Marginal rate of technical substitution

Not to be confused with Marginal rate of substitution.

In microeconomic theory, the marginal rate of technical substitution (MRTS)—or technical rate of substitution (TRS)—is the amount by which the quantity of one input has to be reduced (${\displaystyle -\Delta x_{2}}$) when one extra unit of another input is used (${\displaystyle \Delta x_{1}=1}$), so that output remains constant (${\displaystyle y={\bar {y}}}$).

An isoquant map where Q3 > Q2 > Q1.At any point on any isoquant, the marginal rate of technical substitution is the absolute value of the slope of the isoquant at that point.

${\displaystyle MRTS(x_{1},x_{2})={\frac {-\Delta x_{2}}{\Delta x_{1}}}}$

It can be shown that ${\displaystyle MRTS(x_{1},x_{2})={\frac {MP_{1}}{MP_{2}}}}$, where ${\displaystyle MP_{1}}$ and ${\displaystyle MP_{2}}$ are the marginal products of input 1 and input 2, respectively.

Proof

Let ${\displaystyle Y}$ be our production function.

By taking the total differential of the production function, we obtain the following results:

${\displaystyle \ dY=(\partial Y/\partial x_{1})dx_{1}+(\partial Y/\partial x_{2})dx_{2}}$, or substituting from above,

${\displaystyle \ dY=MP_{1}dx_{1}+MP_{2}dx_{2}}$, or, without loss of generality, the total derivative of the utility function with respect to good x,

${\displaystyle {\frac {dY}{dx_{1}}}=MP_{1}{\frac {dx_{1}}{dx_{1}}}+MP_{2}{\frac {dx_{2}}{dx_{1}}}}$, that is,

${\displaystyle {\frac {dY}{dx_{1}}}=MP_{1}+MP_{2}{\frac {dx_{2}}{dx_{1}}}}$.

Through any point on the indifference curve, ${\displaystyle dY/dx_{1}=0}$, because ${\displaystyle Y=c}$, where ${\displaystyle c}$ is a constant. It follows from the above equation that:

${\displaystyle 0=MP_{1}+MP_{2}{\frac {dx_{2}}{dx_{1}}}}$, or rearranging

${\displaystyle -MP_{1}=MP_{2}{\frac {dx_{2}}{dx_{1}}}}$

${\displaystyle {\frac {MP_{1}}{MP_{2}}}={\frac {-dx_{2}}{dx_{1}}}}$

Along an isoquant, the MRTS shows the rate at which one input (e.g. capital or labor) may be substituted for another, while maintaining the same level of output. Thus the MRTS is the absolute value of the slope of an isoquant at the point in question.

When relative input usages are optimal, the marginal rate of technical substitution is equal to the relative unit costs of the inputs, and the slope of the isoquant at the chosen point equals the slope of the isocost curve (see conditional factor demands). It is the rate at which one input is substituted for another to maintain the same level of output.

See also

• Marginal rate of substitution (the same concept on consumption side)
• Marginal rate of transformation (slope of the production-possibility frontier)

References

• Mas-Colell, Andreu; Whinston, Michael; Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0-19-507340-1.