Question

A monopolist faces a market demand curve given by 

Q=70-P

 a. If the monopolist can produce at constant average and marginal costs ofAC-MC-6, what output level will the monopolist choose to maximize profits? What is the price at this output level? What are the monopolist's profits? 

b. Assume instead that the monopolist has a cost structure where total costs are described by 

C(Q) = 0.25Q² - 5Q + 300. 

With the monopolist facing the same market demand and marginal revenue, what price-quantity combination will be chosen now to maximize profits? What will profits be? 

c. Assume now that a third cost structure explains the monopolist's position, with total costs given by 

C(Q) = 0.0133Q³ - 5Q + 250. 

Again, calculate the monopolist's price-quantity combination that maximizes profits. What will profit be? Hint: Set MC MR as usual and use the quadratic formula to solve the second-order equation for Q.

d. Graph the market demand curve, the MR curve, and the three marginal cost curves from parts (a), (b), and (c). Notice that the monopolist's profit-making ability is constrained by (1) the market demand curve (along with its associated MR curve) and (2) the cost structure underlying production.

Answer 1

The inverse demand would be \(p=70-Q\). The total revenue would be \(T R=p Q\) or \(T R=(70-Q) Q\) or \(T R=(70-Q) Q\) or \(T R-70 Q-Q^{2}\). The marginal revenue would be \(M R=\frac{\partial}{\partial Q}(T R)=70-2 Q\).

(a) The profit would be maximum where the MR is equal to the MC, which is where \(70-2 Q=6\) or \(2 Q=64\) or \(Q=32\). which is the profit maximizing quantity. The corresponding demand price would be \(P=70-Q\) or \(p=70-32=38\) dollars.

The average cost is 6, and hence, \(\frac{T C}{Q}=6\) or \(T C=6 Q\), and hence for 32 units, we have \(T C=6 * 32=192\) dollars. The total revenue would be \(T R=p Q\) or \(T R=38 * 32=1216\). Hence, the profit would be \(\pi=T R-T C=1216-192=1024\) dollars.

(b) The marginal cost would be \(M C=\frac{\partial C}{\partial Q}\) or \(M C=\frac{\partial}{\partial Q}\left(0.25 Q^{2}-5 Q+300\right)\) or \(M C=0.5 Q-5 .\) The profit would be maximum where \(M C=M R\), ie \(0.5 Q-5=70-2 Q\) or \(2.5 Q=75\) or \(Q-30\), which is the profit maximizing output. The corresponding demand price would be \(p=70-30=40\) dollars.

The profit would be \(\pi=T R-C\) or \(\pi=p Q-\left(0.25 Q^{2}-5 Q+300\right)\) or \(\pi=40 * 30-\left(0.25 * 30^{2}-5 * 30+300\right)\) or \(\pi=1200-(225-150+300)\) or \(\pi=825\) dollars.

(c) The marginal cost would be \(M C=\frac{\partial C}{\partial Q}\) or \(M C=\frac{\partial}{\partial Q}\left(0.0133 Q^{3}-5 Q+250\right)\) or \(M C=0.0399 Q^{2}-5 .\) The profit would be maximum where \(M C=M R\), ie \(0.0399 Q^{2}-5=70-2 Q\) or \(0.0399 Q^{2}+2 Q-75=0\) or \(Q=\frac{-2 \pm \sqrt{2^{2}-4 * 0.0399 *(-75)}}{2 * 0.0399}\) or \(Q=\frac{-2 \pm \sqrt{4+11.97}}{0.0798}\) or \(Q=\frac{-2 \pm 3.9962}{0.0798}\) or \(Q=-75.14,25.015 .\) Since \(Q\) can not be negative, the profit maximizing quantity would be \(Q=25.015\). The corresponding demand price is \(p=70-25.015=44.985\) dollars.

The profit would be \(\pi=T R-C\) or \(\pi=p Q-\left(0.0133 Q^{3}-5 Q+250\right)\) or

\(\pi=44.985 * 25.015-\left(0.0133 * 25.015^{3}-5 * 25.015+250\right)\) or \(\pi=1125.3-(208.19-125.07+250)=792.18\) dollars.

(d) The graph is as below.

As can be seen, the profit maximizing quantity depends on the market demand and hence the MR, and the cost function.