Question

Q4. There are two firms A and B in a homogenous product industry. Inverse demand is P = 120 Q where Q is the combined output of the firms. Firm A has a marginal cost of 0 and firm B has a marginal cost of 10. There is an infinite sequence of periods in which firms simultaneously set prices. In this question we will consider whether the following collusive strategies with trigger strategy punish- ments are a subgame perfect Nash equilibrium: both firms set a price of 60 (and split output equally) unless one of the firms has deviated, in which case they play the static Nash equilibrium forever. The discount factor is 6. (a) Assume that the static Nash equilibrium involves the lower marginal cost firm producing all the price equal to the marginal cost of the higher marginal cost firm (so A would take the output at a entire market at a price of 10). What is A's per-period profit in the punishment phase? What is B's profit? (b) What is A's per-period profit when the firms are setting the collusive price of 60? What is B's profit? (c) Find the critical discount factor for firm A not to want to deviate from the collusive equilibrium? Clearly show your working. (d) B's critical discount factor is 0.5. Give a simple verbal explanation for why one of the firm's has higher critical discount factor in this problem. a

Answer 1

There exists two firms producing a homogeneous product having an inverse demand function : P =120- Q where Q_A + Q_B = Q.

Marginal Cost of production for firm A : MC_A = 0

Marginal Cost of production for firm B : MC_B = 10

a) There is an infinite sequence of periods in which the firms engage in strategic behavior by simultaneously setting prices.

Now we consider a collusive strategic behavior of the two firms with trigger strategy punishments i.e., both firms set a price 60 and split their output equally unless one of them deviates from the strategy in which case the two firms play Nash equilibrium forever as a punishment. The Nash Equilibrium strategy is defined as the lower MC firm (in this case firm A as MC_A < MC_B) all the output at a price equal to the higher MC firm (in this case P=10 as MC_B = 10).

So for each period when Nash Equilibrium strategy is chosen by the firms, firm A captures the entire market by producing at a lower cost and selling at a lower price than firm B. Therefore in this case,

firm A charges price P' = 10 and produces the entire market output Q' = Q_A' = 120 - 10 = 110.

The total revenue of firm A per period of punishment : TR_A' = P' $\times$ Q_A' = 10 $\times$ 110 = 1100

The total cost of firm A per period of punishment : TC_A' = MC_A$\times$ Q_A' = 0 $\times$ 110 = 0

The total profit of firm A per period of punishment phase :
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_A' = TR_A' - TC_A' = 1100 - 0 = 1100

Similarly,

in case of a Nash Equilibrium strategy choice by both firm, firm B ends up producing and selling zero output as the lower MC firm captures the entire market by charging a very low price. Therefore, Q_B' = 0

TR_B' = 0 ; TC_B' = 0

The total profit of firm B per period of punishment phase :
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_B' = TR_B' - TC_B' = 0

b) When the firms are setting Collusive prices of 60 then total output produced in market : Q^* = 120 - 60 = 60

Since the market output is divided equally between two firms , outputs of the individual firms: Q_A^* = Q_B^* = (60 / 2) = 30

The per period total revenue of firm A when setting collusive price : TR_A^* = P^* $\times$ Q_A^* = 60 $\times$ 30 = 1800

The per period total cost of firm A when setting collusive price : TC_A^* = MC_A$\times$ Q_A^* = 0 $\times$ 60 = 0

The per period total profit of firm A when setting collusive price :
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_A^* = TR_A^* - TC_A^* = 1800 - 0 = 1800

Similarly for firm B producing same output as A:

TR_B^* = P^* $\times$ Q_B^* = 60 $\times$ 30 = 1800

TC_B^* = MC_B $\times$ Q_B^* = 10 $\times$ 30 = 300

The per period total profit of firm B when setting collusive price :
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_B^* = TR_B^* - TC_B^* = 1800 - 300 = 1500.

c) In order to find out the critical discount rate fir firm A not wanting to deviate from the Collusive equilibrium, we need to compare the present value of firm A profits over all possible time periods for both collusion and deviation strategy choices made by the firm.

WHEN FIRM A CHOOSES THE COLLUSIVE STRATEGY BY CHOOSING P = 60 IN EACH TIME PERIOD STARTING FROM PERIOD 1 :

Present discounted value of Firm A' s profit in period 1 : _A^* = 1800

71

Present discounted value of firm A's profit in period 2 : $\delta$.
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_A^* = $\delta$.1800

Present discounted value of firm A's profit in period 3 :$\delta$. $\delta$
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_A^* = $\delta$².1800

..............................................................................................and it continues up to infinity period.

