# Find the marginal rate of technical substitution for the following production functions. a. q =...

## Question:

Find the marginal rate of technical substitution for the following production functions.

a. {eq}q = 10L^{1/3}K^{2/3} {/eq}

b. {eq}q = 4L^{0.5} + 2K^{0.5} {/eq}

c. {eq}q = \min \left \{ 2K,4L \right \} {/eq}

d. {eq}q = 3L + 8K {/eq}

## Rate of Technical Substitution

The marginal rate of technical substitution is actually the slope given for any type of production function. The slope of the production function is derived by using the marginal product and taking the ratio of each marginal productivity of each input. The ratio of marginal product of labor to that of capital is the marginal rate of technical substitution.

## Answer and Explanation: 1

Become a Study.com member to unlock this answer! Create your account

View this answer

**a.**

The marginal rate of technical substitution:

{eq}MRT{S_{L,K}} = \frac{{M{P_L}}}{{M{P_K}}}{/eq}

{eq}\begin{align*} q =...

See full answer below.

#### Ask a question

Our experts can answer your tough homework and study questions.

Ask a question Ask a question#### Search Answers

#### Learn more about this topic:

from

Chapter 3 / Lesson 13The marginal rate of substitution shows how quickly a person will substitute or replace one product for a different one. Study the definition, formula, and examples of the marginal rate of substitution, how producers use it, and differing quantities.

