THE METHOD
Summarizing the data to be used with this analysis of the equation of exchange we have:
As a matter of convenience the symbol X(N) will be used whenever referring to one of the data elements. That is, Demand Deposits will be called X(1); Savings - X(2), etc. The reason for this convention becomes obvious when writing the equation with all of its elements. For example, the equation as so far developed can be written:N Data Element 1 Demand Deposits 2 Savings 3 Money 4 Investment Credit 5 Consumer Credit 6 Government Receipts 7 Government Borrowing 8 Domestically held National Debt 9 Foreign held National Debt
For an even more abbreviated form of the equation we can write:PQ = Demand Deposits plus Savings plus Money plus Investment Credit plus Consumer Credit plus Government Receipts plus Government Borrowing plus Domestically held National Debt plus Foreign held National Debt; or, PQ = X(1) + X(2) + X(3) + X(4) + X(5) + X(6) + X(7) + X(8) + X(9)
The symbol:9 PQ = SUM X(N) N=1
means simply that each X(N) from 1 to 9 is to be summed or added together.9 SUM N=1
In Chapter IV it was shown that there may be other items that would behave like each of the selected data items. It was then stated that to account for this sympathy effect each of the data items would be multiplied by a constant. There are nine data items so nine constants, C(1) through C(9), are needed. Putting these in the abbreviated equation we have:
The next element to be considered is time. In Chapter IV it was said that to compute the June 1983 price level for comparison with the June 1983 Consumer Price Index both the May value and the change in value during June of each of the data elements would be used. This is accomplished by identifying the month being computed by the symbol "T" for time. The preceding month is then identified by "T-1". Since the effect of the values at the start of the month can be expected to be different from the change during the month an additional set of constants, B(1) through B(9) will be required. Adding these, the equation becomes:9 PQ = SUM C(N)X(N) N=1
In addition to the items measured by the data elements available and those that are sympathetic, there are those not measured. For lack of a measurement of these items, a constant (A) representing the average of the unmeasured items is added to the total. This gives us:9 PQ = SUM [B(N)(X(N,T) - X(N,T-1)) + C(N)X(N,T-1)] N=1
And finally, after dividing by Q, we have the version of the equation of exchange used for this study:9 PQ = A + SUM [B(N)(X(N,T) - X(N,T-1)) + C(N)X(N,T-1)] N=1
Now all that needs to be done is to determine the values for each of the constants and compare the computed values of P with the measured values of the CPI. If the correlation is good it will demonstrate the equation holds true for the limited amount of measured transactions and money represented by the X(N)'s.9 P = {A + SUM [B(N)(X(N,T) - X(N,T-1)) + C(N)X(N,T-1)]} / Q N=1
There are, undoubtedly, better methods to determine the value of each of the constants. But, for this study the simple and tedious method of trial and error was used. It is:
As stated earlier, simple but tedious. This process must be repeated many times. For each repetition the computation of each P involves many multiplications and additions plus some other extraneous computations. With the Timex-Sinclair 1000 computer first used for this analysis some versions of the equation took over a month of continuous operation to arrive at the values of the constants. The correlations included in this revision were repeated on a larger computer which produced more accurate results. But, the computations were still quite time consuming.(1) Select a value for each constant. (2) Compute the value of P for each of the times for which data is available. (3) Change the value of one of the constants. (4) Re-compute the value of P for each of the times available. (5) Compare the results of the first and second values with the appropriate CPI values. (6) a: If the second value is better correlated with CPI use the second value for the constant. b: If the second value shows less correlation use the first value of the constant. (7) Change the value of the same or another constant and repeat until the best value for each constant is determined.