3. The amount of fish caught per week on a trawler is a function of the crew size assigned to operate the boat. Based on past data, the following production schedule was developed:

CREW SIZE (NUMBER OF WORKERS) 2 AMOUNT OF FISH CAUGHT PER WEEK (HUNDREDS OF LBS) 3

3 6

4 11

5 19

6 24

7 28

31

9 33

10 1 34

11 34

12 33

a. Over what ranges of workers are there (i) increasing, (ii) constant, (iii) decreasing, and (iv) negative returns?

b. How large a crew should be used if the trawler owner is interested in maximizing the total amount of fish caught?

c. How large a crew should be used if the trawler owner is interested in maximizing the average amount of fish caught per worker?

4. Consider Exercise 3 again. Suppose the owner of the trawler can sell all the fish caught for $75 per 100 pounds and can hire as many crew members as desired by paying them $150 per week. Assuming that the owner of the trawler is interested in maximizing profits, determine the optimal crew size.

5. Consider the following short-run production function (where L = variable input, Q = output):

Q = 6L2 0.4L3

a. Determine the marginal product function (MPr,).

b. Determine the average product function (APj).

c. Find the value of L that maximizes Q .

d. Find the value of L at which the marginal product function takes on its maximum value.

e. Find the value of L at which the average product function takes on its maximum value.

6. Consider the following short-run production function (where L = variable input, Q = output):

Q = 10L 0.5L2

Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input (L) as it needs at $20 per unit.

a. Determine the marginal revenue product function.

b. Determine the marginal factor cost function.

c. Determine the optimal value of L, given that the objective is to maximize profits.

7. Suppose that a firm’s production function is given by the following relationship:

Q= 2.5/Lk (i.e., Q = 2.5L5K5)

where Q = output

L = labor

K = capital input

a. Determine the percentage increase in output if labor input is increased by

10 percent (assuming that capital input is held constant).

b. Determine the percentage increase in output if capital input is increased by

25 percent (assuming that labor input is held constant).

c. Determine the percentage increase in output if both labor and capital are increased by 20 percent.

8. Based on the production function parameter estimates reported in Table 7.4:

a. Which industry (or industries) appears to exhibit decreasing returns to scale? (Ignore the issue of statistical significance.)

b. Which industry comes closest to exhibiting constant returns to scale?

c. In which industry will a given percentage increase in capital result in the largest percentage increase in output?

d. In which industry will a given percentage increase in production workers result in the largest percentage increase in output?

9. Consider the following Cobb-Douglas production function for the bus transportation system in a particular city:

Q = alP'F^K^

where L = labor input in worker hours

F = fuel input in gallons

K = capital input in number of buses

Q = output measured in millions of bus miles

Suppose that the parameters (a, p2 and (33) of this model were estimated using

annual data for the past 25 years. The following results were obtained:

a = 0.0012 p, = 0.45 = 0.20 = 0.30

a. Determine the (i) labor, (ii) fuel, and (iii) capital input production elasticities.

b. Suppose that labor input (worker hours) is increased by 2 percent next year (with the other inputs held constant). Determine the approximate percentage change in output.

c. Suppose that capital input (number of buses) is decreased by 3 percent next year (when certain older buses are taken out of service). Assuming that the other inputs are held constant, determine the approximate percentage change in output.

d. What type of returns to scale appears to characterize this bus transportation system? (Ignore the issue of statistical significance.)

e. Discuss some of the methodological and measurement problems one might encounter in using time-series data to estimate the parameters of this model.

10. Extension of the Cobb-Douglas Production Function—The Cobb-Douglas production function (Equation 7.16) can be shown to be a special case of a larger class of linear 1 o homogeneous production functions having the following mathematical form:

Q = y[dK p + (1 -9)L Ppv/P

where y is an efficiency parameter that shows the output resulting from given quantities of inputs; 3 is a distribution parameter (0 3 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and v is a scale parameter (v 0) that indicates the type of returns to scale (increasing, constant, or decreasing). Show that when v — 1, this function exhibits constant returns to scale. (Hint: Increase capital K and labor I, each by a factor of X, or K* — (X)K and L* — (X)L, and show that output Q also increases by a factor of X, or Q* = (X)(Q).)

11. Lobo Lighting Corporation currently employs 100 unskilled laborers, 80 factory technicians, 30 skilled machinists, and 40 skilled electricians. Lobo feels that the marginal product of the last unskilled laborer is 400 lights per week, the marginal product of the last factory technician is 450 lights per week, the marginal product of the last skilled machinist is 550 lights per week, and the marginal product of the last skilled electrician is 600 lights per week. Unskilled laborers earn $400 per week, factory technicians earn $500 per week, machinists earn $700 per week, and electricians earn $750 per week.

Is Lobo using the lowest cost combination of workers to produce its targeted output? If not, what recommendations can you make to assist the company?

Case Exercise

The Production Function for Wilson Company

Economists at the Wilson Company are interested in developing a production function for fertilizer plants. They collected data on 15 different plants that produce fertilizer (see the following table).

Questions

1. Estimate the Cobb-Douglas production function Q — alF'K^, where Q = output; L — labor input; K — capital input; and a, and (32 are the parameters to be estimated.

2. Test whether the coefficients of capital and labor are statistically significant.

3. Determine the percentage of the variation in output that is explained by the regression equation.

4. Determine the labor and capital estimated parameters, and give an economic interpretation of each value.

5. Determine whether this production function exhibits increasing, decreasing, or constant returns to scale. (Ignore the issue of statistical significance.)

