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**1.0 Introduction**

This post presents elements of a model of a smoothly reproducing economy, that is, of an economy growing along at the warranted growth rate. I have previously presented a more detailed exposition of a variant of this model. One could add, say, Harrod-neutral technical change to that exposition. I would find it easier to add biased technical change by assuming fixed, not variable, coefficients of production. Perhaps this model reflects conventions and the balance of class forces prevalent in Anglo-American economies after World War II and before the collapse of the Bretton Woods system.

Anyways, I am revisiting this model because, recently, I have noticed another mathematical property of this model. Not only are the determinants of the rate of profits along a warranted growth path independent of the decisions of the workers to save. So is the average stock price of corporations.

**2.0 The Model**

This model abstracts from the existence of government spending and taxation. It also treats foreign trade as negligible. National income is comprised of wages, *W*, and profits, *P*. The rate of profits, *r*, is the ratio of profits to the value of capital goods, *K*, used in producing national income.

**2.1 The Corporate Sector**

I begin with corporations. The corporations own the capital goods and hire the workers to produce output with these capital goods. Corporate managers decided on the level of investment, *I*, to achieve a target growth rate, *g*.

Investment, in this model, is financed by some mixture of retained profits and the issuance of new stock (also known as shares) on the stock market. Corporate managers decide on this mix. Let *s*_{c} be the proportion of profits that are retained to finance new investment. And let *f* be the proportion of investment financed by issuing new shares:

I=s_{c}P+fI

Some algebra yields:

P/K= [(1 -f)/s_{c}] (I/K)

Or:

r= [(1 -f)/s_{c}]g

Thus, the rate of profits consistent with a warranted rate of growth is determined by parameters characterizing decisions made by corporate managers.

**2.2 Finances and Households**

In this model, households do not own capital goods. Rather, corporations own capital goods, and households own stock in these corporations. The ratio of the market value of stock to the value of the capital goods owned by the corporations is called the *valuation ratio*, *v*. The valuation ratio is assumed constant along a warranted growth path. Variations in the valuation ratio reflect short-term speculation. Generally, the valuation ratio is above unity.

Households are divided into two classes in this model, workers and capitalists. Workers receive part of their income in the form of wages. Given a positive savings rate on the part of workers, they also receive dividends and capital gains from their stock. Capitalists do not labor; their households receive all their income from dividends and capital gains. The variable *j* is used to denote the proportion of stocks owned by the workers.

Dividends consist of profits received and not retained by the corporations. By assumption, the value of dividends is then (1 - *s*_{c})*P*. Net investment, *I*, is the increase in the value over a year of the capital goods owned by corporations, while the increase in the value of stocks is *v**I*. But the value of new shares is only *f**I*. The difference, (*v* - *f*)*I*, is the value of capital gains.

The interest rate is the ratio of the returns to financial capital (that is, dividends and capital gains) to the value of stock. With a valuation ratio above unity the interest rate, *i*, falls below the rate of profits. The valuation ratio then becomes:

v= (r-g)/(i-g)

I assume workers typically save at the rate *s*_{w}, and capitalists typically save at the greater rate *s*_{r}. Table 1 shows sources of savings, based on these definitions and behavioral assumptions.

Source | Amount |

Retained Earnings: | s_{c} P |

Capitalist Savings Out of Dividends: | (1 - j)s_{r}(1 - s_{c})P |

Minus Capitalist Consumption Out of Capital Gains: | - (1 - j)(1 - s_{r})(v - f)I |

Worker Savings Out of Wages: | s_{w}W |

Worker Savings Out of Dividends | j s_{w}(1 - s_{c})P |

Minus Worker Consumption Out of Capital Gains: | - j(1 - s_{w})(v - f)I |

In adding up savings, one must be sure not to double-count retained earnings. Corporations decide to save retained earnings, but households can undo this decision by consuming capital gains. Total savings for capitalists, *S*_{r}, are:

S_{r}= (1 -j)s_{r}[(1 -s_{c})P+ (v-f)I]

