NOTES 3: The Aggregate Production Function
Note: I want you to understand the main parts of these notes. These notes are fair game
for quizzes and exams. Some parts of this chapter get “mathy” especially in the proof.
You do not need to understand every step of the proof, but you will need to understand
the basic points I am trying to make - especially when it refers to the marginal product of
labor (MPN) and the marginal product of capital (MPK)!
I.
Overview
When dealing with the supply side of the economy, we need to specify the aggregate
production function (Y).
Eventually, we need to balance the demand side of the
economy (what people buy: Y = C + I + G + NX) with what firms produce (Y =
F(inputs, technology) some production function). But, before we get there, we need to
know how firms produce.
Measuring the supply side of the economy has been the subject of much active research
in macroeconomics during the last two decades. In this class, we specify one aggregate
production function for the economy. In the real world, each firm has its own production
function. Ford takes some inputs and mixes them with some technology to make a
product; Honda also takes some inputs and mixes them with some technology to make a
product. Both make cars. But, Ford may use more labor to make cars and Honda may
use more assembly lines - in the end, they both make cars. We assume that there is only
one 'production function' (defined below) for the ‘aggregate’ economy. Recent research
has tried to relax this assumption - it is very difficult to do. But, this assumption allows
our models to have high predictive power. As we will see, our aggregate production
function matches the aggregate data well AND has all the qualitative predictions that we
believe any individual firm specific production function should have.
Let’s start this discussion by outlining some facts that we know about aggregate
production. Any production function that we posit should be able to match the empirical
aggregate facts. After that, I am going to give you a well-used production function and
show that this production function has the desired properties to match the empirical facts.
II.
Some Preliminaries
A production function is a mathematical relationship between the inputs a firm uses and
the output the firm generates. For this class, we are going to assume that the firm only
uses three broad inputs: labor (workers), capital (machines) and some technology (we are
going to abstract from things like raw materials and land as of now - if it makes you more
comfortable, we will include these in our measure of capital). You should think of
capital as economic capital. Economic capital is simply the integral of all past
investment - net of depreciation. You should think of capital as a flow variable where
today’s capital would equal the capital your firm owned yesterday net of depreciation
plus any new capital you bought today (your investment). Note: We measure labor and
capital like we measure output - in terms of dollars. Our measure of labor, then, is the
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'real' value of wages paid to workers (wage/price) and our measure of capital is the ‘real’
value of machines that we use.
An aggregate production function can look something like:
Y = f(K, N, A)
where f(.) is some mathematical relationship (a function), K is our symbol for the capital
stock, N is our symbol for labor and A is our symbol for our technology measure (much
more on this measure of technology in class).
This is a production function!!!!! Notice, we have two equations that determine Y in our
class. Y = C + I + G + NX (this is the demand side of the economy - what people buy).
Also, we have Y = f(K, N, A) (this is the supply side - what firms produce). As we
learned in week 1, what is bought - by definition - must equal what is produced (again,
once we account for inventory changes). These equations will become the foundation of
our aggregate demand and aggregate supply equations later in the class!!! These
equations will jointly determine both prices and output (and with some additional steps interest rates and wages and other stuff). Prices will adjust in the economy to ensure
that what is demanded is what is produced (store this away; this is one of the key
insights from the class). We will have two equations (demand and supply) and two
unknowns (Y and P) that we can solve. We will discuss the (P) part later in class.
III.
Facts about aggregate production
When specifying our aggregate production function, there are some empirical
relationships that the production function must display. Here are certain things we
believe production functions should display:
a.
Constant Returns to Scale. This simply means that if you double all nontechnology inputs (labor and capital), you should be able to double output. Think
about it. If you have a given level of technology and you have one plant that
produces 10 units of output, shouldn't you be able to produce an identical plant
(same amount of machines and workers) next to the original plant and produce an
additional 10 units of output? Doubling the inputs (making another plant) should
lead to a doubling of the output. We call this concept constant returns to scale.
(This is distinct from concepts such as increasing returns to scale – i.e, if you
double all inputs, you more than double all output). Much existing research
supports this idea that firms are close to constant returns to scale.
