Solution by the Maximum Principle

Transcription

1 Economic Applications per capita variables so that it is formally similar to the previous model. The introduction of the per capita variables makes it possible to treat the infinite horizon version of the new model. Let L(t) denote the amount of labor at time t. Since it is growing exponentially at rate g, we have L(t) = L(0)e gt. Let F(K, L) be the production function which is assumed to be concave and homogeneous of degree one in K and L. We define k = K/L and the per capita production function f(k) as F(K, L) f(k) = = F( K, 1) = F(k,1). (11.7) L L To derive the state equation for k, we note that K = kl + k L = kl + kgl. Substituting for K from (11.1) and defining per capita consumption c = C/L, we get k = f(k) c γk, k(0) = k 0, (11.8) where γ = g + δ. Let u(c) be the utility of per capita consumption of c, where u is assumed to satisfy u (c) > 0 and u (c) < 0 for c > 0 and u (0) =. (11.9) The objective is the total discounted per capita consumption utilities over time. Thus, we { } maximize J = e ρt u(c)dt. (11.10) 0 Note that the optimal control model defined by (11.8) and (11.10) is a generalization of Exercise Solution by the Maximum Principle The current-value Hamiltonian is The adjoint equation is H = u(c) + λ[f(k) c γk]. (11.11) λ = ρλ H k = (ρ + γ)λ f (k)λ. (11.12)

2 11.1. Models of Optimal Economic Growth 293 To obtain the optimal control, we differentiate (11.11) with respect to c, set it to zero, and solve u (c) = λ. (11.13) Let c = h(λ)= u 1 (λ) denote be the solution of (11.13). To show that the maximum principle is sufficient for optimality we will show that the derived Hamiltonian H 0 (k, λ) is concave in k for any λ solving (11.13); see Exercise However, this follows immediately from the facts that u (c) is positive as assumed in (11.9) and that f(k) is concave because of the assumptions on F(K, L). Equations (11.8), (11.12), and (11.13) now constitute a complete autonomous system, since time does not enter explicitly in these equations. Therefore, we can use the phase diagram solution technique employed in Chapter 7. In Figure 11.1 we have drawn a phase diagram for the two equations k = f(k) h(λ) γk = 0, (11.14) λ = (ρ + γ)λ f (k)λ = 0, (11.15) obtained from (11.8), (11.12), and (11.13). In Exercise 11.3 you are asked to show that the graphs of k = 0 and λ = 0 are as shown in Figure The point of intersection of these two graphs is ( k, λ). Figure 11.1: Phase Diagram for the Optimal Growth Model

3 Economic Applications The two graphs divide the plane into four regions, I, II, III, and IV, as marked in Figure To the left of the vertical line λ = 0, k < k and ρ+γ < f (k) so that λ < 0 from (11.12). Therefore, λ is decreasing, which is indicated by the downward pointing arrows in Regions I and IV. On the other hand, to the right of the vertical line, in Regions II and III, the arrows are pointed upward because λ is increasing. In Exercise 11.4 you are asked to show that the horizontal arrows, which indicate the direction of change in k, point to the right above the k = 0 curve, i.e., in Regions I and II, and they point to the left in Regions III and IV which are below the k = 0 curve. The point ( k, λ) represents the optimal long-run stationary equilibrium. The values of k and λ were obtained in Exercise We now want to see if there is a path satisfying the maximum principle which converges to the equilibrium. Clearly such a path cannot start in Regions II and IV, because the directions of the arrows in these areas point away from ( k, λ). For k 0 < k, the value of λ 0 (if any) must be selected so that (k 0, λ 0 ) is in Region I. For k 0 > k, on the other hand, the point (k 0, λ 0 ) must be chosen to be in Region III. We analyze the case k 0 < k only, and show that there exists a unique λ 0 associated with the given k 0. The locus of such (k 0, λ 0 ) is shown by the dotted curve in Figure In Region I, k(t) is an increasing function of t as indicated by the horizontal right-directed arrow. Therefore, we can replace the independent variable t by k as below, and then use (11.14) and (11.15) to obtain d(ln λ) dk [ 1 = λ ] / dλ dk dt dt = f (k) (ρ + γ) h(λ) + γk f(k). (11.16) For k < k, the right-hand side of (11.16) is negative, and since h(λ) decreases as λ increases, we have d(lnλ)/dk increasing with λ. We show next that there can be at most one trajectory for an initial capital k 0 < k. Assume to the contrary that λ 1 (k) and λ 2 (k) are two paths leading to ( k, λ) and are such that the selected initial values satisfy λ 1 (k 0 ) > λ 2 (k 0 )> 0. Since d(lnλ)/dk increases with λ, d ln[λ 1 (k)/λ 2 (k)] dk = d lnλ 1(k) dk d lnλ 2(k) dk > 0, whenever λ 1 (k) > λ 2 (k). This inequality clearly holds at k 0, and by (11.16), λ 1 (k)/λ 2 (k) increases at k 0. This in turn implies that the inequality holds at k 0 + ε, where ε > 0 is small. Now replace k 0 by k 0 + ε

