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Using this result in the third first-order condition: 1224020240515*=⇒=⇒=+llll(2 points) Since , . (2 points) 24=+lN12*=NOptimal consumption can be found by using for example lC15=(the budget constraint can also be used): 180)12(1515****=⇒=⇒=CClC(2 points) Finally, utility (=happiness) can be found as: 4.91)12()180()()(25.0.750*25.0*.750**==⇒=ulCu(4 points) 3
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Izmir Univery of Economics I. Hakan Yetkiner Department of Economics 4. (30 Points) Suppose that utility function of a representative agent is , where is consumption of physical goods and lis consumption of leisure. Suppose that production technology is represented by where u75.025.0lcu=c65.035.02KNy=48=Kis the physical capital stock and is labor. We assume that N24=h, Nlh+=and that there is no government in the economy (use and wπto denote the real wage and profits, respectively).. a) Find the optimal values of c, , , , , lNywπ, and uunder the competitive equilibrium assumption. b) Find the optimal values of , , , , and under the social planner’s solution assumption. Are the results different or same? Why or why not? clNyuThis is a General Equilibrium Model. Let us start from the household’s problem. The Lagrange is: {}πλ−−+−=whwlClL75.00.25C. The first order conditions are: 0C)25.0(075.00.75-=−⇒=∂∂λlCL(Equation 1) 0C)75.0(025.00.25=−⇒=∂∂−wllLλ(Equation 2) (3 points) 00=−−+⇒=∂∂πλwhwlCL(Equation 3) From the first two first-order conditions (i.e., from (1) and (2)), we obtain: 3C)75.0(C)25.0(25.00.2575.0-0.75wlCwll=⇒=−λλUsing this result in the third first-order condition: wlwhlwhwlwhwlwlππππ43184343343+=⇒+=⇒+=⇒+=+. In order to solve the model, we need labor supply, which may be obtained directly from : 24=+lNwNwNlNsssππ43643182424−=⇒⎟⎠⎞⎜⎝⎛+−=⇒−=. We cannot solve the problem unless πis determined. For this, let us look at the production side. From the firm’s profit maximization problem: 4