# 18. Supposex, and x2 are used in fixed proportions such that y = mink,,2×2). At the point x1 = x2…

18. Supposex, and x2 are used in fixed proportions such that y = mink,,2×2). At the point x1 = x2 = 4, the MPs A. 1; 2 and the MP2 is C. 0; 1 D. 2; 0 E. 0; 2 19. A competitive firm has a production function Y= 2L + 10K. If w = \$2 and r= \$8, what will be the minimum cost of producing 50 units of output? A. \$40*** B. \$50 C. \$45 D. [None of the above]

## Solutions

##### Expert Solution

Solution:

18. At the kink, x1 = 2*x2, the point given is x1 = x2 = 4

The given point gives utility = min{4, 2*4} = min{4, 8} = 4

Notice, that with x1 = 4, increasing x2 any beyond 2 (as at kink
4 = 2*x2 , so x2 = 2) will not increase the utility. So, marginal
utility from good 2 is 0 (if you increase x2 by 1 more unit, so x2
= 5, utility is still 4). Notice, however increasing good 1 by 1
unit, will generate utility = 5, so marginal change in utility due
to an additional consumption of good 1 is 1. Thus, correct option
is (B) 1; 0

19. With production function of Y = 2L + 10K, L and K act as
perfect substitutes in production of Y. Note that marginal product
of labor, MPL = 2 and marginal product of capital, MPK = 10, so
marginal rate of technical substitution = MPL/MPK = 2/10 = 0.2

Now, wage to rental rate ratio = w/r = 2/8 = 0.25

Clearly, wage to rental rate ratio = 0.25 > 0.20 = MRTS,
implying that relative marginal cost of hiring labor is more than
relative marginal benefit of labor (or in other words, relative
marginal cost of hiring capital is lower than relative marginal
benefit of hiring capital). So it would be beneficial to hire only
the capital (since capital and labor are perfect substitutes in
production).

Thus, for Y = 50, using the production function, we have 50 =
10*K, K = 5

Total cost = w*L + r*K

TC = 2*0 + 8*5 = \$40

So, correct option is (A) \$40.