This question demonstrates why the CPI may overstate ination in years following the base year and understate it in years prior to the base year. It highlights the value of chain weight- ing approaches. Consider the following economy where individuals only consume apples and bananas. | Prices per unit Year 1 I Year 2 l Apples 2 | 1 | Bananas 2 4 If a consumer buys QA apples and Q3 bananas and the prices are PA and PB, then she must spend: PAQA + PBQB- Suppose all consumers in this economy get utility (happiness) from consuming apples and ba- nanas. The amount of utility that a consumer gets from consuming QA apples and QB bananas is: 1/2 1/2 A B ' Further assume that all consumers have a hard constraint of having 4 units of happiness (utils) every period. Consumers want to spend the smallest (the minimal) amount they can to achieve this constraint of 4 utils. The consumer's problem is given by: min E : PAQA + PBQB (1) QAaQB st. 1 2 1 2 A\" B/ = (2) where QA is the quantity of apples, Q B is the quantity of bananas and PA and P3 are the prices of apples and bananas respectively. So, E is the (least) cost of achieving a given target level (4 utils) of happiness. This is a model grounded measure of the cost of living. We will also use this choice problem to generate quantities consumed in the economy. A) Solve for the quantities QA and Q B the individual would choose to minimize her expendi- ture given prices in Year 1. B) Solve for the quantities QA and Q3 the individual would choose to minimize her expendi- ture given prices in Year 2. C) Calculate the percentage change in expenditures, E, between year 1 and year 2. D) You have in part C calculated how E and, hence, the cost of achieving a given level of happiness changes. How would a conventional CPI index measure the change in the cost of living? Below use the quantities QA and Q3 you have calculated in parts A and B to dene baskets for different base years. Set year 1 to be the base year. Calculate the Consumer Price Index in years 1 and 2 and the rate of ination between years 1 and 2 using this price index. E) Why is the percent change in actual expenditures less than the percent change in presumed expenditures under the CPI? Use this to explain why the CPI may overstate or understate the cost of living. F) Repeat step D) using year 2 as the base for the CPI (and the consumers' choices in that year as the basket). G) Finally compute a geometric average of C'F'Ig'ase:1 and C PIE\";2 for each If (as is done when building chain weighted price measures, see Lecture Notes!) Calculate the ination rate implied by these two geometric averages. Comment on your results