Difference inclusions with delay of economic growth (Q2753279)

The asymptotic behavior of the trajectories describing macroeconomical growth is investigated. The initial phase is the model NEWLINE\[NEWLINE a (K,L,M) = \{ y (K',L', M') \geq {\mathbf 0} \mid K' \leq M , NEWLINE\]NEWLINE NEWLINE\[NEWLINE M' \leq \nu M + I', I' + c L' \leq F(K,L), I' \geq 0 , L' \geq 0 \} , NEWLINE\]NEWLINE where the set-valued mapping \(a\) is generated by the system NEWLINE\[NEWLINE K_{t+2} = \nu K_{t+1} + I_{t+1} , \qquad I_{t+1} +c L_{t+1} = F(K_{t}, L_{t}) , NEWLINE\]NEWLINE (\(Y_{t+1}=F(K_{t},L_{t})\) is a single homogeneous product, \(L\) is labor, \(K\) is capital and \(F\) is a production function). It is supposed that \(a\) is a normal mapping, i.e. \(a\) is superlinear (\(a( {\mathbf x} +{\mathbf y}) \geq a( {\mathbf x}) + a ( {\mathbf y})\), \(a( \lambda x) = \lambda a( x)\) \((\lambda > 0)\)), \(\text{gr } a = \{ ( {\mathbf x}, {\mathbf y})\mid {\mathbf y} \in a ( {\mathbf x}) \}\) is a convex cone, \(a\) is an onto mapping such that \(a ( {\mathbf 0}) ={\mathbf 0}\) and \(({\mathbf y} \in a ( {\mathbf x})) \& ({\mathbf 0} \leq {\mathbf y}' \leq {\mathbf y}) \Longrightarrow {\mathbf y}' \in a ( {\mathbf x})\). The trajectory \(({\mathbf x}_{t})^{ + \infty }_{t=0}\) is a stationary if there exists a constant \(\beta > 0\) (the rate of growth) such that \({\mathbf x}_{t} = {\beta}^{t}{\mathbf x}_{0}\) for all \(t\). The von Neumann rate of growth of the inclusion \(a\) is determined by the equality NEWLINE\[NEWLINE \alpha = \sup \{ \beta \mid\text{ there exists a stationary}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{trajectory with the rate of growth equal to }\beta \}. NEWLINE\]NEWLINE The conjugate mapping \(a^{*}\) is determined by the equality \(a^{*} ({\mathbf g}) =\partial q_{{\mathbf g}}\), where NEWLINE\[NEWLINEq_{\mathbf g}({\mathbf x}) = \max_{ {\mathbf y} \in a ( {\mathbf x}) } [{\mathbf g}, {\mathbf y} ],\quad \partial q = \{ {\mathbf l} |(\forall {\mathbf x}) [ {\mathbf l} , {\mathbf x} ]\} \geq q( {\mathbf x})NEWLINE\]NEWLINE (\([ {\mathbf u}, {\mathbf v} ]\) is the inner product of the vectors \({\mathbf u}\) and \({\mathbf v}\)). A process \(({\mathbf x} , {\mathbf y})\) is productive (\({\mathbf x}= (K,L,M)\), \({\mathbf y} = (K,L_{*},M_{*})\), \(M_{t}\) is an expected at time \(t\) quantity of the capital at time \(t+1\), \({\mathbf y} \in a({\mathbf x})\)) if \(I_{*},L_{*}>0\). NEWLINENEWLINENEWLINEThe structure of admissible processes is characterized as follows: NEWLINENEWLINENEWLINE1. If \({\mathbf g}=(g_{1}, g_{2}. g_{3})\) and \({\mathbf f} \in a^{*}({\mathbf g})\) then there exists \({\mathbf l}=(l_{1},l_{2}) \in \partial F\) such that \({\mathbf f} = (d_{1}l_{1},d_{1}l_{2},d_{3})\), where \(d_{1}= \max \{ g_{2}c^{-1},g_{3} \}\) and \(d_{3}=g_{1} + \nu g_{3}\). NEWLINENEWLINENEWLINE2. If \({\mathbf x} =(K,L,M) \in {\mathbf R}^{3}_{+}\), \({\mathbf y} = (K,L_{*},M_{*}) \in a( {\mathbf x})\), \({\mathbf g}=(g_{1}, g_{2}. g_{3})\), \({\mathbf f} = (f_{1}, f_{2}, f_{3}) \in a^{*}({\mathbf g})\) and \([ {\mathbf f} , {\mathbf x} ] = [ {\mathbf g}, {\mathbf y} ]\) then \(f_{3}=d_{3}\), \(f_{1}K+f_{2}L=d_{1}F(K,L)\). NEWLINENEWLINENEWLINE3. For a four-tuple \(( {\mathbf x}, {\mathbf y}, {\mathbf f}, {\mathbf g})\) if a process \(({\mathbf x} , {\mathbf y})\) is productive and \((g_{2},g_{3}) \neq 0\) then \(g_{2} =c g_{3}\). NEWLINENEWLINENEWLINE4. For a triple \(( \alpha , {\overline{\mathbf x}}_{*}, {\overline{\mathbf p}})\), where \({\overline{\mathbf p}} = ({\overline{p}}_{1}, {\overline{p}}_{2}, {\overline{p}}_{3})\) is a von Neumann equilibrium state of the mapping \(a\) and \(({\overline{\mathbf x}}_{*} , \alpha {\overline{\mathbf x}}_{*})\) is a productive process the following holds: NEWLINENEWLINENEWLINE1) \({\overline{p}}_{1} = ( \alpha - \nu) {\overline{p}}_{3}\), \({\overline{p}}_{2} = c {\overline{p}}_{3}\); NEWLINENEWLINENEWLINE2) \(\alpha = \max _{K,L,M} (F(K,L) + \alpha M)( (\alpha - \nu)K + cL + M)^{-1}\); NEWLINENEWLINENEWLINE3) \(\alpha > \nu\); NEWLINENEWLINENEWLINE4) a point \({\overline{\mathbf x}} = ({\overline{K}},{\overline{L}},{\overline{M}})\) is the von Neumann equilibrium vector if and only if \(\alpha {\overline{K}} = {\overline{M}}\) and the maximum in 2) is attained at \(({\overline{K}},{\overline{L}},{\overline{M}})\); NEWLINENEWLINENEWLINE5) for every von Neumann vector \({\overline{\mathbf x}}\) the process \(({\overline{\mathbf x}} , \alpha {\overline{\mathbf x}})\) is productive. NEWLINENEWLINENEWLINEThe following characteristics of asymptotic behavior of trajectories for strictly superlinear production function \(F\) (i.e. the restriction of \(F\) to any segment with nonzero endpoints placed on different coordinate axes is a strictly concave function) are established. NEWLINENEWLINENEWLINEIf \(F\) is strictly superlinear, \(\alpha\) is the von Neumann rate of growth of \(a\), \({\overline{\mathbf x}} = ({\overline{K}},{\overline{L}},{\overline{M}})\) is the von Neumann vector such that the process \(({\overline{\mathbf x}} , \alpha {\overline{\mathbf x}})\) is productive then for every trajectory \((K_{t},L_{t},M_{t})_{t=1,2, \dots }\) starting from the strictly positive point \((K_{0},L_{0},M_{0})\) exactly one of the following two statements holds: NEWLINENEWLINENEWLINE1) \({\alpha}^{-t}(K_{t},L_{t},M_{t}) \to 0\); NEWLINENEWLINENEWLINE2) \((\exists \gamma >0)({\alpha}^{-t}(K_{t},L_{t},M_{t}) \to \gamma ({\overline{K}},{\overline{L}},{\overline{M}}))\).