CHAPTER 6
PRODUCTION
THEORY AND ESTIMATION
1. The table below presents estimates
of the maximum levels of output possible with
various combinations of two inputs.
Capital (K)
5 11
25 37 47 51
4 10
23 33 41 44
3 8 18 25 30 34
2 5 11 16 20 22
1 1 4 8 10 11
1 2 3 4 5
Labor (L)
Assume that a unit of output sells for
$2 and that the firm currently employs 2 units of capital (K = 2).
(i) What is the marginal product of
labor when L
= 4?
(ii) What is the average product of
labor when L
= 4?
(iii) What is the marginal revenue
product of labor when L = 4? What is the output elasticity of labor when L = 4?
(iv) If the wage rate of labor is $10,
how many units of labor should the firm hire and how many units of output
should it produce?
Solution:
(i) The marginal product of labor when L = 4 is (20 − 16)/(4 − 3) = 4.
(ii) The average product of labor when L = 4 is 20/4 = 5.
(iii) The marginal revenue product of
labor when L
= 4 is (4)($2) = $8. The output elasticity
of labor when L = 4
is 4/5 = 0.80.
(iv) The marginal revenue product of
labor is equal to (5)($2) = $10 when the firm employs L = 3 units of labor. The level of output
when L
= 3 is 16.
2. The table below presents estimates
of the maximum levels of output possible with various combinations of two
inputs.
Capital (K)
5 11
25 37 47 51
4 10
23 33 41 44
3 8 18 25 30 34
2 5 11 16 20 22
1 1 4 8 10 11
1 2 3 4 5
Labor (L)
Assume that a unit of output sells for
$3 and that the firm currently employs 3 units of capital (K = 3).
(i) What is the marginal product of
labor when L
= 4?
(ii) What is the average product of
labor when L
= 4?
(iii) What is the marginal revenue
product of labor when L = 4? What is the output elasticity of labor when L = 4?
(iv) If the wage rate of labor is $12,
how many units of labor should the firm hire and how many units of output
should it produce?
Solution:
(i) The marginal product of labor when L = 4 is (30 − 25)/(4 − 3) = 5.
(ii) The average product of labor when L = 4 is 30/4 = 7.5.
(iii) The marginal revenue product of
labor when L
= 4 is (5)($3) = $15. The output elasticity
of labor when L = 4 is 5/7.5 = 0.67.
(iv) The marginal revenue product of
labor is equal to (4)($3) = $12 when the firm employs L = 5 units of labor. The level of output
when L
= 5 is 34.
3. The table below presents estimates
of the maximum levels of output possible with various combinations of two
inputs.
Capital (K)
5 11
25 37 47 51
4 10
23 33 41 44
3 8 18 25 30 34
2 5 11 16 20 22
1 1 4 8 10 11
1 2 3 4 5
Labor (L)
Assume that a unit of output sells for
$5 and that the firm currently employs 1 unit of capital (K = 1).
(i) What is the marginal product of
labor when L
= 2?
(ii) What is the average product of
labor when L
= 2?
(iii) What is the marginal revenue
product of labor when L = 2? What is the output elasticity of labor when L = 2?
(iv) If the wage rate of labor is $10,
how many units of labor should the firm hire and how many units of output
should it produce?
Solution:
(i) The marginal product of labor when L = 2 is (4 − 1)/(2 − 1) = 3.
(ii) The average product of labor when L = 2 is 4/2 = 2.
(iii) The marginal revenue product of
labor when L
= 2 is (3)($5) = $15. The output elasticity
of labor when L = 2
is 3/2 = 1.50.
(iv) The marginal revenue product of
labor is equal to (2)($5) = $10 when the firm employs L = 4 units of labor. The level of output
when L
= 4 is 10.
4. The table below presents estimates
of the maximum levels of output possible with various combinations of two
inputs.
Capital (K)
5 11
25 37 47 51
4 10
23 33 41 44
3 8 18 25 30 34
2 5 11 16 20 22
1 1 4 8 10 11
1 2 3 4 5
Labor (L)
Assume that a unit of output sells for
$10 and that the firm currently employs 4 units of capital (K = 4).
(i) What is the marginal product of
labor when L
= 5?
(ii) What is the average product of
labor when L
= 5?
(iii) What is the marginal revenue
product of labor when L = 5? What is the output elasticity of labor when L = 5?
(iv) If the wage rate of labor is $80,
how many units of labor should the firm hire and how many units of output
should it produce?
Solution:
(i) The marginal product of labor when L = 5 is (44 − 41)/(5 − 1) = 3.
(ii) The average product of labor when L = 5 is 44/5 = 8.8.
(iii) The marginal revenue product of
labor when L
= 5 is (3)($10) = $30. The output elasticity
of labor when L = 5
is 3/8.8 = 0.34.
(iv) The marginal revenue product of
labor is equal to (8)($10) = $80 when the firm employs L = 4 units of labor. The level of output
when L
= 4 is 41.
5. A firm currently employs 40
production workers and 5 supervisors. The marginal product of the last
production worker employed is 36 units of output per hour and production workers
are paid $8 per hour. The marginal product of the last supervisor employed is 120
units of output per hour and supervisors are paid $20 per hour. Every employee works
40 hours per week.
(i) What is the firm's total labor cost
per week?
(ii) Assume that hours of labor by
supervisors (Ls) is plotted on the vertical axis and hours of labor by
production workers (Lp) is plotted on the horizontal axis. What is the equation
for the firm's isocost line? What are the two intercepts of the isocost line?
