ECON 410Professor TauchenSpring 2008Practice Problems on Production Functions -- Answers1. 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 functionthat gives the amount of capitalrequired to produce Q units ofoutput with L units of labor.[Hint: Solve the productionfunction Q=L2K for K.] Graphat least four points on theisoquant function for Q=100and four points on the isoquantfunction for Q=500. 0204060801001201401601802000 1 2 3 4 5 6 7 8 9 10 11 12Workers/DayUnits Capital/DayAns. 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 productof 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 isQ=F(h,g)=h+g where h denotes thenumber of employees with high schooldegrees and g is the number of employeeswith a GED certificate. Identify thequantity associated with each of theisoquants on the graph to the right. Doesthis production function exhibit constant,increasing, or decreasing returns to scale? Ans. The isoquant closer to the origin isfor output level 2. The next is for outputlevel 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. 012345670 1 2 3 4 5 6 7# High School Graduate Employees/day# GED Employees/dayc. 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 axisand capital on the vertical axis. What does this isoquant map imply about the substitutability between capital and labor?Ans. There is 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