Question

Suppose a monopolist produces alkaline batteries that may have various useful lifetimes \((X) .\) Suppose also that consumers (inverse) demand depends on batteries' lifetimes and quantity (Q) purchased according to the function \\[P(Q, X)=g(X \cdot Q),\\] where \(g^{\prime} < 0 .\) That is, consumers care only about the product of quantity times lifetime: They are willing to pay equally for many short-lived batteries or few long-lived ones. Assume also that battery costs are given by \\[C(Q, X)=C(X) Q,\\] where \(C^{\prime}(X) > 0 .\) Show that, in this case, the monopoly will opt for the same level of \(X\) as does a competitive industry even though levels of output and prices may differ. Explain your result. Hint: Treat \(X Q\) as a composite commodity.

Short answer

Based on the analysis and solution above, it can be concluded that the monopolist and the competitive industry will choose the same battery lifetime, \(X\). This is because the consumers care only about the composite commodity \((XQ)\), allowing both the monopolist and competitive firms to opt for the same battery lifetime. Although the monopolist may restrict output and charge a higher price compared to the competitive market, the optimal battery lifetime will remain the same in both cases.

Step-by-step solution

Demand function

Given the inverse demand function, \(P(Q, X) = g(XQ)\). Since \(g'(XQ) < 0\), it means that the demand function is inversely related to the composite commodity \((XQ)\).

Cost function

The cost function is given by \(C(Q, X) = C(X)Q\). Since \(C'(X)>0\), the cost of production increases with the battery lifetime.

Monopolist's profit function

The monopolist's profit function can be written as: \\[\pi(Q, X) = P(Q, X) Q - C(Q, X),\\] Substitute the given demand and cost functions to obtain the profit function: \\[\pi(Q, X) = g(XQ) Q - C(X) Q.\\]

Optimal values of \(Q\) and \(X\) for the monopolist

To find the optimal values of \(Q\) and \(X\), we will take the partial derivatives of the profit function with respect to \(Q\) and \(X\), and set them equal to zero: \\[\frac{\partial \pi}{\partial Q} = g'(XQ) X Q + g(XQ) = 0,\\] \\[\frac{\partial \pi}{\partial X} = g'(XQ) Q^2 - C'(X) Q = 0.\\] From the first equation, we can find the optimal value of \(Q\) in terms of \(X\) as: \\[Q^*(X) = -\frac{g(XQ)}{g'(XQ)X},\\] Substitute the optimal value of \(Q\) in the second equation to solve for the optimal value of \(X\): \\[g'(XQ)Q^{*2} - C'(X)Q^* = g'(XQ)\left(-\frac{g(XQ)}{g'(XQ)X}\right)^2 - C'(X)\left(-\frac{g(XQ)}{g'(XQ)X}\right) =0.\\] We can simplify this equation to get the optimal value of \(X\): \\[\frac{g(XQ)}{(g'(XQ)X)^2} = \frac{C'(X)}{g'(XQ)X}.\\] Since both sides are equal, the value of \(X\) that maximizes the monopolist's profit will be the same as that chosen by a competitive industry.

Comparison with competitive market output

In a competitive market, the price is equal to the marginal cost. So, the competitive firms will select the value of \(X\) that equates the marginal benefit \((g'(XQ)XQ)\) with the marginal cost \((C'(X))\). As shown in step 4, the monopolist also ends up choosing the same value of \(X\). So, even though the monopolist may charge a higher price and produce a different quantity, the battery lifetime chosen by the monopolist and the competitive industry will be the same. This result can be explained by the fact that consumers care only about the composite commodity \((XQ)\), i.e., they treat the product of battery lifetime and quantity purchased as a single commodity, and this allows both monopolist and competitive firms to choose the same optimal lifetime for the batteries. However, the monopolist may still restrict the output and charge a higher price compared to the competitive market.