Deriving a per capita production function from a general economy production function.

This is an important process for many macroeconomic models, but is crucial for intermediate macroeconomics in the derivation of the Solow model.  Most of these models (including the Solow model) begin with an equation that describes the production function for the economy:

Y=f(K,L)

or

Y=K^aL^1-a

Here a represents the returns to the factor input, and it is traditionally

denoted as a Cobb-Douglas function which multiplies the two inputs together which are then raised to a power.  In the above example, the sum of the two exponents is one, which represents a constant returns to scale production function.  If the sum were greater than one, then it would be an increasing returns to scale production function, and a decreasing returns to scale production function if the sum were less than one.  It is also possible to use numbers in the place of variables, for example both:

Y=K^1/2L^1/2

And

Y=K^1/3L^2/3

Are common examples you will see for production functions.  Now, in order to get these production functions into per capita values, we have to divide the equations by labor, or L.  This sounds very simple, and it is, but the algebra involved may be confusing if you haven’t seen it before. 

First begin with your production function, say:

Y=K^aL^1-a

And divide both sides by labor, or L:

Y/L=(K^aL^1-a)/L

The next step is the confusing one.  Begin with the L variables on the right hand side.  Since we know that 1/L, is equal to L^-1, we can rewrite this term as:

L^1-a-1   or  L^-a

This then gives us:

Y/L=K^aL^-a   or    Y/L=(K/L)^a

This is because K/L is equal to k (or little K, the per capita amount of capital) and we simply carry along the a term instead of a one.  We can now rewrite the above equation, plugging into our little k and little y where appropriate:

y=k^a

And there you go, we have shown the algebra behind getting the per capita production function.  Below is another example using numbers.  Begin with:

Y=K^1/3L^2/3

And divide both sides by labor:

Y/L= (K^1/3L^2/3)/L

Since 2/3-1=-1/3, we can rewrite the above equation as:

Y/L= (K^1/3L^-1/3)

Or (by moving labor on the right hand side to the denominator and removing the minus sign)

Y/L= (K^1/3 /L^1/3)

Now we can factor out the 1/3 and plug in our per capita variables to get:

y=k^1/3

And we end up with our per capita production function.

Remember: It will only be the case that the per capita production function ends up with the exponent on the per capita capital term being equal to what it was in the general production function, if you have a constant returns to scale model, so be careful.