Competition with the Leader
In country B, the market for beer is organized as follows: There is a major brewery, Brewery A, and numerous (potentially very many) microbreweries and craft beer producers. Brewery A is the market leader and sets the price for a bottle that others follow as price-takers. After Brewery A sets the price, each price-taking firm decides how much beer to produce (if any). Any demand not met by these firms is satisfied by Brewery A.
Assume the market parameters are as follows:
- beer is a homogenous product, but the production technologies may differ across firms;
- the demand function is D(p) = 20,000 − 3p ;
- the marginal cost of Brewery A is constant and equal to MCA = 30 ;
- marginal cost of price-taking firm number i is MCi = 3i + qi, where i = 1, 2, 3,..., and qi is the amount of beer produced by firm i
Also, just in case you forgot, 1 + 2 + 3 +... + n = n(n+1)/2.
(a) (5 rp) Determine the supply function of a price-taking firm i. Don't forget to specify the supply level under all prices.
(a) p = MCi gives p = 3i + qi, or qi = p − 3i. Production will be positive only if p > 3i, otherwise it is 0.
(b) (5 rp) How many price-taking firms will produce beer under different prices set by Brewery A?
(b) The minimal price for firm i to produce a positive amount of beer is slightly higher than 3i. If the price is lower or equal to 3, there are no price-taking producers, if it is higher than 3 but less or equal to 6, there will be 1 price-taking producer, and so on. This can be summarized by saying that there will be n = [p/3] firms if p is not divisible by 3, where [x] denotes the closest integer not greater than x, and n = (p/3 − 1) if p is divisible by 3.
(c) (10 rp) What is the profit-maximizing price that firm A should set?
(c) If there are n price-taking firms with positive level of beer production, their total supply is
S = n \times p − 3 (1 + 2 +... + n) = n \times p − 3n(n+1)/2.
Let's assume for simplicity that n = p/3, not [p/3]. Then, S = p^2 /6 − p/2. In this case, the residual demand (after small firms produce their outputs) is RD(p) = 20,000 − 3p − (p^2 /6 − p/2) = 20,000 − p^2 /6 − 2.5p. Given MCA = 30, the profit is equal to
Pr_A = (20,000 − p^2 /6 − 2.5p) \times (p − 30).
Taking the derivative with respect to p, we get
PrA' = 20,000 − p^2 /2 − 5p + 10p + 75 = 20,075 + 5p − p^2 /2.
It equals to 0 under positive p \approx 205.44. Under this price, exactly 68 price-taking firms will enter the market, so we can recalculate residual demand and profit more accurately:
RD(p) = 20,000 − 3p − (68p − 3 \times 68 \times 69/2) = 27,038 − 71p
PrA = (27,038 − 71p) \times (p − 30).
The profit-maximizing price is approximately p \approx 205.41.
(d) (5 rp) You are the economic advisor to the President of Country B. The President believes that to improve consumers' welfare, the government needs to cease operations of Brewery A because it has enormous power in setting prices, which is always bad for consumers. If this policy is implemented, the market becomes perfectly competitive with only price-taking firms (i = 1, 2, 3,... ). What advice would you give regarding such a policy?
(d) Again, let's assume n = p/3. In this case, the equilibrium will be determined by 20,000 − 3p = p^2 /6 − p/2. This is a quadratic equation, its positive solution is slightly less than 339. If it was exactly 339, then 339/3 = 113 firms would enter the market. But it is slightly less, so, the 113th firm will not produce anything, and there will be 112 firms.
Comparing this with (c), we can infer that the situation with the dominant firm is better for consumers (price is lower). So, even though Brewery A sets the price, the consumers are better off with it being in place. The policy does not help consumers and thus should not be implemented.
(e) (5 rp) Now assume that instead of shutting down Brewery A's, the government orders it to become one of the price-taking firms. That is, the market becomes perfectly competitive with Brewery A and some other firms. How many firms will there be in the market in equilibrium? Compare this situation with (c) from the perspective of: (1) market concentration, (2) consumers' welfare.
(e) Brewery A can sell any amount of beer for price 30 or higher. That is, under perfect competition, no firms with necessary prices >30 will operate, that is, there will be only (30/3 − 1) + 1 = 10 firms with positive production, and the equilibrium price will be 30. 9 firms with MC = 3i + qi will produce only 135 bottles of beer, while Brewery A will cover 20,000 − 3 \times 30 − 135 = 19,775 bottles. So, the market concentration is higher than in (c), but the consumers are happier, because the price is lower.