CLASS #3: INFLATION, CONSUMER PRICE INDEX, AND RELATIVE PRICES.

So far we have been talking about nominal variables, i.e., GDP measured in Pesos;...what if you want to know whether Mexico has really grown from 1988 to 1998? Do nominal variables say something about whether Mexicans' living standards have improved in the past ten years?

No! Why? Inflation! What do we really mean when we say ``inflation''?

Inflation has something to do with prices,..which prices? $\Longrightarrow$Theory of Index Numbers

• once upon a time, gold was the numeraire: goods were measured in terms of their relative price with respect to gold
• nowadays, prices are quoted in terms of a different numeraire: money yet, nor money nor gold say anything about the real living standards of individuals

Some commonly used price indices

• GDP Deflator
  
  $$P_{gdp}=\frac{P_{apples}^{98}Q_{apples}^{98} +P_{oranges}^{9... ..._{apples}^{88}Q_{apples}^{98}+P_{oranges}^{88}Q_{oranges}^{98}}$$
  
  
  variable weight price index: use current basket
• CPI
  
  
  $$CPI=\frac{P_{apples}^{98}Q_{apples}^{88}+P_{oranges}^{98}Q_{o... ...pples}^{88}Q_{apples}^{88}+P_{oranges}^{88}Q_{oranges}^{88}}$$
  
  
  fixed weight price index: use base year basket
• Real growth
  
  
  $$g=\ln Y_{1998}-\ln Y_{1997}-(\ln CPI_{1998}-\ln CPI_{1997})$$
  
  

What about new products? Quality changes for existing products?

• Chain weight: change the base every (five) years, calculate the growth rate from year to year in real terms, and use the growth rate to construct a new index
  $g^1 =\ln (P_{apples}^{97}Q_{apples}^{96}+P_{oranges}^{97}Q_{oranges}^{96})\\ ... ...^{97}Q_{apples}^{97}+P_{oranges}^{97}Q_{oranges}^{97}) \\ \\ g^{1+2} =g^1+g^2$
• What exactly is a price index? What do we mean when we say real variables?
• A price index addresses the following question: How much money does one have to spend to enjoy the same amount of happiness (utility) from consumption in each period?
• Utility from consumption of two goods, X (gold) and Y (haircuts), is equal to:
  
  $$u(X,Y)=\frac 1{\gamma ^\gamma (1-\gamma )^{1-\gamma }}X^\gamma Y^{1-\gamma }$$
  
  

• Expenditure:
  
  P+P=Z
  
  

• Definition: The consumption-based (CPI) price index is defined as the minimum expenditure Z is, such that u(X,Y)=1.
• Solve the ``dual'' problem: max utility for given expenditure
  
  $$\max_{X,Y}\frac 1{\gamma ^\gamma (1-\gamma )^{1-\gamma }}X^\gamma Y^{1-\gamma }\mbox{ s.t. }P_XX+P_YY=Z$$
  
  

• To simplify matters, let us define $p\equiv \frac{P_Y}{P_X}$ and $\widehat{Z} \equiv \frac Z{P_X}$: p is the relative price of Y in terms of X, and $\widehat{Z}$ is the budget expressed in terms of the good X.
• the budget constraint becomes:
  
  $$X+pY-\widehat{Z}=0$$
  
  

• substitute the budget constraint into the objective function and solve the problem:
  
  $$\max_{X,Y}\frac{1}{\gamma ^{\gamma }(1-\gamma )^{1-\gamma }} (\widehat{Z}-pY)^{\gamma}Y^{1-\gamma }$$
  
  

• From the first order conditions we obtain:
  
  $$X^*=\gamma \widehat{Z} \mbox{, }Y^{*}=(1-\gamma )\frac{\widehat{Z}}p$$
  
  

• Substituting for the solution (X^*,Y^*) into u(X,Y), we have utility as a function of $\widehat{Z}$ (expenditure)
  $\frac {1}{\gamma ^\gamma (1-\gamma )^{1-\gamma }} X^{*\gamma }Y^{*1-\gamma }\... ...rac{\widehat{Z}^{1-\gamma }}{p^{1-\gamma }}= \frac{\widehat{Z}}{p^{1-\gamma }}$
• CPI= the least expenditure of money that buys you one unit of utility
  
  $$u(X^{*},Y^{*})=\frac{\widehat{Z}}{p^{1-\gamma }}=1$$
  
  
  or, CPI measured in terms of unit of the good X:
  
  $$\widehat{Z}=p^{1-\gamma }$$
  
  
  or, CPI measured in monetary terms:
  
  $$Z=P_X p^{1-\gamma }=P_X^\gamma P_Y^{1-\gamma }$$
  
  

SO WHAT?

• Deeper understanding of what CPI is, and of what economists mean when they say real (consumption, standard of living).
• We can compare real output across countries:Let us measure the CPI in terms of unit of the good X ( $\widehat{Z}$). Since the good X (gold) has the same price across countries, we have a common unit of measurement to assess utility/standards of living: If you give $p_{mex}^{1-\gamma }$ units of gold (tradables) to a Mexican, and $p_{us}^{1-\gamma }$ to a US citizen, the two guys are equally happy.
• Say that P_X is the price of gold expressed in US Dollars. Then, we can also say that if you give $P_Xp_{mex}^{1-\gamma }$ US$ to a Mexican, and $P_Xp_{us}^{1-\gamma }$ US$ to a US citizen, the two guys are equally happy.
• However, if you give the SAME amount of DOLLARS to a Mexican and to a US Citizen, the two are NOT equally happy
• As long as p_mex and p_us are different, that is, as long as the relative price of non tradable goods (haircuts, financial services, maids, cab rides) in terms of tradable goods (gold, cars?, refrigerators?) is different across countries, the standards of living corresponding to a given amount of money (US$, Pesos) are different.
• Summers and Heston's Penn Word Tables.

What have we got?

• we saw different ways of measuring the changes in the price level (CPI, GDP deflator, fixed-weight CPI, chain-weight CPI)
• perhaps more importantly, we tried to understand what are we after when constructing a price index to measure variables in real terms
• we want to measure how living standards of individuals (utility) change over time, or to make cross country comparisons
• for cross country comparison, differences in the price of non-tradable goods may make a lot of difference