Question 2
Consider an industry with 2 firms. The firms' products are perfect substitutes (like the spring water
suppliers in the original Cournot model). Market demand for this good is given by Q(p)=180-2p,
and its inverse is denoted as p(Q)=90-0.5Q.
Each firm chooses an amount of output to produce to maximize its profit. The quantity choices are
made simultaneously. Firm 1's output level is denoted as q_(1); Firm 2's output level by q_(2). Firms are
aware that given q_(1), and q_(2), the price is determined by Q(p)=180-2p=q_(1)+q_(2) condition,
p(q_(1)+q_(2))=90-0.5(q_(1)+q_(2))
The cost functions are the same for both firms: C(0)=0,C(q)=F if q>0. F I sa fixed cost but not a
sink cost, F can be avoided by not producing any output.
Verify that if 1800>=F>1012.5, there will be three Cournot equilibria:
Eq #1: q_(1)=60;q_(2)=60->p=30
Eq #2: q_(1)=90;q_(2)=0->P=45
Eq #3: q_(1)=0;q_(2)=90->P=45
You will accomplish your task by completing steps 1 and 2 below:
Step 1.
Please verify that if F<=1800,q_(1)=60;q_(2)=60 is a Cournot equilibrium. You need to verify that when
q_(1)=60, the profit maximizing output level for firm 2 is q_(2)=60; and when q_(2)=60, the profit
maximizing output level for firm 1 is q_(1)=60.
Step 2.
Please verify that if F>1012.5,q_(1)=90;q_(2)=0 is also a Cournot equilibrium: you need to verify that
when q_(1)=90, the profit maximizing output level for firm 2 is q_(2)=0; and when q_(2)=0, the profit
maximizing output level for firm 1 is q_(2)=90.
Help for step 1 and 2. Finding the best response q_(1) of firm 1 to a given quantity q_(2) of firm 2: Given q_(2),
firm 1 can assure 0 profit by choosing q_(1)=0. But suppose firm 1 chooses q_(1)>0. What would be the
best q_(1) in that case? Note that when we look for the best q_(1) conditional on q_(1) being strictly positive, F
is not going to have an effect on the profit maximizing q_(1). Then, ask: how much profit does firm 1
make when they choose the best q_(1) subject to q_(1)>0 ? If q_(1)>0, then firm 1 's cost is F. If these profits
are negative, best response will be q_(1)=0.
Question 2 Consider an industry with 2 firms. The firms' products are perfect substitutes (like the spring water suppliers in the original Cournot model). Market demand for this good is given by Q(p) = 180 -- 2p, and its inverse is denoted as p(Q) = 90 -- 0.5Q. Each firm chooses an amount of output to produce to maximize its profit. The quantity choices are made simultaneously. Firm 1's output level is denoted as q1; Firm 2's output level by q2. Firms are
p(q1+qz)=90-0.5(q1+qz) The cost functions are the same for both firms: C(0)=0,C(q)=F if q>0.F I sa fixed cost but not a sink cost, F can be avoided by not producing any output.
Verify that if 1800 F > 1012.5, there will be three Cournot equilibria: Eq#1: q1=6O;q2=60> P=30 Eq#2: q1=90;q2=0>P=45 Eq#3: q1=0;q2=90>P=45 You will accomplish your task by completing steps 1 and 2 below:
Step 1. Please verify that if F 1800, q1 = 60; q2 = 60 is a Cournot equilibrium. You need to verify that when q1 = 60, the profit maximizing output level for firm 2 is q2 = 60; and when q2 = 60, the profit maximizing output level for firm 1 is q1 = 60.
Step 2. Please verify that if F > 1012.5, q1 = 90; q2 = 0 is also a Cournot equilibrium: you need to verify that when q1 = 90, the profit maximizing output level for firm 2 is q2 = 0; and when q2 = 0, the profit maximizing output level for firm 1 is q2 = 90.
Help for step 1 and 2. Finding the best response q1 of firm 1 to a given quantity qz of firm 2: Given q2, firm 1 can assure 0 profit by choosing q1 = 0. But suppose firm 1 chooses q1 > 0. What would be the best q1 in that case? Note that when we look for the best q1 conditional on q1 being strictly positive, F is not going to have an effect on the profit maximizing q1. Then, ask: how much profit does firm 1 make when they choose the best q1 subject to q1>0? If q1>0,then firm1's cost is F.If these profits are negative, best response will be q1 = 0.