#### Related to this Question

- Find the Marginal Rate of Technical Substitution for the following production functions: a. q = L^0.5 K^0.5 b. q = L^0.5 + K^0.5 c. q = min{K, L} d. q = L + K
- The production function is Q = K^{.6}L^{.4}, what is the marginal rate of technical substitution?
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = 10L + K.
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = K^2 L.
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long-run production function q = 5L^0.5K.
- Suppose the production function for a firm is given by q = 4L + 2K. If the firm currently has 20 units of capital (K) and 10 units of labor (L), then calculate the marginal rate of technical substitution (MRTSLK).
- Suppose the production function for a firm is given by q = 8L + 2K. If the firm currently has 20 units of capital (K) and 10 units of labor (L), then calculate the Marginal Rate of Technical Substitution (MRTS_(LK)).
- Suppose the production function for a firm is given by q = 4L0.75K0.25. If the firm currently has 10 units of capital (K) and 10 units of labor (L), then calculate the marginal rate of technical substitution (MRTSLK).
- How do you calculate the marginal rate of technical substitution from the production function?
- Derive the marginal rate of technical substitution for the Cobb-Douglass production function Q = cL^{alpha}K^{beta}.
- The production function is Q = K^.6L^.4. What is the marginal rate of substitution of L for K? What is the numerical MRS when K = 5 and L = 10? What is MRS if K = 10 and L = 5?
- Find the Marginal rate of technical Substitution when capital is 10 and labor is 10 if the production function is: F ( L , K ) = 10 K + 2 L 2
- Production function is Q=K^.6L^.4, what is the marginal rate of technical substitution? Show work.
- Determine the Marginal Rate of Technical Substitution for each production function (treat K as the y-input and L as the x-input) q(K, L) = 3KL^2 + 2L + 3K
- Determine the Marginal Rate of Technical Substitution for each production function (treat K as the y-input and L as the x-input) q(K, L) = 3L^2K^3
- Consider the following production function: Q = F(L, K) = L^4 * K^7. A) Does this production function exhibit diminishing or increasing marginal rate of technical substitution of labor for capital? Show your work. B) Find the elasticity of substitution fo
- Determine the Marginal Rate of Technical Substitution for each production function (treat K as the y-input and L as the x-input) q(K, L) = 3K + 2L
- List whether each of the following production function functions has diminishing marginal returns to labor (Y or N). a. Q = 50K + 30L - .5L2, MPL.= 30- L b. Q = L.5K.8 MPL = .5K.8/L.5 c. Q = 2L + K
- The production function for the Gwilmo Firm can be written as Q = 9K^{1/2}L^{1/2}. 1. Graph the isoquant for Q = 1,350. 2. Assume K = 1,600 and L = 225. Calculate the marginal product of L. 3. Assume K = 1,600 and L = 225. Now, decrease L by one unit. By
- For the production function, Q = K^{0.5}L^{0. 5}, the slope of the Q = 100 isoquant when L = 16 is: a. -39.06 b. -4 c. -25 d. -6.25
- A firm's production function is Q = 5L2/3K1/3. a) Does this production function exhibit constant, increasing, or decreasing returns to scale, and why? b) What is the marginal rate of technical substitution of L for K for this production function? c) Wh
- Consider a firm that has a production function f(L, K) = 5L^{1/3}K^{2/3}. What is the expression for the marginal rate of technical substitution, MRTS_{LK} at (L, K)? A. K/L B.10K/L C. K/2L D. 2K/L E. K/10L
- Total Cost Function is given below: TC= 100 + 5 Q + 2.5 Q^2 If the output level; Q = 10 units, find the following:
- For each of the following production functions, find the MRTS (in terms of K and L) and the elasticity of substitution (?) a.) Q = 3K^(1/For each of the following production functions, find the MRTS (
- Find the rate of technical substitution (RTS) for this production functions: a) q=10K^{.6}L^{.3} b) q=7K+3L
- Consider the following production functions and corresponding isoquants. Which isoquant exhibits diminishing marginal rate of technical substitution? 1. Q=10K+5L 2. Q=Min(10K, 5L) 3. Q=K0.6 L0.6 4. Q=K0.5 L0.5
- Write the equations for the marginal product of capital, marginal product of labor, and marginal rate of technical substitution for the long run production function q =12K^0.5 L^0.5.
- Consider the following production function: Q = L^AK^0.45. If this production technology exhibits constant returns to scale, what must be the value of A?
- "Answer the following questions using the production function q=AK^a L^b, where A,a,b0. 1. Does labor exhibit diminishing marginal product if b=0.2? 2. Find the rate of technical substitution (RTS)
- Consider the following production function: Q = AL^aK^b. Assume A is greater than 0. Further assume 0 is less than a is less than 1, and 0 is less than b is less than 1. a. What is the Marginal Produ
- Given the production function q = 6L + 2K, what is the marginal product of labor when capital is fixed at 15?
- Suppose the production function is given by Q = 3K + 4L. What is the marginal product of capital when 5 units of capital and 10 units of labor are employed?
- Suppose a production function is given by Q = 4K + 3L. What is the marginal product of capital when 10 units of capital and 10 units of labor are employed?
- For the production function Q = K^{0.5}L ^{0.5}, the slope at any point on any isoquant will be: a. -2. b. -1 c. \frac{-K}{L} d. \frac{-3K}{2L}
- Suppose the cost function is C(Q) = 50 + Q - 10Q2 + 2Q3. What is the variable cost of producing 10 units?
- Suppose the production function is given by Q = 2K + 5L. What is the marginal product of labor when 15 units of capital and 10 units of labor are employed?
- Given the following graph, briefly respond to the questions. a) For an output level of 40 units, calculate the marginal rate of technical substitution between points A and B. b) For an output level of
