Do economically illiterate slobs use duality theory?

Dunno. I was never that good at micro/math. That's why I'm asking.

The government has a banana machine that converts any number of apples into the same number of bananas. If it starts the machine up we get a new equilibrium, where people will produce more apples and fewer bananas, and/or consume fewer apples and more bananas, compared to the old equilibrium.

Let the price of an apple be 1+r bananas. Assume a competitive equilibrium where the slope of the Production Possibility Frontier (MRT) = 1+r = the slope of the indifference curve (MRS). If r > 0, the government makes a loss from operating the banana machine, and people will be worse off in the new equilibrium, in the sense of having lower utility.

One way to look at that loss is in terms of inefficient movements along the PPF and Indifference curves. Private production is distorted in one direction, and consumption is distorted in the other direction.

A second way to look at that loss is in terms of the lump sum tax the government needs to impose to cover its losses from operating the banana machine. If B is the number of bananas the government produces, the government's loss will be rB bananas, and the government will need to impose a lump-sum tax of rB bananas to pay for operating the banana machine.

If a sophisticated economist says that there is no loss because those rB bananas do not get destroyed, and still get eaten, because they are handed back as subsidies to get people to swap B apples for B(1+r) bananas (B from the banana machine, plus rB from the subsidy), that sophisticated economist is totally missing the point. Utility is lower in the new equilibrium where the government operates the banana machine.

But is rB a good measure of the loss in utility from operating the banana machine? In other words, if the government did not operate the banana machine, and simply imposed a lump-sum tax of rB bananas and threw away those rB bananas, would that cause the same loss in utility?

1. My guess is that if the PPF were linear, or if the indifference curves were linear, the answer would be "yes". Throwing away rB bananas would cause the exact same loss in utility as operating the banana machine to convert B apples into B bananas.

2. My guess is that if technology or preferences were smooth (no kinks), and if the change in B is small, then r.deltaB would be a good measure of the loss in utility from a small increase deltaB in the operation of the banana machine. The loss would be approximately equivalent to throwing away r.deltaB bananas. (Because smooth technology and preferences are linear in the limit as the change gets smaller.)

3. My guess is that if deltaB is large, so that r increases when B increases, r.deltaB is an overestimate of the loss in utility from operating the banana machine. Throwing away r.deltaB bananas would cause a bigger loss in utility. (But are we talking about the r before or after B changes? If we use r before B changes, then my guess is that r.deltaB underestimates the loss in utility.)

But then I don't understand duality theory. (My fault, and my memory's fault, and not my teacher's fault.) Or calculus (should I be integrating the area under the rB curve?). And I'm too old to learn (the depreciation rate on the investment in human capital is too high). But somebody reading this ought to know the answers. Why did God invent micro theorists, if not to answer questions like these?

"Apples" are "beers when young"; and "bananas" are "beers when old"; the "banana machine" is a stock of bonds worth B beers, on which the government pays rB interest financed through lump-sum taxes, and holds the stock of bonds constant over time. Young people buy B bonds, so either consume B fewer beers and/or produce B more beers. Old people sell B bonds, so either produce B fewer beers and/or consume B more beers. Economically illiterate slobs think that the cost of a deficit is the extra future taxes needed to pay the interest r on the extra debt deltaB created by the deficit. I think the economically illiterate slobs are approximately right, and exactly right in continuous time, or with linear technology or preferences.

But if r < 0, so the government makes profits from operating the banana machine, or if r < g, where the banana machine converts one apple into 1+g bananas and so also makes profits, then it would be very different. The loss in utility (r-g).deltaB would be a negative loss, and the banana machine would provide lump-sum subsidies.

And the banana machine can also operate in reverse gear, where the national debt is negative. Student loans are like that.

Update: and a small open economy is an economy where there is a banana machine in mid-ocean that converts 1 apple into 1+r bananas, or vice versa, so the domestic price of apples as 1+r bananas is independent of domestic B, so the economically illiterate slobs are exactly right.

Comments

[Sandwichman]

There are no such machines.

[Nick Rowe]

Sandwichman: of course there are. Canadians load a lot of apples into a ship, it sails out into the ocean, and a few days later it sails back to Canada loaded with bananas. How could that have happened, if banana machines don't exist? Magic?

[Niveditas98 .]

If the young are willing to exchange B beers for B bonds, and the old are willing to exchange B beers for B bonds, doesn't that imply that "beers when young" have the same value as "beers when old"?

[Michael S.]

Good idea, Nick. You could not goad PK into responding to your earlier salvos, but maybe if you make it about trade...

[Nick Rowe]

Niveditas: no. A young person will exchange B beers when he is young for a promise to pay him (1+r)B beers when he is old. (The old person may be consuming more beers than a young person, so the Marginal Utility of a beer when old may be less than the Marginal Utility of a beer when old. Or people may simply be impatient.)

[Nick Rowe]

Michael: funny thing is, it was only after I had written this post that I realised: hang on, this is trade theory! And PK is a trade theorist! Uh oh, I'm in for it now! (I don't do trade theory either.)

[Niveditas98 .]

Nick, how so? Law of one price -- if B beers when young are valued by the market the same as B bonds the same as B beers when old, I don't see what the future has to do with it. This is all happening in the here and now.

[Niveditas98 .]

In fact, we can say more: if the young person gives up B beers in exchange for B bonds, he does so because it makes him better off. If the old person gives up B bonds in exchange for B beers, he does so because it makes him better off too.

