News

Loading...

Monday, May 05, 2008

The (Limited) Usefulness of Formal Economic Models

(This post is dedicated to Robert Wegner.)

As a graduate of New York University, I was forced to learn formal models of the economy, which would have horrified me when I was in college and hooked on Austrian economics. However, even though I never would have gone to NYU had I realized what I was getting into, after the fact I was glad that I had done it. There really are benefits of these formal modeling techniques. This isn't to say the benefits outweigh the opportunity costs; obviously graduate students could be doing something else with their time that might be much more valuable in terms of producing good economists. Nonetheless, in this blog post I want to give an example of the power of formal modeling.

The specific issue is whether a consumption or income tax is less distortionary. As usual, we are going to try to compare apples with apples by insisting on revenue neutrality, and we are also not going to worry about issues of privacy.

Following Rothbard, my intuition had always been that an income tax was better (on these limited grounds). I thought, "The government takes a certain amount out of your income, and then you're free to do what you want with it. If you want to invest it, go ahead. If you want to blow it at the racetrack, that's fine too. The government shouldn't be trying to encourage you to save more than your time preferences indicate."

If you ask me now, though, my answer would be the exact opposite. If we make enough assumptions to render the question meaningful, then it is clearly the case that a consumption tax respects people's intertemporal preferences, whereas an income tax penalizes savings. In this sense, then, a consumption tax is Pareto superior; a consumption tax leaves consumers with more utility than an income tax that yields the same (present value) in tax revenues.

In defense of Rothbard and my earlier self, a lot of proponents of consumption taxes aren't clear on this point. They either imply or explicitly claim that their plan "encourages savings and investment" above what would happen in the absence of any taxation, and that is clearly inefficient. (After all, it would be crazy to force everyone to save 99% at gunpoint, even though this would lead to very high GDP growth.)

But as I said, studying a really simple model led me to reverse my earlier position. Under reasonable assumptions, a consumption tax doesn't alter the consumption/savings tradeoff. Yes, people are obviously worse off because their consumption is lower in every period, but they are not hurt by distortions in the intertemporal tradeoffs they face.

Before continuing, let me give the caveat: Of course you can always come up with ways to disprove any sort of "rule" in these types of analyses. E.g. if people had religious views favoring income taxes, then they would obviously be worse off with a switch to consumption taxes. But if we assume people don't care about the tax system itself, that there are equivalent costs of compliance, etc., and just focus on the incentives on individuals' saving rates and preferences for consumption at different points in time, then the consumption tax is clearly superior.

So here's the verbal reasoning: From the individual's POV, the purpose of saving and investing is simply to consume in the future. So really the issue is consumption now versus consumption later. Without any taxes, people refrain from consumption today until the point at which the marginal utility of present consumption is higher than the present utility of the future consumption which that investment would yield.

Slapping on a consumption tax doesn't alter this tradeoff, it simply lowers the level of consumption in every period. Think of it this way: Suppose you were a farmer trying to decide how much corn to eat and how much to plant for next year's harvest. After you made your decision, what if you learned that the cook who makes your meals is a klutz, and so it takes 10% more corn to make a meal than you previously estimated. With this new information, that shouldn't alter how much of your harvest this year you devote to feeding your family, versus how much you plant again. Yes, the amount you set aside to eat right now yields fewer meals, but that is true of the amount you are planting.

In contrast, an income tax does give you an incentive to consume more in the present than you would have without any taxes. It's as if there are locusts that destroy 10% of the crops before you harvest them. Now that you've got the 90% in your possession, you can either eat them or plant them again, when they again will be subject to the locusts. So the technology for converting potential present meals into future meals has changed; now abstaining from one present meal yields a smaller amount of future meals than it did originally. Thus you consume a larger fraction of your harvest, and plant less for next year.

