Intermediate microeconomics

Professor Yongmin Chen

Topic 2. Consumer Preferences and Utility

The Nature of Consumer Preferences

How consumers make choices is an important question. To answer this, we first need to know the nature of consumer preferences. Suppose there are several market baskets (or commodity bundles), say A, B, C, etc, each of which consists of some combination of goods. A consumer should have some preferences over these baskets. We make the following assumptions about preferences:

(1) Completeness. For any two baskets, the consumer should be able to compare them. In other words, if A and B are any two baskets, the consumer should be able to say whether she likes A better, or B better, or she is indifferent between the two (she equally likes them).

(2) Transitivity. If a consumer prefers A to B, and prefers B to C, then she should prefer A to C.

(3) Nonsatiation. The consumer always prefers more to less of a commodity.

Utility and Utility Functions

The level of satisfaction a consumer receives from a certain choice is often referred to as the utility of that choice. When a consumer prefers A to B, we say she gets higher utility from A than from B. For each consumption bundle, we can assign a numerical value to it which preserves the preference ordering, and we call this correspondence between preference and real values the utility function.

Utility function is defined such that for any baskets A and B: U(A) > U(B) iff the consumer prefers A to B; U(A) = U(B) iff the consumer is indifferent between A and B.

There are two types of preference rankings: ordinal rankings and cardinal rankings. Ordinal rankings give us information about the order in which a consumer ranks baskets, while cardinal rankings give us information about the intensity of a consumer’s preference. Utility is an ordinal ranking of commodity bundles. Thus the utility function for a certain preference system is not unique

Analysis for A Single Good

If a consumer consumes only one good, the utility function can be denoted as U(y), where y is the amount of the good consumed. For instance:

U(y) = 10vy.

Marginal utility is the change in utility caused by an incremental change in the consumption of a good.

MU_Y = DU/DY.

Example: When U(y) = 10vy , we have MU_Y = 5/vy.

Example: The curves of utility function and marginal utility function.

Notice that marginal utility is the slope the utility function.

The Principle of diminishing marginal utility: As consumption of a good increases, the marginal utility of that good will eventually decrease after some point.

Analysis of multiple goods:

Indifference Curves

A useful way to describe a consumer's preference is indifference curves. An indifference curve is the set of points representing the market baskets among which the consumer is indifferent. Alternatively, an indifference curve is the set of points representing the market baskets that give the consumer the same level of satisfaction.

Example. The derivation of indifference curves for a consumer.

Properties of indifference curves:

(1) All indifference curves are downward-sloping.

(2) The further an indifference curve is away from the origin, the higher level it represents of a consumer's satisfaction.

(3) Two indifference curves cannot cross each other.

These properties are derived from our assumptions about consumer's preferences. Explanations in class.

Example. The shape of indifference curves. What happens if the assumption of nonsatiation is violated?

Marginal Rate of Substitution

Example. Utility function with two goods: U(X,Y) = XY. Another one: U(X,Y) = 2XY.

The consumer is usually willing to substitute amounts of one good for another. If you have 5 units of apple and 12 units of orange, you might be equally happy to have 6 apples and 10 oranges instead. In other words, you are willing to give up 2 units of orange in order to have one more unit of apple. In this particular case, we say your marginal rate of substitution (of apple for orange) is 2. In general, we define:

Marginal rate of substitution (of X for Y) is the number of units of Y a consumer is willing to give up to get one more X.

MRS can be measured by the negative slope of the indifference curve. MRS_XY = ‑DY/DX.

In general, MRS varies at different points of an indifferent curve. Two ways MRS can be calculated.

(1) If you know the functional form of an indifference curve, say XY=10, then you can calculate MRS at each point of the indifference curve by obtaining the slope of the indifference curve at each point.

Example. What is MRS_XY if the indifference is given by XY=10?

(2) If you do not know the functional form of the indifference curve, but you do know some points on it, then your calculation can be based on these points. In the apple and orange example above, when you have 5 units of apple and 12 units of orange, your MRS of apple for orange is ‑(10‑12)/(6‑5) = 2.

MRS is usually diminishing. As more apples (X) are consumed, the consumer becomes less

willing to reduce the consumption of another good (Y) for even more apples (X).

When the consumption of one good changes alone, utility changes too. Marginal utility is the change in utility divided by an incremental change in consumption of one good alone.

MU_X = DU/DX; MU_Y = DU/DY.

Example: If U(2,3) = 5, U(2.5,3) = 8, what is MU_x at X=2?

Example: If U(X,Y) = XY, then MU_X = Y, and MU_Y = X.

Example: If U(X,Y) = Ö(XY), then MU_X = (1/2)×Ö(Y/X), and MU_Y = (1/2)×Ö(X/Y)

There is an important relationship between marginal rate of substitution and the ratio of MU_X over MU_Y: MRS_XY = ‑DY/DX = MU_X/MU_Y

That is, MRS_XY is equal to MU_X divided by MU_Y. Intuition: the larger MU_X compared with MU_Y, the more Y you are will to give up to increase the consumption of X.

Example: If MU_X = 6, MU_Y = 3, what is the MRS_XY?

Example: If U(X,Y) = XY, what is MRS_XY? If U(X,Y) = Ö(XY), what is MRS_XY? What do the results imply about the indifference curves for these two utility functions?