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, q^min? (Hint: consider the relationship between AT C and M C when AT C is minimized.) What is the average total cost at q^min?

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^∗.)

Answer 1

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.