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Intermediate Microeconomics

- Choice

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.

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

A

C

E

D

sq. ft.

B

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

A

C

E

D

sq. ft.

B

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?

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?

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?

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?

Optimal Choice Analytically

- 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? - If he has 20 and p1 4 and p2 5, how much q1

will he buy? - If he has 20 and p1 2 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?

Optimal Choice Analytically

- 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?

Optimal Choice Analytically

- 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?

Optimal Choice Analytically

- Now consider again an individual who has

Cobb-Douglas utility U(q1,q2) q1cq2d, 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?

Optimal Choice Analytically

- 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?

Optimal Choice Analytically

- Example
- Consider an individual whose preferences are

captured by U(q1,q2) q10.4q20.6 - 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?

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?

Application Government Funding of Religious

Institutions

- Assume
- Govt grant equals 4,000/yr
- A religious institution has an annual budget of

20,000. - Institutions preferences are captured by

U(qr,qn) qr0.75qn0.25 - What will be institutions spending on religious

and non-religious activity without grant? - How will grant change budget constraint?
- What will be institutions spending on religious

and non-religious activity with grant?

Application Government Funding of Religious

Institutions

- What will this problem look like graphically?

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

A

housing

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

A

housing