Question

An alternative way to study the welfare properties of a monopolist's choices is to assume the existence of a utility function for the customers of the monopoly of the form utility \(=U(Q, X),\) where \(Q\) is quantity consumed and \(X\) is the quality associated with that quantity. A social planner's problem then would be to choose \(Q\) and \(X\) to maximize social welfare as represented by \(S W=U(Q, X)-C(Q, X)\). a. What are the first-order conditions for a welfare maximum? b. The monopolist's goal is to choose the \(Q\) and \(X\) that maximize \(\pi=P(Q, X) \cdot Q-C(Q, X) .\) What are the first-order conditions for this maximization? c. Use your results from parts (a) and (b) to show that, at the monopolist's preferred choices, \(\partial S W / \partial Q>0\). That is, as we have already shown, prove that social welfare would be improved if more were produced. Hint: Assume that \(\partial U / \partial Q=P\) d. Show that, at the monopolist's preferred choices, the sign of \(\partial S W / \partial X\) is ambiguous-that is, it cannot be determined (on the sole basis of the general theory of monopoly) whether the monopolist produces either too much or too little quality.

Short answer

In conclusion, the first-order conditions for a welfare maximum are: $$ \frac{\partial U}{\partial Q} - \frac{\partial C}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial U}{\partial X} - \frac{\partial C}{\partial X} = 0. $$ The first-order conditions for the profit maximization are: $$ \frac{\partial (P \cdot Q)}{\partial Q} - \frac{\partial C}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial (P \cdot Q)}{\partial X} - \frac{\partial C}{\partial X} = 0. $$ At the monopolist's preferred choices, it is shown that \(\partial SW/ \partial Q > 0\), which means that social welfare would increase if the monopolist produced more of the good. However, the sign of \(\partial SW / \partial X\) is ambiguous, as it depends on the change in price with respect to quality (X). Therefore, it is unclear whether social welfare would increase or decrease if the monopolist improved the quality of the good.

Step-by-step solution

Set up the social welfare function

Having utility function U(Q, X) and cost function C(Q, X), the social welfare (SW) function is given by: $$ SW = U(Q, X) - C(Q, X). $$

Finding the first-order conditions

In order to find the first-order conditions for a welfare maximum, we need to take the partial derivatives of SW with respect to Q and X, and equate them to zero. $$ \frac{\partial SW}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial SW}{\partial X} = 0, $$ which gives us the following equations: $$ \frac{\partial U}{\partial Q} - \frac{\partial C}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial U}{\partial X} - \frac{\partial C}{\partial X} = 0. $$ #b. The monopolist's goal is to choose the \(Q\) and \(X\) that maximize \(\pi=P(Q,X) \cdot Q-C(Q, X) .\) What are the first-order conditions for this maximization?#

Set up the profit function

The monopolist's profit function is given by: $$ \pi = P(Q,X) \cdot Q - C(Q, X). $$

Finding the first-order conditions

In order to find the first-order conditions for the profit maximization, we need to take the partial derivatives of profit (π) with respect to Q and X, and equate them to zero: $$ \frac{\partial \pi}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial \pi}{\partial X} = 0, $$ which gives us the following equations: $$ \frac{\partial (P \cdot Q)}{\partial Q} - \frac{\partial C}{\partial Q} = 0 \quad \text{and} \quad \frac{\partial (P \cdot Q)}{\partial X} - \frac{\partial C}{\partial X} = 0. $$ #c. Use your results from parts (a) and (b) to show that, at the monopolist's preferred choices, \(\partial S W / \partial Q>0\).#

Substitute the hint

Use the hint to replace \(\partial U / \partial Q\) in the first welfare condition with \(P\): $$ P - \frac{\partial C}{\partial Q} = 0. $$

