I just spent a little while trying and failing to construct a homework problem for my honors class. Although it didn’t turn out the way I wanted it to, I thought it might serve as a good illustration of how economists (often) think about income redistribution.
The idea is that different people are born with different talents, and that if it were possible for us all to meet in a shadowy pre-birth world (what the philosopher John Rawls called “behind the veil of ignorance”), we’d want to insure against landing in the shallow end of the gene pool — so we’d probably agree that the lucky ones — those with a lot of talent — would help to take care of the rest.
The further idea is that because we’d presumably all have voluntarily signed on to such an agreement, there’s at least a plausible case for enforcing it. (I’ve argued elsewhere that this plausible case does pretty much nothing to justify the actual sorts of redistribution that are practiced by, say, the United States government — but for present purposes, that’s neither here nor there.)
The big problem is to figure out exactly what terms we’d have all agreed on. Jim Mirrlees won a Nobel Prize for a major attack this problem. But I don’t want to ask my college sophomores to digest a Nobel-worthy body of work, so my goal is to construct a sort of baby version of the Mirrlees approach — which I hope might also interest at least one or two blog readers.
Now if governments were omniscient and omnipotent, the problem would be pretty easy — you’d take a whole lot from the rich and give a whole lot to the poor, and you’d forbid talented people to respond by working less.
In practice, though, governments face a lot of constraints. The one I want to focus on is that our talents and/or incomes might be at least partially invisible to the government. You can’t “take from the rich” if you don’t know who the rich are.
One solution is, instead of taking directly from the rich, to tax things that only rich people buy.
So my idea was to imagine that everyone has a natural talent, and an associated natural income, ranging from 0 to 1. You can spend all your income on corn (in whatever quantity you can afford), or you can spend part of your income to buy a car, for a price of 1/2. Obviously, only people with incomes over 1/2 can even consider buying a car, and even some of them might prefer not to.
Now the government levies a tax on cars, equal to some amount T. This raises the price of a car from 1/2 to 1/2+T. All of the tax revenue gets distributed equally to the population. (You can’t just give it to the poor, because you don’t know who they are, but the net effect is still to make the rich poorer and the poor richer.)
Now people look at their new incomes, they look at the new price of cars, some buy cars and some don’t, and as a result they achieve a certain total utility. We want to find the tax rate that maximizes that total utility. (There are good reasons — first made explicit by another Nobel Prize winner, John Harsanyi — to believe that this is what we’d have agreed to do in our state of pre-birth ignorance.)
To make this problem concrete (and this is the point where I expect almost everyone to stop reading), I assumed that your utility is equal to the square root of your corn consumption, multiplied by 2 if you’ve got a car. Given this, and given an income of Y and a car price of P, it turns out that you’ll want to buy a car if and only if Y > 4P/3.
In particular, given a God-given talent of x, which yields income of x, together with a check from the government for an amount we’ll call A, your total income is Y=x+A. And given an initial car price of 1/2, together with a tax of T, the car price is P = 1/2 + T. Putting this together with the condition Y > 4P/3, we see that you’ll buy
a car if and only if x > 2/3 – A + 4T/3. Call this critical value Q.
Now if we imagine that talents are uniformly distributed between 0 and 1, then the fraction of people with talent greater than Q is 1-Q. These are the people who buy cars, and each pays a tax of T, so the average person pays a tax of (1-Q)T. At the same time, everyone (and hence in particular the average person) receives a government transfer of A. In order for the books to balance, we must have (1-Q)T=A.
Now replace the Q in that equation with the definition of Q from two paragraphs back, and you’ve got an equation involving only A and T. Solve this for A, to get A=(T-4T^2)/(3-3T). Put this back into the definition for Q — the critical amount of talent above which people buy cars — and you’ll discover that Q=(T+2)/(3-3T).
A person with talent x between 0 and Q won’t buy a car, and will therefore spend x+A dollars on corn, earning a utility equal to the square root of x+A. A person with talent between Q and 1 buys a car for a price of 1/2+T, spends x+A-1/2-T dollars on corn, and earns a utility of twice the square root of x+A-1/2-T.
We can add these expressions up (where “add up” means integrate!) over all values of x to get an expression for total utility in terms of the tax rate T. Unfortunately, that expression is kind of ugly (for a homework problem I wanted it to be first simple!). We want the tax rate that maximizes this expression, which is an exercise in freshman calculus.
Unfortunately, after all this work, the optimal tax rate turns out to be zero. So considered this a failed attempt to construct an example in which you’d want to tax something that rich people buy in order to transfer income to the poor. I expect some variant should work, but I think I’m done for the day.
“…multiplied by 2 if you’ve got a car”
Isn’t that an odd way to define the utility of a car? Do you want the utility of a car to depend on how much corn you have?