Now we take a sum of all the present discounted value of profits for all possible periods when the two firms operate :

P.V. of
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_A^* = 1800 + $\delta$.1800 + $\delta$².1800 +.................... $\infty$ = 1800 (1 + $\delta$ + $\delta$² + ................$\infty$ )

= 1800/(1-$\delta$)

[using the formula of infinite G.P. Series]

WHEN FIRM A CHOOSES TO DEVIATE FROM COLLUSIVE STRATEGY IN PERIOD 1 BY CHOOSING P < 60 AND THEN IN EACH OF THE LATER PERIODS FACES A PUNISHMENT NASH EQUILIBRIUM STRATEGY CHOICE BY FIRM B:

Suppose, firm A choose to deviate in the 1st period from the collusive strategy choice by setting a price slightly less than 60 (say, 60 - ε ) and ends up capturing the entire product market because all the buyers buy from firm A at a slightly lower price than the price at which firm B offers the product(say, collusive price 60). For all practical purposes the very small amount ε can be ignored without loss of generality we can write, P " = 60

Thus, output produced by firm A : Q_A" = 120 - 60 = 60.

Then Total Revenue of firm A in period 1 : TR_A " = P " $\times$ Q_A" = 60 $\times$ 60 = 3600

Total cost of firm A in period 1 : TC_A " = MC_A $\times$ Q_A " = 0 $\times$ 60 = 0

Present discounted value of Firm A' s profit in period 1 : _A" = TR_A" - TC_A" = 3600 - 0 = 3600

71

After the 1st period, firm a faces punishment from period 2 on wards in the form of trigger strategy Nash Equilibrium played by firm B.

Therefore,

Present discounted value of firm A's profit in period 2 : $\delta$.
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_A' = $\delta$.1100

Present discounted value of firm A's profit in period 3 : $\delta$.$\delta$
71
_A' = $\delta$².1100

.............................................................................................and it continues upto infinity period.

Now we take a sum of all the present discounted value of profits for all possible periods when the two firms operate :

P.V. of
71
_A" = 3600 + $\delta$.1100 + $\delta$².1100 + ......................................$\infty$ = 3600 + 1100 (1 + $\delta$ + $\delta$² + ................$\infty$)

= 3600 + 1100 [ $\delta$ / (1 - $\delta$)

[ using the formula of infinite G.P. series]

Now if we compare the two present discounted value of profits for two strategy choice made by firm A in period 1, the critical discount factor where firm A will have no incentive to deviate can be identified by equating the total Present value of profits :

1800 / (1-$\delta$) = 3600 + 1100 [$\delta$/(1-$\delta$) ]

or, 1800 / (1-$\delta$) =[3600 - 3600. $\delta$ + 1100.$\delta$] /  (1-$\delta$)

or, 1800 = 3600 - 2500.$\delta$ [Multiplying by (1-$\delta$) on both sides as $\delta$$\neq$ 1]

or, 2500.$\delta$ = 1800

or, $\delta$ = 1800 / 2500 = 0.72

So the critical discount rate at and above which firm A will not want to deviate is 0.72 because even if he deviates,he cannot earn a higher payoff or profit for the total infinite number of time periods than what he earns by following the collusive strategy throughout. If he deviates in later periods instead of the first period, then again the same logic applies.

d) The critical discount factor for firm B is 0.5 while that of firm a is 0.72

The disparities are caused by the difference in the Cost of production in two firms. Firm A can produce additional output at zero marginal cost while firm B needs to incur 10 units of cost per unit increase in output production. So, evidently firm A can earn higher payoff or profit by deviating as he can produce output at a much lower cost and capture the entire market simultaneously. Also when the Nash equilibrium is played, firm A earns positive profits while firm B earns zero profit. Now discount rate depreciates income or profits in future periods as value of money reduces  over time. The further away an income is earned in future, the more it is discounted. The only possible way to deter low cost Firm A from deviating and capturing the entire market is through a very high discount rate of future profits.Therefore, firm requires higher discount rate for future profits in order to deter him from deviating as he needs to be compensated for his loss of extra earning he could gain by deviating and that amount is much higher than what B could possibly earn. His opportunity cost of collusion is higher thus his critical discount rate must be higher as well.