Consider a profit-maximizing firm with the production function q = f(x, x2), facing output price p...
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Consider a profit-maximizing firm with the production function q = f(x₁, x2), facing output price p and input prices W₁ and W₂. Suppose the government taxes the firm according to the total cost of input 2, that is, total tax T = tw₂x₂. a. (20 points) Setup the firm's objective as a constrained optimization problem and through the implicit function theorem (by deriving the total derivative) sign the effects of t on the unconditional input demands x₁ and x2. You must clearly show the firm's objective, write the Lagrangian, and find the first- order conditions and the second-order conditions to receive full credit. b. (5 points) Using the envelope theorem, show the effect of t on the maximized profits. Consider a profit-maximizing firm with the production function q = f(x₁, x2), facing output price p and input prices W₁ and W₂. Suppose the government taxes the firm according to the total cost of input 2, that is, total tax T = tw₂x₂. a. (20 points) Setup the firm's objective as a constrained optimization problem and through the implicit function theorem (by deriving the total derivative) sign the effects of t on the unconditional input demands x₁ and x2. You must clearly show the firm's objective, write the Lagrangian, and find the first- order conditions and the second-order conditions to receive full credit. b. (5 points) Using the envelope theorem, show the effect of t on the maximized profits.
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