The laws of diminishing marginal rate of substitution can be explained with the help of the following indifference schedule (Table 5.2) and curve (Fig. 5.5).

The marginal rate of substitution at a point on the indifference curve can be measured by its slope at that point. Consider a small movement down from point ‘A’ to point ‘B’ in indifference curve IC in Fig. 5.4.

Here, a small amount of commodity ‘Y’ say, ?Y, is replaced by an amount of commodity ‘X’, say ?X, without any loss of satisfaction. The slope of the indifference curve at point ‘B’ is therefore equal to ?Y / ?X. This also indicates the rate at which the consumer is willing to substitute commodity ‘X’ for commodity ‘Y’.

Thus,

MRSX,Y = ?Y/?X (By definition)

The number of units of commodity ‘X’ which would compensate the consumer for the loss of one unit of commodity ‘Y’ is called the marginal rate of substitution of ‘X’ for ‘Y’ (MRSX,Y ). It is the rate at which a consumer can exchange a very small amount of ‘X’ (one commodity) for a very small amount of ‘Y’ (other commodity), without affecting the total utility. The ratio ?Y / ?X indicated by the marginal rate of substitution of ‘X’ for ‘Y’ is different at different points on the indifference curve.

Hence, MRS is defined more precisely at a point than on an arc. MRSV y at a point is taken as the negative of the slope of the indifference curve (i.e., slope of the tangent to the indifference curve) at that point. Since the slope of the indifference curve is dY/dX and it is negative, so, MRS is positive. Further, MRSX,Y = – dY/dX and MRSy x = – dX/dY. Thus, these two concepts are symmetrical.

In the above table, five combinations of biscuits and tea cups are given, which provide the same satisfaction to the consumer. We may note that when the consumer has just one cup of tea, he is willing to give up 4 biscuits for an additional cup of tea.

Now, as the number of cups of tea increases, he is prepared to sacrifice fewer and fewer number of biscuits to get each additional cup of tea. This is due to the reason that more of one good and the less of another a consumer has, the units of the second become more important to consumer relative to the units of the first.

The law of diminishing marginal rate of substitution can also be shown diagrammatically. In Fig. 5.5, five combinations (‘A’, ‘B’, ‘C’, ‘D’ and ‘E’) of two goods biscuits (shown along Y-axis) and tea cups (shown along X-axis) have been depicted on the indifference curve IC. It is evident from the figure that, X1 B = X2 C = X3 D = X4 E, as every time one additional cup of tea is consumed by the consumer. Further. AX1 > BX2 > CX3 > DX4. Now.

This implies that successive units of tea substitute for lesser and lesser number of biscuits. Or, the rate of substitution of biscuits for tea cups is declining. Now, we can say that according to the law of diminishing marginal rate of substitution, a consumer is prepared to part with lesser and lesser units of one commodity for each additional unit gain of other commodity.

Since each unit increase in one commodity (say, ‘X’) causes the number of units of other commodity (say, ‘Y’) to decrease by lesser and lesser amount, the numerical value of the slope goes on declining. Therefore, indifference curves are expected to be convex to the origin.