Per capita production functions, math and graphs

Most of the time in economics’ classes you are given the production function and asked to convert it into a per capita production function for further manipulation.  This article is going to go over the economics of production functions and per capita production functions with 5 different examples including graphs showing what the functions look like.  Check out this other post for information on the math behind getting a production function to a per capita or per person production function, using a generic form and some other specific examples. 

Here is the actual question being looked at:

Write each production function below in terms of output per person y= Y/L and capital per person k=K/L. Show what these per person versions look like in a graph with k on the horizontal axis and y on the vertical axis. Assume that A (technology) is some fixed positive number.

(a) Y=K^(1/3)*L^(2/3) and Y= K^(3/4)*L^(1/4) (on the same graph)

To get these equations into per capita production function we need to divide both sides by L.  Again, check out this post for the math behind per capita production function derivation.
The first one is y = k^(1/3) and the second is y=k^(3/4).  

These functions result in diminishing returns to capital per capita with respect to output.  This means that the slope is steep at first, and then flattens out.  The intuition behind this is that the first few units of capital are very useful, but as each person gets more and more capital they are able to get less and less out of it.

 For example, imagine someone digging holes in the ground.  The amount of holes per person is y, and the amount of shovels per person is k.  If the person has 0 shovels, they can dig 0 holes.  If they get 1 shovel, they can dig 5 holes per hour.  If they have 2 shovels, they can dig 8 holes per hour (meaning the second shovel adds 3 holes per hour).  This may be the result of special shovels, like a big flat one, and a short sharp one (for different types of dirt).  However, as you get more and more shovels the return to this increase in capital decreases.  How much would you benefit from having 20 different types of shovels??? Not much…

(b) Y=K

Dividing both sides by L leaves us with y = k, a 45 degree line shown on a graph.  This means that capital has constant returns and does not diminish.

(c) Y=K+AL

Divide both sides by L, this gives us y = k + A, again it will be a 45 degree line, but the intercept on the Y axis will be at A.

(d) Y = K-AL

This equation doesn’t really make sense from an economic perspective, but again we can divide both sides by L to get:

y = k – A, which means that the better technology we have, the less we will be able to produce which doesn’t make sense.  This will again result in a 45 degree line, and the intercept on the Y axis will be at –A.

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