The formula to calculate CPI between a base year (say, Year 0) and another, specified year, say Year 1, is CPI = (Expenditure on basket in Year 1)/(Expenditure on basket in Year 0)×100. Suppose an economy produces only two goods, food (f) and clothing (c). Let quantities and prices in Year 0 be qf0, qc0, pf0 and pc0, respectively, and let us adopt similar notations for Year 1, with the caveat that, for CPI, the quantities remain the same as in Year 0.
a) Calculate, using the notations in the problem, the expenditure on the basket in Year 0, the expenditure on the basket in Year 1, and the CPI.
b) Find the formulas for the share of each of the two goods in the expenditure on the basket in Year 0. Let sf and sc be the two shares. Show that the sum of the two shares is equal to 1. Let us multiply each of these shares by 100 and call them weights, wf and wc. Check that their sum is equal to 100.
c) Using the notations wf and wc for the two weights, re-write the CPI formula.
d) Replace wf and wc in the formula that you just wrote, and compare the result to the formula of definition for CPI. Are they identical? If not, what changes do you need to make in order to make them identical?
e) Did you need to know quantities in Year 1 in order to compute CPI?

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The Answer of The formula to calculate CPI between a base year (say, Year 0) and another, specified year, say Year 1, is CPI = (Expenditure on basket in Year 1)/(Expenditure on basket in Year 0)×100. Suppose an economy produces only two goods, food (f) and clothing (c). Let quantities and prices in Year 0 be qf0, qc0, pf0 and pc0, respectively, and let us adopt similar notations for Year 1, with the caveat that, for CPI, the quantities remain the same as in Year 0.
a) Calculate, using the notations in the problem, the expenditure on the basket in Year 0, the expenditure on the basket in Year 1, and the CPI.
b) Find the formulas for the share of each of the two goods in the expenditure on the basket in Year 0. Let sf and sc be the two shares. Show that the sum of the two shares is equal to 1. Let us multiply each of these shares by 100 and call them weights, wf and wc. Check that their sum is equal to 100.
c) Using the notations wf and wc for the two weights, re-write the CPI formula.
d) Replace wf and wc in the formula that you just wrote, and compare the result to the formula of definition for CPI. Are they identical? If not, what changes do you need to make in order to make them identical?
e) Did you need to know quantities in Year 1 in order to compute CPI?

The Formula to Calculate CPI Between a Base Year (Say