Precautionary taxation vs tax-smoothing - on paying down the debt

Bob Murphy is arguing with Steve Landsburg over whether the debt/GDP ratio should be (slowly, eventually) reduced. So I have to join in. Plus, (with my Carleton colleague Vivek Dehejia) I actually published a paper once on this very topic (unfortunately not available online) (link here thanks to Keshav Srinivasan).

(Just to forestall some comments, this is an argument about paying down the debt over the long run, and not about paying down debt in the middle of a recession).

Steve's argument is based on tax-smoothing. Planning to lower the debt/GDP ratio over time would mean that you would plan to have lower tax rates at some future time (when the debt/GDP ratio is lower) than today. But this would violate Ramsey's principle of optimal taxation, according to which the tax rates on two goods which have the same elasticities of supply and demand should be the same. To minimise deadweight losses for a given amount of tax revenue, we want to equalise the marginal deadweight loss per dollar of tax revenue, not just across two different goods, but across two different time periods. This Barro/Pigou/Ramsey argument for tax-smoothing implies that (provided we expect the future economy to be just a scaled-up version of the present) we should plan to keep the debt/GDP ratio at whatever it is right now. (Barro wrote a paper on this once, but I can't find it. Update: found it.)

I'm going to make one very small and very reasonable change to Steve's (implicit) assumptions. Assume that the future is uncertain. There is some degree of uncertainty over future government spending, or future GDP growth. So Steve's intertemporal first order condition now becomes "Current marginal deadweight cost per dollar of revenue equals expected future marginal deadweight cost per dollar of revenue". (All I have done is added the word "expected"). 

Let "t" be the tax rate, and MDWL(t) be the function that represents Marginal DeadWeight Loss per extra dollar of tax revenue as a function of the tax rate t. Then:

Current MDWL(t) = Expected future MDWL(t)

But this implies current t = expected future t only if MDWL(t) is a linear function (that is also time-invariant because everything scales up). In a world where the future is certain, or where the MDWL function is linear, Steve would be right. We should plan to have future tax rates equal to current tax rates, and so plan to keep the debt/GDP ratio equal to whatever it is right now.

But any reasonable marginal deadweight cost function (like in Steve's quadratic example) will be concave convex in tax rates (or is it convex?, I always get them muddled). So we apply Jensen's Inequality, which tells us that MDWL(t)=Expected future MDWL(t) implies that current t > expected future t. We should plan to have lower tax rates in the future, which means we should plan to have a declining debt/GDP ratio.

The intuition is that slowly paying down the debt is like buying insurance against an uncertain fiscal future, because the benefits of a good surprise aren't as big as the costs of a bad surprise. It's like precautionary saving, only applied to the government debt.

Math appendix: Steve assumes R=At-Bt² where R is tax revenue and t is tax rate. I think that means the area of the deadweight loss triangle is DWL=(1/2)Bt² . We want to find the Marginal Deadweight Loss function, which is defined as MDWL(t) = dDWL/dR = (dDWL/dt).(dt/dR) = Bt/(A-2Bt). And I think that function is increasing at an increasing rate in t (positive second derivative) provided you are on the good side of the Laffer curve, and so is concave convex (or convex, whatever).

Comments

[123]

Bingo. I had the same reaction to Steve's post and I hope he will read what you wrote.

[Sealander]

It's convex.

[Nick Rowe]

Thanks 123 and Sealander. Post edited.

[Keshav Srinivasan]

(I just posted this to Bob Murphy's blog as well.) Nick, don't underestimate the power of the internet. The paper is available here:
http://onlinelibrary.wiley.com/doi/10.1111/j.1467-8454.1995.tb00032.x/abstract

If you don't have access to Wiley, I can email the PDF to you.

[Nick Rowe]

Keshav: good find! Thanks!. Link added.

[Ralph Musgrave]

“Planning to lower the debt/GDP ratio over time..” makes no sense, and for a reason given by Keynes: “Look after unemployment, and the budget looks after itself”.

For example, if the private sector wants to continue accumulating net financial assets (and national debt is the biggest part of NFA) then a government just has to let the national debt expand. Though there is no reason to pay a positive real rate of interest on it. (The real – i.e. inflation adjusted – yield on US, German and UK debt is currently about zero).

Conversely, if the private sector tries to spend its NFA, then government just has to run a surplus, which results in the debt declining.

[Nick Rowe]

Ralph: The private sector's desired accumulation of Net Financial Assets (desired saving minus desired investment) depends on the rate of interest. Desired private NFA accumulation is endogenous, not exogenous. And this is a long run argument, not about fiscal policy at the ZLB.

[Patrick]

If the future were certain - for example people where sure that the population and productivity were increasing in the long run - would this same logic lead one to conclude that increasing debt now was the thing to do?

[Nick Rowe]

Patrick: not really. Under certainty you would want to keep tax rates constant over time, which means debt would be constant over time too (more precisely, the debt/GDP ratio would be constant).

[Frank Restly]

Nick,

I think you, Bob, and Steve are missing something fundamental here:
Question: Why does the federal government collect taxes in the first place?
Answer: To create a demand for its money.

With that in mind you must consider that the federal government in collecting taxes affects the demand for its money. If the federal government creates a demand for its money (by levying taxes) and creates a supply of its money (by borrowing into existence) then the federal government by definition affects the cost of money.

And so, how should the federal government use tax policy to effectively regulate the cost of money?
For that you need to look at tax burden instead of tax rates.

