## Summary

## Introduction

*r=g*/*s _{c}*, (1)

where *r* is the profit rate, *g* is the growth rate of aggregate output and income, and *s _{c}* is the capitalists’ propensity to save.

The equation was derived under the assumption of balanced growth, when the individual wealth of households (capitalists and workers) remains proportional, that is, the wealth inequality remains constant. However, since the 1980s the wealth inequality is actually increasing, particularly in developed countries. In his famous book Piketty (2014) concluded that such growth is a consequence of the inequality *r*>*g *(more precisely,* r*>*g*/*s _{c}*,[1] that is non-compliance with the Cambridge equation). Thus, the balanced growth with a constant level of wealth inequality is a consequence of adhering to the Cambridge equation, and not vice versa.

The Cambridge economists did not consider carefully the case of non-compliance with their equation. Our paper aims to fill this gap. We will study the process of accumulation of individual fortunes in the absence of the assumption of the balanced growth; equation (1) does not have to be true.

[1] Piketty considered the simplified inequality *r>g* as a condition for the growth of inequality, but if you carefully read the text, he had in mind exactly r> *g*/*s _{c}*, and simplified the formula. However, the simplified Piketty inequality does not greatly sin against the truth; the owners of the largest fortunes are not able to consume a significant part of their huge income, thus their saving rate

*s*≈ 1.

_{c}

## Accumulation of individual fortunes

Consider the process of wealth *W _{i}* accumulation of a separate

*i*-th household. Obviously, it will change due to the net savings

*s*of this household having income

_{i}Y_{i}*Y*:

_{i}*W _{i}*(

*t+*Δ

*t*)

*= W*(

_{i }*t*)

*+s*(

_{i}Y_{i}*t*)Δ

*t= W*(

_{i }*t*)

*+ s*(

_{i}*w*(

_{i}*t*)

*+r W*(

_{i}*t*))Δ

*t*, or if Δ

*t→*0,

*dW _{i}*(

*t*)/

*dt= rs*(

_{i}W_{i }*t*)+

*s*(

_{i}w_{i}*t*) (2)

Let’s make a change of variables *X _{i}= W_{i}*(

*t*)/

*Y*(

*t*) and differentiate the value

*X*with respect to time. Taking into account the obvious relation

_{i}*dY*/

*dt=gY*and equation (2), we have:

*dX _{i}*/

*dt*=(1/

*Y*(

*t*))

*dW*(

_{i}*t*)/

*dt*–(

*W*(

_{i}*t*)/

*Y*(

*t*)

*)*

^{2}*dY*(

*t*)/

*dt=*[

*rs*(

_{i}W_{i}*t*)+

*s*(

_{i}w_{i}*t*)]/

*Y*(

*t*)

*–gY*(

*t*)

*W*(

_{i}*t*)/

*Y*(

*t*)

^{2}=

=*s _{i}w_{i}*(

*t*)/

*Y*(

*t*)+[

*rs*]

_{i}– g*X*(

_{i}*t*)

*(3)*

We assume that the population is constant and wages and output grow at the same rate *g*, which means *s _{i}w_{i}*(

*t*)/

*Y*(

*t*)=

*const*. Then the solution of the last differential equation is:

*X _{i}= W_{i}* (

*t*)/

*Y*(

*t*) =

*s*(

_{i}w_{i}*t*)/[

*Y*(

*t*)(

*g–rs*)]

_{i}*+const*(

_{1i}×exp*–t×*(

*g– rs*)) (4)

_{i}The constants in the last equation are determined from the initial conditions:

*const _{1i} =*[

*W*(0)(

_{i}*g–rs*)

_{i }*–s*(0)]/[

_{i }w_{i}*Y*(0)(

*g–rs*)]=[(

_{i}*gW*(0) –

_{i}*s*(0)]/[

_{i}Y_{i}*Y*(0)(

*g–rs*)] (4a)

_{i}If the condition *rs _{i}>g* is satisfied for the

*i*-th household, then the second term on the right side of equation (4) experiences exponential growth over time. Therefore, the equity capital

*W*of this household will grow at a faster rate than output

_{i}*Y*. In this case, we can neglect the first (labor) term of equation (4) in the long term. Then we will say that the second (exponentially growing) term is responsible for the capital component of the accumulation of the individual wealth

*W*.

_{i}On the contrary, if the inequality *rs _{i}<g* holds for the

*i*-th household, then the exponential term on the right side of equation (4) tends to zero at

*t*>>1/(

*g– rs*), so the ratio

_{i}*X*of this household’s capital to income asymptotically tends to a constant value:

_{i}*W _{i}* (

*t*)/

*Y*(

*t*) =

*s*(

_{i}w_{i}*t*)/[

*Y*(

*t*) (

*g–rs*)]=

_{i}*const*(5)

_{2i}The constants are determined from the initial conditions:

*const _{2i}= s_{i}w_{i}*(

*0*)/[

*Y*(

*0*) (

*g–rs*)] (5a)

_{i}According to equation (5), the equilibrium value of the wealth *W _{i}*, accumulated by the

*i*-th household during long-term economic growth, is directly proportional to the propensity to save and wages and does not depend on the initial value of wealth. The stability of the

*W*/

_{i}*Y*ratio means the possibility of a long-term scenario of balanced economic growth with a constant level of wealth inequality. In this case, individual propensities to save

*s*of households can differ from each other over a wide range. However, if they are all constant in the long run, and satisfy the inequality

_{i}*s*, then all the corresponding individual fortunes will grow at the same rate g, which is the growth rate of aggregate income. Therefore, the ratio of individual fortunes of any two households

_{i}<g/r*W*/

_{i}*W*will also remain unchanged. That is, wealth inequality does not increase. In this case, wealth inequality for such households is proportional to inequality in labor income, taking into account individual propensities to save, regardless of initial wealth conditions.

