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Intermediate Microeconomics. Utility Theory. 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.
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Intermediate Microeconomics Utility Theory
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
Utility • Utility is a purely theoretical construct defined as follows: • If an individual strictly prefers bundle A-{q1a,q2a,..,qna} to another bundle B-{q1b,q2b,..,qnb}, then an individual is said to get “a higher level of utility” from bundle A than bundle B. • If an individual is indifferent between a bundle A-{q1a,q2a,..,qna} and another bundle B-{q1b,q2b,..,qnb}, then an individual is said to get “the same level of utility” from bundle A than bundle B. • How is utility related to happiness?
Utility Function • So a utility functionU is just a mathematical function that assigns a numeric value to each possible bundle such that: • If an individual strictly prefers bundle A-{q1a,q2a,..,qna} to another bundle B-{q1b,q2b,..,qnb}, then U(q1a,q2a,..,qna) > U(q1b,q2b,..,qnb) • If an individual is indifferent between a bundle A-{q1a,q2a,..,qna} and another bundle B-{q1b,q2b,..,qnb}, then U(q1a,q2a,..,qna) = U(q1b,q2b,..,qnb) • 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”)
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?
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 beer and peanuts?
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 , 10q10.5 + q2 • Cobb-Douglas Utility: U(q1,q2) = q1aq2b for some positive a and b. • Examples: q10.3q20.7, q1 q2 , q12q23
Other Commonly Used Utility Functions • Given a utility function U(q1,q2) = q10.5 + q2 • Which bundle would be preferred—{25,4} or {4,9}? • Given a utility function U(q1,q2) = q10.5q20.5 • 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.
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 Δq1=0.5 u=6 u=4 1 1.5 q1
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) = q10.5q20.5
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. • 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?
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.
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 andΔq1 are small? q2 4 Δq1 Δq2 u=4 1 q1
Marginal Utility and MRS • Therefore, MRS is both: • The slope of an indifference curve at a particular point, and • The negative ratio of marginal utilities at that particular point. • Should this be surprising?
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?
Interpreting MRS Equations • So consider a Cobb-Douglas utility function U(q1,q2) = q10.25q20.50 • MU1 ? • MU1 ? • MRS? • So how do we interpret this expression for MRS? • What is MRS at {4,4}? • What is MRS at {9, 1}?
Comparing Utility functions • Recall again: • Quasi-linear utility function U(q1,q2) = aq10.5 + q2 • Cobb Douglas utility function U(q1,q2) = q1aq2b(for a, b > 0) • What is general expression for MRS under each specification? • What are key similarities and differences between specifications?
Quasi-linear Cobb-Douglas Comparing Utility functions q2 q2 q1 q1
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?
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.