Couldn’t you just have two goods – one whose utility falls off more sharply than the other; e.g., let your utility be the square root of your corn consumption plus the cube root of your caviar consumption? Wouldn’t that lead to redistribution increasing overall utility?
Continuing the steps you gloss over at the end:
The function for the utility of a person with talent x:
if x
Substituting Q = (T+2)/(3-3T) and A = (T-4T^2)/(3-3T):
if x = (T+2)/(3-3T): sqrt(x+(T-4T^2)/(3-3T)-1/2-T)
Total utility: integrate over x = [0,1]:
integrate(sqrt(x+(T-4T^2)/(3-3T)), x, 0, (T+2)/(3-3T)) + integrate(sqrt(x+(T-4T^2)/(3-3T)-1/2-T), x, (T+2)/(3-3T), 1)
Plugging T=0 into the above equation: ~0.553
Plugging T=0.01 into the above equation: ~0.558
It seems to me that there exist non-zero T which produce more total utility than zero T. Which one of us made a math error?
p 8 or 9: “You can’t just give it to the poor, because you don’t know who they are, but the net effect is still to make the rich richer and the poor poorer” Is this right? Surely the point is to make the poor richer and the rich poorer?.
Any result where the optimal tax rate is zero is not a failure! :)
Aha! Fixing this….
David Sloan: I get total utility = .74357 when t=0. It’s not impossible that I’ve made a mistake.
It looks to me like you forgot to multiply utility by 2 for the people who bought cars.
Apart from the “rich richer and the poor poorer” typo, it seems to me that there may be another error:
Q is talent cutoff for buying a car in the presence of a tax only. In the absence of redistribution, people with a talent below Q will not buy a car, those above will.
But if we add redistribution A, some car buying decisions will change. Some marginal persons will be sufficiently enriched by A that they will end up buying a car after all.
So the sentence that read “A person with talent x between 0 and Q won’t buy a car” and ” A person with talent between Q and 1 buys a car” should be changed to “A person with talent x between 0 and Q-A won’t buy a car” and “A person with talent between Q-A and 1 buys a car.”
I am too lazy to do the math myself, but does this change the outcome?
Eric:
Couldn’t you just have two goods – one whose utility falls off more sharply than the other; e.g., let your utility be the square root of your corn consumption plus the cube root of your caviar consumption? Wouldn’t that lead to redistribution increasing overall utility?
I had a reason for not wanting to do this, and can’t seem to remember what it was — though possibly it was that the calculations became unwieldy. I’ll try it again and see if I can remember why I abandoned it yesterday.
I am interested in why this comes out at zero (if that is confirmed after comment 1).
The utility of the car is a function of the amount of corn you can buy. Why do you not fix the utility of the car at a certain value (say 1/2)? If we were to have a smaller luxury where more than one was possible, we could say the utility was sq rt of the number consumed.
I can sort of see why. Ability 0.5 means you are indifferent between car and corn. Anyone just above this will buy a car and virtually no corn. By doubling the corn utility with a car you ensure that everyone gets a minimum amount of corn, which seems reasonable to avoid starvation.
Perhaps more realistic valuation of corn utility would a non- smooth function of quantity. The first bits give you a very large utility, but then it drops off. Obviously more complicated.
Is this assumption realistic, and if not, is that why it works out at a zero tax rate.
Prof. Landsburg, I think you replied to Eric’s comment under my comment. Is my comment incorrect?
Also, I promise comment 1 wasn’t there when I typed mine.
Sub Specoe AEternitatis: I believe I have this right. Q is the *talent* cutoff for buying a car in the presence of both a tax and redistribution. It’s still (as always) possible I’ve made a mistake, but at the moment this still looks right to me.
Harold:
Ability 0.5 means you are indifferent between car and corn.
The right question is not “when am I indifferent between a car and corn” (it’s not even clear what this means) but “when am I indifferent between buying a car and not buying a car?”. The answer, in the absence of any redistribution, is at talent = 2/3, not 1/2.
Even correcting for all this, I don’t see why your observations would fully explain the zero tax rate. There are plenty of similar situations in which the optimal tax rate is not zero.
> David Sloan: I get total utility = .74357 when t=0. It’s not impossible that I’ve made a mistake.
> It looks to me like you forgot to multiply utility by 2 for the people who bought cars.
I stand corrected on both counts.
“you’ve got an equation involving only [T and A]”
Dirty equations! Tee hee:-)
I had nothing useful to add to the post, but I can make even equations into dirty jokes.
You are right, my mistake was misreading Q as not accounting for the redistribution, but your definition of course already subtracts out A.
