2. (Perfect Competition) Individual firms operating in the market for widgets ha
ID: 1147613 • Letter: 2
Question
2. (Perfect Competition) Individual firms operating in the market for widgets have costs given by C(q) = 1000 + 10q°. The market demand function is given by D(p) = 40, 000 – 100p. (a) [2 points] Provide formulas for a firm's fixed cost, variable cost, average total cost, and average variable cost: FC, VC (9), ATC(q), and AVC (9), respectively. (b) [2 points) The marginal cost function is given by MC (Q) = 20q. Find the individual firm's short-run supply function, q(p). (Note that min{AVC} = 0, and assume all fixed costs are sunk.) (c) [2 points] What is the quantity at which average total cost is minimized, qmin? (Hint: consider the relationship between ATC and MC when ATC is minimized.) What is the average total cost at qmin? (d) [2 points] If there is free entry and exit, then what is the long-run equilib- rium price, p*? How many units does each individual firm produce, q*? How many firms, n*, are needed to meet the demand at p*? (Your answers should be the values of p*, q*, and n*.),Explanation / Answer
(a)
FC = 1,000
VC(q) = 10q2
ATC(q) = C(q) / q = (1,000 / q) + 10q
AVC(q) = VC(q) / q = 10q
(b)
Firm's short run supply function is its MC function:
p = 20q
q = p/20
q = 0.05p
(c)
ATC is minimized when dATC(q) / dq = 0
dATC(q) / dq = (- 1,000 / q2) + 10 = 0
1,000 / q2 = 10
q2 = 1,000 / 10 = 100
q = 10 (= qmin)
When qmin = 10, ATC = (1,000 / 10) + (10 x 10) = 100 + 100 = 200
(d)
In long run equilibrium, p = MC = ATC
Equating MC & ATC,
(1,000 / q) + 10q = 20q
(1,000 / q) = 10q
q2 = 100
q = 10 (= q*)
p* = MC = 20 x q* = 20 x 10 = 200
Industry demand, D(p*) = 40,000 - 100p* = 40,000 - (100 x 200) = 40,000 - 20,000 = 20,000
n* = D(p*) / q* = 20,000 / 10 = 2,000
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