Question

In: Economics

2. Consider a firm with the following production function: Q = K 1/3 L 2/3 (a)...

2. Consider a firm with the following production function: Q = K 1/3 L 2/3

(a) Consider an output level of Q = 100. Find the expression of the isoquant for this output level.

(b) Find the marginal product of labor, MPL. Is it increasing, decreasing, or constant in the units of labor, L, that the firm uses?

(c) Find the marginal product of capital, MPK. Is it increasing, decreasing, or constant in the units of capital, K, that the firm uses?

(d) Use your result in parts (b)-(c) to find the marginal rate of technical substitution, MRTS, for this firm.

(e) Is the MRTS increasing or decreasing in the units of labor, L? What does that imply about the shape of the isoquant?

(f) Given your result in part (d), what can you say about the firm’s ability to substitute one input for another? (g) Assume now that the firm were to increase all inputs by a common factor > 0. What happens to the output that the firm produces? [Hint: check whether the firm’s production function exhibits increasing, decreasing, or constant returns to scale.]

Solutions

Expert Solution

2. (a) The isoquant equation for Q=100 would be as .

In terms of two variable equaiton/function, it would be as or .

(b) The MPL would be as or or or .

We have or , meaning that MPL is decreasing in L, ie MPL decreases as L increases.

(c) The MPK would be as or or or .

We have or , meaning that MPK is decreasing in K, ie MPK decreases as K increases.

(d) The MRTS would be as or or .

(e) We have or , meaning that the MRTS increases as L increases or MRTS is increasing in L (while MRTS would be decreasing in K, also since MRTS is negative, MRTS increasing in L means that MRTS tends to 0 as L increases). According to the founded condition, the isoquant is negatively sloped and is convex to the origin (MRTS can be taken as first derivative of isoquant while derivative of MRTS can be considered as second derivate of isoquant, which is positive).

(f) The firm would substitute 2K/L units of capital for a marginal unit of labor. This means that if firm wants to increase a marginal unit of labor, firm must reduce 2K/L units of capital, in order to remain at the same production level.

(g) The firm's production function is , and increasing inputs by a>0 times, we have or or . This means that the firm have constant returns to scale, ie the output increases by 'a' times for an increase in input by 'a' times.


Related Solutions

Consider production function Q= L^3 * K^4 - L^2 (a) Determine the MRTS L,K for this...
Consider production function Q= L^3 * K^4 - L^2 (a) Determine the MRTS L,K for this production function (b) Does this production function have an uneconomic region? If so, describe the region algebraically. (Hint: your answer will be an inequality like this: K<5L)
Consider the firm with production function given by q = f ( L , K )...
Consider the firm with production function given by q = f ( L , K ) = L ^(1/4) K^(1/4). If w = r = 4, what is the change in the producer surplus when the price increases from $16 to $32? (round your answer to one decimal place if necessary)
Consider a firm whose production is given by Q(K, L) = K^1/3L^1/3, where K and L...
Consider a firm whose production is given by Q(K, L) = K^1/3L^1/3, where K and L are, respectively, the quantities of capital and labour production inputs. Prices of capital and labour are both $1 per unit. (a) Derive and interpret the firm’s demand functions for capital and labour. (b) Derive and interpret the firm’s Long-Run Cost Function. (c) In the long-run, if the firm wished to produce 16 units of output, what quantities of capital and labour would it optimally...
A plant’s production function is Q = L^1/3 K^2/3, where L is hours of labor and...
A plant’s production function is Q = L^1/3 K^2/3, where L is hours of labor and K is hours of capital. The price of labor services, w, is $40 per hour and of capital services, r, is $10 per hour. a. Derive the long-run expansion path. In words describe what the expansion path represents. b. In the short-run, the plant’s capital is fixed at K = 64. Labor, on the other hand, is variable. How much will it cost to...
Consider the production function F(L,K) = L^2/3 K^2/3 . (f) Does this production function exhibit increasing,...
Consider the production function F(L,K) = L^2/3 K^2/3 . (f) Does this production function exhibit increasing, decreasing or constant returns to scale? Explain. (g) Find the total cost, average cost and marginal cost of producing y units of output. Is the average cost increasing or decreasing in y? Is the marginal cost higher or lower than the average cost? Question 2 The production of magic chairs requires only two inputs: seats (S) and legs (L) (no other inputs are required...
3. Consider the production function Q = K2L , where L is labor and K is...
3. Consider the production function Q = K2L , where L is labor and K is capital. a.[4] What is the Marginal Product of Capital for this production function? Is it increasing, decreasing, or constant? Briefly explain or show how you arrived at your answer. b.[4] Does this production function exhibit increasing, constant or decreasing returns to scale? Briefly explain or show how you arrived at your answer. c.[5] If the firm has capital fixed at 15 units in the...
Suppose that the firm has a production function described by q=0 if L≤6 q = −(L^3)/6+K(L^2)+26K...
Suppose that the firm has a production function described by q=0 if L≤6 q = −(L^3)/6+K(L^2)+26K if L>6 Further suppose that we are concerned only in the short run and that the units of capital employed are currently fixed at 8. Also, suppose that labor units are integer. That is, they can only be in terms of whole numbers 1, 2, 3, ... and so on. Suppose that the price of each unit produced by the firm is 2. 1....
The production function of a firm is given by Q(K,L) =15K^(1/4) L^(1/4) . Wage is $3...
The production function of a firm is given by Q(K,L) =15K^(1/4) L^(1/4) . Wage is $3 per unit of labor (L), and rent is $6 per unit of capital (K). (1) The firm’s objective is to produce Q units of output at minimum cost. Write the Lagrangian and derive the FONC. (2) Find the optimal levels of K, L, and λ given Q. (3) Find the minimum cost given Q = 100. Find the firm’s minimum cost functiongiven any Q....
A firm has production function q = 100 L + KL− L^2 − K^2 The price...
A firm has production function q = 100 L + KL− L^2 − K^2 The price of the good is $1. The wage is $10, and the price of capital is $30. Assume that the firm is a price - taker in a perfectly competitive market. a. What will the firm’s profit maximizing choice of capital and labor be? b. Suppose that the firm’s capital is fixed in the short-run and wage rises to $20. What is the firm’s new...
Exercise 3. Consider a firm with the Cobb-Douglas production function Q = 4L^1/3*K^1/2. Assume that the...
Exercise 3. Consider a firm with the Cobb-Douglas production function Q = 4L^1/3*K^1/2. Assume that the firm faces input prices of w = $7 per unit of labor, and r = $10 per unit of capital. a) Solve the firm’s cost minimization problem, to obtain the combination of inputs (labor and capital) that minimizes the firm’s cost of production a given amount of output, Q. b) Use your results form part (a) to find the firm’s cost function. This is...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT