The effect of expected future taxes on current investment: how big?

If the government runs a fiscal deficit now, that may mean that future taxes increase. The expectation of those future taxes may reduce the expected after-tax marginal return to current investment. That may reduce current investment, and may offset some of the effects of the fiscal deficit on aggregate demand. This mechanism for crowding out is distinct from any Ricardian effects on consumption, and distinct from any effect of fiscal deficits on interest rates or monetary policy.

Brad De Long says that Greg Mankiw should have done a back-of-the-envelope calculation to get a sense of how big these effects might be. Brad does his own b-o-t-e calculation, and argues they are very small. I don't like Brad's calculation, and have decided to do my own.

I am not at all good at doing this sort of thing. I'm bad at math and I usually mess things up. But it needs doing, so I'm going to try. All you keen young economics students who know much more math and tech stuff than me ought to be able to fix my mistakes and get a better calculation. It's something for you to play with.

What I'm really interested in is whether this effect could ever be big enough to matter. Or can we safely ignore it? So if I get to any decision where I'm really not sure, I'm going to stack the deck to make this effect bigger. If I stack the deck and still can't make it big enough to matter, then we can safely ignore it. But if I can stack the deck and make it big enough to matter, then we can't safely ignore it. It means it might matter, and we need something better than my back-of-the-envelope calculation.

I am going to start with a standard Cobb-Douglas production function: Y=K^a.L^1-a. The parameter a should be somewhere around 1/3, if the income share of capital is 1/3, and labour's share is  2/3. That's standard too, roughly.

Let R be the rental rate of capital. R is what it costs a firm to hire one unit of K for one year. If we differentiate the production function to find the marginal product of capital, set it equal to R (profit-maximising firms in competitive equilibrium), then play around, we derive the equilibrium capital output ratio as K/Y=a/R.

(Let's stop and check. Brad says the capital stock is nearly 4 times annual output, so if a/R=4, and a=1/3, that means R=1/12, or 8.33%. That looks a bit small, since depreciation should be around 10%, and if the real interest rate is around 5% (on risky real investment), that means R should be around 15%. Hmmm. Anyway, we are at least in the right ballpark. So a machine worth $100,000 can be rented at somewhere between $8,333 and $15,000 per year, to cover depreciation plus real interest plus taxes.)

If K/Y=a/R, that means the elasticity of the desired capital/output ratio with respect to R should be minus one. If R doubles (so it costs firms twice as much to rent bulldozers), K/Y should halve (firms use half as many bulldozers, and more labour, per unit of output).

But what we really need is the elasticity of the desired stock of capital (not the capital/output ratio) with respect to R. If R rises, and K falls, Y will presumably fall too. So if the elasticity of K/Y is minus one, the elasticity of K will be bigger (more negative) than minus one. How much Y falls also depends on how much L falls, which will depend on the elasticity of labour supply.

I'm going to assume that the elasticity of the desired capital stock with respect to R is minus 4/3. (I think this is about what you get if you assume a perfectly inelastic labour supply curve, but the math was too hard for me, though any competent economics student could work it out.)

So a (say) 3% (NOT a 3 percentage point) rise in R should cause a 4% fall in desired K.

How much will a current fiscal deficit increase future R?

Brad assumes a $1 deficit for one year will cause a permanent increase in taxes of $0.05 (he is assuming a 5% real interest rate). I am going to assume that an increased fiscal deficit of 1% of GDP will cause a permanent increase in taxes of 0.05% of GDP. That's maybe a bit high. It could be smaller, if the real interest rate is lower, or if there's positive real growth. But then if the taxes reduce long-run growth, that could work the other way.

How much of that tax increase will fall on the returns to capital? I don't know. Maybe none, maybe one third (if all forms of income are taxed at the same rate), maybe all of it. Depends on future US politics. Let's make the extreme assumption that it all falls on the returns to capital. So taxes on capital increase by 0.05% of GDP. If capital income is 1/3 of GDP, that means capital income gets taxed at an additional 0.15%. And let's also make the assumption about tax incidence that the after-tax rate of interest stays the same, so capital rentals increase by the full amount.

