EC 352: Intermediate Microeconomics, Lecture 7
Economics 352: Intermediate Microeconomics
Notes and Sample Questions
Chapter 7: Production Functions
This chapter will introduce the idea of a production function.
A production process uses inputs such as labor, energy, raw materials and capital to
produce one (or more) outputs, which may be computer software, steel, massages or
anything else that can be sold.
A production function is a mathematical relationship between the quantities of inputs
used and the maximum quantity of output that can be produced with those quantities of
inputs.
For example, if the inputs are labor and capital (l and k, respectively), the maximum
quantity of output that may be produced is given by:
q = f(k, l)
Marginal physical product
The marginal physical product of a production function is the increase in output resulting
from a small increase in one of the inputs, holding other inputs constant. In terms of the
math, this is the partial derivative of the production function with respect to that
particular input. The marginal product of capital and the marginal product of labor are:
∂q
= fk
∂k
∂q
MPl =
= fl
∂l
MPk =
The usual assumption is that marginal (physical) product of an input decreases as the
quantity of that input increases. This characteristic is called diminishing marginal
product. For example, given a certain amount of machinery in a factory, more and more
labor may be added, but as more labor is added, at some point the marginal product of
labor, or the extra output gained from adding one more worker, will begin to decline.
EC 352: Intermediate Microeconomics, Lecture 7
Given some amount of land, a farmer can apply more and more seed, fertilizer and water,
but at some point the marginal product of these inputs will begin to decline. At some
point, the marginal product of water and fertilizer may actually become negative if crops
are drowned or buried in fertilizer.
The concept of diminishing marginal product need not be restricted to traditional
production situations. In a military operation, there may be a fixed set of targets against
which raids may be run. If the most valuable targets are attacked first, there will be
diminishing marginal product (in terms of damage done to the enemy) of raids. Each raid
will be of slightly less net value than the raid before.
In terms of math, diminishing product means that the second derivatives are negative:
∂MPk ∂ 2 q
= f kk < 0
=
∂k
∂k 2
∂MPl ∂ 2 q
= f ll < 0
=
∂l
∂l 2
The book gives this on page 184 and also replaces k (capital) with the number 1 and l
(labor) with the number 2.
It is worth noting that if capital increases the marginal product of labor, which certainly
seems to be true in many situations, then it will be that case that:
∂MPl ∂ 2 q
∂ 2 q ∂MPk
=
= f lk = f kl =
=
>0
∂k
∂l∂k
∂k∂l
∂l
That is, fkl=flk, and these are all generally positive.
Average Product
Average (physical) product of an input is equal to total output divided by the quantity of
the input:
output q f (k , l)
= =
kapital k
k
output q f (k , l)
= =
APl =
labor
l
l
APk =
EC 352: Intermediate Microeconomics, Lecture 7
Example
Imagine the production function q = f(k, l) = 64k2l2 – 2k3l3
We will find the marginal product functions and the average product functions.
∂q
= 128kl 2 − 6k 2 l 3
∂k
∂q
MPl =
= 128k 2 l − 6k 3l 2
∂l
MPk =
q
= 64kl 2 − 2k 2 l 3
k
q
APl = = 64k 2 l − 2k 3l 2
l
APk =
To show this function, imagine that k=1. We can graph the total output, q, as a function
of the labor used:
output
12000
10000
8000
6000
4000
2000
0
0
5
10
15
20
25
labor
A graph showing total output as a function of labor for this example.
Continuing to assume that k=1, we can graph the marginal product of labor:
EC 352: Intermediate Microeconomics, Lecture 7
MPl
800
600
400
200
0
-200 0
5
10
15
20
25
-400
-600
-800
Labor
A graph showing marginal product of labor for this example.
The marginal product of labor increases up to about ten workers and then starts to
decline. Past ten or eleven units of labor there is diminishing marginal productivity of
labor. This could actually be determined analytically by looking at the equation for MPl,
and finding its maximum when k=1:
MPl =
∂q
= 128k 2 l − 6k 3l 2
∂l
∂MPl
= 128k 2 − 12k 3l = 0
∂l
k =1
128 − 12 ⋅ l = 0
l=
128
= 10.6667
12
So, when k=1 the marginal product of labor is maximized at l = 10.6667.
It should be noted that when the marginal product of labor becomes negative, total output
actually starts to fall when more labor is added. This is what negative physical product
means. For a factory example, workers start to trip over each other.
