Notes: Production Functions
Basically we begin with a simple production relationship:
Q = f(N) where Q = Output and N = Input
How are Q and N related? Let’s use L = labor input to keep things simple. Do you
expect Q to increase as L increases? In other words, if more workers are hired, do you
expect output to increase?
The relationship between Q and L is usually direct, or positive. That is…
If L increases then Q increases
If L decreases then Q decreases.
But the exact relationship may have 3 theoretical, functional relationships…
Three Theoretical Production Functions
A. Linear -- demonstrates constant marginal returns, that is, as additional units of input
are added to the production process each additional unit of input adds an equal
amount to the number of units of output/product produced.
ex.
Let L = number of units of input
Q = number of units of product/output
AP = TP/L
MP = TP/L
L
0
TP
0
MP
AP
Q
10
1
10
2
20
10
10
slope constant

10
10
10
3
30
10
10
4
40
1
10
10
10
0
MP = AP
L
10
5
TP
50
1
Note: The slope of the TP is the MP.
B. Increasing -- demonstrates increasing marginal returns, that is, as additional units
of input are added to the production process each additional unit of input adds more
and more to the number of units output/product produced.
L
0
TP
0
1
10
MP
AP
TP
Q
10
10
slope increasing

20
2
30
15
30
3
MP
60
AP
20
40
4
100
25
10
L
0
1
2
C. Decreasing -- demonstrates diminishing marginal returns, that is, as additional units
of input are added to the production process each additional unit of input adds less
and less to the number of units of output/product produced.
ex.
L
0
TP
0
1
MP
AP
Q
200
200
200
slope
decreasing
TP

150
2
350
3
450
175
100
150
75
4
525
131.25
200
AP
MP
0
L
1
Note : MP is the slope of TP. Remember MP = TP
L
The Typical Production Function -- A typical production function includes all the three
theoretical cases presented on the previous page. The firm, whose objective is to
maximize profits, will operate in the area of diminishing returns. This result will be
explained with additional comments from class lecture.
TP, AP
MP (units
of output)
point of diminishing total production

TP
Inflection
point 


point of diminishing marginal product

point of diminishing average product
= SOCIAL OPTIMUM
AP
Input
(units
labor)
0
L1 L2
L3
MP
Production Function
of
3
Three Stages of Production
Stage 1 = 0 – L1 Increasing Returns Stage. Here TP is increasing at an increasing
rate. That increasing rate means that the MP is increasing. In this stage of
production all workers should be hired as each additional worker adds more than the
previous worker to total production.
Stage 2 =
L1 – L3
Decreasing Returns Stage. Here TP is increasing at a
decreasing rate. That decreasing rate means that the MP is decreasing. In this
state of production, aka “the economic region of production”, the firm will locate the
profit maximizing amount of workers to hire.
Stage 3 = Beyond L3 Negative Returns Stage. Here TP is decreasing and the MP
is negative. The firm should not hire workers past L3.
Two Equivalent Decision Rules for the Profit Maximizing Firm
(1) Hire L, the # of workers, such that the firm maximizes TOTAL profits. This occurs
where
Max (Total Value of Output – Total Cost of Input) = Max Total Profit
(2) Hire L so that the marginal value product, MVP, is equal the marginal cost, MC.
That is…
MVP = MC
where MVP = MP x Px
(Px = output price)
Given the PX = Price of the output = $3, and the PL = price of the input, ie. labor =
$75, and the following input and marginal product information, the following relationships
are derived. Assume additionally that each laborer is paid the same amount, that is, the
input cost of each additional worker is the same, or the marginal cost of each worker is
the same, thus PL = MCL = $75.
(1)
L
(2)
MP
(3)
MVP
(4)
MCL
40
$120
$75
35
105
75
0
1
2
30
90
75
25
75
75
20
60
75
16
48
75
3
4
5
6
(5)
Total Value
of Output
(6)
Total Value
of Input
(7)
Total Profit
(5) - (6)
$0
$0
$0
120
75
45
225
150
75
315
225
90
390
300
90
450
375
75
498
450
48
This firm will hire 4 workers in order to maximize total profits.
4
Private Optimum
Determined by hiring the number of units, ie. workers, so that Total Profit is maximized,
and the number of inputs so that the MVP = MCL.
Note : MVP = marginal value product = MP x PX.
production created by an additional worker.
MVP = the value of the additional
Social Optimum -- is defined as the highest output per worker, ie. the largest AP, also
coincides with the point of diminishing average product.