Icosian Reflections

…a tendency to systematize and a keen sense

that we live in a broken world.

Elasticities, revisited

Hannes Malmberg commented on my review of Saez and Zucman in response to a point I made about the elasticity of capital supply:

I think you are confusing demand and supply elasticities of capital.

The revenue calculations hinge on the elasticity of capital supply, i.e., how fast capital supply rise with the interest rate (how much more do people decide to save).

The Piketty spiral, in contrast, hinges on the elasticity of capital demand, i.e., how fast the interest rate fall with increasing capital (i.e., how fast firms and companies switch away from using capital when it gets more expensive).

There is nothing theoretically preventing us from having an almost horizontal capital demand curve, and an almost vertical capital supply curve. In such a world, a capital tax raises a lot of revenue, but increasing the savings propensity increase the capital stock without reducing the interest rate. (...)

I think he is basically right, and I'll partially retract section 3A of my prior post. (The rest of the post basically stands independently.)

"Partially", here, because reconciling the TToI revenue assumptions and the Piketty wealth spiral requires taking a specific stance on the slope of the capital demand curve, which should be considered a nontrivial burden that the authors don't even purport to defend. But we'll get there, hold on.


First, how did I make this mistake? Basically, by forgetting the Intro Econ that I never took, and making an implicit assumption about capital demand.

Specifically, I implicitly assumed (and didn't realize I had assumed) that the capital demand curve was downward-sloping. Why? Probably for the same reason that Bryan Caplan explicitly assumes it:

Part of the reason is admittedly my strong prior. In the absence of any specific empirical evidence, I am 99%+ sure that a randomly selected demand curve will have a negative slope. I hew to this prior even in cases -- like demand for illegal drugs or illegal immigration -- where a downward-sloping demand curve is ideologically inconvenient for me. What makes me so sure? Every purchase I've ever made or considered -- and every conversation I've had with other people about every purchase they've ever made or considered. (...)

In this setting specifically, downward-sloping demand is less about "purchases get gradually less desirable as they get more expensive" than about "the best available projects get gradually less productive as you try to find more of them". Not provable from first principles, no, but definitely the default assumption that I'd start with.

nb: Here, 'goodness of projects' is not a claim about securities prices, though it sounds somewhat like one.

If some projects return more value per unit of required capital than the marginal project, their securities' prices should reflect that premium already (with the gains accrued to whoever originally provided the actually-scarce component). So the securities investor pays more in 'capital' than is required for the project, but the difference accrues as profit to the original entrepeneurs and stakeholders, and the net capital invested is still that of the underlying project.

If you weren't confused before this sidebar and are now confused, best to forget it.

With this assumption, my original analysis goes through -- either wealth taxes make people decide to save and earn less (and so the assumptions about the tax's revenue are overstated), or the return on capital drops when the capital stock grows (and the Piketty spiral is self-limiting).

In pictures:

Here we fix the downward slope of the capital demand curve (implying that people will have fewer uses for capital if it's more expensive), and consider differing elasticities of capital supply. From top to bottom, these plots represent:

  • perfectly elastic supply -- there's a specific rate that will make people decide to earn/save more, and enough similar people to make that rate always prevail
  • somewhat elastic supply -- there are some people who decide to earn and save more if a higher rate of return is available
  • perfectly inelastic supply -- there's a fixed quantity of capital stock; no one is motivated to earn or save more by a higher rate of return

Okay, but how are we supposed to read these plots?

The pink and blue lines trace out hypothesized supply and demand curves for (investment) capital. Without any other effects, they meet at the solid purple dot. When a wealth tax is imposed, the naive projection (current wealth stock times tax rate) predicts revenues of the brown box, but the tax frictions generally cause less capital to be accumulated/invested, so only the gold box is actually collected in taxes once prices and fortunes re-equilibrate.

Then there's the analysis of the Piketty spiral -- the light-blue dashed line posits a world where people save and try to invest more. Then if demand stays the same, supply and demand meet at the hollow purple dot, generally at a lower "price" (rate of return), and with a greater quantity (total amount accumulated/invested, after new supply crowds out other supply). In the top scenario, the new investments simply replace other flexible suppliers of capital, so quantities and prices don't chance.

Among these the scenarios, it's only in the bottom case (i. e., inelastic supply) that the tax doesn't convince investors to earn and save less and reduce its own base (seen as the gold box shifting left). But it's only in the top case (i. e., perfectly elastic supply) that the owners of a growing capital stock can find additional opportunities to invest without the marginal rate of return falling off (seen as the hollow purple dot moving down).


But if we relax the assumptions that demand is downward sloping, we see other possibilities. By varying the slope of the pink line (i. e., the elasticity of capital demand, i. e., the degree to which different uses of capital differ in their expected rates of return), we produce the grid of plots below.

In the case of perfectly elastic capital demand (at the left), we're assuming that there's an arbitrarily large pool of potential investments that promise the same rate of return. In the case of inelastic capital demand (at the right), we're assuming that some investments demand capital regardless of price while no others make sense at potentially market-clearing prices. The effects of taxes are again given by brown and gold boxes, and the effects of increasing savings propensity are again given by the shift from solid to hollow purple dots.

Sure enough, a horizontal demand curve and a vertical supply curve (lower left) imply full tax revenues and a non-self-limiting wealth spiral. (You might notice similar outcomes in the upper-right chart, though that one's politically inconvenient, as it would imply that wealth taxes are incident on the users of capital, not on investors.) Hannes is right, and if you believe those two curves, you can reconcile Saez and Zucman's revenue assumptions and Piketty's runaway-wealth models.


It's a big 'if'.

Those two curves are potentially consistent with each other, but not with other claims that commonly pass without criticism. To pick a recent example, the flat demand curve is inconsistent with this post by Bridgewater's Ray Dalio that I saw circulating in some circles recently:

Money is free for those who are creditworthy because the investors who are giving it to them are willing to get back less than they give. More specifically investors lending to those who are creditworthy will accept very low or negative interest rates and won’t require having their principal paid back for the foreseeable future. They are doing this because they have an enormous amount of money to invest that has been, and continues to be, pushed on them by central banks... (...; emphasis mine)

Now, I'm not claiming that Dalio is an authority here, but his implicit claim is that a surfeit of money exhausts the possible (good) uses for it and pushes up the prices of the limited ones available. Maybe true. The point is that if you believe that, you can't believe that the marginal opportunities for productive investment are plentiful and about equally good, like the flat demand curve claims.

Perhaps even more difficult to believe: The vertical supply curve requires that highly-paid workers, given the opportunity to instead work their gardens and sail their boats for equal pay, would decline and stay at their desks. I mean, I do love my job more than most other things I could be doing, but as for other people, color me...skeptical.


Anyway, I'll bow out before I get myself into trouble again. I agree that inelastic capital supply and perfectly elastic demand are not inconsistent; I was confused and mistaken when I claimed otherwise in my last post. As assumptions, they're strong enough to likely be inconsistent with other things you implicitly believe, but that's a story for another time.