Question

Problem #3: Long-Run Labor Demand and Factor Substitutability Suppose there are two inputs in the production function, labor (L) and capital (K), which carn be combined to produce Y units of out put according to the following production function: Y-30K+10L The firm wants to produce 600 units of out put 1. Draw the isoquant that corresponds to that level of production (600 units) in a graph that has L on the horizontal axis and K on the vertical axis 2. The shape of the isoquant tells us about the relationship between the two inputs in production. How substitutable are L and K in the production of Y? In particular, how many units of L can be replaced by one unit of K wit hout affecting the level of output? 3. Is this isoquant conver (bowed toward the origin)? In class, we said that isoquants are convex under our "standard assumptions. To see which standard assumption violated in this case, hold K fixed at sotne level (for con- venience, suppose K is fixed at zero. Graph Y as a function of L for L-0.5 is the change in Y (ΔΥ) when L increases by l tmit (A.-1)? different from the "standard assumption" about the MPL we made in class? . By look ing at your graph, determine the marginal product of labor (MPL). That is, what 6. How does the marginal product of labor (MPL) change as L increases? How is this 7. Suppose the firm can choose whatever combination of capital (K) and labor (L) it wants to produce 600 units. Suppose the price of capital is $1,000 per machine per week. What combination of inputs (K and L) will the firm use if the weekly salary of each worker is 400?7 8. What if everything is same as in the previous question but the weekly salary of each worker is $300? Now what combination of inputs (K and L) will the firm use to produce its 600 units? 9. (Bonus) What is the (wage) elasticity of labor demand for this firm as the wage falls from $400 to $300?

Answer 1

Production function: Y = 30K + 10L

(1)

When Y = 600, we get

600 = 30K + 10L

60 = 3K + L (Dividing by 10)

When L = 0, K = 60/3 = 20 (Vertical intercept) and when K = 0, L = 60 (Horizontal intercept).

In following graph, Q0 is the isoquant. When production function is linear, isoquants are straight lines touching both axes.

(2)

Since production function are linear, K and L are perfect substitutes. The production process requires 10N units of labor or 30N units of capital for an output level of Q. So,

30 units of K can be substituted for 10 units of L.

1 unit of K can be substituted for (10/30) = (1/3) unit of L. This is the substitution ratio between K and L.

(3)

Since the isoquant is a straight line, it is not convex. A convex isoquant requires a non-constant Marginal rate of technical substitution (MRTS) but for a linear isoquant, MRTS is constant (MRTS = slope of isoquant).

(4)

When K = 0, substituting in production function we get

Y = 10L

When L = 0, Y = 10 x 0 = 0

When L = 1, Y = 10 x 1 = 10

When L = 2, Y = 10 x 2 = 20

When L = 3, Y = 10 x 3 = 30

When L = 4, Y = 10 x 4 = 40

When L = 5, Y = 10 x 5 = 50

The following graph plots the value of Y for values of L.

NOTE: As per Answering Policy, 1st 4 parts are answered.