Optimal allocation of public goods

From: Hal Finney (hal@finney.org)
Date: Wed Mar 12 2003 - 15:01:04 MST

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    I wanted to explain and comment on the paper by Groves and Ledyard
    that I mentioned in another thread, Optimal Allocation of Public
    Goods (http://www.finney.org/~hal/GrovesLedyard.pdf, temporarily).
    This paper shows how a government can provide public goods at what is
    called a Pareto-optimal level, which means in effect that you can't
    improve the situation for anyone unless you make someone else worse off.
    That's generally considered about as good as you can do in economic terms.
    It's what the market achieves for private goods. Being able to do the
    same thing for public goods means that the government is doing its job
    about as well as is theoretically possible.

    First let me describe what is meant by a public good. Public goods are
    one of the main justifications for governments. Basically these are
    goods which can't be provided in an efficient manner by markets because
    of the "free rider" problem. They are goods which people can get benefit
    from even when other people are paying for them. An example would be
    clean air. If the government takes steps to clean the air, everyone
    breathes easier. A private company can't sell outdoor air-cleaning
    technology because few would buy it. Even though the purchaser would
    gain a benefit by making his own air a little cleaner, most of the
    benefit goes to everyone else who also breathes the clean air.

    Another example of a public good is national defense - if your nation
    has a nuclear deterrent then everyone is protected, not just the
    people who paid for it. Many environmental issues are public goods,
    such as ocean pollution or species depletion; likewise, scientific and
    medical research that is public funded. Wei Dai proposed (in effect,
    this is my interpretation anyway) that if copyright can't be protected
    technologically, that information goods may become public goods.
    When someone produces a song or movie or novel, everyone gets it, not
    just the person who paid for it.

    Again, the problem with public goods is that people prefer to be "free
    riders". They would rather that someone else pays for the good and they
    can enjoy it for free. Napster and its offshoots exemplify free riding.
    Of course if nobody pays, there are no goods. In practice, some people
    will still pay, for various reasons. But economically speaking you can
    show that the total amount of the good being produced is less than would
    be collectively desired. In principle you could take some money from the
    consumers and give it to the producers to make more goods, and everyone
    would be happier - the consumers would get more value in new goods than
    they lost in money, and the producers could afford to make more goods.
    This means that the standard outcome is not Pareto optimal and is an
    economic failure.

    Governments try to solve the problem by taxing people and spending the
    money on public goods. They don't do that for music, yet, but they
    do for national defense and the environment and other public goods.
    The problem is, how does government decide how much should be spent
    on each good? How much should we be spending on national defense or
    environmental protection? Everyone has a different opinion. We use
    voting to come up with a crude measure, but representational government
    has many flaws of its own which have been widely analyzed.

    Groves and Ledyard (GL for short) present an alternative approach, which
    is a fixed mathematical formula which could be used to fund public goods.
    Every taxpayer specifies a value for how much of each public good to
    buy on his behalf. Based on these amounts, the government uses a simple
    formula to purchase the public goods, and a somewhat more complex formula
    to lay taxes, which are enough to fund the goods. The government could
    be replaced by a computer, unswayed by lobbyists or special interests,
    which would perform this very mechanical calculation.

    So here's the GL formula in more detail. To start with, each person
    specifies amounts M that they want the government to spend on their
    behalf for each public good. The government will then add up the amounts
    everyone chose, and those sums are how much is spent on each good. Now,
    some people may not want as much to be spent as the sum of everyone
    else's wants, so the M values can be negative. For each good, the M
    values are added, positive and negative, and the final result is the
    amount to be spent on that good.

    That's simple enough so far. The fancy part comes with how the taxes are
    calculated, which also depends on the M values. The taxes have two parts.
    The first part is that each person pays his share of the total amount of
    public spending. For simplicity we'll assume that this is an even split.
    So if there are one million taxpayers, each has to pay one millionth of
    the sum of all the M values in taxes.

