## Validating assumptions

from **Lars Syll**

Piketty uses the terms “capital” and “wealth” interchangeably to denote the total monetary value of shares, housing and other assets. “Income” is measured in money terms. We shall reserve the term “capital” for the totality of productive assets evaluated at constant prices. The term “output” is used to denote the totality of net output (value-added) measured at constant prices. Piketty uses the symbol β to denote the ratio of “wealth” to “income” and he denotes the share of wealth-owners in total income by α. In his theoretical analysis this share is equated to the share of profits in total output. Piketty documents how α and β have both risen by a considerable amount in recent decades. He argues that this is not mere correlation, but reflects a causal link. It is the rise in β which is responsible for the rise in α. To reach this conclusion, he first assumes that β is equal to the capital-output ratio K/Y, as conventionally understood. From his empirical finding that β has risen, he concludes that K/Y has also risen by a similar amount. According to the neoclassical theory of factor shares, an increase in K/Y will only lead to an increase in α when the elasticity of substitution between capital and labour σ is greater than unity. Piketty asserts that this is the case. Indeed, based on movements α and β, he estimates that σ is between 1.3 and 1.6 (page 221).

Thus, Piketty’s argument rests on two crucial assumptions: β = K/Y and σ > 1. Once these assumptions are granted, the neoclassical theory of factor shares ensures that an increase in β will lead to an increase in α. In fact, neither of these assumptions is supported by the empirical evidence which is surveyed briefly in the appendix. This evidence implies that the large observed rise in β in recent decades is not the result of a big rise in K/Y but is primarily a valuation effect …

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.

It seems to me that Rowthorn is closing in on the nodal point in Piketty’s picture of the long-term trends in income distribution in advanced economies.

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 = 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)}*.*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 = 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 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.

“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.” Silly me. I thought the first was caused, at least in part, by what causes the latter – there being no real profitable investments to make. The future of capital is rents – owning things, not making things.