Tax Rate = Federal Receipts / Total Private Sector Income
Tax Burden = Federal Receipts / (Total Private Sector Assets - Total Private Sector Liabilities)

[Frances Woolley]

Nick: "Patrick: not really. Under certainty you would want to keep tax rates constant over time, which means debt would be constant over time too (more precisely, the debt/GDP ratio would be constant)."

Does this result rely on any assumptions about growth rates and the marginal utility of consumption? Suppose, e.g., I'm extremely optimistic about technological progress, and figure that we'll manage how to use nuclear fusion to generate boundless energy with minimal negative environmental consequences within the next couple of decades. Does this mean it's a good idea to run up the debt now, because my children and grandchildren will be able to pay it off with ease?

[Frances Woolley]

Sorry, I didn't see the part about the future economy being just a scaled up version of the present economy. Presumably that takes care of growth?

[Ron Ronson]

Does time preference come into play here ?

If taxpayers prefer consumption now to consumption in the future won't they prefer low tax now to lower taxes in the future , and not want to pay down the debt? The higher the level of uncertainty about the future, it strikes me, the more likely people are to not worry about future tax rates, since there will be way more variables than just that one to factor in, and the higher their current pre-tax income the more secure they will feel.

(Don't understand the maths but just trying to look at at this intuitively)

[Ron Ronson]

post-tax not pre-tax

[Nick Rowe]

Frank: "Tax Burden = Federal Receipts / (Total Private Sector Assets - Total Private Sector Liabilities)"

Not really. The sort of analysis I'm doing here, following Steve Landsburg, assumes distorting taxes, so we are talking about the deadweight loss triangle, aka the excess burden from taxation.

Frances: Yep, the assumption that everything scales up, and elasticities stay the same, means you can think in terms of debt/GDP ratios and tax/GDP ratios, basically abstracting away from growth. It's probably OK as a first approximation.

Ron: Positive time preference makes you lean towards future taxes, but positive interest rates makes you lean towards current taxes. If we assume the marginal rate of time preference equals the rate of interest, those two effects should cancel out.

[Simon van Norden]

Nick, how would the presence of exhaustible natural resources affect your analysis? Do you think that we might want to take those into account before we repeat sweeping generalizations about tax rates (like "Planning to lower the debt/GDP ratio over time would mean that you would plan to have lower tax rates at some future time.")?

Nick, the biggest swings that we've seen in debt/GDP ratios occur as a result of wars. Do you think your framework of analysis is applicable to war finance?

Nick, do you think that, for the purpose of the intertemporal optimization that you are considering, that the govt. should assume that the real rate of interest it pays on its debt will always be zero?

Nick, we know that governments face predictable changes in the demographic structure of their nation's population. We think that these demographic shifts should affect national savings behaviour and (typically) the level of international indebtedness of a nation. Do you think it should affect govt. debt planning?

[Frank Restly]

Nick,

"The sort of analysis I'm doing here, following Steve Landsburg, assumes distorting taxes, so we are talking about the deadweight loss triangle, aka the excess burden from taxation."

An economy can be said to be perfectly efficient when it produces exactly as much as it consumes. Deadweight loss (and a decrease in efficiency) occurs when an economy shifts out of symmetry between supply and demand, either the economy consumes more than it produces (generating an inflationary pressure) or produces more than it consumes (generating a deflationary pressure).

Economists like to plot supply and demand curves in terms of price and quantity of goods and then say that deadweight loss occurs when a third party (typically a government) introduces some type of control (price ceiling, tax, subsidy, etc.) that shifts the price of said good away from where the "free market" would set it.

This analysis ignores the obvious. The price of a good is often quoted in terms of a government sponsored currency. The supply and demand for the good is regulated by the "free market" while the supply and demand for the currency is regulated by the government. And so the concept of deadweight loss or excess burden from taxes can be thought of as how taxes accentuate or stabilize price trends in the market. Tax policy that accentuates price trends (like value added taxes) increase dead weight loss and economic inefficiency.

My own opinion is that tax policy should be used to maximize productivity (Real GDP / Total Debt Outstanding) but I understand the argument that it should be used to maximize efficiency.

Maximizing efficiency would entail that the federal government regulates the cost of money through tax policy to offset any upward or downward movement in the price level. Maximizing productivity would entail that the federal government uses tax policy to both reduce the amount of debt and increase the quantity of goods produced and consumed.

[Alex Godofsky]

An economy can be said to be perfectly efficient when it produces exactly as much as it consumes.

No, it can't, that's silly and your whole premise is silly. If the police prevented anyone from doing any work or going shopping, we would produce exactly as much as we consumed, but it would be absurd to call this "perfectly efficient".

[Nick Rowe]

Simon: I found the paper by Barro that this is an extension/modification of.

Barro's analysis is good for considering things like wars. When an unexpected war hits, you raise taxes by the annuitised cost of the war, and use bond-finance for the rest. This means that tax rates, and the debt/GDP ratio, have a unit root. But if you accept my modification you eliminate that unit root in the debt/GDP ratio.

Exhaustible natural resources would be like a negative war. (Except maybe it might affect the assumption that elasticities stay constant over time, because you could accept a Henry George type argument and put higher tax rates on those goods with a less elastic supply.)

The same for demographics.

This doesn't assume that the real interest rate will be zero. It does assume that future deadweight losses are discounted at the same rate of interest as the rate of interest on government debt, which is an assumption that could be challenged.

Take a very simple example, to get the gist of Barro. Suppose r=5%, zero growth, and every year there is a 50% risk of war which costs W. iid shocks to G, and no other shocks. If there is a war, the debt rises by 0.5W, and taxes rise by 0.025W. If there is no war, debt falls by 0.5W and taxes fall by 0.025W. So both debt and taxes follow a random walk. And I'm saying that Jensen's inequality plus a convex Marginal Deadweight loss function imply it won't be a random walk.

