Intermediate Microeconomics

UTILITYBEN VAN KAMMEN, PHDPURDUE UNIVERSITY

OutlineTo put this part of the class in perspective, consumer choice is the underlying explanation for the demand curve.◦As utility and preferences are discussed, consider how it relates to the demand curve.

Define and describe the problem that rational consumers solve.◦ Define the consumer’s objective function and describe its

properties.◦ Describe the constraints facing a consumer’s choices.

Utility“The satisfaction that a person receives from his or her economic activities.”Two obvious definitions:◦ Good: Something that increases a person’s utility, ceteris

paribus.◦ Examples: food, good books, Pontiac Trans Ams.

◦ Bad: Something that decreases a person’s utility, ceteris paribus.◦ Examples: litter, taking away of a Trans Am.

Ceteris Paribus: “other things being equal.”

Utility FunctionsHow can we formalize (“model”) the concept of utility mathematically?If you can account for all the things that affect a consumer’s utility, a utility function could be specified with all of those things as arguments.◦ Utility is a function of all goods and bads for a given

consumer.

Example𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈= 𝑈𝑈(𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓,𝑔𝑔𝑓𝑓𝑓𝑓𝑓𝑓 𝑏𝑏𝑓𝑓𝑓𝑓𝑏𝑏𝑏𝑏,𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑏𝑏 𝐴𝐴𝐴𝐴𝑏𝑏, 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑙𝑙𝑇𝑇,𝑇𝑇𝑓𝑓𝑇𝑇 − 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑏𝑏− 𝐴𝐴𝐴𝐴 𝑈𝑈𝑇𝑇𝑇𝑇𝑇𝑇𝑏𝑏𝑡𝑡𝑓𝑓𝑇𝑇𝑈𝑈𝑇𝑇𝑈𝑈𝑈𝑈𝑓𝑓𝑇𝑇, 𝑏𝑏𝑠𝑓𝑓𝑙𝑙𝑏𝑏 . . . )As the example illustrates, there are many things that affect utility. ◦ This is something that our model will have to simplify.

PreferencesAnother thing to consider when modeling utility is how each good affects the consumer’s utility.◦ Inevitably, some people will like a given good more than

others. e.g., I like Jelly Belly® jelly beans more than 99% of the population, but I hate frosting—something that many people enjoy.

◦ So in addition to representing a list of all goods and bads, a utility function also has to account for each good’s effect on utility. ◦ Note: this is also something that will have to be simplified to make

the model workable.

PreferencesLastly a utility function should account for the possibility that the marginal utility of a good changes depending on how many we already have . . . ◦think back to the water example and diminishing marginal utility.

To recapitulateModeling utility involves writing a mathematical expression that:

1) contains an exhaustive list of goods and bads,

2) specifies the marginal utility for each combination of those goods and bads, and

3) can be analyzed by a human being using a minimum of calculus.◦Clearly something in the preceding list will have to be compromised.◦My suggestions are #1 and #2.

Properties of Utility FunctionsCompleteness: there is a parable about a donkey that found himself between a trough of oats and a trough of hay and starved to death because he couldn’t decide.

We will assume consumers are always able to say which bundle of consumption they prefer.

You could say that utility is “defined”, in the mathematical sense, for all relevant bundles.

Properties of Utility FunctionsTransitivity: if consumption bundle X is preferred to bundle Y, and bundle Y is preferred to bundle Z, transitivity implies that X is preferred to Z.◦ Analog in geometry: 𝑋𝑋 > 𝑌𝑌 > 𝑍𝑍 → 𝑋𝑋 > 𝑍𝑍.

More of an economic good is, well, better.◦ Goods have nonnegative marginal utility.◦ Consumers prefer more of a good to less, ceteris paribus.

Modeling UtilityTime to abstract away from reality to make some analytical sense of consumers’ preferences.Assumption: a consumer’s utility is a function of only 2 goods: jelly beans and Pabst Blue Ribbon beer.◦ To make this assumption seem less extreme, tell yourself

that the consumer has all the other things that he wants and the other goods he consumes does not change, i.e., ceteris paribus.

The function is: 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 = 𝑈𝑈(𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏,𝑃𝑃𝐽𝐽𝑃𝑃).

Graphing UtilityThere are 3 ways to graph this utility function in 2-dimensional space.

1.Graph utility as a function only of Jbeans, holding the level of PBR constant.

2.Graph utility as a function only of PBR, holding the level of JBeans constant.

3.Graph quantity of one good as a function of quantity of the other good, holding the level of utility constant.

The 1st way of graphing utility

Do jelly beans have diminishing marginal utility in this graph?

The 2nd way of graphing utility

The 3rd way of graphing utility

Note that this function has a negative slope that takes form, Δ𝑃𝑃𝑃𝑃𝑃𝑃Δ𝐽𝐽𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽

.

Indifference curvesThe 3rd method, yields a graph of an indifference curve:◦ “a curve that shows all the combinations of goods

that provide the same level of utility.”

Each point on the curve is a combination of JBeansand PBR, and each point gives the consumer a utility level of 𝑈𝑈0.The consumer is indifferent to all points on this curve, so it is called an indifference curve.

Properties of indifference curvesWhen analyzing utility as a function of two goods:◦Indifference curves slope downward.◦Indifference curves do not intersect.◦Indifference curves are convex (“bowed toward the axis”).◦An increase in utility is shown by an indifference curve that is further from the origin.

Indifference Curves Slope DownwardBegin from some point “A” on the indifference curve, 𝑈𝑈0. Move to a point “B” that is directly to the left of “A” and not on the indifference curve. ◦ Conceptually take away some of the consumer’s jelly beans without

changing his PBR level (Δ𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏 < 0).◦ The consumer will certainly not be indifferent to this change.◦ In order to make him indifferent, you would have to give him some more

PBR (Δ𝑃𝑃𝐽𝐽𝑃𝑃 > 0) and move him up to point “C”.

