In the text we focus attention on the case in which household utility features a finite...

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In the text we focus attention on the case in which household utility features a finite intertemporal substitution elasticity. As an extension we now study the case for which marginal utility is constant. The lifetime utility function (13.57) is replaced by: A (0) E = Lo c(t)e P²dt, where c(t) is consumption per worker. We study the social planning solution to the household optimization problem. The fundamental differential equation for the capital stock per worker, k(t), is: k(t) = f (k(t)) — c(t) — (8+n₁) k(t), (Q13.23) where y(t) = f (k(t)) is output per worker. The production function satisfies the In- ada conditions and there is no technological progress. The solution for consumption must satisfy the following constraints: ē ≤ c(t) ≤ f (k(t)), (Q13.24) where is some minimum consumption level (it is assumed that 0 <c<cGR, where CGR is the maximum attainable "golden rule" consumption level). (Q13.22) (a) Set up the appropriate current-value Hamiltonian and derive the first-order conditions. Show that the solution for consumption is a so-called "bang-bang" solution: for μ(t) > 1 for μ(t) = 1, for µ(t) < 1 where u(t) is the co-state variable. (b) Derive the phase diagram for the model and show that there exists a unique saddle-point stable equilibrium. Show that this equilibrium is in fact reached provided the initial capital stock per worker lies in the interval (k₁, ku). c(t) = C free [ƒ(k(t)) (Q13.25) In the text we focus attention on the case in which household utility features a finite intertemporal substitution elasticity. As an extension we now study the case for which marginal utility is constant. The lifetime utility function (13.57) is replaced by: A (0) E = Lo c(t)e P²dt, where c(t) is consumption per worker. We study the social planning solution to the household optimization problem. The fundamental differential equation for the capital stock per worker, k(t), is: k(t) = f (k(t)) — c(t) — (8+n₁) k(t), (Q13.23) where y(t) = f (k(t)) is output per worker. The production function satisfies the In- ada conditions and there is no technological progress. The solution for consumption must satisfy the following constraints: ē ≤ c(t) ≤ f (k(t)), (Q13.24) where is some minimum consumption level (it is assumed that 0 <c<cGR, where CGR is the maximum attainable "golden rule" consumption level). (Q13.22) (a) Set up the appropriate current-value Hamiltonian and derive the first-order conditions. Show that the solution for consumption is a so-called "bang-bang" solution: for μ(t) > 1 for μ(t) = 1, for µ(t) < 1 where u(t) is the co-state variable. (b) Derive the phase diagram for the model and show that there exists a unique saddle-point stable equilibrium. Show that this equilibrium is in fact reached provided the initial capital stock per worker lies in the interval (k₁, ku). c(t) = C free [ƒ(k(t)) (Q13.25)