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#### In the text we focus attention on the case in which household utility features a finite intertemporal substitution elasticity

###### Economics

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: 0 A(0) = 1° c(t)ept dt, (Q13.22) = 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+nz) 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: is c(t) = f (k(t)), (Q13.24) where 7 is some minimum consumption level (it is assumed that 0 <7 < CGR, where CGR is the maximum attainable "golden rule" consumption level). (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: c = for u(t) > 1 c(t) free for y(t) = 1, (Q13.25) ( f (k(t)) for u(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 (kliku).