Question 5
Problem 6 Repeat Worked-Out Problem 10.4, but assume that in Angela's view, one kg of food this year and one kg of food next year are perfect complements. In other words, assume that her indifference curves are L- shaped, and that the corner of the L in each curve lies on the 45-degree line. How would you interpret her preferences? Worked-Out Problem 10.4 Angela cares only about food this year and food next year. Let's use Fo to stand for food this year and F, to stand for food next year (both in kilograms). Her preferences correspond to the utility function U(Fa, Fi) = Fox Fi. For that utility function, the marginal utility of food this year, MUn, is Fi, and the marginal utility of food next year, MUs, is Fo. (To understand why, refer back to Section 4.4.) Suppose she earns $100 this year and nothing next year. Food costs $1 per kg in both years and the interest rate is 10 percent. How much does she consume this year and how much does she save? Write a formula for her saving as a function of the interest rate, R. Does she save more, less, or the same amount when the interest rate rises? Let's write the marginal rate of substitution for food this year with food next year as MRS.. Using the fact that MRSor = MU./MU, along with MU. = F, and MUs = Fo, we have MRSo = F1/Fo. Because each of Angela's indifference curves has a declining MRS, any bundle on her budget line that satisfies the tangency condition also maximizes her utility. To determine Angela's best choice, we solve two equations in two unknowns. The first equation tells us that the solution must satisfy the tangency condition: Angela's marginal rate of substitution for food this year with food next year must equal the slope of her budget line (times negative one). Using formula (11) along with the fact that food costs $1 per kilogram both this year and next year, we see that the slope of her budget line is simply -(1 + R). Therefore, we can write the tangency condition as follows: Tangency Condition: F,IF, = 1 + R The tangency condition tells us that Angela's food consumption rises at the same rate as the interest rate. The second equation tells us that the solution must lie on the budget line, which we can write as follows: Budget Line: Fo + + R' F1 = 100 Now we solve the two equations for Fo and Fi. The tangency condition implies that F, = (1 + R)Fo. Using that expression to substitute for F, in the equation for the budget line, we obtain 2Fo = 100, or equivalently Fo = 50. So, this year Angela spends $50 and saves $50; next year she spends $50(1 + R). When the interest rate is 10 percent, she consumes 50 kg of food this year and 55 kg next year. The formula for saving, S, is simply S = 100 - F, = 50. Angela's saving doesn't depend on the interest rate
