Question

( Trigger Prices in Games with Uncertainty) Suppose a duopoly is characterized by the following profits: if the two firms collude and charge the joint profit-maximizing price, they each earn a profit equal to 2000 in each period; if the two firms charge the Cournot–Nash price, they each earn a profit equal to 1200 in each period; and if one firm defects while the other charges the joint profit-maximizing price, the firm that defects earns 4000. Now suppose both firms adopt the following strategies: Start by cooperating with each firm charging the joint profit-maximizing collusive price; Continue to sell at the joint profit-maximizing outputs and price unless the other firm increases its output and lowers its price, in which case produce the Cournot–Nash quantity and charge the Cournot–Nash price forever. a. If the probability that the game is played in the next period is 0.99, what discount rates will sustain collusion? b. If the profit under collusion increases, is it easier or harder to sustain collusion?

Answer 1

Let be the minimum discount rate that can sustain collusion.

The collusion will only sustain of the present discounted value of profits from collusion is greater than or equal to the present discounted value of profits fom deviation.

And if either firm deviated it earns higher profit in current period and then from next period they go back to cournot nash equilibrium in whigh both firm earns a profit of 1200.

Profit from collusion = 2000

Profit from deviation = 4000

Profit in nash equilibrium = 1200.

So the present discounted value of profits from collusion will be equal to,

2000 + 0.99×2000 + (0.99)^2×(​​​​​​)^2 + (0.99)^3×()^3×2000 +................

As you can see it's an infinite GP, and the sum of infinite GP is equal to

= a/(1 - r)

Here a = 2000

r = 0.99××2000/2000

r = 0.99×

So putting in the values,

Present value of profit from collusion =

= 2000/(1 - 0.99×​​​​​​) -----------(1)

Now let's calculate the present value of profit from deviation. So if firm deviates from collusion then it will earn a profit of 4000 in current period but from it will earn only 1200 form next period since firms will go back cournot nash equilibrium.

Present value of profit from deviation = 4000 + 0.99××1200 + (0.99)^2 × (​​​​​​)^2 × 1200 + (0.99)^3 × (​​​​​​)^3 × 1200 +..........

Again notice here we have infinite gp from second term onwards with,

a = 0.99××1200

And r = (0.99)^2×(​​​​​​)^2×1200/0.99××1200

r = 0.99×

So the sum of GP will be equal to,

= 0.99××1200/(1 - 0.99×​​​​​​)

And by adding the first term of 4000 we get the present value of profit from collusion as,

Present value of profits from collusion,

= 4000 + 1200×0.99×/(1 - 0.99×​​​​​​) -------(2)

Now let's put the present value of profits from collision > present value of profits from deviation

Equation (1) > equation (2)

2000/(1 - 0.99×​​​​​​) > 4000 + 1200×0.99×/(1-0.99×​​​​​​)

Now we just need to solve for ​​​​​​.

2000/(1-0.99×​​​​​​) = [4000(1 - 0.99×​​​​​​) + 1200×0.99×​​​​​​]/( 1 - 0.99×​​​​​​)

Now cancelling out (1 - 0.99×​​​​​​) on both sides from denominator we get,

2000 > 4000(1 - 0.99×​​​​​​) + 1200×0.99×

2000 > 4000 - 4000×0.99× + 1200×0.99×

(4000×0.99× - 1200×0.99×​​​​​​) > (4000 - 2000)

Taking out and 0.99 common outside from LHS

×0.99 ( 4000 - 1200) > (4000 - 2000)

×0.99 > (4000 - 2000)/(4000 - 1200)

×0.99 > 2000/2800

×0.99 > 0.714

> 0.714/0.99

> 0.7212

So the discount rates that can sustain the collusion is equal to all discount rates greatest than 0.7212..

Now as you can see the minimum discount rate that can sustain collusion is negatively related with the profits in collusion see that profit from collusion that is 2000 enters into numerator with negative sign which is as profits from collusion increases the minimum discount rate that can sustain collision also decreases which means the collusion can sustain for larger range of discount rates.

So if the profits from collusion increases it becomes easier to sustain collusion as the range of discount rates that can sustain collusion becomes larger.