greater than k∗, the gradual movement back up the saddle causes the opposite pattern for all of these variables. 8.5 Shocks to the Rate of Time Preference and to Government Purchases In the absence of any changes in the exogenous variables, convergence along the saddle path is the only story to be told. It is time now to look at such exogenous changes. We will look first at shocks to the rate of time preference ρ and then at shocks to the level of government purchases g. For now, in studying the effects of shocks, we will make the certainty- equivalence approximation: looking at the decisions agents would make if all uncertainty vanished and they were certain to face the expected values of fu- ture variables. In other words, using the certainty-equivalence approximation, one proceeds as if all the agents in a model had perfect foresight. As we proceed to study “perfect-foresight” models, it is important to keep sight of their purpose of providing a certainty-equivalence approximation to stochastic models.4 3This is more difficult to show. Indeed, it is easiest to show using techniques from later on in the book. 4See Chapter ?? for a more detailed discussion of certainty equivalence and the certainty- 8.5. SHOCKS TO THE RATE OF TIME PREFERENCE AND TO GOVERNMENT PURCHASES 91 When looking at the effects of a particular shock, think of it as a realization from a distribution of possible shocks that are equally likely to be positive or negative. Given the certainty-equivalence approximation, the dynamic effects or “impulse response functions” predicted by the perfect foresight model yield the impulse response functions for the stochastic model as well. As shocks repeatedly buffet the economy, the impulse responses from the various shocks all add on top of one other, algebraically. Thus, the dynamics of perfect foresight models are the key to understanding the corresponding stochastic models. When thinking about shocks, it is best to think of shocks as packets of genuine news—of genuinely new information about the future. When using the certainty-equivalence approximation, that new information takes the form of a change in the future expected time path of one or more exogenous variables (and maybe a change in the current value of one or more of the exogenous variables). Sometimes, the expected time path for exogenous variables will be unchanged for some time into the future. In this case, there is a difference between when the shock is unsheathed and when it actually cuts, between when it is unveiled and when it actually hits, between when it is revealed and when it actually materializes. As we will see, a shock begins affecting the economy as soon as it is revealed. An economy with forward-looking agents does not wait until the shock actually materializes to respond. Moreover, even when some change in an exogenous variable happens immediately, what is expected to happen thereafter has a very important influence on the effect of a shock. 8.5.1 Shocks to the Rate of Time Preference (ρ) One broad category of shocks is the category of “preference shocks.” In the Ramsey model, the simplest example of a preference shock that affects behav- ior is a change to ρ (the “degree of impatience,” “utility discount rate,” or “rate of time preference”). Note that a simple multiplication of the felicity equivalence approximation. Chapter ?? investigates how good the certainty equivalence approximation is. One simple generalization is that when only aggregate, economy-wide uncertainty is at issue, the certainty-equivalence approximation is typically quite a good approximation. The certainty-equivalence approximation is often not a very good approxi- mation when the idiosyncratic risk faced by heterogeneous households and firms is at issue. (The law of averages helps to make aggregate uncertainties much smaller as a percentage of mean values than the risks faced by individual households and firms.) In any case, the certainty-equivalence approximation is the foundation on which higher-order approximations will be built, so it is the right place to start. 92 CHAPTER 8. THE RAMSEY MODEL VIA PHASE DIAGRAMS function at every instant by the same positive constant makes no difference at all to optimal choices. For choices to be affected, the balance between felicity at two different dates must be altered. An increase in ρ alters that balance by reducing the importance of future felicity in comparison to present felicity. Just as the k- and λ-isoclines are the key to understanding the dynamics of the model in the absence of shocks, shifts in the k- and λ- isoclines are the key to understanding the response of the model to shocks. Equation (8.5) indicates that k̇ is not affected by ρ. In particular, the k̇ = 0 locus is unaffected. But equation (8.4) indicates that , given k and λ (that is, given a specific location on the phase diagram), the increase in ρ causes an increase in λ̇. In particular, a point on the old λ̇ = 0 locus must have λ̇ > 0. Therefore, the old λ̇ = 0 locus must be to the right of the new λ̇ = 0 locus; in other words, an increase in ρ shifts the λ̇ = 0 locus to the left. (The leftward shift in the λ̇ = 0 locus can also be seen from (8.6). An increase in ρ requires the λ̇ = 0 locus to have a higher marginal product of capital. Capital per worker must be smaller in order to get this higher marginal product of capital.) To begin with, consider an immediate permanent increase in impatience ρ. In a dynamic economic model, any shock has at least two aspects: (a) the arrival of the information that the shock will happen and (b) the alter- ations in the paths of exogenous variables indicated by that information.5 For convenience, we will call the moment at which the information arrives time zero. In the long run, as shown in Figure 8.5.1, a permanent increase in ρ leads to a fall in the steady-state capital stock k∗ and an increase in the steady- state marginal value of capital λ∗. However, at time zero, of k and λ, only the marginal value of capital λ—which is a forward-looking expectation— can jump. The capital stock k—which is a historical variable—cannot jump. Backward induction is the key to deducing what should happen. Since the 5When there is a delay between the arrival of the information and the indicated alteration in the paths of exogenous variables, it is called an anticipated shock. When there is no such delay, it is called an immediateshock. (This