Using the constraint, solve for x as a function of y and ¯ u. Substitute your solution into the objective function. Now you are choosing only one variable, y, to minimize expenditure.

This question demonstrates why the CPI may be a misleading measure of inflation. Go back to Micro Theory. A consumer chooses two goods x and y to minimize expenditure subject to achieving some target level of utility, ¯ u. Formally, the consumer’s problem is
min x,y
E=pxx+pyy s.t. ¯ u=xαyβ Total expenditure equals the price of good x times the number of units of x purchased plus the price of good y times the number of units of y purchased. α and β are parameters between 0 and 1. px and py are the dollar prices of the two goods. All the math required for this problem is contained in Appendix A.
(a) Using the constraint, solve for x as a function of y and ¯ u. Substitute your solution into the objective function. Now you are choosing only one variable, y, to minimize expenditure.
(b) Take the first order necessary condition for y.
(c) Show that the second-order condition is satisfied. Note, this is a one variable problem. (d) Use your answer from part b to solve for the optimal quantity of y, y∗. y∗should be a function of the parameters α and β and the exogenous variables, px,py and ¯ u. Next, use this answer for y∗ and your answer from part a to solve for the optimal level of x, x∗. Note, the solutions of endogenous variables, x∗and y∗in this case, only depend on parameters and exogenous variables, not endogenous variables. (e) Assume α=β=0.5 and ¯ u=5. In the year 2000, px=py=$10. Calculate x ∗ 2000,y∗2000 and total expenditure, E2000. We will use these quantities as our “consumption basket” and the year 2000 as our base year. (f) In 2001, suppose py increases to $20. Using the consumption basket from part e, calculate the cost of the consumption basket in 2001. What is the inflation rate?
(g) Now use your results from part d to calculate the 2001 optimal quantities x∗2001 and y∗2001 and total expenditures, E2001. Calculate the percent change between expenditures in 2000 and 2001.
(h) Why is the percent change in expenditures less than the percent change in the CPI? Use this to explain why the CPI may be a misleading measure of the cost of living.