09 May 2014

The Final Nail in the Coffin: MPK Is Not *Sufficient* For Interest, Either

Austrian School, Capital & Interest, Shameless Self-Promotion 42 Comments

For those who really want to understand what all the fuss is about capital & interest, you need to get a cup of coffee, perhaps even a piece of paper and pencil, and spend 20 minutes reading my latest Mises Canada post. You know I don’t often ask these things of you, but this is really tough stuff and it takes a while to free your mind from its shackles.

In this latest post, what I’ve done is tweak my bird economy to show that even if there are physically productive nets which augment output, and even if everybody gets paid his or marginal product, the rate of return on financial capital can still be zero. An excerpt:

Suppose our hypothetical economy only last for 10 periods, and that everybody has perfect foresight of what’s coming. This time, however, the “bird catch” steadily declines each period, in order to perfectly offset the natural “time preference” in people’s subjective preferences. Thus, in equilibrium, a bird in period 1 trades for a bird in period 2, which trades for a bird in period 3, etc. Yes, *other things equal* people would prefer a present bird to a future bird, but other things aren’t equal: There is going to be very little bird consumption occurring in period 10, so a worker even way back in period 1 would be willing to trade away one of his present birds in order to obtain an airtight claim to someone else’s bird to be delivered in period 10.

Now then, the other wrinkle in our hypothetical economy is that a few workers start out in period 1 in possession of nets. A worker with a net can always catch 1 bird per hour more, than a worker who just uses his bare hands. (It’s true that over time it gets more difficult to catch birds per hour of labor, but workers also get more adept at handling the nets. The advantage of having a net in your possession is always 1 extra bird per hour, for periods 1, 2, …, 9, and 10.)

Notice that in this world, capital goods and labor contribute to total output. (When we explained that the real interest rate on birds would be exactly 0% because of the declining bird catch per period, we had already taken into account the fact that workers would be using the nets optimally.) We are assuming standard competitive markets, meaning that each worker earns the marginal product of his labor. The owners of the nets also earn the marginal product of their capital goods: Either they use the nets themselves and reap the extra output, or they rent the net to another person who then hands over the surplus above what that person would have caught with his bare hands. Either way, the earnings of the net owners are 1 bird/hour of usage.

Finally, to keep the math simple, suppose that all the workers just work for 10 hours each period. (If you want, you can suppose that there is only a 10-hour window each period when the birds fly low to the ground. Or, you can assume that the workers’ preferences for leisure just happen to yield the result that everybody always optimally choose to work exactly 10 hours each period, all things considered.) What can we say about the earnings of the capitalists, i.e. the owners of the nets?

Well, it’s clear that the owners of the nets will receive a flow of “real bird income” of 10 birds per period. If they want, the capitalists can thus eat 10 birds per period more than the workers who only have their raw labor power to generate an income. Now in Piketty’s preferred framework where we calculate a distribution of “national income” between “labor and capital” or between “the workers and the capitalists,” clearly we should say that the capitalists are getting some of the output, while the workers are getting the rest. According to Piketty, the share of income going to the capitalists is the real return to capitalr multiplied by the stockpile of capital β. (Remember he uses the formula α = rXβ.) Since the result is positive–we know the owners of the nets are getting birds each period, while the workers aren’t retaining the full catch–it must be the case that the real rate of return on capital is positive.

Oops, except that it isn’t. If we calculate the market price of the nets in each period, we can see that their owners earn a zero return.

It’s easiest to work backwards. At the end of period 10, the market price of the nets will be 0 birds; nobody would trade even a single bird to buy the net from an owner, because the world ends in period 10.

At the end of period 9, people know that a net still has one period left to contribute to physical output. This contribution will be 10 birds (accruing one period in the future). But since (in this contrived example) we’ve adjusted things on purpose to get birds trading at par across time periods, that means a “bird-in-period-9″ has the same market value–trades one-for-one against–claims to a “bird-in-period-10.” Thus, the market price of the net in period 9 is “10 present birds.” In other words, the owner of a net could sell it outright in period 9 to someone who would hand over 10 birds that could be eaten on the spot.

At the end of period 8, similar logic shows that the spot market price of the net must be 20 birds immediately available. At the end of period 7, the market price of the net will be 30 birds, etc. etc., and at the end of period 1 the market price of the net will be 90 birds. Also note that at the beginning of period 1–which is equivalent to the “end of period 0″ before time begins–the net has the full value of 100 birds, which will be its lifetime contribution to physical output.

Now then, suppose the owner of a net asks his accountant to calculate the rate of return on his financial capital from period to period. What will the answer be?

