Perfect Competition Questions Question 1 Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm’s total cost is given by the equation TC = 100 + q2 + q where q is the quantity of output produced by the firm. You also know that the market demand for this product is given by the equation P = 1000 – 2Q where Q is the market quantity. In addition you are told that the market supply curve is given by the equation P = 100 + Q. a. What is the equilibrium quantity and price in this market given this information? To find the equilibrium set market demand equal to market supply: 1000 – 2Q = 100 + Q. Solving for Q, you get Q = 300. Plugging 300 back into either the market demand curve or the market supply curve you get P = 400. b. The firm’s MC equation based upon its TC equation is MC = 2q + 1. Given this information and your answer in part (a), what is the firm’s profit maximizing level of production, total revenue, total cost and profit at this market equilibrium? Is this a short-run or long-run equilibrium? Explain your answer. From part (a) you know the equilibrium market price is $400. You also know that the firm profit maximizes by producing that level of output where MR = MC. Since the equilibrium market price is the firm’s marginal revenue you know that MR = $400. Setting MR = MC gives you 400 = 2q + 1, or q = 199.5. Thus, the profit maximizing level of output for the firm is 199.5 units when the price is $400 per unit. Using this information it is easy to find total revenue as the price times the quantity: TR = ($400 per unit)(199.5 units) = $79,800. Total cost is found by substituting q = 199.5 into the TC equation: TC = $40,099.75. Profit is the difference between TR and TC: Profit = TR – TC = 79,800 – 40,099.75 = $39,700.25. Since profit is not equal to zero this cannot be a long-run equilibrium situation: it must be a short-run equilibrium situation. c. Given your answer in part (b), what do you anticipate will happen in this market in the long-run? Since there is a positive economic profit in the short run, there should be entry of firms in the long-run resulting in an increase in the market quantity, a decrease in the market price, and firms in the industry earning zero economic profit. d. In this market, what is the long-run equilibrium price and what is the long-run equilibrium quantity for a representative firm to produce? Explain your answer. The long-run equilibrium price is that price that results in the representative firm earning zero economic profit. This will occur when MC = ATC for the representative firm. ATC is just the TC equation divided by q. Thus, 2q + 1 =(100 + q2 + q)/q. Solving for q, q = 10. Plugging 10 in for q into the ATC equation yields the following: ATC = (100 + 102 + 10)/10 = 21. So, when Price equals MR = min ATC = MC = $21, this firm will break even. To see this compute TR for the firm when it produces 10 units and sells each unit for $21: TR = $210. Notice that this is the same as the firm’s TC: thus, the firm earns zero economic profit. e. Given the long-run equilibrium price you calculated in part (d), how many units of this good are produced in this market? To find this quantity you need to substitute $21 (the long-run equilibrium price) into the market demand curve to determine the quantity that the market must produce in order to be in long-run equilibrium. This quantity is equal to 489.5 units. Question 2 The market for study desks is characterized by perfect competition. Firms and consumers are price takers and in the long run there is free entry and exit of firms in this industry. All firms are identical in terms of their technological capabilities. Thus the cost function as given below for a representative firm can be assumed to be the cost function faced by each firm in the industry. The total cost and marginal cost functions for the representative firm are given by the following equations: TC = 2qs2 + 5qs + 50 MC = 4qs + 5 Suppose that the market demand is given by: PD = 1025 - 2QD Note: Q represents market values and q represents firm values. The two are different. a) Determine the equation for average total cost for the firm. ATC for the firm is TC/q, so dividing the total cost equation above by q gives us: ATC = 2qs + 5 + 50/qs b) What is the long-run equilibrium price in this market? (Hint: since the market supply is unknown at this point, it’s better not to think of trying to solve this problem using demand and supply equations. Instead you should think about this problem from the perspective for a firm. Specifically, a long run equilibrium occurs where ATC = MC = Price) In a long-run equilibrium, ATC equals Marginal Cost and profits equal zero. Setting the two equations equal: ATC = 2qs + 5 + 50/qs = 4qs + 5 = MC50/qs = 2qs 50 = 2qs2 25 = qs2 Take the square root of both sides and find: 5 = qs However, the question wants us to find long run prices. We know that the firm produces were Price = MR = MC, so if we can determine the firm’s MC, then we can determine the equilibrium price in the market. We know that: MC = 4qs + 5 And solved for: 5 = qs Substituting: MC = 4(5) + 5 = 25 The equilibrium price in the market is 25. c) What is the long-run output of each representative firm in this industry? We solve for this in the previous part. 5 = qs d) When this industry is in long-run equilibrium, how many firms are in the industry? (Hint: firms are identically sized). Now we should determine the market quantity Q from the market demand curve, given that we know the market price is 25. Market demand is given as: PD = 1025 - 2QD And we know that market price = 25, so: 25 = 1025 - 2QD 1000 = 2QD 500 = QD Since each firm is making 5 units (as we found in parts b and c), there must be 100 firms, since they are all identically sized. Now suppose that the number of students increases such that the market demand curve for study desks shifts out and is given by, PD = 1525 - 2QD e) In the short-run will a representative firm in this industry earn negative economic profits,positive economic profits, or zero economic profits? (Hint: You can solve this without calculation.) The demand curve has shifted to the right. Given what we learned earlier in the semester, we should know that the market price will increase. If market prices …
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