1 intermediate microeconomics choice. 2 optimal choice we can now put together our theory of...

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Intermediate Microeconomics

Choice

2

Optimal Choice

We can now put together our theory of preferences with our budget constraint apparatus and talk about “optimal choice”.

Unlike psychology, which often attempts to understand why particular individuals make particular choices, economic theory is trying to develop a model of what individuals as a whole generally do.

Therefore, at its most basic, economic theory simply assumes individuals choose their most preferred bundle, or equivalently the bundle that gives them the most utility, that is in their budget set.

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Optimal Choice

Consider an individual with a $1000 and spends it on lbs. of food and sq. ft. of housing, where pf = $5/lb and ph = $10/sq. ft.

Budget Constraint depicted to the right. What are intercepts? What is slope?

If his preferences are captured by the indifference curves depicted here, what will be his optimal bundle? Why?

Lbs food

sq. ft.

A

B

C

D

E

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Optimal Choice

* Why is A not “optimal”?

* Why is B not “optimal”?

* Why is C not “optimal”?

* Why is D not “optimal”?

* So what all is true at E?

* What happens if price of food falls?

food

sq. ft.

A

B

C

D

E

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Optimal Choice Does tangency condition always have to hold for

optimum bundle?

Consider goods that are perfect substitutes. e.g. Suppose you are working for Doctors without

Borders. You have 20 beds, malaria patients take a week

to treat, TB patients take two weeks. What does your monthly budget constraint look like?

Your preferences are such that you want to treat as many patients as you can. What do your indifference curves look like?

So how would you optimally allocate your bed slots per month?

What if each Tuberculosis treatment cost took only one week?

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Optimal Choice

Now consider two goods that are perfect complements (i.e. must be consumed in fixed proportions). E.g. I only like coffee if it is 1/2 coffee 1/2

milk. What will my indifference curves look

like? Suppose I had $6, coffee costs $0.50/oz and

cream costs $1.00/oz. What will my budget constraint look

like? What will be my optimal choice?

What if prices were $1/oz for each?

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Demand Function

Demand Function for a given consumer for each good i - the amount consumer chooses to consume of that good given any set of prices and her endowment

qi(p1, p2, m)

In general, demand function will tell how a consumer reacts to changes in prices and endowment.

How would we derive a demand function graphically?

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Optimal Choice Analytically

While graphs are informative, they can be cumbersome, so we often want to solve things analytically.

For a two-good analysis, for each good i, we will want to find a function qi(p1, p2, m) that maps prices and endowment into an amount of that good.

How do we find one of these? Where should we start?

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Consider again an individual who finds q1 and q2 perfect substitutes, or

U(q1,q2) = q1 + q2.

So if he has $20 and p1 = 7 and p2 = 5, how much q1 will he buy? (hint: think about graph)

If he has $20 and p1 = 6 and p2 = 5, how much q1 will he buy?



How would things change if he had $40?

So what is general form of demand function for q1 and q2 given linear utility function?

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Demand functions for Quasi-linear utility

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

endowment $m, prices p1 and p2

Finding demand function is more complicated, but still helps to think about graphically. What two conditions must be true at optimum

bundle given Quasi-linear utility?

How can we use these conditions to find demand functions?

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Demand functions for quasi-linear utility are given by:

Do these demand functions make intuitive sense?* What happens when p1 rises? Falls? How

about a?

* What do these demand functions reveal about why quasi-linear utility functions are not always appropriate for modeling preferences?

1

22

2212

21

22

2

211

4),,(

4),,(

p

pa

p

mmppq

p

pamppq

12


Now consider again an individual who has Cobb-Douglas utility U(q1,q2) = q1

cq2d, who

has $m, and faces prices p1 and p2.

What two conditions must be true at optimum bundle given Cobb-Douglas utility?

How can we use these conditions to find demand functions?

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So with Cobb-Douglas preferences, demand functions will be given by:

Do these demand functions make intuitive sense? What happens when p1 rises? Falls?

What happens when m rises?

Why is it convenient to choose a specification such that c + d = 1?

1211 ),,(

p

m

dc

cmppq

2212 ),,(

p

m

dc

dmppq

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Example: Consider an individual whose preferences

are captured by U(q1,q2) = q10.4q2

0.6

p1 = $2, p2 = $4, m = $20

What is optimal bundle? How would we sketch this graphically?

If p1 changed to $1, how would optimal bundle change? How would graph change?

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Application: Government Funding of Religious Institutions Suppose government is considering giving

grants to religious institutions with the restriction that these funds are used for non-religious purposes only.

Why might advocates for separation of church and state still find this proposal troubling?

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Application: Government Funding of Religious Institutions Assume:

Gov’t grant equals $4,000/yr A religious institution has an annual budget of $20,000. Institution’s preferences are captured by U(qr,qn) = qr

0.75qn0.25

What will be institution’s spending on religious and non-religious activity without grant?

How will grant change budget constraint?

What will be institution’s spending on religious and non-religious activity with grant?

Application: Government Funding of Religious Institutions What will this problem look like graphically?

17

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Application: Social Security Indexing for Inflation This framework can help us think

about issues involved in indexing payments such as social security.

Adjustments in Social Security are currently determined by changes in Consumer Price Index (CPI). CPI is essentially determined by calculating the price of a “basket” of goods.

Some argue that this makes SS increasingly generous over time and therefore should be reformed. Why would they say this?

food

housing

A

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Application: Social Security Indexing for Inflation Chained CPI – recognizes consumers

will change optimal bundle as relative prices change.

Idea is to keep “utility” the same. food

housing

A

1 intermediate microeconomics choice. 2 optimal choice we can now put together our theory of...

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