More on Krugman and Marginal Product, Sans Calculus
By popular request, my new article at The American Conservative makes the same basic point that Karl Smith made back in November, but I think I spelled it out here very clearly to isolate Krugman’s mistake. Contrary to some of you guys in the comments, I don’t think the issue on this one was nominal vs. real, I really think it was marginal vs. inframarginal. I tried to show that by reproducing Krugman’s error in a story involving direct payment in apples. An excerpt:
In terms of our story, we can illustrate the problem by imagining that two more workers show up in town, looking for work picking apples. Through reasoning similar to that above, we conclude that in the new equilibrium, one of the newcomers goes to Smith’s orchard, the other to Jones’s, and they both get paid 10 apples per hour. However—and here is the crucial part—the original workers now only get paid 10 apples per hour, too. Because the four workers are all interchangeable, they have to get paid the same amount, and we know that the marginal product of the new guys is only 10 apples per hour.
Step back now and survey the whole picture. There are a total of 60 apples being harvested every hour among both orchards, while the four workers are only being paid 40 apples total. Thus the orchard owners are enjoying a “surplus” of 10 apples each, for every hour the laborers are at work. Thus, it is not true to say that the workers in this hypothetical world are “getting paid what they put in,” in the sense that Krugman means. Yet even so, the workers are being paid their marginal product, according to textbook price theory.
The Myth of Wartime Prosperity
This ran a while ago but I forgot to blog about it. Half of my piece is just the standard case against World War II “prosperity,” with a link to Steve Horwitz’s co-authored piece drawing on primary sources to show how tough things were on the homefront.
However, after that I take Krugman et al. head-on. I’m not saying I am the first to advance this particular argument, but I don’t think it gets stressed much when people like me typically take on the myth of wartime prosperity:
Yet the Keynesian case is weaker still. Suppose we accept, for the sake of argument, that $1 million of government spending is just as economically significant as $1 million of private investment or private consumer spending. Even so, the World War II era still doesn’t present a model for dealing with an economic depression.
Look again at the figure above. Even on the Keynesians’ own terms, private GDP—the share of the economy devoted to civilian purposes—was lower during the height of the war than during 1933, the very worst year of the Great Depression. When we take into account the increase in population during the decade in between, the impact on the homefront is even more astounding.
Let us suppose, then, that the government today did exactly what Paul Krugman recommends, and engaged in massive government purchases comparable to those during World War II. Yet rather than build tanks and bombers, instead the government today buys socially useful things (in Krugman’s vision) such as roads, bridges, parks, the services of additional police and firefighters, etc. If things turned out today the way they did during the war years, would Americans be happy with the result?
I suggest they would not; they would reject the bargain, even on Krugman’s own terms. Suppose people took today as equivalent to 1941, and then the country proceeded to have similar government spending and economic performance as the official statistics show happened during World War II. That means private economic output would fall a total of 55 percent between now and 2015, or at an annualized rate of about 24 percent per year. Does the average American household right now want to suffer a 24 percent annual drop in their private standard of living, for three years in a row? This would put their standard of living back to around 1984 levels (and again, I’m not even accounting for population changes). Would the average household’s answer be affected if we told them about how many potholes would be filled, and how many new schools would be built during those three years?
But wait, it gets worse. It’s not merely that there would be a 3-year period of extreme penury (in terms of private goods and services), in exchange for those things the government provides. After this burst of Keynesianism—again using World War II as the model, now by comparing 1941 to 1946—the federal government’s gross debt held by the public would have grown by a factor of five. Since the federal debt held by the public right now is about $10 trillion, that means mimicking the World War II experience would yield a federal debt of $50 trillion by 2017. This new fact is on top of the reversion to 1984 levels of private GDP. Again I ask: Is there any amount of new schoolteachers and bridges that would compensate for these two trends?
I Got the Blog Host Blues, Baby I Got the Blog Host Blues
OK apparently what happened is that the business credit card on file with HostGator expired, and they were sending warning emails to a different account. So I called up HostGator today and did a quick manual payment to get the blog back. Then they told me how to go into my account settings and (a) update my credit card on file and (b) change the primary email for my account, so this doesn’t happen again.
Inasmuch as the interface would have shocked Kafka, I am quite sure I didn’t accomplish either of the above objectives.
Krugman, Karl, and Kalculus
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 0<@<1. So what this means is that adding one more unit of capital drives up workers wages, and drives down the rental rate of capital. Yet we know that competitive markets means that the change in total output must accrue entirely as payment to the owner of that last unit of capital. Thus, the total gains in real income to the workers must be exactly counterbalanced by the total losses to the capitalists (including the guy who supplied the latest unit, if we just focus on his supply of all previous units). This 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.
Two Keynesians vs. Two Austrians
Brad DeLong thinks Mario Rizzo is a “psychopath” (actual quote) for recommending that people don’t tip NYC cab drivers; DeLong recommends (obviously tongue in cheek) strip-searching “these people” (by which he means Austrians, I think, and not Italians) whenever visiting one’s house. Brad DeLong also thinks I’m nuts for saying that it would be immoral for the government to take money from people against their will and use it to fund the destruction of a killer asteroid. Daniel Kuehn agrees with both points.
