by Piketty and the need for validating assumptions 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
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Piketty and the need for validating assumptions
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 labor 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-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 does in Capital in the twenty-first century is to predict 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 and Piketty’s two crucial assumptions — β = K/Y and σ > 1 — is of course, from a realist point of view, of rather 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.