So rental rates on capital rise by 0.15% (not percentage points). So, given my elasticity assumption of minus 4/3, the desired capital stock falls by 0.20%. So if the K/Y ratio is 4, this means the desired capital stock falls by 0.80% of current GDP.

So a temporary increase in the deficit of 1% of GDP for one year causes a decline in the desired capital stock of 0.80% of GDP. That's quite large.

Next, we need to get from a fall in desired K to a fall in desired investment, I. As every student of intermediate macro should know, we now run into a serious stock-flow problem in the theory of investment, since K is a stock and I is a flow. How quickly do firms adjust the actual capital stock to the desired? Instantly? Or slowly over time? How slowly?

The standard assumption is that it is costly to adjust the capital stock quickly. So investment does not immediately spike to plus infinity when the desired capital stock exceeds the actual. And investment is usually bounded from below when the desired capital stock is less than the actual, since we usually cannot convert capital goods back into consumption goods, so gross investment cannot be negative.

I assumed the temporary fiscal deficit lasted for one year. I'm also going to assume that the actual capital stock adjusts to equal desired within the same year. That's an assumption that may stack the deck to make this effect bigger than it is.

So if the desired capital stock falls by 0.80% of GDP, gross investment will also fall by the same 0.80% of GDP for one year.

OK. That's definitely big enough to matter. A 1% increase in government spending, maintained for one year, financed by future taxes on capital income, will cause a 0.80% fall in investment spending in the same year. That's big enough to cancel out 80% of the fiscal stimulus.

Sure, I have stacked the deck to make the effect big. That doesn't mean it is big; it just means it might be big. We can't be sure that it's small enough to safely ignore.

This result surprised me. I didn't think it would be as big when I started writing this post.

The two assumptions I made that are really deck-stacking are: that all the extra taxes fall on capital incomes; and that all the adjustment in the desired capital stock happens within one year.

If we assume that the extra taxes fall equally on all sources of income, we can reduce the magnitude of the effect by two thirds.

If we assume it takes 2 years instead of 1 to adjust the actual capital stock to the desired, then we halve the size of the effect, but the effect lasts twice as long. And if at the same time we double the length of the fiscal stimulus, so it too lasts for 2 years instead of 1, everything is the same as in my original calculation.

OK. I always mess something up when I do math. What did I mess up this time? Small things don't matter. Am I out by an order of magnitude somewhere?

Update: After all the helpful comments, and after thinking about it, I've come to the following tentative conclusions:

1. For countries where the interest rate on government debt is low, relative to the growth rate, this effect is too small to worry much about, though it may be more than trivial. Unless people expect the government to stupidly try to pay off the accumulated deficit very quickly and by putting lots of extra taxes on the returns to investment.

2. For countries where the interest rate on government debt is very high, relative to the growth rate, this effect may be big enough to worry about. It's conceivable it might even reverse the sign of the fiscal multiplier. I'm not saying it will, just I can't rule it out.

[There are two things I didn't like about Brad DeLong's calculation:

1. He only looked at the permanent effect on investment; he assumed a 10% depreciation rate, so with a (say) $100 fall in the desired capital stock, depreciation would eventually be $10 lower, and gross investment would eventually be $10 lower. But what matters is the effect on investment during the life of the fiscal deficit spending, which is when firms will be adjusting the actual capital stock down towards the new lower desired level.

2. I really can't follow his method of calculating the elasticity of the desired capital stock with respect to taxes. He argues that a $1 increase in taxes will have a $0.50 marginal deadweight cost, which means a $0.50 loss of productivity, and that loss of productivity will reduce the desired capital stock. But in a simple model of deadweight costs, the size of the deadweight cost triangle will increase in proportion to the square of the tax rate. So if the tax rate is initially zero, for example, the marginal deadweight cost will also be zero, which seems to mean that Brad's method says that the marginal effect of taxes on the level of investment will also be zero, if we start at zero taxes. That can't be right.]