- Suppose that a firm has a production function given by q = 10L0.5K0.6. The firm has 10 units of capital in the short run. Which of the following will describe the marginal product of labor (MPL) for this production function? a. increasing marginal returns
- Suppose that you are given the following cost function C(w,r,Q) = 2w^1/2 r^1/2 Q^3/2 where w is the wage rate for labor, r is the rental rate of capital and q is the output level. (a) Derive the marg
- If a price taking firm's production function is given by q=2\sqrt l, its supply function is given by: a. q = 2pw b. q = p/w c. q = 2p/w d. q = p/2w
- Consider the production function Q = (L^{1/2} + K^{1/2}) ^2 Find out the shape of the production.
- Suppose that the production function takes the form Q = L - 0.9L^2. In addition, marginal revenue is $10 and marginal cost is $1. The optimal number of hours of labor is (blank).
- Suppose that a firm's production function is q = 10 L^0.5 K^0.5. This means that the marginal rate of technical substitution is K/L.The cost of a unit of labor is $20 and the cost of a unit of capital is $80. The firm wants to produce 130 units of output.
- Suppose you are given the following total cost function: TC = 2000 + 15Q - 6Q2 + Q3 , where Q = units of output: Using this function: How much is TFC at an output of 2000 units? At 5000 units? How m
- How do you find the marginal product given a production function?
- Find the returns to scale for the following production functions. Q=X1^.34 X2^.34 Q= (2X1+3X2)^.5 Q=[0.3X1^.5+0.7X2^.5]^2 Q=[min(X1,2X2)]^2
- 2. determine the returns to scale for the following production functions: (a) Q=(L+K)2 (b) Q=(L(1/2)+K(1/2))2 3. A firm's production function is given by q = 5L2/3 K1/3 (a) Calculate APL and MPL. D
- What is the relationship between the marginal products of the factors of production and the marginal rate of technical substitution?
- Determine if each of the following production functions exhibit decreasing, constant, or increasing returns to scale. a. q = 5L0.4K0.5 b. q = min(21, K)
- Consider the production function f(x1, x2) = 2x1 + \sqrt {x2}. What are the marginal products of factors 1 and 2, and the technical rate of substitution at the bundle of inputs (x1, x2)?
- The Calmadum Company has the production function Q = 4,500L + 60L^2 - 0.5L^3. 1. Calculate the L at which Q is at maximum. 2. What is the maximum Q? 3. Calculate the L at which the marginal product of L is a maximum. 4. What is the maximum marginal produc
- Given a classical production function, q = 80 L + 12 L^2 - L^3, find the following: a. The formulas for the marginal product of labor, and the average product of labor b. The level of labor where MP
- Consider the following production function: q = k + l^y , 0< y < 1 Is the production function CRS? Why or why not? Find the cost function and draw it (with respect to q).
- Consider the production function is q = L^0.6 + 4K. A) Starting from the input combination (10,10), calculate the marginal product of adding one worker. B) What is the marginal product of adding anoth
- Suppose that a firm's production function is q = 5x^{0.5} in the short run, where there are fixed costs of $1,000, and x is the variable input whose cost is$1250 per unit. The total cost of producing a level of output q is C(q) = 1,000 + \frac{1250q^2}{25
- List whether each of the following production functions has decreasing, increasing or constant returns to scale: a. Q = Min (2K, L) b . Q = L .5 K .8 c . Q = L .5 + K .5 d . Q = 10 + K + L
- For each of the two-input production functions below, calculate the corresponding cost function, c (w_1, w_2, q). (a) q = x^1/2 _1 x^1/2 _2 (b) q = x_1/2 + \sqrt{x_2} (c) q = x_1 (x_2 - 8) (d) q =
- Determine whether the following production function exhibits increasing, constant or decreasing returns to scale. Q = Min(2K, 2L)
- Determine whether the following production function exhibits increasing, constant or decreasing returns to scale. Q = K + L
- Given the Production Function Q = 72X + 15X^2 - X^3, where Q = output and X = Input a. What is the Marginal Product (MP) when X = 8? b. What is the Average Product (AP) when X = 6? c. At what value of
- Suppose a firm's short-run production function is given by Q = 16L0.80. What is the marginal product of the fourth worker?
- For each of the two-input production functions below, calculate the corresponding cost function, c(wi, w2, q). (a) q = x_1^{1/2}x_2^{1/2} (b) q = x_1/2 + \sqrt{x_2} (c) q = x_1(x_2 - 8) (d) q = [min
- Suppose a firm's production function is given by Q = L1/2*K1/2. The Marginal Product of Labor and the Marginal Product of Capital are given by: MPL = (K^1/2)/(2L^1/2), and MPK =(L^1/2)/(2K^1/2). a) S
- Consider the production function Q = (0.5K^{1/3} + 0.5L^{1/3})^3 . a. Prove that this production function exhibits constant returns to scale. b. Suppose the firms want to minimize the cost of produc
- The production function for a product is given by q = K1/2L1/4 where K is capital, L is labor and q is output. a. Find the marginal products of labor and capital. b. Is the marginal product of labor increasing or decreasing with labor? Is the marginal p
- Suppose that the production function of a firm is given by the equation Q = 2K1/2L1/2, where Q represents units of output, K units of capital, and L units of labor. What is the marginal product of labor and the marginal product of capital at K = 40 and L
- consider a firm that operates with the following production function: q = 2K^2L a. calculate the marginal products of labor and capital (MP_L AND MP_k) b. calculate the marginal rate of technical substitution of labor for capital, MRTS_{LK} Firm faces m
- Consider a production function given by: Q = 27K^{2}L^{0.5} - 2K^{4} A. Let L = 16. Find the level of K at which the marginal product of capital reaches a maximum B. Let L = 16. Find the level of K
- Determine whether the production function exhibits increasing, constant or decreasing returns to scale. Q = L^(0.5) K^(0.5)
- Suppose the cost function is C(Q) = 50 + Q - 10Q2 + 2Q3. What is the total cost of producing 10 units?
