Q2. Monopoly model with linear demand In this course, you will repeatedly solve
ID: 1174239 • Letter: Q
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Q2. Monopoly model with linear demand
In this course, you will repeatedly solve models with linear demand function. We certainly do not imply demand is linear in reality. In fact, demand function is hardly linear in any market. The linearity is assumed for simplicity in deriving analytical solution, and it allows us to capture some useful economics under a complicated model setting. The following monopoly model with linear demand function is an exercise for characterizing equilibrium quality and the resulting welfare measure.
Suppose a monopoly firmfaces the following linear (inverse) demand
P (Q) = a - bQ
Assume the firm produces at a constant marginal cost MC(Q) = c for any Q?0.
a. Set up the firm’s profit maximization problem and derive the optimal output Qm and price P (Qm).
b. Calculate the firm’s profit ?m using the optimal output derived in [a.].
c. Calculate consumer surplus when the firmproduces optimally CSm.
d. Calculate total surplus when the firmproduces optimally Wm.
e. Calculate the efficient level of output Qs and the corresponding total surplus Ws ?
f. Calculate the DWL under monopoly?
Explanation / Answer
(a)
Total revenue (TR) = P x Q = aQ - bQ2
Total cost (TC) = MC x Q = cQ
Profit (Z) = TR - TC = aQ - bQ2 - cQ = (a - c)Q - bQ2
Profit maximization problem is to Maximize profit: Max Z = (a - c)Q - bQ2
Profit is maximized when dZ/dQ = 0
dZ/dQ = (a - c) - 2bQ = 0
a - c - 2bQ = 0
2bQ = a - c
Q (= Qm) = (a - c)/2b
P (= Pm) = a - bQ = a - b x [(a - c)/2b] = a - [(a - c)/2] = (2a - a + c)/2 = (a + c)/2
(b)
TR = [(a + c)/2] x [(a - c)/2b] = (a2 - c2)/4b
TC = c x [(a - c)/2b] = (ac - c2)/2b
Profit = [(a2 - c2)/4b] - [(ac - c2)/2b] = (a2 - c2 - 2ac + 2c2)/4b = (a2 - 2ac + c2)/4b = (a - c)2/4b
(c)
From demand function, when Q = 0, P = a (Reservation price & vertical intercept of demand curve)
Consumer surplus (CS) = Area between demand curve and market price
= (1/2) x [a - {(a + c)/2}] x [(a - c)/2b] = (1/2) x [(2a - a - c)/2] x [(a - c)/2b]
= (1/2) x [(a - c)/2] x [(a - c)/2b] = [(a - c)/4] x [(a - c)/2b] = (a - c)2/8b
(d)
Producer surplus (PS) = Area between supply curve (MC) and market price = Pm x Qm = TR
PS = (a2 - c2)/4b
Total surplus (TS) = CS + PS = [(a - c)2/8b] + [(a2 - c2)/4b] = [(a2 - 2ac + c2)/8b] + [(a2 - c2)/4b]
= (a2 - 2ac + c2 + 2a2 - 2c2)/8b = (3a2 - 2ac - c2)/8b
NOTE: As per Chegg Answering Policy, first 4 parts are answered.
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