Below average

Which measure of income best gauges a nation’s economic success?

To characterize a dataset simply, economists usually figure the average value (the “mean”).  If the Budapest String Quartet dined at the Four Seasons individually for $50, $75, $100 and $25 (that one for a glass of water), then one may measure their average gastronomical pleasure as the mean of their checks -- $250 divided by 4, or $62.50.  In general, the mean for n data points xi in a set is the sum of x1, x2, x3, …, xn, divided by n.

The mean is particularly useful when the likelihood of some number depends on its distance from the mean, regardless of whether it is higher or lower.  For example, if the mean grade in a classroom is a C, then the likelihood of an A, which is two letters higher than the mean, may equal the likelihood of an F, two letters lower than the mean.  In the presence of such symmetry, the mean can tell us a lot.  (And we could learn a lot more if we had a measure of how widely dispersed the grades were.  For example, are many grades Bs and Ds, or do they tend to be As and F’s?  But for now, let’s ignore dispersion).

Not all data follow a symmetric distribution.  In most nations, household income is skewed to the right, because we have a few billionaires.  Their treasures push up the mean income to a point that most of us consider unrealistic for a “typical” family.  It would seem to make more sense to use, as a measure of average income, the figure that is just above 50% of the household incomes (and, of course, just below the other 50%).  This is the median.

A trifling' trillion

The median has other fortes.  In a dynamic economy, the distribution of household incomes may change every few years.  It will remain skewed to the right, but the length of the skew may vary.  (Just wait until we pick up a few trillionaires.)  If we want a summary statistic that will hold true for more than a few years, then the mean won’t do.  The trillionaires will cause it to skyrocket.  The median, however, won’t change as long as it is just below half of the household incomes, regardless of how large some of those incomes may be.

Consequently, a growing gap between mean and median income may tip us off to rising inequality in the distribution of income.  In the United States, mean family income grew much more rapidly than its median counterpart over the Eighties and Nineties, concluded Arnold Katz of the Bureau of Economic Analysis in the U.S. government.  Over the long run – from 1969 to 2009 – mean family income grew 1.08% per year; median family income, only .69%.  Both figures were adjusted for price changes.

In Kazakhstan, mean income grew far more rapidly than in Europe and Central Asia in general for several years up through 2012, when it was $9,780 for KZ, reported the World Bank (the Notes have details).  This raises the possibility that income inequality grew relatively rapidly in Kazakhstan.  Without more data, it is only a possibility.

What does the mean mean?

The median has a hidden weakness:  It cannot reflect uncertainty about the income of a given household.  As we know all too well, the amount of money we make depends partly on chance.  The owner of a Cajun restaurant in New Orleans early in 2005 could not have predicted that a hurricane would blow away her assets.  Surely our notion of average income should take such uncertainty into account.  We may want a summary statistic for the income of each household – and then a statistic that summarizes all of those statistics. 

In that case, the median would not suit us, because it does not reflect that every household income may vary according to circumstance.  It just gives us the income that is at the midpoint of the entire distribution.  Yes, we could compute a median for each household, since its income varies from one scenario to another; but it is not clear how all of these medians would relate to the national one.  For the mean, however, the relationship is crystal clear.  The mean of the distribution is the sum of the means of all the households; it takes individual uncertainty explicitly into account.  Suppose that mean income is $10 for one person and $30 for another.  Then the mean income for this distribution of two people is $40 divided by 2, or $20.

So is mean income a better measure of economic success than median income?  Not necessarily.  Consider an economy of three households.  Family A earns $10,000; B, $20,000; and C, $3 billion.  For the distribution, the median is $20,000, and the mean is just over $1 billion.  Does the high mean mislead us about the economy’s success?  Yes, if C had opportunities that A and B didn’t.  The median would be more accurate.  No, if all three families enjoyed the same opportunities but fortune smiled on C.

The point, of course, is that we should provide both the mean and the median of the income distribution – and let the user choose.  The charge that conservative analysts use the mean in order to overstate the economy’s success is not always justified.  Neither is the corresponding charge against liberals.  –Leon Taylor, tayloralmaty@gmail.com

Notes

1.        The World Bank adjusted its estimate of mean income in Kazakhstan, which is in U.S. dollars, by using a three-year average for the exchange rate and by controlling for differences in inflation rates between Kazakhstan and a composite of the euro area, Japan, the United Kingdom, and the United States.  This is the Atlas method.  It tries to avoid sharp fluctuations in the exchange rate that occur in the short run for reasons unrelated to economic capacity.
2.  To measure the inequality of a national distribution of income, economists widely use the Gini coefficient.  It ranges from 0 to 100, where higher numbers denote more inequality.  (In principle, the coefficient ranges from 0 to 1, but practitioners often multiply this by 100 for easy comparisons.)  The coefficient measures the disparity between a group's share of the population and its share of income.  If x percent of the population always has x percent of national income, then the coefficient is 0, denoting perfect equality.  On the other hand, if 99% of the population has 0% of the nation's income, then the coefficient is nearly 1.