The only possible negative is from the tax of rB beers -- the government transfers rB beers from some people to some other people. This transaction makes the some people worse off and other people better off. The total loss from this is very unlikely to be as much as rB beers, unless the recipients value beers at zero -- this is the difference from your hypothetical about apples and bananas, because the government doesn't actually throw away the rB that it collects in taxes, it gives them to someone else.

[Nick Rowe]

Niveditas: "... this is the difference from your hypothetical about apples and bananas, because the government doesn't actually throw away the rB that it collects in taxes, it gives them to someone else."

If the government uses the banana machine, it doesn't throw away rB bananas either. But the loss in utility is (approximately) the same as if it did not use the banana machine and did throw away rB bananas.

[Niveditas98 .]

I think I was confused by your statement "holds the stock of bonds constant". You're really asking what happens when that stock changes, due to a deficit. In this case, it is true that if the government issues B bonds, takes the beer and simply puts it into a strategic beer reserve for a year, and then returns it, it will have destroyed rB value. But, this isn't what governments ordinarily do when they run deficits, is it?

[Nick Rowe]

Niveditas: The simplest assumption is that B=0 initially, then the government sells B bonds to the young for B beers, and gives those B beers to the old, who drink them, then die. So that first cohort of old clearly benefit. But each period the next cohort of old have to pay a tax of rB beers, to pay themselves the interest on the bonds. And the stock of bonds stays at B forever. How much do future generations lose? Do they lose as much as if the government had issued no bonds, and simply taxed them rB beers when old, and threw those rB beers away?

Have you read my previous post?

[Frank Restly]

Nick,

"The government has a banana machine that converts any number of apples into the same number of bananas. If it starts the machine up we get a new equilibrium, where people will produce more apples and fewer bananas, and/or consume fewer apples and more bananas, compared to the old equilibrium."

Why would the introduction of a banana machine move the economy off the old equilibrium? If an economy is producing just enough apples and just enough bananas, the introduction of such a machine should do nothing.

The loss of utility seems to be a choice rather than an imposition of government. Maybe not a wise choice, but a choice nonetheless. Is a chosen loss of utility still a loss or are we measuring utility improperly?

[Michael Sigman]

Nick,

Say I am a young, healthy, ambitious libertarian. My utility is MY utility! Even in this case,there must be some benefit to getting a (credible) promise from the government to give me some number of beers when I am old, no matter what bad luck may afflict me in the meantime. Granted the cost of mandatory insurance is more than many would pay voluntarily. A few heathy small business owners do seem to pay an extraordinarily high price for mandatory coverage. We might look at Obamacare as the demographically healthy young subsidizing premiums for the demographically less healthy (but pre- Medicare) old. Effectively having the rate "r" imposed by the government for insurance when old versus insurance when young.. But it's also true that young people, even the confident ones, may have some unexpected health catastrophe that they cannot afford, that will be billed at a higher rate to them than to the more powerful foregone insurance company, and that society will have to fund anyway.

There seems to me to be some unpriced public good lurking in the background of these very useful simple models. The government does not turn young healthy apples into overripe bananas just for the perverse joy of crushing the wants and desires of the capable. True it may impose big costs to every cohort if the public good has no value.

Usual caveat: just trying to learn what I am missing. Not pretending I know as much as you.

[Odie]

Would it make a difference when the people only produce apples but would like to have bananas (which they did not until the government started the machine)? Would that not move the equilibrium to the point where they hold the amount of apples and bananas they desire?

[Nick Rowe]

Michael: you may indeed be right. The government may be able to offer intergenerational insurance on favourable terms, and make a profit and increase expected utility by doing so. Roger Farmer (if I understand him correctly) builds models where the government does just that. My simple model here has no uncertainty. To build it in, I would need 4 goods: apples when a good shock hits; apples when a bad shock hits; bananas when a good shock hits, bananas when a bad shock hits. Too hard for me.

Frank: "Why would the introduction of a banana machine move the economy off the old equilibrium?"

Jeez! If you reduce the supply of apples by destroying B apples, and increase the supply of bananas by creating B bananas, of course that will change the equilibrium!

No more comments on this post.

[Nick Rowe]

Odie: "Would it make a difference when the people only produce apples but would like to have bananas (which they did not until the government started the machine)?"

The price of one apple is 1+r bananas, where r > 0. So bananas are cheaper than apples, by assumption.

No more comments on this post.

(I am tempted to join the Razib Khan school of comment moderation.)

[rjs]

OK, what would this be in terms of financial asset operations only?

When the government sells a bond to households, and distributes the proceeds back to households (say as a basic income grant), it does two things:

1. Increases the number of bonds.
2. Changes the ratio of bonds to deposits (except in some very special cases)

In this way, you can argue that the public is pushed off their optimal frontier of asset holdings. But the public is able to reflux that back onto the banking system, for example by selling that bond to the banks in exchange for a deposit, so the public is not really pushed off of their optimal frontier. All that happens is that the total quantity of bonds + deposits increases, and it will decrease back when the public is taxed to redeem the bond. In a flexible price economy, this has no effect. With sticky prices, the income grant will be expansionary and the taxation will be contractionary, so the government can increase total welfare with fiscal stabilization or decrease it with pro cyclical fiscal policies.

It is as if there is a separate machine, opposite the government, that is converting the bananas back to the apples just as the government converts the apples to the bananas. That machine is the banking system when the government intervention is bond financed income grants.