- Suppose the production function for a firm is as follows: q = min (3K, L). (a) Draw the isoquants for q = 3 and q = 6. (b.) Explain whether the production function exhibits constant, increasing or dec
- Suppose the production function for a firm is given by q = 5L0.5K0.25. In the short run, the firm has 16 units of capital. Find the marginal product of labor (MPL). Round to 2 decimal places.
- A. Suppose that a firm's production function is Q = 2L^0.5K^0.5, Derive the isoquant associated with Q = 12 units of output. B. If L = 1, what must K be in order to produce Q = 12 units of output? C
- Are the returns to scale of the following production functions increasing, decreasing, or constant? a) Q = KL/4 b) Q = K + L c) Q = Min(K/6, L/3)
- A firm has the following weekly production function: Q = 20KL - 0.025KL^2. Suppose the firm is in the short-run with K fixed at 20. a. What is the equation for the marginal product? Explain whether the production function is consistent with the Law of Di
- Suppose a firm's short-run production function is given by Q = 4L^{0.8}. If the production function is Q = L^{0.8} K^{0.2}, how many units of capital is it using?
- Consider the production function F(l,K)=3l^{.25}K^{.75} a) Find the cost-minimizing bundle and the long-run total cost if w = 64 and v = 1 and total output = q = 36. b) Change the price of capital to
- Which of the following production functions displays decreasing returns to scale? a) Q = aL + bK^{2} b) Q = aL + bK c) Q = bLK d) Q = cL^{0.2} \times K^{0.5}
- Suppose that a company produces output according to the following production function: Q = 0.5L^2 a) Define and calculate the marginal product of labor. b) Define and calculate the average product of
- Suppose the production function for good q is given by q = 3K + 2L where K and L are capital and labor inputs. Consider three statements about this function: I. The function exhibits constant returns to scale. II. The function exhibits constant marginal p
- Suppose that the production function is Q = L^{2 / 3} K^{1 / 2}. a. What is the average product of labour, holding capital fixed? b. What is the marginal product of labour? c. Determine whether the production function exhibits diminishing marginal product
- Suppose you have the following production function: Q = L^25 K^75, where Q is production, L is the amount of labor used in production, K is the amount of capital. The marginal product of labor is mathematically defined as: partial Q/ partial L. If K is fi
- Suppose a firm's production function is given by Q = L^(1/2)K^(1/2). The Marginal Product of Labor and the Marginal Product of Capital are given by: MP_(L) = K^(1/2)/(2L^(1/2)), and MP_(K) = L^(1/2)/(
- Suppose that the marginal rate of technical substitution of L for K is constant and equal to x. Then A. If the firm substitutes one unit of labor with x units of capital, then production increases.
- Consider a production function of the form with marginal products MP_K = 2KL^2 and MP_L = 2K^2L. What is the marginal rate of technical substitution of labor for capital at the point where K = 25 and L = 5? A. 5 B. 25 C. 15 D. 1
- The production function is given by Q = K^1/4L^1/4. a. Derive the marginal product of capital. Consider a production manager who must produce 200 units. b. Given this, express labor in terms of the needed output (200) and capital (K). c. From this, derive
- Suppose that Poland Spring has the following production function for bottled water: Q = K+10L1/2 What is the marginal rate of technical substitution of labor for capital?
- Suppose there are N firms in a perfectly competitive industry. Each firm is producing output Q using production function Q = K^(0.40) cdot L^(0.60). Which of the following statements must be true about the long-run competitive equilibrium? a. p = AC = MC
- Suppose that the production function of a firm is f(x_1, x_2) = x1 + min ( x_1, 2x_2). Derive (i) Technical rate of substitution (ii) Elasticity of Substitution (iii) Elasticity of scale (iv) Isoq
- Draw representative isoquants for the following production functions and indicated output levels: Q = (x1+x2)^2 for Q= 4 and Q=9 Q = min(2x1+x^2,x1+2x2) for Q = 12 and Q =15.
- Assume that the following production function is given by: Q = 5 K0.8 L0.8 Pk = 10 Pl = 3 A) what is the value of the elasticity of the labor factor (L)? B) what is the value of the elasticity of th
- Use the production function below to answer the following questions: A) Calculate marginal productivity (MP) and put this in the table. B) At what level of employment does diminishing marginal product
- If the production function is Q = K^(1/2) L^(1/2) and capital is fixed at 100 units, then the marginal product of labor (MPL) will be?
- Suppose a firm's production function is given by Q = L^{1/2}*K^{1/2}. The Marginal Product of Labor and the Marginal Product of Capital are given by: MP_L = 1/2L^{-1/2}K^{1/2} and MP_K = 1/2L^{1/2}K
- Following are different algebraic expressions of the production function. Decide whether each one has constant, increasing, or decreasing returns to scale. a. Q = 75L 0.25 K 0.75 b. Q = 75A 0.15 B 0
- Define the following terms as rigorously as you can. (a) Marginal Product (of a factor of production) (b) Opportunity Cost (c) Technical Rate of Substitution
- Suppose the following production function: Q = 10 (K)^{1/3} (L)^{2/3} subject to; W *L + r * K = Cost. a. Suppose that K the amount of capital is K = 8. If this company hires 64 workers (L), calculate the value of Q. b. Determine if this production func
- A firm technology that can be expressed by the following production function: Q=KL^2-L^3 The firm has a fixed amount of capital of [{MathJax fullWidth='false' bar K = 600. a. Derive the Marginal Prod
- A firm has a production function of y = f(L, k) = ( sqrtL + sqrtk)^2 a) Find expressions for the marginal product of labor and capital (b) Find the cost function
- Suppose we know that output in the economy is given by the production function: Y_t = A_t K_t^(1/4) L_t^(3/4) a) Use partial derivative techniques to solve for the marginal product of capital (Remembe