will be our default case, whenever we do not explicitly state that a shock is anticipated.) When the alterations in the paths of exogenous variables are expected to eventually die away to nothing, it is called a temporary shock. If there is any alteration that does not ultimately die away, it is called a permanent shock. When we talk about temporary and permanent shocks without further modification, we will be focusing on the simple case when the alterations in the exogenous variables can be described by simple step functions that move from zero to some other value, remaining constant at that value—and then in the case of temporary shocks, eventually going back to zero. 8.5. SHOCKS TO THE RATE OF TIME PREFERENCE AND TO GOVERNMENT PURCHASES 95 (For clarity, we have also shown the dependence of c, g and k on time explic- itly.) A necessary condition for solving social planner’s problem (8.11) is that for whatever path of extraneous government purchases g that is chosen, the social planner be solving the subproblem max c ∫ ∞ 0 e−ρtu(ct)dt (8.13) s.t. k̇t = f(kt)− δkt − ct − gt. (8.14) Why is this necessary? If ∫∞ 0 e −ρtu(ct)dt can be increased without changing the path of g, then ∫ ∞ 0 e−ρtu(ct)dt + ∫ 0 e−ρtphit(gt) must also be increased since the second term ∫ 0 e −ρtφt(gt) has been left un- changed. Thus, anything that increases ∫∞ 0 e −ρtu(ct)dt without changing the path of g also moves the economy closer to a solution of (8.11). This straight- forward result means that optimization of (8.11) implies optimization of the simpler problem (8.1) or (8.13)we have been emphasizing, conditional on some path of extraneous government purchases g. One can learn something about optimization conditional on a path of ex- traneous government purchases by treating the path of government purchases as if it were exogenous, then optimizing. That is the approach we will take here. The actual paths of g that we will condition on will be most relevant if at any point in time φt is tightly curved as a function of g, indicating a strong necessity for a certain level of purchases, after which more is not so important. This “necessary” level of government purchases can vary over time because of the dependence of phit on time. As with changes in ρ, the key to understanding the response of the model to changes in extraneous government purchases g is to understand the effects of g on the k- and λ-isoclines and on the rest of the economic geography of the phase diagram. All of these effects on the economic geography of the phase diagram can be summarized and compared for both ρ and g by the following direct effect table: each entry in the column for g shows the direct effect of g on that variable for given values of k and λ. The first column shows each variable as a function of k, λ and the exogenous g and ρ. 96 CHAPTER 8. THE RAMSEY MODEL VIA PHASE DIAGRAMS Variable Function Direct Effect of ρ Direct Effect of g c : u′−1(λ) 0 0 y : f(k) 0 0 i : f(k)− u′−1(λ) 0 − w : f(k)− kf ′(k) 0 0 R : f ′(k) 0 0 r : f ′(k)− δ 0 0 k̇ : f(k)− δk − u′−1(λ) 0 − λ̇ : ρ + δ − f ′(k) + 0 Holding k and λ fixed—in other words, staring at a given point on the phase diagram—an increase in extraneous government purchases g has no effect on consumption, output, the real wage, the real rental rate or the real interest rate. The direct effect of extraneous government purchases is limited to a reduction in investment i, with a consequent reduction in capital accumulation k̇. A Permanent Increase in Government Purchases The direct effects on k̇ and λ̇ indicate the way the k- and λ-isoclines shift. In the case of an increase in extraneous government purchases g, capital accu- mulation k̇ is reduced at a given value of k and λ. A point on the old k̇ = 0 locus will have k̇ < 0, and so must be below the new k̇ = 0 locus. Therefore, the k̇ = 0 locus must shift up. Since λ̇ is unchanged at a given value of k and λ, there is no shift of the λ̇ = 0 locus. As shown by Figure 8.5.2, a permanent increase in extraneous government purchases g—financed by lump-sum taxes—leads to a new steady state that is due north of the old steady state. Since no capital accumulation is required, the initial jump in λ upon receipt of the new information can take the economy instantly to the new steady state. With no saddle-path effects, the impact effects and overall steady state effects of a permanent increase in g are identical. 8.5. SHOCKS TO THE RATE OF TIME PREFERENCE AND TO GOVERNMENT PURCHASES 97 Variable Impact Effect Saddle Path Effect Overall Steady State Effect c − 0 − y 0 0 0 i 0 0 0 w 0 0 0 R 0 0 0 r 0 0 0 k̇ 0 0 0 λ̇ 0 0 0 (8.15) It is easy to summarize this table: in the Ramsey model, a permanent increase in government purchases immediately crowds out consumption. It has no other effect on any key variable. Investment i is affected by both λ and directly by g, but these effects must cancel out since i∗ = δk∗ in both steady states, with the same value of k∗. A Temporary Increase in Government Purchases If extraneous government purchases g increase by a certain amount, and ev- eryone knows that they will return to the old value at time T , the isoclines and dynamic arrows on the phase diagram shift for an interim period (0, T ), then return to normal when g returns to its previous value. It is important to realize that, since it is foreseen in advance, the shift back at time T is a dif- ferent kind of shift than the shift at time zero. The marginal value of capital λ depends on expectations of the future. It can jump when those expecta- tions are changed by new information, but not when events are playing out as expected. With g returning to normal, the economy should eventually return to the old steady state. Backward induction is the easiest way to deduce the dynamic general equilibrium that will eventually return the economy to the old steady state. From time T on, the economy follows the old dynamics, and the only way to get back to the once and future steady state will be along the old saddle path. Although λ can jump at time 0 when there is new information, λ cannot jump at time T, since there is no new information then—only the economy playing out its prearranged script. Therefore, the economy must use the initial jump in λ and the dynamics during the interim period to arrive at the old saddle path at time T . After the initial jump in λ, it is easy to mechanically trace out where the