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To have the optimal level of labor input n w,r,y = y

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Unformatted text preview: to have the optimal level of labor input: N ( W,R,Y ) = Y AN φ T ! 1 β + γ βR γW γ β + γ Notice that the optimal K is proportional to the optimal N , K ( W,R,Y ) = γW βR N ( W,R,Y ) = Y AN φ T ! 1 β + γ γW βR β β + γ b) The cost function is given by: C ( W,R,Y ) = WN ( W,R,Y ) + RK ( W,R,Y ) = Y AN φ T ! 1 β + γ " W βR γW γ β + γ + R γW βR β β + γ # = ( Y ) 1 β + γ ( A )- 1 β + γ ( N T )- φ β + γ " W βR γW γ β + γ + R γW βR β β + γ # c) The derivative of the cost function with respect to A and N T are: ∂C ∂A =- 1 β + γ ( Y ) 1 β + γ ( A ) 1 β + γ- 1 ( N T )- φ β + γ " W βR γW γ β + γ + R γW βR β β + γ # < ∂C ∂N T =- φ β + γ ( Y ) 1 β + γ ( A )- 1 β + γ ( N T ) φ β + γ- 1 " W βR γW γ β + γ + R γW βR β β + γ # < d) No. If more productive firms disproportionately locate in NYC, the average cost can be lower in NYC without higher agglomeration externalities. 3. Private Provision of Public Goods: (a) Suppose Falco contributes C F for the concerts and Mozart contributes C M . Then, the total amount of concerts enjoyed by Falco and Mozart are C M + C F . Falco’s maximization problem is: max X F ,C F U = ln X F + ln( C M + C F ) subject to 70 = X F + C F The Lagrange function associated with this problem is: L = log X F + log( C M + C F )- λ ( X F + C F- 70) Taking derivatives and setting them equal to zero, yields: ∂ L ∂X F = 1 X F- λ = 0 ∂ L ∂C F = 1 C M + C F- λ = 0 Eliminating the Lagrange multiplier yields: X = C M + C F Substituting this optimality condition into the budget constraint yields: 70 = 2 C F + C M which yields the following reaction function: C F = 70- C M 2 Since Falco and Mozart have identical preferences and income, we get for Mozart the following reaction function: C M = 70- C F 2 Substituting Mozart’s reaction function into Falco’s yields: C F = C M = 70 / 3 = 23 . 334 Hence there would be 46.667 concerts. (Each individual would also enjoy 46 . 667 units of the private good.) (b) With the government intervention, Falco’s maximization problem changes to: max X F ,C F U = log X + log( C M + C F + 10) subject to 65 = X F + C F Setting up the Lagrange function and computing the first order conditions yields: ∂ L ∂X...
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to have the optimal level of labor input N W,R,Y = Y AN φ...

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