Production functions

A production function shows the relationship between inputs of capital and labor and other factors and the outputs of goods and services.

In macroeconomics, the output of interest is Gross Domestic Product or GDP

The simplest possible production function is a linear production function with labor alone as an input.

For example, if one worker can produce 500 pizzas in a day (or other given time period) the production function would be

Q = 500 L

It would graph as a straight line: one worker would produce 500 pizzas, two workers would produce 1000, and so on.

A linear production function is sometimes a useful, if very rough approximation of a production process -- for example, if we know that wages are $ 1000 a day , we know that the price of a pizza must be at least $ 2 to cover the labor cost of production.

We also note that the 500 represents labor productivity , and if the number increases to 600, it means that labor productivity has increased to 600 pizzas a day.

However, more realistic production functions must incorporate diminishing returns to labor or to any other single factor of production. This may be done simply enough: replace the production function

Q = 500 L

with the production function

Q = 500 L a

where a is any fraction, and you will have a production function which shows the curvature characteristic of diminishing returns.
For example, if we choose a = 0.5, so that we are taking the square root of L, we could compute the following relationships:

Labor Output Marginal
of Labor
100 5,000 50
200 7,071 20.71
300 8,660 15.89
400 10,000 13.40
500 11,180 11.80

Note that the final column, marginal product of labor, shows how much additional output is due to the addition of one more worker, that is, it is given by

Change in output / change in labor

and the change in labor is 100 at each level.

The graph of the production function is given below:

Multifactor production functions

One further step toward reality is to incorporate capital as well as labor as a factor of production. The generalization to a multifactor production function is straightforward:

Q = 50 K a L b

Note that if we had 100 units of capital, and a and b both were equal to 0.5, this production function would be exactly the same as the previous one. Substituting 100 in for K, we would have

Q = 50 (100) 0. 5 L .5 = 500 L .5

Both capital and labor show diminishing returns to increasing any single factor of production, but they may show (and do in this example) constant returns to scale . That is, if you double both capital and labor you will double output. This does not contradict the "law of diminishing returns, " which applies only to increasing a single factor of production.

When we increase all factors of production, we speak of a change in the scale of operations, and we may have:

increasing returns to scale, if the exponents a and b on capital and labor add up to more than one

constant returns to scale, if the exponents a and b add up to exactly one

diminshing returns to scale, if the exponents a and b add up to less than one

As an exercise, fill in the following table, using the production function

Q = 100 K 0.5 L 0.5

100 200 300 400 500
100 ---- ---- ---- ---- ----
200 ---- ---- ---- ---- ----
300 ---- ---- ---- ---- ----
400 ---- ---- ---- ---- ----
500 ---- ---- ---- ---- ----
Note especially what happens along the diagonal as you double both inputs from 100 to 200 and 200 to 400.

Fill in a similiar table, only using the production function

Q = 100 K 0.3 L 0.8

What can you say about returns to scale and the law of diminishing returns in this case?

Answers to the questions
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