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Intermediate Microeconomics

Utility Theory

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Utility

A complete set of indifference curves tells us everything we need to know about any individual’s preferences over any set of bundles.

However, our goal is to build a model that is useful for describing behavior.

While indifference curves are often sufficient for this, they are somewhat cumbersome for some tasks.

Therefore, we will often think of individual preferences in terms of Utility

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Utility

Utility is a purely theoretical construct defined as follows:

1. If an individual strictly prefers bundle A-{q1

a,q2a,..,qn

a} to another bundle B-{q1

b,q2b,..,qn

b}, then an individual is said to get “a higher level of utility” from bundle A than bundle B.

2. If an individual is indifferent between a bundle A-{q1

a,q2a,..,qn

a} and another bundle B-{q1

b,q2b,..,qn

b}, then an individual is said to get “the same level of utility” from bundle A than bundle B.

How is utility related to happiness?

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Utility Function

So a utility function U is just a mathematical function that assigns a numeric value to each possible bundle such that:

1. If an individual strictly prefers bundle A-{q1a,q2

a,..,qna}

to another bundle B-{q1b,q2

b,..,qnb}, then U(q1

a,q2a,..,qn

a) > U(q1

b,q2b,..,qn

b)

2. If an individual is indifferent between a bundle A-{q1

a,q2a,..,qn

a} and another bundle B-{q1b,q2

b,..,qnb},

then U(q1a,q2

a,..,qna) = U(q1

b,q2b,..,qn

b)

We can often think of individuals using goods as “inputs” to produce “utils”, where production is determined by utility function.

* So how do utility functions relate to Indifference curves? (“hill of utility”)

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Constructing a Utility Function

Consider an individual’s preferences over consumption bundles containing different amounts of peanuts (qn) and pretzels (qp).

The person always prefers a bundle that allows him to eat more than less.

Person likes peanuts and pretzels equally, meaning he would always be willing to trade a bundle with one fewer ounce of pretzels for a bundle with one more ounce of peanuts and vice versa.

* What will this person’s indifference curves over bundles containing these two goods look like?

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Constructing a Utility Function

What is a utility function that captures this individuals’ preferences?

* How do we get Indifference Curves from this utility function?

* What is Marginal Rate of Substitution (MRS)?

Would this utility function be a good approximation for your preferences over lemonade and pretzels?

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Other Commonly Used Utility Functions

Two commonly used forms for utility functions are:

Quasi-linear Utility:

U(q1,q2) = aq10.5 + q2

Examples: q10.5 + q2 , 10q1

0.5 + q2

Cobb-Douglas Utility:

U(q1,q2) = q1c q2

d for some positive c and d.

Examples: q10.3

q20.7, q1 q2 , q1

2q23

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Other Commonly Used Utility Functions

Given a utility function U(q1,q2) = q1

0.5 + q2

Which bundle would be preferred—{25,4} or {4,9}?

Given a utility function U(q1,q2) = q1

0.5q20.5

Which bundle would be preferred—{25,4} or {4,9}?

To understand what types of situations they would be appropriate for, let us look deeper at what these functions capture.

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Marginal Utility

In economics, we are generally interested in trade-offs “at the margin”. How does a consumer value a little more

of a particular good?

Consider the ratio of the change in utility (ΔU) associated with a small increase in q1 (Δq1), holding the consumption of other goods fixed, or

Ex: U(q1,q2) = q1q2

At bundle {1,4}, with Δq1 = 0.5

What happens when Δq1 gets really small?

q2

4

1 1.5 q1

u=4

u=6

Δq1=0.5

1

21211

1

),(),(

q

qquqqqu

q

U

45.0

46

5.0

)4*1()4*5.1(

5.0

)4,1()4,5.1(

uu

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Marginal Utility

Marginal Utility of a good 1 (MU1) - the rate-of-change in utility from consuming more of a given good, or

“MU1 - the partial derivative of the utility function with respect to good 1”

