I want to follow-up on my initial thoughts on Krugman saying that rich people don’t add anything (net) to the economy, since I personally am seeing this whole thing much more clearly now, largely thanks to you guys and (believe it or not) my working through a mathematical model that is standard in grad programs. Later on in this post, I’ll even use basic calculus to show just how wrong Krugman has been on this issue. (Don’t tell Walter Block!)

In case you’re just joining the discussion: Paul Krugman for a while has been saying that standard price theory shows that right-wingers are crazy for thinking that entrepreneurs add anything to the economy, above what they take out. He did it here most recently, saying:

So, imagine a Romney supporter named John Q. Wheelerdealer, who works 3000 hours a year and makes $30 million. And let’s suppose that he really does contribute that much to the economy, that his marginal product per hour — the amount he adds to national income by working an extra hour — really is $10,000. This is, by the way, standard textbook microeconomics: in a perfectly competitive economy, factors of production are supposedly paid precisely their marginal product. Now suppose that President Obama has reduced Mr. Wheelerdealer to despair…So Wheelerdealer decides to go…one-third Galt, reducing his working time to just 2000 hours a year… According to marginal productivity theory, this does in fact shrink the economy: Wheelerdealer adds $10,000 worth of production for every hour he works, so his semi-withdrawal reduces GDP by $10 million. Bad! But what is the impact on the incomes of Americans other than Wheelerdealer? GDP is down by $10 million — but payments to Wheelerdealer are also down by $10 million. So the impact on the incomes of non-Wheelerdealer America is … zero. Enjoy your leisure, John!

This is totally wrong. But the question is, what exactly is wrong? A bunch of people in my earlier post listed all sorts of great insights. Word-for-word, here was the best one, from Dean T. Standin: “I wonder if people in third world countries realize that food shortages aren’t bad for them, because they get to keep the money they wanted to spend on food.”

William was astute enough to point out that Karl Smith made (one of) my points back in November 2011, an earlier time that Krugman made this mistake on his blog. Here’s Karl, who was chiding his co-blogger Adam when he (Adam) had conceded too much to Krugman:

Yet we don’t actually need [as Adam seems to think] any deviation from standard neo-classical general equilibrium analysis [in order to prove Krugman wrong]. The problem was [that Paul] stopped with one person, one hour. However, most people suspect that we are dealing with more than a single man-hour here. Let’s review. The underlying assumption here is that the economy is an optimizing machine. And, we know that [at] the optimum, the marginal products of any factor are equal to the factor price and that the cross marginal products are equal to zero. However, this holds only at the optimum. Any deviation away from the optimum will cause the condition to fail and prices and quantities will need to readjust to bring the economy back into line. If that sounds [too] abstract imagine the following claim: Every factor of production is paid its marginal product, including crude oil. So, if crude oil imports to America are restricted the impact on real wages and income for everyone in America would be precisely zero. That doesn’t sound right. Well, it would be right if we were talking about 1 barrel. The effect of 1 barrel of crude oil on the US economy is about the price of the barrel. Restrict the barrel and we lose that positive [effect] but we also lose having to give up what we paid for it. However, as you continue to increase the number of barrels you restrict the marginal product of each barrel rises and the marginal product of most everything else in the economy falls. The way we experience this is that when there is a sharp restriction on imports of oil – because of a war or something – we see the price of crude oil rise and the real incomes of most Americans fall. The same thing would happen with the “job creators.” [Typical Karl grammar mistakes fixed.–RPM]

Okay, so here Karl is making the point I handled under “consumer surplus” in my original post. And of course, Karl is just spelling out what many of you instantly realized too.

Let me pause for a minute and make sure we all see what happened here. Krugman took the mainstream approach of using infinitesimally small changes–such that in a competitive market, a factor is paid exactly what it adds, and thus the buyer is indifferent–and then erroneously applying this knife-edge result to all of the inframarginal units.

So to be clear, Krugman is wrong even on neoclassical modeling grounds. This isn’t so much a problem with “mathematical models,” I would say, as it is a problem with GDP macro models. This isn’t an isolated example, either. In a previous article I claimed that Krugman’s reasoning from national income accounts led him to an absurd conclusion regarding international trade (though I can’t find the article at the moment…).