CREW SIZE (NUMBER OF WORKERS) 2 AMOUNT OF FISH CAUGHT PER WEEK (HUNDREDS OF LBS) 3

3 6

4 11

5 19

6 24

7 28

31

9 33

10 1 34

11 34

12 33

a. Over what ranges of workers are there (i) increasing, (ii) constant, (iii) decreasing, and (iv) negative returns?

b. How large a crew should be used if the trawler owner is interested in maximizing the total amount of fish caught?

c. How large a crew should be used if the trawler owner is interested in maximizing the average amount of fish caught per worker?

4. Consider Exercise 3 again. Suppose the owner of the trawler can sell all the fish caught for $75 per 100 pounds and can hire as many crew members as desired by paying them $150 per week. Assuming that the owner of the trawler is interested in maximizing profits, determine the optimal crew size.

5. Consider the following short-run production function (where L = variable input, Q = output):

Q = 6L2 0.4L3

a. Determine the marginal product function (MPr,).

b. Determine the average product function (APj).

c. Find the value of L that maximizes Q .

d. Find the value of L at which the marginal product function takes on its maximum value.

e. Find the value of L at which the average product function takes on its maximum value.

6. Consider the following short-run production function (where L = variable input, Q = output):

Q = 10L 0.5L2

Suppose that output can be sold for $10 per unit. Also assume that the firm can obtain as much of the variable input (L) as it needs at $20 per unit.

a. Determine the marginal revenue product function.

b. Determine the marginal factor cost function.

c. Determine the optimal value of L, given that the objective is to maximize profits.

7. Suppose that a firm’s production function is given by the following relationship:

Q= 2.5/Lk (i.e., Q = 2.5L5K5)

where Q = output

L = labor

K = capital input

a. Determine the percentage increase in output if labor input is increased by

10 percent (assuming that capital input is held constant).

b. Determine the percentage increase in output if capital input is increased by

25 percent (assuming that labor input is held constant).

c. Determine the percentage increase in output if both labor and capital are increased by 20 percent.

8. Based on the production function parameter estimates reported in Table 7.4:

a. Which industry (or industries) appears to exhibit decreasing returns to scale? (Ignore the issue of statistical significance.)

b. Which industry comes closest to exhibiting constant returns to scale?

c. In which industry will a given percentage increase in capital result in the largest percentage increase in output?

d. In which industry will a given percentage increase in production workers result in the largest percentage increase in output?

9. Consider the following Cobb-Douglas production function for the bus transportation system in a particular city:

Q = alP'F^K^

where L = labor input in worker hours

F = fuel input in gallons

K = capital input in number of buses

Q = output measured in millions of bus miles

Suppose that the parameters (a, p2 and (33) of this model were estimated using

annual data for the past 25 years. The following results were obtained:

a = 0.0012 p, = 0.45 = 0.20 = 0.30

a. Determine the (i) labor, (ii) fuel, and (iii) capital input production elasticities.

b. Suppose that labor input (worker hours) is increased by 2 percent next year (with the other inputs held constant). Determine the approximate percentage change in output.

c. Suppose that capital input (number of buses) is decreased by 3 percent next year (when certain older buses are taken out of service). Assuming that the other inputs are held constant, determine the approximate percentage change in output.

d. What type of returns to scale appears to characterize this bus transportation system? (Ignore the issue of statistical significance.)

e. Discuss some of the methodological and measurement problems one might encounter in using time-series data to estimate the parameters of this model.

10. Extension of the Cobb-Douglas Production Function—The Cobb-Douglas production function (Equation 7.16) can be shown to be a special case of a larger class of linear 1 o homogeneous production functions having the following mathematical form:

Q = y[dK p + (1 -9)L Ppv/P

where y is an efficiency parameter that shows the output resulting from given quantities of inputs; 3 is a distribution parameter (0 3 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and v is a scale parameter (v 0) that indicates the type of returns to scale (increasing, constant, or decreasing). Show that when v — 1, this function exhibits constant returns to scale. (Hint: Increase capital K and labor I, each by a factor of X, or K* — (X)K and L* — (X)L, and show that output Q also increases by a factor of X, or Q* = (X)(Q).)

11. Lobo Lighting Corporation currently employs 100 unskilled laborers, 80 factory technicians, 30 skilled machinists, and 40 skilled electricians. Lobo feels that the marginal product of the last unskilled laborer is 400 lights per week, the marginal product of the last factory technician is 450 lights per week, the marginal product of the last skilled machinist is 550 lights per week, and the marginal product of the last skilled electrician is 600 lights per week. Unskilled laborers earn $400 per week, factory technicians earn $500 per week, machinists earn $700 per week, and electricians earn $750 per week.

Is Lobo using the lowest cost combination of workers to produce its targeted output? If not, what recommendations can you make to assist the company?

Case Exercise

The Production Function for Wilson Company

Economists at the Wilson Company are interested in developing a production function for fertilizer plants. They collected data on 15 different plants that produce fertilizer (see the following table).

Questions

1. Estimate the Cobb-Douglas production function Q — alF'K^, where Q = output; L — labor input; K — capital input; and a, and (32 are the parameters to be estimated.

2. Test whether the coefficients of capital and labor are statistically significant.

3. Determine the percentage of the variation in output that is explained by the regression equation.

4. Determine the labor and capital estimated parameters, and give an economic interpretation of each value.

5. Determine whether this production function exhibits increasing, decreasing, or constant returns to scale. (Ignore the issue of statistical significance.)

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