The first chart below shows one simple way to compare income across countries; it shows net national income (that's the income earned by the various factors of production in the economy, after the loss of used-up capital equipment has been accounted for) in per capita terms. To put them in a common currency to make a cross-country comparison more meaningful, I converted all figures to US dollars using PPP exchange rates, which (in theory) should take into account the differences in relative prices between countries.
This picture shows that, according to national income statistics, the US is definitely the richest country of the handful I've selected for this analysis. However, the gap has closed somewhat over the past 20 years, in part because real per capita income growth was so lousy in the US during the 2000s.
However, I don't think this is nearly sufficient as a comparison, for a couple of reasons. First, it doesn't pass the smell test. Do we really think that the average family in the Netherlands today is less well off than the average family in the US was in 1990? That seems pretty unlikely to me.
Part of what's going on is that, if we're interested in understanding the international purchasing power that residents of each economy had, then we should really use actual exchange rates instead of PPP exchange rates. Actual exchange rates tell us how much stuff someone who gets paid in one currency -- dollars, for example -- could buy if they are shopping in the international marketplace (either as a tourist or an importer) rather than the purely domestic market.
The next chart shows what has happened to the relative purchasing power of people in the same handful of countries. And what this shows is that the continental European countries have indeed become significantly "richer" relative to the US over the past 10 years. This is the result of the strengthening of the euro relative to the dollar over that period, which more than offset the gains enjoyed by the US during the 1990s that resulted from good economic growth and a strong dollar.
What this means is that, for example, a person earning the average income in the US was able to buy about 28% more goods and services internationally in 2010 than they could in 1990. Meanwhile, a person earning the average income in Germany was able to buy about 55% more stuff in 2010 compared to 1990. This suggests that an important part of my subjective perception that Europeans have gotten richer than Americans in recent years is simply due to the weakness of the dollar and strength of the euro. (Note that this is an important facet of the ongoing rebalancing process wherein the US is getting better at living within its means.)
But I think there's another, more fundamental problem with these sorts of simple national income comparisons across countries. In particular, most economists would agree that GDP is actually a pretty poor measure of welfare, if used in isolation. It may be the single best indicator we have of economic well-being, but that doesn't mean it's a particularly good one.
Consider some events that would increase a country's GDP: an epidemic that causes health care spending to rise; an environmental disaster that requires expensive cleanup; a crime wave that leads to increased policing and detention; a decision to build a million houses and then blow them up as a spectacular fireworks display. All of these would boost GDP -- but would that mean that they made the country better off?
Since national income can rise as a result of things that most people would agree are probably not particularly "good", if you're trying to get an understanding of cross-country differences in welfare you will have to use additional measures. Stay tuned for some examples...
UPDATE: To illustrate exactly what I mean, I'm adding one more chart. The graph below shows Net National Income per capita for the same countries in 2008, using actual exchange rates to convert them all into dollars. So the US had about $41,000 in total net income per capita, Germany about $38,000, and the Netherlands a bit over $43,000. Next, I've taken spending data from the OECD's 2008 PPP Benchmark survey, and broken out how much residents of each country spent on health care and education. That's the red portion of each bar. The remainder is what individuals in each country had to spend on everything else. So, for example, on average people spent about $9,500 per capita on healthcare and education in the US in 2008, leaving about $31,500 available for all other types of goods and services.
Interestingly, people in the US actually had less national income left to spend on things other than healthcare and education than in any of the other countries. So even though the US's GDP looks higher than most other countries, a good chunk of that is actually an artifact of the US's high health care costs. Whether US residents receive more health care services than residents of these other countries is another question... but the result is that the US has fewer resources available to spend on the other things that produce well-being for individuals than any of these other countries. And that is certainly consistent with casual observation.
Oh, and as noted by a commenter, we haven't even gotten into issues of income distribution...
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