[Nick Rowe]

Frank: "An economy can be said to be perfectly efficient when it produces exactly as much as it consumes."

No. This ignores I+G+NX in the accounting identity Y=C+I+G+NX. Plus, this says nothing about the level of Y.

Ah, I've just noticed that Alex has beaten me to it.

Look: you are thinking about the effect of taxes on AD. But you are ignoring the fact that monetary policy is used for AD. And you are ignoring the supply-side effect of distorting taxes, which is what this analysis is about.

[Frank Restly]

Alex,

"No, it can't, that's silly and your whole premise is silly. If the police prevented anyone from doing any work or going shopping, we would produce exactly as much as we consumed, but it would be absurd to call this perfectly efficient."

First, I didn't say go shopping, I said consume. Are you saying that a police can reasonably stop ALL production and consumption? Talk about absurd and silly. What are they going to do, stop all agrarian activities - planting and harvesting? How would the police force survive?

[Simon van Norden]

Thanks, Nick; those sound like very useful extensions. And when I mentioned the case of zero real interest rates, my intention (as I think you guessed) was to suggest that in many cases we'd expect some predictable changes in the costs of government finance. That, in turn, implies predictable changes in debt/GDP ratios, as do wars and exhaustible natural resources.

So now that you've bought the idea that changes in real interest rates might matter, let me take that argument one step further. What if the real interest rate on the debt eventually increases with the amount of debt (or debt/GDP ratio, or whatever normalized level of debt you like to think of)? (Do I need to make the argument to you that this assumption is reasonable on both a theoretical and historical basis?) Does this imply that policies which allow debt to follow a unit root are not sustainable, since high levels of debt must eventually imply interest rates that exceed the growth rate of the economy (i.e. the debt explodes)?

[Alex Godofsky]

Frank: it's a reductio ad absurdum, not a serious proposal. The fact that everyone starves to death is the point.

[Nick Rowe]

Simon: yep, there can be foreseeable changes in debt/GDP ratios in Barro's model. The key is that the expectation of the future tax rate equals the current tax rate. If you unexpectedly get news of a bad thing coming in the future, you raise taxes today, don't plan to either raise or lower them in future, so the debt/GDP ratio is falling until the bad thing happens, then it rises, then it stays constant thereafter. (It was hard to explain this in the post.)

Barro's model assumes the government minimises the Present Value of the deadweight losses, subject to the budget constraint. And he uses the same rate of interest in the PV of DWL as in the budget constraint. I *think* (not 100% sure) that means that r doesn't affect the results, because it appears on both sides of the first order conditions. But now you mention it, that does seem to have rather weird implications if we push it to extremes. I can't quite get my head around it.

Barro assumes Ricardian Equivalence as a first-order approximation (the distorting effect of non-lump sum taxes is the only departure from Ricardian Equivalence) and that r > g, so Ponzi-finance is unsustainable. I would prefer an OLG framework where r is an increasing function of debt/GDP, so whether r > or < g depends on debt/GDP.

I can get my head around the OLG framework with lump sum taxes. I can get my head around the Ricardian framework with distorting taxes and uncertainty and non-linear MDWL function (just). I can't get my head around them both at once.

[Frank Restly]

Alex,

In all other sciences efficiency is defined as the ratio of what you put in versus what you get out. For instance the efficiency of a motor can be expressed as the ratio of the electrical power (Amps x Volts) you apply to the motor versus the mechanical power you receive at the shaft (Foot pounds torque x speed of rotation).

It is implied that to have an efficient motor, you have to have a motor. I understand your point about an reductio ad absurdum. But my point is that to have an efficient economy, you first need to have an economy. An economy with no production and no consumption is no economy at all.

It would be like me saying that the motor I imagine sitting in front of me has an efficiency of 100% (no friction losses) even though it does not exist.

To define economic efficiency and then to try to apply it to situations where an economy does not exist is ludicrous.

[Frank Restly]

Nick,

"An economy can be said to be perfectly efficient when it produces exactly as much as it consumes."
"No. This ignores I+G+NX in the accounting identity Y=C+I+G+NX. Plus, this says nothing about the level of Y."

I never said an economy can be 100% efficient. I said that an economy reaches 100% efficiency when it produces exactly as much as it consumes.

If a geographically isolated economy had every raw material in sufficient quantities to meet any demand and sufficient technical expertise to produce any good, then net exports(NX)go away.

If an economy had instantaneous production cycles where there was no time delay between demand for a good and production & delivery of that good, then business investment (I) and consumption expenditures (C) are not distinct from each other.

If an economy had perfect foresight that could anticipate any and all future demands for goods, then government expenditures (G) goes away.

Instead most economies have to deal with material / qualified labor shortages, production cyles of several days to several years, and goods that are overproduced, underproduced, or not produced at all.

[Simon van Norden]

I understand that models are simplifications; they are not expected to capture every nuance of what we see in the real world. However, when we appeal to models to justify policy recommendations, we're typically assuming that the models to which we're appealing capture all the "relevant" factors that we should be worrying about.

Nick, do you have a view on what the minimum set of "relevant" factors is for a model that we use to justify fiscal policy? The discussion above suggests some candidates:
1) MDWL(t)convex
2) shocks to revenues and expenditures caused by war, non-renewable resources, demographics, etc.
3) r = r(debt,GDP) + shocks
4) g varies?

[Alex Godofsky]

Frank:

In all other sciences efficiency is defined as the ratio of what you put in versus what you get out.

