QUESTION 1 1. There are three firms in the industry. All firms have the same discount factor δ < 1. They compete in the style of Bertrand. The demand is as follows : The quantity demanded is Qo = 30 units if p ≤ 10, quantity demanded is 0 if p > 10. Firm 1 has the cost function c(q) = 2q, and Firms 2 and 3 both have the cost function c(q) = 4q. a. Compute the Bertrand Nash equilibrium of the one‐shot game. Firm 1 (the low cost firm) sets p = 3.9999… the other two firms set p = 4.
b. Describe the trigger (grim) strategy that the firms can use to support a collusive outcome in which all firms charge the monopoly price in every period in the (infinitely) repeated game. Start with p =10 and continue with p = 10 as long as all other firms set p = 10 in all past
periods. If any one firm sets a price other than p = 10 in any period switch to the price you
would set in the Bertrand Nash equilibrium of the one-shot game and continue so for ever
after.
c. Compute the range of discount factors for which there is a Nash equilibrium in the repeated game in which the firms use the trigger strategy you described above. We need to worry about the low cost firm. Following the grim strategy brings a profit of 80
in every period. Cheating and setting p =9.99 and then switching to the Bertrand Nash
equilibrium price of the one-shot game brings a profit of 240 in the period when you cheat
and then 60 in each period after.
80
60
> 240 + δ
gives us δ > 8/9
1− δ
1− δ
d. Now, assume that the market demand (Qo) decreases by 10% every period. Compute the range of discount factors for which there is a Nash equilibrium in the repeated game in which the firms use the trigger strategy you described above. 80
60
80
60
gives us δ > 80/81.
> 240 + 109 δ
Replace
with
> 240 + δ
9
1 − 10 δ
1 − 109 δ
1− δ
1− δ
QUESTION 2. Cournot with free‐entry. The timing: In stage 1 firms decide whether to enter. If a firm decides to enter it pays a fixed entry cost of F = 12,5. Stage 2; Those firms that have entered compete in the style of Cournot. The inverse demand is P = 60 − 2Q, all firms have the same cost function c(q) = 10q. Suppose there are n firms in the industry. Write down the payoff function for Firm 1. pq1 –10q1 = [60 –2(q1 + q2 + … qn)]q1 – 10q1 Compute the best response function for Firm 1. We differentiate the payoff function with respect to q1 set equal to zero solve it for q1. 60 –4q1 – 2(q2 + … qn) – 10 = 0 q1 = [ 50 – 2(q2 + … qn) ]/4. Note that since all firms are identical, the Cournot‐Nash equilibrium quantities will be the same. Use this information and the best response function to compute the quantity produced by the individual firm, q*, price, P*, and profits per firm, π*, in the Cournot‐Nash equilibrium. We take q1 = [ 50 – 2(q2 + … qn) ]/4 and let q* = q*1 = q*2 = …= q*n.
This gives us 4q* = 50–2(n–1)q* and q* = 25/(n+1).
With n firm total production is Q* = np* = 25n/(n+1), therefore the equilibrium price is
p* = (50+10n)/(n+1)
2
1 ⎛ 50 ⎞
Finally, each firm makes a profits of ⎜
⎟ . (We use the formula (p* – 10)q* to
2 ⎝ n +1 ⎠
compute profits.)
How many firms will there be in the market in the Cournot‐Nash equilibrium with free‐
entry? Use your answer to part c. 2
The equilibrium number of firms n* is given by
1 ⎛ 50 ⎞
⎜
⎟ = 12.5 which gives us n* = 9.
2 ⎝ n * +1 ⎠
QUESTION 3. Entry deterrence There are two firms: the incumbent and the entrant. Both firms have the same constant marginal cost of 40. ( c(q) = 40q for both firms.) The timeline : First, the incumbent chooses its output; that decision once made, is irreversible. Then, the entrant decides whether to enter. If it enters it pays a fixed cost of entry F = 25, and chooses its output level. The market demand is P = 68 – Q. a. Compute the monopoly output level for the incumbent. P = 54, Q = 14.
b. If the incumbent produces the monopoly quantity will the entrant enter? If so, how much output will it produce? How much profit will the incumbent make in that case? Entrant’s best response to q = 14 is to produce 7 units.
Entrant’s profit function is (68 – Q)qE – 40qE. With 14 units produce by the incumbent this
becomes (54 – qE)qE – 40qE. Differentiate this set equal to 0 solve for qE : we get qE = 7.
Profits for the entrant are 49 bigger than the entry cost so the entrant will enter.
c. Compute the smallest quantity that the incumbent must produce to deter entry. How much profit will the incumbent make in that case? Entrant’s profit function is (68 – qE – qI)qE – 40qE. Differentiate this set equal to 0 solve for
qE : we get qE = (28 – qI)/2. Profits for the entrant will be (28 – qI)2/4 to make this equal to
25 the entry cost the incumbent must produce qI = 18.
d. Based on your findings so far can you determine whether entry deterrence is the best strategy for the incumbent? Entry deterrence is better. The price is higher with entry deterrence and the incumbent has
100% market share. Without entry deterrence the price is lower and the entrant produces less.
QUESTION 4. What is the relationship between the market size and the number of firms (industry structure) when entry costs are exogenous? How does that relationship change when entry costs are endogenous? In your answer please explain the concept of endogenous entry costs. PLEASE READ the endogenous cost section of CHAPTER 14 and consult the lecture notes
posted on the course website.
Additional question QUESTION 3. Consider and industry with inverse demand P = 21 − Q. There are N firms in the industry, all firms have the cost function TC(q) = F + q. With N = 3, each firm will produce 5 units in the Cournot Nash equilibrium. With N = 4, each firm will produce 4 units in the Cournot Nash equilibrium. Use that information in your answers a. What is the increase in the consumer surplus as N goes from 3 to 4? N = 3, q = 5, Q = 15 p = 6 CS = (21-6)x15/2 = 112.5
N = 4, q = 4, Q = 16 p = 5 CS = (21-5)x16/2 = 128
The change in CS = 15.5
b. What is the decrease in the sum of firms’ profits as N goes from 3 to 4? N = 3, q = 5, Q = 15 p = 6 total profits = (6-1)x15 = 75
N = 4, q = 4, Q = 16 p = 5 total profits = (5-1)x16 = 64
The change in total profits = -11
c. What is the profit per firm when there are 4 firms in the industry? 16
d. Suppose there are 3 firms in the industry. Based on your answers to parts a, b, and c, please compute the value of F (the entry cost) for which the entry by one more firm reduces social welfare but is profitable for the entering firm. If F < 4.5 entry by the fourth firm improves total welfare but the fourth firm will enter
whenever F < 16. So if F is bigger than 4,5 and smaller than 16, the fourth firm will ente
even though this reduces total welfare