Chapter 7 3g5i49

CHAPTER 7: PRODUCTION ECONOMICS QUESTION 2 At your current production level, your marginal product of labour is 5, while your marginal product of capital is 7. The wage is $15 per unit, while the per unit cost of capital is 14. Are you operating efficiently? Why or why not?

ANSWER MARGINAL PRODUCT OF LABOUR = 5 WAGE: $15/unit MARGINAL PRODUCT OF CAPITAL = 7 COST OF CAPITAL = 14/unit

The current production level is not operating efficiently because labour ratio is less than capital ratio. Production level operating efficiently when labour ratio is equal to capital ratio.

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CHAPTER 7: PRODUCTION ECONOMICS QUESTION 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 (No. of workers) 2 3 4 5 6 7 8 9 10 11 12

amount of fish caught per week (hundreds of lbs) 3 6 11 19 24 28 31 33 34 34 33

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

crew size (No. of workers) 2 3 4 5 6 7 8 9 10 11 12

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amount of fish caught per week (hundreds of lbs) 3 6 11 19 24 28 31 33 34 34 33

Marginal Product of labour MP = ∆Q / ∆X 0 3 5 8 5 4 3 2 1 0 -1

Average Product of Labour AP = Q / X 1.50 2.00 2.75 3.80 4.00 4.00 3.88 3.67 3.40 3.09 2.75

CHAPTER 7: PRODUCTION ECONOMICS a i) Ranges of workers increasing The returns is increasing when the crew size increase to 5 crew member. It reaches to Maximum marginal product when they have 5 crew member. aii) Ranges of workers constant Nil aiii) Ranges of workers are decreasing returns The decreasing of returns is when the crew size is between 5 to 11 workers. aiv) Ranges of worker are negative returns At stage 3, when the no of crew is 12. b) How large a crew should be used if the trawler owner is interested in maximizing the total amount of fish caught? In order to maximize the total amount of fish caught, the trawler owner should have 11 crew member or we could say when MP = 0. c) How large a crew should be used if the trawler owner is interested in maximizing the average amount of fish caught per person? In order to maximize average amount of fish caught/person, the trawler owner should have 7 crew member where AP max when MP = AP QUESTION 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 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. crew size (No. of workers)

amount of fish caught per week

Marginal Product (MP )

Total Revenue ($)

Marginal Revenue (MR )

Marginal Revenue (MR )

(hundreds of lbs)

MP = ∆Q / ∆L

TR = P x Q

MR = ∆TR / ∆Q

MRP =MP . MR

Marginal Factor Cost (MFC )

2

3

3

225

75

225

150

3

6

3

450

75

225

150

4

11

5

825

75

375

150

5

19

8

1425

75

600

150

6

24

5

1800

75

375

150

7

28

4

2100

75

300

150

8

31

3

2325

75

225

150

9

33

2

2475

75

150

150

10

34

1

2550

75

75

150

11

34

0

2550

75

0

150

12

33

-1

2475

75

-75

150

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CHAPTER 7: PRODUCTION ECONOMICS In order the owner of the trawler is interested in maximizing profits, assume the labour market is perfectlhy competitive, the MF is equal to / $150 per worker. MR

= MF

(where MFC =

)

Therefore optimal input is L = 9 workers because MR

= MF

equal to $150 / worker.

QUESTION 5 Consider the following short-run production function (where L = variable input, Q = output) Q = 6L2 - 0.4L3 a) Determine Marginal Product Function, MPL MPL = ∂Q / ∂L MPL = 12L - 1.2L2 b) Determine the Average Product Function, APL APL = Q / L APL = 6L - 0.4L2 c) Find the value of L that maximize Q Q is maximum when slope equal to 0. MPL = ∂Q / ∂L = 0 12L - 1.2L2 = 0 12L = 1.2L2 12 = 1.2L L = 12 / 1.2 L = 10 d) Find the value of L at which marginal product function takes on its max value. MPL = 12L - 1.2L2 ∂MPL / ∂L= 12 - 2.4L = 0 12 = 2.4L L = 12 / 2.4 L=5 e) Find the value of L at which the average product function takes on its max value. APL = 6L - 0.4L2 ∂APL / ∂L = 6 - 0.8L = 0 6 = 0.8L L = 6 / 0.8 = 7.5

