Home > Uncategorized > Models and the need to validate assumptions

Models and the need to validate assumptions

from Lars Syll

Piketty argues that the higher income share of wealth-owners is due to an increase in the capital-output ratio resulting from a high rate of capital accumulation. The evidence suggests just the contrary. The capital-output ratio, as conventionally measured has either fallen or been constant in recent decades. The apparent increase in the capital-output ratio identified by Piketty is a valuation effect reflecting a disproportionate increase in the market value of certain real assets. A more plausible explanation for the increased income share of wealth-owners is an unduly low rate of investment in real capital.

Bob Rowthorn

Say we have a diehard neoclassical model (assuming the production function is homogeneous of degree one and unlimited substitutability) such as the standard Cobb-Douglas production function (with A a given productivity parameter, and k  the ratio of capital stock to labor, K/L) y = Akα , with a constant investment λ out of output y and a constant depreciation rate δ of the “capital per worker” k, where the rate of accumulation of k, Δk = λy– δk, equals Δk = λAkα– δk. In steady state (*) we have λAk*α δk*, giving λ/δ = k*/y* and k* = (λA/δ)1/(1-α)Putting this value of k* into the production function, gives us the steady state output per worker level y* = Ak*α= A1/(1-α)(λ/δ))α/(1-α)Assuming we have an exogenous Harrod-neutral technological progress that increases y with a growth rate g (assuming a zero labour growth rate and with y and k a fortiori now being refined as y/A and k/A respectively, giving the production function as y = kα) we get dk/dt = λy – (g + δ)k, which in the Cobb-Douglas case gives dk/dt = λkα– (g + δ)k, with steady state value k* = (λ/(g + δ))1/(1-αand capital-output ratio k*/y* = k*/k*α = λ/(g + δ). If using Piketty’s preferred model with output and capital given net of depreciation, we have to change the final expression into k*/y* = k*/k*α = λ/(g + λδ). Now what Piketty predicts is that g will fall and that this will increase the capital-output ratio. Let’s say we have δ = 0.03, λ = 0.1 and g = 0.03 initially. This gives a capital-output ratio of around 3. If g falls to 0.01 it rises to around 7.7. We reach analogous results if we use a basic CES production function with an elasticity of substitution σ > 1. With σ = 1.5, the capital share rises from 0.2 to 0.36 if the wealth-income ratio goes from 2.5 to 5, which according to Piketty is what actually has happened in rich countries during the last forty years.

Being able to show that you can get these results using one or another of the available standard neoclassical growth models is of course — from a realist point of view — of limited value. As usual — the really interesting thing is how in accord with reality are the assumptions you make and the numerical values you put into the model specification.

Professor Piketty chose a theoretical framework that simultaneously allowed him to produce catchy numerical predictions, in tune with his empirical findings, while soaring like an eagle above the ‘messy’ debates of political economists shunned by their own profession’s mainstream and condemned diligently to inquire, in pristine isolation, into capitalism’s radical indeterminacy. The fact that, to do this, he had to adopt axioms that are both grossly unrealistic and logically incoherent must have seemed to him a small price to pay.

Yanis Varoufakis

  1. deshoebox
    January 24, 2022 at 9:59 pm

    From an ordinary person’s point of view (a non-economist, say) the issue is not whether Economist A’s assumptions used to explain the extraordinary growth in the wealth of the ultra-rich during the pandemic are more or less realistic than those. used by Economist B to explain the same thing. The issue is, of course, why we allow billiionaires to exist in the first place and how we have collectively let the system operate in such a way that as hundreds of millions of actual people become poorer, sicker, and hungrier, billionaires just get richer and richer. Needless to say, this issue is not of concern to economists and will not be commented upon by the so-called Real World Economic Review.

    • Garrett Connelly
      January 25, 2022 at 1:05 am

      Hello there deshoebo x; Your points are quite clear even though I disagree with the last idea that RWER does not focus on the plight of those left awash in the excesses of billionaires.

