09 Jan 2013

Interest Does NOT Equal the Marginal Product of Capital, Even in Equilibrium

Economics, Nick Rowe, Shameless Self-Promotion 42 Comments

Oh man, here I’m trying to really be productive. I have even come up with strict limits on my Facebook time. And then Nick Rowe goes and starts posting on capital & interest theory!

Here’s the situation in a nutshell. In mainstream economics, it is commonplace for people to say that in a competitive equilibrium, the interest rate equals the “marginal product of capital.” This is considered to be analogous to the claim that in a competitive equilibrium, the wage rate equals the marginal product of labor. After all, the thinking goes, interest is the price (or opportunity cost) of using another unit of capital, and the marginal return to capital is diminishing, so firms use more and more capital until the point at which r*=MPK (in standard notation). Just like, firms keep hiring workers until w*=MPL.

The only problem with all of this, is that it’s totally wrong. Besides all of the critiques you might raise about formal modeling etc., there is a qualitative sense in which interest does not have anything to do with the “marginal product of capital,” in the way that we can plausibly say that wages are intimately related to the marginal product of labor.

Now the Austrian School has had a bee in its bonnet over this issue since at least Bohm-Bawerk, and his famous critique of “the naive productivity theory of interest.” (I give the details for a layperson here.)

And yet, if you go to the trouble of learning modern techniques for mathematical models of the economy, it seems that r=MPK just pops out of the model. There’s nothing wrong with the math. So what the heck is going on here? Was there a flaw in the Austrian verbal logic, that the precise mainstream guys blew up?

Nope, not at all. What’s happening is that the standard r=MPK result–where MPK is defined as the increment in physical output from an additional input of capital into the production function–crucially assumes that the capital and consumption good are the same physical things, or at least, that they are always physically convertible into each other in a constant ratio.

If you can read the notation, I think I showed very clearly in the Appendix of my dissertation. I had a model with a distinct capital and consumption good, and then derived the expression for the equilibrium interest rate. It wasn’t at all r=MPK, because you had to worry about the changing market value of the capital good vis-a-vis the consumption good. But, if you assumed that they always traded at par against each other (which would of course be true if they were the same thing, as one-good models assume), then my expression reduced to r=MPK.

Nick Rowe is groping towards the same type of result. (I’m not saying “groping” to be disparaging; he is admitting upfront that his modeling is a bit off.) Here are the crucial passages from his post:

Here is the simple aggregate technology macroeconomists often assume:

C + I = F(K,L) where I = dK/dt (I have ignored depreciation for simplicity).

Some economists object to the right hand side of that equation. They complain that it aggregates all labour into one type of labour L. And they complain that it aggregates all capital goods into one type of capital good K.

But I object more to the left hand side of that equation.

It aggregates newly-produced consumption goods C with newly-produced capital goods I. It assumes they are perfect substitutes in production. It assumes the Production Possibilities Frontier between C and I is a straight line with a slope of minus one. It assumes the opportunity cost of producing one more capital good is always and everywhere one less consumption good. It means that the price of the capital good will be always one consumption good. And that means that the (real) rate of interest will always equal the marginal product of capital.

We don’t assume a straight line PPF between two different consumption goods. Why should we assume a straight line PPF between consumption goods and capital goods?

[This approach] also shows what’s wrong with “r = MPK”, in a simple model.

You could add in a second capital good if you like. Just add K2 to F( ), and I2 to H( ), then you get a second equation for Pk2, for R2, and for r as a function of Pk2 and R2. But I don’t think it makes as much difference. The problem is not aggregating capital goods. The problem is aggregating the capital good with the consumption good. [Bold added.]

When I’m not daydreaming about becoming today’s Bobby Darin, I imagine I have tenure down the hall from Nick. We go to lunch, and talk about overlapping generations and then robots that can build copies of themselves.

