Question

Two firms are price-competing as in the standard Bertrand model. Each faces the market demand function D(p)=50-p. Firm 1 has constant marginal cost c1=10 and firm 2 has c2=20. As usual, if one of the firms has the lower price, they capture the entire market, and when they both charge exactly the same price they share the demand equally.

1. Suppose A1=A2={0.00, 0.01, 0.02,...,100.00}. That is, instead of any real number, we force prices to be listed in whole cents. Find a Nash Equilibrium in pure strategies for this game.

2. Find the total output Q=D(p) and the profits of both firms for the NE you found.

Answer 1

1) NE : <P1, P2> = < 20,20>

P1 can't be below 10 , as then firm will have losses

P2 can't be below 20 , bcoz then price will be below MC, so it implies losses

since if P1> P2>20, then both will undercut each other to become lowest price firm & capture entire market

Similarly P1>P2> 20. , Doesn't constitute the NE bcoz profitable deviations exist

P1 = 10 , P2= 20, then firm 1 has incentive to increase price & earn positive profit

Thus P1= P2 = 20 is the NE.

Also P2= 21 is also possible, since if P2 = 20,then profit is zero , but positive market share.

So if P2 = 21 , also imply zero profit , as positive markup over price, but zero market share .

So < P1 = 20, P2 =21> is also one NE

2) if P1= P2 = 20

Then Q= 50-20 = 30

Each produce half of output = 15.

π1 = (20-10)*15 = 150

π2 = 0

If P1 =20 , P2 = 21

So π1 = (20-10)*30 = 300

Firm 1 gets entire Market share

π2 = 0