Comments

[Adam P]

Nick, this:

"And investment is usually bounded from below when the desired capital stock is less than the actual, since we usually cannot convert capital goods back into consumption goods, so gross investment cannot be negative."

is not a usual assumption. We don't generally assume that investment is irreversible, that assumption is sometimes made but it leads to an entirely different theory of investment. You get a real options theory of investment with that assumption.

Adjustment costs can be asymmetric, so it costs more to disinvest than to invest but absolute irreversibility is a special assumption, not the normal case.

[Adam P]

Actually, something else screwy about DeLong's calculation is that if K/Y = 400% and depreciation is 10% then steady state investment is 40% of GDP, way too high.

[Nick Rowe]

Adam: good points.

I don't think your first point will affect my calculation. Unless the current fiscal deficit is so big that desired gross investment actually goes negative.

Your second point will affect my calculations. By quite a bit. If K/Y is (say) 2, then a/R becomes 2 too. If a=1/3, then R=1/6, or 16.6%, which looks much better (10% depreciation, plus 6.6% real interest plus taxes).

And if K/Y=2, that will halve my estimate, from 0.80% to 0.40% of GDP decline in investment. That's a bit closer to my priors (not that my priors were based on anything other than gut feel).

[Simon van Norden]

I like the topic, Nick. As I recall, this got kicked off by a comment of Krugman's to which Mankiw responded and DeLong's post was the response to the response....but I digress.

I'm interested in the assumptions regarding the long-run cost of financing the debt. Mankiw has written about this before. (Laurence Ball, Douglas W. Elmendorf, N. Gregory Mankiw, "The Deficit Gamble", Journal of Money, Credit and Banking, 1998, Vol.30(4), 699-720.) Let's recall Mankiw's description of what they find.

"The historical behavior of interest rates and growth rates in US data suggests that the government can, with high probability, run temporary budget deficits and then roll over the resulting government debt forever....whenever a perpetual rollover succeeds, policy can make every generation better off. This conclusion does not imply that deficits are good policy, for an attempt to roll over debt forever may fail. But the adverse effects of deficits, rather than being inevitable, occur with only a small probability."

Briefly put, Mankiw & co. look at the real interest rate on US government debt and the real growth rate of the economy from 1871 to 1992. They note that on average (and most of the time), the growth rate of the US economy has been greater than the real interest rate on its government debt. This means that the debt to GDP ratio tends to decline over time, even if the government makes no effort to pay the interest on its debt. (The historical statistics in the paper make for interesting reading.)

As a case in point, they describe the US experience from 1945-1975 as a successful Ponzi Gamble. Over this period, US Debt/GDP fell from 115% to 27%. Budget surpluses accounted for a change of 12%; the rest was strong economic growth.

Now, I understand your desire to use pessimistic assumptions. But before you use a 5% real interest rate (in an economy that I think you implicitly assume is not growing), why not measure that against the range of historical experience? I think the public debate needs to consider the range of outcomes for the spread between real interest and real growth rates to understand the implications of fiscal policy choices.

[Nick Rowe]

Simon: Thanks, yes. Krugman to Mankiw to DeLong, and now I'm putting in my twopenceworth. I actually wrote a post on this topic as a theoretical possibility over a year ago, sometime, but never took it very seriously as an empirical possibility. But after reading Brad DeLong's post, I thought I really ought to do a back of the envelope, just to see if my gut was right about this being small beer.

I didn't know about that Ball, Elmendorf, Mankiw paper. Thanks for giving me the gist of it. (Wish I had known about it when I did my old "Do we need a bubble?" post.

I took the 5% from Brad DeLong. He probably chose 5% to be ultra-conservative (and to keep the math simple).

I think what matters is the difference between the interest rate and the GDP growth rate (both real, or both nominal, but same price index in both cases). And that could indeed be 0%, not the 5% I've assumed. If for example it's 2.5% not 5%, then my guesstimate gets cut from 80% to 40%. It should be all proportional.

In which case, probably not a big deal for the US.

What about Greece though? Much higher interest rates on government debt, and a lower long run growth rate (demographics). There's presumably a point at which interest rates on government debt get high enough that deficit spending would be contractionary, through this channel. On the other hand, if interest rates are high because people expect default, they can't also expect the higher taxes that would reduce the return to investment. Or could they, at the margin?? Lump sum default, increase taxes to pay marginal debt? My brain hurts again. It's too hot here.