42 Responses to “Interest Does NOT Equal the Marginal Product of Capital, Even in Equilibrium”

  1. Nick Rowe says:

    Bob: “What’s happening is that the standard r=MPK result–where MPK is defined as the increment in physical output from an additional input of capital into the production function–crucially assumes that the capital and consumption good are the same physical things, or at least, that they are always physically convertible into each other in a constant ratio.”

    Bingo. In a nutshell. Yes.

  2. Daniel Kuehn says:

    I’m groping, but I would think the solution is to have

    C = F(Kf,Lf)
    I = G(Kg,Lg)
    Kg+Kf =K
    Lg+Lf=L
    r = dF/dKf = dG/dKg

    I’m not sure the LHS is anything more than an accounting identity. The problem is still with the RHS, and of course we use the simple version Nick presents because it’s nice and concave, whereas the implied production function I have here is additive and not concave.

    But if we’re really optimizing at the level of consumer and capital goods markets – and not at the economy as a whole – things seem to be concave again, right?

    I think a lot of “microfounded” macro takes short cuts like this and figures that if we’re optimizing we’re microfounded. I don’t think that’s strictly true, but if we were to disaggregate the production function (a very good idea IMO), you’re still going to get basically the same result, right?

    • Daniel Kuehn says:

      Whether the shortcut has other dire consequences as feared by the Post Keynesians and others, I’m not entirely sure.

      But I am less inclined to see why the fact that these things are not homogenous implies that we’re not still dealing with the marginal productivity of capital. If capital has a higher marginal productivity in capital goods production than in consumer goods production, don’t we expect it to move into capital goods production?

      Or am I missing something?

    • Bob Murphy says:

      DK wrote:

      I don’t think that’s strictly true, but if we were to disaggregate the production function (a very good idea IMO), you’re still going to get basically the same result, right?

      No.

      • Bob Murphy says:

        To be more specific, with certain assumptions on robots, I can show that in competitive equilibrium, the real rate of interest is equal to the marginal product of labor. I certainly am not going to teach that to undergrads.

      • Daniel Kuehn says:

        So I’ve been working through your appendix. Would it be correct to say that your point is that the interest rate is equal to the marginal revenue product rather than the marginal product and that the we normally normalize the price of both consumer and producer goods.

        Then as you say above, you have to think about changing relative prices. Which is fine, but when we solve these things don’t we solve for a steady state? Then your spot price minus future spot price term in equation four on page 190 drops out and we have a constant ratio between the normalized consumer good price and the capital good price again.

        Is this all right?

        I guess I’m just curious about what my take-away on all this needs to be. When I assume a steady state (so pi_t = pi_t+1 using your diss notation), and plug (3) into the second part of (2) on page 189, I get i = g1 which seems like it’s what I should get.

        When you express it in terms of the marginal revenue product of capital used in the production of consumption, of course you have to divide by the spot price of capital because as you said the prices are not equal to each other.

        But I guess I just don’t get how it’s wrong to say that the interest rate is equal to the marginal product of capital so long as we’re willing to concede “oh ya – what we actually mean is the marginal revenue product of capital, not the marginal product of capital”.

        I’m worried I’m missing your point entirely, but in case I wanted to take another look at this tomorrow I wanted to leave a comment with where I got myself to as of 3:26 pm :-P

        • Bob Murphy says:

          But I guess I just don’t get how it’s wrong to say that the interest rate is equal to the marginal product of capital so long as we’re willing to concede “oh ya – what we actually mean is the marginal revenue product of capital, not the marginal product of capital”.

          But look–you just illustrated the problem perfectly. In your summary, you didn’t mention the crucial caveat that it applied only to the steady state. That’s a much stronger requirement than mere equilibrium.

          • Bob Murphy says:

            Oh, I’m on a phone call right now and maybe I misread you DK. I have to think about it later. You’re saying r=MRP in general, not just in steady state?

            • Daniel Kuehn says:

              No I think you’re right that you need the steady state for the pi_t minus pi_t+1 to drop out, then you have the price ratio (1 divided by pi_t) if you express it in terms of the marginal product of capital used to produce consumption, and of course you don’t have that when you look at marginal product of capital used to produce capital because you have pi_t in the numerator and the denominator.

