Optimal control of one-sector economy under random variation of fixed capital and labor resources

The problem of optimal control of one-sector economy under random variation of fixed capital and labor resources is considered. The state of the economy is characterized by such variables: the fixed capital K(t), the labor resources L(t), the non-productive con sumption C(t), and also the production function Y(K, L). In a stochastic case change of these sizes is described by the equations К = uY(K,L) - XK{t) + o_KKl,_K{t), K{ 0) = K₀>0, С = pC + (1 - u)Y(K,L), C(0) = 0, (1) L = vL + а_£Д_£(0,£(0) =L₀> 0, where X (> 0) is the depreciation rate, p (> 0) is the discount rate, ^K(t) and ^t(t) denote the standard independent white Gaussian noises, ctk and <7l are the coefficients of volatility correspondingly. Here u is the control parameter defining the share of the product produced, which is used to increase in fixed capital. Then size 1 - u defines part of a product which goes on increase in non-productive consumption. It is obvious that 0 < u < 1. For research variables are entered: k(t) = K(t)/L(t) is the capital-labor ratio and c(t) = C(t)/L(t) - the non-productive consumption falling on one worker. For these variables on the basis of Ito's formula from (1) the equations turn out k = -y.k+uF(k) + o_Kk^_K(t)-a_Lk^_L(t), k(0) = k₀, с = 5c + (1 -u)F{k)-a_Lc^_L(t), c(0) = 0. Here | = X + v + ctkctl - ctl², 5 = p - v + ctl², ka = K0/L0, F(k) = Y(K, L)/L is the labor productivity (a gross product on one worker). Further it is supposed that | > 0 and 5 > 0. In this study the production function of Kobb-Douglas is considered, that is Y(K, L) = AK^aL^e, where A denotes the scale of rate of production (A > 0), a is the elasticity coefficient on fixed capital, в is the elasticity coefficient on labor resources. From here F(k) = Ak^a. It is supposed that the planned period of production [0,T] is given and rather great. The task consists in finding on the time interval [0, T] for system (2) of such control u(t) satisfying the condition 0 < u < 1 at which the average value of с(Т) is maximum. This problem is solved by making use of a method of dynamic programming. Bellman's function s(k,c;t,T) is considered, which is an average value of с(Г) provided that the process proceeds on the interval [t, T] with initial conditions k(t) = k and с^) = с. The solution of a task we obtain by solving Bellman's equation. This solution consists in the fact that the interval [0, T] by the points t1 and t2 (0 < t1 < t2 < T) is divided into three intervals: [0, t1], [t1, t2], and [t2, Т]. The interval [0, t1] corresponds to the output to the highway, the interval [t1, t2] - to the highway (if it exists), the interval [t2, Т] - to the final stage (to a descent from the highway). The highway is the time site, where special control uoc is used. On the highway k = koc = const, and ,1-a = _u = »k_oc^кос _s^, аос 4. 5 + |l ^F (koc ⁾ If k(0) < koc, then u = 1 on the interval [0, t1] and u = 0 on the interval [t2, Т]. As a result, it turns out that the control structure is determined by values of t1 and t2. The moment t1 depends only on the initial condition k(0) and doesn't depend on a stochastic component. The moment t2 doesn't depend on the initial condition, but depends on the volatility coefficients ok and ctl. Thus with growth of these coefficients the interval length [t2, T] and average value of non-productive consumption decrease.

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