Production function

Topic 4 page 1
Topic 4: The Production Function
Chapter 8
1) The Production Function
2) Properties of Production Technologies
No free lunch, non-reversibility, free disposability
3) Changing Factors of Production in the Short and Long
runs: diminishing returns, MRTS
4) The Short-Run Production Function
Total Product, Average Product, Marginal Product

Topic 4 page 2
Objective: To examine how firm and industry supply
curves are derived.
Introduction: Up to this point we have examined how the
market demand function is derived. Next, we will
examine the supply side of the market. We will explore
how firms minimize costs and maximize productive
efficiency in order to produce goods and services. By
effectively combining labour and capital, the firm
develops a production process with the objective of
efficient resource allocation and cost minimization. The
firm is assumed to produce a given output at minimum
cost.

Topic 4 page 3
The Production Function
Definitions:

Topic 4 page 4
Factors of Production: are factors used to produce output.
Example: Labour
Capital - machines
-buildings
Land
Natural resources

Topic 4 page 5
State of technology: consists of existing knowledge about
method of production.
The quantity that a firm can produce with its factors of
production depends on the state of technology.
The relationship between factors of production and the output
that is created is referred to as the production function.
“The production function describes the
maximum quantity of output that can be
produced with each combination of factors
of production given the state of technology.”

Topic 4 page 6
For any product, the production function is a table, a graph or
an equation showing the maximum output rate of the product
that can be achieved from any specified set of usage rates of
inputs.
The production function summarizes the characteristics of
existing technology at a given time; it shows the
technological constraints that the firm must deal with.
Some common assumptions in regard to technology:
No free lunch: without inputs there are no outputs
Non-reversibility: Cannot run the production process in
reverse.
Free disposability: Can throw away the excess without any cost
or using more inputs.
Convexity: ?

Topic 4 page 7
Model Assumption: The quantity produced per period is
‘q’, and the two factors of production are labour and
capital.
Notation: The Production Function of the Firm:
q=f(L, K)
where f , the function, describes the relationship
between the inputs L, K and the output each different
combination produces per period.

Topic 4 page 8
Some important production functions:
1) The Leontief technology (fixed-proportions
technology)
Q(X1,X2) = min(aX1, bX2)
Q(K,L) = min(1/6L,K)

Topic 4 page 9
2) The Cobb-Douglas technology

Topic 4 page 11
3) Perfectly Substitutable input production functions:
Q(K,L) =aK + bL

Topic 4 page 13
Changing Factors of Production in the
Short and Long Runs
We must make a distinction between the short and long run.
In the short run, a firm is able to change some of the factors
of production, but at least one factor is fixed.
In the long run, all factors of production can be varied.

Topic 4 page 16
The Short-Run Production Function
Model: consider the simplest case where there is one input
whose quantity is fixed and one input whose quantity is
variable.
Suppose that the fixed input is the number of machines
(capital) and the variable input is labour.
In the short run the firm cannot change the number of
machines quickly without incurring a high cost.
With one fixed input the short-run production function shows
how total output changes as the variable factor changes.

Topic 4 page 17
The Total, Average and Marginal Product of Labour
Total Product Function: expresses the relationship between
the variable input and the total output.
→The total product function of labour: TPL: shows the
various amounts of output that is produced when the amount
of labour is varied with a given fixed amount of capital.

Topic 4 page 18
The diagram illustrates what occurs when the amount of the
variable input increases.

Topic 4 page 19
The total product (output) increases when the amount of
labour increases, holding the amount of capital fixed at K0.
Quantity increases initially at an increasing rate, but
eventually quantity increases at a decreasing rate when
more labour is employed.
Algebraically:


2
2 0
TP L K
L
L ( , )
*
 at first;
and then eventually,


2
2 0
TP L K
L
L ( , )
*
 (becomes negative).
At some point, adding more labour units no longer increases
output.

Topic 4 page 20
We can derive the average and marginal product function
of labour from the total product function.
The average product function, APL, measures output per
unit of labour:
L 0
L 0
Total product of labour
Average product of labour
Number of labour units
TP (L,K )
AP (L,K )
L


Average product of labour is the measure of productivity of
labour.
Average product of labour, APL, measures output per unit of
labour.

Topic 4 page 21
It is the slope of a ray drawn from the origin to any point on
the TPL function.
The average product of labour for any given level of
employment is equal to the slope of a straight line drawn from
the origin to the total product function at that employment
level.
Generally, the APL increases at first as labour is increased.
I.e. the output per worker increases initially.
Further increases in labour reduce APL.
APL declines when employment increases.

Topic 4 page 22
The marginal product of an input is the addition to total
output resulting from the addition of the last unit of the input
when the amount of other inputs used is held constant.
The marginal product function of labour, MPL, measures
the change in quantity due to a change in the labour input, or
the slope of the total product function of labour:
L 0
L 0
in total product of labour
Marginal product of labour
in number of labour units
TP (L,K )
MP (L,K )
L
L
TP
KL






 


Topic 4 page 23
There is a distinct relationship between marginal
product and average product:
When: MP>AP, AP is increasing
MP<AP, AP is decreasing
MP=AP, AP is constant and at a maximum

Topic 4 page 24
The law of diminishing returns describes the eventual
decline in the marginal product of the variable factor as the
variable factor increases with other factors held constant.
The law of diminishing returns applies only to situations
where one factor is increasing and the other factors are fixed.
“The law of diminishing marginal returns:
if equal increments of an input are added,
and the quantities of other inputs are held
constant, the resulting increments of product
will decrease beyond some point; that is, the
marginal product of the input will diminish.”

