Marginal Rate of substitution

In economy, the marginal rate of substitution (TMS) measurement the variation of the consumed quantity of a good Y which is necessary, along a Courbe of indifference, to compensate for an infinitesimal variation of the consumed quantity of a good X. the marginal rate of substitution calculates the way in which one substitutes for the margin a product by another. If the marginal rate of substitution remains identical, the goods are perfectly substitutable (simplified example of oil and natural gas). If one wants a little bit more of the product there (in ordinate), it is necessary to give up much product X (in X-coordinate).

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Calculation

That is to say a function of utility U

$\backslash \; U=F\; (X,\; there)$

Where U is the utility of the consumer, X and there the consumed quantities of the goods and F the function of utility.

That is to say:

$\backslash \; Um\_\; \{X\}\; =dU/dx$

$\backslash \; Um\_\; \{there\}\; =dU/dy$

By differentiating the equation from the function of utility, one obtains the following result:

$\backslash \; dU=F\; (X)\; dx\; +\; F\; (there)\; Dy$

$\backslash \; dU=\; (dU/dx)\; \backslash \; Delta\; X\; +\; (dU/dy)\; \backslash \; Delta\; there$

$\backslash \; dU=Um\_\; \{X\}\; \backslash \; Delta\; X\; +\; Um\_\; \{there\}\; \backslash \; Delta\; there$

Like of the = 0 for all curves of indifferences (because U   =  C , with C a constant), it follows that:

$\backslash \; F\; (X)\; dx\; +\; F\; (there)\; dy=0$

$\backslash \; \backslash \; Rightarrow\; -\; (dy/dx)\; =F\; (X)\; /F\; (there)$

and

$\backslash \; Um\_\; \{X\}\; \backslash \; Delta\; X\; +\; Um\_\; \{there\}\; \backslash \; Delta\; y=0$

$\backslash \; \backslash \; Rightarrow\; -\; (\backslash \; Delta\; there\; \backslash \; Delta\; X)\; =Um\_\; \{X\}\; /Um\_\; \{there\}$

Where $F\; (X)$, or $\backslash \; frac\; \{of\; the\}\; \{dx\}$, represents the marginal utility of the good $x\; (Um\; \{X\})$ and $F\; (there)$, or $\backslash \; frac\; \{of\; the\}\; \{Dy\}$, represents the marginal utility of the good $y\; (Um\; \{there\})$. Moreover, $-\; \backslash \; frac\; \{Dy\}\; \{dx\}\; =TMS\_\; \{2/1\}$, therefore $TMS\_\; \{2/1\}$ equalizes less the slope of the curve of indifference. From where:

$\backslash \; TMS\_\{y/x\}=Um\_x/Um\_y$

Reference

• Microeconomics (3rd edition) by Pindyck, and Rubinfeld (1995)

See too

• Curve of indifference
• Theory of the consumer
• Micro-economics

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