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Answers without the blur. Sign up and see all textbooks for free! Problem 10

Expert-verified  ### Microeconomic Theory: Basic Principles and Extensions

Book edition 9 Edition
Author(s) Walter Nicholson # 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.)

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.
See the 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: $$\Pi(Q, X) = P(Q, X)Q - C(Q, X)$$ =

$$\Pi(Q, X) = g(XQ)Q - C(X)Q$$ 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#

## Step 2: 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#

## Step 3: 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. ### Want to see more solutions like these? 