Question

A monopolist faces a market demand curve given by \\[Q=70-p.\\] a. If the monopolist can produce at constant average and marginal costs of \(A C=M C=6,\) what output level will the monopolist choose to maximize profits? What is the price at this output level? What are the monopolist's profits? b. Assume instead that the monopolist has a cost structure where total costs are described by \\[ C(Q)=0.25 Q^{2}-5 Q+300 \\] With the monopolist facing the same market demand and marginal revenue, what price-quantity combination will be chosen now to maximize profits? What will profits be? c. Assume now that a third cost structure explains the monopolist's position, with total costs given by \\[ C(Q)=0.0133 Q^{3}-5 Q+250 \\] Again, calculate the monopolist's price-quantity combination that maximizes profits. What will profit be? Hint: Set \(M C=M R\) as usual and use the quadratic formula to solve the second-order equation for \(Q\)

Short answer

Answer: The output level, price, and profit for each scenario are as follows: a) Output Level (Q) = 32, Price (p) = 38, Profit = 1024 b) Output Level (Q) = 20, Price (p) = 50, Profit = 800 c) Output Level (Q) = 15, Price (p) = 55, Profit ≈ 600.04

Step-by-step solution

Calculate the Marginal Revenue Function

To find the marginal revenue function, we first need to find the total revenue function. From the demand curve, we can express price as a function of quantity: \(p = 70 - Q\). Total Revenue (TR) is price multiplied by quantity: \(TR = p * Q = (70 - Q) * Q\). Now differentiate TR with respect to Q to find the Marginal Revenue function: \(MR = \frac{d(TR)}{dQ} = 70 - 2Q\).

Solve for the Output Level (Q)

In each scenario, we will set the marginal cost equal to the marginal revenue and solve for Q. a) \(MC = 6\), so we have \(6 = 70 - 2Q \Rightarrow Q = 32\). b) To find the MC function, we first need to find the total cost function. Differentiate the total cost function with respect to Q: \(MC = \frac{d(C(Q))}{dQ} = 0.5Q - 5\). Now, set the MC equal to the MR: \(0.5Q - 5 = 70 - 2Q \Rightarrow Q = 20\). c) Differentiate the total cost function with respect to Q to find the MC function: \(MC = \frac{d(C(Q))}{dQ} = 0.04Q^{2} - 5\). Set MC equal to MR and solve the second-order equation for Q: \(0.04Q^{2} - 5 = 70 - 2Q \Rightarrow 0.04Q^{2} + 2Q - 75 = 0\). Use the quadratic formula to solve for Q: \(Q = \frac{-2 \pm \sqrt{(-2)^{2} - 4(0.04)(-75)}}{2(0.04)}\). The two possible solutions for Q are 15 and 125, but since the demand function has a maximum quantity of 70, we choose Q = 15.

Calculate the Price at the Output Level

Use the demand function to calculate the price at the chosen output level. a) \(p = 70 - 32 = 38\) b) \(p = 70 - 20 = 50\) c) \(p = 70 - 15 = 55\)

Calculate Total Cost and Profit

Calculate the total cost at the chosen output level using the corresponding total cost function and then calculate the monopolist's profit using the formula Profit = Total Revenue - Total Cost. a) \(Total\_Cost = MC * Q = 6 * 32 = 192\), \(Total\_Revenue = p * Q = 38 * 32 = 1216\), \(Profit = 1216 - 192 = 1024\) b) \(Total\_Cost = 0.25(20)^{2} - 5(20) + 300 = 200\), \(Total\_Revenue = 50 * 20 = 1000\), \(Profit = 1000 - 200 = 800\) c) \(Total\_Cost = 0.0133(15)^{3} - 5(15) + 250 = 224.9625\), \(Total\_Revenue = 55 * 15 = 825\), \(Profit = 825 - 224.9625 = 600.0375\) Results: a) Output Level (Q) = 32, Price (p) = 38, Profit = 1024 b) Output Level (Q) = 20, Price (p) = 50, Profit = 800 c) Output Level (Q) = 15, Price (p) = 55, Profit ≈ 600.04

Key concepts

Market Demand Curve

The market demand curve represents the relationship between the price of a good and the quantity demanded by consumers at each price level. It typically slopes downward, indicating that as the price decreases, consumers are willing to purchase more of the good, and vice versa. In the context of monopolist profit maximization, the demand curve is crucial because it directly influences the monopolist's pricing decisions.

For example, in the given exercise, the market demand curve is illustrated by the equation \(Q = 70 - p\). This concise linear equation shows that if the price \(p\) increases by one unit, the quantity demanded \(Q\) decreases by the same amount, emphasizing the inverse relationship between price and quantity demanded. The monopolist uses this demand curve to determine the price that consumers are willing to pay for a certain quantity of goods and, subsequently, to calculate total revenue and profit.

Marginal Cost

Marginal cost (MC) is a concept that refers to the additional cost of producing one more unit of a good. It is derived from the total cost function particular to a firm's production technology. In essence, the MC is crucial for decision-making processes as it signifies the increase in total costs when output is increased by one unit. Understanding MC is vital for a monopolist when determining the most profitable quantity to supply.

In the exercise, different cost structures were presented. For instance, when the monopolist faces constant average and marginal costs of \(AC = MC = 6\), the marginal cost is constant, irrespective of the number of goods produced. However, with more complex cost structures like \(C(Q) = 0.25Q^2 - 5Q + 300\), the marginal cost isn't constant but rather depends on the output level and must be calculated by deriving the total cost function with respect to quantity \(Q\). The monopolist aims to equate this MC with marginal revenue to determine the output level that maximizes profits.

Marginal Revenue

Marginal Revenue (MR) indicates the additional revenue a firm gains when selling one more unit of a good. For a monopolist, understanding the MR is essential because it is used to determine the profit-maximizing level of output. The revenue will increase with each additional unit sold up to a certain point; beyond this point, MR can become negative if producing more leads to a decrease in overall revenue.

In the context of the exercise, the marginal revenue function was derived from the market demand curve. By setting price as a function of quantity from the demand equation \(p = 70 - Q\) and finding total revenue (\text{TR = p * Q}), the MR function is calculated by differentiating TR with respect to \(Q\). For the given demand curve, the MR function was found to be \(MR = 70 - 2Q\). A profit-maximizing monopolist will continue to produce units until MR equals MC. This condition forms the backbone of the monopolist's profit maximization strategy and is central to the calculations in the exercise.