To do penance, I worked out the math myself. To not do too much penance, I just plugged it in to Mathematica:
In[1]:= unc [x] = Sqrt[x+A]
Out[1]= Sqrt[A+x]
In[2]:= uwc [x] = 2*Sqrt[x+A-1/2-T]
Out[2]= 2 Sqrt[-(1/2)+A-T+x]
In[3]:= Q = Solve[unc[x]==uwc[x],{x}][[1,1,2]]
Out[3]= 1/3 (2-3 A+4 T)
In[4]:= A = Solve[A==(1-Q)*T,{A}][[1,1,2]]
Out[4]= (-T+4 T^2)/(3 (-1+T))
In[5]:= utot = Integrate[unc[x],{x,0,Q}, Assumptions -> { T >= 0, T { T >= 0, T < 1/4 }]
Out[5]= (-4 Sqrt[2+4 T]+2 T (Sqrt[(4+3/(-1+T)) T]-2 Sqrt[2+4 T]+4 T (-Sqrt[(4+3/(-1+T)) T]+Sqrt[2+4 T])))/(9 Sqrt[3] (-1+T))-(1/(9 (-1+T)^2))Sqrt[2/3] (Sqrt[1+2 T]-3 Sqrt[(-1+T) (-3+T (7+2 T))]+T (7 Sqrt[(-1+T) (-3+T (7+2 T))]+T (-3 Sqrt[1+2 T]+2 T Sqrt[1+2 T]+2 Sqrt[(-1+T) (-3+T (7+2 T))])))
In[6]:= Plot[utot,{T,0,1/4}]
The plot shows the same result you got. Maximum utility at T=0 of about (4 Sqrt[2/3])/9 – 1/9 Sqrt[2/3] (1 – 3 Sqrt[3]) or 0.7435700478, monotonically declining to about 0.68 at T=1/4.
I notice that the ASCII representation of utot in In[5] got truncated. The actual formula in Mathematica is the full and correct one.
Here, a figure might help explain what’s going on:
http://imgur.com/82m3pnO
You’ve got three populations to worry about in your calculations:
1) Those who can’t ever buy a car
2) Those who could buy a car without the tax and can’t with the tax
3) Those who can buy a car even with the tax
Group 1 is always better with the tax (they get an additional A of income for some extra utility). Group 3 is always worse off with the tax (they get taxed and lose some income and utility). Group 2 could go either way, depending on how high the tax is, how expensive cars are, and the relative utilities of corn and cars.
With the equations as you’ve described them, cars are *so* valuable that the loss of utility from the tax to the car-buyers and the lost opportunity of buying cars vastly outweighs the very small additional utility to those who can’t afford cars at all.
I think it’s still possible to get an optimal non-zero tax, you just need to make cars less overwhelmingly valuable – give them a utility that doesn’t depend on corn so highly, I would expect.
Great work on the visualization Brett. That makes total intuitive sense to me that the tax = 0 is optimal because SL made cars so overpowered. The analogy kind of works; if I were behind the veil of ignorance and knew that in the slim case that I’m born Bill Gates I would be so ecstatically happy with the large consumption that it would outweigh the more likely chance that I’m not Bill Gates then indeed I might prefer taxes of zero.
But this isn’t how the real world works, at least according to introspection. Again placing myself behind the veil of ignorance, I would rate getting that fourth yacht as quite a bit less utility enhancing than getting that first loaf of bread every week, not the other way around, and so would favor taxes to redistribute it in that way somehow.
#13 – I was probably unclear. I was trying to understand why you defined car utility as a function of corn utility – i.e. having a car doubles the utility you get from the corn you can buy. Corn utility is sq rt of consumption. Cars could also be sq rt of consumption, but they are so expensive that no one can afford more than one car, so the utility of one car could be just a multiple of corn utility – say equal to the amount of corn you could get with income 1/2. Then if you had income of 1/2, you would be indifferent between buying a car or buying corn. Intuitively this seems wrong, as you would need some corn to survive. I was speculating if this is the reason for giving cars the utility described.
“Now if we imagine that talents are uniformly distributed between 0 and 1”
Another factor of the real world is that income (or talent) is not distributed anything like evenly. Very few earn huge amounts and the vast majority earn relatively little.
Brett’s visualisation is very useful. The little area between the red and blue lines at the bottom (where red is on top) is much smaller than the area between the red and blue lines at the top. Thus loss of utility at the top is greater than gain at the bottom. Thanks
The problem may simply be that you can have only one car.
You’d probably get a more realistic outcome if you change the model to allow unlimited car ownership but with steeply declining utility.
Or maybe not….
We often hear in the news that the top 1% of people have some large percentage of the wealth in the world, so I would have used a different distribution of income instead of just having it be uniformly distributed from 0 to 1.
David Grayson:
I would have used a different distribution of income instead of just having it be uniformly distributed from 0 to 1.
That’s because, relative to me, you’d have placed more emphasis on getting the right answer and less emphasis on creating a feasible homework problem.