[Simon van Norden]

To tempt you to read the paper, here's a another quote from Ball, Mankiw and Elmendorf....

"If future economic growth turns out to be low....the government will eventually be forced to raise taxes. But, using historical data on growth rates and interest rates, we estimate that the probability of this outcome is only 10 or 20 percent. With a probability of 80 or 90 percent, a debt of the size inherited from the 1980s will never force the government to raise taxes higher than they otherwise would have been."

They spend some time arguing that this occurs despite the fact that the US economy seems to be dynamically efficient, then present a couple of models in which such Ponzi Gambles are usually successful. Both raise welfare by providing households with a safe asset (government debt.)

I have not heard Mankiw stressing such arguments lately. Nor have I heard Elmendorf stress them since he became head of the CBO. Perhaps I just have not been paying attention....

[Simon van Norden]

If anyone wants the original article and has access to JSTOR, you can find it at http://www.jstor.org/stable/2601125

[himaginary]

Nick:"If we assume it takes 2 years instead of 1 to adjust the actual capital stock to the desired, then we halve the size of the effect, but the effect lasts twice as long. And if at the same time we double the length of the fiscal stimulus, so it too lasts for 2 years instead of 1, everything is the same as in my original calculation."

DeLong seems to assume one year of fiscal stimulus, thirty years of tax increase(*), ten years of adjustment of the actual capital stock to the desired(**).

(*)"...each $1 of infrastructure spending now lowers after-tax future incomes in total by a present value of $1.50--amortizing over the long-term future, by $0.05 for each future year..."

(**)"With standard rates of depreciation, investment in any one year is at most one-tenth of the desired capital stock."

I think reading this would help to understand the rationale behind that assumption.

BTW, about the elasticity of the desired capital stock with respect to R:

From Y=K^a*L^(1-a) and K/Y=a/R,
K^(1-a)=a*L^(1-a)/R
Therefore,
log(K) = -1/(1-a)log(R) + log(L) + 1/(1-a)log(a)
So elasticity of K with respect to R is -1/(1-a).
If a is 1/3, this is -3/2.

[Nick Rowe]

himaginary: Good work!

I am kicking myself for having forgotten to take logs. I used to know that! dlogK/dlogR is the elasticity of K with respect to R. Of course!

So the elasticity is 3/2, not 4/3 as I assumed. That increases my estimate from 80% offset to 90%. (That's still a high-end estimate, of course).

I've been re-thinking my assumption (and Brad's assumption) that a 1% of GDP deficit causes a permanent 0.05% of GDP tax increase. It might not be permanent. As you say, it might be amortised over (say) 30 years, or less. And that's important. The assumption we make here will have a big effect on the results. If people expect the government to pay off the debt quickly, then the expected tax increase can be large, even if the real interest rate minus the growth rate is small.

I still don't get what Brad meant by "With standard rates of depreciation, investment in any one year is at most one-tenth of the desired capital stock.". With 10% depreciation (reasonable) investment in steady state (no growth) will be one-tenth of K. But it will be less than that in the transition to the new steady state, if the desired capital stock is lower.

The Paul Krugman post you link to doesn't help me understand this, since that post is all about Ricardian Equivalence, which is about the effect of expected future taxes on current *consumption*, not investment.

Good work!

[Adam P]

Nick, that statement you quote of DeLong's, that's where I got the idea he was asserting a turnpike property. If the optimal investment has the turnpike property then investment just goes to its new steady state value and stays there and the actual capital stock takes a very long time to adjust.

[Adam P]

Took a quick look at Blanchard and Fisher, I didn't look at the math at all so don't quote me here, but they have a diagram of the turnpike property that shows a large intial reaction to investment and then it settles quickly to its long run level. Thus K moves quickly towards he steady state at first but once near it it takes a long time to get all the way there.

This makes sense to me, when you are far from the new steady state the cost of a suboptimal capital stock makes paying larger adjustment costs worthwhile, once near the optimal value the adjustment costs dominate so you minimize them by converging very slowly.