Welcome to economics.

[Frank Restly]

Nick,

"Look: you are thinking about the effect of taxes on AD. But you are ignoring the fact that monetary policy is used for AD. And you are ignoring the supply-side effect of distorting taxes, which is what this analysis is about."

I don't see it that way. Monetary policy is used to set the pretax cost of money. Fiscal policy is used to set the after tax cost of money.

Whether money is borrowed to fund consumption (AD) or borrowed to fund production (AS) is at the choice of the borrower. And so I don't pidgeon hole monetary policy as only an aggregate demand control, and taxes as only an aggregate supply control.

It is easy to come to the conclusion that monetary policy affects consumption more than production since producers have equity financing available to them, but that does not mean that production is completely unaffected by monetary policy. Likewise consumption is not totally indifferent to changes in tax policy.

The way I see it, there are really two costs of money - the pretax cost of money set by monetary policy, and the after tax cost of money set by the fiscal authority. And so I am not sure what you mean by distorting taxes. If you mean taxes that amplify changes in the price level (increase deadweight losses), then I understand. If you are talking about something else, then please elaborate.

[Nick Rowe]

Simon: that's a hard question. I think your list is a good one. I would change "G varies" to "the marginal benefits of G varies" (OK, same thing). I might make a distinction between Government investment and consumption? The biggish things missing from your list is maybe something about intergenerational equity, in some sort of OLG setup, plus the Samuelson 58 idea that if r < g a Ponzi is sustainable.

I find it very hard to think of all those things at once, and tend to take them a couple at a time.

Frank: It would take me hours to go through your comment line by line to explain all the many ways I think you are looking at this wrong. And I've just come off a full day teaching, so sorry.

[Simon van Norden]

Nick: But I said g ... not G! ;-)

That is, I was suggesting that we should worry about possible variation in economic growth rates/trends.

Yeah, intergenerational equity might be a good addition.

Despite the evidence in Ball, Elmandorf and Mankiw, I'd still be tempted to rule out sustainable Ponzi Games....can't shake the feeling that banking on a strategy like that is a Bad Idea.

[Frank Restly]

Nick,

Okay how about a simple one. How do you distinguish between a tax that distorts and a tax that does not? In my mind, a tax that distorts is a tax that amplifies a shift in the pricing of goods and a tax that mitigates a shift in the pricing of goods would not be considered to be distorting.

[Nick Rowe]

Simon: Aha! Yes, growth rates will matter a lot.

I too am very uneasy about banking on a Ponzi scheme. But I'm also very uneasy about letting a possible free lunch go uneaten, even if we won't know whether it was free or not until long after we've eaten it. The Trill perpetuity post is as far as I have got towards a solution.

Frank: no, how about this simple one: you tell me what economics you have learned. Then I will know where you are coming from, so I'm not wasting my time and yours. Because "In my mind, a tax that distorts is a tax that amplifies a shift in the pricing of goods and a tax that mitigates a shift in the pricing of goods would not be considered to be distorting." is totally wrong in my mind.

Start right at the beginning, with a supply and demand diagram. Can you analyse the effect of a $1 per apple tax? Identify and interpret the deadweight cost triangle? Know the relation between the area of that triangle and elasticities of supply and demand?

[Bob Murphy]

I too am very uneasy about banking on a Ponzi scheme. But I'm also very uneasy about letting a possible free lunch go uneaten, even if we won't know whether it was free or not until long after we've eaten it.

Nick when I called you a kung fu master on my blog, I was joking. You don't actually have to speak in fortune cookie language.

[Frank Restly]

Nick,

Adding a dollar tax to an apple will cause fewer apples to be bought at a higher price UNLESS that dollar tax either funds the production of additional apples or funds the consumption of additional apples. If that dollar tax is used to subsidize the production of oranges or increase the consumption of oranges then the relative costs of apples and oranges will change. And here I would agree, tax policies that change the relative prices of goods can be considered distortionary. As for analyzing the effect of a $1 per apple tax? You can't look at the tax in isolation and say that it will have this effect or that. Tax revenue does not disappear down a rabbit hole never to be found again.

If we assume linear demand and supply elasticity:
Qd (P) = kd * P + qd
Qs (P) = ks * P + qs

Demand elasticity: Ed = P * kd / ( kd * P + qd )
Supply elasticity: Es = P * ks / ( ks * P + qs )

The first vertex of the dead weight loss triangle is the "free market equilibrium" point:
Q1 = Qd(P1) = Qs(P1)
P1 = ( qs - qd ) / ( kd - ks )

The second vertex is the intersection of the demand curve and the supply curve shifted up by $1
Q2 = Qd(P2) = Qs(P2) + 1
P2 = ( qs - qd + 1) / ( kd - ks )

The third vertex is the intersection of the supply curve and a vertical line at the quantity Q2
Q3 = Qs(P3) = Qs(P2) + 1
P3 = ( ks * P2 + 1 ) / ks

Using a bit of algebra you can find the area of the triangle. From that you can derive a relationship between the elasticities of the supply and demand curves and the area of the deadweight cost triangle. I imagine you are looking for an answer like the area of the deadweight loss triangle increases with higher demand elasticity and lower supply elasticity. Likewise the area of the deadweight loss triangle decreases with lower demand elasticity and higher supply elasticity.

Hopefully that is enough to convince you that I have a little bit of knowledge about economics. If you want an algebraic solution to the actual area of the triangle I would be happy to email it to you - linear supply and demand only :-) The calculus needed for nonlinear supply and demand is pretty deep.