_{j}## Two scenarios of wealth accumulation (*rs*_{i}<g or *rs*_{i}>g)

_{i}<g

_{i}>g

The first scenario, when the condition *rs _{i}<g* is satisfied for all households, is socially fair, but it looks unlikely in a capitalist society. Indeed, any capitalist, even having a large initial capital, will subsequently squander most of it due to the low profit rate. Simply put, the capitalist will turn into a worker. Therefore, there will be no incentive to invest for profit, which is the driving force of capitalism. Making a profit from a goal turns into a hobby that does not have a fundamental impact on the welfare of the entrepreneur.

Nevertheless, such a scenario (balanced growth when *rs _{i}<g*) is theoretically possible. It may seem that it contradicts the conclusions of Kaldor and Pasinetti. From their point of view,

*rs*must be observed for the balanced growth. However, there is no contradiction, because in the case we are considering here, all households receive both capital income and wages. In the model of Cambridge economists, the capitalists do not receive a salary. Then, if the condition

_{i}=g*rs*is met, they will go bankrupt in the long run and disappear as a class. However, while receiving wages, at

_{i}<g*rs*the capitalists will also disappear, by turning into workers.

_{i}<gSo, in the case when the condition *rs _{i}<g* is satisfied for all households, there will be a bias in income distribution in favor of workers. Capitalists will have no economic incentive to entrepreneurship.

Let us consider the second scenario, when the condition *rs _{i}>g* is satisfied for a number of capitalist households. This means that their unconsumed income

*s*of such households exceeds the level of

_{i}rW_{i}*gW*, required for the balanced growth of their wealth. Consequently, the individual fortunes of these capitalists will grow faster than the total income. Therefore, their wealth will grow relative to the fortunes of workers, for which the condition

_{i}*rs*will be satisfied. After all, the wealth of workers will grow at the same rate as their wages (if their propensity to save is unchanged), which, in turn, is equal to the rate of growth of the total income. Balanced growth is impossible; inequality will inevitably rise. Total income is distributed primarily in favor of the capitalists.

_{i}<g

## Concluding remarks

We considered two scenarios: the first, when *rs _{i}*<

*g*for all households, and the second, when

*rs*>

_{i}*g*for the largest capitalists’ fortunes. In the first scenario, there is a bias in the distribution of income in favor of workers. The profit of the capitalists is insufficient for a balanced business development. In the second scenario, the profits of the capitalists exceed their immediate development needs, which causes the accelerated growth of their wealth. For this scenario, the income distribution has a bias in favor of the capitalists. Then, the both scenarios do not look perfect, because they do not provide a fair (without distortions) distribution of total income.

The Kaldor-Pasinetti Cambridge equation (*rs _{i}*=

*g*) corresponds to the border between the two considered scenarios. Hence, it describes the case when the total income is distributed fairly between workers and capitalists. A balanced growth take place in this case, when the wealth of workers and of capitalists grow at the same pace. The Cambridge equation draws a line below which (

*rs*<

_{i}*g*for all households) capitalists die out as a class in the long term. Above this limit (

*rs*>

_{i}*g*for a number of capitalists) the growth of inequality is inevitable. This is precisely the meaning of the Kaldor-Pasinetti equation from our point of view.

It is clear that exact observance of the equation is hardly possible for all capitalists. However, a relatively fair distribution of total income is possible only if the growth rate of the largest fortunes is limited by the value of GDP growth rate.

Let us note that the Cambridge equation means the equality of not consumed profits and investments in the real sector of the economy necessary for balanced growth, *s _{c}rK=gK*. This equality is the same as the equality of savings and investment, if we adhere to the classical hypothesis (capitalists invest their profits and workers consume their wages; all capital

*K*belongs to the capitalists). Matching savings and investment is essential for sustainable growth according to the Keynissian approach (Keynes, 1936).

In our opinion, the Cambridge equation corresponds to the optimal and fair scenario for economic growth. This equation declares the proportionality between profit and investment (equivalent to the proportionality of the profit rate and the growth rate). It is interesting that similar proportions of the interest rate and the growth rate were obtained earlier as a result of the search for the optimal (maximizing consumption) path of growth. This is Phelps’ golden rule (Phelps 1961) and Ramsey’s formula (Ramsey, 1928).

## References

Kaldor, N. (1955-1956), “Alternative theories of distribution”, *The Review of Economic Studies*, 23, 94-100.

Kaldor, N. (1963) “Capital Accumulation and Economic Growth”. In Friedrich A. Lutz and Douglas C. Hague, eds., Proceedings of a Conference Held by the International Economic Association. London: Macmillan.

Keynes J. M., 1936. “The General Theory of Employment, Interest, and Money”, New York: Harcourt.

Pasinetti L. (1962) “Rate of Profit and Income Distribution in Relation to the Rate of Economic Growth”, Review of Economic Studies 29(4), 267–79.

Pasinetti L., (2000) “Critique of the Neoclassical Theory of Growth and Distribution”, *Banca Nazionale del Lavoro Quarterly Review*, 53(215), 383—431.

Phelps, E. S. (1961) “The Golden Rule of Accumulation: A fable for growthmen”, *American Economic Review,* 51, 638-643.

Piketty T., 2014. “Capital in the 21st Century” (Cambridge, MA: Harvard University Press).

Ramsey F. P., 1928. “A Mathematical Theory of Saving”, *The Economic Journal*, Vol. 38, No. 152 (Dec., 1928), pp. 543-559

Ricardo, D., [1817], (1951), *On the Principles of Political Economy and Taxation*, P. Sraffa, ed. with the collaboration of M. Dobb, “*The Works and Correspondence of David Ricardo*”, Cambridge, Cambridge University Press.