Calculate the short run total and average cost curves -

Question

Question 1. Frisbees are produced according to the production function:

q = 4 K + L

where q = output of Frisbees per hour

K= capital input per hour

L = labor input per hour

(a) If K=10, how much L is needed to produce 100 Frisbees per hour?

(b) If K=20, how much L is needed to produce 100 Frisbees per hour?

(c) Graph the q=100 isoquant letting y-axis be capital (K) and x-axis be labor (L).

(d) What is the RTS along this isoquant? Explain why the RTS is the same at every point on this isoquant.

(e) Graph the q=200 isoquant. Describe the shape of the entire isoquant map.

(f) If the wage rate (w) is $1 and the rental rate on capital (v) is $1, what cost- minimizing combination of K and L will the manufacturer employ for q=100?

(g) If the wage rate (w) is $1 and the rental rate on capital (v) is $1, what cost- minimizing combination of K and L will the manufacturer employ for q=200?

(h) What is the manufacture's expansion path?

Question 2. Venture capitalist Sarah purchases two firms to produce widgets. Each firm produces identical products and each has a production function given by q_i = √(K_iL_i) , where i = 1, 2

The firms differ, however, in the amount of capital equipment each has. In particular, firm 1 has K_i = 25, whereas firm 2 has K₂ =100. Rental rates for K and L are given by w = v = $1.

(a) What is the marginal product of labor ( MP_L1 ) for firm 1?

(b) What is the marginal product of labor ( MP_L2 ) for firm 2?

(c) If Sarah wishes to minimize short-run total costs of widget production, how would output be allocated between the two firms?

(d) Given that output is allocated between the two firms, calculate the short run total and average cost curves.

Question 3. Please draw isoquants that show:

(a) constant returns to scale,

(b) increasing returns to scale,

(c) decreasing returns to scale.

In your graph, you need to label capital (K) on the y-axis and labor (L) on the x- axis. You can assign isoquant value randomly.

Question 4. Given the Cobb-Douglas production function q = K^aL^b such that

MP_K = aK^a-1L^b, MP_L = bK^aL^b-1. If this production function exhibits constant returns to scale (a+b=1), show that:

(a) Both marginal productivities are diminishing.

(b) The marginal rate of technical substitution (RTS) for this function is giving by RTS = bK/aL

(c) The function exhibits a diminishing RTS.

Question 5. Given the Cobb-Douglas production function q = 100 K^0.5L^0.5. Rental rates for K and L are given by w = v = $15. Total cost =$900.

(a) Please find the quantity of labor and capital that the firm should use in order to maximize output.

(b) What is this level of output?