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 better14 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 con￾straint: 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