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CHAPTER 7: PRODUCTION ECONOMICS QUESTION 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. MPL = ∂Q / ∂L MPL = 10 - L MRP = MPL x PL MRP = (10 - L) x 10 MRP = 100 - 10L b) Determine the marginal factor cost function. Tc = PL x L Tc = 20L MFC = ∂Tc / ∂L MFC = 20 c) Determine the optimal value of L, given that the objective is to maximise profits. MRP = MFC 100 - 10L = 20 L = (100 - 20)/10 L=8

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CHAPTER 7: PRODUCTION ECONOMICS

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CHAPTER 7: PRODUCTION ECONOMICS CASE EXERCISE 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 following table).

Plant

Output, Q (000 tons)

Capital, K ($000)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

605.3 566.1 647.1 523.7 712.3 487.5 761.6 442.5 821.1 397.8 896.7 359.3 979.1 331.7 1064.9

18891 19201 20655 15082 20300 16079 24194 11504 25970 10127 25622 12477 24002 8042 23972

Labor, L (000 worker hrs) 700.2 651.8 822.9 650.3 859 613 851.3 655.4 900.6 550.4 842.2 540.5 949.4 575.7 925.8

Ln Q

Ln K

Ln L

6.405724203 6.338770741 6.472500842 6.260919001 6.568499171 6.18929029 6.635421483 6.092440464 6.710644905 5.985949366 6.798721358 5.884157694 6.886633782 5.804230946 6.970636177

9.846440897 9.86271764 9.9357127 9.621257259 9.918376165 9.685269352 10.09385995 9.35045008 10.1646973 9.222960403 10.15120664 9.431642228 10.08589244 8.992433087 10.08464176

6.551366009 6.479737766 6.712834687 6.477433795 6.755768922 6.418364936 6.746764593 6.485245736 6.803061208 6.310645287 6.736017516 6.292494637 6.855830206 6.355586692 6.830658229

1) Estimate the Cobb-Doughlas production function Q = αLβ1Kβ2 where Q = output, L = labor input, K = capital input; and α, β1 and β2 are parameters to be estimated SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations

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0.973748198 0.948185553 0.939549812 0.089655141 15

CHAPTER 7: PRODUCTION ECONOMICS ANOVA df Regression Residual Total

Intercept Ln K Ln L

SS 2 1.765119513 12 0.096456533 14 1.861576046

Standard Coefficients Error -4.754736609 0.805779683 0.415207154 0.134516805 1.07800389 0.249334664

MS F 0.882559757 109.7978206 0.008038044

Significance F 1.93511E-08

t Stat P-value -5.90078989 7.24392E-05 3.086656374 0.009420678 4.323521936 0.000989946

Lower 95% -6.510379718 0.122120214 0.534750325

Upper 95% -2.999093499 0.708294095 1.621257454

Ln Q = α + β1K + β2L Ln Q = -4.755 + 0.415K + 1.078L Q = eα Kβ1 Lβ2 Q = e-4.755 K0.415 L1.078 α = -4.754 β1 = 0.415 β2 = 1.078 2) Test whether the coefficient of capital and labor are statistically significant. Capital P-value = 0.009; t = 4.324 which means Capital is significant at 1% in identifying Output. There is positive relationship between labour input and output Labor P-value = 0.0009; t = 3.087 which means Labor is significant at 1% in identifying There is positive relationship between capital input and output Adjusted R square = 0.94, which means 94% of output is explained by Capital and Labour 3) Determine the % of variation in output that is explained by the regression equation. When labour input increase 1%, output will be increased by 1.078% When capial input increase 1%, output will be increased by 0.414% 4) Determine the labor and capital estimated parameters, and give economic interpretation of each value. Labour parameter = 1.078; which means 1% increase in labour input, result in 1.078% increase in total production output. Capital parameter = 0.415; which means 1% increase in lcapital input, result in 0.415% increase in total production output.

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CHAPTER 7: PRODUCTION ECONOMICS 5) Determine whether this production function exhibits increasing, decreasing or constant returns to scale. (Ignore the issue of statistical significance) 0.415 + 1.078 = 1.493 1.493 is > 1 = increasing return to scale.

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