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

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.

Step-by-step solution

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: \(\Pi(Q, X) = P(Q, X)Q - C(Q, X)\) =

\(\Pi(Q, X) = g(XQ)Q - C(X)Q\) Now we need to find the optimal values of \(Q\) and \(X\) that maximize the profit function. #Step 2: Find the First Order Conditions for the Monopolist's Profit Maximization Problem#

We will need to find the partial derivatives of the profit function with respect to \(Q\) and \(X\), so that we can set them equal to 0 and solve for the maximization problem's first-order conditions. \(\frac{\partial \Pi}{\partial Q} = g'(XQ)XQ + g(XQ) - C(X) = 0\) \(\frac{\partial \Pi}{\partial X} = g'(XQ)Q^2 - C'(X)Q = 0\) After obtaining the first-order conditions, we'll proceed to solve them for \(Q\) and \(X\). #Step 3: Solve the First Order Conditions for the Monopolist's Optimal Choice of X#

We'll now solve the first-order conditions for \(Q\) and \(X\). We can use the second equation to express \(C'(X)\): \(C'(X) = g'(XQ)Q\) We know that \(g'<0\) and \(C'(X)>0\), therefore the choice of \(X\) must be such that \(g'(XQ)Q\) is positive. In other words, the monopolist will choose \(X\) in such a way that the marginal cost reduction due to lifetime improvement is equal to the marginal revenue reduction due to a higher product of quantity and lifetime. This is the monopolist's optimal choice of \(X\). #Step 4: Formulate the Profit Maximization Problem for the Competitive Industry#

Now, let's find the equilibrium for the competitive industry. In a competitive market, each firm's price is equal to the marginal cost of production, hence: \(P(Q, X) = C'(X)\) Substitute the inverse demand function: \(g(XQ) = C'(X)\) #Step 5: Comparing the Monopolist's and Competitive Industry's Choices of X#

Comparing the two equilibria, we can observe that both the monopolist and the competitive firms have the same condition for their choice of \(X\): \(g'(XQ)Q = C'(X)\) This implies that the monopolist and the competitive industry will produce at the same level of \(X\) even though their output and price might be different. The reason behind this result is that both types of firms would choose the level of lifetime for their batteries that balances the trade-off between the higher revenue from increasing the lifetime \((X)\) and the higher cost of improving the battery's quality. Since this trade-off depends only on the batteries' qualities and not on the market structure, the optimal choice of \(X\) is the same for both the monopolist and competitive firms.

Key concepts

Monopoly

A monopoly exists when a single firm dominates the entire market for a particular product or service. Unlike competitive markets with many sellers, a monopolistic market has only one provider who can significantly influence prices and output levels.
This means the monopolist can set prices higher than what would prevail under perfect competition, often leading to greater profits. However, monopolies face certain constraints, such as demand elasticity and costs of production, which influence their pricing and production decisions.
In the context of alkaline batteries, our monopolist must consider how changing the lifetime of the batteries \(X\) impacts consumer demand, taking advantage of their ability to influence prices.

Competitive Industry

A competitive industry comprises many firms, each producing a similar product, leading to a situation where no single firm can influence market price.
In such industries, firms are price takers, meaning they must accept the market price as given. Their goal is to produce where price equals marginal cost (MC), maximizing efficiency.
For the competitive industry producing batteries, the equilibrium condition is when the price of batteries equals the marginal cost of producing them. Unlike monopolies, competitive firms cannot set prices above this level as they would lose customers to competitors. This competitive dynamic prompts firms to be more efficient and innovative.

Marginal Cost

Marginal cost (MC) is the additional cost incurred by producing one more unit of a good. It plays a crucial role in decision-making for both monopolies and competitive firms.
For the monopolist in our problem, the decision on battery lifetime involves balancing the added cost of this improvement with the potential revenue benefits.
In the profit maximization condition, the monopolist sets the marginal cost of improving battery lifetime equal to the marginal revenue obtained from it.

• In competitive industries, firms produce where their price equals marginal cost, ensuring no extra costs exceed the revenue from selling one more unit.
• This ensures that resources are used efficiently, benefiting consumers with better prices and product quality.

Inverse Demand Function

The inverse demand function describes how the price of a good is related to its quantity demanded. It is a crucial tool for analyzing consumer behavior and setting optimal prices.
The demand for batteries in this scenario is based on a function that considers the total product of quantity and lifetime \(X \cdot Q\). This reflects consumers' preferences for either numerous short-lived batteries or fewer long-lived ones.
For a monopolist, understanding this relationship helps inform strategic decisions about production quantities and battery lifetimes, crucial for profit maximization. In competitive industries, the inverse demand function helps firms understand market dynamics and respond accordingly to consumer needs.