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Rate at which a consumer can replace one good with another while maintaining the same utility

In economics, the marginal rate of substitution (MRS) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels (assuming no externalities), marginal rates of substitution are identical. The marginal rate of substitution is one of the three factors from marginal productivity, the others being marginal rates of transformation and marginal productivity of a factor.

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.

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