A

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
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
Product 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

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

LABOR CAPITAL | 100 | 200 | 300 | 400 | 500 |

100 | ---- | ---- | ---- | ---- | ---- |

200 | ---- | ---- | ---- | ---- | ---- |

300 | ---- | ---- | ---- | ---- | ---- |

400 | ---- | ---- | ---- | ---- | ---- |

500 | ---- | ---- | ---- | ---- | ---- |

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?

Answers to the questions

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