Suppose that the market demand curve for mineral water is given as Q=100−10PQ=100−10P and marginal cost is fixed at $4$4. Find the equilibrium price and quantity in each type of different market structure. Show your calculation.

a) Monopoly,

b) Cournot duopoly,

c) Stackelberg duopoly,

d) Bertrand duopoly (MRMR is fixed at the level of MCMC),

e) Perfect competitive market (MRMR is fixed at the level of MCMC).

Given:

Q=100−10P⇒P=10−0.1QMC=$4TC=4QQ=100−10P⇒P=10−0.1QMC=$4TC=4Q

a) To obtain the equilibrium price and quantity under monopoly, equate the MR to the MC.

The total revenue and marginal revenue are:

TR=10Q−0.1Q2MR=10−0.2QTR=10Q−0.1Q2MR=10−0.2Q

The equilibrium price and quantity are:

MR=MC10−0.2Q=40.2Q=6Q∗=30unitsP∗=10−0.1(30)=$7MR=MC10−0.2Q=40.2Q=6Q∗=30unitsP∗=10−0.1(30)=$7

b) Assume 2 firms under cournot competition.

The profit functions for the firms are:

π1=PQ1−TC1=10Q1−0.1Q2Q1−0.1Q21−4Q1=6Q1−0.1Q2Q1−0.1Q21π2=PQ2−TC2=10Q2−0.1Q2Q1−0.1Q22−4Q2=6Q2−0.1Q2Q1−0.1Q22π1=PQ1−TC1=10Q1−0.1Q2Q1−0.1Q12−4Q1=6Q1−0.1Q2Q1−0.1Q12π2=PQ2−TC2=10Q2−0.1Q2Q1−0.1Q22−4Q2=6Q2−0.1Q2Q1−0.1Q22

The best response functions for both the firms are:

∂π1∂Q1=6−0.1Q2−0.2Q1=0⇒Q1=30−0.5Q2∂π2∂Q2=6−0.1Q1−0.2Q2=0⇒Q2=30−0.5Q1∂π1∂Q1=6−0.1Q2−0.2Q1=0⇒Q1=30−0.5Q2∂π2∂Q2=6−0.1Q1−0.2Q2=0⇒Q2=30−0.5Q1

Substitute the value of Q2 in Q1:

Q1=30−0.5(30−0.5Q1)0.75Q1=15Q1=20unitsQ2=30−0.5(20)=20unitsQ1=30−0.5(30−0.5Q1)0.75Q1=15Q1=20unitsQ2=30−0.5(20)=20units

The equilibrium price and quantity are:

Q∗=Q1+Q2=40unitsP∗=10−0.1(40)=$6Q∗=Q1+Q2=40unitsP∗=10−0.1(40)=$6

c) Substitute the value of Q2 in the profit function of firm 1:

π1=6Q1−0.1Q2Q1−0.1Q21=6Q1−0.1Q1(30−0.5Q1)−0.1Q21=3Q1+0.05Q21−0.1Q21=3Q1−0.05Q21π1=6Q1−0.1Q2Q1−0.1Q12=6Q1−0.1Q1(30−0.5Q1)−0.1Q12=3Q1+0.05Q12−0.1Q12=3Q1−0.05Q12

The first order condition with respect to Q1:

∂π1∂Q1=3−0.1Q1=0Q1=30unitsQ2=30−0.5(30)=15units∂π1∂Q1=3−0.1Q1=0Q1=30unitsQ2=30−0.5(30)=15units

The equilibrium price and quantity are:

Q∗=Q1+Q2=45unitsP∗=10−0.1(45)=$5.50Q∗=Q1+Q2=45unitsP∗=10−0.1(45)=$5.50

d) Given:

MR=MC=$4Q=100−10(P1+P2)MR=MC=$4Q=100−10(P1+P2)

The profit functions for the firms are:

π1=P1Q−TC1=100P1−10P2P1−10P21−4(100−10P1−10P2)=140P1−10P1P2−10P21+40P2π2=P2Q−TC2=100P2−10P2P1−10P22−4(100−10P1−10P2)=140P2−10P1P2−10P22+40P1π1=P1Q−TC1=100P1−10P2P1−10P12−4(100−10P1−10P2)=140P1−10P1P2−10P12+40P2π2=P2Q−TC2=100P2−10P2P1−10P22−4(100−10P1−10P2)=140P2−10P1P2−10P22+40P1

The best response functions for both the firms are:

∂π1∂P1=140−10P2−20P1=0⇒P1=7−0.5P2∂π2∂P2=140−10P1−20P2=0⇒P2=7−0.5P1∂π1∂P1=140−10P2−20P1=0⇒P1=7−0.5P2∂π2∂P2=140−10P1−20P2=0⇒P2=7−0.5P1

Substitute the value of P2 in P1:

P1=7−0.5(7−0.5P1)0.75P1=3.5P1=$4.66P2=$4.67P1=7−0.5(7−0.5P1)0.75P1=3.5P1=$4.66P2=$4.67

The equilibrium price and quantity are:

P∗=P1+P2=$9.33Q∗=100−10(9.33)=6.7unitsP∗=P1+P2=$9.33Q∗=100−10(9.33)=6.7units

e) The equilibrium price and quantity under perfect competition can be calculated by substituting price to MC:

P=MC10−0.1Q=40.1Q=6Q∗=60unitsP∗=$4