Problem_Set_4-1 (1)

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Problem Set 4 Microeconomic Theory due November 12, 2021 at 9:50AM Read all problem statements carefully. Show all of your work to receive full credit. Part 1 Q1: Consider the short-run cost function C ( y ) = y 2 + 4. Write down the equations for the following: Average total cost function: Average variable cost function: Average fixed cost function: Marginal cost function: Short-run supply function: 1

Q2: On the graph below, label the total cost, ATC, AVC, AFC, and MC curves. Highlight the firm’s supply curve. 2

Consider this graph of a natural monopoly scenario, wherein market demand is small enough relative to the cost of production that even the most efficient firm cannot ”break even” without pricing above marginal cost. Q3: When competitive price p P C and corresponding output level y P C result in a loss for the monopolist, what can you deduce about the monopolist’s returns to scale at output level y P C ? Explain your answer using what you know about the relationships between the different cost curves. 3

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Part 2 Suppose there are 2,000 potential consumers in the Chestnut Hill market for tablet computers. All consumers have identical preferences over tablets (T) and expenditures on other consumption goods (X), represented by the following utility function: U ( T, X ) = 500 T (1 − T ) + X Q4: Find the total market demand for T (call it T M ) as a function of the price of a tablet ( p T ). Q5: Write down the equation for the market demand curve (with T M on the horizontal axis and p T on the vertical axis). 4

Suppose there is only one firm operating in the market for T . The firm uses workers (labor-hours L ) and capital (machine-hours C ) to produce T using the following production function: f ( L, C ) = 1 200 √ L · C For the next quarter, the firm’s capital usage is fixed at C = 1 , 000, with an hourly rental rate of $ 1. Suppose the hourly wage of a worker in this industry is $ 25. Q6: Derive the firm’s short-run total cost as a function of output T . Q7: Find the profit-maximizing price p ∗ and quantity y ∗ chosen by the monopolist (round to the nearest whole number). 5

Q8: Calculate the monopolist’s total profit (round to the nearest whole number) and the percentage markup the monopolist charges over the competitive equilibrium price. Part 3 Suppose the monopolist pays a consultant to conduct market research into demand for tablet computers. The consultant finds that potential buyers can be sorted into two groups: students and professors. Students have lower incomes than professors and are thus more sensitive to changes in the price of a tablet. Specifically, the firm estimates the following demand curves for each group: P = 1200 − 0 . 75 Q Profs P = 200 − 0 . 25 Q Students Q8: If the monopolist can differentiate between customers who are students and customers who are professors, which price should they charge each group? Assume the firm has the same total cost function you used in Part 2. Explain the difference in the prices facing students and professors using the concept of price elasticity of demand. 6

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Q9: Fill in the table in the Excel spreadsheet attachment EstimatedMarket.xlsx with the total quantity of tablets demanded by the student and professor groups combined at each of the listed prices. This should generate a scatter plot of price-quantity combinations and a trend line. Attach a pdf screenshot of your table and graph to your as- signment submission. Q10: Suppose the monopolist is unaware of the true market demand schedule and uses the equation of the trend line you found in Q9 as their estimated demand curve (round all numbers in the equation to the nearest 0.1). What is the price the monopolist will charge for a tablet, and what will be their total profit? Hint: Find the quantity and price the firm will choose based on the estimated demand, and then find the actual quantity of tablets they will sell at their chosen price based on the true demand schedule you found in Q9. Q11: Now, suppose YOU are the economist that put together the data on student and professor demand. Assume the market for this type of research is competitive. How much do you think you should charge the firm for your work? 7

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