[Nick Rowe]

rjs: "OK, what would this be in terms of financial asset operations only?"

No. Let's keep it very simple. Flexible prices, no money, and certainly no accursed banks. Just apples and bananas. Very simple micro trade theory. What I need is someone who understands duality theory, and indirect utility functions. All the stuff I have forgotten from grad skool micro.

[Niveditas98 .]

Nick, thanks for the pointer to the other post. There are definitely some interesting ideas here to think about, and I think I am learning something.

Regarding your question, I can't help but feel that any model that implies that the two societies in your question (i.e. one has debt of B and taxes itself rB to pay itself bond interest, the other taxes itself rB and throws away the tax) are even approximately equally well off, must have some flaw, because resources are only redistributed in the first instance, while they are destroyed in the second.

I have some issues understanding the Euler equation referred to in the other post -- as I understand it, the equation relating marginal utility of consumption in the two periods to the rate of interest is derived assuming that non-investment income and the interest rate are fixed. Doesn't the equation have to change form if the interest rate is not exogenous but determined in a market? I am not familiar with these concepts, so that might be a dumb question. But I feel like something is missing here -- in the real world, the young don't just buy debt because they have to in order to maintain some equilibrium in the government debt market. They want to invest money so they can consume more than they produce in retirement. Savings and investment should arise even in the absence of the government -- if they didn't I'm not sure how the government would be able to issue debt in the first place.

[Niveditas98 .]

From your previous post, why don't we consider model #3 instead of model #4? It seems more realistic, and better suited to the existence of debt -- in model #4, everyone is producing what he himself would like to consume, so introducing debt moves you away from optimality, because no-one actually wants to lend. Whereas in model #3, everyone is better off with the existence of debt. In the long run, you get taxed 50r/(1+r) when you're young, you buy 50/(1+r) bonds when you're young, and you consume 50. When you're old, the 50/(1+r) bonds mature at 50 -- financed by taxing the next generation 50r/(1+r) and selling them 50/(1+r) bonds -- and you consume 50. Debt stock is constant at B = 50/(1+r), each cohort pays rB in taxes, and they're better off than if there were no debt but the government just taxed the young rB?

[Nick Rowe]

Niveditas: "I am not familiar with these concepts, so that might be a dumb question."

Ignorant, but not stupid.

In a competitive market, each individual buyer and seller of apples takes the price of apples as exogenous when deciding how many apples to buy or sell. But in equilibrium the price of apples adjusts so that total quantity demanded equals total quantity supplied. The rate of interest is a price. ("But who actually adjusts the price in a competitive market?" is a good question, but not on-topic for this post.)

"...must have some flaw, because resources are only redistributed in the first instance, while they are destroyed in the second."

Take an extreme example of linear preferences: suppose

Utility = 1.05xConsumption when young + consumption when old

In this case, you would need to pay 5% interest to make people indifferent to postponing consumption. If you paid each individual 5% interest to lend you 100 beers when young, and then taxed all the other individuals 5 beers when old so you could pay that interest, their utility would drop by 5 utils, which is the same amount it would drop if you just taxed them 5 beers when old and threw those beers away.

"From your previous post, why don't we consider model #3 instead of model #4?"

Model #3 has a negative interest rate with no debt. Model #4 has a 0% interest rate with no debt. They aren't really different models, just 2 points on a continuum.

"...so introducing debt moves you away from optimality, because no-one actually wants to lend."

They will want to lend if you offer them a positive interest rate. The higher the interest rate, the more they will want to lend.

"the 50/(1+r) bonds mature at 50"

Keep it simple. You buy a 50 bond for 50 beers, and it matures and pays 50(1+r) beers.

Level with me: how much economics have you studied? How come you recognise an Euler equation, but also ask very basic questions?

[JKH]

Super post. Very interesting to think about this – especially the analogy - even if one is not completely familiar with the duality/utility.

I certainly can’t answer the questions, except this seems intuitively yes (obvious?) at least as a stand-alone:

“should I be integrating the area under the rB curve?"

Right now, I’m struggling and stuck on the following:

“But is rB a good measure of the loss in utility from operating the banana machine?”

+

“if r < g, where the banana machine converts one apple into 1+g bananas and so also makes profits, then it would be very different. The loss in utility (r-g).deltaB would be a negative loss, and the banana machine would provide lump-sum subsidies”

I’m confused. Why the change from r to (r – g)? This seems discontinuous at r = g in the comparison somehow.

[Nick Rowe]

JKH: Thanks!

Integrating the area under the rB curve feels right to me too. But I'm trying to remember whether the area under the demand curve is an exact measure of consumer surplus, and the difference between compensating variation and equivalent variation. Stuff that micro theorists are supposed to know.

"Why the change from r to (r – g)? This seems discontinuous at r = g in the comparison somehow."

In a world where growth is zero, the natural question to ask is: how big is the loss of utility per period from debt, if the debt is constant over time? If r < 0, the tax is negative, and utility increases.

In a world where growth is positive (or negative), the natural question to ask is: how big is the loss of utility per period from debt, if the debt/GDP ratio is constant over time? If r < g, the tax is negative (the government issues more debt each period than is required to pay the interest), and utility increases.

[rjs]

1. I also have a hard time wrapping my head around an endowment OLG model in which r != g = population growth rate. Don't you need production to get an r different from g? In all of your endowment economy examples there is no possibility of crowding out investment and so debt is never going to be a burden unless the government screws up the sequence of transfers. I'm not saying it's possible to come up with a model in which the tax burdens are incorrectly levied, but that point has already been made.

