The Marginal Rate of Substitution (MRS) | Indifference Curve | Microeconomics

The slope of an indifference curve at a particular point is known as the marginal rate of substitution (MRS). It measures the rate at which the consumer is just willing to substitute one commodity for the other.

Let us suppose we take a little of good 1, ∆x₁, away from the consumer. Then we give him a little of good 2, ∆x₂ which is just sufficient to enable him to stay on the same indifference curve, so that he is just compensated in welfare terms, i.e., he is neither better-off nor worse off as a result of the change, i.e., after the substitution of x₂ for x₁.

The ratio ∆x₂/∆x₁ is known as the MRS. It is known as the desired rate of commodity substitution, i.e., it is the rate at which the consumer is willing to substitute x₂ for x₁ and vice-versa so that he is just as well-off after this substitution as he was before.

If we think of ∆x₁ as being a ‘very small’ or a ‘marginal’ change then the rate ∆x₂/∆x₁ measures the MRS of good 2 for good 1. As we make ∆x₁ smaller and smaller ∆x₂/∆x₁ approaches the slope of the indifference curve as shown in the Fig. 4.13.

In expressing the ratio ∆x₂/∆x₁ we treat both the numerator and the denominator as being small numbers as describing marginal changes from the original consumption bundle (x₁, x₂). The MRS, which is the slope of an indifference curve, is the rate of which the consumer is just willing to substitute a little more of x₂ for a little less of x₁.

The MRS is a negative number because of monotonic preferences which implies that indifference curves must have a negative slope. The main reason for this is that the whole indifferences curve approach is based in the law of substitution which suggests that the consumption of one commodity (x₁) is always at the expense of the other (x₂).

The MRS as the Exchange Ratio:

Let us suppose the consumer is given an opportunity to exchange x₁ for x₂ or x₂ for x₁ while he is currently consuming some bundle (x₁, x₂) at a “rate of exchange” of R. We assume that his preferences are monotonic and convex.

In this case, if the consumer gives up ∆x₁ units of x₁ he can get R ∆x₁ units of x₂ in exchange. Alternatively, if he gives up ∆x₂ units of ∆x₂ he can get ∆x₂/R units of x₁. This means that the consumer is getting an opportunity to move to any point along a line with slope-R which passes through (x₁, x₂) as shown in Fig. 4.14.

In this case the consumer is allowed to trade the good at an exchange rate R, which implies that he can move along a line with slope – R.

If the consumer moves up and to the left from (x₁, x₂) he is exchanging x₁ for x₂ and if he moves down and to the right he exchanges x₂ for x₁. For each type of movement, the exchange rate is E. Since exchange always involves sacrifice or trade-off, i.e., giving up one good in exchange for other, the exchange rate E corresponds to the slope of – R.

For the consumer to buy the bundle (x₁, x₂) the budget line whose slope is the ratio of the two prices p₁ and p₂ has to be tangent to the highest attainable indifference curve. At the point of tangency the slope of the exchange line, -R, must be equal to the slope of the indifference curve at (x₁, x₂) or the MRS, at which the consumer is just on the margin of trading or not trading.

At any exchange rate other than the MRS, the consumer would want to trade one good (x₂) for another (x₁). But if the price ratio equals the MRS, the consumer is in equilibrium. At any other price ratio, the budget line would cut the indifference curve and thus enable the consumer to move to a preferred point on a higher indifference curve.

This point may be explained a little more. Any time the budget line crosses the indifference curve, there must lie some points above the indifference curve and will thus be preferable to (x₁, x₂).

Another Interpretation of the MRS:

The MRS is actually the desired rate of commodity substitution, i.e., the rate at which the consumer is willing to substitute one good for the other while staying on the same indifference curve. It shows how much of good to the consumer is willing to pay for one extra unit of good one, i.e., MRS is the demand price of x₁ in terms of x₂.

So the slope of the indifference curve measures the marginal willingness to pay. This interpretation of MRS makes enormous good sense if x₂ represents money implying the consumption of “all other goods”. Thus if x₂ is money, then it can be spent on all other goods.

In this sense the MRS of x₂ for x₁ measures how many rupees the consumer is just willing to sacrifice on other goods in order to consume one extra unit of x₁. Differently put, the MRS measures the marginal willingness to give up rupees in order to consume a little more of x₁. But giving up these rupees is just like paying rupees in order to consume a little more of x₁.

It may be noted that the MRS just measures the amount of x₂ that the consumer is willing to pay for a marginal amount of consumption of x₁. It depends on the consumer’s preferences. How much he has actually to pay for some given amount of extra consumption will depend on the market price of the good.

In this context we may note a related point. A consumer is not ready to pay the same price for a large change of consumption as he is for a small (marginal) change. At the end, how much the consumer actually ends up paying for a good depends on his preferences for that good and the prices that he faces.

Behaviour of the MRS:

We often make use of the concept of MRS to describe the shape of indifference curves. In fact, the type of indifference curves we get while describing choices faced by consumers depends on the behaviour of the MRS. For example, if two commodities are perfect substitutes, the MRS is -1 throughout. In case of neutral goods, the MRS is infinite throughout.

If two goods are perfect complements, the MRS is either zero or infinite and nothing in between.

An indifferences curve, as we have already noted, is downward sloping due to the assumption of monotonicity of preferences. In other words, indifferences curves must have a negative slope because the consumer has to reduce the consumption of one good in order to get more of another. This is the essence of the law of substitution. If this law is violated the indifference curve approach breaks down.

Convexity:

However, the assumption of monotonicity cannot explain the convexity of indifference curves. For indifference curves to be strictly convex, the MRS — the slope of the indifference curve — has to diminish (in absolute value) as the consumer gets more and more x₁. Thus indifference curves are convex to the origin due to diminishing MRS.

This simply means as the consumer gets more and more of x₁ its marginal significance to him falls, i.e., the amount of x₁ that he is ready to sacrifice for an additional unit of x₂ increases as the amount of x₁ increases. Interpreted in this way, convexity of indifference curves is a very natural occurrence.

It says that the more the consumer gets of x₁, the more he is willing to sacrifice some of it in order to get x₂. However, this assumption is not universally applicable. In case of some pairs of goods this assumption might not hold. Much depends on the relation between the goods as also on the consumer’s tastes and preferences.