Question

Consider a firm with monopoly power that faces the demand curve $$P=100-3 Q+4 A^{1 / 2}$$ and has the total cost function $$C=4 Q^{2}+10 Q+A$$ where \(A\) is the level of advertising expenditures, and \(P\) and \(Q\) are price and output. a. Find the values of \(A, Q,\) and \(P\) that maximize the firm's profit. b. Calculate the Lerner index, \(L=(P-M C) / P\), for this firm at its profit- maximizing levels of \(A, Q,\) and \(P\)

Short answer

First define the profit function and substitute the given price function. Differentiate the profit function with respect to \(Q\) and \(A\), and set the derivatives to zero. Solve the resulting equations for \(Q\) and \(A\), which result in the values that maximize profit. Find the optimal price by substituting these optimal values into the price equation. Finally, calculate the Lerner Index \(L=(P-MC)/P\) using the optimal values for \(P\) and \(MC\).

Step-by-step solution

Define Profit Function

First define the profit function. Profit (\(π\)) is defined as total revenue (\(TR\)) minus total cost (\(TC\)). The total revenue is price (\(P\)) times quantity (\(Q\)), and the total cost is given as \(C=4Q^2+10Q+A\). Substitute \(P = 100-3Q+4\sqrt{A}\) into the profit function to get: \(π(Q, A) = (100-3Q+4\sqrt{A})Q - (4Q^2 + 10Q +A)\).

Differentiate Profit Function

Next, take the partial derivatives of the profit function with respect to \(Q\) and \(A\). Setting these derivatives equal to zero yields: \(\frac{\partial π(Q, A)}{\partial Q} = 0\) and \(\frac{\partial π(Q, A)}{\partial A} = 0\). Solve the system of equations for \(Q\) and \(A\).

Find optimal values for Q and A that maximize profit

The solutions for \(Q\) and \(A\) will provide the values that maximize the firm's profit. Substituting these values into the price equation \(P=100-3Q+4\sqrt{A}\) will yield the optimal price, \(P\).

Calculate the Lerner Index

Given the equation \(L=(P-MC)/P\), substitute the optimal values for \(P\) and \(MC\) to get Lerner Index. Where \(MC\) is the derivative of the cost function with respect to \(Q\) at the optimal value of \(Q\). This is \(\frac{d C(Q,A)}{d Q}\).

Key concepts

Profit Maximization

Understanding how firms with monopoly power maximize profit is crucial for economics students. Profit maximization in a monopoly setting involves setting production levels in a way that revenue is maximized while costs are kept as low as possible. The profit \( \pi \) can be expressed as total revenue (TR) minus total costs (TC). Calculating this involves determining both revenue and cost functions and then finding the level of production that maximizes profit.

A monopolistic firm's revenue function is tied directly to its demand curve since the price it can charge depends on how much product it decides to sell. For the firm in question, the profit function becomes \( \pi(Q, A) = (100 - 3Q + 4\sqrt{A})Q - (4Q^2 + 10Q + A) \), showcasing the relationship between output, advertising expenditure, and profit. To maximize profit, the firm needs to calculate the optimum values of output (Q) and advertising expenditures (A) that lead to the highest profit.

This is typically achieved through calculus, by taking the partial derivatives of the profit function with respect to Q and A, and setting them to zero. These critical points indicate the maximum or minimum of the function, which in this context will help determine the profit-maximizing output and advertising expenditure levels.

Lerner Index

The Lerner Index is a measure of a firm's monopoly power, indicating the degree to which a firm can mark up prices above marginal costs. It is calculated as \( L = \frac{P - MC}{P} \). A higher Lerner Index value indicates a greater market power, as it reflects the firm's capacity to set prices significantly above costs without losing customers.

In the exercise, calculating the Lerner Index involves determining the optimal price \( P \) and marginal cost \( MC \), once the firm's profit-maximizing output \( Q \) and advertising expenditure \( A \) levels are found. The marginal cost is the additional cost of producing one more unit and can be obtained from the derivative of the total cost function with respect to Q at the optimal level of Q. The firm's ability to price above marginal cost reflects its market power and is critical in assessing the welfare impacts of monopoly in the market.

Partial Derivatives

Partial derivatives are used in the optimization of functions of multiple variables, such as the profit function in our monopoly example. They can be likened to slopes for multivariable functions, telling us which way a function increases or decreases when we change one variable while keeping the others constant.

In our firm's problem, the partial derivatives of the profit function with respect to Q and A are needed to find the values that maximize profit. Setting the partial derivatives equal to zero and solving them simultaneously gives us the critical points. These critical points, particularly where both partial derivatives are zero, indicate possible local maximums or minimums. Analyzing these points helps in determining the best levels of output and advertising expenditure to maximize profits for the firm. Proficiency in calculating and interpreting these derivatives is essential for students studying economics and business.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a good and the quantity of the good that consumers are willing and able to purchase. For monopolies, the demand curve is also the firm's average revenue curve.

Understanding the shape and position of a demand curve is key in price and output decisions for a monopolistic firm. In our case study, the demand curve equation \( P = 100 - 3Q + 4\sqrt{A} \) indicates how price \( P \) changes as output \( Q \) and advertising expenditure \( A \) vary. The demand curve slopes downwards, illustrating the typical inverse relationship between price and quantity: as price decreases, quantity demanded usually increases, and vice versa. Knowledge of the demand curve is essential when assessing the market and making decisions about production levels, pricing, and advertising strategies.