Question

Please do you know how to solve these questions?:

Consider a duopoly with a demand curve given by P = a –bQ, where a and b are positive constants and Q is the total production by the two firms. Firms sell identical goods and have an identical constant marginal cost of production c. Fixed costs are equal to zero. We assume firms choose quantities simultaneously (Cournot competition).

a. Obtain the first order condition of profit maximization for each firm. Use graphical analysis and economic intuition to explain what they represent. [30%]

b. Obtain the profit maximizing quantity for each firm. Explain what they represent using game theory concepts. [20%]

c. Demonstrate using relevant graphical analysis and economic intuition that the results obtained in b are not a Pareto Optimum for the firms involved. [20%]

d. How would the graphical analysis in part a change if Firm A had a fixed cost of production? 

Answer 1

a)

inverse demand function for market $p =a -b(q_1+q_2)$

marginal cost = c for each firm

total cost $= cq$

each firm maximizes her profit by choosing quantity given by other firm given level of output (Cournot competition)

$max \space Profit = Pqi-Cqi$

for firm one gives q2 as constant maximize profit by choosing q1

$max\space q_1 (a-b(q_1+q_2))q_1-cq_1$

by the first-order condition from differentiating wrt q1 gives $a - 2bq_1 -bq_2-c =0 eq_1$

similarly, from maximizing firm 2 profit given q1

$max\space q_2 (a-b(q_1+q_2))q_2-cq_2$

by the first-order condition from differentiating wrt q2 gives

$a-2bq_1 -bq_1 -c =0 eq_2$

these equation 1 and 2 known as reaction function that interaction in the graph gives the equilibrium solution.

B) by solving eq 1 and 2 simultaneously gives

$q_1* = \frac{a-c}{3b}$

$q_2* = \frac{a-c}{3b}$

$p= \frac {(a+2c)}{3}$

profit to each $firm =\frac{ (a-c)^2}{9b}$ this equilibrium represent Nash equilibrium.

C) if both firm make collusion than optimal output to maximize profit is

$max\space pq-cq$

$max \space (a-bq)q-cq$

$q* = \frac{(a-c)}{2b} \space and\space p =\frac{ (a+c)}{2}$

$profit = \frac{(a-c)2}{4b}$

collusion profit is higher than aggregate profit of each firm.

D) if the firm has fixed cost, then there is no change in the reaction function is graph mathematically as still reaction function describe by the same eq 1 and 2. but as profit is now

$profit=\frac{ (a-c)2}{9b}-fix ed\space cost >0$

this term should always positive, so given fixed cost if profit is negative, then best for from to produce zero output with zero profit instead -ve profit.