      Although Lars Syll’s mathematical poetry may not be in your style of analysis, his insights are intriguing. For example, I slogged through Piketty’s work and appreciated it yet did not clearly see the source of my unease with that work until reading the math and the included quotes posted by Lars Syll.

      I see you as correct about everything else. Please describe an alternate reality that can help cure the ills you point to.

      I propose removing advertising from business expense to free speech and adjusting market prices to include all product and service costs as well as their public benefits. Caloric money would also be helpful, in my mind. What do think we should do?

  2. Dave Raithel
    January 25, 2022 at 1:15 pm

    “With σ = 1.5, the capital share rises from 0.2 to 0.36 if the wealth-income ratio goes from 2.5 to 5, which according to Piketty is what actually has happened in rich countries during the last forty years.” Did it? Or that it did is indeterminable (epistemically)? Or that it did happen is coincident to Piketty’s explanation why, since some other scheme with numbers will describe the same outcome?

  3. Dave Raithel
    January 25, 2022 at 1:19 pm

    “With σ = 1.5, the capital share rises from 0.2 to 0.36 if the wealth-income ratio goes from 2.5 to 5, which according to Piketty is what actually has happened in rich countries during the last forty years.” Did it? Or whether it did is indeterminable (epistemically)? Or that it did is coincidental to Piketty’s explanation why, since some other scheme with numbers will also explain it?

  4. January 26, 2022 at 1:42 pm

    Presumably, there is a law of diminishing marginal returns. This may also apply to capital. There more capital there is relative to GDP, the lower the yields are likely to be. That is because in the end, only effective demand can make capital profitable. And so, interest rates go down in capital accumulates, and as a consquence, existing capital assets may rise in value due to discounting. This can be a transitory if interest rates are somehow a reflection of future yields on captal.

    Low interest rates may be unfavourable for the wealty on aggregate in the long run. The following thought experiment explains why.

    Assume that you buy a 10-year government bond yielding 4% when the interest rate is 4%. The bond is worth € 1,000, yields € 40 in interest each year, and after ten years, the € 1,000 is returned. Over the 10-year period, you will receive € 1,400 in total. If the interest rate is 4%, then the present value of the bond is € 1,000. Now suppose that immediately after you buy the bond, the interest rate drops to 0%. The present value of the bond suddenly rises to € 1,400, a gain of € 400. That is the result of discounting. But if you keep the bond until the end, you still receive € 1,400 in total over the 10-year period.

    If you reinvest this money, the interest rate matters. If the interest rate is still 0% after ten years, and you buy another 10-year bond, you will still have € 1,400 after twenty years. But if the interest rate never had dropped to zero and had remained 4% all the time, you could have reinvested the proceeds of the bond at 4% interest, and you would have ended up with € 2119 after 20 years. In the short term, lower interest rates may make the rich richer, but in the long run, the opposite is true. That is because interest is capital income, and the rich own most capital.

  5. Gerald Holtham
    January 26, 2022 at 10:20 pm

    Aggregate production functions do not really explain anything. There is no reputable theory underlying them so they are just an exercise in fitting convenient functional forms to data. They are no more than a rather arbitrary means of organising data in order to talk about it. You make a battery of assumptions about the nature of “technical progress” the rate of scrapping or depreciation of capital and the functional form of the production relationship and then use the degrees of freedom you have to estimate the parameters of your model. The fact is a different combinations of those assumptions yields different parameters but the model can give very similar results. A few aggregate data series cannot discriminate; there is observational equivalence.. This is illustrated by Rowthorn’s point; different assumptions can give the same result in terms of profit share. We have to look at a lot more and more finely-grained data to discriminate.
    Lars nearly hits the nail on the head when he says the results can be obtained by “one or another” of the standard neoclassical models. In fact they can be obtained with one AND another of the standard models.. and yet another one again. Varoufakis makes a fair point, a touch harshly. The approach does not stop you asking important questions but it certainly doesn’t answer them.

