ECON 410

Professor Tauchen

Spring 2008

Practice Problems on Production Functions -- Answers

1. A firm's production function is Q=F(L,K)= L2K.

a. Compute the marginal product of labor and the marginal product of capital. Does this production function satisfy diminishing marginal returns?

Ans. The marginal product of labor is 2LK, and the marginal product of capital is L2. This production function does not satisfy diminishing marginal returns. The marginal product of labor is an increasing function of L. The marginal product of capital does not vary with K. If the production function satisfies diminishing marginal returns, then the MP of labor should decline with L and the MP of capital should decline with K.

b. Determine an expression for the marginal rate of technical substitution between labor and capital. Compute the marginal rate of technical substitution between capital and labor at the following capital and labor combinations: (10 labor , 1 capital); (5 labor , 4 capital);

(2 labor , 25 capital); (1 labor , 100 capital). Does the production function satisfy diminishing marginal rate of technical substitution? Explain.

Ans. The MRTS between labor and capital is the marginal product of labor divided by the marginal product of capital, which for this production function is 2LK/ L2 =2K/L The MRTS between labor and capital for the input allocations listed are 2/10, 8/5, 50/2, 200/1.

Notice that each of these input bundles is on the isoquant for 100 units of output. As the amount of labor falls, the MRTS between labor and capital becomes larger and larger. This is exactly the required condition for diminishing MRTS between capital and labor.

c. Find the isoquant function that gives the amount of capital required to produce Q units of output with L units of labor. [Hint: Solve the production function Q=L2K for K.] Graph at least four points on the isoquant function for Q=100 and four points on the isoquant function for Q=500.

Ans. The isoquant function is K=Q/ L2. Input combinations on the isoquant for 100 include (2,25), (4,6.25), (5,4), and (10,1), Input combinations on the isoquant for 500 include (2,125), (4,31.25), (5,20), and (10,5).

2. A firm’s production function is Q=F(L,K) = L.3 K.5 . Find expressions for the marginal product of labor, the marginal product of capital, and the marginal rate of technical substitution between capital and labor.

Ans. The Marginal Product of Labor is .3 L-.7 K.5 . The Marginal Product of Labor is a declining function of the amount of labor. This production function satisfies decreasing marginal product.

The marginal product of capital is .5 L.3 K-.5. The production function also satisfies the assumption of decreasing Marginal Product of Capital.

The MRTS between labor and capital is the ratio of the Marginal Product of Labor and the Marginal Product of Capital.. (And, now we can apply some of the work with exponents from the very beginning of the semester.) This ratio can be simplified as .6 K/L.

3. a. A firm uses two types of labor inputs – those with high school diplomas and those with GED certificates. The firm’s output is Q=F(h,g)=h+g where h denotes the number of employees with high school degrees and g is the number of employees with a GED certificate. Identify the quantity associated with each of the isoquants on the graph to the right. Does this production function exhibit constant, increasing, or decreasing returns to scale?

Ans. The isoquant closer to the origin is for output level 2. The next is for output level 4 and the third for output level 6.

When the input bundle doubles from (1,1) to (2,2), output increases from 2 to 4. Since output exactly doubles when all inputs double, the technology exhibits constant returns to scale.

b. What does the shape of the above isoquants imply about the substitutability between high school graduate labor and GED labor?

Ans. The inputs are perfect substitutes. One high school graduate is equivalent to a GED graduate in production.

c. Answer the questions in part a for the production function Q=F(h,g)=(h+g)2 . The isoquant map is shown below.
Ans. The isoquant closest to the origin is for 4 units of output, the next for 16, and the third for 36. As the inputs double, for example from (1,1) to (2,2), the output more than doubles. The production function exhibits increasing returns. / d. Answer the same questions as in a for the production function Q=F(h,g)=(h+g).5 .
Ans. The isoquant closest to the origin is for 2.5 =1.414 units of output, the next for 2, and the third for 6.5. As the inputs double, for example from (1,1) to (2,2), the output increases but does not double. The production function exhibits decreasing returns.

4. Now let’s consider the isoquant map that shows the # of UNC MBA graduates employed per day on the horizontal axis an the number of Duke MBA graduates employed per day on the vertical axis. As you might expect, the UNC graduates are better trained. Indeed, each UNC graduate accomplishes exactly twice as much per day as a Duke graduate. Construct a graph on which you show the number of UNC graduates employed per day on the horizontal axis and the number of Duke graduates employed per day on the vertical axis. Describe the shape of the indifference curves.

Ans. The isoquants are straight lines but have a slope of –2 rather than –1. The input combination (1,0) yields the same output as the combination (0,2).

5. Now let’s consider a Leontief type isoquant map. Each of the indifference curves is L-shaped with the corners of the L at the points (1,1) , (2,2), ..... with labor measured on the horizontal axis and capital on the vertical axis. What does this isoquant map imply about the substitutability between capital and labor?

