Garett Jones  

Redistributing from Capitalists to Workers: An Impossibility Theorem

How I Raise My Children... Dr. Henderson in Africa...
There's an old story about a mathematician asking Paul Samuelson for one idea in economics that was simultaneously true and not obvious.  Samuelson's answer is here.  Today, I've got another: The Chamley-Judd Redistribution Impossibility Theorem.  

Chamley and Judd separately came to the same discovery: In the long run, capital taxes are far more distorting that most economists had thought, so distorting that the optimal tax rate on capital is zero.  If you've got a fixed tax bill it's better to have the workers pay it.  You can search the web for details and qualifications of their result, here's Yglesias:

The standard "classic" result in this field is, in fact, that an optimal system would have no taxation of investment income. 

He turns to a collection of counterarguments by Piketty and Saez, concluding:

That's a bit of an exotic argument, but if you want to undermine the standard approach there you have it. 

There, indeed, you do. 

Why isn't Chamley-Judd more central to economic discussion? Why isn't it part of the canon that all economists breathe in?  Why isn't it in our freshman textbooks?  Part of the reason is surely mood affiliation--it's an uncomfortable result for some to talk about as evidenced by the handwringing I see in most textbook treatments (exception here, big PDF, p.451). The result can't be waved away as driven by absurd assumptions: It's not too fragile, it's too solid. It's OK to teach Real Business Cycles since we all know (or "know") that the Federal Reserve and aggregate demand really drive things in the short run.  But to tell people that if we care about the long run, the tax on capital income--on interest, profits, dividends--should be zero?  And to have only "exotic" counterarguments?  Let's just leave that for the more advanced courses....

But another reason is just that the proof is too opaque.  I've worked it out a few times and while I can see the answer, I don't really get it the way I do with comparative advantage or the equity premium puzzle or Arrow's Impossibility Theorem.  

So I decided to take a first step at trying to fix that.  First, let me sum up a key implication of Chamley-Judd: 

Under standard, pretty flexible assumptions, it's impossible to tax capitalists, give the money to workers, and raise the total long-run income of workers.    

Not, hard, not inefficient, not socially wasteful, not immoral: Impossible. 

If you tax capital income and hand all of the tax revenue to workers, then in the long run (or the "steady state") you'll wind up with a smaller capital stock. And since workers use the capital stock to earn their wages, the capital tax pushes down their wages.  

So far so obvious, standard supply-side stuff. At this point, you're probably guessing that sometimes the taxes you hand to workers are more than the fall in wages, sometimes it's all depends on the assumptions, depends on the tax rate, depends on this or that.  But the magic of Chamley-Judd is that they proved that "fall in wages > rise in transfer" is a pretty stable result...hence the need for "exotic" counterarguments.  

Rational workers would rather have the extra machines to work with rather than a transfer from a tax on capital, thank you very much. 

Rather than offer you a proof of the result, I offer you an Excel simulation of two economies, one where capitalists pay taxes to workers, and another where the society is taxless. I use standard assumptions, only the usual number of rabbits going into the hat.  Check and see which economy gives the highest total income (wages + transfer payments) to the working class.  Here are a few examples; note that wages fall by more than the rise in the transfer payment: It's impossible to redistribute total income to workers:


You can tinker with the numbers yourself in the Excel simulation: Change the degree of patience in the society (that changes the savings rate and hence the long-run capital stock) or change the size of the transfer payment to the proletariat (don't make it too big or it'll be impossible to pay the bill).  Every time you change the numbers, the Excel simulation tells you whether the Chamley-Judd result holds: It either reports "C-J Vindicated" or "C-J Fail."  Let me know if you can make their theorem fail.  

I wrote up a derivation of the result here, two pages, some algebra, and a link to an extra Excel simulation. I welcome further attempts to popularize Chamley-Judd; Landsburg does a good job here

One big lesson I draw from Chamley-Judd: Good economic policy doesn't try to do things that are impossible.  And if the world works roughly the way Chamley and Judd assume it does, a long run policy that redistributes total income from capitalists to workers is impossible. 

Comments and Sharing

COMMENTS (31 to date)
Jason Sorens writes:

What about the Przeworski and Wallerstein model under which taxes on consumption out of investment with a closed economy allows redistribution from capital to labor? (Now, maybe that's what is meant by "exotic counterarguments"! I would agree that the ancillary policies needed to make this policy effective would be unworkable.)

