elaborate on the idea that riz2 is the general form for Cobb-Douglas preferences ons(e ke it look different. It's a good idea to calculate the MRS for a few representations of the Cobb-Douglas utility function in class so that people can see how to do them and, more importantly, that the MrS doesnt change as you change the representation of utility The example at the end of the chapter, on commuting behavior, is a very nice one. If you present it right, it will convince your students that utility is an operational concept. Talk about how the same methods can be used in marketing surveys, surveys of college admissions, etc. The exercises in the workbook for this chapter are very important since they drive home the ideas. A lot of times, students think that they understand some point, but they don't, and these exercises will point that out to them. It is a good idea to let the students discover for themselves that a sure-fire way to ell whether one utility function represents the same preferences as another is to compute the two marginal rate of substitution functions. If they don't get this idea on their own, you can pose it as a question and lead them to the answer Utility A. Two ways of viewing utility 1. old way a) measures how“ satisfied” you are 1)not operational 2)many other problems a)summarizes preferences b)a utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers at is, u(a1, r2)>u(y1, y2)if and only if(a1, r2)>(y1, y2) d)only the ordering of bundles counts, so this is a theory of ordinal tility 1)op 2) gives a complete theory of demand B. Utility functions are not unique 1. if u(a1, I2)is a utility function that represents some preferences, and f() is any increasing function, then f(u(r1, r2))represents the same preferences 2. why? Because u(r1, r2)>u(y1, y2) only if f(u(a1, a2))>f(u(y1, y2)) 3. so if u(r1, r2)is a utility function then any positive monotonic transfor- mation of it is also a utility function that represents the same preferences C. Constructing a utility function 1. can do it mechanically using the indifference curves. Figure 4.2 2. can do it using the "meaning " of the preferences D. Examples 1. utility to indifference curves a)easy -just plot all points where the utility is constant 2. indifference curves to utility 3. examples a) perfect substitutes - all that matters is total number of pencils, so
Chapter 4 11 elaborate on the idea that xa 1xb 2 is the general form for Cobb-Douglas preferences, but various monotonic transformations (e.g., the log) can make it look quite different. It’s a good idea to calculate the MRS for a few representations of the Cobb-Douglas utility function in class so that people can see how to do them and, more importantly, that the MRS doesn’t change as you change the representation of utility. The example at the end of the chapter, on commuting behavior, is a very nice one. If you present it right, it will convince your students that utility is an operational concept. Talk about how the same methods can be used in marketing surveys, surveys of college admissions, etc. The exercises in the workbook for this chapter are very important since they drive home the ideas. A lot of times, students think that they understand some point, but they don’t, and these exercises will point that out to them. It is a good idea to let the students discover for themselves that a sure-fire way to tell whether one utility function represents the same preferences as another is to compute the two marginal rate of substitution functions. If they don’t get this idea on their own, you can pose it as a question and lead them to the answer. Utility A. Two ways of viewing utility 1. old way a) measures how “satisfied” you are 1) not operational 2) many other problems 2. new way a) summarizes preferences b) a utility function assigns a number to each bundle of goods so that more preferred bundles get higher numbers c) that is, u(x1, x2) > u(y1, y2) if and only if (x1, x2) (y1, y2) d) only the ordering of bundles counts, so this is a theory of ordinal utility e) advantages 1) operational 2) gives a complete theory of demand B. Utility functions are not unique 1. if u(x1, x2) is a utility function that represents some preferences, and f(·) is any increasing function, then f(u(x1, x2)) represents the same preferences 2. why? Because u(x1, x2) > u(y1, y2) only if f(u(x1, x2)) > f(u(y1, y2)) 3. so if u(x1, x2) is a utility function then any positive monotonic transformation of it is also a utility function that represents the same preferences C. Constructing a utility function 1. can do it mechanically using the indifference curves. Figure 4.2. 2. can do it using the “meaning” of the preferences D. Examples 1. utility to indifference curves a) easy — just plot all points where the utility is constant 2. indifference curves to utility 3. examples a) perfect substitutes — all that matters is total number of pencils, so u(x1, x2) = x1 + x2 does the trick
