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.
Worthwhile Canadian Initiative 
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.
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.
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.
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.
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?
r = (50+B)/(50-B) – 1 = F(B)
Anti-derivative is -2(b+50 ln(b-50) ). From 0 to 10: approx. 2.3144
U(B=10) = approx 7.78322
U = approx 7.824
They are close. 4 is a little bigger.
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?
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.
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.
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.
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.
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.
Nick,
I was thinking of the slope of rB rather than the slope of the MRT curve
Maybe that’s still wrong anyway
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.
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.
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.
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.
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.
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.
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.
“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?
Or maybe a better question is:
Is (1 + r) as a debt accumulation factor due to interest analogous to an MRT?
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.
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?
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!