The Firm and Production
Overheads
Nature of the firm
Neoclassical firm A neoclassical firm is an organization
that controls the transformation of inputs
(resources it owns or purchases)
into outputs
(valued products that it sells),
and earns the difference between
what it receives in revenue,
and what it spends on inputs (costs).
Business firm
A business firm is an organization,
owned and operated by private individuals,
that specializes in production.
Production systems, goods, services and
factors
A production system or technology is a description
of the set of outputs that can be produced
by a given set of factors of production or inputs
using a given method of production
or production process.
Production Technologies
The technology set (technology for
short) for a given production process is
defined as the set of all input and
output combinations such that the
output vector y can be produced from
the given set of inputs x
A factor of production (input) is a good or
service that is employed in the production
process.
A product is a good or service that is
the output of a particular production
process.
Expendable factors of production are
raw materials, or produced factors that
are completely used up or consumed
during a single production period.
Capital is a stock that is not used up
during a single production period,
provides services over time, and
retains a unique identity.
Capital services are the flow of
productive services that can be obtained
from a given capital stock during a
production period.
They arise from a specific item of capital
rather than from a production process.
It is usually possible to separate the right to use
services from ownership of the capital good.
Revenue
Revenue is the total income that comes from
the sale of the output (goods and services)
of a given firm or production process.
Revenue  R(p, y)  py
m
Revenue  R (p , y)   Σ
j 1
pj y j
Cost
Cost is the value of all factors of production
used by the firm in producing
a given level of output,
whether a single product or multiple products.
If the input bundle used by a firm for a particular process is
(x1, x2, . . . xn) , and wi is the price of the ith input, then
n
cost  C(x , w)  C  Σ wi xi
i 1
Profit
The profit from a given production plan
is the revenue obtained from the plan
minus the costs of the inputs used to implement it
m
n
π  Σ pj yj Σ wi xi
j 1
n
i 1
π  p yΣ wi xi
i 1
π  p yw1 x1 w2 x2
Objectives of the firm
We usually assume that firma exists to make money
Such firms are called for-profit firms.
Given this assumption we can set up the firm level
decision problem as maximizing the returns from
the technologies controlled by the firm
taking into account
• the demand for final consumption goods,
• opportunities for buying and selling
factors (or products) from other firms
• the actions of other firms in the market
Objectives of the firm
In a perfectly competitive market, this means
the firm will take prices as given,
and choose the levels of inputs and outputs
that maximize profits
Purely competitive markets
When buyer or sellers in a market are
not able affect the price
of a product, we say that the market is
purely competitive, or just, competitive.
Why firms?
Gains from specialization
One man draws out the wire, another straightens it,
a third cuts it, a fourth points it,
a fifth grinds it at the top for receiving the head:
to make the head requires three distinct operations;
to put it on is a [separate] business,
to whiten the pins is another;
it is even a trade by itself to put them into the paper;
and the important business of making a pin is, in this manner,
divided into about eighteen distinct operations,
which, in some manufactories, are all performed by distinct hands.
Examples of gains from specialization
Assembly lines
Machines needed more than on person (2-person saw)
Learning by doing and improved skills
Learning by doing and economies of size
Lower transactions costs
Transactions costs are the time and other costs
required to carry out and enforce the terms of market
exchanges (transactions)
Examples
Coordination of production
Lower transportation costs
Lower cost of price discovery or negotiation
Lower costs of making and enforcing contracts
Avoiding hold-up problems and opportunistic behavior
Reduced risk
Larger firms may be able to reduce income
risk by diversification
Diversification is the process of reducing risk
by spreading sources of income among different
alternatives
Problems with firms
Agency problems with employees
Lack of market discipline
Communication problems
We describe the technological possibilities
for the firm by its technology set
(technology for short)
For a given level of inputs, x,
we call this set the Production Possibility Set
We denote the set of all feasible
input-output combinations by T
just as we denote the set of all outputs producible
with a given level of inputs x, by P(x)
A particular element of the technology set
is called a production plan and we write
(x, y)  T
Some input and output combinations (x, y)
may not be elements of T
Such combinations are said to be infeasible
Production Functions
The production function is a function
that gives the maximum output attainable
from a given combination of inputs.
f (x)  max [y: (x, y) ε T]
y

max [y]
y ε P( x )
The production function really only makes sense
with one output
y1
P(x)
y2
The Production Possibility Set with One Output
y
P(x1)
x1
x
The Production Function
Output -y
y = f (x1, x2, x3, . . . xn )
350
300
250
200
y
150
100
50
0
0
2
4
6
8
10
Input -x
12
Examples
y (bushels) = f (land, tillage, seed, fertilizer, … )
y 
α1 α2
Ax1 x2

