4. Consider a duopoly with firms that offer homogeneous products where each has
ID: 1142135 • Letter: 4
Question
4. Consider a duopoly with firms that offer homogeneous products where each has constant mar- Letting P and p2 be the prices of firms 1 and 2, respectively, the demand function of firm 1 is specified to be ginal cost of c. Let D(P) denote market demand. Firms make simultaneous price decisions. if p, P2 If firm I's price is lower than firm 2's price, then all consumers buy from it, so that its demand equals market demand. If both firms charge the same price, then they equally split market demand. If firm 1's price is higher than firm 2's price, then all consumers go to firm 2. Firm 2's demand function is similarly defined. Each firm chooses prices to maximize its profit. a. Show that both firms pricing at marginal cost is a Nash equilibrium. b. Show that any other pair of prices is not a Nash equilibrium. Suppose that we limit firms to choosing price equal to c, 2c, or 3c. c. Compute the payoff/profit matrix. d. Derive all of the Nash equilibrium price pairs.
Explanation / Answer
4.c) Case-1(Firm1)
Here we consider that the marginal cost of both firms is c.
Demand functions of firm1 has been specified as:-
D1(p1,p2)=p1 if p1<p2
D1(p1,p2)=p1/2 if p1=p2
D1(p1,p2)=0 if p1>p2
The Total Revenues for firm1 under three scenarios can be written as:-
TR1=p1q1 if p1<p2
TR1=P1^2/2 if p1=p2
TR1=0 if p1>p2
Total Cost for firm 1=cq1
Total Profit functions of firm under all 3 scenarios:-
TP1=p1q1-cq1 if p1<p2
TP1=p1^2/2-q1c=q1(p1/2-c) if p1=p2
TP1=0-q1c=-cq1 if p1>p2
Based on the profit maximizing condition of the firm,we can state:-
p1=Marginal Cost of firm 1(MC1)
p1=c
Now,plugging the value of p1 in the above equation into the 3 Total Profit functions for firm1,we get:-
TP1=p1q1-cq1 if p1<p2
TP1=cq1-cq1
TP1=0
TP1=p1^2/2-q1c if p1=p2
TP1=c^2/2-cq1
TP1=(c^2-2cq1)/2
TP1=-cq1 if p1>p2
TP1=-cq1
Hence,notice that the TP1 for firm 1 is maximum when p1=p2 and price is equal to marginal cost.
Case-2(Firm 1)
Now,we consider the MC1 to be 2c.
Total Cost for firm 1:TC1=2cq1
Total Profit functions of firm under all 3 scenarios:-
TP1=p1q1-2cq1 if p1<p2
TP1=p1^2/2-2q1c if p1=p2
TP1=(p1^2-4q1c)/2
TP1=0-2q1c=-2cq1 if p1>p2
Based on the profit maximizing condition of the firm,we can state:-
p1=MC1
p1=2c
Now,plugging the value of p1 in the above equation into the 3 Total Profit functions for firm1,we get:-
TP1=2cq1-2cq1 if p1<p2
TP1=2cq1-2cq1
TP1=0
TP1=(p1^2-4q1c)/2 if p1=p2
TP1=((2c)^2-4q1c)/2
TP1=(4c^2-4q1c)/2
TP1=4c(c-q1)/2
TP1=2c(c-q1)
TP1=-2cq1 if p1>p2
TP1=-2cq1
Again,notice that the TP1 for firm 1 is maximum when p1=p2 and price is equal to marginal cost.
Case-3(Firm 1)
Now,we consider the MC1 to be 3c.
Total Cost for firm 1:TC1=3cq1
Total Profit functions of firm under all 3 scenarios:-
TP1=p1q1-3cq1 if p1<p2
TP1=p1^2/2-3q1c if p1=p2
TP1=(p1^2-6q1c)/2
TP1=0-3q1c=-3cq1 if p1>p2
Based on the profit maximizing condition of the firm,we can state:-
p1=MC1
p1=3c
Now,plugging the value of p1 in the above equation into the 3 Total Profit functions for firm1,we get:-
TP1=p1q1-3cq1 if p1<p2
TP1=3cq1-3cq1
TP1=0
TP1=(p1^2-6q1c)/2 if p1=p2
TP1=((3c)^2-6q1c)/2
TP1=(9c^2-6q1c)/2
TP1=3c(3c-2q1)/2
TP1=3c(3c-2q1)/2
TP1=-2cq1 if p1>p2
TP1=-3cq1
Case-1(Firm 2)
Here we consider that the marginal cost of both firms is c.
