Marginal rate of technical substitution

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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 ($-\backslash Delta\; x\_2$) when one extra unit of another input is used ($\backslash Delta\; x\_1\; =\; 1$), so that output remains constant ($y\; =\; \backslash bar\{y\}$).

$MRTS(x\_1,x\_2)\; =-\backslash frac\{\backslash Delta\; x\_2\}\{\backslash Delta\; x\_1\}\; =\; \backslash frac\{MP\_1\}\{MP\_2\}$

where $MP\_1$ and $MP\_2$ are the marginal products of input 1 and input 2, respectively.

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.