Question

Two firms A and B control the market for mouth guards and face a market demand curve for mouth guards given by

Q = 320 − 4P

where Q is the total sales of mouth guards. The short-run cost curves of the two firms are given by

SRTC_A = 2090 + (q²A)/4

SRTC_B = 1045 + (q²B)/2

1. a) If the two firms agree to collude in order to maximize their joint profits, what will be output of each firm and the price of mouth guards? What will be the profit of each firm?

2. b) If instead the two firms act as price takers, what will be the output of each firm and the price of mouth guards? What now will be the profit of each firm?

3. c) What is the deadweight loss associated with the collusive outcome?

4. d) Show that at the collusive outcome, each firm has an incentive to

   cheat by raising its output.

Answer 1

Answer 2

a) If the two firms agree to collude to maximize their joint profits, they will act as a single monopolistic entity in the market. In a collusive outcome, they will set the market quantity and price jointly to maximize their combined profits.

To find the collusive output and price, we need to equate the market demand and total quantity supplied by the two firms:

Market Demand: Q = 320 - 4P Total Quantity Supplied by Firms A and B: QA + QB

Since the two firms are colluding, they will act as a single entity to maximize their joint profit, so they will have the same quantity supplied, i.e., QA = QB = Q.

Equating the market demand and total quantity supplied: 320 - 4P = Q

Now, let's find the profit-maximizing output and price: Total Cost for Firm A: CTA = 2090 + (q2A)/4 Total Cost for Firm B: CTB = 1045 + (q2B)/2

Total Cost for the colluding entity: CT = CTA + CTB CT = (2090 + (Q^2)/4) + (1045 + (Q^2)/2) CT = 2090 + (3/4)Q^2

To maximize joint profits, the colluding entity will set marginal cost equal to marginal revenue: Marginal Revenue (MR) = Market Demand (P)

The total revenue (TR) for the colluding entity is: TR = P * Q TR = P * (320 - 4P) TR = 320P - 4P^2

Marginal Revenue (MR) is the derivative of total revenue with respect to quantity (Q): MR = d(TR)/dQ MR = 320 - 8P

Setting MR equal to P to find the price: 320 - 8P = P 320 = 9P P = 320/9 P = 35.56 (approximately)

Now, plug the price back into the demand equation to find the collusive output: Q = 320 - 4P Q = 320 - 4(35.56) Q = 320 - 142.24 Q = 177.76 (approximately)

So, each firm will produce 177.76 units, and the price of mouth guards will be $35.56.

Profit for each firm: Profit = Total Revenue - Total Cost Profit = (P * Q) - CT Profit = (35.56 * 177.76) - (2090 + (3/4)(177.76^2)) Profit = 6331.58 (approximately)

b) If the two firms act as price takers, they will independently maximize their profits, taking the market price as given. In a competitive market, the price will be determined by the market demand and supply. To find the output and price when they act as price takers, we equate the market demand and total quantity supplied by the two firms:

Market Demand: Q = 320 - 4P Total Quantity Supplied by Firms A and B: QA + QB

Each firm will maximize its profit independently by setting its marginal cost equal to the market price (P). Since the cost curves are given by SRTCA = 2090 + (q2A)/4 and SRTCB = 1045 + (q2B)/2, the marginal cost for each firm will be the derivative of its total cost with respect to quantity (q):

Marginal Cost (MC) for Firm A: MC = d(CTA)/dq = qA/2 Marginal Cost (MC) for Firm B: MC = d(CTB)/dq = qB

Setting marginal cost equal to the market price to find the output: For Firm A: qA/2 = P qA = 2P

For Firm B: qB = P

Now, substitute the expressions for qA and qB into the total quantity supplied equation: QA + QB = 2P + P = 3P

Equate the total quantity supplied (3P) to the market demand (320 - 4P) to find the market price (P): 3P = 320 - 4P 7P = 320 P = 320/7 P = 45.71 (approximately)

Now, plug the price back into the expressions for qA and qB to find the output of each firm: qA = 2P qA = 2 * 45.71 qA = 91.42 (approximately)

qB = P qB = 45.71 (approximately)

So, when the two firms act as price takers, Firm A will produce approximately 91.42 units, Firm B will produce approximately 45.71 units, and the price of mouth guards will be approximately $45.71.

Profit for each firm: Total Cost for Firm A: CTA = 2090 + (q2A)/4 Total Cost for Firm B: CTB = 1045 + (q2B)/2

Profit for Firm A: ProfitA = (P * qA) - CTA ProfitA = (45.71 * 91.42) - (2090 + (91.42^2)/4) ProfitA = 3863.15 (approximately)

Profit for Firm B: ProfitB = (P * qB) - CTB ProfitB = (45.71 * 45.71) - (1045 + (45.71^2)/2) ProfitB = 825.04 (approximately)

c) The deadweight loss associated with the collusive outcome is the difference between the total surplus in a competitive market and the total surplus under collusion.

In a competitive market, the quantity supplied by each firm is equal to the market quantity (Q) divided by the number of firms (n). In this case, n = 2 (Firm A and Firm B).

Competitive market output per firm: qC = Q/n = 320/2 = 160

The competitive market price is the market demand when Q = 2qC: Pc = 320 - 4(2qC) = 320 - 4(2 * 160) = 320 - 128 = 192

Total surplus in a competitive market: Total Surplus = Consumer Surplus + Producer Surplus Consumer Surplus = (1/2)(Q - Pc) * Q = (1/2)(320 - 192) * 320 = 5,120 Producer Surplus = (1/2)(Pc) * Q = (1/2)(192) * 320 = 30,720

Total Surplus in a competitive market = 5,120 + 30,720 = 35,840

Total surplus under collusion: Total Surplus = (P * Q) - CT = (35.56 * 177.76) - (2090 + (3/4)(177.76^2)) = 6331.58

Deadweight Loss = Total Surplus in a competitive market - Total Surplus under collusion Deadweight Loss = 35,840 - 6331.58 = 29,508.42 (approximately)