Optimal сontrol of one-sector economy under random variation labor funds

The problem of optimal control of one-sector economy under random variation labor funds is considered. The state of the economy is characterized by two variables: capital-labor ratio k(t) and non-productive consumption per an employee c(t) along with the production function F(k) being the gross product made per one unit of time. In this study the Kobb-Douglas production function is considered, that is F(k) = Ak^a, where A denotes the scale of rate of production (A > 0), a is the elasticity coefficient on fixed assets. Variables k(t) and c(t) satisfy the following equations: k = uF-yJx+ok^t), A-(O)=A-₀, с = 8c + (f - u)F, c{ 0)=0, where | is the depreciation rate, 5 is the discount rate (| > 0, 5 > 0), ^(t) denotes the standard white Gaussian noise (or ^(t) = dro(t)/dt, where (T-t> (1 - u)F(k)dt. 0 The objective of this study is to find such control u(t) on the interval [0, T] for which the average value of с(Т) reaches its maximum. This problem is solved using a dynamic programming method. Bellman's function s(k;t,T) is introduced; s(k;t,T) is an average value of value с(Т) provided that process continues on time interval [t, T] with an initial condition k(t) = k and on this interval is applied optimal control. For this function Bellman's equation is specified. The formulated objective is a solution of the Bellman's equation. This solution consists that an interval [0, T] by points t1 and t2 (0 < t1 < fc < T) breaks 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, T] - to the final stage (to a descent from the highway). On the highway, there is k = koc = const, and kl-* = , u = „ =- ^ S + ^ ^ос F (k_oc) If k(0) < koc, then u is equal to 1 on the interval [0, t1]; u is equal to 1 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 initial condition k(0) and doesn't depend on a stochastic component, the moment t2 doesn't depend on initial condition, but it depends on a. Thus, the greater a, the less length of the interval [t2, Т] and average value of non-productive consumption.

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