A 10pts define a competitive equilibrium for this

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(a) (10pts) Define a Competitive Equilibrium for this economy. ANSWER: A Competitive Equilibrium is a set of endogenous variables ( ˆ C, ˆ N s , ˆ N d , ˆ π, ˆ T, ˆ w ) s.t. given exogenous variables ( z, K, G, h ): 1. Given ( ˆ w, ˆ π, ˆ T ), ( ˆ C, ˆ l ) solves consumer’s problem: max C,l u ( C, l ) s.t. C = ˆ w ( h - l ) + ˆ π - ˆ T C 0 , 0 l h with ˆ N s = h - ˆ l . 2. Given ( z, K ), ˆ N d solves firm’s problem: max N d zF ( K, N d ) - wN d s.t. N d 0 3. Government budget constraint balances: ˆ T = G 4. Market clearing: ˆ N s = ˆ N d (= ˆ N ) ˆ C + G = zF ( K, ˆ N ) (b) (10pts) Find the competitive values ( ˆ C, ˆ N s , ˆ N d , ˆ π, ˆ T, ˆ w ) considering u ( C, l ) = C 1 / 2 l 1 / 2 , 4
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Y = zK 1 / 2 ( N d ) 1 / 2 , h = 24, z = 1, K = 49 and G = 0. ANSWER: First, notice that G = 0 implies that ˆ T = 0. From the consumption market clearing, we have: ˆ C = 7 ˆ N 1 / 2 From consumer’s optimal condition: u 2 u 1 = ˆ w 1 2 ˆ C 1 / 2 ˆ l - 1 / 2 1 2 ˆ C - 1 / 2 ˆ l 1 / 2 = 1 2 7 ˆ N - 1 / 2 ˆ C = 15 2 ˆ N - 1 / 2 ˆ l Using the time constraint, we have that ˆ l = h - ˆ N = 24 - ˆ N . Putting previous two optimal conditions together, we have an equation only on ˆ N : 7 ˆ N 1 / 2 = 7 2 ˆ N - 1 / 2 (24 - ˆ N ) ˆ N = 1 2 (24 - ˆ N ) 3 ˆ N = 24 ˆ N = 8 Thus, ˆ C = 7(8) 1 / 2 = 14 2. We may use firm’s optimal condition to find ˆ w : ˆ w = 7 2 (8) - 1 / 2 = 7 4 2 = 7 8 2 And, finally, profits is equal to: ˆ π = 7(8) 1 / 2 - 7 2 (8) - 1 / 2 8 = 7(8) 1 / 2 - 7 2 (8) 1 / 2 = 7 2 (8) 1 / 2 = 7 2 (c)
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