This suggest you're right here.

[Nick Rowe]

Adam: that makes intuitive sense to me too, with convex adjustment costs. (The nearest I have been to a turnpike theorem is that I once drove through Rochester NY, on the back roads ;-) Sorry.)

[himaginary]

"The Paul Krugman post you link to doesn't help me understand this, since that post is all about Ricardian Equivalence, which is about the effect of expected future taxes on current *consumption*, not investment."

I linked to that post just to show that Krugman and DeLong are thinking in terms of one-shot stimulus accompanied by tax increase for an extended period.

As for depreciation, I think it's more simple story. If current capital stock is 100, and it depreciates 10 each year, it will disappear entirely in ten years if you don't invest at all. If you invest 10 each year, then the capital stock will remain steady 100. If you want the capital stock to be 90 in ten years, you will decrease each-year investment by 1, i.e., from 10 to 9. And, as in DeLong's example, if you want the capital stock to decrease by 0.1 to be 99.9 in ten years, you will decrease each-year investment by 0.01, i.e., from 10 to 9.99.

[Simon van Norden]

More Mankiw (from his Feb. NYT op-ed at http://www.nytimes.com/2010/02/14/business/economy/14view.html)

"...Even in the long run, a balanced budget is too strict a standard. Because of technological progress, population growth and inflation, the nation’s income and tax base grows over time. If the government’s debts grow at or below that pace, servicing the debt will not become a major problem. That means the government can run budget deficits in perpetuity, as long as they are not too large.

Recent history illustrates this principle. From 2005 to 2007, before the recession and financial crisis, the federal government ran budget deficits, but they averaged less than 2 percent of gross domestic product. Because this borrowing was moderate in magnitude and the economy was growing at about its normal rate, the federal debt held by the public fell from 36.8 percent of gross domestic product at the end of the 2004 fiscal year to 36.2 percent three years later."

[Nick Rowe]

himaginary: OK, I understand you now. Yes, the assumption of a one-year fiscal deficit followed by multi-year tax increase makes sense.

On investment and the desired capital stock: But if we think taxes will rise next year, the desired capital stock for next year will be lower. And, without adjustment costs, we would want to reduce investment immediately (this year) by the full amount, to bring next year's actual stock equal to the desired stock.

Simon: but we have to be careful here. With a growing GDP, the sustainable deficit that keeps debt/GDP constant over time is positive. OK. But suppose we start at that steady state deficit, and then have a temporarily increased deficit for 1 year. We would still need to increase future taxes by (r-g) times the accumulated deficit, just to keep the debt/GDP ratio at the new higher level. And by more than that in the short run, if we wanted to bring the debt/GDP ratio back down to where it was originally.

[Simon van Norden]

So I've been wondering what real interest rate is required for a pessimistic analysis of the cost of deficit finance.

TIPS yields are currently 1.79 and 1.85 for the 20 and 30 year bonds, so that's the US govt.'s real cost of borrowing. However, if the government merely aims to stabilize its debt/GDP ratio (as Mankiw argues), DeLong's calculations should use the above real rate minus the real growth rate of the US economy. We want to take a pessimistic view of that growth rate; how pessimistic should we be?

The BEA publishes annual real GDP back to 1929. If we look back over the past 80 years, the US has never had a 20-year average growth rate below 2.52% (2.69% if we average over 30 years.) For comparison, the average growth has been 3.65%.

As you note, the costs are directly proportional to the real interest we assume. Given present TIPS yields, we can assume that long-term growth in the US will be well below anything anyone alive today has ever witnessed (say, 2%) and still get costs of zero.

I think these are the kinds of calculations that are driving Krugman nuts at present.

As an aside on the calculations, those are geometric averages of annual growth rates in real GDP. Yes, the sample includes the Great Depression. However, I was surprised to find that the period with those lowest historical growth rates doesn't include the Great Depression: the lowest growth rates are precisely those we've seen over the most recent past.

[Simon van Norden]

Nick: Sorry....I guess you posted while I was composing.