Also, even if the government changes the relative price of one good to another via some tax policy, the private sector is not precluded from re-establishing the original relationship through barter or some other means of exchange. Using your example, suppose that an one apple is priced the same as one orange. Now suppose the federal government adds a 10% tax on apples and a 10% subsidy on oranges. An apple now costs 1.22 oranges. But the original one to one relationship can be re-established through barter presuming that enough people do not like the arrangement.

The whole concept of distortive tax policies relies on a presumption of some normal tax policy or a policy of no taxes at all. If a government collects no taxes at all, then the notion of measuring the price of any good relative to a government's currency goes out the window. And so I chose to define distortive tax policy to mean policy that fails to properly regulate the price level. And no, monetary policy alone cannot handle all possibilities.

[Nick Rowe]

Frank:

1. "Q2 = Qd(P2) = Qs(P2) + 1" should be Q2 = Qd(P2) = Qs(P2 - 1)

With P on the vertical axis, and Q on the horizontal, you shifted the supply curve horizontally by one unit, and should have shifted it vertically by one unit. You had the government take away 1 apple, rather than take away $1 per apple sold.

2. " I imagine you are looking for an answer like the area of the deadweight loss triangle increases with higher demand elasticity and lower supply elasticity." should be the area of the deadweight loss triangle increases with higher demand elasticity and *higher* supply elasticity.

3. "You can't look at the tax in isolation and say that it will have this effect or that. Tax revenue does not disappear down a rabbit hole never to be found again."

If the tax revenue did disappear down a rabbit hole then the triangle alone would not measure the deadweight loss. We would have to add the rectangle in too. When we say the triangle is a measure of the deadweight loss, we assume the tax revenues are not wasted, because we count them as a benefit.

4. "Also, even if the government changes the relative price of one good to another via some tax policy, the private sector is not precluded from re-establishing the original relationship through barter or some other means of exchange."

That's called "tax evasion". Yes, if people can costlessly evade taxes, then taxes will not create distortions.

5. "The whole concept of distortive tax policies relies on a presumption of some normal tax policy or a policy of no taxes at all."

No. It does indeed rely on some presumptions (like no externalities). If the taxes were lump-sum then there would be no distortions in this sense.

Frank: level with me. I have just spent 20 minutes teaching you for free the ECON 1000 stuff I am teaching for a living. So you owe me an open answer. My guess is that you have a background in physics/engineering/whatever, have spent some time reading MMT blogs, and spent an hour or so last night Googling supply, demand, and taxes. Either confirm my guess or correct it.

[Nick Rowe]

BTW: In your math, you had the government impose a lump sum tax of one apple. Which would create zero distortions. No deadweight loss triangle.

Your (mathematician's) response will be: "But what's the difference between shifting a linear supply curve vertically or horizontally???" The difference is whether the marginal benefits equal marginal costs of an extra apple. Which is what the whole idea of tax distortions is all about.

[Frank Restly]

Nick,

Yea you are right. Engineering. And not just MMT. I have tried to read a gammit of economic b-logs. I find MMT informative because it stays away from theory and sticks to how credit based currencies actually work. And, when I need some technical information I use Wikipedia. Google is there if Wikipedia does not have an answer.

I originally arrived at this website when it was linked by another b-log by Dave Glassner (www.uneasymoney.com).

Thank you for the correction on shifting the supply curve. I was alternating between watching the Chicago / San Francisco football game last night and typing a response.

"The whole concept of distortive tax policies relies on a presumption of some normal tax policy or a policy of no taxes at all. No. It does indeed rely on some presumptions (like no externalities). If the taxes were lump-sum then there would be no distortions in this sense."

The way that I understand your definition of nondistortive lump sum taxes are that they are either a tax on goods produced in a previous period (for instance property taxes) or that they are a tax on goods that are not resource limited (for instance a voting or polling tax). But even that definition misses the possibility of overly onerous taxes that indirectly affect the supply and demand for goods. Even if a government focuses on taxing goods produced in a previous period, it can tax them to the point that it starts to affect the supply and demand for those goods. Why buy a house if the government is going tax the bejesus out of it when I could rent instead? Does not demand curve shifting as a result of anticipated taxation also result in deadweight loss even if the tax is lump sum?

That is why I focused on the effect of taxation on the price level and came to the conclusion that if the federal government is to regulate the price level measured in its own currency, and if monetary policy cannot accomplish this on its own, then that leaves tax policy. Hence I would say that distortive taxes are taxes that amplify changes in the price level and nondistortive taxes are taxes that dampen changes in the price level.

[Bob Murphy]

Frank Restly it was good that you came clean. When Nick Rowe starts interrogating people, if they prevaricate he knows. And then they regret it.

[Alex Godofsky]

Frank, you are trying to combine some very different effects that may interact but can and should be examined in isolation first. An engineering example... it's like bringing up the structural stresses on a wing when you are learning about airfoils in your fluid dynamics class.

Here are some of the concepts you need to throw out so you can understand what economists mean by distortionary taxation: money, sticky prices/wages, credit constraints, income effects, time. Once you understand tax distortion in that context, you can add those things back to your model, one by one, and see what they do.

[Determinant]

Nick:

In Frank's defence, he, like me, sees the Central Bank as A TOOL, not THE TOOL. That's the way engineers think, if the current tool doesn't work to our satisfaction, we get a new one, or invent it.

Which is the one key assumption you always make that I have trouble agreeing with. The rest seems to flow from that. I admit I don't get it.

So fiscal/tax policy has micro aspects and politics. No tool is ever perfect.

[Nick Rowe]

OK Frank.