2. With production, you can argue that some investment is crowded out by selling the bond to the young, because the young buy the bond in lieu of buying real capital, which they "should" do in order to save. But then Barro's argument enters that the future tax obligation will cause them to not save less.

You only need to care about future generations when the tax burden falls on the young. When it falls on the old, you care about yourself (when old) when you buy the bond and so there is no crowding out. That would be a 1 period bond. You agree that in the one period bond case, where the tax is levied on the old, that no crowding out of investment occurs?

If so, how is rolling over a sequence of 1 period bonds different from selling your consol?

[rjs]

Or, to say it another way, the government selling the bond also comes with a tax obligation, which shifts the preference curves so that there is no change. This assumes a 1 period bond which is sold to the young (who are the savers) and the taxes are paid next period by the old (the current young).

If you buy that, for the case of the 1 period bond. Then we can iterate to get the consol case.

[Nick Rowe]

rjs:

1. a) here's one example. People have an endowment of 40 beers when young, and 60 beers when old. If r=0%, everyone would like to borrow 10 beers so they could smooth their consumption. That creates an excess demand for personal loans from each other, so r rises until the excess demand for loans drops to zero.

b) A second example. People have an endowment of 50-50, but are just impatient, or can't handle as much booze when they get old.

This is very basic theory of interest rates. If people were indifferent about when they consume beers, interest rate would always be 0% (unless you hit a non-negativity constraint), regardless of investment opportunities.

2. "You only need to care about future generations when the tax burden falls on the young."

If you assume Barro-Ricardo equivalence: if the government gives a transfer to old me, and will tax my kids when old, and their kids when old.... to pay for it, I will save the whole of the transfer so my kids and grandkids.... are not made poorer

[Nick Rowe]

rjs: It makes no difference (in my model with no uncertainty) whether the government debt is one period bonds or consuls.

[Niveditas98 .]

Nick, re economics I know -- formally nothing beyond a couple of undergrad courses a long time ago. I just read up on topics that interest me. I don't "recognize" the Euler equation, I read other posts that called it that, and then looked it up on the web :) I set the bonds to mature at 50 because that's how much the old people need in order to consume, and it seemed more natural to tax the young, rather than tax the old, which effectively just reduces the interest rate they get.

I think I see what you mean by a negative interest rate -- the young would clamor to lend funds even at r < 0. But on the other hand, isn't the natural rate of interest supposed to equal the rate of time preference, for equilibrium?

I do see a paradox in my example though, since effectively all I've done is take 50 from everyone when they're young and give it back when they're old. At a positive rate of interest, that should be equivalent to destroying value. Yet everyone appears to better off.

[Nick Rowe]

Niveditas: OK, so you have big gaps in your knowledge, but are learning fast.

"At a positive rate of interest, that should be equivalent to destroying value. Yet everyone appears to better off."

I think you made an arithetic mistake somewhere. Or maybe assumed the government simply *gave* the young people the bonds?

[Majromax]

@Nick:

I think your original post is slightly unclear, which gave rise to Frank and Odie's misunderstanding. You introduce the government's fruit machine, but you don't explicitly state that the government independently chooses how much to use it.

At first reading, before realizing the allegory with intergenerational transfer, it sounds as if the government is making this machine available for public use, or alternately the government is only using it when there is profit in it.

Regarding the taxes to pay for old-beer-bonds, I think it does make a difference whether the young or old are taxed. In the fruit example, if the government wants to distribute X bananas then collecting X bananas in taxes will have different utility effects than collecting X apples in taxes and operating the machine.

I am also unsure that one can simply apply Barro-Ricardo equivalence. Wouldn't that imply that the utility function is actually a combination of young-beer, old-beer, and kid-utility? That suggests the equilibrium interest rate would be different than under a non-equivalence model.

[Matthew]

To answer the general question "Is consumer surplus a measure of welfare?" the answer is no, except in special cases where wealth effects don't matter. However, your intuitions are mostly correct.

In the case of linear technology, the government is taking (1+r)B worth of bananas and converting them into B bananas. By forcing people to use the government's less efficient production technology, rB worth of bananas are destroyed. Consumers would be willing to pay up to rB bananas to end this policy. So, this has welfare significance when production is linear (in fact, rB is the equivalent variation).

More generally welfare depends on the production technology, which affects how r changes. Under normal assumptions, this would mean rB understates welfare loss because the government's technology isn't just inefficient, but actually makes the private sector technology less efficient as well by causing movement up the marginal cost curve for apples.

But you lose me when you get to the dynamic example. The welfare loss of the apple/banana example arises because the government's banana machine is less efficient than private sector technologies. If instead we had assumed that the government machine yields 1+r bananas per apple, then except for a couple possible corner cases, the market would just trade back to the original equilibrium as if the government didn't exist and welfare effects are zero--ie, riccardian equivalence. In the dynamic example, it's not clear why the government's bond-selling technology would be so much less efficient than the private sector's. You are literally proving here that if the government's transaction cost of issuing bonds is substantially higher than the private sector's then riccardian equivalence does not hold.

[Nick Rowe]

Majro: Frank and Odie have a history of sidetracking comment threads.

Collecting X bananas in tax has the same effect as collecting X/(1+r) apples in tax, if you adjust the speed of the banana machine to cause the same net destruction of apples and creation of bananas.