12 Chapter Highlights 1)can use any monotonic transformation of this as well, such as b)perfect complements- what matters is the minimum of the left and right shoes you have, so u(a1, r2)=minfal, r2) works c)quasilinear preferences - indifference curves are vertically parallel Figure 4.4 1)utility function has form u(a1, I2)=v(a1)+a2 Cobb-Dougl glas preferences. Figure 4.5 1)utility has form u(r1, I2)=zfrs 2)convenient to take transformation f(u)=ut and write erg +e 3 or rfa2, where a=b/(b+c) E Marginal utility extra utility from some extra consumption of one of the goods, holding the fixed 2. this is a derivative, but a special kind of derivative a partial derivative 3. this just means that you look at the derivative of u(a1, r2) keeping r2 fixed treating it like a constant a)if u(a1, I2)=51+a2, then MU1=au/ax1=1 b)if u( rar2, then MU1 5. note that marginal utility depends on which utility function you choose to represent preferences a) if you multiply utility times 2, you multiply marginal utility times 2 b)thus it is not an operational concept c)however, MU is closely related to MRS, which is an operational co concept 6. relationship between MU and MRS a)u(a1, a2)=k, where k is a constant, describes an indifference curve b)we want to measure slope of indifference curve, the MRS c)so consider a change(dr1, dr2) that keeps utility constant. Then MU1dx1 MU2dx2=0 dx1+-dx2=0 MUI e)so we can compute MRS from knowing the utility function F Example 1. take a bus or take a car to work? 2. let I1 be the time of taking a car, yi be the time of taking a bus. Let az be cost of car. etc 3. suppose utility function takes linear form U(I,., In)=B1I1+.+BnIn 4. we can observe a number of choices and use statistical techniques to estimate the parameters Bi that best describe choice 5. one study that did this could forecast the actual choice over 93% of the 6. once we have the utility function we can do many things with it a)calculate the marginal rate of substitution between two characteristics 1)how much money would the average consumer give up in order to get a shorter travel time? b)forecast consumer response to proposed changes c)estimate whether proposed change is worthwhile in a benefit-cost sense
12 Chapter Highlights 1) can use any monotonic transformation of this as well, such as log (x1 + x2) b) perfect complements — what matters is the minimum of the left and right shoes you have, so u(x1, x2) = min{x1, x2} works c) quasilinear preferences — indifference curves are vertically parallel. Figure 4.4. 1) utility function has form u(x1, x2) = v(x1) + x2 d) Cobb-Douglas preferences. Figure 4.5. 1) utility has form u(x1, x2) = xb 1xc 2 2) convenient to take transformation f(u) = u 1 b+c and write x b b+c 1 x c b+c 2 3) or xa 1x1−a 2 , where a = b/(b + c) E. Marginal utility 1. extra utility from some extra consumption of one of the goods, holding the other good fixed 2. this is a derivative, but a special kind of derivative — a partial derivative 3. this just means that you look at the derivative of u(x1, x2) keeping x2 fixed — treating it like a constant 4. examples a) if u(x1, x2) = x1 + x2, then MU1 = ∂u/∂x1 = 1 b) if u(x1, x2) = xa 1x1−a 2 , then MU1 = ∂u/∂x1 = axa−1 1 x1−a 2 5. note that marginal utility depends on which utility function you choose to represent preferences a) if you multiply utility times 2, you multiply marginal utility times 2 b) thus it is not an operational concept c) however, MU is closely related to MRS, which is an operational concept 6. relationship between MU and MRS a) u(x1, x2) = k, where k is a constant, describes an indifference curve b) we want to measure slope of indifference curve, the MRS c) so consider a change (dx1, dx2) that keeps utility constant. Then MU1dx1 + MU2dx2 = 0 ∂u ∂x1 dx1 + ∂u ∂x2 dx2 = 0 d) hence dx2 dx1 = −MU1 MU2 e) so we can compute MRS from knowing the utility function F. Example 1. take a bus or take a car to work? 2. let x1 be the time of taking a car, y1 be the time of taking a bus. Let x2 be cost of car, etc. 3. suppose utility function takes linear form U(x1,...,xn) = β1x1+...+βnxn 4. we can observe a number of choices and use statistical techniques to estimate the parameters βi that best describe choices 5. one study that did this could forecast the actual choice over 93% of the time 6. once we have the utility function we can do many things with it: a) calculate the marginal rate of substitution between two characteristics 1) how much money would the average consumer give up in order to get a shorter travel time? b) forecast consumer response to proposed changes c) estimate whether proposed change is worthwhile in a benefit-cost sense
