Flummoxed Brad DeLong answers Krusell & Co.

6 Jun, 2014 at 08:50 | Posted in Economics | 5 Comments

flummoxedKrusell and Smith favor a deprecation rate of 10%/year – and I genuinely do not understand why they think it is appropriate. We are not, after all, dealing with short-run business-cycle fluctuations in which the pieces of the capital stock that vary are made up mostly of inventories and machines here. We are talking about land, very durable buildings, powerful property rights and the ability to summon the police to protect them – claims over future output that do not, I think, erode away at anything like 10%/year …

Starting around 1980, Piketty argues, the North Atlantic shifted out of its Social-Democratic Era and is now moving into a new configuration, with increasingly-concentrated wealth, savings no longer reduced by highly progressive capital taxation and fear of expropriation, and slower rates of population and labor productivity growth. Piketty expects the consequence to be a rise in the savings rate back to Belle Époque levels and a return to the capital intensity and inherited-wealth dominance of those days. In my view, the next questions are two:

Would this be a good thing? More savings and wealth accumulation by the rich that increase the capital intensity of the economy increase real wages for the working class and the poor, no? Here I think the answer is perhaps – and I think this is what the debate over Piketty should be about. Unfortunately, that is not the debate we are having …

Can this happen? And Krusell and Smith and company are saying: no, it cannot. As the capital-output ratio rises, the desire to consume wealth pushes the gross savings rate goes down and the fact that capital depreciates at 10%/year pushes the net savings rate down much further, and so there are no macroeconomic forces in play that could push the wealth-to-annual-net-income ratio far up above its current value of 300%.

But if the savings rate necessarily falls as the wealth-to-annual-net-income ratio rises, why was the (gross) savings rate half again as high back before World War I when the economy was wealth-dominated as it is today? And from where comes the 10%/year depreciation rate assumption?

What we clearly have here is a failure to communicate. And I really, really do not think that it is the result of a failure to try on Thomas Piketty’s part.

Brad DeLong

[h/t Jan Milch]


  1. Why does depreciation occur? Why is a car driven off the lot suddenly more than 10% depreciated? What in fact are the amortization rates as practiced under Federal Tax Codes? 10% may in some ways be an underestimate, and for other goods, value may increase, a negative depreciation. Piketty does stipulate that real estate values in the US increase at only a 3-4% per year which when compared to maintenance costs of houses and most buildings would lead to a depreciation, a cost. No one believes this but it is real. When markets are flourishing, goods move at an accelerated rate, liquidity becomes most relevant, but calculated value becomes irrelevant. Theoretically, inflation, wear-and-tear and other psychological factors may propel sales, and tax breaks, but the depreciation rate on major goods: land, buildings, vehicles, machines, computers, is certainly greater than .1% or 1% which are so low as to become irrelevant whereas everyone understands depreciation as significant.

  2. P. 231 in Piketty’s “Capital in the 21st century”

    “It was not until the 1970s that Solow’s… model definitively carried the day
    If one rereads the exchanges in this controversy with the benefit of hindsight, it is clear that the debate… did more to cloud economic thinking than to enlighten it. There was no real justification for the suspicions of the British.”

  3. Piketty doesn’t really talk that much about the Solow model in the book, but let’s do a back of the envelope analysis based on that model and say we have that diehard neoclassical model — and assuming both the production function is homogenous of degree one and unlimited substitutability –such as the standard Cobb-Douglas production function

    y = Ak^α.

    With a constant investment λ out of output y and a constant depreciation rate δ of the “capital per worker” k, the rate of accumulation of k,

    Δk = λy– δk,

    then equals

    Δk = λAk^α– δk.

    In steady state (*) we have

    λAk*^α= δk*,


    λ/δ = k*/y*


    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= A^(1/(1-α)) (λ/δ))^(α/(1-α)).

    Assuming we have technological progress that increases y with a growth rate g, we get

    dk/dt = λy – (g + δ)k,

    which in the Cobb-Douglas case gives

    dk/dt = λk^α– (g + δ)k,

    with steady state value

    k* = (λ/(g + δ))1/(1-α)

    and 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 close to 8.

    (And as far as I can get, 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.)

    • Lars, David Ruccio made a good point i think: in RWER on US poverty rate, actual and simulated, 1959 – 2012: ”


      “One of the points Thomas Piketty makes in his new book is that mainstream economists enshrined as “laws” of capitalist development certain “facts” that only had relevance during the immediate postwar decades. see :https://anticap.files.wordpress.com/2014/06/poverty-rate.jpg

      “These so-called laws included constant capital and labor shares and declining inequality. We now know they were no more than artifacts of a particular period of capitalist development for some countries (including the United States). Things began to change radically in the mid- to late-1970s for those same countries (again, including the United States).

      The same is true, as it turns out, of the relationship between economic growth and poverty. As the Economic Policy Institute
      explains growth and poverty here As the Economic Policy Institute in explains (in pdf) see:

      Click to access Raising-America’s-Pay-2014-Report.pdf

      and “Economic growth used to be associated with significant poverty reductions, but since the 1970s the benefits of aggregate growth for lowering poverty have largely stalled. The figure compares the actual poverty rate with a simulated poverty rate based on a model of the statistical relationship between growth in per capita gross domestic product (GDP) and poverty that prevailed between 1959 and 1973. The model forecasts poverty quite accurately through the mid-1970s. Since then, the actual poverty rate stopped falling and has instead fluctuated cyclically within 4 percentage points above its trough in 1973.

      However, the simulated poverty rate shows that if the relationship between per capita GDP growth and poverty that prevailed from 1959 to 1973 (wherein poverty dropped as the country, on average, got richer) had held, the poverty rate would have fallen to zero in the mid-1980s. Therefore, broadly shared prosperity could have led to a near eradication of poverty in the United States, but it did not.

      So, the next time someone exclaims that the solution to poverty is more economic growth, explain to them that trickle-down economics, even if it was valid grosso modo for the immediate postwar period, has not worked for those at the bottom of the distribution of income for many decades.

      And it certainly doesn’t!!”

  4. Thomas Pickety: A Lot Of My Crtitics Haven’t Even Read My Book http://www.huffingtonpost.com/2014/06/06/thomas-piketty-book_n_5459635.html

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