"I am also unsure that one can simply apply Barro-Ricardo equivalence. Wouldn't that imply that the utility function is actually a combination of young-beer, old-beer, and kid-utility?"

Yes. And yes it can affect things. But I'm not assuming Ricardian Equivalence here.

[Nick Rowe]

Matthew: Ah! Thanks for coming!

"Under normal assumptions, this would mean rB understates welfare loss because the government's technology isn't just inefficient, but actually makes the private sector technology less efficient as well by causing movement up the marginal cost curve for apples."

I'm still thinking about that one. If we think of the banana machine as a fact of nature, operated by insects eating a fixed number of apples and excreting bananas, wouldn't private apple and banana producers be responding in an efficient manner, reducing the losses in utility?

"In the dynamic example, it's not clear why the government's bond-selling technology would be so much less efficient than the private sector's. You are literally proving here that if the government's transaction cost of issuing bonds is substantially higher than the private sector's then riccardian equivalence does not hold."

I'm ignoring Ricardian Equivalence, assuming each generation is selfish in an OLG model. The government has a technology that is not available to the private sector, because it can make trades on behalf of all unborn future generations. But whether it is acting in their interests if it uses that technology is what we want to find out.

[Nick Rowe]

Back later.

[Matthew]

Under "normal" assumptions, production experiences diminishing returns, but individuals prefer some of both apples and bananas to extremes of one versus the other. So they respond to the machine/insects by increasing apple production to partially offset the machine/insect's production of bananas. But because of diminishing returns, they can't convert the extra bananas into as many apples as they could if technology were linear (the MRT is locally 1+r and increases for each extra apple). Hence, fewer apples & less balanced bundle=decreased welfare relative to linear technology.

[Nick Edmonds]

Two points.

First, this assumes we have a good equilibrium to start with, in particular no output gap.

Secondly, I think it is wrong to equate the change in debt with the deficit, due to the effect of inflation (unexpected, if we want to treat expected r as given). In general, we cannot assume unexpected inflation is uncorrelated with any deficit, but especially so if we are assuming no output gap.

[Market Fiscalist]

If all you knew was that the government had run the machine for a while but you didn't know what the relative price of apples and bananas was , then you would have no idea if the result was good or bad compared to before.

If you were a free market type you may conclude that if the machine was capable of making people better off then it would be doing so anyway in private hands therefore the government running the machine could never be good.

If you believe that the machine can only be operated by the govt then you may conclude that there is a chance that the machine can increase utility , especially if you have faith in the government's ability to only run the machine when it is useful to do so.

It would probably be incorrect though to conclude that running the machine never mattered because you always end up with the same qty of fruit, and that the only issue is who eats it.

[Nick Rowe]

Matthew: Aha! I think I've got it.

With diminishing returns to apple production and banana production, the PPF is curved. The slope is 1+r, and the slope increases as the government increase B and we move along the PPF. There is r *before* B increases, and r *after* B increases.

rB will underestimate the loss if we measure r *before* B increases.

rB will overestimate the loss if we measure r *after* B increases.

(With a linear PPF, r will stay the same, so it makes no difference, and we get an accurate measure.)

[Nick Rowe]

Nick E:

1. True. Always start with the simplest case, if people are muddled to begin with.

2. That depends on what monetary policy targets.

MF: if you didn't know what the world would be like with no debt, you wouldn't know if the existing level of debt was better or worse than no debt. But you would (in principle) be able to tell if the debt was too high or too low at the margin.

"If you believe that the machine can only be operated by the govt then you may conclude that there is a chance that the machine can increase utility , especially if you have faith in the government's ability to only run the machine when it is useful to do so."

Though if the economists advising the government didn't understand the burden (+ or -) of the debt, you wouldn't have much faith!

[Niveditas98 .]

Nick, thanks for the kind encouragement :)

I don't see any obvious arithmetic error. To put concrete numbers in, say the young pay 5 beers in taxes, and buy bonds with 45 beers, and consume the remaining 50 beers. When they're old, the bonds mature at 50 beers, which they consume (the old aren't taxed). The flow of beer balances out in each period.

These people are clearly better off than if the government simply taxed the young 5 beers and threw them away. The interest is apparently positive, with a 45 beer investment yielding 50 beers. But if you forget that 5 of the beers that you gave up when young were labeled "taxes", then you give up 50 beers when young and get 50 beers when old. That's a zero rate of return.

[Nick Rowe]

Niveditas: that's not adding up right. They produce 50 years when young, and 50 beers when old. So young people can't pay 5 taxes, lend 45, and drink 50. Because they only produce 50. You are making the 50 beers they produce when old travel back in time.

[Nick Rowe]

Nive: but if they *did* produce 100 beers when young, and 0 beers when old, then yes. A national debt *would* make them better off. Because they can't save beers for their old age with no bonds to buy. That's my case #3.

[Niveditas98 .]

Nick, right I was talking about case #3, which you said works out because interest rate is negative.

But how do I interpret my example numbers to see that the interest rate is not 5 on 45, which looks positive, but some negative number?

[Niveditas98 .]

Or do you mean that the interest rate was negative before the issuance of debt, not after?

[Nick Rowe]

If you assume U=Log(Cy)+Log(Co), then in equilibrium: 1+r= Co/Cy

Where Cy = consumption when young, and Co = consumption when old.

If they produce 100 when young and 0 when old, then r would be -100% with no debt, and 0% with a debt of 50.