Ghost of Christmas Past writes:

The biggest problem with Chamley and Judd is their mutual simplifying assumption of infinite-lived economic actors who want only to maximize society's total long-run income. In fact people have short lives and "want" (in an evolutionary sense) to maximize their own inclusive fitness. A tax scheme optimized to freeze existing status relationships (as by exempting capital income from taxation) will not maximize inclusive fitness for most people.

(Also, what's the deep difference, if any, between return to capital (land, capital goods, whatever) and return to human capital? Shouldn't the optimal rate of taxation on labor income also be zero? Why should one pay tax on wages for pulling a rickshaw but not tax on fees for whipping one's horse to pull a hansome cab?)

Jim writes:

And what if the tax is a Brown tax?

[comment edited with permission of commenter--Econlib Ed.]

ThomasH writes:

I'm having trouble understanding what taxation of "capital" or "labor" is. Is this the same as the standard proof that only "consumption" and not "income" should be taxed? Does it imply that transfers to workers should be in the form of capital assets so that the long run solution everyone is equally a "worker" and a "capitalist?"

Garett Jones writes:

Ghost, you're right: If i may take a glibly Hansonian angle, a problem with Chamley-Judd is that is assumes redistribution is about raising the income of workers.

Thomas: The story is that any permanent tax on capital reduces capitalists' incentive to save, so the workers' wages are crippled. Even if you gave the taxed capital each year to workers---as capital--the tax-induced disincentive to save means the capital stock is shrinking too much to help workers in the long run.

And yes, in my PDF I note that everything is capital...implications for another day....

david writes:

Intro long-term-growth courses invariably cover OLG and therefore cover dynamic inefficiency, which immediately generates the trivial possibility of welfare-improving capital taxation, because people accumulate excess capital to meet their desire to save.

Granted, dynamic inefficiency doesn't appear to hold in the empirics, but I don't see why you find that the argument needs to be 'exotic'.

As Samuelson argued way back then, it is the infinitely-lived household that is highly exotic, and dynamic inefficiency is annoyingly difficult to stamp out, because there is no apparent market failure. It's not incomplete markets, it's not incomplete contracting, it's not government failure. All trade is free and unimpeded. Yet...

You can put modern variations on the intuition of dynamic inefficiency, such as Waldman's model, which dodges the empirical problems that Sameulsonian dynamic inefficiency faces.

Bryan Willman writes:

Transfering the returns on capital is the wrong "solution".

The "solution" is to create a different organizational structure in which the worker "buys into" the venture (not often used today, but possible.) So the worker earns labor returns, and also capital returns which said worker may apply as makes sense (more investment when the net present value of expected return on said investment is positive.)

But this is really saying we need better ways to organize "The Firm".

Davidt writes:

"And since workers use the capital stock to earn their wages, the capital tax pushes down their wages."

Ah that is the rub,
"Capital is used to create more opportunities for workers to earn more." This is not true today. How do you explain the flat worker wages for decades? The reason is the capital is being used to replace workers by machines. If the worker is not replaced a less skilled worker is employed cheaper and the machine guides the worker or limits the workers mistakes.

If the masses do not have expectations or excitement of a future for themselves or their children no religion or police force will control the streets for long. Those who can not see a way forward for themselves and see the rich and powerful do not have to obey the Civil Laws or Society's Expectations of behavior will surely follow the lawless behavior of the elite.

I would say that the French Revolution came about because of the Royalty believing a fulfilling life with expectations of a better future was a waste of assets to use accomplish for commoners and peasants. I would bet if the French could have modeled their future a revolutions scenario would not have been included. The same is true today. Our models are made to satisfy rich patrons of College Departments and to be acceptable to an audience that could not imagine a modern American Revolution.

Models use assumed scenarios and are not useful because the crisis comes from the problems never considered or intentionally thrown out to smooth data and outcomes.

If we made Economist liable for their mistakes that cause pain and suffering for their model’s mistakes like engineers and architects are for their constructs I bet we would have better models. Maximizing GDP is not the same as the greatest good for the greatest number of Citizens. Knowingly to Plan or modeling an economy to result in maximizing wealth at the expense of the vast majority people in the county should be considered an act of torturing, maiming or the killing of individuals on a mass scale.

Ken B writes:

Is there a good link for the Ramsey result? Good meaning good for a non economist.

Josh S writes:

Bryan, most retirement funds today are based on capital ownership. If the goal is for common laborers to earn more money than entrepreneurs and other risk-takers, or more money than engineers, accountants, and other people with highly scarce skills, well, there's lots of other economic theory to explain why that never happens.

James A. Donald writes:

Analogously, suppose you wanted to benefit consumers of bread at the expense of producers of bread. Obviously a tax on producers and a subsidy of consumers will not have the desired result, and neither will price control, as Venezuela is discovering yet again.