    Then there is a compensation factor in the taxes, which works like this.
    For each good, calculate the mean of the M values for everyone else,
    and call that u. Calculate the standard deviation of the M values for
    everyone else, and call that s. Then each person has to pay an additional
    tax whose value is (M-u)^2 - s^2. This is summed for each public good.

    Now, let me explain how this works. The idea is that (M-u)^2 is a measure
    of how far your own M value is from the average, u. The farther you are
    from the average, the more tax you'll have to pay. However we subtract
    s^2, aka the variance, which is high for goods where there is a wide
    spread of M values and low for goods where there is a narrow spread.
    So if you are far from the mean, the penalty is not so bad for goods
    where there is a high variance, which makes sense because then everyone
    else is far from the mean too. The worst case is where almost everyone
    agrees about the M value, meaning that the variance is low, but you're
    way different, so your (M-u)^2 value is high. Then you have to pay the
    biggest penalty.

    Another thing to note is that (M-u)^2 - s^2 can be negative. In that
    case it reduces your tax instead of adding to it. This will be true for
    people whose M value is exceptionally close to the average. For them,
    (M-u)^2 is less than s^2 and so this term reduces their tax for that good.
    And a final thing to note is that this correction term sums to zero
    across all taxpayers, so the total amount of money raised by the taxes
    comes only from the first part of the taxation, which was the simple
    split of the expenditures. Therefore the government's budget is balanced.

    The main point of the GL system is that people whose M values are far
    from the average pay a penalty, and people whose M values are very close
    to the average get a bonus. Everyone has to take this into consideration
    when choosing their M values.

    Ideally, the social task of each person choosing M values happens
    collectively, with full information. Imagine that everyone gets a
    little controller with a dial they can turn to set their M value, which
    remember is the amount "extra" (positive or negative) they want spent on
    a particular public good. In real time, a computer is constantly adding
    up the total M values and is displaying two things to each person: the
    total amount which will be spent on the public good, and the amount of
    tax the person is going to have to pay based on his current dial setting.
    Turning the dial to the right slightly increases the total amount which
    will be spent (not much, though, if there are thousands or millions of
    voters), and also changes the amount of tax which that person will have
    to pay. The tax will be minimized when the dial is set at roughly the
    mean value of everyone else's choice, and will go up as the knob turns
    left or right from that position. Each person has to trade off their
    desire to influence the amount spent on the public good against their
    willingness to pay taxes to make their voice heard.

    That's basically how the GL system works. It's a little complicated,
    especially the taxation formula. And keep in mind that all the public
    goods have to be done at once, so you could increase one good's M value
    while decreasing another and keep your taxes in balance, that kind
    of thing. But I think with some computer help it might be manageable.
    In practice there are probably some few goods you care about and a
    larger number that you don't. For the second category you can just
    set a policy to minimize your tax burden associated with those goods,
    and focus your energy on trying to influence the spending on the goods
    you are interested in.

    It may seem that the GL system is fundamentally unfair, in that it in
    effect rewards conformists and punishes dissidents. The farther you
    are from the mean, relative to everyone else, the more you have to pay.
    But in a way, the problem is more with the nature of public goods
    than the GL system itself. Since they will be consumed collectively,
    and purchased in that same way, people who depart from the consensus
    get penalized. Those who have very different opinions about spending
    from everyone else can't have their preferences win. In order for this
    effect to be reflected in the economics, this must be manifest as a cost
    laid on dissenters. Each person can then make their own decision about
    how much of a cost they are willing to pay to try to influence society
    away from its average value.

    If we do have to have public goods, then it seems to me that a
    technologically based system like this would be worth experimenting with.
    It may not be able to replace all of the functions of government (like
    setting policies and deciding what activities should be legal), but for
    the purposes of funding public goods, it would have many advantages.
    With the aid of computer interfaces and possibly some simplifications
    to the full formulas in the GL paper, it might be practical to begin
    using such a system within a few years.

    Hal



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