Suppose Laura has five units of both x and y. Calculate a) the utility level, b) the marginal. utility when Îx = 1, and
Consumer Choice ECN205 Spring 2017 Wake Forest University Instructor: McFall 1. Amanda’s preferences between waffles (w) and pancakes (p) are shown below. w
I3 I2 I1 p a) Pick three points on I1 and plot those points on the graph. (You’ll have to come up with the bundles.) b) Pick a point on I2 and plot it. Why is the bundle you plotted on I2 preferred to the bundles you plotted on I1? c) Why can’t these indifference curves cross each other? 2. Suppose Laura has five units of both x and y. Calculate a) the utility level, b) the marginal utility when Δx = 1, and c) the marginal utility when Δy = 1, under the following utility functions: 1) U = x0.5y0.5, 2) U = 3x + 2y, and 3) U = min(0.2x, y). How does this answer change when she has ten units of x and five units of y? 3. Graph separately the utility functions you see above, making at least three utility levels for each function. 4. Given three utility functions, U = (xy)0.5, U = x0.6y0.4, and U = x0.4y0.6, a) write dU/dx, b) dU/dy, and c) calculate the difference in marginal utility of x at x = 5 and x = 10. (Set y = 1.) Why is MUx different across the three utility functions? Why is MUx always smaller when x = 10 compared to x = 5? (Explain with economic intuition, not mathematical intuition.) 5. Given the three functions shown above, what is the marginal rate of substitution at x = y = 5, if Δx = -1. What about if Δy = -1? For each utility function, what happens to MRS at x = 10 and y = 2.5? 6. Justin can buy either units of pizza (p) or burritos (b). The price of pizza (Pp) is $9/unit. The price of a burrito (Pb) is $6/unit. He has $36/week to spend on these goods. (And he will spend it all because he can’t save it.)
a) Write his budget constraint. b) Identify two different bundles that he can consume given the prices and income constraint he faces currently. c) What is the marginal rate of transformation of pizza that he currently faces? d) What happens to the marginal rate of transformation when the price of pizza increases? e) Given the original prices, what happens to his budget constraint when his income increases to $45/week? 7. Let’s ponder maximizing utility under a budget constraint. Assume a person has a budget constraint in which income is $40/week, the price of X (px) is $4/unit, and the price of y (py) is $5/unit. First, assume that this person has a utility function of U = x0.5y0.5. Calculate the utility level when she has 2.5x and 6y, a bundle that lies on her budget constraint. Now calculate the marginal rate of substitution at this bundle as the ratio of marginal utility of y over the marginal utility of x. (MRS = -MUy/MUx) How does the marginal rate of substitution you calculated compare to the marginal rate of transformation at 2.5x and 6y? Now, calculate the utility level she has when she consumes 5x and 4y. How does the marginal rates of substitution compare to the marginal rate of transformation at 5x and 4y? Finally, calculate the utility level when she consumes 8x and 1.6y. What is the marginal rate of substitution compared to the marginal rate of transformation 8x and 1.6y? (Hint: In all three situations, the marginal rate of transformation is always -4/5x/y, and the marginal rate of substitution of x should be smaller than this when the initial bundle is considered.) 8. Graph the three points, the budget constraint, and the utility functions you just pondered in question 7. 9. Suppose that Chris has a utility function in which U = 5x + 3y. Assume he can consume fractional units of x and y. If the marginal rate of transformation is -2y/3x, what should he do in order to maximize utility?
