## Piketty and reasonable estimates of depreciation rates (wonkish)

10 Jun, 2014 at 10:22 | Posted in Economics | Comments Off on Piketty and reasonable estimates of depreciation rates (wonkish)Thomas Piketty emails:

“We do provide long run series on capital depreciation in the “Capital is back” paper with Gabriel [Zucman] (see http://piketty.pse.ens.fr/capitalisback, appendix country tables US.8, JP.8, etc.). The series are imperfect and incomplete, but they show that in pretty much every country capital depreciation has risen from 5-8% of GDP in the 19th century and early 20th century to 10-13% of GDP in the late 20th and early 21st centuries, i.e. from about 1%[/year] of capital stock to about 2%[/year].

Of course there are huge variations across industries and across assets, and depreciation rates could be a lot higher in some sectors. Same thing for capital intensity.

The problem with taking away the housing sector (a particularly capital-intensive sector) from the aggregate capital stock is that once you start to do that it’s not clear where to stop (e.g., energy is another capital intensive sector). So we prefer to start from an aggregate macro perspective (including housing). Here it is clear that 10% or 5% depreciation rates do not make sense.”

No, James Hamilton, it is not the case that the fact that “rates of 10-20%[/year] are quite common for most forms of producers’ machinery and equipment” means that 10%/year is a reasonable depreciation rate for the economy as a whole–and especially not for Piketty’s concept of wealth, which is much broader than simply produced means of production.

No, Per Krusell and Anthony Smith, the fact that “we conducted a quick survey among macroeconomists at the London School of Economics, where Tony and I happen to be right now, and the average answer was 7%[/year” for “the” depreciation rate does not mean that you have any business using a 10%/year economy-wide depreciation rate in trying to assess how the net savings share would respond to increases in Piketty’s wealth-to-annual-net-income ratio.

Who are these London School of Economics economists who think that 7%/year is a reasonable depreciation rate for a wealth concept that attains a pre-World War I level of 7 times a year’s net national income? I cannot imagine any of the LSE economists signing on to the claim that back before WWI capital consumption in northwest European economies was equal to 50% of net income–that depreciation was a third of gross economic product…

Some of the critics of Piketty obviously think that the issue is theoretical and that somehow he has misspecified the standard growth model. Now, Piketty doesn’t really talk that much about the standard (Solow) growth 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 (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 = Ak** ^{α}* , 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,

*Δ*

*k =*

*λ*

*y*

*–*

*δ*

*k,*equals

*Δ*

*k =*

*λ*

*Ak*

^{α}*–*

*δ*

*k*. In steady state (*) we have

*λ*

*Ak**

^{α }*=*

*δ*

*k*,*giving

*λ/δ = k*/y**and

*k* = (*

*λ*

*A/*

*δ)*Putting this value of k* into the production function, gives us the steady state output per worker level

^{1/(1-α)}.*y* = Ak**

^{α}*= A*

^{1/(1-α)}*(*

*λ*

*/*

*δ))*

^{α}

^{/(1-α)}*.*

*A*ssuming 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 = k*) we get

^{α}*dk/dt = λy – (g + δ)k,*which in the Cobb-Douglas case gives

*dk/dt = λk*with steady state value

^{α}– (g + δ)k,*k* = (λ/(g + δ))*and capital-output ratio

^{1/(1-}^{α}^{) }*k*/y* = k*/k**If using Piketty’s preferred model with output and capital given net of depreciation, we have to change the final expression into

^{α}= λ/(g + δ).*k*/y* = k*/k**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.

^{α}= λ/(g + λδ).Being able to show that you can get the Piketty 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.

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