[Niveditas98 .]

Ok, thanks.

I tried to work out a detailed example in a slightly different model: Suppose there are just two infinitely long-lived people, Peter and Paul, and no government. Normally, they each brew 100 beers per period and consume it themselves. What will happen if in one period, something goes wrong with Paul's brewery and he produces no beer in just that period?

Assumptions: their utility function is log c_0 + 1/(1+n) log c_1 + 1/(1+n)^2 log c_2 + ..., where n = 1/9 (to make numbers look nice), and Paul's income is zero in period 0.

Assuming my math was worked out correctly, I conclude that Paul will borrow 45 beers from Peter in the first period, and will in each subsequent period pay 10 beers in interest, rolling the debt over indefinitely. In general, I get that the interest rate will be 2n, and the sum borrowed will be 50/(1+n).

I have to dig into this a bit more though -- my method was to assume that some amount B is borrowed and interest rB is paid on it forever, then maximize (individually) Peter and Paul's lifetime utility assuming a fixed rate of interest r, then note that the two optimum B's are equal only for a certain value of r, which ties down everything.

However, if instead I try to do it assuming a sum B is borrowed and then in each period B1 is paid back (with no assumption on the relationship between B1 and B), I can't see how to get a unique solution.

[Nick Rowe]

Nive:

"Assuming my math was worked out correctly, I conclude that Paul will borrow 45 beers from Peter in the first period, and will in each subsequent period pay 10 beers in interest, rolling the debt over indefinitely. In general, I get that the interest rate will be 2n, and the sum borrowed will be 50/(1+n)."

Sounds plausible. You've rediscovered Milton Friedman's Permanent Income hypothesis.

"I have to dig into this a bit more though -- my method was to assume that some amount B is borrowed and interest rB is paid on it forever, then maximize (individually) Peter and Paul's lifetime utility assuming a fixed rate of interest r, then note that the two optimum B's are equal only for a certain value of r, which ties down everything."

Yep. You solved for the competitive equilibrium r. (Strictly, that only makes sense if there are 1,000 Peters and 1,000 Pauls, but that's OK.)

"However, if instead I try to do it assuming a sum B is borrowed and then in each period B1 is paid back (with no assumption on the relationship between B1 and B), I can't see how to get a unique solution."

Yep. Because now you have one Peter and one Paul and you are trying to solve for the bilateral monopoly-monopsony equilibrium, which is not unique. Just like Greece vs Eurozone right now. (Google "Nash Bargaining solution" for one vaguely plausible attempt to say what happens.)

Are you a physics or maths guy?

[rjs]

Nick,

Ahh, OK, thanks.

This is totally offtopic. I think you can get a lower bound problem, even in real terms. E.g. suppose that we are in a corn economy, in which the old lend corn to the young. The young plant the corn, tend to the fields, and then both the old and young eat. The young can save some corn for the future and lend it when they are old.

Now, there is always the option for the old to just keep the corn and not lend it out, so that the young starve. They would only lend it out if there was a positive return. There is a zero bound!

[Niveditas98 .]

Nick, thanks! Now I won't try to break my head working out an impossible problem. I'm a maths guy.

[Michael Sigman]

Is the deluded sophisticated economist making the same mistake as if he said " there are no benefits to free trade, because even if I produce bananas at a comparative disadvantage, all the apples and bananas I produce will be eaten...Who cares if the opportunity cost of a banana produced is one home grown apple or 1/(1+r) mid ocean apples"?

[Nick Rowe]

You assumed that Paul issued a perpetuity (consol), so your r=2n solution is the infinite period interest rate on a perpetuity. If you instead assume that Paul issues a sequence of one-period bonds, r will be very high in the first period, and will be r=n in all following periods.

[rjs]

Nick,

This is totally unrelated (:P) -- but it seems to me that you can get a real zero bound problem.

E.g. suppose that it's a corn economy, in which the old lend their corn to the young (as capital). After the young work the field, both the old and young eat. Here, the old could always choose to not lend their corn and eat it instead -- the young would starve.

[Nick Rowe]

OK Niveditas, since you're a math guy, here's a job for you:

Take my model #4 in my previous post.

U=log(Cy)+log(Co)

Cy=50-B

Co=50+B

1+r=Co/Cy

1. r is a function of B. Solve for that function r=F(B)

2. Solve for the integral of F(B) between B=0 and B=10 (for example). Call the answer X [edited to fix typo/mento]

3. Calculate U if B=10 (Lifetime Utility if B=10)

4. Calculate U if Cy=50-X/2 and Co=50-X/2 (Lifetime Utility if the government throws away X beers)

5. See if your answers to 3 and 4 are the same.

[Nick Rowe]

rjs: And the old have no incentive to lend, because they will be dead by the time the loan is repaid, right?

That's like having an endowment of 0 when young and 100 when old!

But that's not a ZLB problem. The interest rate would be very high.

[rjs]

Heh, well, I'm assuming the flow of time is:

lending/borrowing --> production/leisure --> consumption --> go to next period

So the old lend at the beginning of the production period and eat the food at the end.

As far as I can tell, the young need to work for a period and can only consume their output at the end also. Why must the old eat their output at the beginning?

[rjs]

1. r = (50+B)/(50-B) - 1 = F(B)
2. Anti-derivative is -2(b+50 ln(b-50) ). From 0 to 10: approx. 2.3144
3. U(B=10) = approx 7.78322
4. U = approx 7.824
5. They are close. 4 is a little bigger.