P on the vertical axis, and Q on the horizontal. Draw (linear) demand and supply curves. Suppose they cross at P=$10 and Q=100. Now stick a $1 vertical wedge between the supply and demand curves, on the left of where they intersect. Suppose the price the buyer pays, gross of tax, Pb, is now $10.50. And the price the seller gets, net of tax, Ps, is now $9.50. And that Q falls to 90.

See the rectangle that is $1 high (the tax wedge) and 90 apples long, and so has an area of $90? That is the government's tax revenue. That is NOT what we mean by the deadweight loss from taxes. That's not a loss at all. It's just a transfer from buyer and seller to the government. It's only a loss if the government spends it on something totally useless, and throws it down a rabbit hole.

See the triangle that is $1 high and 10 apples long, and so has an area of $5? THAT is what we mean by the deadweight loss from taxes. What does that triangle mean? It means the losses from unexploited gains from trade. The height of the demand curve represents the Marginal Benefit of an extra apple to the buyer. The height of the supply curve represents the Marginal Cost of the extra apple to the seller. When the government imposes a $1 tax per apple traded, MB=$10.50, and MC=$9.50. Total benefits minus total costs is not being maximised, because MB > MC. There are potential gains to trade of an extra 10 apples that are not being realised, because the government would want a cut bigger than the gains to those trades.

That triangle DID just disappear down a rabbit hole. (In ECON 1000 we normally talk about it disappearing into thin air, or some such metaphor, but rabbit hole works fine.)

This post is about that triangle. A whole sequence of triangles like that over time. And about how to minimise the total sum of the areas of those triangles. Subject to keeping the sum of the areas of those rectangles equal to a certain amount. Under uncertainty.

This is a very hard post, that presumes a lot of previous economics. It presumes the reader already understands the difference between the triangle and the rectangle.

An example of a lump sum tax would be where, instead of making everyone pay $1 for each apple traded, the government made everyone pay $90 regardless of how many apples were traded. The rectangles would be exactly the same size. But there would be no triangle for the second tax.

From my experience teaching, engineers can learn economics very quickly and easily. BUT you can't just wander in and assume you already know it. You don't. There really is more to it than you think there is. Grab an intro textbook, and read it through. then I won't have to spend an hour explaining what we mean when we talk about deadweight loss triangles.

[Frank Restly]

Nick,

Okay,

In your example PQ at market equilibrium equals 100 Apples * $10.00 per apple = $1,000.00
Adding a tax wedge shifts the intersection of the supply and demand curves to a new PQ equal to 90 Apples * $10.50 per apple = $945.00
The amount of revenue the federal government collects is equal to 90 Apples * $0.50 per apple = $45.00
The area of the triangle must then be $1000.00 - $945.00 - $45.00 = $10.00

And so even though the $45.00 that the federal government collects in tax revenue is not lost, the remaining $10.00 is lost.

BUT, this makes the assumption that tax policy is to be used to maximize P*Q. Hence the term dead weight loss implies that maximizing PQ is what you are trying to do with tax policy. What if instead you are just trying to maximize Q? This is the point I have been trying to make, the whole concept of gains and losses from tax policy depends on what you are trying to do with it.

Being an economic outsider means that I tend to take economic terms at face value. And so the concept of deadweight loss from taxes may have a very specific definition in economic circles (loss of nominal GDP), but I would say why should a government focus on nominal GDP in making tax policy?

Suppose as an economic policy advisor you are trying to maximize productivity instead. Productivity (on a macroeconomic scale) can be expressed as follows:

Debt * Velocity = Real GDP * (1 + Inflation Rate)
Productivity = Real GDP / Debt = Velocity / ( 1 + Inflation Rate )

Now what sort of tax policy would you create that would maximize this? It will be totally different from tax policy that minimizes losses of nominal GDP.

[Alex Godofsky]

Frank:

BUT, this makes the assumption that tax policy is to be used to maximize P*Q. Hence the term dead weight loss implies that maximizing PQ is what you are trying to do with tax policy. What if instead you are just trying to maximize Q? This is the point I have been trying to make, the whole concept of gains and losses from tax policy depends on what you are trying to do with it.

No. Our social welfare function is the size of the region between the supply and demand curves, not P*Q. That's the total surplus, that's what is diminished by the deadweight loss.

Oh, I just realized why you might have gotten this confused - the geometry just happens to make the size of that triangle ALSO equal to P*Q. But if the demand curve topped out at $12 (i.e. no one would ever buy an apple for more than $12) then the size of the surplus would be less than P*Q, but our welfare loss would be unchanged.

[Reverend Moon]

Nick,
You said
"The height of the supply curve represents the Marginal Cost of the extra apple to the seller."

Is that true? Why would it be true in reality? Wouldn't it require the assumption of diminishing marginal productivity to be true (and possibly perfect competition)? So that profit is maximized when MC=MR? Are those assumptions a reasonable reflection of reality?
Sorry for the possibly naive question

[Nick Rowe]

Frank: "The amount of revenue the federal government collects is equal to 90 Apples * $0.50 per apple = $45.00"

No. The tax is $1 per apple. The tax revenue is $1 per apple x 90 apples = $90. The price the buyer pays rises 50 cents to $10.50, and the price the seller gets, net of tax, falls 50 cents to $9.50. (I have assumed the supply and demand curves have the same elasticity).

"BUT, this makes the assumption that tax policy is to be used to maximize P*Q."

No. That is not what I am assuming. (If I wanted to do that I would want a tax or subsidy that lead to a point on the demand curve at which elasticity = 1.

"What if instead you are just trying to maximize Q?"

