Money Mischief_ Episodes in Monetary History - Milton Friedman [36]
Note: Dependent variable: nonmonetary demand for silver (in millions of ounces); independent variables: world income, real prices of silver and gold.
Estimating a hypothetical price level using the linear equation is mathematically far more tractable than using the log equation, and that reinforces the theoretical consideration in favor of the linear equation (that the silver available for nonmonetary use out of current production can be negative). Hence, from here on I use only the linear equation.
c. Equating Supply and Demand. Equating equations (5) and (9), and rearranging the terms:
(10) UMDSH = SPROD - EWMDS - 58.28 - 2.13WZ - 0.88RPGH + 66.21RPSH.
To simplify, let k2 equal all the terms on the right-hand side of equation (10) except the last, and let x equal the hypothetical real price of silver that is our objective. All of these are also functions of time. However, given our assumptions up to this point, we have estimates of the values of k1 and k2 for all the years from 1874 to 1914.
In terms of these symbols, we can rewrite equation (7), using equation (6), as
Equating equations (10) and (11) and simplifying:
Equation (12) is now in the form of a straightforward quadratic equation except for the troublesome presence of the term including x(t — 1) in the denominator. This x(t — 1) is one of the unknowns that we are trying to determine. As a first approximation, assume that the real price of silver does not change from year to year, in other words, that x(t) equals x(t — 1). That assumption converts equation (12) into the simplified equation (13), which involves only the current year's value of the unknown x, although it does involve the prior year's value of k1, via substituting Δk1 for k1(t) — k1 (t — 1).
The solution to this equation is a first approximation to x.
For a second, third, and succeeding approximations, we can return to equation (12) and replace x(t — 1) by the prior approximation estimate. The successive approximations do converge, though rather slowly. The main changes are not in the level or general pattern but, rather, in the year-to-year movements. However, each approximation involves losing one value at the beginning of the series. 1 stopped with the eleventh approximation, at which point 1884 is the first year for which there is an estimate. For earlier years, I used the earlier approximations, beginning with the third for 1876, the year in which the silver standard would have been adopted.* Given this estimate of the real price of silver, it is necessary only to divide the legal price by the real price to estimate the hypothetical price level under a silver standard. The resulting estimate of the hypothetical price level for the United States is plotted in Figure 3 of chapter 3.
d. Gold-Silver Price Ratio. Since we have already estimated the hypothetical price of gold, it is trivial to get the hypothetical price ratio of gold to silver. The result is plotted in Figure 2 of chapter 3, along with the actual and legal gold-silver price ratio. The actual gold-silver price ratio under a U.S. silver standard would almost surely have fluctuated much less than our estimates of the hypothetical gold-silver price ratio, given the arbitrary assumptions and inevitable measurement errors that affect our estimates and the extent to which they have been affected by the monetary uncertainty of the period.
These estimates suggest that, if the United States had returned to a bimetallic standard in 1879 and had stayed on it consistently, the market gold-silver price ratio would have remained roughly equal to or only slightly above the U.S. legal price ratio—just as for close to a century the market ratio remained roughly equal to the legal price ratio in France. (Table 1 gives the numerical values for the curves plotted in Figures 2, 3, and 4 of chapter 3.)
5. An Even More Sophisticated Estimate. In principle, it would be possible to get a fully simultaneous solution for both the real price of silver and the real price of