Let me address something that bothered me for a while, even though the above reasoning seemed impeccable. In intermediate micro, professors love to go over indifference curves to show that an income tax is better for the consumer than an excise tax on a particular good. E.g. if there are beer and donuts with their market prices, it is better to take $10 from the consumer and let him spend the remaining $90 however he wants, rather than placing a unit tax on beer (let's say) such that in equilibrium the consumer buys enough of that good to give $10 in tax revenue to the government. The intuition is that not only is the consumer $10 poorer, but under the excise tax beer is made artificially more expensive, and so the individual's options get a double-whammy from the government.

But that's not what's happening here. Specifically, it's not true that you have a "given" amount of income, and then the government let's you spend it on present versus future consumption. If you save and invest, then your future income is higher than it otherwise would be. So that's one difference from the typical intermediate micro demonstration (with a fixed endowment of income that the consumer is allocating between two goods).

The other difference is that with a consumption tax, the government isn't slapping a tax on one possible good, while leaving the other untaxed. Again, the whole purpose of saving is to consume in the future (or to allow your heirs to consume). So there's no distortion, encouraging individuals to save more than they otherwise would (the way there was in an excise tax on beer, which would encourage spending on donuts).

========

Now why did I title this blog post the way I did? Because I was only led to the above realizations through working out a very simple model, with two periods and a consumer with utility function U = ln(C1) + ln(C2), and interest rate of 50%, and wage income in each period of $100. I slapped on a 50% income tax and had the consumer optimize, and calculated how much (in PDV in period 1) revenue the government would collect. Then I figured out what rate a consumption tax would have to be in each period to yield the same (PDV in period 1) revenue to the government. In the second approach, the consumer's utility (from the after-tax consumption) is higher than with the after-tax stream of consumption under the first setup. Also, in the consumption tax approach the gross amount of saving is unaffected by the tax; i.e. it is the same whether the consumption tax rate is 0 or positive. This isn't true with the income tax.

Another benefit of using the model: When I first tried to solve for the revenue neutral consumption tax rate, it came out to something like 116%. So I thought I had messed something up, because that was clearly crazy. But then I realized that no, there is nothing illogical (though grossly immoral of course!) about consumption tax rates being higher than 100%. If your wage income is $100, you can, say, save $10, consume $30, and pay $60 in taxes to the government, if the consumption tax rate is 200%. That would be awful, but it's not unsustainable the way a 200% income tax would be.

So the lesson for Austrian purists? I think that formal models allow us to check our intuitions on complex things, such as comparing different tax regimes. Of course you can't rest with the model results; if the results are surprising, you need to figure out whether it's because you made a bogus assumption, or because your intuition is wrong.

I know Rothbard has argued that this is superfluous at best; why translate economics into formal symbols, get a result, and then translate back into English? But the answer here is the same as for why they do this in symbolic logic: Because sometimes the argument is very complex, and you might make a mistake in your reasoning if you try to do it in English.

I never would have come to the understanding that I can now give in purely Austrian terms, had I not known how to create a very crude neoclassical model. I would still think that consumption taxes distort intertemporal decisions vis-a-vis consumer preferences.

One last thing: If I still haven't sold you, consider comparative advantage. If you haven't worked through a ridiculously crude two-good, two-country numerical example, then I submit you probably don't really "see" why free trade makes all countries richer. Sure, that Ricardian model doesn't prove anything, but a simple numerical example illustrates the principles very efficiently. And that's why everybody I've ever seen teach this--including people at Mises University--rely on simple numerical examples.

7 comments:

  1. Wow Bob,

    I never thought I'd ever have a column dedicated to me. I think I am going to print it out and frame it.

    As for comments on the column, I am much too tired to do it justice tonight. Though even in my current sleepy state, a couple of things do come to mind :)

    I'll be back tomorrow or Wednesday with a full commentary. It's the least I can do for a column dedicated to me.

    ReplyDelete
  2. OK, but let's not get ahead of ourselves. I call this a blog post. I write columns for LRC and Mises.org.

