Sometimes referred to as *variable factor proportions*, law of diminishing returns states that as equal quantities of one variable factor are increased, while other factor inputs remain constant, ceteris paribus, a point is reached beyond which the addition of one more unit of the variable factor will result in a diminishing rate of return and the marginal physical product will fall.

## Example

A common example is hiring more people on a factory floor. Given that the capital on the floor (production machines) is held constant, increasing from one employee to two is, theoretically, going to more than double production. This is increasing returns.

At some point, say, 50 employees, increasing the number of employees by two percent (from 50 to 51) would increase output by 2 percent. This is constant returns.

Look further along the production curve, say, 100 employees, floor space is getting crowded, and workers are getting in each other’s way. Increasing the number of employees by 2 percent (from 100 to 102) would increase output by less than 2 percent. This is diminishing returns.

Through each of these examples, the floor space and capital of the factor remained constant. I.e. all other inputs were constant.

## Mathematics

Signify {\displaystyle Output=O\ ,\ Input=I\ ,\ O=f(I)}

Increasing Returns: {\displaystyle 2*f(I)<f(2*I)}

Constant Returns: {\displaystyle 2*f(I)=f(2*I)}

Diminishing Returns: {\displaystyle 2*f(I)>f(2*I)}

### Link with Output Elasticity

Start from the equation for the Marginal Product: {\displaystyle {\Delta Out \over \Delta In_{1}}={{f(In_{2},In_{1}+\Delta In_{1})-f(In_{1},In_{2})} \over \Delta In_{1}}}

To demonstrate diminishing returns, two conditions are satisfied; marginal product is positive, and marginal product is decreasing.

Elasticity, a function of Input and Output, {\displaystyle \epsilon ={In \over Out}*{\delta Out \over \delta In}}, can be taken for small input changes. If the above two conditions are satisfied, then {\displaystyle 0<\epsilon <1}.^{[5]}

This works intuitively;

- If {\displaystyle {In \over Out}} is positive, since negative inputs and outputs are impossible,
- And {\displaystyle {\delta Out \over \delta In}} is positive, since a positive return for inputs is required for diminishing
*returns*

- Then {\displaystyle 0<\epsilon }

- {\displaystyle {\delta Out \over Out}} is relative change in output, {\displaystyle {\delta In \over In}} is relative change in input
- The relative change in output is smaller than the relative change in input; ~input requires increasing effort to change output~

- Then {\displaystyle {\delta Out \over Out}/{\delta In \over In}={In \over Out}*{\delta Out \over \delta In}=\epsilon <1}

## Returns and costs

There is an inverse relationship between returns of inputs and the cost of production, although other features such as input market conditions can also affect production costs. Suppose that a kilogram of seed costs one dollar, and this price does not change. Assume for simplicity that there are no fixed costs. One kilogram of seeds yields one ton of crop, so the first ton of the crop costs one dollar to produce. That is, for the first ton of output, the marginal cost as well as the average cost of the output is $1 per ton. If there are no other changes, then if the second kilogram of seeds applied to land produces only half the output of the first (showing diminishing returns), the marginal cost would equal $1 per half ton of output, or $2 per ton, and the average cost is $2 per 3/2 tons of output, or $4/3 per ton of output. Similarly, if the third kilogram of seeds yields only a quarter ton, then the marginal cost equals $1 per quarter ton or $4 per ton, and the average cost is $3 per 7/4 tons, or $12/7 per ton of output. Thus, diminishing marginal returns imply increasing marginal costs and increasing average costs.

Cost is measured in terms of opportunity cost. In this case the law also applies to societies – the opportunity cost of producing a single unit of a good generally increases as a society attempts to produce more of that good. This explains the bowed-out shape of the production possibilities frontier.

## Justification

### Ceteris Paribus

Part of the reason one input is altered *ceteris paribus*, is the idea of disposability of inputs.^{[6]} With this assumption, essentially that some inputs are above the efficient level. Meaning, they can decrease without perceivable impact on output, after the manner of excessive fertiliser on a field.

If input disposability is assumed, then increasing the principal input, while decreasing those excess inputs, could result in the same ‘diminished return’, as if the principal input was changed certeris paribus. While considered as ‘hard’ inputs, like labour and assets, diminishing returns would hold true. In the modern accounting era where inputs can be traced back to movements of financial capital, the same case may reflect constant, or increasing returns.

It is necessary to be clear of the ‘fine structure’^{[2]} of the inputs before proceeding. In this, ceteris paribus is disambiguating.

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