1
3
5 x1
1
4
x2
1 3
y  x  x
30
2
y  α1 x  α2 x 2  α3 x 3
 10 x  20x 2 0.60 x 3
The short run and the long run
The short run and the long run have to do with
what is fixed for a given decision problem
The short-run is a time period brief enough that the firm
can vary some, but not all,
of its inputs in a costless manner.
The long-run is a time period long enough that the firm
can vary all of its inputs in a costless manner
If there are costs associated with
varying the level of an input
we say that the firm experiences adjustment costs
Fixed inputs
A input whose quantity remains constant,
regardless of how much output is produced
in the current decision period
is called a fixed input
Variable inputs
A variable input is an input whose usage changes
as the level of output changes
in the current decision period
Fixed, variable, and sunk costs
Fixed costs are those costs that the firm
is committed to pay for factors of production,
regardless of the firm's current decisions
Suppose x2 = 10 and w2 = $50.
If x2 is fixed, then fixed cost = $500
Suppose
C(y)  100 6y 0.4y 2 .02y 3
Cost
Example of fixed costs
160
140
120
100
80
60
40
20
0
FC
0 2 4 6 8 10 12 14 16 18 20
Output - y
Sunk costs
The portion of fixed cost that is not recoverable
if the firm liquidates, is called sunk cost
Example of a pizza restaurant
Sub-lease of land or a building
Sell off tables and chairs
Specialized pizza oven
Fixed cost = sunk cost + avoidable fixed cost
Variable costs
Variable costs are those costs that are affected
by the firm's actions in the current period
Variable costs occur because of the decision
to purchase additional factors or factor services
for use in production.
n1
TVC  Σ wi xi
i 1
n1 is the number of variable inputs
Example of variable cost
Variable Cost
Cost
500
400
300
VC
200
100
0
0
5
10
15
20
25
Output - y
30
Fixed costs
Fixed costs are those costs that the firm
is committed to pay for factors of production,
regardless of the firm's current decisions
n
TFC  Σ wi x̄i
i n1
The bar over x denotes it is fixed
Inputs n1 - n are all fixed
Total costs
The sum of fixed cost and variable cost
is called fixed cost.
TC  TVC
 TFC
n1
n
i 1
i n1
 Σ wi xi  Σ wi xi
n
 Σ wi x i
i 1
Example
x1 = cooks
x2 = brats
x3 = brat buns
x4 = grills
x5 = brat turners
x6= charcoal
Variable
Cooks, brats, buns, charcoal
Fixed
n = 6 n1 = 4
Grills, brat turners
Variable and fixed cost
TVC  w1 x1  w2 x2  w3 x3  w4 x4
TFC  w5 x5  w6 x6
Example of total cost
Cost
500
FC
VC
TC
400
300
200
100
0
0
5
10
15
20
25
Output - y
30
Production in the short run
Total (physical) product - TP (TPP)
Total product (y) is the maximum quantity
of output that can be produced
from a given combination of inputs.
It is the value of the production function
y = f (x1, x2 , . . . , xn )
Example numerical function
y  f(x1 ,x2 ,x3 ,,xn)
 f(x1 , x2 )

2
2
1
200x1 
20x2 40x1 200 x1 x2 20x2
3
2x1
3
x2
Story
y - bales of hay hauled per hour
x1 - number of laborers hauling hay
x2 - number of tractor-wagon combinations
Data with 1 tractor and wagon
Input 1
x1
labor
0.00
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
Input 2
x2
wagons
0.00
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Total
Product
y (TPP)
bales
--38.0
144.0
306.0
512.0
750.0
1008.0
1274.0
1536.0
1782.0
2000.0
2178.0
2304.0
2366.0
2352.0
Output - y
Graph of total product
Total Product of Input 1 - x2 = 1
2400
2100
1800
1500
1200
900
600
300
0
y
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Input - x1
Marginal (Physical) Product
Marginal (physical) product is the increase in
output that results from a one unit increase in
a particular input
In discrete terms or average terms
the marginal product of the ith input is given as
Δy
y 1
y0
MPi 

1
0
Δxi
xi 
xi
where y1 and x1 are the level of output and input after
the change in the input level and y0 and x0 are the levels
before the change in input use.
For small changes in xi, the marginal product
is given by the derivative

f(x)

y
MPi 


xi

xi
Example calculations
Change x1 from 4 to 5
Change x1 from 1 to 2
Input 1
x1
labor
0.00
1.0
2.0
3.0
4.0
5.0
6.0
Δy
750 512
MP 