The accountant will say, “At the beginning of period 1, your net had a market value of 100 birds. During that period, you rented it out to earn a gross return of 10 birds. However, even though the net was physically in perfect condition when the worker gave it back to you after his shift, by the end of period 1 the market price of the net had fallen to 90 birds. Thus your total wealth went from 100 birds at the start of the period, to 10+90 = 100 birds at the end of the period. You just treaded water financially speaking; you converted the form of your wealth from being 100% concentrated in a net to being 90% in a net and 10% in a stockpile of birds. But your total wealth–measured in market value–didn’t go up. Your rate of return was zero.”

If you go to the link, you’ll also see I included a quote from Mises on this stuff. I figured if it’s Mises Canada then Bohm-Bawerk shouldn’t be getting all the love.

42 Responses to “The Final Nail in the Coffin: MPK Is Not *Sufficient* For Interest, Either”

  1. khodge says:

    I didn’t know that this blog came with homework assignments!

  2. Transformer says:

    Piketty’s fundamental thesis appears to be that if r > g then you get rising inequality. It seems quite conceivable that in an Austrian model this could happen (for example in the ERE g =0, but r >o).

    Assume society is divided between workers who spend all their wages each period and capitalists who save out of income and own all the capital. Time preference increases and (lets assume) this leads to negative g and higher r (until we reach a new equilibrium). If capitalists spend their income each period on durable consumer goods like mansions and boats etc they indeed become richer while workers gets poorer (as g is falling and r has increased).

    Would non-capitalists living inside this Austrian model feel justified at some point in saying “screw this – lets tax the rich and build some things that benefit the rest of us” ?

    • guest says:

      Not if I’m a mansion-builder.

      Or the mansion-builder’s preferred place to shop.

      Or one of the shop owner’s suppliers.

      … Etc.

    • guest says:

      “… lets tax the rich and build some things that benefit the rest of us” ?

      The whole point of producing is to supply those who can pay you what you want.

      If you don’t have what producers want, there’s no reason to produce for “the rest of you”:

      Myths and Facts About Big Business

  3. Philippe says:

    Bob, isn’t there deflation going on in this model?

    The interest rate is zero, but birds are becoming more valuable over time, as the supply of birds decreases. (In fact this deflation is why the interest rate is zero).

    So if you have deflation, then the real return to capital is positive, not zero.

    • Bob Murphy says:

      Bob, isn’t there deflation going on in this model?

      No, there’s no money, to keep things in “real” terms. The “basket of consumer goods” is a bird, the only thing that can be consumed.

      If you wanted to introduce money, you could. You would find that once you adjusted for the “bird price level,” the real interest rate would be zero. A present unit of consumption in period 1 trades 1-for-1 for a unit of consumption in period 2.

      • Philippe says:

        but the birds become more valuable in real terms over time, which is why the people in your model accept a zero interest rate.

        “A present unit of consumption in period 1 trades 1-for-1 for a unit of consumption in period 2.”

        Yes, but that unit of consumption in period 2 has to be more valuable than a unit of consumption in period 1. Otherwise why do people accept a zero interest rate?

        • Tel says:

          I thought the “official” measure of deflation was that it was only related to prices. If goods are not exchanged for money then you don’t have prices.

          Surely you aren’t saying there’s some fundamental aggregate quantity in an economy that might be the basis of inflation / deflation ?

          • Philippe says:

            no, I’m just pointing out that in Bob’s model, birds become more valuable over time, which explains why there is a zero interest rate and not a positive interest rate.

            There is a positive real return on capital in Bob’s model because the birds in future periods are worth more than birds in the present.

            This is similar to a deflationary scenario in which the interest rate is zero, but money gains value over time.

      • Bob Murphy says:

        Philippe,

        If I walk through this with you and convince you that the real rate of interest is indeed zero in this bird/net example, are you going to then say, “Oh wow, OK, I guess you can have MPK>0 but r=0. That has made me rethink my whole framework of how capital and interest relate to each other.” ?

        Because if my convincing you of that will just make you think up another reason that Piketty is “basically right,” then there’s really no point to it.

        • Philippe says:

          Bob,

          “If I walk through this with you and convince you that the real rate of interest is indeed zero in this bird/net example”

          Please do.

          The reason I’m disagreeing with you is not because I think Piketty’s right no matter what. I agree with many of the criticisms made by James Galbraith, for example. But Galbraith recognizes that his criticisms of Piketty’s approach to capital don’t in themselves undermine his main thesis, regarding rising inequality resulting from the return on wealth being higher than the growth rate of the economy.

          • Bob Murphy says:

            Wait a second Phillipe, but you didn’t explicitly agree to this part:

            are you going to then say, “Oh wow, OK, I guess you can have MPK>0 but r=0. That has made me rethink my whole framework of how capital and interest relate to each other.” ?

            I’m not saying you have to agree that the 1% deserve their wealth, or that you have to favor a flat tax.

