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

# The Marginal Rate of Substitution (MRS)

## Marginal Rate of Substitution (MRS)

The marginal rate of substitution is the rate at which one commodity can be substituted for another without affecting total satisfaction. The MRS is given by the slope of the Indifference curve. To explain the concept of MRS, let us suppose that a consumer consumes only two goods X and Y. The utility function of the consumer is given as:

U = ⨏(X,Y)
Where X and Y are substitutes

Let us assume that the consumer substitutes X for Y such that his total utility remains the same. When the consumer sacrifices some units of Y, his stock of Y decreases by ΔY and he loses a part of his total utility. His loss of utility may be expressed as;

- ΔY. MUy

On the other hand, as a result of substitution, his stock of X increases by ΔX. His gain of utility from X equals + ΔX. MUx:

The total utility remains the same only when - ΔY. MUy = ΔX. MUx

MRSxy = -( ΔY/ΔX ) =MUx/MUy

Here, ΔY/AX is simply the slope of the indifference curve, which gives the MRSxy when X is substituted for Y.

Similarly, ΔX/ΔY gives MRSyx when Y is substituted for X.

Symbolically,
MRSyx = - ( ΔX/ ΔY) = MUy/MUx

### Principle of Diminishing MRS

The marginal rate of substitution refers to the amount of one good that an individual is willing to give up for an additional unit of another good while maintaining the same level of satisfaction. For example, the marginal rate of substitution of good X for good Y (MRSxy) refers to the amount of Y that an individual is willing to exchange per unit of X by maintaining the same level of satisfaction.

The marginal rate of substitution of X for Y diminishes as more and more units of good X are substituted for Y. In other words, as a consumer has more and more units of goods X, he is willing to forgo fewer and fewer units of goods Y. The principle of diminishing MRS can be explained with the help of the following table (schedule) and diagram:

 Combination Goods-X Goods-Y MRSxy A 1 12 - B 2 8 4 C 3 5 3 D 4 3 2 E 5 2 1

In the above table, the rate of substitution of good X for good Y is shown. In the beginning, as the consumer moves from A to B, he is ready to give up 4 units of Y for the one extra unit of good X. In this process, his level of satisfaction remains the same. It follows that one unit gain in X fully compensates him for the loss of 4 units of Y. At this stage, the marginal rate of substitution of X for Y is 4.

When the consumer moves from combinations B to C, C to D, and D to E, he is willing to sacrifice 3 units, 2 units, and 1 unit of goods Y for the addition of one unit of X. Hence, the marginal rate of substitution of X for Y is 3, 2 and 1 respectively, So the table clarifies that the rate of the substitution is diminishing.

In the above diagram, starting at point A the individual is willing to give up 4 units of Y for one additional unit of X and reaches point B on IC. Thus, MRSxy = 4. Between points B and C, MRSxy = 3, between C and D, MRSxy= 2, between D and E, MRSxy = 1 Thus, MRSxy declines as the consumer moves down on the indifference curve. As a result, IC is convex to the origin.

### Why does MRS Diminish? Reasons for Diminishing Marginal Rate of Substitution

The following three factors are responsible for diminishing MRS:

l. The particular want is satiable:
The want for a particular commodity is satiable. As the consumer has more and more units of a commodity, the intensity of his want for that good goes on falling. Due to this reason, when the stock of a commodity, say X, increases with the consumer, he is willing to forgo less and jess of another commodity, say Y, for every increase in X.

In the beginning, when the consumer's stock of good Y is relatively large and his stock of good X is relatively small, the consumer's marginal significance for good Y will be low while his marginal significance for good X will be high. Therefore, the consumer will be willing to give up a larger amount of Y for a unit increase in good X.

But as the stock of good X increases the marginal significance for good X will diminish. On the other hand, as the stock of good Y decreases, the marginal significance for good Y will go up. As a result, as the individual substitute more and more of X for Y he is prepared to give up less and less of Y for a unit increase in X.

2. Goods are not a perfect substitute for each other :
The marginal rate of substitution diminishes because goods are not perfect substitutes for each other. If the two goods are perfect substitutes, the indifference curve will be a straight line with a negative slope. Since goods are not perfect substitutes, the marginal utility attached to the additional quantity of a commodity decreases faster in relation to the other commodity whose total quantity decreases.

Therefore, when the quantity of one commodity (let, X) increases, the quantity of the other (let, Y) decreases. It becomes increasingly costlier for the consumer to sacrifice more units of Y for one unit of X. Therefore when a consumer is required to sacrifice an additional unit of Y, he will require increasing units of X to maintain the level of his satisfaction. As a result, the MRS diminishes.

3. Increase in the quantity of one good does not increase the want satisfying of the other:
The principle of diminishing marginal rate of substitution will hold good only if the increase in the quantity of one good does not increase the want satisfying power of the other. If with the increase in the stock of good X, the want satisfying power of good Y increases, the greater and greater amount of good Y will be required to be given up for a unit increase in good X. As a result, the consumer satisfaction remains the same.