(iii) Assume that the firm's isoquants
are smooth curves and that labor hours can be varied continuously. Is the firm
producing the maximum level of output given its current level of cost? If it
is, explain how you can tell. If it isn't, explain what it should do to
increase output.
Solution:
(i) Total labor cost = (40)(40)($8) +
(5)(40)($20) = $16,800
(ii) Isocost: Ls
= (16800/20) − (8/20) Lp
= 840 − 0.40Lp
Vertical intercept: 16800/20 = 840
Horizontal intercept: 16800/8 = 2,100
(iii) The firm is not producing the
maximum possible level of output given its current level of cost. The marginal
rate of technical substitution with the current combination of inputs is 36/120
= 0.30, while the absolute value of the slope of the isocost line is 0.40. The
isoquant is flatter than the isocost line, so more supervisor hours and fewer
production worker hours should be employed. Also, the marginal product per
dollar spent on supervisors is 120/20 = 6 and that for production workers is
36/8 = 4.5, which leads to the same conclusion.
6. A firm currently employs 25
production workers and 4 supervisors. The marginal product of the last
production worker employed is 50 units of output per hour and production workers
are paid $10 per hour. The marginal product of the last supervisor employed is 160
units of output per hour and supervisors are paid $40 per hour. Every employee works
40 hours per week.
(i) What is the firm's total labor cost
per week?
(ii) Assume that hours of labor by
supervisors (Ls) is plotted on the vertical axis and hours of labor by
production workers (Lp) is plotted on the horizontal axis. What is the equation
for the firm's isocost line? What are the two intercepts of the isocost line?
(iii) Assume that the firm's isoquants
are smooth curves and that labor hours can be varied continuously. Is the firm
producing the maximum level of output given its current level of cost? If it
is, explain how you can tell. If it isn't, explain what it should do to
increase output.
Solution:
(i) Total labor cost = (25)(40)($10) +
(4)(40)($40) = $16,400
(ii) Isocost: Ls
= (16400/40) − (10/40) Lp
= 410 − 0.25Lp
Vertical intercept: 16400/40 = 410
Horizontal intercept: 16400/10 = 1,640
(iii) The firm is not producing the
maximum possible level of output given its current level of cost. The marginal
rate of technical substitution with the current combination of inputs is 50/160
= 0.3125, while the absolute value of the slope of the isocost line is 0.25.
The isoquant is steeper than the isocost line, so more production worker hours
and fewer supervisor hours should be employed. Also, the marginal product per
dollar spent on supervisors is 160/40 = 4 and that for production workers is
50/10 = 5, which leads to the same conclusion.
7. A firm currently employs 45
production workers and 6 supervisors. The marginal product of the last
production worker employed is 50 units of output per hour and production workers
are paid $10 per hour. The marginal product of the last supervisor employed is 150
units of output per hour and supervisors are paid $30 per hour. Every employee works
40 hours per week.
(i) What is the firm's total labor cost
per week?
(ii) Assume that hours of labor by
supervisors (Ls) is plotted on the vertical axis and hours of labor by
production workers (Lp) is plotted on the horizontal axis. What is the equation
for the firm's isocost line? What are the two intercepts of the isocost line?
(iii) Assume that the firm's isoquants
are smooth curves and that labor hours can be varied continuously. Is the firm
producing the maximum level of output given its current level of cost? If it
is, explain how you can tell. If it isn't, explain what it should do to
increase output.
Solution:
(i) Total labor cost = (45)(40)($10) +
(6)(40)($30) = $25,200
(ii) Isocost: Ls
= (25200/30) − (10/30) Lp
= 840 − 0.33Lp
Vertical intercept: 25200/30 = 840
Horizontal intercept: 25200/10 = 2,520
(iii) The firm is producing the maximum
possible level of output given its current level of cost. The marginal rate of
technical substitution with the current combination of inputs is 50/150 = 0.33
and the absolute value of the slope of the isocost line is also 0.33. Also, the
marginal product per dollar spent on supervisors is 150/30 = 5 and that for
production workers is 30/10 = 3, which leads to the same conclusion.
8. A firm wants to minimize the cost of
producing 2,800 units of output per week. It has hired a production engineer to
identify alternative production technologies that will accomplish this goal.
The production technologies use the different combinations of capital (K) and labor (L) that are listed below.
K 100 90 80
70 60 50
40 30 20
10 8 7
L 8 9 11 14 18
23 30 40
55 80 90
100
Assume that the rental price of capital
is $5 and the wage rate of labor is $4. Determine the minimum cost of producing
2,800 units of output and then show how the combination of inputs that yield
the minimum cost can be determined using the marginal approach.
Solution:
The total cost of production for each
of the combinations listed is given below.
K 100 90 80 70 60 50 40 30 20 10 8 7
L 8 9
11 14 18 23 30
40 55 80 90 100
TC 532 486 444 406 372 342 320 310 320 370 400
435
The minimum cost combination is K = 30 and L = 40. The slope of the isocost line is −4/5
= −0.80. The marginal rate of technical substitution between the point defined
by K = 50 and L = 23 and the point defined by K = 40 and L = 30 is (50 − 40)/(30 − 23) = 1.4.
Since this is greater than 0.80, the
next point should be considered. The marginal rate of substitution between the
point defined by K = 40
and L
= 30 and the point defined by K = 30 and L = 40 is (40 − 30)/(40 − 30) = 1. Again,
this is greater than 0.80, so the next point should be considered. The marginal
rate of substitution between the point defined by K = 30 and L = 40 and the point defined by K = 20 and L = 55 is (30 − 20)/(55 − 40) = 0.67.
This is less than 0.80, so the point of tangency is closest
to K
= 30 and L = 40.
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