[reason]

Nick,
it seems to me you are comparing different people's utility and calling them the same. Maybe the government has a different utility curve than you do?

[JKH]

Nick,

“With diminishing returns to apple production and banana production, the PPF is curved. The slope is 1+r, and the slope increases as the government increase B and we move along the PPF. There is r *before* B increases, and r *after* B increases.”

Just thinking out loud:

The calculus here seems to correspond to that for the price of a “0 coupon” bond at a continuously compounding rate of interest – so the price curve from issuance to maturity is nicely convex in a similar way. The slope increases continuously (Same hold more roughly for a coupon bond between payments dates I believe).

Maybe visualizing the interest on the bond as being the accrual of discount on a zero coupon bond is one way of simplifying the analysis in general. At any point in time, the accrual looks almost like a second bond on top of the first principal amount – and the tax issue at that point in time can be split between doing nothing and just letting the interest keep accruing as a growing bond, versus the decision to tax the interest alone (keeping the outstanding principal amount of debt constant), versus the decision to tax the principal and any outstanding interest due.

[Nick Rowe]

rjs: "4. U = approx 7.824"

One of us made an arithmetic mistake (probably me).

Because you say: log(50-X/2)+log(50-X/2)= 7.824 (where X=2.3144)

And I said in my previous post: log(50)+log(50)=7.824

We are both using natural logs, right?

I make it log(50-X/2)+log(50-X/2)= 7.729

And, your number for X looks a little bit too big for me. Because X is the integral of r(B), where r and B both start at 0, and r rises to 0.5 and B rises to 10, so rB rises from 0 to 5, and r is a very concave (convex?) function of B. But that might be just my eyeball math getting it wrong.

Or maybe I'm mentally confused.

[Nick Rowe]

reason: "it seems to me you are comparing different people's utility and calling them the same."

Yes and no.

Here, I am adding the same individual's utility when young + (maybe subjectively discounted) utility when old (his lifetime utility).

We might (or might not) want to add the utility of a young person + the utility of an old person, both alive at the same time.

[Nick Rowe]

JKH "Just thinking out loud"

That analogy doesn't work.

Suppose you have 100 acres of identical land, and 1 acre can grow either 1 apple or 1 banana. Put quantity of apples produced on the horizontal axis and quantity of bananas produced on the vertical axis, and you draw a downward-sloping line that hits the axes at 100 apples and 100 bananas. That's the PPF.

Now change the example so the land is all spread out north-south, and the northern land is better at apples and worse at bananas, and vice versa for the southern land, and you get a curved PPF that's bowed out. The slope of the PPF is the Marginal Rate of Transformation (opportunity cost) of one good into the other good. The slope changes as we move along the PPF. The more apples we produce, the bigger the opportunity cost of extra bananas lost per extra apple produced.

[Niveditas98 .]

Nick, in my example, I think Paul has to issue a consol. After period 0, both of them want to keep their consumption constant, and their production is constant, so the only possible payments between them must also be constant.

Have to run to work, so don't have time to read your question in full just now, but here's a calculation of debt impact in scenario #4.

Assume lifetime utilities of log c_y + 1/(n+1) log c_o. Suppose there is debt of B/(1+r), interest of rB/(1+r) (and hence taxes of the same amount), and assume B changes by a small amount dB.
Assume the young are the ones who pay taxes. Then c_y = Y-B, c_o = Y+B for flows to balance.

Reduction in utility from the additional debt is
dB/(Y-B) - 1/(n+1) dB/(Y+B) = ( Y+B - 1/(n+1) (Y-B) )/(Y^2-B^2) dB = ( Y n/(n+1) + B (n+2)/(n+1) )/(Y^2-B^2) dB

If n = 0, this is 2B/(Y^2-B^2) dB

This doesn't seem to depend on what interest rates or taxes actually are, only on what the time rate of preference is and the amount of existing debt (as long as you measure debt by its maturity value rather than face). I think this makes sense, in that it shouldn't matter whether the government chooses to call your payments "debt" or "taxes". This would all work through exactly the same if the entire B were just called taxes: the government taxes you B when you're young and gives it to the older generation, and there's no debt.

[JKH]

Nick,

I was thinking of the slope of rB rather than the slope of the MRT curve

Maybe that's still wrong anyway

[Roger Sparks]

This seems to be an excellent analogy: "Now change the example so the land is all spread out north-south, and the northern land is better at apples and worse at bananas, and vice versa for the southern land, and you get a curved PPF that's bowed out. The slope of the PPF is the Marginal Rate of Transformation (opportunity cost) of one good into the other good. The slope changes as we move along the PPF. The more apples we produce, the bigger the opportunity cost of extra bananas lost per extra apple produced."

I think this is an example illustrating the Ricardo marginal-land-rent-theory where prices of commodity determine how much marginal land is used for production. Your model has the marginal land break-even point moving north or south depending upon the relative price of each commodity.

[Nick Rowe]

Roger: thanks! That's the example I use to teach my students why PPFs are concave. (The example where there are two factors of production, with variable proportions, with apples being land-intensive and bananas being labour-intensive, is both more complicated and less general.)

"Your model has the marginal land break-even point moving north or south depending upon the relative price of each commodity."

Exactly. And we can use it to illustrate Ricardian Comparative Advantage, where Comparative Advantage is a matter of degree. The further north the land, the bigger the comparative advantage in apples compared to bananas, relative to land at a given latitude.

