A: Suppose a firm employs labor and capital k to produce output x using a homothetic,

A: Suppose a firm employs labor ℓ and capital k to produce output x using a homothetic, decreasing returns to scale technology.
(a) Suppose that, at the current wage w, rental rate r and output price p, the firm has identified A = (xA, ℓA, kA) as its profit maximizing production plan. Illustrate an isoquant corresponding to xA and show how (ℓA, kA) must satisfy the conditions of cost minimization.
(b) Translate this to a graph of the cost curve that holds w and r fixed — indicating where in your isoquant graph the underlying input bundles lie for this cost curve.
(c) Show how xA emerges as the profit maximizing production level on the marginal cost curve that is derived from the cost curve you illustrated in (b).
(d) Now suppose that the government taxes labor — causing the cost of labor for the firm to in- crease to (1 + t)w. What changes in your pictures — and how will this effect the profit maximizing production plan?
(e) What happens if the government instead imposes a tax on capital that raises the real cost of capital to (1 + t)r ?
(f) What happens if instead the government imposes a tax on both capital and labor — causing the cost of capital and labor to increase by the same proportion (i.e. to (1 + t)w and (1 + t )r .)
(g) Now suppose the government instead taxes economic profit at some rate t < 1. Thus, if the firm makes pre-tax profit π, the firm gets to keep only (1 − t)π.
B: Suppose your firm has a decreasing returns to scale, Cobb-Douglas production function of the form x = Aℓαkβ for which you may have previously calculated input and output demands as well as the cost function. (The latter is also given in problem 12.4).
(a) If you have not already done so, calculate input demand and output supply functions. (You can do so directly using the profit maximization problem, or you can use the cost function given in problem 12.4 to derive these.)
(b) Derive the profit function and check that it is correct by checking whether Hotelling’s Lemma works.
(c) If you have not already done so, derive the conditional input demand functions. (You can do so directly by setting up the cost minimization problem, or you can employ Shephard’s Lemma and use the cost function given in problem 12.4.)
(d) Consider a tax on labor that raises the labor costs for firms to (1 + t)w. How does this affect the various functions in the duality picture for the firm?
(e) Repeat for a tax on capital that raises the capital cost for the firm to (1+t)r.
(f) Repeat for simultaneous taxes on labor and capital that raise the cost of labor and capital to (1+t)w and (1+t)r.
(g) Repeat for a tax on profits as described in part A(g).