Inflation is difficult to measure because it represents the percentage change over time of a nonexistent economic variable--the price level 'P_t'.

%ΔP_t= (P_t - P_t-1)/P_t-1

Unlike GDP or other national income measures, no single observable measure exists to represent the aggregate price level. Thus economists rely on a price index based on some well-defined market-basket of goods as a proxy to measure the level of prices and changes in prices over time.

The most common measure of inflation is that of the Consumer Price Index or 'CPI' as calculated by the Bureau of Labor Statistics (the BLS). This particular index is based on the prices of a basket of goods which represents the purchasing behavior of some average urban consumer. The CPI, also known as the Laspeyres Index, is calculated using a weighted average of current to past price ratios for this basket of goods:

CPI_t  =  Σw_i,t[P_i,t / P_i,o]    (1)

These weights 'w_i,t' are based on the expenditure patterns of the consumer in a base period (currently 1982-84) reflecting the importance of each item relative to the overall level of consumer expenditure in that base period or:

  P_i,oQ_i,o
w_i  =  ---------------   (2)
 Σ [P_i,oQ_i,o]

thus

      Σ[P_i,tQ_i,o]
CPI_t  =  ---------------   (3)
       Σ[P_i,oQ_i,o]

where 'Q_i,o' represents the quantity of the i^th good consumed in the base time period (t = 0), 'P_i,o' represents the price of the i^th good in the base time period, and 'P_i,t' represents the price of the same good in the current time period 't'.

A measure of inflation is then developed by computing the percentage change in CPI from one time period to the next:

                     CPI_t - CPI_t-1
π_t = %Δ.(CPI) =     --------------        (4)
                       CPI_t-1

It is important to note that the CPI is not a perfect measure of the price level or changes in the price level. Because this index is computed using base-period quantities (reflecting buying behavior and preferences in the base year), it does not allow for substitution among goods as relative prices change. For example, it might be that the overall rate of inflation is 5%. However, within that value some goods might be rising by 3-4% and other goods by 6-7%. Consumers will attempt to soften the effects of increasing prices on household budgets by substituting away from the relatively more expensive goods and towards the relatively cheaper good. This behavior is not captured in the CPI.

A second problem with the CPI is that it does not allow for changes in product quality over time. It may be that prices are rising due to improved quality of the good being purchased such that this good does not have to be replaced as often. Quality changes can also show up in the size of the good in question. Over the past generation, housing prices have been rising. But during this same period of time, the average size of a housing unit (in terms of square footage, number of bedrooms and baths, size of the garage and lot) has also increased.

Finally, the CPI does not allow for the inclusion of new goods and services as they emerge into the market place. A fixed basket of goods based on 1982/84 preferences ignores DVD players, PDA's, cell phones, audio CD's and many other goods that perhaps lead to improvements in living standards or life style.

A common use of this measure of inflation is to add an inflation premium to interest rates to allow for expectations about future inflation. As stated above inflation erodes the purchasing power of money over time. An individual lending money in an inflationary environment will be repaid in dollars which possess less purchasing power upon maturity of the debt contract. An inflation premium is often built in to nominal interest rates protect against this loss of purchasing power. However, at the time the debt contract is developed the inflation premium is based on expected rates of future inflation. If these expectations differ from actual inflation rates during the life of the debt contract either the lender or borrower can be adversely affected.

The inflation premium represents the difference between nominal interest market rates 'i_market' (i.e., those interest rates published in the paper or posted on the wall at a bank) and the desired real rate of interest 'r*' which usually reflects the rate of real economic growth (the amount of reward that should accrue to the lender for lending to a productive economy). Thus the nominal rate of interest (holding risk constant) on a short-term debt contract (one year or less) is developed as follows:

i_market = r* + E[π_t]        (5)

where 'E[π_t]' represents the expected rate of inflation. At the termination of the debt contract an ex-post real rate of interest 'r' can be developed as follows:

r = i_market - π (6)

A one year loan (N = 1) with the following terms:

• Principal 'P' = $1000, and
• nominal rate of interest 'i' = 5%.