Demand functions of firm 2 has been specified as:-
D2(p1,p2)=p2 if p1<p2
D1(p1,p2)=p2/2 if p1=p2
D1(p1,p2)=0 if p1>p2
The Total Revenues for firm 2 under three scenarios can be written as:-
TR2=p2q2 if p2<p1
TR1=p2^2/2 if p1=p2
TR1=0 if p2>p1
Total Cost for firm 2=cq2
Total Profit functions of firm under all 3 scenarios:-
TP2=p2q2-cq2 if p2<p1
TP2=p2^2/2-q2c=q1(p1/2-c) if p1=p2
TP2=0-q2c=-cq2 if p2>p1
Based on the profit maximizing condition of the firm,we can state:-
p2=Marginal Cost of firm 2(MC2)
p2=c
Now,plugging the value of p1 in the above equation into the 3 Total Profit functions for firm2,we get:-
TP2=p2q2-cq2 if p2<p1
TP2=cq2-cq2
TP2=0
TP2=p2^2/2-q2c if p1=p2
TP2=c^2/2-cq2
TP2=(c^2-2cq1)/2
TP2=-cq2 if p1>p2
TP2=-cq2
Hence,notice that the TP2 for firm 2 is maximum when p1=p2 and price is equal to marginal cost.
Case-2(Firm 2)
Now,we consider the MC2 to be 2c.
Total Cost for firm 1:TC2=2cq2
Total Profit functions of firm 2 under all 3 scenarios:-
TP2=p2q2-2cq2 if p2<p1
TP2=p2^2/2-2q2c if p1=p2
TP2=(p2^2-4q2c)/2
TP1=0-2q2c=-2cq12 if p2>p1
Based on the profit maximizing condition of the firm,we can state:-
p2=MC2
p2=2c
Now,plugging the value of p2 in the above equation into the 3 Total Profit functions for firm1,we get:-
TP2=2cq2-2cq2 if p2<p1
TP2=2cq2-2cq2
TP2=0
TP2=(p2^2-4q2c)/2 if p1=p2
TP2=((2c)^2-4q2c)/2
TP2=(4c^2-4q2c)/2
TP2=4c(c-q2)/2
TP2=2c(c-q2)
TP2=-2cq2 if p1>p2
TP2=-2cq2
Again,notice that the TP2 for firm 2 is maximum when p1=p2 and price is equal to marginal cost.
Case-3(Firm 2)
Now,we consider the MC2 to be 3c.
Total Cost for firm 2:TC2=3cq2
Total Profit functions of firm 2 under all 3 scenarios:-
TP2=p2q2-3cq2 if p2<p1
TP2=p2^2/2-3q2c if p1=p2
TP2=(p2^2-6q2c)/2
TP2=0-3q2c=-3cq2 if p2>p1
Based on the profit maximizing condition of the firm,we can state:-
p2=MC2
p2=3c
Now,plugging the value of p2 in the above equation into the 3 Total Profit functions for firm 2,we get:-
TP2=p2q2-3cq2 if p2<p1
TP2=3cq2-3cq2
TP2=0
TP2=(p2^2-6q2c)/2 if p1=p2
TP2=((3c)^2-6q2c)/2
TP2=(9c^2-6q2c)/2
TP2=3c(3c-2q2)/2
TP2=3c(3c-2q2)/2
TP2=-2cq2 if p2>p1
TP2=-3cq2
d) When the MC1 and MC2 are c
p1=c and p2=c
Both firms would choose to charge a price equal to the price of the other firm.
q1=p1/2/q2=p2/2
q1=c/2/q2=c/2
Thus,the Nash Equilibrium price pairs for both firm 1 and firm 2 are both c or p1=p2
When MC1 and MC2 are 2c
In this case also,observe that both firms maximize profit when they charge the same price per unit of product or p1=p2 as demonstrated in part c).
Therefore,the Nash Equilibrium price pairs would be again p1=p2=2c
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