EconS 301 Intermediate Microeconomics. Review Session #3

Transcription

1 EconS 0 ntermediate Microeconomics Review Session #. A consumer purchases two goods, food ( ) and clothing ( ). Her utilit function is given b U(, ) +. The marginal utilities are + and. The price of food is, the price of clothing is, and the consumer s income is. a) What is the equation for the demand curve for clothing? Setting up the tangenc condition implies + ( + ) Substituting this result into the budget line implies + ( + ) (+ ) + + b) s clothing a inferior good in this case? Since the amount of clothing purchased will increase as income increases, as noted b the demand curve, clothing is a normal good, and not an inferior good.

2 . onsider a consumer who purchases two goods, and. The consumer s utilit function is U(, ) with and. n addition, the demand curve for is given b. Assume initiall that the consumer s income is $0, the price of is $8, and the price of is $. a) rom the given information determine ) the utilit maimizing amount of, ) the utilit maimizing amount of, and ) the total utilit at the utilit maimizing bundle. Using the demand curve for we can find the utilit maimizing amount of. 0 () 80 Since each unit of costs $, the consumer will spend $(80) $80 on. This leaves $80 to spend on. Since costs $8, the utilit maimizing amount of is therefore 0 units. At this bundle, total utilit is U 0(80) 800. b) Now assume the price of increases to $. Recompute the values from part a) at the new price. Again, using the demand curve for we can find the utilit maimizing amount of. 0 () 0 Since each unit of costs $, the consumer will spend $(0) $80 on. This leaves $80 to spend on. Since costs $8, the utilit maimizing amount of is therefore 0 units. At this bundle, total utilit is U 0(0) 00. c) Determine the decomposition basket that identifies the substitution and income effects as the consumer moves from the optimal basket in part a) to the optimal basket in part b).

3 To determine the decomposition basket we note that this basket must satisf two conditions. irst, the decomposition basket must have total utilit the same as at the initial prices, U(, ) 800. Second, the decomposition basket must satisf the tangenc condition at the new prices. This implies 8 These two conditions give two equations in two unknowns: 800 and. Substituting the second equation into the first and solving implies 00. and 00.. d) dentif the substitution and income effects as the consumer moves from the initial consumption basket to the final consumption basket. nitiall, the consumer chose to consume 80 units of and at the final basket the consumer chose to consume 0 units of. Therefore, the substitution effect is the initial amount less the amount in the decomposition basket, 80.. and the income effect is the amount in the decomposition basket less the final amount,. 0.. Notice that these two amount sum to the total change of 0 units.. Karl s preferences over hamburgers (H) and beer (B) are described b the utilit function: U(H,B) min(h,b). His monthl income is dollars, and he onl bus these two goods using his income. Denote the price of hamburgers b H and for beer B. a) Derive Karl s demand curve for beer as a function of the eogenous variables. Recall that goods that are compliments have utilit functions in the min functional form, where the indifference curves are L shaped. And, from the utilit function, we know Karl s optimal bundle will alwas be such that H B. That is, he will alwas consume in the ratio of hamburger for ever /rds beer, or an other combination following that ratio such that H B is satisfied. f this were not true then he could decrease the consumption of one of the two goods, staing at the same level of utilit and reducing ependiture. Also, at the optimal bundle, it must be true that H H + B B. Solving the first condition for H, that is H/B, and substituting into the second we get

4 B(.H + B ) which implies that the demand curve for beer is given b, B (. H + B ) ) arina bus two goods, food and clothing, with the utilit function U +. Her marginal utilit of food is + and her marginal utilit of clothing is. She has an income of 0. The price of clothing is. a) Derive the equation representing arina s demand for food, and draw this demand curve for prices of food ranging between and. + Tangenc: / /. ( + )/ / > +. (Eq ) Budget Line: +. > + 0. (Eq ) Substituting (Eq ) into (Eq ): Thus, independent of. rom the budget line, we see that + () 0, so the demand for is /. Demand for food b) alculate the income and substitution effect on arina s consumption of food when the price of food rises from to, and draw a graph illustrating these effects. nitial Basket: rom the demand for food in (a), /, and. Also, the initial level of utilit is U + () +. inal Basket: rom the demand for food in (a), we know that /, and. And the final level of utilit is, U () + 9. Decomposition Basket: Must be on initial indifference curve, with U + (Eq ) Tangenc condition satisfied with final price: / /. ( + )/ / > +. (Eq ) Eq can be written as ( + ). lugging Eq into the rewritten Eq we have ( + ), and thus,. Also, b Eq,. So the decomposition basket is,. ncome effect on : final basket decomposition basket -. Substitution effect on : decomposition basket initial basket -.

5 lothing nitial U 7 inal U Decomp Basket inal BL nitial Basket inal Basket Decomp BL nitial BL nc Effect - Subst Effect - ood c) Determine the numerical size of the compensating variation (in monetar terms) associated with the increase in the price of food from to. + () + (). So she would need an additional income of (plus her actual income of 0). The compensating variation associated with the increase in the price of food is -.