            All I’m saying is, my example was offered to show that MPK>0 does *not* guarantee a positive real interest rate, even when all factors get paid their marginal product.

            Your objection (thus far) to my example is that you think I’m calculating r incorrectly.

            So, I’m happy to walk through it with you, but only if you agree that yes, r=0 in this example would indeed mean that the typical neoclassical understanding of capital and interest is woefully deficient.

            So do you agree? Or are we just arguing over a calculation that has no real significance one way or the other?

            • Philippe says:

              I’m not an expert on neoclassical economics – far from it – so if I agree that it’s wrong that doesn’t mean all that much really.

              I can agree to think carefully about your argument, and to not let my own biases cloud my judgement.

              And if I think your argument is correct then yes, I will say that I think it is correct.

              I’m genuinely interested in understanding your argument and your model, so if you could walk me through it I would definitely appreciate that.

    • Tel says:

      The interest rate is zero, but birds are becoming more valuable over time, as the supply of birds decreases. (In fact this deflation is why the interest rate is zero).

      You are muddled up, an interest rate of zero implies that the birds have equal value (in a subjective sense) over time. That is to say, scarcity of birds exactly offsets personal time preference such that a bird in any period can be exchanged for a bird in any other.

      Come to think of it, if money did exist in Bob’s world then the Regression Theorem would work in reverse. Think about it, if I sell 1 bird in period 1 for $100 and then I get to period 10 and I still have the $100, would anyone sell me a bird?

      Well in period 10 they can’t eat the $100 and they know the world is ending so they can’t spend it in period 11, it seems unlikely they would be willing to sell.

      But knowing that $100 is useless in period 10, it would also be difficult to buy a bird in period 9 as well… this steps back all the way to period 1 and so the exchange value of the money becomes zero in all periods.

      • Philippe says:

        “an interest rate of zero implies that the birds have equal value (in a subjective sense) over time.”

        No it doesn’t. Birds are more scarce in future time periods so their value increases. This is why in Bob’s model people are willing to give you 1 bird today for a promise of 1 bird in the future.

        In other words people are willing to give up 1 bird in the present in order to get a more valuable bird in the future.

        • Tel says:

          This time, however, the “bird catch” steadily declines each period, in order to perfectly offset the natural “time preference” in people’s subjective preferences. Thus, in equilibrium, a bird in period 1 trades for a bird in period 2, which trades for a bird in period 3, etc.

          The stipulation is exactly that all birds trade one for one through any period, he even spells it out.

        • Philippe says:

          “The stipulation is exactly that all birds trade one for one through any period”

          Yes I know. The point is that birds are more scarce in future time periods, so one bird in a future time period is more valuable that one bird in the present.

          Hence I would be willing to give you one bird now, if you promise to give me one bird in the future. Future birds are more valuable, so there is a positive real return on capital.

          • Tel says:

            OK, define “valuable” then. What do you mean by it?

            Who is doing the valuation?

            • Philippe says:

              Bob says the supply of birds, or the ‘bird catch’ declines over time. But the population doesn’t decline, nor does the demand for birds.

              So supply of birds goes down, demand for birds doesn’t change, value of birds goes up, right?

              • Tel says:

                Who is doing the valuation?

              • Philippe says:

                the people in Bob’s model.

                “Yes, *other things equal* people would prefer a present bird to a future bird, but other things aren’t equal: There is going to be very little bird consumption occurring in period 10, so a worker even way back in period 1 would be willing to trade away one of his present birds in order to obtain an airtight claim to someone else’s bird to be delivered in period 10.”

              • Tel says:

                The people are doing the valuation, and those same people are willing to exchange one bird in any period against one bird in any other period, right?

                Thus we conclude these people must have decided that all birds in all periods have equal value. What other valuation can be used here?

              • Philippe says:

                if you lend 1 bird to someone you would normally charge interest.

                In other words, you give someone one bird today, and they promise to pay you, say, 2 birds in the future.

                Now why would you lend someone 1 bird today, and only ask to be repaid 1 bird in the future?

                Why would you lend a bird at zero interest?

                Because that 1 bird in the future is more valuable, because birds in the future are more scarce. So that increase in value represents the real return on the loan.

                In Bob’s example people’s ‘time preference’ is exactly offset by the increasing scarcity of birds. As such, the interest rate is zero, so you give one bird today and get one bird back in the future.

              • Philippe says:

                say for example you know that there is going to be deflation in the future, so the value of money will increase.

                You might be willing to lend $1000 to someone today, on the agreement that they repay you $1000 in the future. You don’t charge interest because the value of money will increase, so the $1000 you are receive in the future will be worth more than the $1000 you lend today.

