Econ 301 Name Problem Set 5 Key 10 October 2007 Perloff, chapter 5 1. George the Gourmet consumes only two goods, pizzas and hamburgers. His utility is determined solely by his consumption of these two goods. The table below gives some of the combinations of goods along some of his indifference curves. His indifference curves have all the properties that well-behaved indifference curves should have. U 0 U 1 U 2 U 3 Pizzas Burgers Pizzas Burgers Pizzas Burgers Pizzas Burgers 10 1 11 2 12 3 14 5 4 2 7 3 7 5 10 6 2 5 4 5 5 7 6 10 1 10 2 12 3 12 4 17 George s allowance is 32 dollars a week, and he spends all this money each week on consumption of these two goods. A pizza costs 4 dollars, and a burger costs $3.20. a) How many pizzas and burgers, respectively, will George consume per week? Solve the problem graphically. Given George s budget, his utility is maximized when he consumes 4 Pizzas and 5 Burgers.
b) Work out at least two points along George s demand curve for burgers, and use these points to sketch his demand curve. Recall that the demand curve describes the relationship between price and quantity demanded. To construct the demand curve, we can experiment with various burger price levels, holding all else equal (i.e. holding preferences, income and the price of other goods constant). As we experiment with new burger prices, the budget constraint in our preference space will change. From the budget constraint corresponding to the new price, we can determine the new quantity demanded. If we repeat the exercise for many price levels, we can construct George s demand curve for burgers. c) Suppose that the government feels that George should be able to afford to buy 10 pizzas per week, and wants to decide how to bring this about. One option being considered is to give a subsidy to pizza producers that will lower the price of pizzas sufficiently that George can buy 10 pizzas per week using his current income. How much would the government need to subsidize each pizza in order to achieve this? How much would it cost to subsidize 10 pizzas in this fashion? For George to be able to afford 10 pizzas, the price of pizzas must be at most $3.20 his $32 income is exhausted in this case. Since the current price of pizza is $4.00, then government must subsidize the $0.80 difference between the prices. It would cost the government a total of $8.00 to subsidize these 10 pizzas. (Alternatively, you could calculate what the price of pizza would need to be to induce George to actually buy 10 pizzas.)
d) An economist suggests that if the government simply gave that money (i.e., the money needed to subsidize 10 pizzas) directly to George, rather than to the pizza producers, he would probably be better off. Is the economist right? Solve the problem graphically. An $8.00 payment increases George s income to $40. Graphically, we can see how the income subsidy allows George to achieve a higher utility than the price change offered by the government in part (c).
2. The federal government provides poor households with food stamps. These are vouchers that can be exchanged for food; if a food stamp has a nominal value of, e.g., 100 dollars, it can be used to buy 100 dollars worth of food. Using a standard utility theory framework, with a consumer choosing between two goods Food and Other stuff try to draw indifference maps for a) a consumer who is worse off with a food stamp with a nominal value of 100 dollars than she/he would be if the government gave her/him 100 dollars in cash, b) a consumer who is just as well off with a food stamp with a nominal value of 100 dollars as she/he would be if the government gave her/him 100 dollars in cash, a) This is one example of preferences that result in a strict preference for cash over food stamps. Any indifference mapping that results in choices of positive other good consumption will lead to a similar conclusion. (Alternatively, you could draw a single budget constraint with preferences that suggest that the individuals would prefer to spend less than $100 on food. Then, the individual is worse off with the food stamps than with the cash being forced to spend at least $100 on food limits his/her options and leaves him/her on a worse indifference curve than the one he/she could reach with $100 in cash.)
b) If the consumer spends all his money on food, then he is indifferent between food stamps and cash he can use both with equal ease at the grocery store. (Alternatively, you could draw a single budget constraint with preferences that suggest that the individual wants to spend at least $100 on food anyway. Then, the individual may use the food stamps for food and the leftover cash for other goods.)
3. 4. Suppose the demand curve for bus travel is downward sloping, and the income elasticity of demand for bus travel is negative. (i) (ii) Design an indifference curve-budget line diagram showing the substitution and income effects created when the price of bus travel falls. In your diagram, place bus travel on the horizontal axis and all other goods on the vertical axis. How you can tell from your diagram that the income elasticity of demand for bus travel is negative? Explain. (i) Bus travel is an inferior good, so the substitution and income effects must be in opposite directions. Since the demand curve for bus travel is downward sloping, the substitution effect must be larger than the income effect. This situation is shown in the accompanying diagram, where point A is the initial optimum and point B is the final optimum.
(ii) Consider points C and B in the diagram. The parallel shift in the budget line shows a rise in income, but quantity demanded is falling from C to B. Therefore, income and quantity demanded are moving in opposite directions, so bus travel is an inferior good. The income elasticity of demand for inferior goods is negative.