9 Some Remarks about Units

In mathematical analysis units do not play a big role, but in technical and physical contexts, however, very well. In particular, equations can be checked to see if there are always in a sum20, or on the left and right hands of an equation, the same units apply. Is the balance of the units not correct, then there is probably an inconsistency21 in the formula. So we have to control units at some important points.

Here we look at the just derived basic equations and by writing them down in their pure units, which seemingly do not fit:

   (9.1 a)

and

   (9.1 b).

The still vague termsandwe do not have to look at more closely, because if they are used, they will be defined in the appropriate units. The apparent problem is, however, that GDP generally is in currency units per year (Cur / y), the capital stock is measured in the contrast as total assets = liabilities in currency (Cur). For this we must remember that we always expect here on an annual basis , ie years. Therefore, to be written in full:

   (9.2 a)

and

   (9.2 b).

Now the units equations are correct:

   (9.3 a)

and

   (9.3 b)


Because of the unity=1 year, you can simply omit the factor on an annual basis. If we, however, compute on different time basis, approximately monthly, then we will have to introduce such a correction factor if necessary. It should be noted also that the unity of a percentage, eg the interest rate is not truly percent, but percent per year. So mathematically. In the lingo of the general, even among economists, the correct unit is often omitted, as of course is usually assumed an annual basis. Capital is a stock variable, measured in monetary units, such as € or $. The gross domestic product GDP, however, is a current size, measured in € or $ per year. In purely formal terms the capital coefficient K/Y is not then dimensionless, but has the less meaningful unit year. Here we need some thoughts regarding the appropriate unit of GDP. Because the denominator of K/Y should also include a stock size. Basically in the denominator, there should be the totality of tradable goods and services. So not only the current GDP Y this year, but also the sum of the past few i-years GDP which is still tradable with depreciation :

   (9.4) ?

It is now, so that the calculated annual GDP is actually already above this sum. Because the actual GDP contains also all of GDP that has been generated by the trading of used items. So, about second-hand-automobiles or real estate or the flea market, etc. In addition, it also contains as an important side note, the proportion of GDP which is introduced by the banking industries itself. For this reason, the current GDP value is justified as well as a stock to use in units of euros or dollars, and the capital coefficient can thus be regarded as indeed dimensionless.