Turnpike principle in a problem of management onesectoreconomy in the presence of restrictions on saving and consumption

On a class of linearly-homogeneous production functions (F (K, L)) research of a problem ofoptimum control by one-sector economy (Y(t) = F (K(t), L(t)) = ƒ (t) + I (t) +C (t) + N(t)) in theshort run t Ўф[0,T] in the presence of restrictions on saving I (t) and consumption C (t) taking intoaccount industrial expenses ƒ (t) and tax deductions N (t) is resulted. Having accepted as criterionof an optimality specific consumption with discounting for all planned period, we come to back ofa problem of optimum control:k(t) = (1− Ґг)(1− u)s(t) f (k(t)) − Ґнk(t), t Ўф[0,T], k(0) = k0, k(T) ЎГ kT > 0,J=Ґг)(1−u)(1−s(t)) f(k(t)) exp{−Ґдt} dt Ўж 0ЎВ s₀ ЎВ s(t) ЎВ s₁ ЎВ1,Ґн = Ґм + Ґл, Ґм > 0, Ґл > 0, Ґд > 0, 0 ЎВ Ґг < 1, 0 ЎВ u < 1,where s (t) - norm of saving, ƒ - factor of amortization of a fixed capital, ƒ - norm consumptionof materials of material inputs, u - the tax rate on profit and a labour force L(t) = L0 exp{Ґлt} ,L0 > 0 , ƒ > 0. The basic result is formulated in the form of the «Turnpike Theorem».Theorem 1. For sufficiently large control time T the solution to the problem has the followingform.1. Time interval [0, T] is divided into three intervals, i.e. [0,T] = [0,T)Ўъ[T,T]Ўъ(T,T ] .2. The control s(t)Ўф{s1;s0;s} , i.e. are a piecewise continuous with three possible values.3. On the turnpike time interval t Ўф[T,T] s (t) = s, and fixed capital per worker k(t) preserversthe constant value k .4. On the initial time interval t Ўф[0,T) when the economy is brought to the turnpike,s(t) = s₁ , if k₀ < k , and s(t) = s₀ , if k₀ > k∗ , is increased or decreased, respectively, k (t) fromk₀ to k .5. On the final time interval t Ўф(T∗∗,T] when the economy is brought off the turnpike forsatisfying the condition of economic horizon k(T) = k_T , s(t) = s₁ , if k_T > k, and s(t) = s₀ ifkT < k∗ , is increased or decreased, respectively, k ∗ (t) from k ∗ to kT .The formulas determining s ∗ , k ∗ , T, ∗ T ∗∗ , and also values of criterion of quality on the optimumsolution are received. The «Golden Rule of Saving», defining how produce made by economyY ∗ (t) = F(K(t),L(t)), where K (t) = kL(t) as the sum of incomes from a fixed capitalYK∗ (t) and labour force YL∗(t) , is distributed on a turnpike between saving I ∗ (t), consumptionC ∗ (t), tax deductions N ∗ (t) and material inputs ƒ ∗ (t). The results are specified for the case ofCobb-Douglas production function and is made analyze results.

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