Question

Suppose a firm uses capital(K) and labor(L) to produce widgets according to the production function Q= K1/4L1/4 = (KL)1/4, where Q is output. The goal in this problem is to find thefirm's total and marginal cost functions. In order to derive the costfunction, the first step is to find the location of thecost-minimizing bundle on any given isoquant. To get a numericalanswer, recall that the slope of an isoquant is -MPL/MPK(K is on the verticalaxis). Usingcalculus, it can be shown that this slope formula reduces to-K/L. Forexample, consider the input bundle with K= 8, L= 2 on the isoquant with Q= 2(note that 2= 161/4). The formula indicates that the slope at this point is-4 =-8/2. Infact, the formula tells us that regardless of which isoquant an input bundle ison, the slope will be-4 as long as the ratio of K to L is4:1. Similarly, the isoquant slope will be-1 whenever the ratio of K to L is1:1.

In answering the followingquestions, use integer values only(don't enter unneeded decimalpoints).

a) Suppose that the wage is$2 per hour and that capital also rents for$2 per hour. The slope of the isocost line is then equal to negative ___. Using resulting slope value along with the above formula for the isoquantslope, compute thefirm's optimal input ratio at anycost-minimizing point(recall that the isoquant and isocost slopes are equal at a tangencypoint). This ratio is ___ (enter a singleinteger). Now suppose that the firm wants to produce one widget(Q =1). Using your previous result on the optimal inputratio, you can compute the input levels that the firm uses to produce this output. These levels are K= ___ and L ___. The cost of this input bundle is $ ___, which givesC(1) (the cost functionc(Q) evaluated atQ=1).

b) Thecost-minimizing input bundle when Q= 2 has K= ___ and L = ___. The cost of this bundle is ___, which givesC(2).

c) Repeat this calculation for Q= 2,3,...,10, indicating your resultsbelow:

Q K L C(Q)

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c) Compute average cost(AC) at each level of output by dividingC(Q) byQ, indicating your resultsbelow:

Q AC

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d) The next step is to find thefirm's marginal cost function. MC is defined in the usualway: it's the cost of an extra unit of output given a particular starting level of Q. Forexample, marginal cost at Q= 1, orMC(1), equalsC(2) -C(1). MC(2) =C(3) -C(2), and so on. Using the results of part(b), compute marginal cost for all the different outputlevels, showing the resultsbelow:

Q MC

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e)) On a piece ofpaper, graph both the MC and AC functions. They are

A. horizontal lines

B. U-shaped curves

C. upward-sloping straight lines

D. convex, upward sloping curves