Chapter 9, Problem 4CACQ

(A)

To determine

The reaction function of the follower is to be ascertained.

Explanation of Solution

  $P=16000-4QwhereQ=Q_L+Q_F$

  $P=16000-4Q_L-4Q_F$

  $C_L\left(Q_L\right)=4000Q_L$

  $C_F\left(Q_F\right)=6000Q_F$

Reaction function of follower

  $M{R}_F=M{C}_F$

  $\frac{dTRF}{dQF}=\frac{dCF(QF)}{dQF}$

  $\frac{dP\times QF}{dQF}=\frac{d6000QF}{dQF}$

  $\frac{d(16000-4QL-4QF)QF}{dQF}=6000$

  $\frac{d\left(16000Q_F-4Q_L{Q}_F-4Q_F{}^2\right)}{dQF}=6000$

  $16000-4Q_L-8Q_F=6000$

  $Q_F=\frac{1}{8}\left(10000-4Q_L\right)$

  $Q_F=1250-\frac{1}{2}{Q}_L$

Therefore, reaction function of the follower is $Q_F=1250-\frac{1}{2}{Q}_L$

(B)

To determine

The equilibrium level of output for both the leader and follower is to be calculated.

Explanation of Solution

For finding the equilibrium output,putting the follower reaction function value of Q_F in P

  $P=16000-4Q_L-4Q_F$

  $P=16000-4Q_L-4(1250-\frac{1}{2}{Q}_L)$

  $P=16000-4Q_L-5000+2Q_L$

  $P=11000-2Q_L$

Now, $T{R}_L=P\times Q_L$

  $=\left(11000-2Q_L\right)\times Q_L$

  $=11000Q_L-2Q_L{{}^2}_{}$

Putting $M{R}_L=M{C}_L$

  $11000-4Q_L=4000$

  $7000=4Q_L$

  $Q_L=1750units$

Now, put the value of Q_L in reaction function of follower.

  $Q_F=1250-\frac{1}{2}{Q}_L$

  $Q_F=1250-\frac{1}{2}\times 1750$

  $Q_F=375units.$

Therefore , Q_L = 1750 units and Q_F = 375 units.

(C)

To determine

The equilibrium market price is to be ascertained.

Explanation of Solution

Equilibrium market price

  $P=16000-4Q_L-4Q_F$

  $P=16000-4\times 1750-4\times 375$

  $P=16000-7000-1500$

  $P=\$7500.$

Therefore, equilibrium market price is $7500.

(D)

To determine

The profits of both leader and follower is to be calculated.

Explanation of Solution

Profit of the leader is $T{R}_L-T{C}_L$

  $Profit=P\times Q_L-4000Q_L$

  $=7500\times 1750-4000\times 1750$

  $=13125000-7000000$

  $=\$6125000$

Profit of the follower is $T{R}_F-T{C}_F$

  $Profit=P\times Q_F-6000Q_F$

  $=7500\times 375-6000\times 375$

  $=2812500-2250000$

  $=\$562500$ .

Therefore , profit of the leader is $6125000 and profit of the follower is $562500.