Download Profit Maximization

The Profit Maximizing Decision
for the Variable Input
J. F. O’Connor
An Alternative Perspective
• One approach is to ask what is the level of
output at which profit is maximized? We
already did this.
• An alternative is to ask what is the level of
employment of the variable input at which
profit is maximized? That is the approach
that is followed here.
Recall
• Recall the production function, which gives
the Total Product and the Average and
Marginal Products. These are given in the
table and the graphs.
The Production Function
L
0
1
4
9
16
25
36
49
64
81
100
121
Q
0
40
80
120
160
200
240
280
320
360
400
440
AP
40.0
20.0
13.3
10.0
8.0
6.7
5.7
5.0
4.4
4.0
3.6
MP
20.0
10.0
6.7
5.0
4.0
3.3
2.9
2.5
2.2
2.0
1.8
Total Product Curve
450
400
350
Output
300
250
200
150
100
50
0
0
20
40
60
80
Labor
100
120
140
Unit Product Curves
20
18
16
14
AP
Output
12
10
8
6
4
MP
2
0
0
20
40
60
80
Labor
100
120
140
Total Revenue Product
• For each amount of input, how much
revenue is received? The total product
multiplied by the price of the output is the
answer. It is called the Total Revenue
Product (TPR). It is
TRP(L) = P*TP(L)
• It is plotted in the following graph and has
the same shape as the TP curve. (Why?)
Total Revenue Product Curve
2500
Dollars per period
2000
1500
1000
500
0
0
20
40
60
80
Labor per period
100
120
140
Unit Revenue Products
• The Average Revenue Product gives the
number of dollars of revenue per unit of the
variable input employed. It is
ARP(L) = TPR(L)/L = P*AP(L)
• The Marginal Revenue Product is the
change in TRP when the variable input is
changed by one unit. It is
MRP(L) = [TRP(L1)-TRP(L0)/[L(1)-L(0)]
= P*MP(L)
• How do the shapes compare with AP and
MP
Unit Revenue Product Curves
50
45
40
35
$/unit
30
25
20
ARP
MRP
15
10
5
0
0
20
40
60
80
Labor
100
120
140
Revenue Products
L
0
0.25
1
4
9
16
25
36
49
64
81
100
121
Q
0
20
40
80
120
160
200
240
280
320
360
400
440
TRP
0
100
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
ARP
0
MRP
0
50.0
40.0
33.3
28.6
25.0
22.2
20.0
18.2
50.0
33.3
25.0
20.0
16.7
14.3
12.5
11.1
10.0
9.1
Profit Maximization
• Profit = TRP(L) – (wL + FC).
where wL + FC is called Total Factor Cost
• We want the level of employment of labor,
L, at which profit is maximized. Find it
from the table or the graph. Profit in the
graph is the vertical distance between the
TRP curve and the TFC line. L*= 100.
Calculating Profit
L
TRP
0
0.25
1
4
9
16
25
36
49
64
81
100
121
FC
0
100
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
wL
300
300
300
300
300
300
300
300
300
300
300
300
300
TFC
0
2.5
10
40
90
160
250
360
490
640
810
1000
1210
300
302.5
310
340
390
460
550
660
790
940
1110
1300
1510
Profit
-300
-202.5
-110
60
210
340
450
540
610
660
690
700
690
Total Reveue Product Curve
and TFC
2500
2000
Dollars
1500
1000
500
0
0
20
40
60
80
Labor
100
120
140
Marginal Thinking
• If one is contemplating a given level of
employment, say L = 20, should one use
one more unit of labor? It depends?
• If the addition to revenue is greater than the
addition to cost, the answer is yes. The
addition to revenue from employing another
unit of labor is the marginal revenue
product while the addition to cost is the
wage rate.
• On the graph, the MRP is the slope of the
TRP and the wage rate is the slope of the
TFC.
• At L =20, MRP>w Therefore, using more of
the variable input will increase profit.
Using more of the input will increase profit
until we get to L=100. Beyond that point,
MRP < w. A necessary condition for profit
maximization is MRP(L) = w.
• The marginal thinking is easier to follow on
the per unit graphs.
L
0
1
4
9
16
25
36
49
64
81
100
121
TRP
0
200.0
400.0
600.0
800.0
1000.0
1200.0
1400.0
1600.0
1800.0
2000.0
2200.0
ARP
MRP
200.0
100.0
66.7
50.0
40.0
33.3
28.6
25.0
22.2
20.0
18.2
100.0
50.0
33.3
25.0
20.0
16.7
14.3
12.5
11.1
10.0
9.1
w
10
10
10
10
10
10
10
10
10
10
10
10
Unit Revenue Product Curves
and Wage Rate
50
45
40
35
$/unit
30
25
20
ARP
MRP
15
10
5
0
0
20
40
60
80
Labor
100
120
140
Profit Maximizing Conditions
• At L*=100,
MRP(100) = w
MRP is decreasing
ARP is greater than w
The third condition ensures that total
revenue exceeds the expenditure on the
variable input.
The Firm’s Demand for Labor
• What would happen if the price of labor
went to $15 per unit? The firm would want
to hire about 45 units of labor.
• Key point is that the firm moves along the
MRP curve as the price of the input varies.
The firm’s demand curve for the variable
input is the Marginal Revenue Product
curve
Factors Affecting Demand for the
Variable Input
• Price of the output
• Marginal product of the input