 238
Δx1
5 4
Δy
144 38
MP 

 106
Δx1
2 1
Input 2
x2
wagons
0.00
1.0
1.0
1.0
1.0
1.0
1.0
Product
y (TPP)
bales
--38.0
144.0
306.0
512.0
750.0
1008.0
Using calculus
y  f (x1 ,x2 ,x3 ,,xn)
 1
2
200x1 20x2 40x1
2
200x1 x2  20x2
3
2x1
3
x2

y
2
 
200  80x1  200x2 6x1

x1

y
 
200 (80)(2)  (200)(1) (6)(22)

x1
 
200 160  200 24
 136
Graphical representation
Output - y
Marginal Product of Input 1 - x2 = 1
300
250
200
150
100
50
0
-50 0
MPP 1
2
4
6
8
10
12
14
Input - x 1
Average (physical) product
An average measure of the relationship between
outputs and inputs is given by the average product,
which is just the level of output divided by
the level of one of the inputs
f(x)
y
APi 

xi
xi
Example calculations
Average product at x1 = 5
Average product at x1 = 2
Input 1
x1
labor
0.00
1.0
2.0
3.0
4.0
5.0
6.0
y
750
AP1 (5) 

 150
x1
5
y
144
AP1 (2) 

 72
x1
2
Input 2
x2
wagons
0.00
1.0
1.0
1.0
1.0
1.0
1.0
Product
y (TPP)
bales
--38.0
144.0
306.0
512.0
750.0
1008.0
Graphical representation
Output - y
Average and Marginal Product of Input 1
300
MPP 1
250
APP 1
200
150
100
50
0
-50
2
4
6
8
10 12 14
Input - x1
Discussion of marginal (physical) product
Increasing returns
When the marginal product rises (increases)
as an input rises, we say that the marginal product
of the input is increasing
When there are increasing returns,
an additional unit of the input causes a larger increase
in output than the previous unit.
When marginal product is increasing,
total product is increasing at an increasing rate
Data with 1 tractor and wagon
Input 1
x1
labor
0.00
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
Input 2
x2
wagons
0.00
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Total
Product
y (TPP)
bales
--38.0
144.0
306.0
512.0
750.0
1008.0
1274.0
1536.0
1782.0
2000.0
2178.0
2304.0
2366.0
2352.0
Average
Product
APP
Average
Marginal
Product
A MPP
Marginal
Product
--38.00
72.00
102.00
128.00
150.00
168.00
182.00
192.00
198.00
200.00
198.00
192.00
182.00
168.00
--38.00
106.00
162.00
206.00
238.00
258.00
266.00
262.00
246.00
218.0
178.0
126.0
62.0
-14.0
--74.00
136.00
186.00
224.00
250.00
264.00
266.00
256.00
234.00
200.00
154.00
96.00
26.00
-56.00
Graphical representation
Output - y
Total Product of Input 1
2400
2100
1800
1500
1200
900
600
300
0
y
0
2
4
6
8
10
12
14
Input - x1
Output - y
Average and Marginal Product of Input 1
MPP 1
300
APP 1
250
200
150
100
50
0
-50
2
4
6
8
10
12
14
Input - x1
Diminishing returns
When the marginal product falls (decreases)
as an input rises, we say that the marginal product
of the input is diminishing
When there are diminishing returns,
an additional unit of the input causes a smaller
(but positive) increase in output than the previous unit
When marginal product is decreasing,
(but positive) total product is increasing
at a decreasing rate.
The law of diminishing returns
The law of diminishing (marginal) returns states
that as we continue to add more of any input
(holding the other inputs constant),
its marginal product will eventually decline.
Examples
fertilizer
hay wagons
counter workers
college administrators
Negative returns
When marginal product is negative,
output actually falls with the addition
of another unit of the input
Examples
fertilizer
water on a plant
cooks in a kitchen
Average, total and marginal product
1. When the marginal curve is positive, the total curve will be rising
2. When the marginal curve is rising, the total curve will be rising
at an increasing rate (becomes steeper)
3. When the marginal curve is positive but falling, the total curve
will be rising at a decreasing rate (becomes flatter)
4. When the marginal curve is greater than the average curve,
the average curve is rising
5. When the marginal and average curves are equal,
the average curve does not change
(is usually at a maximum or minimum point)
6. When the marginal curve is less than the average curve,
the average curve is falling
7. For a production function MP and AP intersect
at the maximum of APP
Intuition for average-marginal relationship
Cumulative GPA
Average test scores
The End