    • yoshinorishiozawa
      February 3, 2022 at 4:48 am

      I support Gerald. Aggregate production functions do not really explain anything.

      Invalidity of aggregate production functions explanations is demonstrated about a half century ago by two famous papers:
      (1) Anwar Shaikh (1974) Laws of production and laws of algebra: The humbug production functions. Review of Economics and Statistics 56(1): 115-20.
      (2) Simon, Herbert A. (1979) On parsimonious explanations of production relations. Scandinavian Journal of Economics 81(4): 459-74.

      Aggregation of production functions requires too strong hypothesis about the set of them. The use of aggregate production functions does not make sense. This is persuasively proved by
      (3) Felipe, Jesus and McCombie, John S. L. (2013) The Aggregate Production Function and the Measurement of Technological Change: “Not Even Wrong.’ Edward Elgar.

      So, all growth theories based on aggregate production functions are logically invalid. Those theories comprise classical growth theory of Solow type and new or endogenous growth theory of Paul Romer and Robert Lucas type. It means that almost all neoclassical growth theories are invalid.

      The problem for us (those who are not satisfied by neoclassical growth theories) is to develop an alternative theory that does not draw on aggregate production functions. For a first step of such attempts, see my paper:
      Shiozawa, Y. (2020) A new framework for analyzing technological change, Journal of Evolutionary Economics 30: 989-1034.

    • yoshinorishiozawa
      February 6, 2022 at 2:15 pm

      Technological change (or technical progress) is an essential prime mover of economic growth. However, this question is not properly treated even among Post Keynesian economists. Most often they assume so-called Kaldor-Verdoon law given by a linear function of the rate of accumulation per head. However, Marc Lavoie in Post-Keynesian Economics: New Foundations (2014, p.429) put it:

      This relationship allowed Kaldor to avoid distinguishing between a move along a production function and a shift of the production function, a distinction that Kaldor (1957 p.595) and all those within the Cambridge tradition reject (Rymes, 1971). Still, we see right away that there is no difference between: (i) Kaldor’s linear technical progress function; (ii) the Cobb-Douglas production function in its dynamic version with appropriate restriction; and (iii) the dynamic version of the national accounts. …

      This shows again the weakness of instrumentalism. The same mathematical expressions may arise from different theories: prediction would be helpless in identifying the right one. The above also shows that one should beware of the successes of Kaldor-Verdoon’s law for precisely the same reason that led us to question the apparent successes of the neoclassical production function. It could also be an artefact.

      Post Keynesian theories that link technological change and economic growth are still far from a satisfactory state. Referring to the Necessity of finding an appropriate analytical framework for dealing with technical change and economic growth (Point 8 of the list he considers characteristic features of the alternative paradigm in Postlude: fighting for independence, Luisi Pasinetti remarks that “this concern with technical progress was not pushed far enough.” (Keynes and Cambridge Keynesians, 2007, p.233)

      It is necessary to build a new theory that links technological change and economic growth in a causal and historical way. My paper indicated in the above post is only a first step toward such an attempt.

  6. John deChadenedes
    January 26, 2022 at 10:42 pm

    I suggest a thought experiment: If a large truck has stopped with its front tire on top of your foot and two passersby – two economists, for example – are arguing about whether the truck’s tires are inflated to the regulation 80 pounds or only to 75, how long would it be before you start hoping the truck will drive over both of them when it finally moves off your foot?

  7. Gerald Holtham
    February 7, 2022 at 6:49 pm

    Ha ha.
    There is actually some useful empirical work on growth determinants in modern economics though it doesn’t throw light on income distribution. Look at the complexity and product space work of the Harvard Growth Lab, led by Ricardo Hausmann. It has nothing to do with aggregate production functions or what’s in simple text books.

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