Taking the two changes together we have a negative change in jelly beans and a positive change in Pabst, i.e., a negative slope.

What the consumer has effectively done is “substitute” PBR for jelly beans and move upward along the indifference curve.

Indifference curves do not intersectThis is something that would make an economist’s head explode.

Look at the point where the two curves on the next slide intersect; that would be two different levels of utility for the same bundle.◦The same bundle of consumption cannot give the consumer two

different levels of utility!!!◦ I know; it would freak me out, too.

What Kind of Messed up Preferences are These?

Indifference curves are convexThe slope of an indifference curve is called the marginal rate of substitution “MRS”.Indifference curves get “flatter” as you move from left to right along them.◦ Getting flatter means the slope is decreasing.

So the MRS is decreasing as you go from left to right.The quantity of JBeans the consumer is willing to substitute for PBR is continually decreasing as you go from left to right.◦ This is a result of diminishing marginal utility, which we will examine

more later.◦ For now suffice it to say that indifference curves are convex.

Variation in utility (graphically)

Note: U2 > U1 > U0.

Marginal Rate of Substitution“The rate at which a consumer is willing to reduce consumption of one good when he or she gets one more unit of another good. The slope of an indifference curve.”◦ Abbreviation: MRS

Diminishing MRS: Diminishing marginal utility leads consumers to prefer balanced consumption bundles to skewed bundles. This gives indifference curves their convex shape.

Particular PreferencesPreferences affect the shape of indifference curves. Consider the following cases.◦A useless good.◦An economic bad.◦Perfect substitutes.◦Perfect complements.

A Useless Good

The consumer cannot increase utility by consuming more of the useless good.

A Bad and a Good

MRS is positive; increasing the bad requires that you increase the good to stay at a given level of utility.

Perfect Substitutes

Perfect substitutes have a constant marginal rate of substitution. Their indifference curves are linear.

Perfect complements; “Hi, Bob”

Goods are consumed in fixed proportions. Each good has zero marginal utility. Unless you have more of the other good, you can’t increase utility.

Utility maximizationThe goal of the consumer is to maximize utility. In the absence of scarcity, he would choose to do so by consuming an infinite quantity of all goods.◦ But goods are scarce.

A consumer maximizes utility subject to a budget constraint.

Budget constraint“The limit that income places on the combinations of goods that a consumer can buy.”◦ Each consumer has a finite capacity (income) for

purchasing goods.◦ In the two-good model, he allocates his income between

the two goods as follows:Income = [Expenditure on X] + [Expenditure on Y]

◦ Expenditure on each good equals the quantity of the good multiplied by its price:Income = I = PX*X + PY*Y

Budget constraint graphically

The vertical intercept is 𝐼𝐼𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃

, and the horizontal intercept is 𝐼𝐼𝑃𝑃𝐽𝐽𝑃𝑃𝐽𝐽𝐽𝐽𝐽𝐽𝐽𝐽

.

Price ratioThe intercepts of the budget constraint are found by setting the quantity of one good to zero:

𝐼𝐼 = 𝑃𝑃𝑃𝑃𝐽𝐽𝑃𝑃 ∗ 𝑄𝑄𝑃𝑃𝐽𝐽𝑃𝑃 + 𝑃𝑃𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏 ∗ 0 → 𝑄𝑄𝑃𝑃𝐽𝐽𝑃𝑃 =𝐼𝐼

𝑃𝑃𝑃𝑃𝐽𝐽𝑃𝑃𝐼𝐼 = 𝑃𝑃𝑃𝑃𝐽𝐽𝑃𝑃 ∗ 0 + 𝑃𝑃𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏 ∗ 𝑄𝑄𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏0 → 𝑄𝑄𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏 =

𝐼𝐼𝑃𝑃𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏

The slope of the budget constraint is the ratio of the two prices:◦Rise over run implies:

𝐼𝐼𝑃𝑃𝑃𝑃𝐽𝐽𝑃𝑃

÷𝐼𝐼

𝑃𝑃𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏=𝑃𝑃𝐽𝐽𝐽𝐽𝑙𝑙𝑇𝑇𝑇𝑇𝑏𝑏𝑃𝑃𝑃𝑃𝐽𝐽𝑃𝑃

Constrained Utility MaximizationIf the consumer has a budget constraint, he consumes a combination of two goods that gets him on the highest indifference curve possible.◦ This occurs where the marginal rate of substitution equals the price

ratio:𝑀𝑀𝑃𝑃𝑀𝑀 = 𝑃𝑃𝑇𝑇𝑈𝑈𝑃𝑃𝑙𝑙 𝑃𝑃𝑇𝑇𝑈𝑈𝑈𝑈𝑓𝑓.

Graphically this occurs where the budget constraint and indifference curve are tangent to one another.

Constrained utility maximization

Why is this Optimal?Consider the alternatives:◦ The consumer can get more utility than 𝑈𝑈0 by consuming more of

each good.◦ The consumer cannot attain utility 𝑈𝑈2 because he doesn’t have

enough income.

Alternative utility levels

Only U1 is attainable and optimal.

Further examplesConstrained utility maximization for perfect substitutes, perfect complements.Consumers facing the same budget constraints make different optimal choices because of differences in preferences.If it makes you uncomfortable to consider a utility function of only two goods, you can think of the good on the vertical axis as a “bundle of all other goods” with a price equal to the index of prices of the goods in the bundle.

ConclusionBy simplifying the number of goods and assuming a functional form for utility, it is simple to predict what an individual will consume, given prices.

How will their consumption change when prices change?

The next topic we will discuss is how utility maximization produces a demand curve.