No. We don't want to do that either. If I wanted to do that, I would put on a very large subsidy, which would also create a very large deadweight loss, just like a very large tax (it's symmetric).

What I want is for Q to equal Q*, where Q* is defined as that Q at which the Marginal Benefit of an extra apple equals the Marginal Cost of an extra apple. I do not want Q < Q*, nor do I want Q > Q*. Either would create a deadweight loss triangle.

If there are no externalities from apples (like pollution), and if there is no market power (monopoly or monopsony), then Q* is at the point where the original supply and demand curves cross. Because the height of the demand curve is MB, and the height of the supply curve is MC. A $1 per apple tax means that Q is now where MB(Q')-MC(Q')=$1, so MB is not equal to MC. The integral between MB and MC from Q' to Q* is the deadweight cost triangle.

This is very basic first year microeconomics. This what we mean when we talk about tax "distortions". If the market equilibrium were not in this sense "efficient", then we wouldn't call them "distortions". For example, if there is a negative externality to apples, then a "Pigou" tax would reduce Q, and we would want it to reduce Q. Pigou taxes cause triangle benefits, not costs.

This stuff is absolutely fundamental to economics. It is in the intro text. You can even disagree with it, if you like. But you have to understand it. Otherwise you won't have a hope of understanding posts like this, or lots of other posts.

(I am saying basically the same thing as Alex is saying above.)

[Nick Rowe]

Rev Moon: If the seller has monopoly power, the supply curve does not exist. (Because in that case the profit maximising quantity to sell depends not only on price but also on elasticity of demand.) If the seller is a competitive firm, (a price-taker), then MR=P, and setting MC=MR to maximise profit means setting MC=P. So yes, the MC curve is the supply curve.

MC=Wage/MP (assuming labour is the variable input, and assuming the firm takes Wage as given). Diminishing MP means increasing (upward-sloping) short run MC curve. If the MC curve is always downward-sloping we will not have perfect competition. We have "natural monopoly".

If we have monopoly power, the equilibrium Q is below the Q* at which MB=MC, so there is a deadweight loss triangle from monopoly. It we add a tax on top of that, the deadweight loss triangle gets even bigger. Taxes create a deadweight loss trapezoid(?) in that case.

[Frank Restly]

Nick,

(Taking deep breath)...

Market equilibrium equals 100 Apples * $10.00 per apple = $1,000.00
The amount of revenue the federal government collects is equal to 90 Apples * $1.00 per apple = $90.00
The consumer pays 90 * $10.50 per apple = $945.00
The producer receives 90 * $9.50 per apple = $855.00

Even if that tax revenue was returned directly to the consumer or producer split evenly, the tax wedge created a lost market opportunity for both buyer and seller.
The consumer pays 90 * $10.50 per apple - $45.00 tax rebate = $900.00
The producer receives 90 * $9.50 per apple + $45.00 subsidy = $900.00

However, if both the consumer and producer have forehand knowledge of the tax rebate and subsidy then wouldn't they migrate back towards market equilibrium (buying and selling 100 apples)? Consumer would buy 100 apples at $10.50 each knowing he will receive a $50 rebate and producer sells 100 apples at $9.50 each knowing he will receive a $50 subsidy. Obviously the situation would be different if the rebate and subsidy went to orange consumers and producers thus creating a change in the relative prices of apples and oranges. It seems to me that tax wedges only work in only one way - changing the relative prices of goods. Tax policy that changes the relative value of all goods relative to the government's currency is simply tax policy that gives monetary policy a helping hand.

"If there are no externalities from apples (like pollution), and if there is no market power (monopoly or monopsony), then Q* is at the point where the original supply and demand curves cross."

But there is monopoly power. The federal government is the monopoly provider of its currency. The only way to determine where the original supply and demand curves cross is to switch to a barter system, determine where the markets set the relative values of all goods, and then switch back. Otherwise, it is impossible to tell whether previously enacted tax and monetary policy have shifted supply and demand curves away from their original positions. The simplest example would be the federal government levying such a heavy tax on an industry that it disappears completely.

What you have shown me is great stuff and I thank you for it.

"You can even disagree with it, if you like. But you have to understand it."
I believe that I understand it. But you are correct, I don't agree with all of it.

[Determinant]

This is why I never contribute to MMT threads. I do not understand the stuff and have no interest in trying. It just makes my eyes glaze over.

I'll take the fast version, if any.

[Nick Rowe]

Frank: I think you are maybe getting it.

"However, if both the consumer and producer have forehand knowledge of the tax rebate and subsidy then wouldn't they migrate back towards market equilibrium (buying and selling 100 apples)?"

If there were only one buyer and one seller, then yes. As soon as they realised what was going on, they go straight back to trading 100 apples. But with a population of 33 million, I know I will only get back one 33 millionth of any extra dollar I pay in taxes. I ignore it. I free ride. Unless I'm altruistic. But if everyone were altruistic, we wouldn't need taxes. Voluntary contributions to the government would work. We probably wouldn't really need a government anyway.

The government has a monopoly on force, and that's what enables it to tax. It doesn't need money to tax. It could tax us 10% of the apples we trade, so we pay taxes in apples.

Determinant: this isn't MMT. This is standard micro. MMT is macro.

[Determinant]

Sorry, that was my unclear point. I am completely baffled by MMT and never contribute to threads on them.

It is an example of this engineer knowing his limits. I prefer policy threads.

[Frank Restly]

Nick,

"The government has a monopoly on force, and that's what enables it to tax. It doesn't need money to tax. It could tax us 10% of the apples we trade, so we pay taxes in apples."

A government does not have a monopoly on force of will. There are plenty of historical examples of governments being toppled by their own citizenry by force.

