NOTES 3: The Aggregate Production Function
Note: I want you to understand the main parts of these notes (ie, these notes are fair game for quizzes, etc). Some parts get “mathy” when I prove things later in this handout. 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. Eventually, we need to balance the demand side of the economy (what people buy : Y = C + I + G + NX) with what firms produce (Y = 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, assuming one production function still 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.
In the lecture notes for the beginning of Topic 2, I start by defining the aggregate production function. Today, I am going to take a different route. I am going to outline 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. That means if you bought a machine this year - it is investment this year. But, that machine also adds to the 'capital stock'. The capital stock is the value of all existing machines that your firm owns. Note: We measure labor and capital like we measure output - in terms of dollars. Our measure of labor, then, is the 'real' value of wages paid to workers 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
Ok, 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 non- technology 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 - ie, if you double all inputs, you more than double all output). Much existing research cannot reject 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 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. 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 4th 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 (ie, so putting them together, marginal product means additional output). The 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).
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 (ie, 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) ensures the 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. That says that if we double N and K, Y will double (holding A constant). 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. Lets rewrite the above equation where we doubled the inputs:
2Y = A (2).3K.3(2).7N.7 (just separate out the 2's)
2Y = A K.3N.7 2.32.7 (just re-organize the equation by putting the term with the 2's in them next to each other).
2Y = A K.3N.72.3 + .7 (add the exponents on the 2's).
2Y = A K.3N.7 2 (simplify the expression 2 .3 + .7 = 2 1 = 2 )
2Y = 2Y (realize that A K.3N.7 = Y : this is by definition)
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, is the MPN is the change in Y that you get from the change in N. Or, mathematically: MPN = DY/DN!
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 = DY/DN = .7 A K.3 N -.3 = .7 A (K/N).3
likewise, MPK = DY/DK = .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 h(labor) for the labor elasticity.
h(labor) = % change in Y / % change in N = (DY/Y)/ (DN/N) (simply re-writing, we get)
h(labor) = (DY/DN) (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 (ie, DY/DN), we get:
h(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 h(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>.
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 = DY/DN = .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. 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 = DY/DN = .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. Likewise, this means that the derivative of MPK with respect to either A or N is also positive. The CD production function satisfies this criteria.