Isoquants and the Marginal Rate of Technical Substitution
We can draw production functions in the same way that we draw consumers’ utility
functions by drawing out a curve along which output is constant, an isoquant, that is
analogous to a consumer’s indifference curve along which utility is constant.
EC 352: Intermediate Microeconomics, Lecture 7
Imagine, for example, that q = kl. We can graph the following isoquant:
A graph of the isoquant 60=kl.
The slope of the isoquant is the marginal rate of technical substitution and represents the
rate at which one input may be traded off for another, holding total output constant. This
is analogous to a consumer’s marginal rate of substitution.
Simply looking at the isoquant above and taking the derivative gives us:
MRTS = −
dk
dl q = 60
which can be tied back to the original production function as:
q = kl = 60
60
k=
l
MRTS = −
dk
60
=
dl q = 60 l 2
However, this can also be tied back to the original production function by the following
relationship:
MRTS =
MPl
MPk
EC 352: Intermediate Microeconomics, Lecture 7
Returns to Scale and the Homogeneity of the Production
Function
This is a fancy way of saying, “What happens to output if we double all inputs?”
If we double all inputs and output doesn’t double, there are decreasing returns to scale.
If we double all inputs and output exactly doubles, there are constant returns to scale.
If we double all input and output more than doubles, there are increasing returns to scale.
Decreasing returns to scale means that the production function is homogeneous of degree
less than one.
Constatn returns to scale means that the production function is homogeneous of degree
one.
Increasing returns to scale means that the production function is homogeneous of degree
more than one.
Again, homogeneity means that if you multiply all input by a factor of t, output is
increased by tk, and k is the degree of homogeneity.
So if we have homogeneity of degree zero, increasing the quantities of inputs doesn’t
change the output:
f (tk , tl ) = t 0 ⋅ f (k , l) = f (k , l)
So if we have homogeneity of degree 0.5, increasing the quantities of inputs will increase
the output, but by less than the increase in inputs:
f (tk , tl ) = t 0.5 ⋅ f (k , l) = f (k, l)
f (2k,2l ) = 2 0.5 ⋅ f (k , l) = 2 ⋅ f (k, l)
So if we have homogeneity of degree 1, increasing the quantities of inputs will increase
the output by exactly the same factor as the inputs increased. Doubling the quantities of
inputs, for example, will double the output.
EC 352: Intermediate Microeconomics, Lecture 7
f (tk , tl ) = t1 ⋅ f (k , l) = t ⋅ f (k , l)
f (2k,2l ) = 21 ⋅ f (k , l) = 2 ⋅ f (k , l)
So if we have homogeneity of degree 2, increasing the quantities of inputs will increase
the output by more than the inputs increased. Doubling the quantities of inputs, for
example, will quadruple the output.
f (tk , tl ) = t 2 ⋅ f (k, l)
f (2k,2l ) = 2 2 ⋅ f (k , l) = 4 ⋅ f (k, l)
One interesting fact that comes out of the analysis in the book is that the marginal product
of either capital or labor depends on the capital to labor ratio. That is, lots of capital will
increase the marginal productivity of labor. This can explain differences in marginal
productivity of labor between countries or industries. When labor has more capital to
work with, it will be more productive.
Practice Problems
Find the marginal product of labor and capital for each of the following production
functions.
1. q = kl
2. q = k0.5l
3. q = k0.3l0.5
4. q = k + l – k0.2l0.2
5. q = 5kl – k2l2
In each of the production functions given above, how does the marginal product of labor
change as the quantity of capital rises?
Which of the functions given above exhibit diminishing marginal product of labor?
Which of the functions given above exhibit diminishing marginal product of capital?
Calculate the cross partials (fkl and flk) for each of the functions given above. Are these
cross partial derivatives positive, negative or ambiguous?
For each of the production functions given above, find the formulas for the average
product of capital and the average product of labor.
EC 352: Intermediate Microeconomics, Lecture 7
In each of the above functions, how does the average product of labor change as the level
of capital employed rises?
Calculate the marginal rate of technical substitution for each of the functions given
above.
For each of the production functions given below, calculate the degree of homogeneity
and state whether it exhibits increasing, decreasing or constant returns to scale.
6. q = k0.5l0.5
7. q = k0.3l0.7
8. q = 5k0.6l0.4
9. q = k + k0.2l0.8 + 5l
10. q = k0.6l0.6
11. q = kl
12. q = k0.4l0.3
13. q = (k0.3 + l0.3)1.5