4 Economic Applications The optimal control shown in Figures 11.2 and 11.3 assumes 0 < x s < N. It also assumes that T is large so that the trajectory will spend some time on the turnpike and Q is large so that x s N Q/β. The graphs are drawn for x 0 > x s and x s < N/2; for all other cases see Sethi (1974c). Figure 11.2: Optimal Trajectory when x T > x s Figure 11.3: Optimal Trajectory when x T < x s

5 11.3. A Pollution Control Model A Pollution Control Model In this section we shall describe a simple pollution control model due to Keeler, Spence, and Zeckhauser (1971). We shall describe this model in terms of an economic system in which labor is the only primary factor of production, which is allocated between food production and DDT production. Once produced (and used) DDT is a pollutant which can only be reduced by natural decay. However, DDT is a secondary factor of production which, along with labor, determines the food output. The objective of the society is to maximize the total present value of the utility of food less the disutility of pollution due to the DDT use Model Formulation We introduce the following notation: b v b v = the total labor force, assumed to be constant for simplicity, = the amount of labor used for DDT production, = the amount of labor used for food production, P = the stock of pollution at time t, a(v) = the rate of DDT output; a(0) = 0, a > 0, a < 0, for v 0, δ C(v) = the natural exponential decay rate of DDT pollution, = f[b v, a(v)] = the rate of food output; C(v) is concave, C(0) > 0, C(b) = 0; C(v) attains a unique maximum at v = V > 0; see Figure Note that a sufficient condition for C(v) to be strictly concave is f 12 0 along with the usual concavity and monotonicity conditions on f, g(c) = the utility of consumption; g (0) =, g 0, g < 0, h(p) = the disutility of pollution; h (0) = 0, h 0, h > 0. The optimal control problem is: { } max J = e ρt [g(c(v)) h(p)]dt 0 (11.26)

6 Economic Applications Figure 11.4: Food Output Function subject to P = a(v) δp, P(0) = P 0, (11.27) 0 v b. (11.28) From Figure 11.4 it is obvious that v is at most V, since the production of DDT beyond that level decreases food production as well as increases DDT pollution. Hence, (11.28) can be reduced to simply v 0. (11.29) Solution by the Maximum Principle Form the current-value Lagrangian L(P, v, λ, µ) = g[c(v)] h(p) + λ[a(v) δp] + µv (11.30) using (11.26), (11.27) and (11.29), where and The optimal solution is given by λ = (ρ + δ)λ + h (P), (11.31) µ 0 and µv = 0. (11.32) L v = g [C(v)]C (v) + λa (v) + µ = 0. (11.33)