Any production function we use should be able to replicate constant returns to
scale.
b.
Elasticities and Cost Shares: Existing research has shown that the elasticity of
output with respect to changes in labor tends to be about .7 and that the elasticity
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of output with respect to changes in capital tend to be about .3. This comes from
empirical research. Elasticity is the percentage increase in Y (dependent variable)
resulting from a 1% increase in X (independent variable). That means if labor
(independent variable) increases by 1%, output (dependent variable) tends to
increase by 0.7% - holding K constant.
Additionally, we find that the share of labor costs out of total GDP is about 70%.
That means out of every $1 of GDP, 70% of that income flows to workers (as
wages). Capital costs shares are the remaining 30%. These numbers are simply
generated by taking labor income and dividing it by GDP or taking capital income
and dividing it by GDP. The flows to capital owners include dividends, profits,
interest payments, and rents.
Any production function we generate should be able to match these empirical
facts.
c.
The
There are diminishing marginal returns to labor or capital individually.
Suppose you do not double both capital and labor, but instead only double labor.
If there are diminishing returns to labor individually, you will less than double
output. Furthermore, for each additional labor added to a fixed amount of capital,
the extra amount of output that is generated gets smaller and smaller over time.
This is just common sense (and a central tenant from micro). After some point,
all production processes have diminishing returns to both capital and labor
individually. Suppose you have a room with 3 computers and 2 employees.
Adding a third employee gets you a lot of extra output. Adding the 4 th worker
will probably get you more total output, but likely not as much additional output
as adding the 3rd worker. The 100th extra employee probably will not provide any
additional value if there are only 3 computers. The more employees you add, the
less additional (extra or marginal) amount of production that you would get
holding the computers constant. Diminishing returns refers to the marginal
product of labor (the extra output you get from adding additional workers).
Marginal - in economics - refers to additional or extra. Product refers to output
(i.e., so putting them together,
marginal product means additional output).
marginal product of labor is the additional output you get from adding one more
unit of labor - holding the other inputs constant.
Any production function we posit should display diminishing marginal returns to
both labor and capital.
d.
Complementarity between A, K and N. This just means that the extra amount of
output that you get from the 4th worker (example above) will be higher if you
have 4 computers than if you had 3 computers. The higher the fixed level of
capital (or technology), the higher the marginal product of labor. Labor is more
productive if you have more machines (and vice versa).
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IV.
The production function.
Any production function that we come up with should display the above four
characteristics (a-d). There are many production functions that could satisfy these
qualities. The common one used in economics is the Cobb-Douglas (CD) production
function. It was developed in the mid-20s by Cobb and Douglas to predict farming
production. It is simple and easy to use AND it matches the above facts.
The (CD) production function:
Y = A * K a N 1-a
where Y, A, K and N are defined above and (a) is parameter that gives the relative weight
to capital and labor in the economy. This production function is flexible enough to match
the data (i.e., we can choose (a) so as to match empirical facts). Additionally, the fact
that the exponents on K and N sum to 1 (a + 1 – a = 1) ensures constant returns to scale
(it is a math thing that I will illustrate below - as an experiment, try putting in exponents
that do not sum to 1 - prove to yourself that constant returns to scale will NOT hold in
this case - you can use the proof I use below).
To match the data, we will set (a) = 0.3 (this was the empirical fact we discussed above).
As a result, our production function will look like:
Y = A * K 0.3 N 0.7
Now, let's show that this production function matches the above 4 facts (a-d).
a)
Constant Returns to Scale. If we double N and K holding A constant, Y will
double. This is an easy proof. We want to show that:
2 Y = A (2K).3 (2N).7
(all we did here is double Y, K and N - what we want to do now is see if that equation
holds. Let’s rewrite the above equation where we doubled the inputs:
2Y = A (2).3K.3(2).7N.7
2Y = A K.3N.7 2.32.7
2Y = A K.3N.72.3 + .7
2Y = A K.3N.7 2
2Y = 2Y
(just separate out the 2's)
(just re-organize the equation by
putting the term with the 2's in them
next to each other).
