1 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.

3 Optimal Choice Consider an individual with a $1000 and spends it on lbs. of food and sq. ft. of housing, where p f = $5/lb and p h = $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

4 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

5 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?

6 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?

7 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 q i (p 1, p 2, 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?

8 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 q i (p 1, p 2, m) that maps prices and endowment into an amount of that good. How do we find one of these? Where should we start?

9 Optimal Choice Analytically Consider again an individual who finds q 1 and q 2 perfect substitutes, or U(q 1,q 2 ) = q 1 + q 2. So if he has $20 and p 1 = 7 and p 2 = 5, how much q 1 will he buy? (hint: think about graph) If he has $20 and p 1 = 6 and p 2 = 5, how much q 1 will he buy? If he has $20 and p 1 = 4 and p 2 = 5, how much q 1 will he buy? If he has $20 and p 1 = 2 and p 2 = 5, how much q 1 will he buy? How would things change if he had $40? So what is general form of demand function for q 1 and q 2 given linear utility function?

10 Optimal Choice Analytically Demand functions for Quasi-linear utility U(q 1,q 2 ) = aq q 2, endowment $m, prices p 1 and p 2 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?

11 Optimal Choice Analytically Demand functions for quasi-linear utility are given by: Do these demand functions make intuitive sense? * What happens when p 1 rises? Falls? How about a? * What do these demand functions reveal about why quasi-linear utility functions are not always appropriate for modeling preferences?

12 Optimal Choice Analytically Now consider again an individual who has Cobb-Douglas utility U(q 1,q 2 ) = q 1 c q 2 d, who has $m, and faces prices p 1 and p 2. What two conditions must be true at optimum bundle given Cobb-Douglas utility? How can we use these conditions to find demand functions?

13 Optimal Choice Analytically So with Cobb-Douglas preferences, demand functions will be given by: Do these demand functions make intuitive sense? What happens when p 1 rises? Falls? What happens when m rises? Why is it convenient to choose a specification such that c + d = 1?

14 Optimal Choice Analytically Example: Consider an individual whose preferences are captured by U(q 1,q 2 ) = q q p 1 = $2, p 2 = $4, m = $20 What is optimal bundle? How would we sketch this graphically? If p 1 changed to $1, how would optimal bundle change? How would graph change?

15 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?

16 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(q r,q n ) = q r 0.75 q n 0.25 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

18 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

19 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