# The Production Function

It makes sense to think that each worker will be more productive (Y/N will be higher) if that worker has more capital equipment to work with (K/N is higher). We start with a *production function:*

????=????(????,????)

Here, K is capital, and N is (the number of) workers. We will assume Cobb-Douglas production function. Most of the results will hold for the general functional forms as well. However Cobb-Douglas will be easier to work with and it has some nice properties.

????=????^{????}????^{????}

For example, let θ = 0.25, γ = 0.75, K = 81, and N = 16. Then

????=81^{0.25}×16^{0.75}=24

The Cobb-Douglas is popular due to certain useful properties.

## Useful Property 1

Note that this function implies that if either K = 0 or N = 0 then Y = 0. We say that inputs ate indispensable for production.

## Useful Property 2

Use the product rule

This is called the **growth accounting equation**: it accounts for the growth rate in GDP in terms of the growth rates of inputs. For example:

???? = ????^{0.25} ????^{0.75}

Then our growth accounting equation becomes:

Assume

Then

So output grows at 1.25% of which 0.50% is due to the growth in K and 0.75% due to the growth in N.

## Useful Property 3

Let’s figure out what θ and γ mean. Use the product rule again:

So θ is the elasticity of output with respect to capital. It answers this question, “If K increases by one percent while L is kept constant, by how many percent will Y increase?” Similarly, assume only labor changes and solve for γ:

So θ is the share of capital in GDP (whether nominal or real) and γ is the share of labor in GDP (whether nominal or real).

If we assume that K and L are the only inputs in the country, then output needs to be exhausted:

This is called the exhaustion property! Multiply through by Y to see this from another angle:

???? × ???? + ???? × ???? = ????

So real factor income equals real GDP, as it should be (recall the three faces of GDP: production = factor income = expenditure). How cool is that? This is one reason we want the two exponents to add to one.

???? + ???? = 1

Therefore, henceforth we will assume γ = 1 – θ and write the Cob-Douglas as:

????=????^{????}????^{1−????}

## Useful Property 4

When the exponents of Cobb-Douglas add to one, we will have constant returns to scale. If you increase the scale of operations by λ, output will increase λ-fold:

(????????)^{????}(????????)^{1−????}=????^{????+(1−????)}????^{????}????^{1−????}=????????

For example, if you double the inputs, output will double. This assumption will enable us to relate (Y/N) to (K/N). This is another reason that we like the exponents to add to one.

## Useful Property 5

Okay, the wait is over. It is now time to relate Y/N to K/N. Divide Cobb-Douglas by N:

The fact that (Y/N) can be so neatly related to (K/L) is another reason we want the exponents to add to one. Define Y/N ≡ y and K/N ≡ k.

????=????^{????}

????=????^{0.25}

This is called per-worker production function. Use the product rule again to see the growth rates:

This is also a growth-accounting equation. It accounts for the growth in per-worker GDP. Using the previous numbers (and the quotient rule for k):

Δ????/???? = 0.2 5× ( 2%−1% )= + 0.25%

So GDP per worker (average labor productivity) grows at a rate of 0.25% per year and that is solely due to the growth in capital per worker. The situation is a bit depressing. If you increase capital per worker by 1%, output per worker will increase by only 0.25%. For example, suppose k increases by 3% per year, then:

Δ????/????= + 0.25 × (3%−1%) = 4% + 0.50% = 0.50%

The growth rate of GDP per worker increases from 0.25% per year to 0.50% per year, an increase of only 0.25%.

## Useful Property 6

You can also show diminishing returns to k. The most pedestrian way to show this is to give each worker additional capital and see how y increases. We see below that y increases but at a decreasing rate (Δy/Δk decreases). The following table and graph assume θ =0.25.

**Exercise:** Show that if ΔN/N increases, ΔY/Y will increase but Δy/y will decrease.

You can also prove diminishing returns directly and precisely by taking the derivative of the per-worker production function:

So an increase in k will cause an increase in y but a decrease in the rate of increase in y (the slope of production function). This property will prove critical in our discussion of the growth models below.

## Useful Property 7

The quantity Δy/Δk is the marginal product of capital. We just learned that an increase in k causes a reduction in MPK. Generally:

Funny thing is, you might complain that this is not really MPK, because MPK means the effect of an increase in K on Y given N. But the above is the effect of an increase in (K/N) on (Y/N). Are the two really the same thing?When the production function is Cobb-Douglas, then they are. To see this:

So the two results are the same.

## Recap of the main points:

- Real GDP (Y) is an index of all the final goods and services produced in an economy. It has been increasing over time. This increase is called economic growth.
- However, population has also been increasing over time. GDP per capita is more relevant for the wellbeing of the people. Actually it has also been increasing over time. So the relevant economic question is what causes GDP per capita to increase.
- We decided that GDP per capita will increase if workers in the country become more productive and produce more goods and services. In other words, if average labor productivity (Y/N = y) increases.
- But what causes y to increase? Our first hunch was that it must be capital per worker (K/N = k). If each worker gets to work with a larger amount of tools, machines, and equipment, that worker will be more productive and will produce more stuff (y↑). As a result the nation as whole will consume more stuff (Y per capita ↑).
- The relationship between k and y is captured by a per –worker Cobb Douglas production function y = kθ.
- In this function θ is the share of capital income in GDP (corporate profit, interest income, rent, etc.). For the U.S. it is about 0.25 to 0.30.
- This function exhibits diminishing returns to capital. If k increases, y will increase, but at a lower rate. So k↑ → y↑ but MPK↓.
- So now the question is what causes k to increase.

? → k↑ → y↑ → Y/POP ↑