Using this result in the third first order 20 240 5

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Using this result in the third first-order condition: 12 240 20 240 5 15 * = = = + l l l l (2 points) Since , . (2 points) 24 = + l N 12 * = N Optimal consumption can be found by using for example l C 15 = (the budget constraint can also be used): 180 ) 12 ( 15 15 * * * * = = = C C l C (2 points) Finally, utility (=happiness) can be found as: 4 . 91 ) 12 ( ) 180 ( ) ( ) ( 25 . 0 .75 0 * 25 . 0 * .75 0 * * = = = u l C u (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 l is consumption of leisure. Suppose that production technology is represented by where u 75 . 0 25 . 0 l c u = c 65 . 0 35 . 0 2 K N y = 48 = K is the physical capital stock and is labor. We assume that N 24 = h , N l h + = and that there is no government in the economy (use and w π to denote the real wage and profits, respectively).. a ) Find the optimal values of c , , , , , l N y w π , and u under 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? c l N y u This is a General Equilibrium Model. Let us start from the household’s problem. The Lagrange is: { } π λ + = wh wl C l L 75 . 0 0.25 C . The first order conditions are: 0 C ) 25 . 0 ( 0 75 . 0 0.75 - = = λ l C L (Equation 1) 0 C ) 75 . 0 ( 0 25 . 0 0.25 = = w l l L λ (Equation 2) (3 points) 0 0 = + = π λ wh wl C L (Equation 3) From the first two first-order conditions (i.e., from (1) and (2)), we obtain: 3 C ) 75 . 0 ( C ) 25 . 0 ( 25 . 0 0.25 75 . 0 -0.75 wl C w l l = = λ λ Using this result in the third first-order condition: w l w h l wh wl wh wl wl π π π π 4 3 18 4 3 4 3 3 4 3 + = + = + = + = + . In order to solve the model, we need labor supply, which may be obtained directly from : 24 = + l N w N w N l N s s s π π 4 3 6 4 3 18 24 24 = + = = . We cannot solve the problem unless π is determined. For this, let us look at the production side. From the firm’s profit maximization problem: 4
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