Fixed amount of Land for Oil Drilling: Suppose that your oil company is part of a competitive industry and is using three rather than two inputs—labor ℓ, capital k and land L —to produce barrels of crude oil denoted by x. Suppose that the government, due to environmental concerns, has limited the amount of land available for oil drilling—and suppose that it has assigned each oil company L̅ acres of such land. Assume throughout that oil sells at a market price p, labor at a market wage of w and capital at a rental rate r — and these prices do not change as government policy changes.

A: Assume throughout that the production technology is homothetic and has constant returns to scale.

(a) Suppose that, once assigned to an oil company, the company is not required to pay for using the land to drill for oil (but it cannot do anything else with it if it chooses not to drill.) How much land will your oil company use?

(b) While the 3-input production frontier has constant returns to scale, can you determine the effective returns to scale of production once you take into account that available land is fixed?

(c) What do average and marginal cost curves look like for your company over the time frame when both labor and capital can be varied?

(d) Now suppose that the government begins to charge a per-acre rental price q for use of land that is assigned to your company, but an oil company that is assigned L acres of land only has the option of renting all L acres or none at all. Given that it takes time to relocate oil drilling equipment, you cannot adjust to this change in the short run. Will you change how much oil you produce?

(e) In the long run (when you can move equipment off land), what happens to average and marginal costs for you company? Will you change your output level?

(g) How much will you produce now compared to the case analyzed in (d)?

(h) Suppose that under this alternative policy the government raises the rental price to q′. Will your company change its output level in the short run?

(i) How do long run average and marginal cost curves change? If you continue to produce oil under the higher land rental price, will you increase or decrease your output level, or will you leave it unchanged.

(j) True or False: The land rental rate q set by the government has no impact on oil production levels so long as oil companies do not exit the industry. Explain. (Hint: This is true.)

B: Suppose that your production technology for oil drilling is characterized by the production function x = f (ℓ,k,L) = Aℓ^αk^βLγ where α + β + γ = 1 (and all exponents are positive).

(a) Demonstrate that this production function has constant returns to scale.

(b) Suppose again that the government assigns L acres of land to your company for oil drilling, that there is no rental fee for the land but you cannot use the land for any other purpose. Given the fixed level of land available, what is your production function now? Demonstrate that it has decreasing returns to scale.

(c) In exercise 13.2, you were asked to derive the (long run) cost function for a 2-input Cobb- Douglas production function. Can you use your result — which is also given in equation (13.35) of exercise 13.5 — to derive the cost function for your oil company? What is the marginal cost function associated with this?

(d) Next, consider the scenario under which the government charges a per-acre rental fee of q but only gives you the option of renting all L acres or none at all. Write down your new (long run) cost function and derive the marginal and average cost function. Can you infer the shape of the marginal and average cost curves?

(e) Does the (long run) marginal cost function change when the government begins to charge for use of the land in this way?

(f) Now suppose that the government no longer requires your company to rent all L̅ acres but instead agrees to rent you up to L̅ acres at the land rental rate q. What would your conditional input demands and your (total) cost function be in the absence of the cap on how much land you can rent?

(g) From now on, suppose that A = 100, α = β = 0.25, γ = 0.5, L̅ = 10,000. Suppose further that the weekly wage rate is w = 1000, the weekly capital rental rate is r = 1000 and the weekly land rent rate is q = 1000. At what level of output x̅ will your production process no longer exhibit constant returns to scale (given the land limit of L̅)? What is the marginal and average cost of oil drilling prior to reaching x̅ (as a function of x)?

(h) After reaching this x̅, what is the marginal and average long run cost of oil drilling (as a function of x)? Compare the marginal cost at x̅ to your marginal cost answer in (g) and explain how this translates into a graph of the marginal cost curve for the firm in this scenario.

(i) What happens to x̅ as q increases? How does that change the graph of marginal and average cost curves?

(j) If the price per barrel of oil is p = 100, what is your profit maximizing oil production level?

(k) Suppose the government now raises q from 1,000 to 10,000. What happens to your production of oil? What if the government raises q to 15,000?