Compare the conditions

Compare the first-order conditions for profit maximization and welfare maximization with respect to Q. In this case, we have: $$ \frac{\partial (P \cdot Q)}{\partial Q} - \frac{\partial C}{\partial Q} = 0, $$ which means, $$ (\frac{\partial P}{\partial Q} \cdot Q + P) - \frac{\partial C}{\partial Q} = 0, $$ or, $$ \frac{\partial P}{\partial Q} \cdot Q + (P - \frac{\partial C}{\partial Q}) = 0. $$ Since the quantity consumed is positive (\(Q>0\)) and the price-demand curve is downward sloping (\(\partial P / \partial Q < 0\)), we get, $$ \frac{\partial SW}{\partial Q} = \frac{\partial \pi}{\partial Q} - \frac{\partial P}{\partial Q} \cdot Q > 0 $$ #d. Show that, at the monopolist's preferred choices, the sign of \(\partial S W / \partial X\) is ambiguous.#

Compare the conditions

Compare the first-order conditions for profit maximization and welfare maximization with respect to X. We have: $$ \frac{\partial (P \cdot Q)}{\partial X} - \frac{\partial C}{\partial X} = \frac{\partial U}{\partial X} - \frac{\partial C}{\partial X}. $$

Analyze the sign of \(\partial S W / \partial X\)

The sign of \(\partial SW / \partial X\) is the same as the sign of: $$ \frac{\partial (P \cdot Q)}{\partial X} - \frac{\partial U}{\partial X}. $$ Since the change in price with respect to quality (X) is unknown, it is not possible to determine whether the sign of this expression is positive or negative. Therefore, the sign of \(\partial SW / \partial X\) is ambiguous.

Key concepts

First-Order Conditions

The first-order conditions are critical to finding an optimum point—either a maximum or a minimum—of a function, especially when we are dealing with economics and optimization problems. These conditions are based upon setting the partial derivatives of the function to zero. In the context of welfare maximization, the social welfare function will have first-order conditions derived by equating its partial derivatives with respect to quantity (\(Q\) and quality (\(X\) to zero. This results in a system of equations that help determine the optimal levels of output and quality that maximize social welfare.

For the monopolist profit maximization, a similar approach is adopted. The first-order conditions involve differentiating the monopolist's profit function with respect to quantity and quality and setting these partial derivatives equal to zero. These resulting equations help the monopolist determine the profit-maximizing levels of output and quality. In simpler terms, the first-order conditions are like solving a puzzle by finding the right levels of output and quality that 'fit' the objective of maximizing either welfare or profit.

Social Welfare Function

A social welfare function is a mathematical representation used to aggregate individual utilities into a measure of societal well-being. In our exercise, it relates to how changes in quantity (\(Q\) and quality (\(X\) of a good affect the happiness of consumers and the overall social welfare. The function is typically written as the total utility derived from consumption minus the cost of providing the goods, representing the net social benefit. Understandably, a social planner would want to maximize this social welfare function to achieve the best possible outcome for society.

When we think about the goal of social welfare maximization, it's like aiming for the best possible scenario where the needs and happiness of all consumers are balanced with the cost of production. This is a delicate balancing act that seeks to provide the optimal amount of goods and services, not just for the benefit of individual consumers, but for society as a whole.

Monopolist Profit Maximization

The goal of a monopolist is quite different from that of a social planner. When we discuss monopolist profit maximization, we're focusing on the strategies used by a monopoly to choose the quantity (\(Q\) and quality (\(X\) of a product in such a way that their profit is maximized. This involves setting a price for their product that reflects both the market's demand and the costs incurred in production.

It's important for students to appreciate that a monopolist operates under different incentives compared to competitive firms. Since a monopolist faces no competition, they can manipulate prices and production levels to their advantage. This can lead to scenarios where the quantity produced is less, and the price is higher than what would be observed in a competitive market, hence reducing the overall social welfare, which is demonstrated in our exercise.

Partial Derivatives

To dig deeper into the math, partial derivatives play a pivotal role in understanding how a function changes as each of its variables is altered independently. In our context, they are used to determine the change in social welfare and the monopolist’s profit as the quantity (\(Q\) and quality (\(X\) of the good change.

Conceptually, consider a partial derivative as a magnifying glass that focuses on the impact of changing one specific factor while holding others constant. For students grappling with calculus, picturing these partial derivatives can be likened to observing just one ingredient in a recipe—seeing how changing that one ingredient alters the taste of the dish, without changing anything else. It's this granular approach that allows economists and planners to dissect the complex relationship between various economic factors and outcomes.