1 β γ βr γw γ β γ notice that the optimal k is

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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 # < 0 ∂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 # < 0 d) No. If more productive firms disproportionately locate in NYC, the average cost can be lower in NYC without higher agglomeration externalities.
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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.)
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(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 F = 1 X F - λ = 0 L ∂C F = 1 C M + C F + 10 - λ = 0 which implies that X F = C M + C F + 10
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