In this article we will discuss about the concept of marginal rate of substitution, explained with the help of suitable diagram and examples.

#### The Marginal Rate of Substitution (MRS):

Before establishing the four properties of ICs, first elaborate the idea of MRS. Marginal rate of substitution of good X for good Y (MRSX, y) at any point in the commodity space, is defined to be the quantity of good Y that the consumer is willing to forego for getting an additional (or the marginal) unit of good X, his level of utility remaining the same.

For example, if the consumer’s level of utility remains unaffected when at any point in the commodity space, he foregoes 3 units of good Y for getting an additional unit of good X, then his MRSX Y would be equal to 3 at the said point.

It is very important to note here that MRS is defined at some point in the commodity space. Since any point in this space is also a point on some IC, it may be said that MRS is defined at a point on an IC.

#### Geometrical Interpretation of MRSX,Y:

From the definition of MRSX Y, it is clear that the substitution between the goods takes place provided the consumer’s level of utility remains unaffected, i.e., here as a result of substitution between the goods, the consumer is moving from one point in the commodity space to another, along one of his ICs.

Suppose now that in Fig. 6.3(a), the consumer moves from point A(x1, y1) to a very close point B(x2, y2) along any one of his ICs, and hence substitutes x2 – x1 of good X for y1 – y2 of good Y. Therefore, by definition, MRSX,Y at the point A on the IC would be

Therefore, MRS at any point on an IC is the numerical slope of the IC at that point can be obtained.

#### The MRS in Mathematical Terms:

Suppose that the utility function of the consumer is given by (6.1). Then the MRS of the good Q) for the good Q2 may be obtained in mathematical terms in the following way.

The total differential of the utility function (6.1) is:

dU = f1dq1 + f2dq2 …..(6.3)

where f1 and f2 are the partial derivatives, or, the rates of change of U w.r.t. q1 and q2, respectively, q2 and q1 remaining constant. (6.3) gives the total change in utility, (dU), is (approximately) equal to change in q1 i.e., dq1, multiplied by f1plus the change in q2, i.e., dq2, multiplied by f2.

Here remember that in the cardinal analysis f1 and f2 are defined as the marginal utilities of the goods Q1 and Q2. While it may retain this definition in the present ordinal analysis, do not forget that the partial derivative of an ordinal utility function cannot have any cardinal significance although its sign has an ordinal meaning.