    ReplyDelete
  3. I know Rothbard has argued that this is superfluous at best; why translate economics into formal symbols, get a result, and then translate back into English? But the answer here is the same as for why they do this in symbolic logic: Because sometimes the argument is very complex, and you might make a mistake in your reasoning if you try to do it in English.

    This is an argument I once tried to make with some Austrian groupies in a discussion of Man, Economy and State...the only result of which was that I got labeled as something between a blasphemer and a heretic.

    ReplyDelete
  4. Okay, Bob,

    First, I am not sure why you dedicated this post to me, in the sense that our prior discussion about models was about empirical models that specifically use empirical data to make forecasts. You in fact supply empirical data for that model in the form of a chart to show the supposed historical relationship between PPI and CPI. Indeed, in that discussion I asked "anonymous" what was the theoretical underpinnings for the empirical model. I received no answer to this point.

    In your current discussion, you are creating a mental model without empirical data. Thus, I must reject the basis of your dedication to me, it is a different type of model that we are now discussing.

    However, I must reject this mental model, as well, and believe that the attempt to put the model in terms of formal symbols has sapped the question of its essence and thus results in your reaching a faulty conclusion by eliminating key factors which should be considered. In other words, because you have chosen to formalize the argument in terms of mathematical symbols you have been forced to eliminate important factors that do not fit easily into mathematical symbol form.

    You pose the question:"The specific issue is whether a consumption or income tax is less distortionary."

    You then eliminate the following from being taken under consideration:

    First in general: "If we make enough assumptions to render the question meaningful"

    Translation "If we make enough assumptions to render the question in a way that will fit in a symbolic equation

    You then go on to eliminate the life from your mental construct:

    "Before continuing, let me give the caveat: Of course you can always come up with ways to disprove any sort of 'rule' in these types of analyses. E.g. if people had religious views favoring income taxes, then they would obviously be worse off with a switch to consumption taxes. But if we assume people don't care about the tax system itself, that there are equivalent costs of compliance, etc., and just focus on the incentives on individuals' saving rates and preferences for consumption at different points in time, then the consumption tax is clearly superior."


    And you end up playing a mental parlor game that leads to an incorrect conclusion.

    People don't care about the type of tax system.

    Why would you take this out of your mental construct?

    You assume ...that there are equivalent costs of compliance

    And Bob, with this last point, you completely cut off the life of your model with reality to make it mathematically pretty.

    There is a BIG difference in compliance costs and I would wage that for most high income earners, they know damn well that they are going to pay a lot less tax under an income tax system such as we have.

    Why so? A consumption tax leaves very little wiggle room (other than savings) to reduce the tax bite. You go into a store and the consumption tax is immediately applied (Naturally,who actually pays the sales tax depends on the elasticity of supply). But, with an income tax all types of strategies can be employed to reduce "income" and show it as a business expense.

    Thus, while in your formal mathematical model you reach Pareto supriority under a consumption tax, in the real world high net worth individuals would most assuredly choose an income tax that they can monkey with, rather than a consumption tax which provides them little wiggle room.

    Yes, I know you assumed in your models that the two taxes would result in equal amounts of tax revenue, but again you are just taking the meat out of the problem. The consumption tax is obviously a more efficient collection method than the income tax. Thus an individual seeking to maximize his utility, would prefer the income tax. The exact opposite conclusion your mathematical model would suggest.

    You have sacrificed knowledge for the sake of an elegant, but faulty mathematical model.

    ReplyDelete
  5. Thus, I must reject the basis of your dedication to me, it is a different type of model that we are now discussing.

    You are assuming you know why I dedicated it to you. I clearly was correct that you would find my post completely useless; I stand vindicated in my (sarcastic) dedication.

    The consumption tax is obviously a more efficient collection method than the income tax. Thus an individual seeking to maximize his utility, would prefer the income tax. The exact opposite conclusion your mathematical model would suggest.