              So steady state seems like something people are pretty up front about.

              What they’re not always explicit about is the marginal revenue product rather than just the marginal product.

              On that point I definitely agree you have a point.

              I’m worried I’m missing something about exactly how significant that point is.

          • Daniel Kuehn says:

            The other thing that’s interesting to me about your appendix is to think about it in Keynesian terms about pushing the amount of capital to the point that it is non-scarce.

            He thinks that drives the interest rate to zero, but that makes some implicit assumptions about how fast the marginal product of capital drops relative to the price of capital as the amount of capital increases.

            I’m wondering if he discusses those sorts of issues and assumptions or not.

    • Nick Rowe says:

      Daniel: your first 4 equations are fine. Your 4th equation does not follow from the first two. Unless the price of I in terms of C is always one. But it won’t be.

      Think: what units do we measure w in? Units of C per unit of L.

      What units do we measure r in? 1/years. And there’s a different r for different goods.

      Gotta go teach.

  3. Major_Freedom says:

    My two cents:

    Period 1:

    Total money spending = $1000
    Consumer goods spending = $100
    Capital goods spending = $900
    Total money costs (book value of capital) = $900
    Total profit = Total spending – Total costs = $100
    Average rate of profit (marginal efficiency of capital) = $100 / $900 = 11%

    Period 2:

    Total money spending = $1000
    Consumer goods spending = $200.
    Capital goods spending = $800
    Total money costs (book value of capital) = $800.
    Total profit = Total spending – Total costs = $200
    Average rate of profit (marginal efficiency of capital) = $200 / $800 = 25%

    Period 3:

    Total money spending = $1000
    Consumer goods spending = $300.
    Capital goods spending = $700
    Total money costs (book value of capital) = $700.
    Total profit = Total spending – Total costs = $300
    Average rate of profit (marginal efficiency of capital) = $300 / $700 = 43%

    Period 8

    Total money spending = $1000
    Consumer goods spending = $800.
    Capital goods spending = $200
    Total money costs (book value of capital) = $200.
    Total profit = Total spending – Total costs = $800
    Average rate of profit (marginal efficiency of capital) = $800 / $200 = 400%

    Period 9

    Total money spending = $1000
    Consumer goods spending = $900.
    Capital goods spending = $100
    Total money costs (book value of capital) = $100.
    Total profit = Total spending – Total costs = $900
    Average rate of profit (marginal efficiency of capital) = $900 / $100 = 900%

    Period 10

    Total money spending = $1000
    Consumer goods spending = $1000.
    Capital goods spending = $0
    Total money costs (book value of capital) = $0.
    Total profit = Total spending – Total costs = $1000
    Average rate of profit (marginal efficiency of capital) = $1000 / $0 => infinity

    —————————

    The marginal efficiency of capital is (total money spending – total money costs) / (book value of capital).

    Rates of interest are constrained to, i.e. determined by, the rates of profit, in “equilibrium”.

    The more capital intensive an economy becomes, the more will productive expenditures rise relative to total expenditures. The more productive expenditures rise relative to total expenditures, the lower the rates of profit, and thus the lower the rates of interest will be.

    The less capital intensive an economy becomes, the more will productive expenditures fall relative to total expenditures. The more productive expenditures fall relative to total expenditures, the higher the rates of profit, and thus the higher the rates of interest will be.

  4. RPLong says:

    The neo-classical approach uses an extremely simplifying assumption in order to discuss an entirely different question. Your Mises piece is good because it shows that the Austrian viewpoint is still relevant. I personally wouldn’t go as far as saying that the neo-classical equation is wrong, but I would say that it only applies to situations in which the price of time with respect to capital is very close to the marginal return on that same piece of capital.

    …which is a glorified way of saying Close enough for rock and roll!