Chapter 5 Choice This is the chapter where we bring it all together. Make sure that students understand the method of maximization and dont just memorize the various special cases. The problems in the workbook are designed to show the futility of memorizing special cases, but often students try it anyway. The material in Section 5. 4 is very important-I introduce it by saying"Why should you care that the MRS equals the price ratio? The answer is that this allows economists to determine something about peoples' trade-offs by observing market prices. Thus it allows for the possibility of benefit-cost analysi The material in Section 5.5 on choosing taxes is the first big non-obvious result from using consumer theory ideas. I go over it very carefully, to make sure that students understand the result, and emphasize how this analysis uses the techniques that we've developed. Pound home the idea that the nalytic techniques of microeconomics have a big payoff--they allow us to answer questions that we wouldnt have been able to answer without these techniques If you are doing a calculus-based course, be sure to spend some time on the appendix to this chapter. Emphasize that to solve a constrained maximization problem, you must have two equations. One equation is the constraint, and one equation is the optimization condition. I usually work a Cobb-Douglas and a perfect complements problem to illustrate this. In the Cobb-Douglas case, the optimization condition is that the MRS equals the price ratio. In the perfect complements case, the optimization condition is that the consumer chooses a bundle at the corner A. Optimal choice 1. move along the budget line until preferred set doesnt cross the budget set 2. note that tangency occurs at optimal point- necessary condition for optimum. In symbols: MRS=-price ratio=-P1/p2 3. tangency is not sufficient. Figure 5.4 a)unless indifference curves are convex
Chapter 5 13 Chapter 5 Choice This is the chapter where we bring it all together. Make sure that students understand the method of maximization and don’t just memorize the various special cases. The problems in the workbook are designed to show the futility of memorizing special cases, but often students try it anyway. The material in Section 5.4 is very important—I introduce it by saying “Why should you care that the MRS equals the price ratio?” The answer is that this allows economists to determine something about peoples’ trade-offs by observing market prices. Thus it allows for the possibility of benefit-cost analysis. The material in Section 5.5 on choosing taxes is the first big non-obvious result from using consumer theory ideas. I go over it very carefully, to make sure that students understand the result, and emphasize how this analysis uses the techniques that we’ve developed. Pound home the idea that the analytic techniques of microeconomics have a big payoff—they allow us to answer questions that we wouldn’t have been able to answer without these techniques. If you are doing a calculus-based course, be sure to spend some time on the appendix to this chapter. Emphasize that to solve a constrained maximization problem, you must have two equations. One equation is the constraint, and one equation is the optimization condition. I usually work a Cobb-Douglas and a perfect complements problem to illustrate this. In the Cobb-Douglas case, the optimization condition is that the MRS equals the price ratio. In the perfect complements case, the optimization condition is that the consumer chooses a bundle at the corner. Choice A. Optimal choice 1. move along the budget line until preferred set doesn’t cross the budget set. Figure 5.1. 2. note that tangency occurs at optimal point — necessary condition for optimum. In symbols: MRS = −price ratio = −p1/p2. a) exception — kinky tastes. Figure 5.2. b) exception — boundary optimum. Figure 5.3. 3. tangency is not sufficient. Figure 5.4. a) unless indifference curves are convex
b)unless optimum is interior 4. optimal choice is demanded bundle b)want to study how optimal choice the demanded bundle-changes as price and income change m/pl if P1 p2: 0 2. perfect complements: I1=m/(p1+p2). Figure 5.6. 4. discrete goods. Figure 5.7 R 3. als and bads a)suppose good is either consumed or not b)then compare(1, m-pi)with(0, m) and see which is better 5. concave preferences: similar to perfect substitutes. Note that tangency doesn't work. Figure 5.8 6. Cobb-Douglas preferences: a1= am/pl. Note constant budget shares, a budget share of good 1 C. Estimating utility function 1. examine consumption data 2. see if you can“ft” a utility function to 3. e. g, if income shares are more or less constant, Cobb-Douglas does a good ob 4. can use the fitted utility function as guide to policy decisions 5. in real life more complicated forms are used, but basic idea is the same D. Implications of MRS condition 1. why do we care that MRS=-price ratio? 