A government has a monopoly on its system of laws. What enables it to tax is people accepting the need for legal proceedings (an impartial 3rd party). And that third party does not work for free. Because lawyers (like anyone else) do not like to be paid in an asset that loses value over time (or rots like apples), the government requires that taxes be paid in its own currency so that it can regulate the value of that currency with respect to all other goods.

Of course this is the ideal case. Most governments also get involved in international diplomacy, social insurance, and a lot of other things.

[Determinant]

I heartily recommend Noah Smith's blog Noahpinion.

He has the most wonderful list of EconoTrolls (with pictures!).

http://noahpinionblog.blogspot.ca/2012/09/econotrolls-illustrated-bestiary.html

Find your own picture, I know I'm on there. :)

[Frank Restly]

Yeah,

I was kind of perturbed that Friedman, Keynes, Minsky, Marx, Ludwig von Mises were all included but Fisher is absent. And so maybe this is how Fisherites see themselves:

http://www.youtube.com/watch?v=yHFDa9efCQU

And this is how the world sees them:

http://www.youtube.com/watch?v=zDAmPIq29ro

[Reverend Moon]

Nick,
Sorry that's not what I was asking. I understand what dead weight loss is. I see you qualified diminishing marginal returns with short run. Is the short run defined as the amount of time required to increase capital? I'll do my own internet study since it's not the subject of this thread and I haven't paid tuition here and don't want to spend too much time highlighting my ignorance. I'm asking why diminishing marginal returns rather than what is it (think Sraffa). It just doesn't jibe with my perception of reality. It seems like the exception rather than the rule unless you assume that full capacity utilization is the normal course of operation (I have other questions too but will stop here). Cheers.

[Nick Rowe]

Rev: There are two distinctions:

1. Diminishing marginal product (holding some inputs fixed and varying other inputs), vs diminishing returns to scale (varying all inputs together.

2. At the firm-level vs at the economy-wide level.

Take agriculture, for example. An individual farm might have diminishing marginal product of labour (holding land fixed) but constant returns to scale (varying land and labour together). But at the level of the whole economy, different types of land will be in fixed supply, so the supply curves for different types of food could be upward-sloping. The economy-wide PPF will not be a straight line, even if there are constant returns to scale, because some land (and labour) has a comparative advantage in producing some goods, while other land (and labour) has a comparative advantage in producing others, so supply curves will slope up at the economy-wide level.

If you think there are increasing returns to scale at the level of the firm, you simply ditch perfect competition and replace it with (say) monopolistic competition. I usually think in terms of monopolistic competition, with perfect competition an OK simplification for some purposes (it's just a limiting case anyway).

[Peter N]

You give MDWL(t) = dDWL/dR = (dDWL/dt).(dt/dR) = Bt/(A-2Bt). This = 0 at t = 0, goes to +infinity at t = A/2B and then reappears at -infinity and climbs to -1/2 at t = infinity. At t = A/2B,the loss = 1/2 Bt**2 = 1/2 B * (A/2B)**2 = A**2/8B. assuming I haven't made a math error somewhere, what does this behavior mean?.

I sympathize with Frank, because real tax policy seems always to be concerned more with externalities to your model. If this weren't the case we wouldn't have distortions like tax expenditures. These should all have a net deadweight loss, and their justification would be positive externalities. I don't know whether one should wish it were true or fear it (since it would be off to the races). Marijuana taxes, carbon taxes, Obamacare penalty taxes, estate taxes, carried interest - the US congress has only two solutions to any problem - make it a crime or distort the tax system (other than for things like declaring national strawberry week).

[Nick Rowe]

Peter N: I haven't checked your math, but it sounds right. As you increase t, and approach the top of the Laffer curve (where R is at a maximum), the MDWL should approach infinity. That's because dR/dt approaches zero (and then goes negative), while dDWL/dt keeps on increasing.

[Peter N]

This is very like an optimal prudent leverage VAR calculation that you'd see with a bank, if you assume that the government spends as much as is prudently possible (that is what is either necessary or has a positive net ROI). This isn't government as we know it, but it fits your model in not questioning the inherent value of government spending and only looking at the result. It doesn't, however, include Ricardian equivalence overtly, though it could possibly sneak in through the ROI calculation. As a nonbeliever in Ricardian equivalence voodoo, I don't much care.

The hardest part of optimal debt is obviously dealing with risk and uncertainty. You have ugly low probability high risk events with arguably low NPV (sort of like the question of how much you should pay to prevent a 10 meter sea level rise 150 years from now?), but God help you if you stumble into one.

There's also the problem that a government's debts serve as peoples' safe assets and low information cost collateral (Triffin dilemma, social security trust fund...).

[Frank Restly]

Peter N,

Because of the way credit markets are structured, the private markets bear both bankruptcy risk(cash flow is less than cost of servicing debt) and solvency risk (present value of assets is less than present value of liabilities) where the government bears neither.

A truly "safe" asset would work to offset one of those risks. Government bonds are safe in the sense that repayment is guaranteed. But because the interest rate paid by the government is always less than that paid by private borrowers, private borrowers are disadvantaged by the legal requirement to pay taxes.

Government equity could fit the bill as a "safe" asset from a solvency point of view. While the return on investment is not guaranteed to any one buyer, the net present value of the asset is know with a high degree of certainty.

The article above discusses whether the debt to GDP ratio should be reduced and if so,how quickly? Before answering that question I think you have to ask, why should the federal government sell debt at all?

[Steven E Landsburg]

Nick: You're exactly right of course. Thanks for posting this.