Friday, January 6, 2017

How Should We Measure Real Savings?

Suppose that at 12:01AM on 1 January I had \$114, and at 11:59PM on 31 December I have \$180. Obviously, I spent less than my income, saving \$66.

On the other hand, at the end of 2015, \$430.89 could be exchanged for one bitcoin; a year later, one bitcoin ran \$966.30. Thus, on 1 January I had Ƀ0.2646 and on 31 December only Ƀ0.1863. Obviously, I overspent my income by Ƀ0.0783.

So which is it? Did I save or dissave over the course of the year? Let us back up a bit.

The saving discussed above we might call “comprehensive.” It is simply my change in wealth over the period. But this wealth is nominal— measured in terms of currency, rather than real goods and services that such wealth might purchase.

Suppose that at 12:01AM on 1 January I had wealth sufficient to purchase 100 pounds of apples, and at 11:59PM on 31 December I had wealth sufficient to purchase 150 pounds of apples. Obviously, I had saved an amount equivalent to 50 pounds of apples. Neither does it matter what the price of apples was on 1 January, nor does it matter what the price of apples was on 31 December. My savings of 50 pounds of apples was “real”— literally comparing pounds of apples to pounds of apples.

At 12:01AM on 1 January, 50 pounds of apples runs \$57. At 11:59PM on 31 December, 50 pounds of apples runs \$60. Clearly, my \$66 saved does not correspond in a direct way to my 50 pounds of apples saved.

The problem, of course, is that (due to inflation) \$114 on 1 January is not real in the same way that \$114 on 31 December is real. It makes no sense to take my 31 December nominal wealth of \$180 and subtract my 1 January nominal wealth of \$114 to get \$66 in real savings. Rather, if we wish to report real savings in terms of dollars, we must choose a consistent price for apples.

Real savings is the change in inflation-adjusted stocks of wealth $$ S^{\left(p\right)}_t=\frac{W_t}{P_t}-\frac{W_{t-1}}{P_{t-1}}=\frac{{W_t}\times{p}/{P_t}-{W_{t-1}}\times{p}/{P_{t-1}}}{p} $$ where $p$ is the common price chosen. Presented in EOP prices ($p=P_t$) real savings comes to $$ P_tS^{\left(P_t\right)}_t=W_t-W_{t-1}+W_{t-1}-W_{t-1}\frac{P_t}{P_{t-1}}=W_t-W_{t-1}-\pi_tW_{t-1} $$ where $\pi_t$ is inflation over the period.

EOY Price Level EOY Nominal Wealth EOY Real Wealth
EOY 2015 Prices EOY 2016 Prices
2015 57 114 114 120
2016 60 180 171 180
Note: change over 2016 66 57 60

We may describe our 50 pounds of apples saved as either 57 “1-January dollars” or 60 “31-December dollars.” We may even describe real savings in terms of bitcoin:

EOY Price Level EOY Nominal Wealth EOY Real Wealth
EOY 2015 Prices EOY 2016 Prices
2015 0.1323 0.2646 0.2646 0.1242
2016 0.0621 0.1863 0.3969 0.1863
Note: change over 2016 -0.0783 0.1323 0.0621

Unlike changes in nominal wealth, changes in real wealth make intuitive sense and are consistent between choices of denomination. At the end of 2016, Ƀ0.0621 could be exchanged for \$60; at the end of 2015, Ƀ0.1323 could be exchanged for \$57. This is because we have employed a consistent set of prices for both periods and denominations. We cannot convert nominal savings measured in dollars to nominal savings in bitcoin because we have not employed a consistent rate of exchange.

To answer our original question, then, the observed savings are real despite the fall in nominal bitcoin wealth. Further, note that real savings is not equal to inflation-adjusted nominal savings; if inflation over the period is zero, then real savings over the period— expressed in EOP dollars— is equal to nominal savings over the period.

Finally, if “comprehensive” savings is defined this way, then real “comprehensive” income (given period-average consumption prices $p_t$) follows naturally as real “comprehensive” savings plus real (inflation-adjusted) consumption: $$ pY^{\left(p\right)}_t=pS^{\left(p\right)}_t+\frac{p}{p_t}C_t=\frac{p}{P_t}\left(W_t-W_{t-1}-\pi_tW_{t-1}\right)+\frac{p}{p_t}C_t $$ noting that this is not equal to inflation-adjusted nominal “comprehensive” income $$ pY^{\left(p\right)}_t\neq\frac{p}{p_t}\left(W_t-W_{t-1}+C_t\right)=\frac{p}{p_t}Y_t $$

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