Question

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3. A firm producing hockey sticks has a production function given by f(11, 12) = 21112 In the short run, the firm's amount of input two is fixed at Tz. 3.1 Calculate the firm's short-run total cost curve as a function of y, w1, W2, 72. 3.2 Suppose that I2 = 100, the price for input one is w1 = 4, and the price of input two is w2 = 1. Draw a graph of the short-run average cost with output y on the horizontal axis. Indicate on the plot the short-run average cost for the firm if it produces 25 hockey sticks, 50 hockey sticks, 100 hockey sticks and 200 hockey sticks. 3.3 Calculate the long-run total cost of hockey stick production. (That is, form the cost-minimization problem for this firm, find the contingent demand functions, and the cost function when the firm is free to vary both inputs.) 3.4 For w. = 4 and we = 1, graph the long-run total cost curve for hockey stick production with output y on the horizontal axis. Show that the long-run cost is an envelope for the short-run curves computed in 3.1 by examining values of T2 of 100, 200, and 400.

Answer 1

3.1. Total short run cost   

TSRC = W1.01 + W2.07

The production funaction can be rewitten as

f(11, 12) = y = 2V11.12

11 = y^/4..17

: TSRC(y, T2, W1, W2) = w1.y?/4.72 + w2.In

- 2 vaga y = 2 5a 100 Tu= 20 vau 2 = 100) SRTC = Wie & Wor SRTC = 4.2 +16.00 (0,24, K2=100, w =1) SRTC = 4 x1 +100 Mouginal cost = d SRTC = 4 dal Average short run tobal cost = SRTC = 4 +100 24 SRTC of inoo 25 50 100 200 g 3.3) cost anonization a Obiective is to minimize cost subject to the production function where is fixed L = won + Wq Az I sit i = 2 x X2

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3.3 CONTINUED

Contingent demands

yW2 11(, w1, W2) = 2./wi

12(], w1, W2) = y vw1 2 Vw2

Substitute the contingent demand function in the total cost curve to get the exact value of TC

TC = 9T1.02

To find the exact values of the contingent demands and total costs we must substitute the values of w1, w2 and total output.