So what is expression for marginal utility of good 1 given utility functions at some bundle-{q1, q2}? U(q1,q2) = q1 + q2

U(q1,q2) = q10.5

q20.5

1

21211

),(),(

q

qquqqMU

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Ordinal Nature of Utility

There is a constraint with this concept of marginal utility as a way to measure how much someone values “a little more” of a good, and it has to do with ordinal nature of utility. https://www.youtube.com/watch?v=4xgx4k83zzc

Consider again the person who saw peanuts and pretzels as perfect substitutes. As we saw, his preferences over these two goods were captured

by

U(qn,qp) = qn + qp

Could his preferences also be captured by

U(qn,qp) = 5(qn + qp) ?

What is MU of a little more peanuts under each utility function?

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Ordinal Nature of Utility

Utility function is constructed to summarize underlying preferences. Therefore, no new information in utility function. So generally more than one utility function can capture a given

set of preferences.

Since preferences were strictly ordinal, so must be the utility function. Utility level of one bundle is only meaningful in as much as

it is higher, lower, or the same as another bundle. How much higher isn’t very informative in itself since it is

hard to compare to other things.

This also means Marginal Utility isn’t necessarily very informative in and of itself.

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Marginal Utility

Instead, consider the following thought exercise:

• Suppose we increase individual’s q1 by “a little bit” (Δq1),

• How much q2 would he be willing to give up for this more q1?

• For small (Δq1), individual’s change in utility will be approximately

Δq1 * MU1(q1,q2)

• Therefore, we would have to decrease some Δq2 large enough such that:

Δq1*MU1(q1,q2) + Δq2*MU2(q1,q2) = 0

or

What does this mean if both Δq2 and Δq1 are small?

q2

4

1 q1

u=4

Δq1

Δq2

),(

),(

212

211

1

2

qqMU

qqMU

q

q

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Marginal Utility and MRS

Therefore, MRS is both:1. The slope of an indifference curve at a

particular point, and

2. The negative ratio of marginal utilities at that particular point.

Should this be surprising?

),(

),(),(

212

21121 qqMU

qqMUqqMRS

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Ordinal Nature of Utility and MRS

Recall again that the following utility functions both capture the same underlying preferences:

u(qq,q2) = q1 + q2

v(qq,q2) = 5(q1 + q2)

As discussed previously, issues regarding preferences revolve around an individual’s willingness-to-trade one good for more of the other.

Therefore, one way to confirm whether or not two utility functions represent the same underlying preferences is to determine whether they result in the same MRS at every bundle. Is this true with above utility functions?

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Interpreting MRS Equations

So consider a Cobb-Douglas utility function U(q1,q2) = q1

0.25q20.50

MU1 ?

MU1 ? MRS?

So how do we interpret this expression for MRS? What is MRS at {4,4}?

What is MRS at {9, 1}?

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Comparing Utility functions

Recall again: Quasi-linear utility function

U(q1,q2) = aq10.5 + q2

Cobb Douglas utility function

U(q1,q2) = q1cq2

d (for c, d > 0)

What is general expression for MRS under each specification?

What are key similarities and differences between specifications?

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Comparing Utility functions

Quasi-linear Cobb-Douglas

q1

q2

q1

q2

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Cobb-Douglas vs. Quasi-linear Utility Functions For comparisons between bundles with the

following two goods, would Quasi-linear or Cobb-Douglas utility function be more appropriate?

pizza and beer?

composite good vs. pencils?

composite good vs. clothes?

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Utility functions and MRS

So this is primary benefit of modeling preferences in terms of utility: It gives us an easily manipulable mathematical

tool that captures the basic underlying structure of preferences inherent in indifference curves.

Recall our discussion of MRS in the context of indifference curves. We could only describe MRS at any given point

by approximating the slope of Indifference curve.

With utility function, we can easily calculate MRS at any given bundle.

Given MRS is key to thinking about an individual’s willingness to make trade-offs, this will be a useful tool.