7 Economic Applications Figure 11.5: Phase Diagram for the Pollution Control Model Observe that the assumption h (0) = 0 implies that the graph of (11.39) passes through the origin. Differentiating these equations with respect to λ and using (11.37), we obtain as the slope of (11.38), and dp dλ = a (v) dv δ dλ > 0 (11.40) dp dλ = (ρ + δ) h (P) < 0 (11.41) as the slope of (11.39). Using (11.35), (11.36), (11.40), and (11.41), we can draw (11.38) and (11.39) in the (λ, P)-space as shown in Figure The intersection point ( λ, P) of these curves denotes the equilibrium levels for the adjoint variable and the pollution stock, respectively. From arguments similar to those in Section , it can be shown that there exists an optimal path (shown dotted in the figure) converging to the equilibrium ( λ, P). Given λ c as the intersection of the P = 0 curve and the horizontal axis, the corresponding ordinate P c on the optimal trajectory is the related pollution stock level. The significance of P c is that if the existing pollution stock P is larger than P c, then the optimal control is v = 0, meaning no DDT is produced.

8 11.4. Miscellaneous Applications 303 Given an initial level of pollution P 0, the optimal trajectory curve in Figure 11.5 provides the initial value λ 0 of the adjoint variable. With these initial values, the optimal trajectory is determined by (11.27), (11.31), and (11.37). If P 0 > P c, as shown in Figure 11.5, then v = 0 until such time that the natural decay of pollution stock has reduced it to P c. At that time the adjoint variable has increased to the value λ c. The optimal control is v = φ(λ) from this time on, and the path converges to ( λ, P). At equilibrium, v = Φ( λ) > 0, which implies that it is optimal to produce some DDT forever in the long run. The only time when its production is not optimal is at the beginning when the pollution stock is higher than P c. It is important to examine the effects of changes in the parameters on the optimal path. In particular, you are asked in Exercise 11.7 to show that an increase in the natural rate of decay of pollution, δ, will increase P c. That is, the higher is the rate of decay, the higher is the level of pollution stock at which the pollutant s production is banned. For DDT, δ is small so that its complete ban, which has actually occurred, may not be far from the optimal policy. Here we have presented a very simple model of pollution in which the problem was to choose an optimal production process. Models in which the control variable to determine is the optimal amount to spend in reducing the pollution output of an existing dirty process have also been formulated; see Wright (1974) and Sethi (1977d). For yet other related models, see Luptacik and Schubert (1982), Hartl and Luptacik (1992), and Hartl and Kort (1996a, 1996b, 1996c, 1977) Miscellaneous Applications The number of papers which apply control theory to problems in economics and management science is now so large that it is impossible to cover them in detail within the confines of a single book. We satisfy ourselves by listing selected references with a brief indication of their contents. For control theory applications to economics, see Tu (1969) and Southwick and Zionts (1974) for optimal educational investments, Kamien and Schwartz (1971b) for limit pricing and uncertain entry, Treadway (1970) for adjustment costs in the theory of competitive firms, Vousden (1974) for international trade, Harris (1976) for money demand

9 Economic Applications 11.8 A variation of the optimal capital accumulation model with stationary population, known as Ramsey s model, is: { } max J = [u(c) B]dt 0 subject to where is the so-called Bliss point, k = f(k) c δk, k(0) = k 0, B = sup u(c) > 0 c 0 lim u[c(t)] = B t so that the integral in the objective function converges, and lim t u [c(t)] = 0. See Ramsey (1928). (a) Show that the optimal capital stock trajectory satisfies the differential equation u (f(k) δk k) k = B u(f(k) δk k). (b) From part (a), derive Ramsey s rule d[u (c(t))] dt = u (c(t))[δ f (k(t))]. (c) Assume u(c) = 2c c 2 /B and f(k) = αk, where α δ := β > 0 and β < B/k 0 < 2β. Show that the optimal feedback consumption rule is c (k) = 2βk B and the optimal capital trajectory k is given by k (t) = 1 β [B (B βk 0)e βt ].