ECON 300 Homework 2 Due Wednesday, October 20 th 1. Jack s preferences for jeans (J) and shirts (S) are given by the utility function: a. Derive a formula for the marginal utility of jeans b. Derive a formula for the marginal utility of shirts c. Derive a formula for the marginal rate of substitution of shirts for jeans d. Draw 1 indifference curve that shows Jack s preferences (Hint: start with some point, then figure out what other points on the same indifference curve would be) I ll start with the point (3,3). At this point, U(3,3)=27 All other points on this indifference curve must also have U(J,S)=27 So when S=1, When S=2, When S=3, Alternatively, you could solve this equation for S and graph that equation:
e. Jack has $150 to spend on clothes every month. The price of jeans is $50 and the price of shirts is $20. What is Jack s marginal rate of transformation between jeans and shirts? f. Set up the Lagrangian and used constrained optimization to determine how many jeans and shirts Jack will buy. Re-write (1) as Re-write (2) as Plug (4) into (3) Jack will buy 1 shirt and 5 pairs of jeans. 2. Many water utilities charge customers in block pricing, where the price charged increases as you consume more water. For instance, say Sarah s utility charged her the following rates for water: $3/hundred cubic feet (HCF) for the first 15 HCF consumed, $4/HCF for the second HCF consumed and $5/HCF for any water consumed over 30 HCF. Sarah has $200 a month to spend on water and food (we ll measure in meals). Let s say meals cost $5 each. Draw Sarah s budget curve for food vs. water. If Sarah spent all her money on food, she could buy $200/$5 = 40 meals, so the intercept on the food axis if 40. If Sarah spent all her money on water, it would cost her $45 for the first 15 HCF (15*$3/HCF) and $60 for the second 15 HCF (15*$4/HCF). So she would spend $105 on the first 30 HCF. That would leave $95 to buy water at $5/HCF, so she could buy 19 more HCF. Thus if she spent all her money on water, she could buy 15 + 15 + 19 = 49 HCF. We know the slope of the budget constraint will change at 15 and 30 HCF, since the price of water gets more expensive. If water is on the x-axis, the slope will get steeper since the slope is. At w=15, the remaining income of 155 buys 31 meals. At w=30, the remaining income of $95 would buy 19 meals. So we can plot the points where the slope changes at (15,31) and (30, 19).
3. We ve talked about 2 special cases of goods, perfect substitutes and perfect complements. For both of these types, a. Give a personal example of goods that are perfect substitutes and perfect complements For me, I m indifferent to Heintz vs. Hunts brand ketchups, so those would be perfect substitutes. My bike lock and the key to it are perfect complements, since they are only useful if I have both. b. Draw an indifference curve for each that goes through the point (3,3) c. Draw an example of a budget constraint and show where the utility-maximizing bundle is
d. Make up a utility function for each type Perfect substitutes: Perfect complements: Chapter 4 problems: 8, 14, 16, 26, 33, 39 8. 14. The 10,000 gallon limit won t affect Max s optimal bundle if that bundle includes water consumption of less than 10,000 gallons, since the limit would not be binding (as seen on left-hand graph). If Max was consuming more than 10,000 gallons, then the limit will be binding and Max will have to consume less water. This will put him on a lower indifference curve, since if consuming less water brought him greater utility, he would have already been consuming less water. The highest indifference curve he can hit is when he s consuming 10,000 gallons, since he would have greater utility from more water consumption, this is as high as his water consumption (and utility) can be.
16. Since consumers in both cities maximize their utility, we know that the marginal rate of substitution must be equal to the marginal rate of transformation for both groups of customers. In Boston consumers pay twice as much for avocados as tangerines, so The marginal rate of substitution of avocados for tangerines is utility, the MRS equals the MRT,. In San Diego, the price of avocados is half the price of tangerines, so. When consumers are maximizing, so the MRS of Boston consumers is -1/2 while the MRS of San Diego consumers is -2. So consumers in San Diego have the higher MRS, in that they will give up more avocados to get an additional tangerine. 26. The slope of the budget line if you don t join the pool is You will consume where your indifference curve is tangent to this budget line, shown by A on the graph. The slope of the budget line if you do join the pool is. Thus to be indifferent between joining and not joining the pool, you would have to be on the same indifference curve as before, but with it tangent to a budget line with a slope of -5, as shown at point B. Since the price of other goods is 1, you can buy Y goods if you don t join the pool and Y-F goods if you do, the fee to join F will be the size required to make a budget line with slope of -5 be tangent to the same IC. 33. Yes, those two utility functions do give the same amount of utility. We are really only interested in the rank of utility values, not the actual values. Squaring and adding a constant to each possible value of U(Z,B) would not change the rank of the various utilities. 39.