[Nick Rowe]

Nive: "Nick, in my example, I think Paul has to issue a consol. After period 0, both of them want to keep their consumption constant, and their production is constant, so the only possible payments between them must also be constant."

Issuing a consol would be one way to do it. Or he could issue a one-period bond, at a high interest rate the first year, then in the second and subsequent years keep rolling over a bigger one-period bond, with a lower interest rate, paying the interest each year. You get the same consumption streams either way.

The only advantage to thinking about it the second way is you get to see the one-period interest rate being higher in the first period than in later periods.

[Nick Rowe]

JKH: in equilibrium, 1+r will equal the slope of the PPF (which is MRT) and also equal the slope of the indifference curve (which is MRS).

The slope of r with respect to B (how much r changes when B changes by one unit) is equal to the *change* in the MRT and MRS.

[Roger Sparks]

In the apple-banana-north-south example, an increased price of one commodity would result in an increased price of the second (the likely result of decreased production of the second).

Now what happens if government decides to convert apples into bananas with a newly invented machine? The conversion is only possible in one direction, apples to bananas.

The demand (and price) for apples would increase but supply of bananas would increase. The banana price would likely decrease. The relative price change would move location of the break-even price toward the south, resulting in less bananas grown and more apples grown.

Government would sell the apple-to-banana product at the same price as grown bananas. Government has effectively moved the break-even point south, thereby increasing apple production and decreasing banana production.

The apple-to-banana machine would cost the government something to operate. Government could tax apple growers (makes sense since their prices have increased). If government taxed banana growers, they would be hit with a tax and suffer decreased prices from increased supply, a double whammy.

But if apple growers pay a tax on their increased prices, that increases the cost of growing apples, which moves the break-even line back towards the north.

Finally, what would be the relative price between apples and bananas if the government machine ran? What would the utility factor be, and how would it change? Both questions are beyond the scope of this comment.

[Nick Rowe]

Roger: if we ignore money (which we should to keep it simple) there is only one price in this economy. It's the barter price of apples in terms of bananas (or its reciprocal). That price would rise if the government uses the apple machine to convert apples into bananas. The margin of cultivation would move south, and people would consume more bananas and fewer apples. Utility would fall if the banana machine made a loss (and would rise if it made a profit). That last point isn't obvious, but follows from the First Theorem of Welfare Economics.

[Nick Rowe]

I'm a thickie. This is really simple. The *marginal* burden of the debt is the rate of interest.

Let lifetime utility be V = U(Cy) + U(Co), where Cy=Ey-B and Co=Eo+B, where Ey and Eo are endowments when young and old.

Then dV/dB = -U'(Cy) + U'(Co) where U' is the derivative of U

And 1+r = U'(Cy)/U'(Co)

So r = [U'(Cy) - U'(Co)]/U'(Co)

If we put a lump sum tax of T on old people and throw away T beers, then dV/dT = -U'(Co) (I'm using the envelope theorem)

So (dV/dB)/(dV/dT) = (dT/dB) holding V constant = -r

So if we take the integral of r with respect to B we get the burden of the debt.

I think that's right.

But don't trust me.

[JKH]

"So if we take the integral of r with respect to B we get the burden of the debt."

Seems very intuitive, quite aside from the utility math. It is the deficit, holding revenues and taxes momentarily unchanged (i.e. holding the primary deficit momentarily at zero?).

Is the rate of interest the marginal compensation for disutility? That also seems intuitive to me.

And there must be an easy intuitive translation of this in reverse to the banana machine, which is what I've been struggling with.

Is the growth in the deficit due to the interest rate analogous to an MRT?

[JKH]

Or maybe a better question is:

Is (1 + r) as a debt accumulation factor due to interest analogous to an MRT?

[Nick Rowe]

JKH: "Is the rate of interest the marginal compensation for disutility?"

It's the marginal compensation for the disutility of postponing consumption of one beer by one period, yes.

If the price of bananas is 5% less than the price of apples, that 5% is the marginal compensation for switching consumption from apples to bananas.

In an intertemporal context, MRT is one plus the physical rate of return on real investment projects.

[Niveditas98 .]

Nick, yeah, I get the same result if I work it out in scenario #4 (working out details, since I don't yet trust myself to see the intuition).

The incremental (throw-away) tax on the young that is equivalent in disutility terms to additional dB debt is 2B/(Y+B) dB. The interest rate is 2B/(Y-B), so the burden is r/(1+r) dB, which is basically the same as what you get (our dB's are different by a factor of (1+r)).

The tax T that produces the same total utility as debt B is just given by solving
ln(Y-T) + ln(Y) = ln(Y-B) + ln(Y+B)
which gives
T = B^2/Y

To express this as \int r/(1+r) dB, I'd have to work out what the interest rate is in the presence of both taxes and debt, I think, and then integrate along the curve of constant utility from (0,B) to (T,0). This doesn't seem to be an easier way of figuring out the equivalent tax burden?

[Nick Rowe]

Nive: "The interest rate is 2B/(Y-B), so the burden is r/(1+r) dB, which is basically the same as what you get (our dB's are different by a factor of (1+r))."

Yep, a $100 bond that pays a $5 coupon when old is equivalent to a $105 zero-coupon bill that you buy for $100 when young. One adds on interest at the end, and the other subtracts interest off at the beginning. Same thing.

I think we've basically got this question sorted. Well done!