Capital Gains Taxes: Taxes on capital g0ains are applied to income earned on investments that return a profit or “capital gain” and not on income derived from labor. To the extent that such capital gains taxes are taxes on the return on capital, they will impact the rental rate of capital in ways we will explore more fully in a later chapter. For now, we will simply investigate the impact of a capital-gains-tax-induced increase in the rental price of capital on firms within an industry.
A: Suppose you are running a gas station in a competitive market where all firms are identical. You employ weekly labor ℓ and capital k using a homothetic decreasing returns to scale production function, and you incur a weekly fixed cost of F.
(a) Begin with your firm’s long run weekly average cost curve and relate it to the weekly demand curve for gasoline in your city as well as the short run weekly aggregate supply curve assuming the industry is in long run equilibrium. Indicate by x∗ how much weekly gasoline you sell, by p∗ the price at which you sell it, and by X∗ the total number of gallons of gasoline sold in the city per week.
(b) Now suppose that an increase in the capital gains tax raises the rental rate on capital k (which is fixed for each gas station in the short run). Does anything change in the short run?
(c) What happens to x∗, p∗ and X∗ in the long run? Explain how this emerges from your graph.
(d) Is it possible for you to tell whether you will hire more or fewer workers as a result of the capital gains tax-induced increase in the rental rate? To the extent that it is not possible, what information could help clarify this?
(e) Is it possible for you to be able to tell whether the number of gasoline stations in the city in- creases or decreases as a result of the increase in the rental rate? What factors might your answer depend on?
(f) Can you tell whether employment of labor in gasoline stations increases or decreases? What about employment of capital?
B: Suppose that your production function is given by f (ℓ, k) = 30ℓ0.4k0.4, F = 1080 and the weekly city-wide demand for gallons of gasoline is x(p) = 100, 040 − 1, 000p. Furthermore, suppose that the wage is w = 15 and the current rental rate is 32.1568. Gasoline prices are typically in terms of tenths of cents — so express your answer accordingly.
(a) Suppose the industry is in long run equilibrium in the absence of capital gains taxes. Assuming that you can hire fractions of hours of capital and produce fractions of gallons of gasoline, how much gasoline will you produce and at what price do you sell your gasoline? (Use the cost function derived for Cobb-Douglas technologies given in equation (13.45) in exercise 13.5 (and remember to add the fixed cost).)
(b) How many gasoline stations are there in your city?
(c) Now suppose the government’s capital gains tax increases the rental rate of capital by 24.39% to $40. How will your sales of gasoline be affected in the new long run equilibrium?
(d) What is the new price of gasoline?
(e) Will you change the number of workers you hire? How about the hours of capital you rent?
(f) Will there be more or fewer gasoline stations in the city? How is your answer consistent with the change in the total sales of gasoline in the city?
(g) What happens to total employment at gasoline stations as a result of the capital gains tax? Explain intuitively how this can happen. Do you think this is a general result or one specific to the set-up of this problem?
(h) Which of your conclusions do you think is qualitatively independent of the production function used (so long as it is decreasing returns to scale), and which do you think is not?
(i) Which of your conclusions do you think is qualitatively independent of the demand function, and which do you think is not?