Friday, October 17, 2014

r > g In A Steady State

1.0 Introduction

This post presents a model of distribution that Luigi Pasinetti developed. It is one of a family of models. Other important models in this family were developed by Richard Kahn, Nicholas Kaldor, and Joan Robinson. These models have been extended in various ways and presented in textbooks. One can see this family as having extended work by Roy Harrod, and as being related to the work of Michal Kalecki and even of Karl Marx.

2.0 The Model

2.1 Definitions

Consider a simple closed economy with no government. All income is paid out in the form of either wages or profits:

Y = W + P,

where W is total wages, P is total profits, and Y is national income. Total savings is composed of savings by workers and by capitalists, where capitalists are a class whose members receive income only from profits:

S = Sw + Sc

S is total savings. Sw is workers' savings, and Sc is capitalist savings. Profits are also split into two parts:

P = Pw + Pc,

where Pw is returns on the capital owned by the workers, and Pc is the return on the capital owned by the capitalists. The behavior assumption is made that both workers and capitalists save a (different) constant proportion of their income:

Sc = sc Pc
Sw = sw (W + Pw)

sc is the capitalists' (marginal and average) propensity to save. sw is the workers' (marginal and average) propensity to save. The propensities to save are assumed to lie between zero and one and to be in the following order:

0 ≤ sw < sc ≤ 1

Workers' savings are assumed to be insufficient to fund all the investment occurring along a steady-state growth path.

The value of the capital stock is divided up into that owned by the workers and by the capitalists:

K = Kw + Kc,

where K is the value of the capital stock, Kw is the value of the capital stock owned by the workers, and Kc is the value of the capital stock owned by the capitalists

2.2 Steady State Equilibrium Conditions

Along a steady-state growth path, in this model, all capital earns the same rate of profits, r:

r = P/K = Pc/Kc = Pw/Kw

It follows from the above set of equations that the ratio of the profits received from the workers to the profits received by the capitalists is equal to the ratio of the value of capital that each class owns:

Pw/Pc = Kw/Kc

Likewise, one can find the ratio of total profits to the profits obtained by the capitalists:

P/Pc = K/Kc

The analysis is restricted to steady-state growth paths where the value of the capitalists' capital and the value of the workers' capital is growing at the same rate:

S/K = Sc/Kc = Sw/Kw

The ratio of profits to savings is the same for the economy as a whole and for workers:

P/S = (P/K)/(S/K) = (Pc/Kc)/(Sc/Kc) = Pc/Sc

Or, after a similar logical deduction for workers:

P/S = Pc/Sc = Pw/Sw

Along a steady-state growth path, planned investment, I equals savings:

I = S
2.3 Deduction of the Cambridge Equation

The following is a series of algebraic substitutions based on the above:

P/I = P/S = Pc/Sc = Pc/(sc Pc) = 1/sc

Or:

P = (1/sc) I

The share of profits in national income is determined by the savings propensity of the capitalists and the ratio of investment to national income:

(P/Y) = (1/sc) (I/Y)

Recall that the rate of profits is the ratio of profits to the value of capital:

r = P/K = (1/sc) (I/K)

Recognizing that I/K is the rate of growth, g, one obtains the famous Cambridge equation:

r = g/sc

As long as the capitalists consume at least some of their income, the rate of profits is greater than the rate of growth along a steady-state growth path. And along such a path the share of income going to profits will be constant.

3.0 Discussion

If one assumes given investment decisions, the Cambridge Equation tells us what rate of profit is compatible with a steady state growth path in which the expectations underlying those investment decisions are satisfied.

Consider two steady states in which the same rate of growth is being obtained. Suppose that along one path workers have a higher propensity to save. Within broad limits, this greater willingness to save among workers has no effect on determining either the share of profits in income or the rate of profits. Only the capitalists' saving propensity matters for the steady state rate of profits, given the rate of growth. Would a capitalist economy have a tendency to approach such a growth path, given a sufficient length of time? I think such stability would entail the evolution of institutions, conventions, the labor force, and what is seen as common sense, including among dominant political parties.

The above model might have some relevance to current political economy discussions elsewhere.