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 is slope? Given the set of indifference curves depicted here, what will be his optimal bundle? 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. Tuberculosis patients saved vs. AIDS patients saved (and you only care about current lives saved) Suppose you had budget of $10,000 and each AIDS treatment cost $1000 while each Tuberculosis treatment cost $500. How would you allocate your money? What if each Tuberculosis treatment cost $2000? * How do we show this graphically?

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 gives the amount she consumes of that good given any set of prices and her endowment. In general, demand function will tell how a consumer reacts to changes in prices and endowment. How would we characterize a demand function graphically?

Optimal Choice Analytically While graphs are informative, 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) = aq11/2 + 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 the 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? * 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) = q1aq2b, 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 the 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? What happens when a rises relative to b?

Optimal Choice Analytically Example: Consider an individual whose preferences are captured by U(q1,q2) = q12q23 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 find this proposal troubling?

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) = qr0.75qn0.25 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: Taxes Suppose you are a government official wanting to raise $R from each person. Let there be two goods in the world and you have two taxing options: Impose a tax equal to $t on each unit of good 2 (a sales tax), where t is set so that this tax will raise $R from each person. Make everyone pay R directly out of their endowment (lump sum tax). Which is a better plan?

Application: Taxes Key is to consider what each does to a person’s budget constraint. How does an individual’s budget constraint change with the sales tax? How does the individual’s budget constraint change with lump sum tax? Homework problem analyzes this issue further.

Application: Indexing This framework can help us think about issues involved in indexing payments such as social security. Suppose Social Security is such that for the current generation, the typical retiree was captured by the picture to the right. In the coming years, the government knows prices will rise, but the price of food will rise more than housing. If the gov’t raised SS payments to a level such that the typical retiree could always just afford bundle A, would this make the typical future retiree the same, worse, or better off than current retirees? food A housing