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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. • 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 • 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 again my preference over Tostitos and Doritos. • What is a utility function can captures my preferences over these goods? • How do we get Indifference Curves from this utility function? • Why wouldn’t this utility function be a good approximation for my preferences over Coke and Chips?
Other Commonly Used Utility Functions • Two other commonly used generic forms for utility functions are: • Quasi-linear Utility: U(q1,q2) = q1a + q2 for 0 < a < 1. • Example? • Cobb-Douglas Utility: U(q1,q2) = q1aq2b for some positive a and b. • Example? • Do they exhibit Diminishing MRS? • To understand what types of situations they would be appropriate models for, let us look deeper at MRS.
Marginal Utility • 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 • What happens when Δq1 gets really small? q2 q’2 Δq1 u=8 u=4 q’1 (q’1+Δq1)
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 general expression for marginal utility of good 1 for following utility functions? • U(q1,q2) = aq1 + bq2 • U(q1,q2) = q1aq2b • U(q1,q2) = q1a+ q2
Marginal Utility • So what is the value of marginal utility of good 1 at the bundles {4, 1} and {4, 4} given the following utility functions? • U(q1,q2) = 4q1 + q2 • U(q1,q2) = q13q22 • U(q1,q2) = q10.5 + q2
Comparing Utility functions • Quasi-linear – MRS depends only on quantity of q1 • Cobb-Douglas – MRS depends on quantity of both goods. q2 q2 q1 q1
Ordinal Nature of Utility • There is a major constraint with this concept of marginal utility as a way to measure how much someone values “a little more” of a good. • A Utility function is constructed to summarize underlying 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 informative. • This also means marginal utility is not very informative in and of itself. • e.g. What does it mean behaviorally that marginal utility equals 16?
Marginal Utility 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 much 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 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?
Interpreting MRS Equations • Consider the following generic Cobb-Douglas utility functions: U(q1,q2) = q1aq2b U(q1,q2) = aq1 + bq2 • MU1 ? • MU1 ? • MRS? • So what will be value of MRS for the following? • MRS for U(q1,q2) = q10.4q20.6 at {4,4}? • MRS for U(q1,q2) = 5q1 + q2 at {9, 1}? • How do we interpret these values?
Ordinal Nature of Utility and MRS • Consider the following utility functions: u(qq,q2) = q10.2q20.4 v(qq,q2) = q13q26 • What is the expression for MRS for each? • What does this imply? • How is this possible?
Marginal Utility and MRS • 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 each 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 important. • In general, modeling preferences in terms of utility functions helps allows us to capture general aspects of preferences in a very manageable way.