Really what Krugman has done here, is akin to the fallacy underlying the water-diamond paradox. So back when Karl Smith and Daniel Kuehn were calling Krugman the modern-day Bastiat, they were right: Krugman writes very eloquent essays that are bereft of modern subjective value theory, just like the old Frenchman could do.

There’s something else going on here, though, and it’s the point I was trying to make with the labor/capital distinction in my first post. For specificity, let’s use a very simple example of a Cobb-Douglas production function, the workhorse of first year grad programs in mathematical economics. It looks like this:

Y = K^(@) * L^(1-@), where 0 < @ < 1. The ^ sign stands for exponentiation; it means "raised to the power of." The @ is supposed to be an alpha, but I am too lazy to try to use better characters here. The * stands for simple multiplication. Okay the neat thing about the Cobb-Douglas production function is that it's easy to work with mathematically, and yet it obeys lots of properties that make economic sense. For example, the first derivative of Y with respect to either K or L is positive, but the second derivative is negative. So adding more capital or labor gives you more output, but there is diminishing marginal returns. The other neat thing is that if you assume a competitive market for capital and labor, so that each gets paid its marginal product, then the owners of capital and labor collectively get paid enough to buy the entire product. In other words, K times the first derivative of Y with respect to K, plus L times the first derivative of Y with respect to L, equals Y. (This is fairly easy to check, for those of you who are good with calculus but have never worked through the Cobb-Douglas production function.) So let's ask the question: If we add an infinitesimal unit of capital to this economy, what will be the impact on workers' real incomes? First let's calculate the wage rate. It's the first derivative of Y with respect to L, in other words: w = (1-@) * K^(@) * L^(-@) Since it's a competitive market (by assumption), we know every period that each unit of labor gets paid w, as defined above. Now to find the impact of an infinitesimal increase of K on workers' wages, we just take the derivative of w with respect to K, getting: dw / dK = @ * (1-@) * K^(@-1) * L^(-@) Note that every piece of the above expression is greater than zero. Hence, their product is greater than zero, meaning that real wages rise with an infinitesimal increment of more capital. But wait a second! We are assuming that the person who supplies that extra (infinitesimally small) unit of capital, gets paid the entirety of the increment in total output, Y. So how can it be that supplying this extra unit of capital, also causes the workers (supplying the same total amount of labor) to receive higher real wages?

The answer, of course, is that the other suppliers of capital get less. Specifically, here’s the rental rate of capital:

r = dY/dK = @ * K^(@-1) * L^(1-@)

NOTE: Mainstream economists will be tempted to call this “the real interest rate,” but no it isn’t. That is only true if you assume that the units of physical capital are the same thing as units of physical consumption goods. If you think that’s a harmless assumption to make, then I am going to start calling w “the real rate of interest” and justify this absurd statement by positing a model with reproducing robots. (For more on this pedantic but crucial point, see the Appendix of my dissertation.)

Now then, what happens to the rental rate of capital when we increase the stock of capital by an infinitesimal unit?

dr / dK = (@-1) * @ * K^(@-2) * L^(1-@)

Notice that this expression is negative, because 0This is the element of truth in what Krugman was saying. If we just focus on an infinitesimal unit of output, then withdrawing it from production will not affect the total of incomes (and here I’m talking real income) earned by all other factors.

Now, Karl Smith (as well as you guys in the comments last time) pointed out that this breaks down once we start withdrawing more than an infinitesimal unit.

But, I want to make one final point: Even if we focus on just that infinitesimal unit, it’s not the case that every single factor owner’s income is unaffected. Rather, all Krugman could prove was that the total income to everybody else was unchanged.

As we’ve seen in the specific case of a Cobb-Douglas function, but which probably generalizes under most (reasonable) assumptions, adding a unit of Factor X will drive down Factor X incomes, while increasing payments to Factor Y.

Thus, to continue with Krugman’s analysis, we can say: Yes, if WheelerDealer cuts back one hour of his work effort because of Obamanomics, total incomes to the rest of the country are unaffected. However, there is a redistribution of this (constant) total away from middle- and lower-skilled workers, and into the pockets of the other fatcats. The competition the other tycoons faced from WheelerDealer just went down by one unit, so their services, on the margin, are now that much more valuable, and hence they command a higher real income.

I hope Keynesian bloggers like Karl continue to point out this problem respectfully. Krugman has been beating this drum for at least 8 months now, perhaps much longer. I would hate for the readers of the NYT to be misled on basic price theory.