  5. Ken B says:

    ” robots that can build copies of themselves”

    Von Neuman forgot to considers pairs of robots, who do this with regularity, and some evident pleasure.

    • Daniel Kuehn says:

      I think we’re meshing Bob’s capital discussion with Gene’s reductionist materialism discussion now…

  6. Matt Tanous says:

    “And yet, if you go to the trouble of learning modern techniques for mathematical models of the economy, it seems that r=MPK just pops out of the model. There’s nothing wrong with the math. So what the heck is going on here?”

    Didn’t Rothbard explain why it is that the rate of profit would be uniform in the ERE, and this would be the same as the interest rate? As I understood Man, Economy, and State (and please correct me if I missed something), profit IS interest + uncertainty, and thus without uncertainty, profit = interest, and all interest and profit rates would be equal.

    • skylien says:

      I read it currently. And he says that in the ERE there is no profit or loss at all. Profit refers to anticipating changing preferences/conditions correctly which you describe as uncertainty. Since that does not happen in the ERE there is only interest income due to time preference.

      • Matt Tanous says:

        Right. I meant “profit” in the sense of total return. In other words, the MPK. (Do I have that right?)

        So, in equilibrium, the MPK would equal interest, because the only return the capitalist investor would get would BE interest.

        • skylien says:

          Yes. That is the way I understand it.

    • Daniel Kuehn says:

      There is no consideration of some kind of risk premium in Bob’s appendix so I doubt this is what he has in mind.

      • Matt Tanous says:

        There is no risk premium in the (obviously unattainable state of the) ERE, because there is no uncertainty. You know precisely what amount of interest return you will get.

  7. Nick Rowe says:

    Let R be the annual real rental on a machine (just as W is the annual real rental of a worker). Measured in units of the consumption good.

    Then assuming perfect competition and profit-maximising firms:

    R=MPK (and W=MPL) where MPK is the extra output of consumer goods per extra machine.

    Let the price of machines in terms of the consumption good be Pk.

    Let r be the real rate of interest, measured in terms of consumption goods.

    Ignore depreciation for simplicity, so an old machine is identical to a new machine.

    In equilibrium, the relation between r, R, and Pk is given by:

    r = R/Pk + annual percentage rate of increase of Pk.

    Pk is not (except in steady state) constant over time.

    Pk is not always and everywhere 1, (even in the steady state) unless one unit of the consumption good can always be converted into one machine.

    Therefore, r =/= MPK.

    • Bob Murphy says:

      I know you know this, Nick, but it’s also useful to point out that saying stuff like “interest rate equals the amount of extra output produced by an extra unit of capital” at first sounds funny, because the interest rate is a % whereas MPK is in units of good1/good2.

    • Gene Callahan says:

      “R=MPK (and W=MPL) where MPK is the extra output of consumer goods per extra machine.”

      This looks wrong to me. Doesn’t MPK have to account for the *price* of the machine? If a machine that costs a thousand hams produces one ham per year I would think we have a much lower MPK than if the machine costs ten hams.

      Then I think we get an indifference curve that touches the PPF just where r = MPK. Because that’s how the price of the machine gets set, after all.

      • Major_Freedom says:

        Seems like in his attempt to refute the “consumer goods = capital goods” fallacy, Rowe took it for granted.

        One consumer good PER one machine requires “one consumer good” and “one machine” to be commensurate variables. But making them commensurable in this way is the very thing being argued against!

  8. Justin Merrill says:

    Ah, capital theory. The price of capital is determined by its productivity, which determines profitability, which determines the interest rate, which determines asset prices, which determines profitability in the production function…Cambridge capital controversy all over again.

  9. Adrian Gabriel says:

    Nick Rowe is clearly wasting his time with this mainstream macro statist goldygook. It sounds like a bunch of mathematical formulas applied to ever-changing human action. Interest is the result of human preference, and as Rothbard so adamantly described, was the Marginal Efficiency of Capital. See here: https://mises.org/Error/Http404/?aspxerrorpath=/rothbard/mes/chap6a.asp#_ftn5

    Keynes was too blind to reality and thus made up names for time-preference that he had no idea how to include them in his interconnected unrealistic models.