The World Bank reports a coefficient of 29 for Kazakhstan in 2009.  By comparison, the coefficient was 40.1 for Russia that year -- and 54.7 for Brazil, a country famous for income inequality.  Gini coefficients elsewhere in Central Asia are fairly low:  36.2 in Kyrgyzstan and falling; 30.8 in Tajikistan.

Good reading

Bulmer, M.G.  Principles of statistics.  Dover.  1979.  A genial introduction.

Katz, Arnold J.  Explaining long-term differences between Census and BEA measures of household income.  Bureau of Economic Analysis.  2012.  bea.gov 

World Bank.  World Development Indicators.  worldbank.org

Unreal

 Do low interest rates signal trouble?

Some shock waves from Europe’s economy reach as far as Kazakhstan.  Two weeks ago, the European Central Bank announced a negative interest rate for regional banks holding their deposits there.  These banks will have to pay for the privilege of lending their money to the ECB.

In a sense, negative interest rates have been common around the world since the financial crisis of 2008.  This refers to interest payments measured in terms of the goods and services that they can purchase – that is, to “real” interest rates.  In economics, “real” refers to quantities as opposed to money.  A real interest rate of 5% indicates that the lender can buy 5% more of autos and bottles than he could have done by spending the principal rather than lending it.   When the real interest rate is negative – say, -3% -- then the lender loses purchasing power despite the interest paid to him.  He can buy 3% less of goods than he could have done by foregoing the loan.

The ECB’s novelty was a negative money interest rate. I’ll explain.

Depositors to a bank are its creditors, since it can put their dollars to work by lending them out or by purchasing assets like government bonds.  (Indeed, a loan is an asset, since the lender can expect it to pay off.)  The ECB is no exception, although it does not make commercial loans as a rule.  A negative money interest rate of, say, 1% means that banks must pay the ECB 1% of their deposits there.  This is in addition to the loss of purchasing power that the depositors suffer by leaving their money at the ECB rather than spending it.

Why might this affect Kazakhstan?  Because the market for financial assets is global.  All assets enable their holders to consume more tomorrow by consuming less today.  If you buy a 5% bond for $10,000, then you give up $10,000 of spending today – on a long Caribbean cruise, perhaps – in order to earn $10,500 for spending next year (add a few lobster dinners to that cruise).  The rate of return on any financial asset – we might as well just call it the “interest rate” – tends to converge on a common global rate, because the assets market is fast-moving (“liquid,” if you will).  If the common rate is 5%, then a 6% asset will quickly attract buyers, pushing up its price and thus reducing its rate of return.  (Confused?  See the Notes.)

The Bankers’ Ball 

Today in the West, most interest rates are close to zero, largely because the central banks in the wake of the financial crisis printed money as if tomorrow would never come.  Increasing the supply of anything will lower its price; otherwise, the additional units can’t be sold.  The price of holding a tenge is the interest rate, since you must give up the interest payment if you choose not to lend out the money.

Adjusted for inflation, inflation rates tend to be higher in Kazakhstan than in the West, probably because people view investment here as risky and demand payment for the risk.  By adopting a negative interest rate, the European Central Bank is pressuring rates in the post-Soviet world to fall, since investors will bid up the price of post-Soviet assets.  For a while, the post-Soviet banks will smile, since Westerners are lending them more dollars than before.  (To the Westerner, these loans are assets.)  But the banks, giddy with new dollars, may make the same mistake that they did in 2007 – lending out the dollars recklessly.  If the financed projects fail, then the banks will be unable to repay Westerners.  Once again, the banks will flirt with insolvency.

Of course, the banks will assure us that they lend with the utmost prudence, that any shortfall of dollars is temporary.  That’s what they said in 2007, too.   –Leon Taylor, tayloralmaty@gmail.com           

Notes

1.  Consider a bond that costs $10,000 to buy and that pays off $500 in interest each year.  Its annual rate of return is $500 / $10,000, or 5%.  If its price rises to $20,000, then its rate of return will fall to $500 / $20,000, or 2.5%.  In general, the price of an asset relates inversely to its rate of return:  When one rises, the other falls.

              

Reference

Jack Ewing and Neil Irwin.  European Central Bank breaks new ground to press growth.  The New York Times.  June 5, 2014.