This note is to give a simple formalism to some material that we implicitly understand when listening to the news media. This basic formalism is then applied to several areas.

Consider a statement like GDP "grew at 5% this year, 3% real and 2% inflation." The statement is obviously based on the idea that the 3% and 2% are additive. In keeping with a traditional economic literature that defines money variables with capital letters and deflated or real variables with small letters, define real GDP as gdp = GDP/P where P is our symbol for "the" price level when, in fact, there are several broad price indices such as the CPI (consumer price index), PPI (producers price index) and (Greenspan's favorite) the PCE (personal consumption expenditure price index).

We want to make extensive use of the general mathematical point (demonstrated as the logarithmic time derivative in calculus texts) that

the percent change of a ratio is the percent change of the numerator minus the percent change of the denominator.

Applied to the equation gdp = GDP/P, the result is

where hatted variables denote percent changes like P-hat = (P_t - P_t-1)/P = ΔP/P

Just as gdp is found as a ratio, gdp=GDP/P, we can also look at the equation as saying that GDP = gdp×P. Another statement of the general algebraic relation is that the percent change a product (gdp×P) is the sum of the percent changes -- i.e.,

One only has to look at equations (1) and (2) to recognize that they are two ways of saying the same thing. Ie, the statement (1) that the percent change of a ratio (GDP/P) is the difference in the growth rates of the numerator and denominator is equivalent to the statement (2) that the percent change of a product (P×gdp) is the sum of the percent changes.

Consider real wages defined as w = W/P. This implies that w-hat = W-hat - P-hat which just formalizes what we already knew, that real wages don't rise unless W-hat is greater than inflation (P-hat).

When examining percent changes over time, we often need to pay attention to the units in which time is measured. This is facilitated by defining hatted variables, like P-hat, Debt-hat, .   .   .   , Q-hat with expressions like

or, after taking limits, where the notation in the numerators, the P'(t), Debt'(t) and Q'(t), represent the SLOPES of the trajectories of the variables against time. So hatted-variables -- or growth rates -- can be seen as ratios, the ratio of the slope (against time) over the level of the variables. (They can also be seen as the derivatives of logarithms (dlnx(t)= [dx/dt]/x(t) = x'/x.)

If Ê = slope-of-E(t)/E, then a constant slope implies a non-constant Ê and a constant Ê implies a non-constant slope. Suppose, for example, that E(t) were linear in t as expressed by the equation of a straight line such as E(t) = a + bt. Since b (a constant) is the slope of the line, then Ê = b/E(t) is a variable that changes over time.

If we represented inflation as ΔP/P, it appears that the numerical value of inflation is independent of the units of time but we know that inflation of 12%/yr, restated in mos, gives a different value, viz., 1%/mo. By recognizing that inflation is the ratio of the slope of P(t) against time, we build the desired time dimension into P-hat.

The distinction between ΔDebt and Debt'(t) is especially interesting since we want to relate the term "Deficit" to the change in Debt. If we use discrete time analysis and define Deficit as Debt_t - Debt_t-1, it would appear that Deficit has no time dimension and is, therefore, a stock rather than a flow variable. But, in fact, a federal deficit (say, the U.S. federal government) of $480bil/year can be equivalently stated as $40bil/mo or $1.3bil/day. The value of a (household, firm, government) deficit is sensitive to the units in which time is measured -- it's a flow not a stock variable. To recognize this characteristic of our measures of budgetary deficits, we will define Deficit = Debt'(t) which is the limit of ΔDebt/Δt as Δt goes to zero. In plain English, a deficit is the change of the level of Debt over some period of time. Thinking about a plot of Debt(t) against t, a "deficit" is the slope of the Debt(t) curve.

While we are on the topic of the distinction between the level of Debt (a stock variable) and its change per unit of time, the Deficit, we should note the relation between Debt-hat (the rate of growth of Debt) and the Deficit, viz., Debt-hat = Debt'(t)/Debt = Deficit/Debt.

Note the difference between the frequency of data collection and the annual units in which the results are often expressed. GDP data is gathered quarterly and is often reported at an annual as well as a quarterly rate. Similarly, the inflation rate is measured monthly but often reported at an annual rate. Journalists usually report the numbers as they are presented by the authorities. One authority might report inflation as .5% one month and another authority might report that it is 6% (for the same month) and the journalists will let their readers figure out the implicit time units.

Notice that inflation rates have a time dimension (in the denominator) just as interest rates have a time dimension (in the denominator). If the interest rate on credit card debt is 12%/yr, that's obviously 1%/month. If one wants to express the real interest rate on a loan, the inflation rate that is netted out has to be measured in the same time units as the interest rate. If inflation is .2%/mo or 2.4%/yr and if the nominal interest rate (i) is 1%/mo or 12%/yr, then the real rate, r = i - P-hat, is One can add/subtract percentage figures if and only if their time dimensions are the same. Theoretically speaking, it is irrelevant whether one chooses to measure time in small (nanoseconds) or large (light-years) units as long as one is consistent throughout the equation. But units do matter to end users. Financial markets strongly prefer annualized numbers.

Annualization vs Frequency: Suppose a person is paid a wage each month of $4,000. The frequency of payment can, of course, be distinguished from the time unit in which the wage is quoted. To say that the person makes $48,000/yr, even if they do not work an entire year at $4k/month, is to annualize the numerical value of the reported wage.

Annualization is performed most regularly for interest rates:

• Consider overnight Fed Funds that is reported an an annual rate (like 1%/yr following the June03 Federal Reserve decision) despite the fact that interest on overnight money is paid daily -- the actual interest paid is 1/360th of the (product of the size of the loan and the) interest rate quoted on an annual basis.
• Consider 3 month Eurodollars (the London Interbank Offer Rate so widely used to index contracts (such as poor country debt) that Bloomberg puts it on the 2nd line of their primary rate quotation page. Although it may seem strange to find US home mortgages indexed to a London rate, the 6 month LIBOR has nevertheless been used in ARMs (adjustable rate mortgages). The actual, total 90-day or 180-day LIBOR is one-fourth or one-half of the quoted LIBOR which is always annualized.