Bob Murphy is too easy on Krugman

The other day I posted on an Adam Ozimek critique of Krugman, arguing that it is not a "free market position" to see revenue maximization as a goal of fiscal policy.

There was actually a lot more to that Krugman post that I didn't read closely or think about in any detail at the time, but Bob Murphy revisists it today and, I think, let's Krugman off too easy. At issue is this passage from Krugman:

"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; not only does the president waste money by doing things like feeding children, he says mean things about some rich people, which is just like the Nazis invading Poland, or something. 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!"

All the tax stuff follows that passage. In the fine tradition of Carl Menger, Bob Murphy then proceeds to tackle the issue with neoclassical marginalism, choosing a Cobb Douglas production function. To review that exercise:

Y = K^aL^1-a with a positive and less than one.

Taking factor prices equal to marginal products (maybe not the best assumption for CEOs, but that's the exercise proposed by Krugman and executed by Bob):

w = (1-a)K^aL^-a
r = (a)K^a-1L^1-a

With r taken to be the CEO's compensation in this case (suppose we're just combining managerial labor and production labor). So what happens to production worker wages and CEO wages when Mr. Wheelerdealer withdraws labor?:

dw/dK = a(1-a)K^a-1L^-a > 0
dr/dK = (a)(a-1)K^a-2L^1-a < 0

Krugman is wrong in an interesting way. Mr. Wheelerdealer's withdrawing from the labor market actually reduces production wages because of the cross marginal products (when you reduce the capital to labor ratio labor becomes less productive). Not Econ 101, but certainly Econ 201.

Everything Bob says up this point is fine. This is what I don't like (bolding is added by me):

"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."

Krugman is abusing an accounting identity and Bob is letting him get away with it too easy.

Just because income equals expenditure does not mean that national income remains unchanged when you rearrange production or demand! This is the whole point of Keynes's paradox of thrift. Just because income is still equaling investment doesn't mean that those quantities don't shift around. What it means is that if they shift around they're going to shift around together. That's all the accounting identity Krugman appeals to says.

Accounting identities are funny like that.

Krugman can say "if we don't pay Mr. WheelerDealer his $10 million then that's $10 million more in our pocket", but I could just as easily respond "but if Mr. WheelerDealer doesn't spend his $10 million, that's $10 million less that he puts in our pocket when he spends it". It's the exact same accounting identity logic.

What we need to know is how output responds.

So, what happens to output? Let's work with the accounting identity that Krugman introduces. Since income equals expenditures:
wL + rK = K^aL^1-a

so:

(1-a)K^aL^-aL + (a)K^a-1L^1-aK= K^aL^1-a

Uh oh! Everybody see where this is going? No?:

 K^aL^1-a = K^aL^1-a

Ooops!

This is exactly why Say's Law is dumb and you need to be careful with accounting identities! As Keynes pointed out, Say's Law holds for any number of equilibria.

This is where the model gets very unsatisfying, but it's illustrative at the very least. It's unsatisfying because we don't have optimizing factors here: we have Ls and Ks that we foist on a production function. What we'd ideally have is a supply and demand function for L and for K, and propose some exogenous shock ("Obamanomics" is suggested above) that reduces the supply of K. Instead of that, we just declare what K and L are. But we can think through this without fleshing out the model. The supply schedule of K shifts left, so K is indeed reduced (as shown above). The price of K increases (as shown above). This changes the productivity of L, which is a function of K so in the invisible factor markets the demand for L shifts to the left. This reduces w (as shown above).

[UPDATE: This is why doing the math is important. See Starving Economist's comment below for why the struck through sentence isn't necessarily right. Bob Murphy is still right to work through how the change in Mr. WheelerDealer's labor supply impacts production worker productivity and the change in relative wages. That sets in motion a couple things, including the reduced marginal productivity of production workers. As Starving Economist also points out, it will lead to a substitution towards production workers. What's the net change in labor demand? We'd need more information. I'm still not so sure about this Krugman claim that taxing the rich won't change total income. But yes, there are substitution effects in play. That's also illustrated in the graphic post I have next - notice that has both an income and substitution effect operating.]

What's not shown above because we just foist Ls and Ks on a production function is of course that if the demand for L shifts left and nothing happens to supply L also declines. You need to have a sense of what's going on in the structural model.

So wait a minute: L declines and K declines as a result of Obamanomics? That means Y has to decline.

As Keynes once warned Roosevelt: "even wise and necessary Reform may, in some respects, impede and complicate Recovery. For it will upset the confidence of the business world and weaken their existing motives to action, before you have had time to put other motives in their place."

Now, is scaring CEOs the biggest risk we're facing?

No, I don't personally think so.

But it has total income costs aside from the marginal productivity effects.

It's also worth noting that this is all in the marginalist framework. If you don't think this is the best way of talking about CEO salary - if you think instead that maybe a bargaining arrangement or some kind of game-theoretic set-up is the better way of talking about CEO salary it doesn't have to have all the bad effects on production wages that Bob rightly highlights in the neoclassical framework.

Note to mainstream-haters: Mainstream does not mean all neoclassical marginalism all the time. Game theory and bargaining models are very mainstream.