                Although you lend $1000 today in exchange for $1000 in the future, clearly this does not mean that $1000 today and $1000 in the future are worth the same.

              • Philippe says:

                **so the $1000 you receive in the future will be worth more than the $1000 you lend today.

              • Major-Freedom says:

                Philippe:

                Why not just hold onto the bird? What is the gain to lending a bird ay zero bird interest rate?

                Note that it is not an answer to say that your one bird may die in the inyerim. The bird example is to be understood as a rough proxy to a monetary economy, and dollars don’t notmally die (unless they’re frb promises, but that is an unneeded complication at this point).

              • Philippe says:

                the cost and risk involved in storing the bird/money perhaps.

              • Major-Freedom says:

                Phillippe,

                The birds are a proxy for money. The reasons you give should not be based on actual birds, but rather money.

              • Philippe says:

                same thing goes for money. Storing money involves costs and risks.

              • Major-Freedom says:

                Philippe:

                But that isn’t why people lend money. For the borrower would also be faced with such risks.

              • Philippe says:

                In Bob’s model the borrower is assumed to be ‘risk-free’. The promise to repay the sum lent is a cast-iron promise. Kind of like a government which promises to repay money which it can print.

              • Philippe says:

                in reality this does actually happen.

                People or organizations with a lot a money prefer to buy US T-Bills than leave their money in a bank (i.e. lend their money to a bank – which is risky above a certain amount), or pay to store their cash in a vault (which is expensive and also risky).

  4. Tel says:

    You just treaded water financially speaking; you converted the form of your wealth from being 100% concentrated in a net to being 90% in a net and 10% in a stockpile of birds. But your total wealth–measured in market value–didn’t go up. Your rate of return was zero.

    By the way, that still seems to fit the rule of the interest rate being locked by the first derivative of the Irving Fisher diagram. If you think about it, prevailing interest rates are set at 0% by the way you have structured the world, and in period 1 the capital earns 10% but also depreciates 10% so if you did the diagram with inclusion of capital depreciation (and probably you should do it that way) you get back to a gradient of -1 (i.e. what goes in also comes out) and then you are back with 0% interest rate.

    You are going to probably argue that the depreciation on the nets is not physical (they are still in good condition at period 10) but I would argue firstly that it is physical depreciation given that the world as a whole is “depreciating” by getting closer to ending, just because the damage is not visible doesn’t change anything; but secondly depreciation could happen for whatever reason and the diagram would be the same.

  5. Cody S says:

    Phillipe,

    I am currious on the future/present one-bird transaction:

    Who is the person who, knowing that one bird next year is worth more than one bird today, decides to accept a bird today in exchange for their own future bird? Are we to assume that some people (in this case, 50%) just like transactions in which they lose value?

    And would not such an assumption have some interesting ramifications in, say, pricing, wage, on up to macro theory?

    • Philippe says:

      “Are we to assume that some people (in this case, 50%) just like transactions in which they lose value?”

      it’s just like paying interest on a loan. You are willing to repay more than the sum you initially borrow, because you prefer to have the money today, rather than save it up over time.

      • Cody S says:

        So, then one person values “one bird today” more than he values “one bird tomorrow”, is what you are saying.

        Tel is right. The reason “one bird today” is exchangeable for “one bird tomorrow” is because the value of the birds is exactly the same. You told him that the exchange happens because the bird tomorrow is more valuable; then, you told me that the exchange happens because the bird today is more valuable.

        There are two people transacting: a person buying a bird tomorrow with a bird today and a person buying a bird today with the promise of a bird tomorrow.

        If the interest (even in bird,) charged tomorrow for a bird today is 0 birds, then a bird tomorrow is worth exactly as much as a bird today.

        One side of the transaction may be anticipating deflation; by the same argument you make, the other side is anticipating inflation. Just because they are expecting both inflation and deflation, does not mean that either has to happen: both definitely don’t.

        You seem to have been assuming that expectations of deflation on one side of a transaction mean deflation has to occur, or that pricing only happens in one direction, or maybe both of those.

        • Philippe says:

          think about it in terms of money.

          Say $1 buys one bird today, and two birds a year in the future, due to deflation.

          Say I lend you $1, and charge you $0 nominal interest. You agree to repay me $1 in one years time.

          We calculate the real rate of return by subtracting the rate of inflation from the nominal rate of return. In this case the rate of inflation is negative, so even though the nominal rate of return is zero, the real rate of return is positive.

          $1 today exchanges for $1 a year from now, because $1 a year from now buys more than $1 today.

        • Tel says:

          You told him that the exchange happens because the bird tomorrow is more valuable; then, you told me that the exchange happens because the bird today is more valuable.

          Ahhh but big government is smarter than both parties to every transaction, that’s why these things must not be left up to the free market.

          😉

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