Babydrink is a profit-maximizing monopolist that produces infant formula. The de
ID: 1199887 • Letter: B
Question
Babydrink is a profit-maximizing monopolist that produces infant formula. The demand curve facing the firm is given by P = a - bQ, where both a b are positive parameters. This firm's total cost function is linear in quantity with constant marginal cost c per unit of output and fixed costs of F. Remembering that profit pi = TR - TC, determine this firm's profit function. Maximize the profit function, and calculate the optimal levels of price and output. Now assume that the government imposes a tax t per unit of output sold by the firm. Using the same procedure as in part (a), obtain the new profit-maximizing levels of price and output. The government decides to choose t so that it maximizes the revenues it can obtain from taxing Babydrink's output. Remembering that tax revenues equal the tax rate times the tax base, or Q, in this example, what level of t should the government choose?Explanation / Answer
P = a - bQ
Total cost, TC = F + cQ
(a)
Total revenue, TR = P x Q = aQ - bQ2
Profit, Z = RE - TC = aQ - bQ2 - F - cQ
Profit is maximized when dZ / dQ = 0
a - 2bQ - c = 0
2bQ = a - c
Q = (a - c) / 2b
At this level, P = a - bQ = a - [b x (a - c) / 2b] = a - [(a - c) / 2] = (a + c) / 2
(b) With tax of t, Demand function becomes
P = a - bQ - t
TR = P x Q = aQ - bQ2 - tQ
Profit, Z = aQ - bQ2 - tQ - F - cQ
Maximizing,
dZ / dQ = a - 2bQ - t - c
2bQ = a - c - t
Q = (a - c - t) / 2b
P = a - bQ - t = a - t - [b x (a - c - t) / 2b] = a - t - [(a - c - t) / 2]
= (2a - 2t - a + c + t) / 2
= (a + c - t) / 2
(c)
Since P = a - bQ - t,
Q = (a - t - P) / b
Tax revenue, T = t x Q = (at - t2 - Pt) / b
Tax revenue is maximized when dT / dt = 0
(a - 2t - P) / b = 0
a - 2t - P = 0
t = (a - P) / 2
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