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 in order 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 \\[T C=.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 \\[T C=.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 \(Q\) d. Graph the market demand curve, the \(M R\) curve, and the three marginal cost curves from parts \((\mathrm{a}),(\mathrm{b}),\) and \((\mathrm{c}) .\) Notice that the monopolist's profit-making ability is constrained by (1) the market demand curve (along with its associated \(M R\) curve) and (2) the cost structure underlying production.

Short answer

Based on the given market demand curve \\(Q = 70 - P\\) and three cost structures, the monopolist's output, price, and profit under each cost structure can be summarized as follows: Cost Structure A: - Optimal Output: \\(Q = 32\\) - Price: \\(P = 38\\) - Profit: \\(\pi = 1024\\) Cost Structure B: - Optimal Output: \\(Q = 30\\) - Price: \\(P = 40\\) - Profit: \\(\pi = 750\\) Cost Structure C: - Optimal Output: \\(Q \approx 27.8\\) - Price: \\(P \approx 42.2\\) - Profit: \\(\pi \approx 675.4\\) These results can be visualized using the demand curve (\\[P = 70 - Q\\]), the MR curve (\\[MR = 70 - 2Q\\]), and the MC curves for each cost structure.

Step-by-step solution

Find Marginal Revenue from Demand Curve

Given the market demand curve \\[Q = 70 - P\\], we can solve for price as a function of quantity: \\[P(Q) = 70 - Q\\] To find the total revenue (TR) as a function of quantity, multiply price by quantity: \\[TR(Q) = P(Q) \cdot Q = (70 - Q)Q\\] Now, find the marginal revenue (MR) by taking the derivative of total revenue with respect to quantity: \\[MR(Q) = \frac{dTR}{dQ} = 70 - 2Q\\]

Step 2a: Find Optimal Output for Cost Structure A

Given that AC = MC = 6, we can equate this to the marginal revenue to find the optimal output: \\[6 = 70 - 2Q \\] Solve for Q: \\[Q = \frac{70 - 6}{2} = 32\\]

Step 3a: Find Price and Profit for Cost Structure A

Using the demand function, find the price at the optimal output level: \\[P(32) = 70 - 32 = 38\\] Total revenue is: \\[TR(32) = 38 \cdot 32 = 1216\\] Total cost is: \\[TC(32) = AC \cdot Q = 6 \cdot 32 = 192\\] And profit is: \\[\pi = TR - TC = 1216 - 192 = 1024\\]

Step 2b: Find Optimal Output for Cost Structure B

The total cost function is given by: \\[TC(Q) = 0.25Q^2 - 5Q + 300\\] To find the marginal cost, differentiate the total cost function: \\[MC(Q) = \frac{dTC}{dQ} = 0.5Q - 5\\] Now equate this to the marginal revenue and solve for Q: \\[0.5Q - 5 = 70 - 2Q\\] \\[2.5Q = 75\\] \\[Q = 30\\]

Step 3b: Find Price and Profit for Cost Structure B

Find the price at the optimal output level: \\[P(30) = 70 - 30 = 40\\] Total revenue is: \\[TR(30) = 40 \cdot 30 = 1200\\] Total cost is: \\[TC(30) = 0.25(30)^2 - 5(30) + 300 = 450\\] Profit is: \\[\pi = TR - TC = 1200 - 450 = 750\\]

Step 2c: Find Optimal Output for Cost Structure C

The total cost function is given by: \\[TC(Q) = 0.0133Q^3 - 5Q + 250\\] To find the marginal cost, differentiate the total cost function: \\[MC(Q) = \frac{dTC}{dQ} = 0.0399Q^2 - 5\\] Now equate this to the marginal revenue and solve for Q using the quadratic formula. Rearrange the equation like so: \\[0.0399Q^2 - 2Q + 5 = 0\\] We get the positive root as the quantity: \\[Q \approx 27.8\\]

Step 3c: Find Price and Profit for Cost Structure C

Find the price at the optimal output level: \\[P(27.8) \approx 70 - 27.8 = 42.2\\] Total revenue is: \\[TR(27.8) \approx 42.2 \cdot 27.8 \approx 1173.16\\] Total cost is: \\[TC(27.8) \approx 0.0133(27.8)^3 - 5(27.8) + 250 \approx 497.76\\] Profit is: \\[\pi = TR - TC = 1173.16 - 497.76 \approx 675.4\\]

Graph Demand Curve, MR Curve, and MC curves

To graph the demand curve, plot the function \\[P = 70 - Q\\]. To graph the MR curve, plot the function \\[MR = 70 - 2Q\\]. For cost structure A, graph the constant MC curve at \\[MC = 6\\]. For cost structure B, graph the MC curve \\[MC = 0.5Q - 5\\]. For cost structure C, graph the MC curve \\[MC = 0.0399Q^2 - 5\\]. Visualizing these curves will help to understand the monopolist's profit-making ability constrained by both market demand and cost structures.