Marginal Rate of Substitution

We now introduce the concept of the marginal rate of substitution of meals for films.

The marginal rate of substitution of meals for films is the quantity of films the consumer must sacrifice to increase the quantity of meals by one unit without changing total utility.

Since consumers prefer more to less, an additional meal tends to increase utility. To hold utility constant, when one meal is added the consumer must simultaneously sacrifice some quantity of the other good (films). The marginal rate of substitution tells us how many films the consumer could exchange for an additional meal without changing total utility. The marginal rate of substitution must be a negative number: it is the ratio of a negative change in the quantity of films to the positive change in the quantity of meals when utility remains unaltered.

Suppose the student begins with a bundle comprising 5 films and no meals. Having already seen 4 films that week, the student probably does not enjoy the fifth film very much. With no meals, the student is very hungry. The utility of this bundle is low: being so hungry, the student cannot really enjoy the films anyway. To obtain the same amount of utility the student could give up a lot of films in exchange for a little food. The marginal rate of substitution is a large negative number and we say that it is high.

Suppose instead that the student is consuming a large number of meals but seeing few films. To obtain the same amount of utility, the student will be reluctant to sacrifice much cinema attendance to gain yet another meal. The marginal rate of substitution is a small negative number and we say that it is low. When the ratio of films to meals is already high the marginal rate of substitution of meals for films is high. It makes sense to sacrifice abundant films for scarce meals. Conversely, when the ratio of films to meals is already low the marginal rate of substitution of meals for films is low. It does not make sense to sacrifice scarce films for yet more meals.

Economists believe that this common-sense reasoning about tastes or preferences is very robust. It will hold in a wide range of circumstances. Indeed, it is sufficiently plausible that it can become a general principle, the third assumption we need to make about consumer tastes. It is called the assumption of a diminishing marginal rate of substitution.

Consumer tastes exhibit a diminishing marginal rate of substitution when, to hold utility constant, diminishing quantities of one good must be sacrificed to obtain successive equal increases in the quantity of the other good.

For example, our student might be indifferent between, that is, be equally happy with, bundle X = (6 films, 0 meals), bundle Y = (3 films, 1 meal), and bundle Z = (2 films, 2 meals). Beginning from bundle X, a move to Y sacrifices 3 films for 1 meal, but a further move from Y to Z sacrifices only 1 film for 1 extra meal. Such tastes satisfy the assumption of a diminishing marginal rate of substitution.

These three assumptions - that consumers prefer more to less, can rankalternative bundles according to the utility provided, and have tastes satisfying a diminishing marginal rate of substitution - are all we shall require. It is now convenient to show how tastes can be represented as indifference curves.

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