Chapter 5 problems: 5, 9, 10, 19, 30, 32, 34 5. a. The substitution effect causes her to buy more clothing when the price of clothing decreases. Clothes are now relatively less expensive in terms of foregone food than they used to be, Michelle will want to shift her consumption to include more clothes and less food. b. When the price of clothes decreases, it increases Michelle s real income since her buying power has increased. If clothes are a normal good, the income effect will cause Michelle s consumption of clothes to increase. If clothes are an inferior good, then the income effect will cause her consumption of clothes to decrease. The income effect is ambiguous in this case since we only know the total affect. On the one hand, it could be that both the substitution and income effects increase consumption (i.e. clothes are a normal good). On the other hand, it could be that clothes are inferior but the substitution effect is stronger than the income effect, so that the total effect would be an increase in consumption. 9. During his first year, Ximing spends $400 on textbooks. For the second year, Ximing s dad s offer of $80 would increase the amount Ximing has to spend on books by 20% (b/c 80/400=0.20). While that would exactly cover the increase in textbook prices if all book prices rose 20% (leaving Ximing the same before and after the price increase), in this case used textbooks only increase by 10%. That means Ximing will substitute away from new books towards used books, and he therefore would not need a full 20% increase in income to buy all the books he would need. He will be better off since his dad didn t consider the substitution effect and thus overestimated the amount it would require cover textbook price increases. 10. Jean is equally well off as she was before the cost of living adjustment. That is because with perfect complements, there is no substitution effect. If the price of cream increases by more than the price of coffee (as shown below), Jean will not be better off by consuming more coffee and less cream since she uses these goods together only. The COLA is designed to maintain Jean s current consumption, so she maintains her current utility.
19. The constraint on Bessie only working only 8 hours decreases her utility, since she would have chosen to work more (the constraint is binding). Working 8 hours puts her on IC 2,which brings less utility than IC 1.
30. Ice cream IC 1 IC 2 BL 1 BL 2 Pie Price P 1 P 2 Demand Pie Q 1 Q 2 If ice cream (I) and pies (P) are perfect complements for Olivia, then her L-shaped indifference curves will be tangent to budget lines where. This can be substituted into the budget constraint,. 32.
34. Before the tax, Steve s optimal utility is when he consumes 60 packs of cigarettes and 30 hamburgers. With the tax, After the tax, Steve s optimal utility is when he consumes 60 packs of cigarettes and 20 hamburgers.
4. Grant earns $20/hour and gets an allowance of $100/week. He has 168 hours per week to spend on work and leisure (L). The price of consumption goods (C) is $1. a. Graph Grant s budget constraint. C C = 3460 Slope = -20 L = 0 Kink @ H = 0 L = 168 L b. Suppose that Grant s utility function is U(L,C)=LC. Solve for Grant s optimal choice of labor supply, leisure and consumption. Illustrate your answer in a diagram including the budget constraint and the indifference curve corresponding to his optimal choice.
(1) Budget Line: C = 3460 20L (2) MRS = P L /P C => C/L = 20 => C = 20L (1) and (2) => 20L = 3460 20L => 40L = 3460 L* = 3460/40 86.5 H* = 168 86.5 = 81.5 C* = 86.5*20 = 1730 C C = 3460 L = 0 L* = 86.5 H* = 81.5 Kink @ H = 0 L = 168 L b. Suppose Grant can earn time and a half for all work beyond 40 hours. Graph his budget constraint now. Kink is @ H = 40, L = 128, C = 100+20*40 = 900 Cmax = 900 + 128*30 = 4740 C C = 4740 L = 0 Slope = -30 Kink @ L = 40, H = 128, C = 800 Slope = -20 Slope = -20 Kink @ H = 0, L = 168 L d. Suppose instead that if Grant earns more than $300 he is subject to a tax of 10% on all labor earnings beyond that $300. Graph his budget constraint (drop the assumption from part c). Kink @ WH = 300, 20H = 300, H = 15, L = 168 15 = 153, C = 300 + 100 = 400 (Point A ) Cmax = 400 + 18*153 = 3154 W when H < 15 = 20 W when H 15 = 20*.9 = 18
This is a progressive tax like the US income tax. C C = 3154 L = 0 Kink @ A Slope = -18 Slope = -20 Kink @ H = 0 L = 168 L