Brett at al. may be right. The answer to the problem posed is skewed by a likely unrealistically high utility payoff to owning a car. Cars are just so very good for their owners’ utility that any tax on them would be counter-productive.
In fact, do these utility functions indicate that total utility will be maximized not by taxing cars, but by subsidizing them from the general population?
Unfortunately, you cannot apply the exact same formulation with negative A and T, because if A becomes negative, some people will consume negative corn and the resulting imaginary utility is hard to interpret. Instead, I tweaked the model to replace A and T with B (a per-valorem tax on corn consumption) and S (a flat subsidy to each car bought).
The result is here: http://imgur.com/WmAP4mo . In words, indeed, an 11% tax on everybody’s corn consumption to pay for a car subsidy for the rich, would improve total utility to about 0.752646 (as compared to 0.74357 in the no-tax case).
PS: A more legible Mathematica version of the orignal problem is at http://imgur.com/jEeg01V .
Sub Specie: My first reaction was that this can’t be right because it both transfers income in the wrong direction *and* creates a deadweight loss, so there’s no possible source of social gain. But this reaction was too hasty, because marginal utility of income is much higher for those with cars, so transferring to those with cars is actually a transfer in the *right* direction. This is NOT (contrary to your assertion) because cars are “just so very good for their owners’ (total) utility”, but because those with cars enjoy corn so much more that we want to make them richer to take advantage of that.
If it were just a matter of cars being very good for their owners, individuals would have all the right incentives to acquire the optimal number of cars on their own. Instead, what’s unusual in this example is that an extra dollar is actually worth *more* in utility terms to a rich person with a car than to a poor person without a car, because that dollar goes to buy corn that has its marginal utility magnified by the car.
But that’s a quibble. I believe you’ve nailed this. Thanks.
To put this a different way, you have inadvertently created an indirect utility function which is not concave everywhere. Let U(x) be optimized utility as a function of income x, in the absence of any tax. Then U(.25)=.5, U(.5)= .71, U(.75)=1, a violation of concavity, so in this society you prefer a coin flip between .25 and .75 to .5 for sure. U actually has a kink at 2/3 and is concave on either side, making the desired direction of redistribution ambiguous.
I suppose when adopting Harsanyi-Rawls, we should pause to think about how sure we are that U from behind the veil of ignorance is really concave. Perhaps, for instance, you consider existence for the lowest quarter of the population to be so dreary that you would just as soon die if you drew x in [0,.25]. In that case U is flat on [0,.25] (hence not globally concave) and in fact (at least for small transfers) any transfer to the very poor decreases total utility. Hmm, this started as a thought experiment but might just be my view. I should try to take into account that behind the veil of ignorance, I would not have been spoiled by the pampered lifestyle which results in my finding existence worthless if demoted to [0,.25] (the typical person at .1 surely finds value in moving to .2), but it is extraordinarily difficult to compensate for experiences which began in infancy.
Jonathan Weinstein: Several great points here. Thanks.
#28. Interesting points. The view from behind the veil is a bit of a problem. It assumes you can reasonably assess the value of certain situations. It is observable that people do not generally kill themselves when they reach a certain level of poverty, so it is reasonable to assume that people generally do not consider life so dreary they would prefer to die. There is no reason to expect that you or I would be any different.
There is a parallel with how people think after becoming disabled. Many people say things like “I would rather die than live like that”, but when they are actually in the situation they assumed would be intolerable, they find compensations and that life is still worth living. It has been said that after a disabling incident people “bounce back” or re-set to their previous levels of happiness, but that seems to be be overstated. However, it does seem that there is some re-setting.
http://ftp.iza.org/dp2208.pdf
In the above discussion paper, I think there is an illustration of the difficulties in this area. The terms utility, happiness and wellbeing are all used to describe the same variable (V)
Professor Landsburg: I remember you saying the common wisdom that it’s efficient to tax inelastic goods more heavily was misleading, and it’s usually most efficient to tax all goods equally*. I hear some people advocate a Land Value tax as efficient because the supply of land is inelastic. Do you buy that logic?
*I often use words like “good” or “best” to mean efficient when talking econ, but that would be especially dumb here since this blog post is about optimizing for something other than efficiency.
Can’t we all just get along – like Scandinavia? Perhaps not.
A year ago, Daron Acemoglu (MIT), James A. Robinson (Harvard), and Thierry Verdier (Paris School of Economics) published “Asymmetric Growth and Institutions in an Interdependent World” arguing – well, kinda what Landsburg has been arguing.
Summary: Yes, Scandinavians enjoy a big social safety net and arguably the highest median living standards, while achieving growth rates similar to the US. How do they do this? Simple — they piggy-back on the technological externalities created by the ball-breaking economic climate of the US.
In short, no, we can’t all be Scandinavia; someone has to break the balls and drive the innovations.