At the time the loan is made, the price of a common commodity 'Gasoline' (P_gas) is equal to $1.00/gal. In real terms the lender is providing the borrower with the purchasing power equivalent to 1000 gallons of gasoline.

If the price of gasoline had risen to $1.07 (a 7% rate of inflation) then the purchasing power of the repayment would have been equal to 981 ($1050/$1.07) gallons of gasoline. In this case the lender provided the opportunity for the borrower to acquire 1000 gallons of gasoline and at the termination of the loan the borrower repaid to the lender the ability to acquire only 981 gallons. An unexpectedly high rate of inflation had had an adverse impact on the lender -- a negative real rate of return.

If E[π(_t)] is greater than π_t then 'r' will exceed 'r*' to the benefit of lenders (real returns to lending greater than desired and perhaps greater than the rate of real economic growth) as shown by the following operation --  substituting (5) into (6) we have:

r = r* + E[π ] - π

If the opposite is true, then benefits will accrue to the borrower.

During the 1980's, many economists have felt that the real rate of interest was abnormally high (i.e., in excess of 2.5-3%). This may be explained in part due to the inflationary expectations that built up in the late 1970's and early 1980's. Nominal interest rates have taken these expectations into account. The effects of these inflationary expectations differing from the actual rate of inflation can be seen in the table below where the annualized 6-month T-bill rate is used as a measure of the market interest rate:

Note the anticipated real rate of interest (r*) is based on an average of the actual rate of real  economic growth over the previous three years.

Over time, changes in market interest rates may be attributed to changes either in the real desired rate 'r* or due to changes in inflationary expectations. Changes in the desired real rate reflects the behavior in the market for loanable funds. If the supply of these funds (public and private savings) exceeds the demand for these funds (public and private borrowing) then the desired rate should fall in reaction to a surplus of these funds. In periods of economic growth the opposite is true. The growing economy is sustained in part by increased borrowing activity for inventory investment and investment in new capital stock to allow for increased production to meet growth in aggregate demand.

Changes in inflationary expectations tends to be a more complicated matter. One may hypothesize that current inflationary expectations are based on the history of past actual rates of inflation. A formal model that may help in understanding the development of these expectations is that of the Adaptive Expectations model. This model is based on the notion that economic agents slowly adapt to a changing inflationary environment. This may have been the case in the late 1960's and early 1970's. During the 1960's, the inflation rate was relatively low in the 2-4% range. Basically, during this period time inflation was not considered to be a major economic problem. Thus in the next decade when actual inflation began to creep up towards the double-digits, many individuals and institutions were surprised. Forecasts of future inflation (based on recent historical experience) consistently lagged behind an accelerating actual rate of inflation.

In the early to mid-1980's the actual rate of inflation was de-accelerating,  a phenomenon known as disinflation. During this period, economic agent's expected rates of inflation were greater than what actually occurred. These agents were slow to adapt thus putting upward pressure on ex-post real interest rates.

A different price index, known as the GDP Deflator or the Paasche Index, is constructed using current expenditure shares [to represent the spending habits as reflected in current GDP via Q(i,t)] and is defined by the following equation:  

P_t  =  Σ [P_i,tQ_i,t] / Σ[P_i,oQ_i,t]

where 'Q_i,t' represents the quantities produced and sold of the i-th good in the current time period 'P_i,t' represents the current price of that i-th good and 'P_i,o' represents the base (1996) price of that same good. This measure can be interpreted as the ratio of actual spending in the current year (NGDP) and the level of expenditure on that same quantity of goods if prices had not changed (RGDP) -- spending in base-year prices.

• Adaptive Expectations
• Consumer Price Index (CPI)
• Deflation
• Desired Real Rate of Return
• Disinflation
• Inflation
• Inflationary Expectations
• Inflation Premium
• Laspeyres Index
• Nominal Interest Rate
• Paasche Index
• Real Interest Rate
• T-bill Rate