Topic 4 page 25
Output
Labour
Output/L
Labour
TPL
Average
Product
Marginal
Product

Topic 4 page 29
The Long-Run Production Function
In the long run, all factors of production are variable.
Substitution Among Factors
Similar to the notion of substituting between goods to
maintain constant utility along an indifference curve, firms
usually can produce the same output quantity by substituting
between factors of production.
The important question that needs to be addressed is:
“What combination of factors should be used to
produce this output?”

Topic 4 page 30
This question is difficult to answer because there is more than
one way to produce the product.
This can be illustrated with the aid of isoquant analysis.
The amount of capital is on the vertical axis and number of
labour units is on the horizontal axis.
The curve with an output of ‘q0’ is called an isoquant.
 “Iso” means equal
 “quant” means quantity.

Topic 4 page 31
The amount produced is the same along the isoquant. The
points along the isoquant q0 represent the different factor
combinations that can produce q0 units per period.
“An isoquant shows the different combinations of factors of
production that can produce a given quantity of output.”
Capital
K1
K2 q0 (Isoquant)
0 L1 L2 Labour

Topic 4 page 32
The Marginal Rate of Technical Substitution
The marginal rate of technical substitution (MRTS) measures
the rate of substitution of one factor for another along an
isoquant.
“The marginal rate of technical substitution is the rate at
which a firm can substitute capital and labour for one another
such that the output is constant.”
MRTSKL
constant




K
L q
where  
K L is the slope between two point on an isoquant.
Note: An isoquant cannot have a positive slope.

Topic 4 page 33
An increase in one factor of production causes output to
increase. Hence this increase in one factor must be offset by
a decrease in other factor in order to keep output at the same
level.
As the firm moves along the isoquant from left to right, the
slope increases. The firm substitutes labour for capital, but at
a diminishing rate. When this occurs there is a diminishing
marginal rate of technical substitution.

Topic 4 page 34
Returns to Scale
In general, the level of a firm’s productivity changes as the
quantity produced by the firm changes.
Returns to scale refers to the percentage change in output to a
percentage change in inputs.

Topic 4 page 38
Three Cases:
1) When the percentage increase in inputs is smaller than the
percentage increase in output, there are increasing returns
to scale.
2) When the percentage increase in inputs leads to the same
percentage increase in output, there are constant returns
to scale.
3) When the percentage increase in inputs is larger than the
percentage increase in output, there are decreasing
returns to scale.

Topic 4 page 39
Capital
q5
q4
q3
K1
q2
K0 q1
q0
0 L0 L1 Labour
The returns to scale can be measured along the ray from the
origin.
Here the capital to labour ratio is given.

Topic 4 page 40
There is a relationship between returns to scale and the
spacing of the isoquants.
When there are increasing returns to scale, the isoquants are
bunched closer together.
When there are decreasing returns to scale, the isoquants are
farther apart.

Topic 4 page 41
Returns to Scale and the Cobb-Douglas Production
Function
A common production function is the Cobb-Douglas
production function.

Topic 4 page 42
Algebraically:
q=ALa
Kb
where A, a and b are constant and greater than zero.
To determine the returns to scale for this function, we could
change labour and capital by a factor ‘m’ and then determine
if output changes by more than, equal to or less than ‘m’
times.
q=A (mL)a
(mK)b
q=Ama
La
m b
K b
q=ma+b
[ALa
K b
]

Topic 4 page 43
Since originally output was q=ALa
Kb
, we can determine if
output will increase by either less than m times if a+b<1
(because ma+b
<m), by exactly m times if a+b=1 or by more
then m times if a+b>1.

Topic 4 page 44
The MRTS and MP of Both Factors of Production
The MRTS and marginal product of labour and capital are
related.
Suppose the firm decides to increase the amount of capital it
employs, holding the amount of labour constant. Output will
increase by the amount qK because of the increase in capital
K .
The increase in output is approximated by
 
q MP K
K K


Topic 4 page 45
If the firm holds capital constant and decreases the amount of
labour it employs, output decreases by qL , where the
amount of labour decreases by L.
The decrease in output is approximated by
 
q MP L
L L
 .
Along a given isoquant, output must be constant.
If the firm increases capital by K , labour must decrease by
an amount L such that: qK + qL =0 in order to remain
on the same isoquant.

Topic 4 page 46
Substituting in these expressions, the condition becomes:
MP K MP L
K L
 
  0
Rearranging such that we have an expression for the MRTS
in terms of the marginal products of the two factors:
MRTS
K
L
MP
MP
KL
L
K
  


The MRTS of K for L equals the negative of the ratio of the
marginal product of L and K.

Production function

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