Consider the perfectly competitive industry for gadgets where there are initially 10 identical firms producing. Each firm has the same MC and TC curves and these curves are given by the following equations where TC is total cost, q is the amount of gadgets produced by a firm, and MC is the marginal cost:
TC for each firm: TC = 4 + 5q + q2
MC for each firm: MC = 5 + 2q
You also know that the market demand curve is given by the following equation where P is the price per gadget and Q is the total amount of gadgets demanded in the market:
Market Demand for Gadgets: Q = 17 – P
- Given this information, the first thing you need to do is to determine the short-run equilibrium. But, you cannot do this until you find the market supply curve. This will take some work: you will need to have some conceptual knowledge (the market supply curve is the horizontal summation of the firms’ MC curves) and some creativity (you will need to figure out how to get a second point that lies on the individual firm’s MC curve and then extrapolate from this point to the market supply curve). Find the market supply curve and show the work you did to find this curve.
- Given the information and the market supply curve you found in (a), compute the short-run equilibrium market quantity (Q), the short-run equilibrium price (P), and the short-run level of production for a representative firm (q). Show your work.
- Given your work in (b), determine the level of short-run profits for a representative firm in this industry? Given your calculation will the representative firm produce in the short-run? Explain your answer. Given your calculation what do you predict will happen in this industry in the long-run? Explain your answer.
- Calculate the long-run equilibrium price (P’), long-run equilibrium market quantity (Q’), the long-run equilibrium firm quantity (q’), and the level of profits in the long-run for a representative firm. Show your work. In the long-run how many firms will be in this industry?
Answer:
A
we are given the firm’s MC curve: MC = 5 + 2q. This equation tells us that the firm’s MC curve has a y-intercept of 5. Since the firm’s level of production when MC = 5 is 0 units we can extrapolate that when the y-intercept of the market supply curve is 5, the market quantity produced is 0 units. Now, we need another point on the market supply curve. So, return to your MC curve and pick a quantity (for example, q = 1) and compute the MC for this quantity: thus, if q = 1, MC = 7. If one of the firms has MC of $7 when it produces a quantity of one unit, this implies that if the market price is $7 then there will be a total of 10 units (since there are ten identical producers in this market). So, now you have two points on the market supply curve: (Q, MC) = (0, $5) and (10, $7). Use these two points to find the linear market supply curve:
P = 5 + (1/5)Q.
B
To find the short-run market equilibrium we need to find where the market demand curve intersects the market supply curve. Thus, 5 + (1/5)Q = 17 – Q or Q = 10. Plug Q = 10 into either the market demand curve or the market supply curve to find the equilibrium short-run market price: P = 17 – 10 = $7 or P = 5 + (1/5)(10) = $7. To find the short-run level of production for the firm, set MR = MC or 7 = 5 + 2q or q = 1. With ten firms, each producing one unit of output that will provide us with the market output of 10 units.
C
Profits for a representative firm are equal to total revenue (TR) minus total cost (TC). TR = P*q = ($7 per unit of output)(1 unit of output) = $7. TC = 4 + 5(1) + (1)(1) = $10. Short-run profits for a representative firm are equal to -$3.
To determine whether this firm will produce in the short-run or not we need to see if the TR is at least large enough to cover the variable costs of production. From the TC curve we can see that fixed costs are equal to $4 (if q = 0, then TC = FC). So, variable costs (which vary as the level of output varies-a lesson I hope you saw in problem 1 of this homework) can be written as VC = 5q + q*q and since q =1 that tells us VC for the representative firm equal $6. So, the TR of $7 is greater than the VC of $6, so this firm will choose to produce in the short-run because its revenues are sufficient to cover its variable costs of production.
Given that short-run economic profits for the representative firm are negative, we can anticipate that some firms will exit the industry in the long-run. This will cause the market supply curve to shift to the left and holding everything else constant result in a higher market equilibrium price, a smaller market equilibrium quantity, and a larger level of production from those firms that remain in the industry.
D
To find the long-run equilibrium we start by recognizing that in the long-run the representative firm will produce at that level of output where it earns 0 economic profit. This will occur where MR = MC = ATC. We know the MC and we can find the equation for ATC. So, let’s start by equating these two curves: MC = 5 + 2q and ATC = 4/q + 5 + q. Thus, 5 + 2q = 4/q + 5 + q or q = 4/q. We can rearrange this to get q*q = 4 or q = 2 (in this example we don’t need to worry about the possibility that q = -2: a firm cannot produce a negative level of output). When the firm produces 2 units of output, we get MC = 5 + 2(2) = $9. Remember that MC = MR = ATC for the firm in long-run equilibrium: this implies that if MC = $9, then MR = $9. If MR = $9, this implies that the market equilibrium price is also $9. Use this price and the market demand curve to find the market quantity: P = 17 – Q or Q’ = 8 gadgets. With each firm producing 2 gadgets and a total production of 8 gadgets, this implies that there are 4 firms in the industry at the long-run equilibrium. So, six firms exited the industry in the long-run, P’ = $9, Q’ = 8 gadgets, and q’ = 2 gadgets. For the representative firm in the long-run, TR = ($9 per gadget)(2 gadgets) = $18 and TC = 4 + 5(2) + (2)(2) = $18: therefore, long-run profits for the firm are equal to zero.
