Question

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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 output according to the following production function: Y = 30K + 10L The firm wants to produce 600 units of output. 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 without affecting the level of output? 3. Is this isoquant conver (bowed toward the origin)? 4. In class, we said that isoquants are convex under our "standard assumptions." To see which standard assumption is violated in this case, hold K fixed at some level (for con- venience, suppose K is fixed at zero). Graph Y as a function of L for L 0, , 5. . 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 By looking at your graph, determine the marginal product of labor (MPL) is the change in Y (AY) when L increases by 1 unit (AL 1)? 5. different from the "standard assumption" about the MPt we made in class? 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?

Answer 1

Y = 30K + 10L

(1)

When Y = 600,

600 = 30K + 10L,or

60 = 3K + L

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

In following graph, Q0 is above isoquant. For a linear production function, isoquants are straight lines touching both axes.

(2)

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

30 units of K can replace 10 units of L.

1 unit of K can replace (10/30) = (1/3) unit of L. This is the substitution ration between K and L.

(3)

Since the isoquant is a straight line, it is not convex. A convex isoquant requires a non-constant MRTS but for a linear isoquant, MRTS is constant where MRTS is the slope of isoquant.

(4)

When K = 0,

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 isoquant Q0 which is a straight line from origin.

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