    You really crack me up, Mr. Wegner. If you had stopped after pointing out all of my question-begging assumptions, that would have been fine. Annoying perhaps, but perfectly legitimate.

    But after you criticize me for all the assumptions needed in order to reach my conclusion, you do the exact same thing.

    What if people like their privacy? Then "clearly" they maximize utility from a national sales tax that allows the IRS to be blown up.

    So once again, you fault the specificity of my conclusions, only to offer your own version that is equally dependent on assumptions that aren't true in general.

    I.e. you criticize my model (which you are perfectly correct in doing), and then offer your own equally baseless model and don't even realize you are doing it.

    ReplyDelete
  6. Oh baby, I got you here, Bob.

    My point was not that you can't make assumptions, only that mathematical models tend to sap the juice out of a discussion. There's nothiing wrong with assumptions, but they must have (at least) a reasonable possibilty of being based on reality. Yours do not, e.g. "People don't care about the type of tax system", and "there are equivalent costs of compliance."

    I contend it was your desire to put these in mathematical form that forced you to make these life extracting assumptions.

    Somehow, I find it hard to swallow that my assumption that individuals want to pay a lower tax (and in turn give up a tiny amount of privacy)is "baseless"--since I fall into this category, myself. Thus, my very existence refutes your assertion. However, I sincerely doubt I am alone.

    Let's put it into a mathematical equation so you can understand it.

    Where P = value of privacy.

    I = potential income tax paid

    S = potential sales tax paid

    You thus state the possibility that


    P > S-I and the opposite, S-I > P is "baseless".

    LOL

    My contention is that:

    If S-I > Y then Y > P

    If S-I = 10 cents, then it is likely that P > S-I.

    However.

    If S-I = $100,000 then it is likely for many that S-I > P.

    BTW:

    Thanks for recognizing the correctness of the rest of my critique:

    If you had stopped after pointing out all of my question-begging assumptions, that would have been fine. Annoying perhaps, but perfectly legitimate.

    ReplyDelete
  7. We offer WoW power leveling,World of Warcraft power leveling,Warhammer Online Power Leveling & Warhammer Power Leveling & Warhammer Online Gold,if you want buy cheap wow power leveling & honor power leveling,please come here to choose. you could find any kinds of powerleveling you want. Please remember,we are your online game helper. Please remember,we are your online game helper.please come here to choose. you could find anything you want.wow power leveling,wow power leveling,wow power leveling,wow power leveling,wow power leveling,wow powerleveling,wow powerleveling,wow powerleveling,wow powerleveling,wow powerleveling,Warhammer Online Power Leveling,war leveling,Warhammer leveling,Warhammer Power Leveling,Warhammer Online Gold,Warhammer Gold,WAR Power leveling,WAR Gold,world of warcraft power leveling,world of warcraft power leveling,world of warcraft power leveling,world of warcraft power leveling,world of warcraft power leveling,world of warcraft powerleveling,world of warcraft powerleveling,world of warcraft powerleveling,world of warcraft powerleveling,world of warcraft powerleveling,wow gold,wow gold,wow gold,wow gold,wow gold,world of warcraft gold,world of warcraft gold,world of warcraft gold,world of warcraft gold,world of warcraft gold,AOC Power Leveling,AGE OF CONAN Power Leveling,Warhammer Online Power Leveling,Warhammer Power Leveling,Warhammer Online Gold,Warhammer Gold,2 Moons Dil,MapleStory Mesos,Maple Story Mesos,MS Mesos,WARHAMMER ONLINE GOLD,Cheap WARHAMMER ONLINE GOLD,RuneScape Gold,RS Gold,RuneScape Money,RS Money,SilkRoad Gold,SilkRoad Online Gold,SRO Gold,EVE ISK,EVE Online ISK,Gaia Gold,2 Moons Dil,WOTLK power leveling,AOC Gold,AGE OF CONAN Gold,LOTRO Gold,Lord of the Rings Online Gold

    ReplyDelete