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
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
with the production function
where a is any fraction, and you will have a
production function which shows the curvature characteristic of
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 |
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
and the change in labor is 100 at each level.
The graph of the production function is given below:
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
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
Fill in a similiar table, only using the production function
What can you say about returns to scale and the law of diminishing returns in this case?