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 Consider a firm whose technology is given by the production function: y = f (K,L) = (N+1)L + 2√K. Assume that input prices for labour, L, and capital, K, are w = 1 and r = 2 respectively. a)Find the firm’s short run total cost function assuming the firm cannot hire more workers nor fire its only worker, i.e. L = 1 is fixed. 

b)Find the firm’s short run marginal cost, average variable cost and average fixed cost. Draw the three of them in the same graph. What is the short-run supply of the firm?

c)Find the firm’s long run total cost function, i.e. when all factors are variable. 

d)Find the firm’s long run marginal cost and average cost. Draw them in the same graph. What is the long-run supply of the firm?

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Related questions

What is the difference between a firm’s shutdown point in the short run and its exit point in the long run?

In the short run, a firm's shutdown point is the minimum point on the

A. average variable cost curve, while in the long run, a firm cannot exit

B. marginal cost curve and in the long run, a firm's exit point is the minimum point on the marginal cost curve

C. average total cost curve and in the long run, a firm's exit point is the minimum point on the average total cost curve

D. average variable cost curve, while in the long run, a firm's exit point is the minimum point on the average fixed cost curve

E. average variable cost curve, while in the long run, a firm's exit point is the minimum point on the average total cost curve.

 

A firm has the following short-run production function: Q 5 50L 1 6L2 2 0.5L3 where Q 5 Quantity of output per week L 5 Labor (number of workers) a. When does the law of diminishing returns take effect? b. Calculate the range of values for labor over which Stages I, II, and III occur. c. Assume each worker is paid $10 per hour and works a 40-hour week. How many workers should the firm hire if the price of the output is $10? Suppose the price of the output falls to $7.50. What do you think would be the short-run impact on the firm’s production? The long-run impact?

3. A firm has a long-run production function:   

(a) In the short run, K = 81 is fixed. Find the short-run production function. What is the marginal product of labor in the short-run? 

(b) If w = 10 and r = 15.24, find the short-run cost function. To do this, first, figure out how many workers are needed to produce Q units of output, and then add up the (fixed) SR cost of capital and the cost of labor as a function of Q. How many workers are needed to produce Q = 10?

(c) Write down equations for an average total cost (ATC) and average variable cost (AVC) as a function of Q. What is ATC when Q = 10? What as AVC when Q = 10?

(d) Write down the equation for the marginal cost (MC) as a function of Q. What is the marginal cost of producing Q = 10 in the short run?

(e) Verify that MC = w*MPL, evaluating the marginal product of labor from part (3a) at the number of workers needed to produce 10 units of output.

(f) In the long run, both capital and labor are variable. Find the marginal product of labor and the marginal product of capital. Find the marginal rate of technical substitution. On a properly-labeled graph, draw an isoquant for Q = 10.

(g) Does the production function exhibit increasing, decreasing, or constant returns to scale?

(h) Find the lowest cost method of producing Q = 10 (two unknowns: K and L). To do this, equate the MRTS with the input price ratio w r (one equation) and use the isoquant for Q = 10 as the second equation.

(i) Use the fact that the cost, in the long run, is wL + rK to find the LR cost of producing Q = 10. What is the long-run average cost (LRAC) at Q = 10?

(j) (Hard) More generally, use the MRTS and the production function to find the cost of producing any quantity Q. To do this, repeat the above exercise but keep Q as an (unknown) variable, and solve for K and L as functions of Q.

(k) On a properly labeled graph, depict ATC from part (3b) and LRAC from part (3j). Why is LRAC everywhere below ATC?

(l) What is the production level at which SRAC is minimized?

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