Chapter 20
Factor Markets—Demand for the Variable Input
Marginal Productivity and Factor Demand
1. Marginal Benefit versus Marginal Cost—Once Again
All economic decisions are about comparing costs and benefits—and usually are about comparing
marginal costs and marginal benefits:
The consumer, in deciding whether to buy an additional unit of a product, compares the cost of buying
the additional unit (marginal cost) to the benefit or utility obtained from that extra unit (marginal
benefit). Since the individual consumer is a price taker, the marginal cost is equal to the price of the
product. Marginal benefit is the (subjective) utility or satisfaction the consumer receives from consuming
that additional unit.
The producer, in deciding whether to hire (to buy the services) of an additional employee, compares the
cost of hiring the additional unit of the variable input to the benefit received from that extra input unit.
The benefit is the contribution of the extra unit of input to the firm’s revenue.
1.1. Cost of hiring additional unit of variable input—The Wage Rate
The cost of hiring the additional unit of variable input (labor) is the wage the firm pays for that input.
Assuming that the firm is operating in a perfectly competitive labor market, the price the firm pays to
hire that additional labor input is the market wage rate, W.
1.2. Benefit of Hiring Additional Unit of Variable Input—Value of Marginal Product (VMP)
The reason the firm hires the additional unit of labor is to increase its output. The additional output
produced by the extra unit of labor is the marginal product of labor (MPL). The firm sells this extra
output, the marginal product, at the prevailing market price. The revenue received from selling this
marginal product is price times the marginal product, or the value of the marginal product (VMP). Thus,
VMP of an input is the value of the additional output generated by employing one more unit of that
input.
VMP = P × MPL
2. The Optimal Input Rule—Does the VMPL Justify Hiring the Additional Worker?
The firm will hire the additional variable input as long as the value of marginal product is greater than
the cost of hiring the additional worker, the wage rate. The optimal rule is when VMP = W.
Consider the following example:
Chapter 20—Demand for Factor Inputs
page 1 of 6
Example
Bob’s perfectly competitive firm is operating in the schmoo market. Let Bob’s total product and
marginal product functions be represented by the following:
Total product:
Marginal product:
Q = 20L – L²
MPL = 20 – 2L
The total product and marginal product schedules are shown in the following table.
L
0
1
2
3
4
5
6
7
8
9
10
Q
0
19
36
51
64
75
84
91
96
99
100
MP
20
18
16
14
12
10
8
6
4
2
0
The graphs are as follows:
Total Product
Q = 20L – L²
Marginal Product
MPL = 20 – 2L
The value of marginal product is determined as the product of price of the good (the output) times the
marginal product of the additional unit of the variable input. Suppose the market equilibrium price of
schmoo is P = $20. Then the value of marginal product is,
Chapter 20—Demand for Factor Inputs
page 2 of 6
VMP = P∙MPL = $20(20 – 2L) = 400 – 40L
Now assume the market wage rate is W = $200. The firm would hire the additional unit of the variable
input as long the value of the output produced by the additional variable input (VMP) is greater than the
cost of hiring the additional unit (W). If the cost of hiring exceeds the value of the additional output, the
firm would not hire the additional variable input. The optimum situation is where VMP is equal to W.
VMP = W
400 – 40L = 200
40L = 200
L=5
The optimum units of labor hired is 5 units. This is also shown in the following table and the graph
below.
Units
of labor
L
0
1
2
3
4
5
6
7
8
Marginal
product of
labor
MPL = 20 ‒ 2L
20
18
16
14
12
10
8
6
4
VMP = P × MPL
P = $20
VMP = 400 – 40L
$400
360
320
280
240
200
160
120
80
Wage
W
$200
200
200
200
200
200
200
200
200
To see why L = 5 units is the optimum units of variable input hired consider the following diagram. At
W = $200 and L = 5 the firm’s total variable cost is TVC = $200 × 5 = $1,000. The firm’s net gain is shown
as the area of the triangle above the W line and below the VMP curve. The area of this triangle is
($400 ‒ $200)(5) ∕ 2 = $500.
Any fewer units of variable input would lead to a smaller net gain.
Chapter 20—Demand for Factor Inputs
page 3 of 6
Value of Marginal Product and the Wage Rate
3. The VMPL represents the firm’s demand schedule for labor
The downward sloping VMPL shows the inverse relationship between the wage rate and units of labor
demanded or hired. VMPL slopes downward because the curve is derived from the marginal product of
labor, MPL, which slopes downward because of the law of diminishing marginal product.
The firm will employ more workers if the wage rate falls, and reduce employment when the wage rate
rises. The change in the wage rate causes a movement along the VMP curve.
3.1. Shift in the Demand Curve
There are three main causes of the shift in demand-for-labor curve:
Chapter 20—Demand for Factor Inputs
page 4 of 6
3.1.1. Change in the price of the product
The firm would hire more workers at the same wage rate if the price the firm’s product rises. This
increase in hiring is shown as the shift in the VMP.
Units
of labor
L
0
1
2
3
4
5
6
7
8
MPL = 20 ‒ 2L
20
18
16
14
12
10
8
6
4
P = $20
VMP = 400 – 40L
$400
360
320
280
240
200
160
120
80
P = $25
VMP₁ = 500 – 50L
$500
450
400
350
300
250
200
150
100
Wage
W
$200
200
200
200
200
200
200
200
200
3.1.2. Change a) in Supply of Other Inputs, and b) in Technology
Generally, availability of other inputs that complement the labor input, or improvements in technology,
improve labor productivity. The impact of the improvement in labor productivity can be shown as the
change in the production function and in the marginal product curves. This is shown in the following
diagram. An improvement in technology causes the marginal product of labor to shift to the right.
Chapter 20—Demand for Factor Inputs
page 5 of 6
Total Product
Q₀ = 20L – L²
Q₁ = 26L – L²
Marginal Product
MP₀ = 20 – 2L
MP₁ = 26 – 2L
Now, holding price of the product at P = $20, the demand for labor shifts to right, shown as the shift
from VMP₀ to VMP₁. As a result of the improvement in technology, the firm would hire more variable
inputs at the going wage rate (W = $200). With the new VMP the optimum units of the variable input is
L = 8.
VMP₀ = 400 – 40L
VMP₁ = 520 – 40L
VMP₁ = 520 – 40L
W = 200
VMP₁ = W
520 – 40L = 200
40L = 320
L=8
Chapter 20—Demand for Factor Inputs
page 6 of 6