Question

In: Economics

Coldsprings Tavern produces burgers according to the following production function:Q=K^1/3 E^2/3.Coldsprings will produce 400 burgers when...

Coldsprings Tavern produces burgers according to the following production function:Q=K^1/3 E^2/3.Coldsprings will produce 400 burgers when the hourly wage is $10 and the rental price of capital is $20.

a.Find the cost-minimizing bundle of employment and capital.b.Now suppose that the market wage falls to $8 per hour, and at this set of input prices, Coldsprings Tavern produces 450 burgers. Find the cost-minimizing bundleof employment and capital.

c.Solve for the numerical values of the scale and substitution effects for both employment and capital. [Hint: First find the cost-minimizing bundle of employment and capital when the wage rate is $10 per hour and q=450

Solutions

Expert Solution

Q = K^1/3E^2/3

Total cost (C): C = wN + rK

Cost is minimized when MUE / MUK = w / r

MUE = Q / E = (2/3) x (K / E)^1/3

MUK = Q / E = (1/3) x (E / K)^2/3

MUE / MUK = 2 x (K / E)

(a) Q = 400 = K^1/3E^2/3

MUE / MUK = 10/20 = 1/2

2K / E = 1/2

E = 4K

Substituting in production function,

400 = K^1/3(4K)^2/3

400 = 4^2/3K^1/3K^2/3

400 = 4^2/3K

K = 400 / 4^2/3 = 158.74

E = 4 x 158.74 = 634.96

(b) Q = 450 = K^1/3E^2/3

MUE / MUK = 8/20 = 2/5

2K / E = 2/5

2E = 10K

E = 5K

Substituting in production function,

450 = K^1/3(5K)^2/3

450 = 5^2/3K^1/3K^2/3

450 = 5^2/3K

K = 450 / 5^2/3 = 153.9

E = 5 x 153.9 = 769.5

MUE / MUK = 10/20 = 1/2

2K / E = 1/2

E = 4K

Substituting in production function,

450 = K^1/3(4K)^2/3

450 = 4^2/3K^1/3K^2/3

450 = 4^2/3K

K = 450 / 4^2/3 = 178.58

E = 4 x 178.58 = 714.32

For employment,

Total effect (TE) = 769.5 - 634.96 = 134.54

Substitution effect (SE) = 769.5 - 714.32 = 55.18

Scale effect = TE - SE = 134.54 - 55.18 = 79.36

For capital,

TE = 153.9 - 158.74 = - 4.84

SE = 153.9 - 178.58 = - 24.68

Scale effect = - 4.84 + 24.68 = 19.84

Rahul Sunny answered 1 year ago

Next > < Previous

Related Solutions

The technology available allows to produce a good according to the production function F(L,K)=[(L-3)K]1/3 if L≥3...

The technology available allows to produce a good according to the production function F(L,K)=[(L-3)K]1/3 if L≥3 and K≥0, and F(L,K)=0 if L<3. The market demand for this good is D(p)=90/p. The prices of labor and capital are w=1 and r=9, respectively. A. (12 points) Describe the cost minimization problem of a firm with this technology, and calculate its conditional factor demands. Also, calculate the firm's total, average and marginal cost functions. B. (10 points) Calculate la supply of a competitive...

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...

4. A firm has the following production function:y = L 1/3 K 1/2 . 2 (a)...

4. A firm has the following production function:y = L 1/3 K 1/2 . 2 (a) Does this production function exhibit increasing, decreasing, or constant returns to scale? Prove. (b) Suppose in the short run, capital is fixed at K = 100. Assuming that the output and factor prices are p, w, and r respectively, find firm’s factor demand for labor. What will the effects be when w, r, and p increase? Explain your results intuitively. (c) Now, suppose the...

A firm produces output according to the following function: q = f (L, K) = L^1/2...

A firm produces output according to the following function: q = f (L, K) = L^1/2 K^1/4 . The cost of labor is $8 per hour and the rental cost of capital is $2 per hour. a. Determine the returns to scale for this function. b. Suppose the firm wishes to produce at cost $96. How much capital and how much labor does the firm employ? c. Derive the short-run cost function with optimal amount of K from part b....

Consider a firm with production function given by q = f(E, K) = E^(1/4)K^(1/4), where E...

Consider a firm with production function given by q = f(E, K) = E^(1/4)K^(1/4), where E is number of workers and K is capital. The price of labor is w = 4 and price of capital is r = 8. The price of the output that firm produces is 20. The firm can adjust both its inputs. 1) Derive an expression for MRT S for this firm. 2) The firm wants to produce q0 units of output. What would be...

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...

In the classic model given the following: Production function: Y=3 K^.5 L^.5 Labor (L) is 400...

In the classic model given the following: Production function: Y=3 K^.5 L^.5 Labor (L) is 400 units Capital (K) is 100 units Taxes (T) are 200 Government Spending (G) is 100 Marginal Propensity to Consume is .6 Investment is determined by the following function: I(r) = 1000- 100r where r is real interest rate. 1. a.) full formula, Y=c(Y-T)+I+G, What are the equilibrium values of C, I and r? b.) How much are Public savings, Private savings and National Savings?...

A firm produces output according to the production function: Q= F(K, L) = 4K + 8L.

A firm produces output according to the production function: Q= F(K, L) = 4K + 8L. a. How much output is produced when K= 2 and L = 3? b. If the wage rate is $60 per hour and the rental rate on capital is $20 per hour, what is the cost-minimizing input mix for producing 32 units of output? c. If the wage rate decreases to $20 per hour but the rental rate on capital remains at $20 per hour, what is the...

Suppose Canada’s aggregate production function is given by the following: Y = (K1^3)(N2^3) Variables are deﬁned...

Suppose Canada’s aggregate production function is given by the following: Y = (K*1^3)(N*2^3) Variables are deﬁned as they were in class. Suppose the savings rate in Canada is 33.33% (s = 1 3) and the depreciation rate is 15% (δ = 0.15). Assume that Canada is not currently experiencing any technological change. a) Calculate the steady-state level of capital per worker and output per worker in Canada’s economy. b) Determine the annual growth rate of output per worker in Canada....

Consider a Solow economy with the following production function F(K,N) = zK^(1/3)N^(2/3) and parameters d =...

Consider a Solow economy with the following production function F(K,N) = zK^(1/3)N^(2/3) and parameters d = 0.05, s = 0.2, N0 = 100 and z = 1.0. Suppose K = 300 in period 0 and the unit period is one year. In contrast to the standard Solow model, we assume that the population growth rate n is no longer exogenous but rather endogenous and determined by (1 + n) = N’/N = g(C/N) = (C/N)^3 as it is the case...

Subjects