Hi, I'd like your help with this excercise (13.9) , it's from the book "Microeconomics: An intuitive approach with calculus", chapter 13: Production Decisions in the Short and Long Run. I already did (a) but I'm a little confused about the others. It is a long excercise, but I'd really appreciate any help. Thanks in advance.

B. Suppose that your production technology for oil drilling is characterized by the production function...(description is better in the pictures).

b. Suppose again that the government assigns acres of land to your company for oil drilling,

and 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 two-input

Cobb-Douglas production function. Can you use your result, which is also given in equation

(13.45) 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 but

only gives you the option of renting all acres or none at all. Write down your new (long-run)

cost function and derive the marginal and average cost functions. 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 acres but

instead agrees to rent you up to acres at the land rental rate . 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 , , , . Suppose

further that the weekly wage rate is , the weekly capital rental rate is ,

and the weekly land rent rate is . At what level of output will your production

process no longer exhibit constant returns to scale (given the land limit of )? What is the

marginal and average cost of oil drilling prior to reaching (as a function of )?

h. After reaching this , what is the marginal and average long-run cost of oil drilling (as a

function of )? Compare the marginal cost at 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 as increases? How does that change the graph of marginal and average

cost curves?

j. If the price per barrel of oil is , what is your profit-maximizing oil production

level?

k. Suppose the government now raises from 1,000 to 10,000. What happens to your production

of oil? What if the government raises to 15,000?

How do longrun 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? 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. (Hint This is true.) B. Suppose that your production technology for oil drilling is characterized by the production function X : 1T6 , LL) : AtkaLy where a: + B + y : l (and all exponents are positive). a. b. Demonstrate that this production function has constant returns to scale. Suppose again that the government assigns I acres of land to your company for oil drilling, and that there is no rental fee for the land but you cannot use the land for any other purpose. Given the xed level of land available, what is your production function now? Demonstrate that it has decreasing returns to scale. In exercise 13.2, you were asked to derive the (longrun) cost function for a twoinput CobbiDouglas production function. Can you use your result, which is also given in equation (13.45) of exercise 13.5, to derive the cost function for your oil company? What is the marginal cost function associated with this? Next, consider the scenario under whiclLie government charges a peracre rental fee of q but only gives you the option of renting all L acres or none at all. Write down your new {longrun) cost function and derive the marginal and average cost functions. Can you infer the shape of the marginal and average cost curves? Does the l[longirun) marginal cost function change when the government begins to charge for use of the land in this way? Now suppose that the government no longer requires your company to rent all I acres but instead agrees to rent you up to Z 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? Chapter 13. Production Decisions in the Short and Long Run g. From now on. suppose thatA 100, a: ,8 0.25, y 0.5. I 10,000. Suppose further that the weekly wage rate is w = 1,000. the weekly capital rental rate is r = 1,000, and the weekly land rent rate is g : 1.000. At what level of output } will your production process no longer exhibit constant returns to scale (given the land limit of Z]? What is the marginal and average cost of oil drilling prior to reaching 3 (as a function of X)? h. After reaching this X. what is the marginal and average longrun cost of oil drilling (as a function of X)? Compare the marginal cost at ito your marginal cost answer in (g) and explain how this translates into a graph of the marginal cost curve for the rm in this scenario. i. What happens to i'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 protmaximizing 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? 13.10 Policy and Business Application: MMmum Wage LaborSubsidy. Suppose you run your business by using a homothetic, decreasing returns to scale production process that requires minimum wage labor 5 and capital Irwhere the minimum wage is w and the rental rate on capital is r. A. The government, concerned over the lack of minimum wage jobs, agrees to subsidize your employ ment of minimum wage workers. effectively reducing the wage you have to pay to (l shit (where 0