# Marginal product of labor

The marginal product of labor (Engl. Marginal product of labor or MPL ) deals with the change in output variation of the input labor in the micro- and macroeconomics . Under marginal product (also marginal) refers to the increase in output with increasing input of a factor by one unit. One such input ( production factor), the increase of which can lead to such an increase in the output volume , is the labor factor. The marginal product of labor is consequently calculated by increasing the input factor labor - at which all other production factors are kept constant - to the point at which an additional unit of labor no longer leads to an increase in the output quantity. This work unit can be, for example, an additional worker or an additional hour worked.

## Explanation and derivation

The productivity indicator is always an indicator of performance and is calculated from the quotient of output and input.

${\ displaystyle \ mathrm {Productivity {\ ddot {a}} t} = {\ frac {\ text {Output}} {\ text {Input}}}}$

Labor productivity therefore indicates the relationship between output and the labor input required for it. The marginal productivity of the labor factor is therefore the change in output when the units of work used vary. The marginal product of labor thus describes the contribution of the labor factor in the production process.

From a mathematical point of view, the marginal product of a production factor is always the first partial derivative of the respective production function according to this factor.

For the marginal productivity of labor we get:

${\ displaystyle MPL: = {\ frac {\ partial Y} {\ partial L}}}$

The multiplied with the price marginal product ( ) is called also limit product or value marginal product of labor. This term indicates how much the last worker hired contributes to sales. ${\ displaystyle {\ tfrac {\ partial Y} {\ partial L}} \ cdot p}$

## Examples

### Production function under income law

This probably oldest production function is based on observations in agriculture and was formulated by Turgot (1727–1781) as the law of decreasing land yield (also yield law , picture). The s-shaped curve is characteristic of this function. Up to the maximum there is a positive but always decreasing marginal yield. I.e. here: The marginal product of labor becomes smaller with every additional unit of work. If the maximum is exceeded, the marginal yield or marginal product even turns negative. The additional use of work is then no longer beneficial to the production process, but harmful. The old adage "Many cooks spoil the broth" sums this up.

### Numerical example for the income law

Here production is to be assumed in which only the production factor labor ( ) is variable. All other factors of production are not changed. Assume a constant amount of capital ( ) of 10. This shows how the amount of output increases (if at all) when the input of the factor labor increases. ${\ displaystyle L}$${\ displaystyle K}$${\ displaystyle Q}$

Example:

job capital output Marginal product
0 10 0 /
1 10 10 10
2 10 30th 20th
3 10 60 30th
4th 10 80 20th
5 10 95 15th
6th 10 95 0
7th 10 90 (-) 5

As can be seen from the table, the marginal product of labor initially increases with every additional labor input. It reaches its maximum of 30 with an additional labor input of 3. Then it decreases again and can even decrease again with increasing input amount. If you z. For example, assuming additional workers hired, one can conclude that the first newly hired workers are more beneficial than the last hired.

This fact can be illustrated very nicely: Five workers can work better on an assembly line than two workers, but 10 workers can get in each other's way.

### Cobb-Douglas production function

Another well-known production function is the Cobb-Douglas production function , which was developed in 1928. Also called CD production function for short.

${\ displaystyle Y = c \ cdot (K ^ {a} \ cdot L ^ {b})}$.
• ${\ displaystyle Y}$: Production amount
• ${\ displaystyle c}$: Factor. If it is not constant, but increases over time, then technical progress can be mapped. As a factor before the entire production function, as here, ( = time) represents Hicks -neutral technical progress.${\ displaystyle c}$${\ displaystyle c (t)}$${\ displaystyle t}$
• ${\ displaystyle K}$: Capital stock
• ${\ displaystyle L}$: Labor input

where and , ${\ displaystyle a> 0}$${\ displaystyle 0

so z. B .:${\ displaystyle Y = F (K, L) = 0 {,} 5 \ cdot K ^ {0 {,} 4} \ cdot L ^ {0 {,} 6}}$

The most important difference to the income law function is that the CD function has no maximum. Consequently, every additional unit of a production factor leads to an increase in the output rate.

With this production function there is a higher yield with every additional labor. However, the amount of the increase decreases.

Since the marginal product of labor is the first partial derivative of the production function according to the labor factor, we get:

${\ displaystyle {\ frac {\ partial Y} {\ partial L}} = 0 {,} 3 \ cdot K ^ {0 {,} 4} \ cdot L ^ {- 0 {,} 4}}$

## Application and meaning

The marginal product of labor is an important reference point for various decisions associated with the labor factor.

### Marginal productivity

The marginal productivity is decisive for the marginal product of a factor. The marginal value product (WGP) is calculated from the product of this marginal productivity and the price of the output.

${\ displaystyle {\ text {WGP}} = \ mathrm {Marginal productivity {\ ddot {a}} t} \ cdot {\ text {Price}}}$

This results in the marginal value product of labor:

${\ displaystyle {\ text {WGP}} = {\ frac {\ partial Y} {\ partial L}} \ cdot {\ frac {w} {p}}}$

The marginal value product plays a major role because it is usually used to determine the market price of a factor. So here is the wage rate for human labor. With an optimal remuneration, the wage should correspond to this WGP.

It can be concluded from this that the marginal product of labor also plays a major role in collective bargaining.

### Labor demand

Given a given production function and a given real wage (quotient of wages and prices ), the labor demand of a polypolistic company can be calculated. A company would continue to hire new workers until the next additional employee would no longer generate a profit. This is the case when the marginal product of labor has fallen so far that the additional income just corresponds to the wage rate.

## literature

• Dirk Diedrichs, Marco Ehmer, Nikolaus Rollwage: Microeconomics. 3rd edition, reprint. WRW-Verlag Rollwage, Cologne 2005, ISBN 3-927250-71-6 .
• Robert S. Pindyck, David L. Rubinfeld: Microeconomics. 4th edition. R. Oldenbourg Verlag, Munich et al. 1998, ISBN 3-486-22358-5 .
• Robert S. Pindyck, David L. Rubinfeld: Microeconomics. 6th edition. Pearson Studium, Munich et al. 2005, ISBN 3-8273-7164-3 .

## Footnotes

1. a b c d R. S. Pindyck, DL Rubinfeld: Microeconomics. 4th edition. 1998.
2. ^ A b D. Diedrichs, M. Ehmer, N. Rollwage: Microeconomics. 3. Edition. 2005.
3. ^ N. Gregory Mankiw : Macroeconomics. Translated from American English by Klaus Dieter John. 5th, revised edition. Schäffer-Poeschel, Stuttgart 2003, ISBN 3-7910-2026-9 , p. 61.