At first, I was going to make a quick blog post about the oddity of this stance. Rizzo is recommending something that is not actually theft, but at worst is really uncool. But DeLong and Kuehn are so horrified at ripping off NYC cab drivers–regardless of what other injustices Rizzo thinks he would thereby be protesting–that it’s fine to label Rizzo a psychopath for such views.
On the other hand, taking money from millions of people with the ultimate sanction of putting them in cages if they refuse–by hypothesis, these people don’t want to hand the money over voluntarily–in order to destroy an asteroid is so obviously a fine thing to do, that DeLong and Kuehn call me nuts for objecting to it. (Technically, DeLong calls me “unbalanced” while Kuehn uses the term “nuts.”)
As I say, the above dichotomy doesn’t interest me. I don’t care what an odd view of property rights and ethical views that it entails.
Rather, here is what I want to think about: Suppose Brad DeLong had the opportunity to stiff a NYC cab driver, and use the money saved in order to spare the planet from a killer asteroid?
Potpourri
==> David R. Henderson, clearly jealous of my notoriety, points out that he debated Paul Krugman when I was an undergrad.
==> Scott Sumner and von Pepe go toe-to-toe. Von Pepe tells Sumner that the word “market” doesn’t mean what he thinks it means.
==> After that warm-up, for something even more surreal: Joseph Fetz and Daniel Kuehn arguing with David Friedman about Adam Smith. (!)
==> Tell me I don’t look intimidating in this promo. I wouldn’t debate me either.
Why Don’t I Spend All My Time Reading About Church History?
Thorn-in-my-side Ken B. made a comment earlier in the week that I’ve seen in several places, in varying forms. So I will quote him here, but realize the point goes beyond Ken B.’s particular question. Here’s Ken:
So a question Bob: I refuted the historicity of the story of the woman taken in adultery, showing it does not belong in the Gospel of John.
I cited several links.
What steps have you taken to investigate the issue?
You certainly have not directly addressed the issue on this blog.
When I first arrived on this board I made points about Jesus of Nazareth as an apocalyptic prophet, quite unlike the stained-glass Jesus.
In particular I argued that the early sources conflict, that these sources are best seen as representing the beliefs of divergent faith communities rather than as accurate history, and that the ‘stained-glass Jesus’ is a strained attempt to harmonize the synoptics, John, and Paul.
These are mainstream views of critical biblical scholarship.
I cited several well-known scholars and provided links to several books.
You admitted you had not read enough to refute me.
What concrete steps have you taken to determine if or where I erred?
In what way, in short, has your behavior here not vindicated Hitch’s insight?
Note that your past snappy dismissive comments count against you here.
I submit that this is a terrible argument. If any of you who read this when Ken B. first wrote it, were nodding your heads in agreement at how Christians “don’t really believe this stuff after all,” then it shows you are very biased when it comes to theological arguments.
As far as Ken B.’s recommendations in particular, the short answer is, I trust him as far as I can throw him. In case you are not familiar with Ken B.’s rhetorical style, here is something very recent. The beauty of this one is that it’s self-contained; you can just read a few lines to see what I am talking about.
To set the context, someone had pointed me to the debate between that Spanish professor and Krugman. Naturally, every Austrian who’s told me about it, thought the Spanish guy crushed Krugman. Here’s what I said in reaction:
[Bob Murphy:]FWIW, I watched the first 10 minutes and thus far I can see why Krugman would think, “I’ve answered every one of these points, numerous times, on my blog. This guy doesn’t even know my position.”
I haven’t watched the rest of it yet, but I am prepared to say, “Krugman won that debate.” Don’t get me wrong, I’m not yet declaring this, because I have to see what Krugman says in response.
(I didn’t volunteer this because it would look so petty. “Oh boy, this guy knocks out Krugman first, and now Bob is jealous.”)
Then about 10 minutes later Ken B. writes the following, in reference to someone’s theory about what Krugman meant when he complained that Pedro (the Spanish guy) was pulling rank:
I wasn’t sure what Kruggers was referring to, since Pedro said many nice things about him, but this [the other commenter’s theory] has to be what PK was talking about. And Krugman is right on the substance too.
Maybe that’s why Murphy says PK won the debate.
This is par for the course, for Ken B., and it’s why twice now I’ve literally had to just stop talking to him in the comments on posts. He makes carelessly false statements of others’ positions, and he’s smart enough to make me think it’s a big joke to him.
So, do I drop other things in my life to go read books that Ken assures me will blow up my religious faith? Absolutely not, and I don’t feel any need to apologize for this.
However, let’s generalize it, and not make it about Ken B. Gene Callahan, for example, has sent me at least one historical book, and I would love to read it. But, during the last 3 months, I have not made time to do so.
