Question

2. Suppose that the price of labor is $20 (w = $20) and the price of capital is $40 (r = $40). A firm is able to pay the total cost of $200 to employ labor (L) and capital (K). The firm's production function is given as Q = LK, where Q is units of output, L is units of labor, and K is units of capital. (a) Construct the firm's isocost function and graph it on a two-dimensional plane. (Note: Place L on the horizontal axis and K on the vertical axis.) (b) Obtain the relative price ratio, w/r, and determine how much of K must be given up when one more unit of Lis hired, holding the total cost constant. (c) Obtain six integer combinations of Land K when Q = 12, and indicate them on the graph obtained in part (a). Which input combination(s) can the firm employ?

Answer 1

(a)

Isocost: C = wL + rK

200 = 20L + 40K

10 = L + 2K

When L = 0, K = 10/2 = 5 (vertical intercept) & when K = 0, L = 10 (horizontal intercept).

In following graph, XY is isocost line.

| 23 16 10

(b)

In least cost combination, MPL/MPK = w/r = $20/$40 = 1/2 = 0.5

MPL = $\dpi{80} \partial$ Q/$\dpi{80} \partial$L = K

MPK = $\dpi{80} \partial$ Q/$\dpi{80} \partial$K = L

MPL/MPK = K/L = 1/2

L = 2K

It means that 0.5 units of K must be given up to use one additional unit of L.

(c)

Q = LK = 12

6 bundles of (L, K) are: A(1, 12), B(2, 6), C(3, 4), D(4, 3), E(6, 2) and F(12, 1).

Cost of bundle A = 20 x 1 + 40 x 12 = 20 + 480 = 500 (exceeds budget and cannot be employed)

Cost of bundle B = 20 x 2 + 40 x 6 = 40 + 240 = 280 (exceeds budget and cannot be employed)

Cost of bundle C = 20 x 3 + 40 x 4 = 60 + 160 = 220 (exceeds budget and cannot be employed)

Cost of bundle D = 20 x 4 + 40 x 3 = 80 + 120 = 200 (equals budget and can be employed)

Cost of bundle E = 20 x 6 + 40 x 2 = 120 + 80 = 200 (equals budget and can be employed)

Cost of bundle F = 20 x 12 + 40 x 1 = 240 + 40 = 280 (exceeds budget and cannot be employed)