Question

In: Economics

2) Mary opens up a shop making flip-flops. Let: q(L,K) = sqrt(L)sqrt(2K) w = price of...

2) Mary opens up a shop making flip-flops. Let:

q(L,K) = sqrt(L)sqrt(2K)

w = price of labor per unit = $5

r = price of capital per unit = $15

p = price of flip-flops per unit= $5

a) Suppose Mary contracts on 50 units of flip-flop making machinery (kapital)… what is her optimal Labor demand in the short run i.e. what quantity of labor (L) should she hire if capital is fixed at K = 50?

b) Suppose wage increases to $10 per unit… how does her optimal short run labor demand change (i.e. what is the new optimal short run L)? Intuitively… what change resulted and why?

c) Mary is now planning for the long run… she must make 1000 flip flops – use the substitution method to write her problem as a single variable optimization (hint she wants to spend as little as possible subject to her production being at least q=1000). Let w=5 and r=15 still. DO NOT SOLVE…

Solutions

Expert Solution

2. The production function is given as . The cost of production would be or . The total revenue would be or .

(a) In the short run, K is fixed at 50. The production function would be or . The cost would be or . The profit would be or or . The profit would be maximum where or or or or or . The optimal labor demand would hence be 25 units.

(b) If the wage increases, then the cost function changes to be , and the profit would be in the short run. The profit would be maximum where or or or or . If the wage increases, the optimal labor demand would be 6.25 units. As the wage increases, the labor demand decreases, because increase in wages increases the cost, and decreases the profit. The relation between wage and marginal labor product is that . As price remains constant, an increase in wage require marginal product of labor to increase, which would require the labor input to decrease.

(c) In this case, both K and L are variable, and since output is given, the cost should be minimized with the output constraint. The problem would be as below.

Minimize

Subject to

The Lagrangian function in this case would be , and solving its FOCs would give us the required labor and capital demand.


Related Solutions

Assume that output is given by Q(L,K)=50 K^0.5 L^0.5 with price of labour L = w...
Assume that output is given by Q(L,K)=50 K^0.5 L^0.5 with price of labour L = w and price of capital K = r 1.If capital in the short run is fixed at K what is the short-run total cost? 2.Write the values for the derivatives of the Total cost with respect to w and r. Does Shephard’s lemma hold in this case?
q(L,K)=L+2K=100w=30v=40find, graph, and explain cost-minimizing solution (L*,K*)
q(L,K)=L+2K=100w=30v=40find, graph, and explain cost-minimizing solution (L*,K*)
Assume that output is given by Q(L,K)=50L^0.5K^0.5 with price of labour L = w and price...
Assume that output is given by Q(L,K)=50L^0.5K^0.5 with price of labour L = w and price of capital K = r a Use the primal formulation of minimising costs to obtain the demand for Labour L and capital K 2 b Using the values of L & K obtained above, verify whether the output Q equals the one given in the question by eliminating the values of w and r. Are the primal and dual problems leading to the same...
Multiple Choice: 1. Suppose the firm's production process is given by Q = 2K^(1/2)*​L. If K=16...
Multiple Choice: 1. Suppose the firm's production process is given by Q = 2K^(1/2)*​L. If K=16 and L=8 what is the marginal productivity of capital? a) 1 b) 2 c) 5 d) 6 e) 8 2. Which of the following is not an assumption we make about perfectly competitive markets? a) Firms are price-takers b) Firms sell identical products c) Firms earn positive profit in the short-run but zero profit in the long-run d) Firms can freely enter or exit...
Multiple Choice: 1. Suppose the firm's production process is given by Q = 2K^(1/2)*​L. If K=16...
Multiple Choice: 1. Suppose the firm's production process is given by Q = 2K^(1/2)*​L. If K=16 and L=8 what is the marginal productivity of capital? a) 1 b) 2 c) 5 d) 6 e) 8 2. Which of the following is not an assumption we make about perfectly competitive markets? a) Firms are price-takers b) Firms sell identical products c) Firms earn positive profit in the short-run but zero profit in the long-run d) Firms can freely enter or exit...
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...
6. Suppose the production function is given by Q=1/20*L^1/2*K^1/4;price of labor(w) = 0.50 and price of...
6. Suppose the production function is given by Q=1/20*L^1/2*K^1/4;price of labor(w) = 0.50 and price of capital (r) = 4: The market price for the output produced is P= 560: (a) Short-run production: ii. Write down this firm's LONG-RUN cost minimization problem. [Note: In long-run, nothing is fixed, so the firm can choose both labor and capital optimally to minimize its cost.] iii. Solving the long-run cost minimization problem, we get the long-run cost function C(Q) = (12)*(5Q)^4/3: Find out...
For the following production functions, find the returns to scales. 1. F(K,L)=K^0.3L^0.7 2. F(K,L)=2K+L 3. F(K,L)=KL...
For the following production functions, find the returns to scales. 1. F(K,L)=K^0.3L^0.7 2. F(K,L)=2K+L 3. F(K,L)=KL 4. F(K,L)=K^0.2L^0.3 An explanation on how to do this, would be appreciated!
QUESTION 3 A firm's production function is Q = 2 KL, with MP L = 2K...
QUESTION 3 A firm's production function is Q = 2 KL, with MP L = 2K and MP K = 2L. The wage rate is $4 per hour, and the rental rate of capital is $5 per hour. If the firm wishes to produce 100 units of output in the long run, how many units of K and L should it employ? a. K = 6.33; L = 7.91. b. K = 2; L = 2. c. K = 4;...
2. A firm’s production function is given by q= L^1/2+ K. The price of labour is...
2. A firm’s production function is given by q= L^1/2+ K. The price of labour is fixed at w = 1, and the price of capital is fixed at r = 8. a. Find the firm’s marginal rate of technical substitution. b. Suppose both labour and capital can be varied by the firm, and that the firm wishes to produce q units of output. Use the answer to (a) to find the cost-minimising amounts of labour and capital (as functions...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT