(Perfect Competition) Individual firms operating in the market for widgets have
ID: 1146481 • Letter: #
Question
(Perfect Competition) Individual firms operating in the market for widgets have costs given by C(q) = 1000 + 10q2. The market demand function is given by D(p) = 10, 000 100p.
A) Provide formulas for a firm’s fixed cost, variable cost, average total cost, and average variable cost: F C, V C(q), AT C(q), and AV C(q), respectively.
B) The marginal cost function is given by M C(q) = 20q. Find the individual firm’s short-run supply function, q(p). (Note that min{AV C} = 0, and assume all fixed costs are sunk.)
C) What is the quantity at which average total cost is minimized, qmin? (Hint: consider the relationship between AT C and M C when AT C is minimized.) What is the average total cost at qmin?
D) 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
The cost structure is C(q) = 1000 + 10q2. Demand function is D(p) = 10, 000 100p.
A) Fixed cost is the constant term in the cost function and here it is FC = 100. The variable part is variable cost and it is given by VC = 10q2. Now ATC or average total cost is per unit cost and it is TC/Q or 1000/q + 10q. Similarly, AVC is VC/Q and here AVC = 10q.
B) The marginal cost function is given by M C(q) = 20q. Since minimum AVC is 0, we start the supply function from Q = 0. Hence supply function (which is the rising MC curve) is MC = P = 20q or q = P/20 simplified to q = 0.05P
C) The quantity at which average total cost is minimized, qmin is the one at which ATC = MC
1000/q + 10q = 20q
1000/q = 10q
10q^2 = 1000
q^2 = 100
qmin = 10. Hence the quantity at which average total cost is minimized is 10 units. Now ATC is minimum where q is 10. Hence minimum ATC is 1000/10 + 10*10 = $200.
D) If there is free entry and exit, then then the long-run equilib- rium price, p is actually the minimum of ATC. Hence, $200 is the long run price. At this price, each individual firm produce 10 units.
This part of the question is ambiguous because demand function is Q = 10000 - 100P and P is 200. This gives demand Q = 10000 - 100*200 = -10000 which is not correct. This means, either the demand function is incorrectly given or the cost function is misinterpreted.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.