2. if everyone faces the same prices, then everyone has the same local trade-off between the two goods. This is independent of income and tastes. 3. since everyone locally values the trade-off the same, we can make policy judgments. Is it worth sacrificing one good to get more of the other? Prices serve as a guide to relative marginal valuations E. Application-choosing a tax. Which is better, a commodity tax or an income tax? 1. can show an income tax is always better in the sense that given any commodity tax, there is an income tax that makes the consumer better off. Figure 5.9 a)original budget constraint: Pif1+ p2:12=m b)budget constraint with tax: (p1 +t) 1+P2 I2=m c)optimal choice with tax: (P1 +t)ri+p2 r*=m d)revenue raised is trl e) income tax that raises same amount of revenue leads to budget con- straint: PlC1+ p2C2=m- tar1 1)this line has same slope as original budget line 2)also passes through(ri, I? 3)proof: Pizi+p2r=m-txl 4)this means that(ai, as) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x1,x2) 3. caveats a) only applies for one consumer for each consumer there is an income tax that is better
14 Chapter Highlights b) unless optimum is interior. 4. optimal choice is demanded bundle a) as we vary prices and income, we get demand functions. b) want to study how optimal choice — the demanded bundle – changes as price and income change B. Examples 1. perfect substitutes: x1 = m/p1 if p1 < p2; 0 otherwise. Figure 5.5. 2. perfect complements: x1 = m/(p1 + p2). Figure 5.6. 3. neutrals and bads: x1 = m/p1. 4. discrete goods. Figure 5.7. a) suppose good is either consumed or not b) then compare (1, m − p1) with (0, m) and see which is better. 5. concave preferences: similar to perfect substitutes. Note that tangency doesn’t work. Figure 5.8. 6. Cobb-Douglas preferences: x1 = am/p1. Note constant budget shares, a = budget share of good 1. C. Estimating utility function 1. examine consumption data 2. see if you can “fit” a utility function to it 3. e.g., if income shares are more or less constant, Cobb-Douglas does a good job 4. can use the fitted utility function as guide to policy decisions 5. in real life more complicated forms are used, but basic idea is the same D. Implications of MRS condition 1. why do we care that MRS = −price ratio? 2. if everyone faces the same prices, then everyone has the same local trade-off between the two goods. This is independent of income and tastes. 3. since everyone locally values the trade-off the same, we can make policy judgments. Is it worth sacrificing one good to get more of the other? Prices serve as a guide to relative marginal valuations. E. Application — choosing a tax. Which is better, a commodity tax or an income tax? 1. can show an income tax is always better in the sense that given any commodity tax, there is an income tax that makes the consumer better off. Figure 5.9. 2. outline of argument: a) original budget constraint: p1x1 + p2x2 = m b) budget constraint with tax: (p1 + t)x1 + p2x2 = m c) optimal choice with tax: (p1 + t)x∗ 1 + p2x∗ 2 = m d) revenue raised is tx∗ 1 e) income tax that raises same amount of revenue leads to budget constraint: p1x1 + p2x2 = m − tx∗ 1 1) this line has same slope as original budget line 2) also passes through (x∗ 1, x∗ 2) 3) proof: p1x∗ 1 + p2x∗ 2 = m − tx∗ 1 4) this means that (x∗ 1, x∗ 2) is affordable under the income tax, so the optimal choice under the income tax must be even better than (x∗ 1, x∗ 2) 3. caveats a) only applies for one consumer — for each consumer there is an income tax that is better
b) income is exogenous -if income responds to tax, problems nly looked at demand side Appendix- solving for the optimal choice 1. calculus problem-constrained maximization 3. method 1: write down MRS=Pi/p2 and budget constraint and solve. 4. method 2: substitute from constraint into objective function and solve. 5. method 3: Lagranges method ngan:L=u(a1, I2)-Ar b)differentiate with respect to T1, I2, A c)solve equations 6. example 1: Cobb-Douglas problem in book 7. example 2: quasilinear preferences a)max u(an)+r2 s.t. P1Z1+I2 b)easiest to substitute, but works each way
Chapter 5 15 b) income is exogenous — if income responds to tax, problems c) no supply response — only looked at demand side F. Appendix — solving for the optimal choice 1. calculus problem — constrained maximization 2. max u(x1, x2) s.t. p1x1 + p2x2 = m 3. method 1: write down MRS = p1/p2 and budget constraint and solve. 4. method 2: substitute from constraint into objective function and solve. 5. method 3: Lagrange’s method a) write Lagrangian: L = u(x1, x2) − λ(p1x1 + p2x2 − m). b) differentiate with respect to x1, x2, λ. c) solve equations. 6. example 1: Cobb-Douglas problem in book 7. example 2: quasilinear preferences a) max u(x1) + x2 s.t. p1x1 + x2 = m b) easiest to substitute, but works each way