    “Modern textbook writers have attempted to skirt this problem by using a one-good model. In all such models, questions of value, which may be affected by changes in the rate of interest, simply do not arise. Value productivity and physical productivity are indistinct; productivity is modelled as the rate of increase in the quantity of the good. The phenomenon of interest is being analogized once again to sheep that reproduce or to plants that grow. But, as Professor Rothbard often reminds us, the rate of interest is a ratio of values, not of quantities. This modeling technique unavoidably conflates growth rates with interest rates and fails thereby to shed any light on the phenomenon of interest.” – Jeffrey Herbener

    A more appropriate description of what the interest rate is in a Misesian praxeological sense is the pure time preference theory of interest. Herbener again explains it best:

    “If the factor payment is made sooner than the revenue is received from the sale of the output produced, then the payment is discounted because of time preference. This discount of future money relative to present money is interest and determines the pure, or time-preference, rate of interest. Because all exchange of present money for future money of the same time structure involves time preference, the pure rate of interest is uniform across all such intertemporal exchange. It follows that all present goods that generate future money will have their prices determined by discounting the future money by the rate of interest to obtain the equivalent amount of present money. This process of capitalization results in a uniform rate of interest as the difference between the present money spent to acquire factors of production and the future money obtained from selling the output produced. Prices, so determined, are the basis for economic calculation, which permits entrepreneurs to appraise the lines of production and investment that people find most valuable.”

    http://mises.org/daily/6240/Action-Time-and-the-Market

    • Gene Callahan says:

      “Nick Rowe is clearly wasting his time with this mainstream macro statist goldygook.”

      1) The word is gobbledygook.
      2) When something goes over my head, I am tempted to be dismissive of it as well.

  10. Tel says:

    So the way I see it is you have a plot of land (that’s capital), and we are talking agriculture here and you grow the same crop every year and prices are always the same, and interest rates are always the same. Everything is always the same every year.

    So the land is a capital good, and it returns $1000 per week in profit once you work out the ongoing costs and how much you can sell your harvest for. You think about taking out a loan to buy that land. In the marginal case you don’t want the repayments to be much higher than $1000 per week, so that puts an upper bound on how much money you can take out on the loan, but if you offer much less than that upper limit then someone else might buy the land in a competitive market.

    You end up borrowing whatever amount of money has interest payments of $1000 per week (or close to it) and you take a very long time to pay the loan. But when you sell the land, same thing happens to the next guy, so you pay off your loan then. All perfectly balanced.

    It aggregates newly-produced consumption goods C with newly-produced capital goods I. It assumes they are perfect substitutes in production. It assumes the Production Possibilities Frontier between C and I is a straight line with a slope of minus one. It assumes the opportunity cost of producing one more capital good is always and everywhere one less consumption good.

    Yeah, well in the case of land, we don’t have newly-produced capital goods. The capital stock is what it is. Come to think of it… can you have both an equilibrium and accumulating capital base at the same time?

    It is a dynamic equilibrium with constant growth rate or something? Then you have to work out the overall growth rate and scale everything with exponential functions and stuff (you know looks like a straight line when plotted on semi-log paper). I think there’s some basic confusion about how we are modelling “equilibrium” here and what we mean by that.

    OK, I’m confused, and the rest of you can do the decent thing and pretend this is totally non-obvious and hard to get, so I feel good about it.

      • Tel says:

        OK, so first you have a pure non-growth model, where interest rates are essentially arbitrary (whatever preference dictates), and a then you have a steady-growth model (at 1% or whatever) where interest rates are preferences plus adjustments for capital costs.

        I think if you are going to do a “steady-growth” model, you kind of have to also have the population growing at the same rate, in order to consume the extra potatoes made by the new land, else it isn’t equilibrium.

