Marginal rate of substitution

Simple mathematical analysis

Assume the consumer utility function is defined by ${\displaystyle U(x,y)}$, where U is consumer utility, x and y are goods. Then the marginal rate of substitution can be computed via partial differentiation, as follows.

Also, note that:

${\displaystyle M{U}_x=\mathrm{\partial }U/\mathrm{\partial }x}$

${\displaystyle M{U}_y=\mathrm{\partial }U/\mathrm{\partial }y}$

where ${\displaystyle M{U}_x}$ is the marginal utility with respect to good x and ${\displaystyle M{U}_y}$ is the marginal utility with respect to good y.

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

${\displaystyle dU=(\mathrm{\partial }U/\mathrm{\partial }x)dx+(\mathrm{\partial }U/\mathrm{\partial }y)dy}$, or substituting from above,

${\displaystyle dU=M{U}_{x}dx+M{U}_{y}dy}$, or, without loss of generality, the total derivative of the utility function with respect to good x,

${\displaystyle \frac{dU}{dx}=M{U}_x\frac{dx}{dx}+M{U}_y\frac{dy}{dx}}$, that is,

${\displaystyle \frac{dU}{dx}=M{U}_x+M{U}_y\frac{dy}{dx}}$.

Through any point on the indifference curve, dU/dx = 0, because U = c, where c is a constant. It follows from the above equation that:

${\displaystyle 0=M{U}_x+M{U}_y\frac{dy}{dx}}$, or rearranging

${\displaystyle -\frac{dy}{dx}=\frac{M{U}_x}{M{U}_y}}$

The marginal rate of substitution is defined as the absolute value of the slope of the indifference curve at whichever commodity bundle quantities are of interest. That turns out to equal the ratio of the marginal utilities:

${\displaystyle MR{S}_{xy}=M{U}_x/M{U}_y}$.

When consumers maximize utility with respect to a budget constraint, the indifference curve is tangent to the budget line, therefore, with m representing slope:

${\displaystyle m_{\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{f}}=m_{\mathrm{b}\mathrm{u}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{t}}}$

${\displaystyle -(MR{S}_{xy})=-(P_x/P_y)}$

${\displaystyle MR{S}_{xy}=P_x/P_y}$

Therefore, when the consumer is choosing his utility maximized market basket on his budget line,

${\displaystyle M{U}_x/M{U}_y=P_x/P_y}$

${\displaystyle M{U}_x/P_x=M{U}_y/P_y}$

This important result tells us that utility is maximized when the consumer's budget is allocated so that the marginal utility per unit of money spent is equal for each good. If this equality did not hold, the consumer could increase his/her utility by cutting spending on the good with lower marginal utility per unit of money and increase spending on the other good. To decrease the marginal rate of substitution, the consumer must buy more of the good for which he/she wishes the marginal utility to fall for (due to the law of diminishing marginal utility).

Diminishing Marginal rate of Substitution

An important principle of economic theory is that marginal rate of substitution of X for Y diminishes as more and more of good X is substituted for good Y. In other words, as the consumer has more and more of good X, he is prepared to forego less and less of good Y.

It means that as the consumer's stock of X increases and his stock of Y decreases, he is willing to forego less and less of Y for a given increment in X. In other words, the marginal rate of substitution of X for Y falls as the consumer has more of X and less of Y. That the marginal rate of substitution of X for Y diminishes can also be known from drawing tangents at different points on an indifference curve.

Using MRS to determine Convexity

When analyzing the utility function of consumer's in terms of determining if they are convex or not. For the horizon of two goods we can apply a quick derivative test (take the derivative of MRS) to determine if our consumer's preferences are convex. If the derivative of MRS is negative the utility curve would be concave down meaning that it has a maximum and then decreases on either side of the maximum. This utility curve may have an appearance similar to that of a lower case n. If the derivative of MRS is equal to 0 the utility curve would be linear, the slope would stay constant throughout the utility curve. If the derivative of MRS is positive the utility curve would be convex up meaning that it has a minimum and then increases on either side of the minimum. This utility curve may have an appearance similar to that of a u. These statements are shown mathematically below.

${\displaystyle \frac{dMR{S}_{xy}}{dx}<0\text{ Non Convexity of Utility Function}}$

${\displaystyle \frac{dMR{S}_{xy}}{dx}=0\text{ Weak Convexity of Utility Function}}$

${\displaystyle \frac{dMR{S}_{xy}}{dx}>0\text{ Strict Convexity of Utility Function}}$

For more than two variables, the use of the Hessian matrix is required.