I agree that "We would still need to increase future taxes by (r-g) times the accumulated deficit, just to keep the debt/GDP ratio at the new higher level." We're just talking about what a reasonable (or pessimistic) value for (r-g) is.

Above, I've tried to make the case that (r-g)=0 is pretty pessimistic for the US right now. Do you think otherwise? If so, I'm curious to hear the reasons.

[Nick Rowe]

Simon: on reflection, I think you're right. Though "pessimism" is ambiguous in this context. Does it mean high r or low g? Anyway, I think it unlikely that r will be much above g, even if it is above g at all in the near future. Yes, i think that's probably the strongest case one could make that we shouldn't worry about this sort of effect, at least in the US, and Canada too. But maybe Greece or Ireland is very different?

[Nick Rowe]

Changing the subject totally, and going totally off-topic:

The sample size of comments on this post is now big enough for a little experiment. Skim through the comments above. Now skim through the comments on Brad DeLong's post on the same subject (linked in my post above). See the difference between the two sets of comments? I see a massive difference. It's big, and it's ugly. It's stuck in my mind ever since. Do you see what I see?

[Adam P]

With power utility (exponent p) a log-linearization gives r = b + p*g +(1/2)*p^2*var(g).

b = subjective discount factor
var(g) = subjective variance of growth rate (g is growth rate of consumption)

Right now I think you could make a good case for r < g based on a high risk adjustment, a high perceived variance of g.

Essentially the risk-adjusted r is higher than g but the actual r is lower.

[Nick Rowe]

Adam: I think that might work quite well for the interest rates that consumers and investors face, at least those who are borrowers on the margin. And it's a puzzle to me that interest rates on government bonds could be lower than the growth rate. (I'm thinking back to my old post on infinite asset prices!). Maybe the resolution to the paradox is that there's a liquidity premium on government bonds. But if that resolution is correct, then maybe that liquidity premium depends on the total value of bonds outstanding. In which case, if a temporary deficit increases r, we have to take that effect into account when we work out how much future taxes increase as a result of the temporary deficit.

That probably wasn't very clear.

[Simon van Norden]

Nick: I agree that the US is currently very different from many other interesting countries in this regard. Particularly from those countries that don't have the "exhorbitant priviledge" of borrowing from foreigners in their own currency.

As for interest rates on government bonds being lower than the growth rate .... well, Mankiw and Co. document that historically this is the typical state of affairs for the USA. (They also present a couple of toy models that reconcile this with dynamic efficiency; my brief look at them suggested that they relied on the US govt. being able to provide a "scarce" "safe" asset, but I'll defer to a modeling expert on that.)

[Jon]

Seems reasonable that there should be knees in the flow around the rate of depreciation and, in the opposite direction, unused capacity in the capital goods industry.

[himaginary]

[email protected]:58PM:"On investment and the desired capital stock: But if we think taxes will rise next year, the desired capital stock for next year will be lower. And, without adjustment costs, we would want to reduce investment immediately (this year) by the full amount, to bring next year's actual stock equal to the desired stock."

I think DeLong already answered this question in reply to your comment in his blog (although apparently you are not satisfied with it):
"I think your model--with more rapid adjustment and a steeper fall in short-term investment than an immediate fall to the new steady-state investment level--requires that news that there will be fiscal stimulus produce a sharp fall in the stock market, and news that there will be fiscal austerity produce a sharp rise in the stock market. I don't think it's coherent to claim that we live in that world...

Even if you have full within-the-year adjustment of the capital stock--a complete no adjustment costs Jorgenson model--you still get $1 of deficit spending producing $0.85 of net fiscal impetus."

[email protected]:53PM:"(r-g)=0 is pretty pessimistic for the US right now"
Well, that's what we thought about our country, around twenty years ago in Japan. (I know, I know, we have rapidly aging population, a bunch of structural problems, but still...)

[email protected]:40PM:"Do you see what I see?"
I suppose everyone sees it.

[Simon van Norden]

Just one more data point.....

The June minutes from the FOMC meeting are out, which include their economic projections. They estimate the long-run growth rate in real GDP (g) to be 2.5-2.8% annually. That's a long way from 1.8% (r).