        It occurs to me that you are essentially saying there is no Rational Economic Man.

        I mean, not just to point out that REM is an idealization (we all know that), and thus an imperfect model of real people, but you are saying that REM is a fundamentally inadequate model to describe the situation at all because how do you figure out personal time preference in a rational way?

        • Tel says:

          Also, if time preference is a personal whim, then you can’t even have equilibrium at all (because people change their minds).

  11. Tel says:

    Hey totally off topic, but thanks for the Great Depression class. I think I’m a lot more clued up on the whole Great Depression history now.

    Also totally off topic, but I note that geeky programming stuff sometimes pops up around here. Does anyone want to team up for MIT Battlecode ? It just started for the year, and there’s potential for a bit of Alpha male adrenaline rush outsmarting some college students. Go about your regular business if this means nothing to you (hint: it’s a bit like Star Trek n that, but better).

  12. Gene Callahan says:

    “What’s happening is that the standard r=MPK result–where MPK is defined as the increment in physical output from an additional input of capital into the production function–crucially assumes that the capital and consumption good are the same physical things, or at least, that they are always physically convertible into each other in a constant ratio.”

    OK, as I said above, I think *physical” convertibility is not the relevant matter: it is the pricing of the capital good. And how is it priced? By the discounted value of the consumer good flows it will produce in the future. And how are those discounted? By the interest rate. And in equilibrium, that sets r = MPK.

    And this formulation is not at all what Bohm-Bawerk was critiquing. It is not that productivity sets the interest rate, it is that we value future productivity according to our time preference, and thus price capital goods so that the marginal productivity will equal the interest rate. Anything else allows pure arbitrage opportunities, which, in equilibrium, can’t exist. (If r MPK, you lend money and short the capital good.)

    • Major_Freedom says:

      Seems like when Callahan is talking about capital theory, I don’t want to pull my hair out.

      Epiphany?

      • Gene Callahan says:

        Major, I been a tryin’ tell Bob he got this interest rate thing wrong for ten years now.

        • Major_Freedom says:

          Gene, I don’t get it. You’re saying Bob should understand that interest should be understood in monetary terms, and yet his PhD dissertation was about a monetary, not physical output, theory of interest.

          What am I missing?

    • Gene Callahan says:

      That last bit got mangled by the comment machine: it should read:

      If r > MPK, you lend money (or whatever else can be lent in the model) and short the capital good. If r < MPK, you borrow money and buy the capital good.

  13. Adrian Gabriel says:

    Gene, is this Jevons speak? I never remember any Austrian writing this statist talk of r=MPK, let alone there being an equilibrium price ever.

    “It is not that productivity sets the interest rate, it is that we value future productivity according to our time preference, and thus price capital goods so that the marginal productivity will equal the interest rate. Anything else allows pure arbitrage opportunities, which, in equilibrium, can’t exist.”

    We don’t price capital goods according to the interest rate, capital goods are priced due to time preference, which is indeed a result of subjective value. The pure rate of interest is the going rate of time dis­count, the ratio of the price of present goods to that of future goods. Thus interest is something that is not derived from the concrete, heterogeneous capital goods, but from the generalized investment of time. Capital is not an independently productive factor. Capital goods are vital and of crucial importance in production, but their production is, in the long run, imputable to land, labor, and time factors. Further­more, land and labor are not homogeneous factors within them­selves, but simply categories of types of uniquely varying factors. Each land and each labor factor, then, has its own physical fea­tures, its own power to serve in production; each, therefore, re­ceives its own income from production, as will be detailed below. Capital goods too have infinite variety; but, in the ERE, they earn no incomes. What does earn an income is the conversion of future goods into present goods; because of the universal fact of time preference, future satisfactions are always at a discount compared to present satisfactions. This is where I think using “models” of which you describe, go contrary to the important Austrian elements such as time preference. Your assumptions are quite rash in suggesting that capital goods can be grouped together as is suggested in r=MPK, for this equation will never exist in realistic human action.

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