If we think of workers as demanding capital, the result is obvious. Capital will cost reflecting supply and demand, and anything that harms suppliers means less capital which will mean that workers will "pay more" for capital - which is to say, lower wages and higher profits.

Tim writes:

@Bryan Wilman

Isn't your solution really just syndicalism?

And how would you respond to the obvious criticisms of syndicalism (namely that it makes business nearly impossible to adapt)?

Khem Y. Panha writes:

[Comment removed for irrelevance.--Econlib Ed.]

c141nav writes:

What is the optimal tax rate on labor?

Glen Smith writes:

If labor is not earning a surplus, they have nothing to be taxed. If they have a surplus that could either be taxed or be contributed to increasing the capital stock, they still have nothing that should be taxed (if you argue that labor will not use their surplus to increase capital stock, you are arguing that the capitalist is a thief). If labor's profit is below 0, then transfer via taxes is simply the politically connected capitalist transferring his/her costs to another capitalist.

Garett Jones writes:


The optimal tax on labor is whatever you need to pay for the (exogenous) cost of government.

At some level, it's trivial: If it's wasteful (indeed, impossible in the long run) to tax capital, and the only alternative is to tax labor, then tax labor....

Aside: One bigger insight of optimal tax theory that escaped me for years is that a tax on labor is the same as a tax on consumer spending.

quadrupole writes:

I can believe that you cannot in the long term increase labors income by taxing capital and redistributing it to labor.

But I'm pretty sure in the long term you can increase the income of people who either *don't* work, or who work at something for which there is little demand by taxing capital. And that's the rub politically.

quadrupole writes:


There's another option: tax consumer consumption.

Taxing consumer consumption does not discourage investment, but *does* treat all sources of the resources that fund that consumption (labor or capital) equally (and in my opinion) fairly.

Ed Conard made a point similar to this in his book last year, Unintended Consequences, that the largest beneficiaries of capital are workers and consumers, not the capitalists.

He keeps repeating it too in his frequent TV appearances. Especially when he's on with Joe Stiglitz.

John writes:

If Impatiences is a measure of the required return to capital then labor is much more interested in having a very patient society with very low rates of return to capital.

For instance, if I put 10% in cell F2 GNP is 5, with w = 2.50, and the untaxed capital base is 250. If we transfer 20% then GNP drops to 4.76 and w = 2.48, and Capital's share drops to 2.28.

If, however, I set F2 = .01, or 1%, then in the untaxed world capital base is 2500 and w = 25, as does the share to Capital. If 20% is now transferred the capital base drops to 2459.84, w = 25 (actually it drops to 24.9984 but...) and the share to Capital drops to 24.8.

So those that own labor, but not capital prefer, a world where capital has very low time preferences to support a large capital base and transfers from capital.

While labor has some interest in the non-transfer case, I don't think under any discount rate will the transfer overcome the decline in base that supports w. That may be a function of the model though doesn't seem too unexpected.

What is clear is that it's really Capital that is against the transfer case as it's income falls farther than Labors due to the transfer.

If we're dealing in a world where the class shares of GNP are divided up among members of different sized classes, and Labor is the larger class the relative results might be seen as more equitable even if the pie has shrunk.

Tim writes:

"If we're dealing in a world where the class shares of GNP are divided up among members of different sized classes, and Labor is the larger class the relative results might be seen as more equitable even if the pie has shrunk."

I would say that often times that's the trade-off those in favor of larger market interventions are trying to make. They'd rather have more equality even when it means less prosperity overall, and often even when it means less prosperity on average.

John writes:

Garett, you have one spreadsheet in the paper and another in the blog. In both everyone is better off without government.

Here you show that labors share of GNP drops more than the amount transferred. As I point out above the impact of the transfer become negligible as the discount rate declines. If we change the discount rate from 10% to 5% in the above table w = 5.00 and with the .2 transfer w = 4.99.

In the spreadsheet you have linked in the paper you talk about capital paying taxes versus labor paying taxed -- but here taxes simply drop out of the picture. Where do they go?

In this second spreadsheet showing the labor taxes using the same parameters, r=.1 and g=.2 again the capital base dominates and w declines more when capital is taxed than when labor is taxed and K-base remains at the untaxed level. But, again, where's the money go. in the labor taxed world w=2.30 versus the 2.28 when capital is taxed but no transfers exist. Under the capital taxed but taxes are paid to labor w=2.48.

Taken together it's not clear to me that a strong argument against taxing capital has been made by the analysis.

