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.

L Q A P M PM P

0 0

1 4 0 4 0 . 0 2 0 . 0

4 8 0 2 0 . 0 1 0 . 0

9 1 2 0 1 3 . 3 6 . 7

1 6 1 6 0 1 0 . 0 5 . 0

2 5 2 0 0 8 . 0 4 . 0

3 6 2 4 0 6 . 7 3 . 3

4 9 2 8 0 5 . 7 2 . 9

6 4 3 2 0 5 . 0 2 . 5

8 1 3 6 0 4 . 4 2 . 2

1 0 0 4 0 0 4 . 0 2 . 0

1 2 1 4 4 0 3 . 6 1 . 8

The Production Function

Total Product Curve

0

50

100

150

200

250

300

350

400

450

0 20 40 60 80 100 120 140

Ou

tpu

t

Labor

Unit Product Curves

0

2

4

6

8

10

12

14

16

18

20

AP

MP

0 20 40 60 80 100 120 140

Ou

tpu

t

Labor

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

0

500

1000

1500

2000

2500

0 20 40 60 80 100 120 140

Do

lla

rs p

er

pe

rio

d

Labor per period

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

0

5

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40

45

50

ARPMRP

0 20 40 60 80 100 120 140

$/u

nit

Labor

L Q T R P A R P M R PM R P

0 0 0 0 0

0 . 2 5 2 0 1 0 0

1 4 0 2 0 0

4 8 0 4 0 0 5 0 . 0

9 1 2 0 6 0 0 3 3 . 3

1 6 1 6 0 8 0 0 5 0 . 0 2 5 . 0

2 5 2 0 0 1 0 0 0 4 0 . 0 2 0 . 0

3 6 2 4 0 1 2 0 0 3 3 . 3 1 6 . 7

4 9 2 8 0 1 4 0 0 2 8 . 6 1 4 . 3

6 4 3 2 0 1 6 0 0 2 5 . 0 1 2 . 5

8 1 3 6 0 1 8 0 0 2 2 . 2 1 1 . 1

1 0 0 4 0 0 2 0 0 0 2 0 . 0 1 0 . 0

1 2 1 4 4 0 2 2 0 0 1 8 . 2 9 . 1

Revenue Products

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.

L T R P F C w L T F C P r o f i t

0 0 3 0 0 0 3 0 0 - 3 0 0

0 . 2 5 1 0 0 3 0 0 2 . 5 3 0 2 . 5 - 2 0 2 . 5

1 2 0 0 3 0 0 1 0 3 1 0 - 1 1 0

4 4 0 0 3 0 0 4 0 3 4 0 6 0

9 6 0 0 3 0 0 9 0 3 9 0 2 1 0

1 6 8 0 0 3 0 0 1 6 0 4 6 0 3 4 0

2 5 1 0 0 0 3 0 0 2 5 0 5 5 0 4 5 0

3 6 1 2 0 0 3 0 0 3 6 0 6 6 0 5 4 0

4 9 1 4 0 0 3 0 0 4 9 0 7 9 0 6 1 0

6 4 1 6 0 0 3 0 0 6 4 0 9 4 0 6 6 0

8 1 1 8 0 0 3 0 0 8 1 0 1 1 1 0 6 9 0

1 0 0 2 0 0 0 3 0 0 1 0 0 0 1 3 0 0 7 0 0

1 2 1 2 2 0 0 3 0 0 1 2 1 0 1 5 1 0 6 9 0

Calculating Profit

Total Reveue Product Curve

and TFC

0

500

1000

1500

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2500

0 20 40 60 80 100 120 140

Do

lla

rs

Labor

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 T R P A R P M R P w

0 0 1 0

1 2 0 0 . 0 2 0 0 . 0 1 0 0 . 0 1 0

4 4 0 0 . 0 1 0 0 . 0 5 0 . 0 1 0

9 6 0 0 . 0 6 6 . 7 3 3 . 3 1 0

1 6 8 0 0 . 0 5 0 . 0 2 5 . 0 1 0

2 5 1 0 0 0 . 0 4 0 . 0 2 0 . 0 1 0

3 6 1 2 0 0 . 0 3 3 . 3 1 6 . 7 1 0

4 9 1 4 0 0 . 0 2 8 . 6 1 4 . 3 1 0

6 4 1 6 0 0 . 0 2 5 . 0 1 2 . 5 1 0

8 1 1 8 0 0 . 0 2 2 . 2 1 1 . 1 1 0

1 0 0 2 0 0 0 . 0 2 0 . 0 1 0 . 0 1 0

1 2 1 2 2 0 0 . 0 1 8 . 2 9 . 1 1 0

Unit Revenue Product Curves

and Wage Rate

0

5

10

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45

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ARPMRP

0 20 40 60 80 100 120 140

$/u

nit

Labor

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