6. Optimize the Cobb-Douglas production function given the following parameters. The maximum about of money available to spend is $1, 600 where the price of K = 12 and the price of L = 6. That is P = 12 and P = 6. The function is given as q = K0.4L0.6. Using the Lagrangian method, what are the optimal values of Ko and Lo?

6. Optimize the Cobb-Douglas production function given the following parameters. The maximum about of money available to spend is $1, 600 where the price of K = 12 and the price of L = 6. That is P = 12 and P = 6. The function is given as q = K0.4L0.6. Using the Lagrangian method, what are the optimal values of Ko and Lo?

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter9: Production Functions

Section: Chapter Questions

Problem 9.9P

See similar textbooks

Similar questions

Returns to scale in production: Do the following production functions exhibitincreasing, constant, or decreasing returns to scale in K and L? (Assume Ais some fxed positive number.)(a) Y = K1/2L1/2(b) Y = K2/3L2/3(c) Y = K1/3L1/2(d) Y = K + L(e) Y = K + K1/3L1/3 (f ) Y = K 1/3L2/3 + A (g) Y = K 1/3L2/3 − A
A bitcoin miner, Alex, needs only electricity (E) and computer (K) to mine bitcoin. Assume that the production function for his bitcoin business is of Cobb-Douglas type,?(?,?)=???? with?+?< 1, resulting in strictly convex isoquants and is the same in South Korea and USA. Suppose that, similar to the podcast, the price per unit of electricity is higher in South Korea than in USA. Suppose that the price of a computer is the same in both countries. i. Determine whether it is more expensive to mine one bitcoin in South Korea than in USA based on the above assumptions by using appropriate diagram and explain your answer. Please keep electricity on the horizontal axis and computer on the vertical axis while drawing your diagram.
A firm’s production function is given by q = min{M, L1/2}, where M is the number of machines and L is the amount of labor that it uses. The price of labor is $1and the price of machines is $2 per unit. a. Is this technology decreasing return to scale? Show your argument. b. Is it true the statement that if average production cost is increasing, a firm’s technology exhibits the decreasing return to scale. (If yes, show the proof. If not, show a counter example) c. Is it true that if a firm’s technology exhibits the decreasing return to scale, then average production cost is increasing? (If yes, show the proof. If not, show a counter example)
Suppose that the production function takes the form X = min(10L, 5K) and that a competitive firm faces a wage rate of £60 per week and a weekly capital rental of £32. (a) How much must the firm spend to produce 100 units of output, and what is the average cost of production when X = 100? (b) What is the incremental cost of producing the 101st unit of output? (c) What happens to the cost of producing 100 units of output if the wage rate and the rental cost of capital rise by 25 per cent each? What happens to the average and marginal cost? (d) What happens to the cost of producing 100 units of output if the wage rate increases by £1, or if the cost of capital increases by £1?
A firm has two opportunities for a new plant location, one is in China and the other is inMexico. The firm's production function is given by q = L 0.5 K 0.5 , In China, the cost of laboris w=$15 and the cost of capital is r=$5. In Mexico, w=$10 and r=$10. The firm wants toproduce 100 units of output. Which location should the firm choose for their new plant?Explain why.Note: Please round the optimal amounts of capital and labor at each location to the nearest whole number when making your calculations.Hint: cost-minimization rule.
explain your anwer with clearly and fully labeled graphs. The change in the optimal capital-labor ration if both inputs are perfect complements inproduction and both their prices increase by an identical percentage. Assume the totalcost before and after the change in input prices remains the same.
1. Given the following production function Q = 20L2K and the unit prices of labor and capital to be Birr 48 and 12 respectively, then a. What combination of labor and capital maximizes output with a cost of Birr 720? b. What is the maximum output? c. Show the output maximizing condition graphically.
Does the value of λ change if the budget changes from $4600 to $5600?What condition must a Cobb-Douglas production function q = cKαW β satisfy toensure that the marginal increase of production is not affected by the size of thebudget?
Question: Orla manages a loom that produces flags (F) using thread (T) and dye (D) as inputs. Herproduction function is given by: Q(T,D) = (T1/2 D1/2)1/2*For this problem, assume F, T, and D are infinitely divisible so you don’t need to worryabout restricting to whole-number answers. a.) Does Orla’s production function exhibit increasing, constant, or decreasing returns toscale? Explain. b.) Set up Orla’s cost-minimization problem to find the lowest-cost combination of inputsrequired to produce a specific level of output (bar Q) given factor prices PT and PD. (Note: You can write this either as a minimization subject to constraints or in Lagrangian form. *You do not need to solve it.)
A firm produces output according to a production function:Q = F(K,L) = min {6K,2L}.a. How much output is produced when K = 2 and L = 3?unit(s)b. If the wage rate is $45 per hour and the rental rate on capital is $25 per hour, what is the cost-minimizing input mix for producing 6 units of output?Capital: Labor: c. How does your answer to part b change if the wage rate decreases to $25 per hour but the rental rate on capital remains at $25 per hour? Capital decreases and labor increases. Capital and labor increase. It does not change. Capital increases and labor decreases. Only typed answer
Which of the following statements is true? (a) If a firm is experiencing decreasing returns, it will face increasing opportunity costs and hence will have a convex production possibility frontier. (b) If a firm has a steep average total cost curve, then it will have fairly low fixed costs as a proportion of total costs and would incur only a low cost penalty if output declines. (c) A firm with a concave production function at all levels of output would face a convex total cost function at all levels of output. (d) If a firm faces diminishing returns, then at low levels of output, the average variable cost effect would dominate, but at higher levels of output, the average fixed cost effect would dominate.
A firm produces output according to a production function:Q = F(K,L) = min {6K,2L}.a. How much output is produced when K = 2 and L = 3? unit(s)b. If the wage rate is $30 per hour and the rental rate on capital is $10 per hour, what is the cost-minimizing input mix for producing 6 units of output?Capital: Labor: c. How does your answer to part b change if the wage rate decreases to $10 per hour but the rental rate on capital remains at $10 per hour? (Choose one that is the best answer) Capital decreases and labor increases. It does not change. Capital increases and labor decreases. Capital and labor increase.