Question

Consider a market where inverse demand is given by P = 40 − Q, where Q is the total quantity produced. This market is served by two firms, F1 and F2, who each produce a homogeneous good at constant marginal cost c = $4. You are asked to analyze how market outcomes vary with industry conduct: that is, the way in which firms in the industry compete (or don’t). First assume that F1 and F2 engage in Bertrand competition. 1. (6 points) What will be the equilibrium price, and the profits for each firm? Next assume that F1 and F2 engage in perfect collusion: that is, they both agree to charge a common price P to maximize the sum of their profits. 2. (6 points) What price will the firms set, and what total quantity will be produced? Now assume instead that F1 and F2 engage in Cournot competition: that is, they compete by choosing quantities rather than prices. 3. (6 points) Suppose that F1 believes that F2 will produce a given quantity q2. Show, by analyzing F1’s profit-maximizing output choice, that F1’s best response is to produce the quantity q1 = 36−q2 2 . [Hint: For full credit, you should derive this result from the relevant optimality conditions; don’t just plug in to the formula from lecture.] 4. (6 points) Using the fact that in equilibrium F1 and F2 should both produce the same quantity, find the Nash equilibrium quantity q ∗ produced by each firm. Based on this, find the equilibrium market price. [Hint: in general, q ∗ = a−c 3 . You may use this formula to check your answer, but for full credit you should derive the result yourself.] Finally, suppose that each firm in this industry emits one ton of carbon per unit produced. The government is considering a carbon tax of $6 per ton, which will increase each firm’s effective marginal cost by $6, to a final marginal cost of c = $10 per unit produced. You are asked to analyze how this proposed carbon tax will impact consumer prices. 5. (10 points) What will be the new equilibrium price if F1 and F2 compete a la Bertrand? If F1 and F2 compete a la Cournot? If F1 and F2 are perfectly collusive? [Note: for this part, it is fine to use formulas given in lecture.] 6. (6 points) How does the change in price induced by the tax relate to the nature of competition in the market? Briefly discuss.

Answer 1

1. Demand curve p=40-q

marginal cost c=4$

If both firms engage in bertrand competition, both will keep lowering their price compared to the other to capture the entire market. This will keep going on until both start The price becomes equal to the mc=4$ and their profits become 0

Thus P=4=40-q

q=36 and each firm will produce 36/2=18 units for a price of 4$ and their profits will be 4*18-4*18=0

2. Now for perfect collusion, they will maximize the joint profit and then

Joint profit = pq-cq where q is the total quantity produced by both firms

Profit = pq-cq= (40-q)q -4q

For maximum profit, we differentiate it and equate to 0, we get

40-2q-4=0

q=18

Thus the total quantity produced in this case will be 18 (9 by each firm) and the price will be 40-q=40-18=22

The profits will be 22*18-4*18= 324 (162 for each firm since it will be split in half)

3. For cournot equilibrium

Let q be the total quantity and q1 the quantity produced by f1 and q2 the quantity produced by f2.

Suppose f1 assumes f2 will produce q2

We have the price

p=40-q= 40-q1-q2

Profit for f1 = pq1-cq1 = (40-q1-q2)q1-4q1

For maximum profit, we differentiate it and equate it to 0, we get

40-2q1-q2-4=0

q1= (36-q2)/2= 18 - q2/2

Hence proven, that with profit maximization output choice for f1 given that f2 chooses to produce q2

is q1=(36-q2)/2

4. Using the fact that in equilibrium F1 and F2 should both produce the same quantity (we can also prove it by q1=(36-q2)/2 which we calculated above and q2=(36-q1)/2 which we can calculate as firm 2's profit maximization equation and solve them at the same time)

We have

q1=18-q1/2

3q1/2=18

q1=2*18/3 = 12

Therefore q1=q2=12

Therefore the nash equilibrium quantity produced by each firm will be q* = 12

and the equilibrium market price = 40-12-12=16

5. If a carbon tax of 6$ per unit is added, the new marginal cost = 10$

Now under Bertrand competition, the equilibrium price to consumer will become 10$ as explained in 1.

Under cournot equilibrium, we have

Profit for f1 = pq1-cq1 = (40-q1-q2)q1-(4+6)q1 =(40-q1-q2)q1-10q1

For maximum profit, we differentiate it and equate it to 0, we get

40-2q1-q2-10=0

q1= (30-q2)/2= 15 - q2/2

and similarly q2 = (30-q1)/2= 15 - q1/2

solving them at the same time gives q1=q2 and so

3q1/2=15

q1=10 and so q2=10

The equilibrium price will thus become 40-10-10=20

If both are perfectly collusive we have

Joint profit = pq-cq where q is the total quantity produced by both firms

Profit = pq-cq= (40-q)q -10q

For maximum profit, we differentiate it and equate to 0, we get

40-2q-10=0

q=15 which is the total quantity produced (7.5 by each firm)

Price = 40-q=40-15=25

6. The change in price induced by the tax is related to the nature of market, when the tax is put, the prices increased. It increases by equal to the tax in bertrand competition (Both from 4 to 10$), it increased by half of the tax in perfect collusion (tax was 6$ but prices went from 22 to 25) and increased from 16 to 20$(Less than the tax) in cournot equilibrium. The changes in price were highest under bertrand competition, then under cournot competition and then under perfect collusion.

Hope it helps. Do ask for any clarifications if required.