Since consumers are assumed to be seeking utility maximization, they will prefer combinations of goods and services that give them higher levels of total utility over combinations that give them lower levels of total utility. They will be indifferent among combinations that give exactly the same level of total utility. In order to depict the indifference-curve model using two-dimensional graphs we restrict the analysis to combinations of only two goods. This simplification does not limit our conclusions, however: you may rest assured that this model can be extended to any number of goods, using mathematical techniques, and the major conclusions stay the same. Thus we assume that the consumer may consume any combination of two particular goods. Some of these combinations will be preferable to others, and some combinations will give the same level of total utility, leaving the consumer indifferent among those combinations. Now imagine these combinations on a graph, with quantities of one product measured along the horizontal axis and quantities of the other product measured on the vertical axis.
■ Definition: An indiffereneeycurve is defined as a locus of points in a graph representing combinations of two products (or any two variables, such as risk and return) that give the same tot a futility to a particular person.
Example: Let us consider a particular consumer’s demand for two particular products. Suppose a college student buys hamburgers and milkshakes for lunch and we wish to know what weekly combination of hamburgers and milkshakes would maximize her utility from those two products. Figure 3-1 shows hamburgers on the vertical axis, measured in physical units, and milkshakes on the horizontal axis, also measured in physical units. The lines I\ and I 2 are two of the student’s indifference curves. The combinations of hamburgers and milkshakes represented by indifference curve / 1 , such as five burgers/one shake and three burgers/two shakes, gives her the same level of expected utility. Combinations that lie to the right and above indifference curve /) promise a higher level of utility, and combinations that lie to the left and below that curve promise a lower level of utility.
For each point in the product space represented by Figure 3-1 there will be a series of other points among which the student is indifferent. Consider point C on indifference curve / 2 , which represents two burgers/four shakes. Starting from this point, we could find the other points on indifference curve / 2 by a process of questioning the student about her preference or indifference between other combinations of burgers and shakes.
Suppose we ask the student which would she prefer: two burgers and four shakes (point C) or three burgers and three shakes? If she prefers the latter combination, it is
Milkshakes per Week
FIGURE 3-1. Indifference Curves between Products
evident that the extra burger in the latter combination more than compensates her in terms of utility for the shake that was taken away. If we then offer this consumer the choice of the initial combination C, or of two and one-half burgers and three shakes, and she expresses preference for the initial combination, we know that the extra half burger does not compensate her for the loss of the one shake. Continuing this process, we would find a new combination of burgers and shakes for which the consumer is indifferent when faced with a choice of this combination or with the combination represented by point C. Suppose this occurs at point D, which is two and three-quarters burgers and three shakes. When confronted with this choice, the student says that she has no preference between the two combinations, that either one is as good as the other. Hence points D and C are on the same indifference curve. By the same process we could generate a multitude of combinations that this consumer feels are identical (in terms of utility derived) to points D and C and which, therefore, also lie on indifference curve / 2 .
Since we could start this process from any combination of hamburgers and milkshakes, it follows that there is an indifference curve passing through every point in the figure. We have shown simply two of the infinite number of indifference curves that represent this particular consumer’s taste and preference pattern between the two products. To the right of those curves shown there will be curves that depict progressively higher levels of utility; and to the left and below indifference curve /, there will be curves that depict progressively lower levels of utility. * * 3
The Properties of Indifference Curves. Indifference curves showing combinations of goods have the following four properties. First, higher curves are preferred to lower curves, because we assume the consumer always prefers more to less. Second, indifference curves are negatively sloped throughout , for the same reason. To stay at the same level of total utility when the quantity of one good is increased necessarily requires a reduction in the quantity of the other good, if the consumer always prefers more of each good to less of each good (other things being equal). Thus the curve has a negative slope. 4 Third, indifference curves neither meet nor intersect, because we assume that the consumer’s preferences are consistent. If two curves did intersect, it would imply that there are some combinations that are equal but are simultaneously inferior (or superior) to each other! Finally, indifference curves are convex from below, because of the assumption of diminishing marginal utility for each product. This is best explained in terms of the marginal rate of substitution.
Definition: The marginal rate of substitution (MRS) is defined as the amount of one product that the consumer will be willing to give up for an additional unit of another product, in order to remain at the same level of utility. The proviso that the consumer remains at the same level of utility makes it clear that the MRS refers to a movement along a particular indifference curve. By convention we define the MRS between two products for a movement down a particular indifference curve. Thus the MRS is the ratio of the amount given up of the product on the vertical axis, to the one-unit increment of the product on the horizontal axis.
The MRS is thus equal to the slope of an indifference curve at any point on that curve, since it is defined in terms of the vertical rise (or fall) over the horizontal run. Symbolically, and in terms of our example,
MRS A " . 2 - _»
AM (3-1)
where AH is the decrement to hamburger consumption necessary to maintain utility at the same level given the one-unit increase in milkshake consumption, AM. (The Greek letter A, uppercase delta, is the conventional symbol for denoting a change in a variable.)
Since convexity of an indifference curve means that the slope will decrease progressively as we move from left to right along the indifference curve, it also means that the MRS will diminish progressively as we move down each indifference curve. In terms of Figure 3-1 the MRS between points B and A on indifference curve 7) is the ratio BE/EA = 2. The MRS for the next (third) milkshake per week is equal to the ratio AG/GF = 1. Observing Figure 3-1 you can see that the MRS for the fourth milkshake is approximately 0.5 and that the MRS continues to diminish as the consumer exchanges more hamburgers for milkshakes along indifference curve /,.
Note: The MRS declines because it is equal to the ratio of the marginal utility of the product on the horizontal axis divided by the marginal utility of the product on the vertical axis. That is,
MRS
MU m
MU h
(3-2)
To appreciate this, note that the movement from point B to point A along indifference curve I\ in Figure 3-1 left the consumer at the same level of utility after substituting one milkshake for two hamburgers. The marginal utility attached to the milkshake received must have been equal to the marginal utility given up by sacrificing two hamburgers. Thus the ratio of the marginal utility of milkshakes to the marginal utility of hamburgers is equal to two. Similarly, from point A to point F, total utility stays constant as the consumer gives up one more hamburger for one more milkshake. Thus the marginal utilities attached to the two products must be equal, and their ratio is
equal to one. Since the MRS is equal to the ratio of the MU of the product being acquired to the MU of the product given up, and since we have assumed diminishing marginal utility for all products, it is clear that MRS must diminish, since MRS is the ratio of a numerator that is falling and a denominator that is rising, as we move down along any given indifference curve. Thus the assumption of diminishing marginal utilities causes indifference curves to be convex to the origin. 5