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.)

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.)

Accepted Answer

Yes, the monopolist and competitive industry will choose the same level of lifetime (X) for the batteries, even though their output and price might be different. This is because both types of firms would choose the level of lifetime that balances the trade-off between the higher revenue from increasing the lifetime (X) and the higher cost of improving the battery's quality, which depends only on the batteries' qualities and not on the market structure.

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Microeconomic Theory: Basic Principles and Extensions

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Step by Step Solution

Step 1: To find the equilibrium for the monopolist, we need to write their profit maximization problem. The profit function for the monopolist can be written as: \(\Pi(Q, X) = P(Q, X)Q - C(Q, X)\) =

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