So, does this mean I don’t really believe in the afterlife? I mean, if I claim that my soul depends on getting these things right, then how can I choose to work an extra hour and pay my mortgage this month, rather than reading more from a book on religion, if there’s even a 1 in a million chance that I’m worshipping the wrong God?
The answer is pretty simple, and any economist should get it: People make decisions on the margin. Look, there are all sorts of comments people leave on blog posts, talking about vaccines causing autism, 9/11 was an inside job, cell phones cause brain tumors, etc. etc. If I choose to work an extra hour and pay down my mortgage, does that mean I like my house more than my kid? That I am naive about my government? That I must not really prefer to avoid cancer, after all?
For what it’s worth, I have made inquiries to people whom I really respect about these sorts of historical claims. And their answers were enough to reassure me that confident statements such as “there really is no historical evidence that Jesus existed” (not saying Ken B. made that claim, but others do here, repeatedly) are balderdash. They have indeed given me further reading, but I haven’t gotten time yet to see for myself.
One final point: I imagine people might say, “Ah, but your soul is infinitely valuable. So marginal analysis breaks down.” No, I don’t think this is right. Jesus gave the command to go and spread the gospel. If His disciples ignored that, and instead spent all their days studying history books and theological tracts, they would not be striving for the best way to please God, which is what their religious beliefs say is the highest end in life. So you can look at it from a secular or a heavenly viewpoint. Either way, spending every waking moment studying books, trying to become ever surer of your personal belief in Christ is not what a good Christian necessarily should do. If you feel called by God to a monastery, okay maybe you should do that. But it’s not what all Christians need to do, by virtue of their professed metaphysical views.
Entrepreneur Hero Worship and Calculus
How’s that for an obscure post title? I’m referring to Krugman’s accusation that right-wingers are hypocrites. (I hope you were sitting down for that one.) Their latest abomination refers to their claim that people with high incomes are engines of job creation and therefore we don’t want the government to drop the hammer on them. What’s the problem? Krugman argues that this standard right-wing line contradicts basic micro, if you think markets work:
The Romney fundraiser in the Hamptons continues to inspire much justified hilarity. Matt Yglesias has fun with whining rich people complaining that they are the engine of the economy, pointing out that quite a few of the whiners make their money in ways that arguably does very little for growth…
There is, however, an even broader critique of the whole keep taxes low on jobcreatorsenginesoftheeconomy [sic–RPM] thing — it doesn’t make sense even when the rich really earn their money….
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 Galt. Well, actually just one-third Galt, reducing his working time to just 2000 hours a year so he can spend more time with his wife and mistress.
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!
OK, there’s a catch: Wheelerdealer pays taxes, and when he chooses to work less, he pays less taxes. So there is in fact a welfare loss to the rest of America, but it comes only through the fact that revenue falls.
…
So there’s a huge contradiction in the whole position of the self-regarding rich — a contradiction that I’m quite sure bothers them not at all. More champagne?
I get where Krugman is coming from here, but is this right? Is it really the case that standard micro suggests–in the absence of externalities and other frictions–that whether you live in a society of Thomas Edisons and the Beatles, or in a society of Billy Madisons and Herman’s Hermits, the only change you’d notice is in the amount of services the government could afford to provide you?
It seems to me this can’t be right, but I don’t know if Krugman is making a basic mistake, or whether there actually is a contradiction in standard micro (or at least the way we think about it). Three things come to mind:
(1) Even in the standard framework, if every factor gets paid its marginal product, it’s still the case that adding one more unit of factor X can raise the marginal productivity and hence the return to factor Y. E.g. it would be wrong to say, “Lots of people think workers are richer because of the introduction of more tools and equipment, but this is silly. The producers of hammers and forklifts get paid their marginal product, so obviously workers don’t see higher wages because of their existence.”
Now I remember vaguely from Kirzner’s history of economic thought class that this was a big deal back in the day, because you had to worry about whether the total incomes to factor owners in this approach would exhaust / exceed total output. But for sure, I think Krugman is glossing over this crucial part of the argument.
(2) What about consumer surplus? That doesn’t seem to be getting its due regard here. If Bill Gates had never existed, Krugman is arguing that the rest of us would have the same real income (except for lower government services). Even if that were true (which I’m not sure it is), we wouldn’t have the same utility right?
(3) Moving away from standard neoclassical models and into Austrian territory, one way of thinking about what entrepreneurs do is that they create new production functions. I.e. they use their brilliant minds to discover/create opportunities that previously didn’t exist. How in the world can we say that this insight forces us to say that either (a) the market fails or (b) entrepreneurs don’t shower benefits on others?
In any event, it is standard in Human Action for Mises to say that the capitalists/entrepreneurs reap the initial gains of their actions, but that eventually competition showers those blessings on the rest of humanity. I sort of understand the way in which Krugman could argue, “Ah, so you guys believe in market failure,” but that seems like a really odd way of putting things from an Austrian perspective.
Recent Comments