Problem 3
a. \(A\) consumer is willing to trade 3 units of \(x\) for 1 unit of \(y\) when she has 6 units of \(x\) and 5 units of \(y .\) She is also willing to trade in 6 units of \(x\) for 2 units of \(y\) when she has 12 units of \(x\) and 3 units of \(y .\) She is indifferent between bundle (6,5) and bundle \((12,3) .\) What is the utility function for goods \(x\) and \(y\) ? Hint: What is the shape of the indifference curve? b. \(A\) consumer is willing to trade 4 units of \(x\) for 1 unit of \(y\) when she is consuming bundle (8,1) She is also willing to trade in 1 unit of \(x\) for 2 units of \(y\) when she is consuming bundle (4,4) She is indifferent betwecn these two bundles. Assuming that the utility function is CobbDouglas of the form \(U(x, y)=x^{\alpha} y^{\beta},\) where \(\alpha\) and \(\beta\) arc positive constants, what is the utility function for this consumcr? c. Was there a redundancy of information in part (b)? If yes, how much is the minimum amount of information required in that question to derive the utility function?
Problem 4
Consider the following utility functions: a. \(U(x, y)=x y\) b. \(U(x, y)=x^{2} y^{2}\) c. \(U(x, y)=\ln x+\ln y\) Show that cach of these has a diminishing \(M R S\) but that they exhibit constant, increasing, and decreasing marginal utility, respectively. What do you conclude?
Problem 7
The Phillic Phanatic always cats his ballpark franks in a special way; he uses a foot-long hot dog together with preciscly half a bun, 1 ounce of mustard, and 2 ounces of pickle relish. His utility is a function only of these four items and any extra amount of a single item without the other constituents is worthless. a. What form does PP's utility function for these four goods have? b. How might we simplify matters by considering PP's utility to be a function of only one good? What is that good? c. Suppose foot-long hot dogs cost \(\$ 1.00\) each, buns cost \(\$ 0.50\) cach, mustard costs \(\$ 0.05\) per ounce, and pickle relish costs \(\$ 0.15\) per ounce. How much does the good defined in part (b) cost? d. If the price of foot-long hot dogs increases by 50 percent (to \(\$ 1.50\) each), what is the percentage increase in the price of the good? c. How would a 50 percent increase in the price of a bun affect the price of the good? Why is your answer different from part (d)? f. If the government wanted to raise \(\$ 1.00\) by taxing the goods that PP buys, how should it spread this tax over the four goods so as to minimize the utility cost to PP?
Problem 8
Graph a typical indifference curve for the following utility functions and determine whether they have convex indifference curves (that is, whether the \(M R S\) declines as \(x\) increases). a. \(U(x, y)=3 x+y\) b. \(U(x, y)=\sqrt{x \cdot y}\) c. \(U(x, y)=\sqrt{x}+y\) \(\mathrm{d} . U(x, y)=\sqrt{x^{2}-y^{2}}\) e. \(U(x, y)=\frac{x y}{x+y}\)
Problem 10
Imagine two goods that, when consumed individually, give increasing utility with increasing amounts consumed (they are individually monotonic) but that, when consumed together, detract from the utility that the other one gives. (One could think of milk and orange juice, which are fine individually but which, when consumed together, yield considerable disutility.) a. Propose a functional form for the utility function for the two goods just described. b. Find the \(M R S\) between the two goods with your functional form. c. Which (if any) of the general assumptions that we make regarding preferences and utility functions does your functional form violate?
Problem 12
Consider the function \(U(x, y)=x+\ln y .\) This is a function that is used relatively frequently in economic modeling as it has some useful properties. a. Find the \(M R S\) of the function. Now, interpret the result. b. Confirm that the function is quasi-concave. c. Find the equation for an indifference curve for this function. d. Compare the marginal utility of \(x\) and \(y .\) How do you interpret these functions? How might consumers choose between \(x\) and \(y\) as they try to increase their utility by, for example, consuming more when their income increases? (We will look at this "income effect" in detail in the Chapter 5 problems.) e. Considering how the utility changes as the quantities of the two goods increase, describe some situations where this function might be useful.