Ans. Thereis no substitutability between capital and labor. A firm that does not want to pay unnecessarily for labor and capital operates at the corner of the L-shaped isoquants. Beginning from such an input combination, the firm cannot substitute labor for capital (or substitute capital for labor) and keep output constant. If the amount of labor is increased, the firm cannot reduce the amount of capital and keep output constant. Likewise, there is no amount of capital that compensates for the loss of a unit of labor (in the sense that output remains constant).

6. Answer the following questions for the short-run production function shown to the right.

a. For what labor quantities is marginal product positive? negative?

Ans: MP is positive if TP is increasing and negative if TP is decreasing. On the graph below, TP is increasing up to 12 units and decreasing beyond 12 units. Thus MP is positive for labor quantities below 12 and negative for labor quantities above 12.

b. For what labor quantities is the MP curve upward sloping? Downward sloping?

Ans. For small amounts of labor the total product is increasing at an increasing rate. For example, output increases very little when the amount of labor increases from 0 to 1 unit. Output increases by far more when the labor quantity increases from three to four units.

For higher input quantities, output increases at a decreasing rate. The increase in output is greater for an increase in labor from 7 to 8 units than for 10 to 11 units.

The switch from increasing at an increasing rate to increasing at a decreasing rate occurs at about five units of labor. That point is marked with a dot on the graph below.

c. For which labor quantities is the AP curve upward sloping? Downward sloping?

Ans. To determine the AP at L units of labor construct the straight line from (0,0) to the point on the TP curve for L units of labor. The slope of the line is the AP of labor. Several of these lines are shown on the graph below. The steepest of the lines is for 8 units of labor. Average product is increasing up to eight units of labor and decreasing for larger labor quantities.

d. At what labor quantity are MP and AP identical?

Ans. The maximum of AP is at eight units of labor. For any marginal and average relationship, the average and marginal are the same at the maximum of the average. Thus AP and MP are the same at eight units of labor.

7. A firm’s production function is f(L,K) = LaKb where the parameters a and b are both positive.

a. Provide an expression for f(λL,λK). Also provide an expression for λf(L,K).

b. For what values of a and b does f(λL,λK) necessarily equal λf(L,K) for any λ>1? Explain.

Answers: a. Mathematicians would describe the function f as having the property that its first argument (regardless whether the first argument is simple or complicted) is raised to the power a and then multiplied times the second argument raised to the b. The first argument of f(λL,λK) is λL so in evaluating f(λL,λK), the entire first argument is raised to the power a. Thus,

f(λL,λK) = (λL)a (λK)b.

Using the properties of exponents, the above expression is

λa+b La Kb.

So, f(λL,λK) = λa+b La Kb.

The second part of the question asks us to evaluate λf(L,K). Given the function f,

λf(L,K) = λ La Kb.

b. f(λL,λK) equals λf(L,K) if λa+b equals λ which holds if λ=1. Thus the production function has CRS if a+b =1.

8. We have 50 units of capital and 100 units of labor that can be used in the production of widgets. We can either use all 50 units of capital in one plant or we can divide up the inputs among five different plant locations. In each location, the production of widgets is f(L,K) where (L,K) denotes the input use at the location.

a. Suppose that the production of widgets exhibits constant returns to scale. Is this production of widgets in one plant using 50 units of capital and 100 units of labor greater than, equal to, or less than the total production from five plants each using 10 units of capital and 20 units of labor? Explain.

Answer: The output from 50 units of capital and 100 units of labor in one plant is f(100,50). The output from one plant with 10 units of capital and 20 units of labor is f(20,10). The output from 5 plants each using 10 units of capital and 20 units of labor is 5f(20,10).

Let’s rewrite f(100,50) as f( 5 20, 5 10). If the production function has CRS,

f(λ ∙ 20, λ ∙ 10)= λf(20,10) for all values of λ>1.

Since the above relationship holds for all λ>1, the relationship certainly holds for λ=5 or

f( 5 ∙ 20, 5 ∙ 10)= 5f(20,10).

The output is the same from one plant using all of the inputs as from 5 plants each using 10 units of capital and 20 of labor.

b. Answer the question above assuming that the production of widgets exhibits increasing returns to scale.

Similarly, if the production function exhibits IRS,

f( 5 ∙ 20, 5 ∙ 10) > 5f(20,10).

The output is larger from one plant using all of the inputs than from 5 plants each using 10 units of capital and 20 of labor. With an IRS production function, there is an advantage to having one firm produce the output rather than a number of smaller plants.

c. Answer the question above assuming that the production of widgets exhibits decreasing returns to scale.

If the production function exhibits DRS,

f( 5 ∙ 20, 5 ∙ 10) < 5f(20,10).

The output is smaller from one plant using all of the inputs than from 5 plants each using 10 units of capital and 20 of labor. With a DRS production function, there is an advantage to having many firms produce use the input rather than having one firm use all of the inputs.