(add the exponents on the 2's).
(simplify the expression 2 .3 + .7 = 2 1
=2)
(realize that A K.3N.7 = Y : this is by
definition)
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The proof holds. Doubling labor and capital with this production function will cause
output to double.
b)
What is the marginal product of labor (MPN)? What is the marginal
product of capital (MPK)? From the definition above, the MPN is the
additional amount of output you get from a given increase in labor. Another way
to say this: the MPN is the change in Y that you get from the change in N. Or,
mathematically: MPN = Y/N!
This is the derivative of Y with respect to N.
To get the MPN, take the derivative of the production function with respect to N:
Y = A * K 0.3 N 0.7
so, MPN = Y/N = .7 A K.3 N -.3 = .7 A (K/N).3
Likewise, to get MPK, take the derivative of the production function with respect
to K:
MPK = Y/K = .3 A K-.7 N .7 = .3 A (N/K).7
Labor elasticity is the percentage change in output that results from a percentage
change in labor (notice the MPN is a unit change in output from a unit change in
labor).
I will use (labor) for the labor elasticity.
(labor) = % change in Y / % change in N = (Y/Y)/ (N/N) (simply rewriting, we get)
(labor) = (Y/N) (N/Y) <<Remember - to get a percentage change in a
variable, you take the change in that variable over time relative to some initial
amount - think about the inflation formula!!!)
The elasticity of labor is a function of the MPN, N and Y. Substituting in the
MPN (i.e., Y/N), we get:
(labor) = .7 A (K/N).3 N / Y = (.7 A (K).3 (N).7) / Y = .7 (remember Y = A (K).3
(N).7 - so, the Y on the bottom will cancel out with the A (K).3 (N).7 on top).
So, the (labor) = .7 which matches empirical data (this is why we choose .3 and
.7 as our exponents - because they match the data)! <<You can do the same
exercise for the capital share>>.
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c.
Diminishing MPN: This one is easy. This just says that as you hire more
workers, the extra output you get from each additional new worker will fall - or as
N increases the MPN will fall. The definition of the MPN is MPN = Y/N
= 0.7 A (K/N).3.
As you see from the equation, as N increases (it is in the denominator), the MPN
will fall! In math terms, diminishing MPN implies that the derivative of MPN
with respect to N is negative (for math people, fNN<0). The Cobb-Douglas
production function predicts a diminishing MPN that we observe empirically and
which we believe to be intuitively true.
The same analysis would hold true as to why there is diminishing marginal
productivity of capital (MPK).
d.
Complementary between K and A and the MPN: As I stated above - this just
means that the MPN is higher if A or K increases OR the MPK is higher if N or A
increase. The 4th worker to enter is more productive if there are more machines it is rather straight-forward.
This result for the Cobb-Douglas Production function is straight forward. Look at
the MPN.
MPN = Y/N = .7 A K.3 N -.3 = .7 A (K/N).3
If A increases or if K increases, the MPN will increase. This shows the
complementary relationship between A, K and the MPN (the same thing will hold
true for the MPK). In math terms, this means that the derivative of MPN with
respect to either A or K is positive (for math people, fNK >0 or fNA > 0) .
Likewise, this means that the derivative of MPK with respect to either A or N is
also positive (for math people, fKN >0 or fKA > 0) The CD production function
satisfies this criteria.
V.
Summary
We are trying to represent a production function that matches aggregate empirical
facts. We start with the facts! After we have the facts, we try to find a production
function that matches these facts. One that matches both the qualitative (constant
returns to scale) and quantitative (elasticities) facts is the Cobb-Douglas
production function. In this note, I proved that the Cobb-Douglas production
function matches these aggregate facts. I want you to be aware of these aggregate
facts and how the Cobb-Douglas satisfies these conditions. For most of this note,
I focused on the MPN (the marginal product of labor). I did that for illustrative
purposes. You should be equally familiar with the MPK (the marginal product of
capital). I am coming into lecture two assuming that you have read these notes.
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