John writes:

@Tim you might be interest to know that the person who first got me thinking about the relative versus absolute reward idea was Jim Buchanan from Public Choice fame. It's not an a priori conclusion that a reduced pie with a different distribution is an inferior outcome.

My point was that, 1) this approach is really only convincing to those who largely already agree and 2) under not too unrealistic conditions the cost of embracing the tax capital and transfer is low to labor owners but not to capital owners.

Steve writes:

To get their results in the basic neoclassical model, it's necessary to 1) limit the initial tax rate on capital income and 2) impose commitment so the government cannot re-optimize in the future. Without these restrictions a capital levy achieves the first-best outcome. Tax capital enough to build up a stock of debt owed by the private sector to the government. Interest payments then finance the exogenous stream of government spending with no need for distortionary taxation. I think it's these artificial restrictions that prevent the result from being part of the canon. That and the effects of capital taxation on wealth distribution across generations in richer setting.

Garett Jones writes:

Steve: The 'capital levy' you speak of is just a lump sum tax, and those are nearly impossible as politics, and actually impossible in a world of renegotiation. 'One time' levies get baked into savers' expectations, and then they stop being levies.

An unfair example: Chavez imposed repeated 'one time' levies on oil investors...investors quickly realized they weren't 'one time' and treated it as what it really was: a progressive tax. Megan shows how they responded: with less investment, eventually yielding less output to tax and redistribute...

acarraro writes:

I don't really understand why the capitalist propensity to invest is 1. I don't think that's what is observed in real life. Obviously if income is strictly a function of capital and investment is strictly a function of return on capital, you get the result you quote.

I don't think it's very general though... Capitalist could consume the entire income they receive and your result is invalidated. You just defined this away...

Plus in the real world, capital return and wage return are hopelessly mingled. If I build a factory and the factory is profitable, is it due to my work in spotting a proper market opportunity or is it simply the return to capital?

Real interest rates are more or less zero now. So almost the entire capital return is risk compensation, which in my mind is really a labour return in allocating capital to good investments.

I think comparative advantage is a much more interesting result...

Steve writes:

Garrett: Agreed, but given the set-up of the Chamley-Judd model, all that's required is an income stream sufficient to finance an exogenous stream of government expenditures. One capital levy (or wealth tax, or lump sum tax, or whatever) does the trick, and with rational expectations everyone realizes that the government will have no further need to tax, ever. Introduce stochastic changes to government spending and things can get even weirder. Chari, Christian and Kehoe show that it can be optimal to tax capital income at a zero rate on average, but have capital tax rates (or taxes on any assets that are inelastically supplied in the very short run) fluctuate wildly (from positive to negative) in response to shocks.

Valerie Keefe writes:

Chamley-Judd is dependent upon an equal distribution of resources and no competition for status goods at the outset. To wit, it's very elegantly simple crap. Gordon Gecko's going to maximize return no matter what the ad velorem rate is, so you may as well have a rate that maximizes income. But hey, because Diamond-Saez is "exotic" it must be wrong, because heaven knows a seventy-trillion-dollar global economy is an exceedingly simple thing. Also, Chamley-Judd ignores capital-biased technological change.

This is as depressing as when I have to patiently explain how weak monopsony and labour inelasticity due to income effect mean that a too-low minimum wage can result in too-low employment (and if you need some macroeconomic data, you can look at the US and CANADA, which have dramatically different MW-GDP/hr worked ratios and see who leads in working-age employment). Die zombie idea, die!

ThomasH writes:

I think I see what my problem with the idea of taxing "labor" or "capital" is. There really is no consumption by those who receive "capital" income so the issue of whose consumption to tax cannot arise. The implication of the model is that for any level of g, the optimal tax on consumption must raise g for consumption not to decline. Of course in a model in which government expenditure is neither reinvestment nor redistribution, nor consumption it is not clear what g could possibly be for.

Michael M writes:

As a tax lawyer, who knows a bit about how the tax system works. I think it is generally impossible to distinguish income from capital and income from labor, and any attempt to tax the two at different rates creates other problems.

stone writes:

When I first read "capital tax" I thought you meant a tax on asset value rather than on "income from capital". I agree that taxing profits is distorting. However a tax on asset value such as the "capital tax" proposed by Michal Kalecki is the least distorting type of tax there is. As Michal Kalecki wrote in 1943

“ the inducement to invest in fixed capital is not affected by a capital tax because it is paid on any type of wealth. Whether an amount is held in cash or government securities or invested in building a factory, the same capital tax is paid on it and thus the comparative advantage is unchanged.”

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