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.

Key concepts

Inverse Demand Function

In a monopoly market, understanding the inverse demand function is crucial. It's the relationship between the price consumers are willing to pay and the quantity of a good. For the problem at hand, the inverse demand function is defined as \(P(Q, X) = g(XQ)\). Here, \(g\) is a function that inversely relates to the composite commodity \(XQ\). When the notation states \(g'(XQ) < 0\), it indicates a downward-sloping demand curve. This means as the quantity of the composite commodity increases, the price that consumers are willing to pay decreases. This concept highlights the idea that consumers do not differentiate between the number of alkaline batteries purchased and the product of lifetime with quantity, as long as the total utility remains constant.

Cost Function

The cost function in a monopoly determines how the production expenses change as the production elements vary. For this case, the cost function is presented as \(C(Q, X) = C(X)Q\). This implies that the total cost is dependent on both the cost function of \(X\) and the quantity \(Q\) produced. The condition \(C'(X) > 0\) tells us that as the battery lifetime \(X\) increases, so does the cost. This reflects a realistic scenario where improving product quality or features, like battery life, tends to drive up production costs. Understanding cost behavior is critical for the monopolist to make optimal production and pricing decisions.

Profit Maximization

Profit maximization is the monopolist's main goal. It involves adjusting output and pricing to achieve the highest possible profit. The monopolist's profit function is given by \(\pi(Q, X) = g(XQ)Q - C(X)Q\). By taking the partial derivatives with respect to \(Q\) and \(X\), and setting them to zero, we find the values that optimize profit. These calculations are central in determining how much of the product to produce \(Q^*(X)\) and what the optimal battery lifetime \(X^*\) should be. The ultimate aim for the monopolist is to balance between maximizing revenue and minimizing costs.

Composite Commodity

The concept of a composite commodity plays a vital role in understanding consumer behavior in this monopoly. Consumers perceive the product as a combination of battery lifetime \(X\) and quantity \(Q\), reflected in the equation \(XQ\). This simplifies consumer choice, as they focus on the overall utility provided by this composite, rather than its individual components. It indicates that consumers are indifferent towards how this utility is obtained, whether through many short-lived batteries or fewer long-lived ones. For the monopolist, this composite nature offers a strategic point to determine optimal production and pricing strategies.

Competitive Industry Comparison

In comparing a monopoly to a competitive industry, a critical observation is that, despite differences in pricing and production levels, both can choose the same optimal battery lifetime \(X\). In a competitive market, firms set prices equal to marginal costs. The problem shows that in both market types, the selected lifetime maximizes utility from the composite commodity \(XQ\). The only divergence lies in output quantity and price charged. Monopolists are likely to produce a lower quantity and set higher prices due to lack of competition. Meanwhile, competitive industries, driven by consumer surplus optimization, will adjust output to meet price at marginal cost, ultimately benefiting consumers with lower prices and higher overall utility.