Total savings for workers, *S*_{w}, are:

S_{w}=s_{w}W+js_{w}[(1 -s_{c})P+ (v-f)I]

Along a warranted growth path, investment is always equal to savings. The following equation is based on the components in Table 1:

I=s_{c}P+ (1 -j)[s_{r}(1 -s_{c})P- (1 -s_{r})(v-f)I]

+s_{w}W+j[s_{w}(1 -s_{c})P- (1 -s_{w})(v-f)I]

A bit of algebra allows the investment-savings equality to be restated:

I=s_{c}P+S_{r}+S_{w}- (v-f)I

The last term (that is, capital gains) is subtracted to avoid double-counting.

Another condition of a warranted growth path in this model is that the corporate sector, capitalist households, and workers continue to endure. This condition requires that the rate of growth of the book-value of the capital goods held by the corporations, the rate of growth of the value of the stock held by capitalists, and the rate of growth of the value of the stock held by the workers all be equal. Thus, the rate of growth of the value of the stock held by capitalists is:

g=S_{r}/[(1 -j)vK]

The rate of growth of the value of the stock held by workers is:

g=S_{w}/(jvK)

This completes the exposition of the equations I need for my point here.

**2.3 Some Algebra**

I now report on some algebraic manipulations of these equations. The condition that the value of the stock held by capitalists and workers grows at the same rate yields the following condition:

S_{w}=S_{r}[j/(1 -j)]

Substituting in the investment-savings equality, one can obtain:

I=s_{c}P+ [S_{r}/(1 -j)] - (v-f)I

Or, by expanding the definition of capitalist savings:

I=s_{c}P+s_{r}[(1 -s_{c})P+ (v-f)I] - (v-f)I

Regrouping yields:

[1 + (1 -s_{r})(v-f)]I= [s_{c}+s_{r}(1 -s_{c})]P

Dividing through by the book value of the capital goods owned by the corporations, one obtains:

r= {[1 + (1 -s_{r})(v-f)]/[s_{c}+s_{r}(1 -s_{c})]}g

Equating for the value of the rate of profits previously found, one obtains an expression for the valuation ratio in terms of model parameters:

v= {[s_{r}(1 -s_{c})]/[s_{c}(1 -s_{r})]} - {s_{r}/[s_{c}(1 -s_{r})]}f

Notice the parameters on the right-hand-side characterize either corporate decisions or the decisions of capitalist households. The saving propensities of the workers do not enter into it. The more that corporations finance investment by issuing shares, instead of using retained earnings, the lower the valuation ratio is along a warranted growth path. If the proportion of profits distributed in dividends lies below the proportion of investment financed by issuing new stock, a smaller capitalist savings propensity is associated with a higher valuation ratio. In some sense, capitalists get what they spend.

**3.0 Conclusions**

This post has outlined some necessary properties of a warranted growth path in a model containing:

- Corporations, a capitalist class, and a class of workers.
- A stock market, in which ownership shares in the corporations are bought and sold.
- A growth rate determined by decisions of the corporate managers.

In this model, the decisions of the corporate manager as to the growth rate, retained earnings, and finance obtained by issues of new stock determine the rate of profits consistent with a warranted growth path. These decisions of the corporate managers, along with the savings propensities of the capitalists, determine the ratio of the price of stock to the book value of the capital goods owned by the corporations. A fortiori, these decisions also determine the interest rate. Within the limits where a warranted growth path exists, the savings propensities of the workers have no effect on the growth rate, the rate of profits, the price of stock, the interest rate, or the functional distribution of income. The savings decisions of the workers do affect, however, the personal distribution of income and the proportion of stock owned by the workers.

**Appendix: Variable Definitions**

*K*is the book value of the capital goods, in numeraire units, owned by the corporations.*I*is investment, in numeraire units.*P*is corporate profits, in numeraire units.*S*_{r}is capitalist savings, in numeraire units.*S*_{w}is worker savings, in numeraire units.*f*is the proportion of investment financed by issuing new stock (also known as shares).*g*is the warranted rate of growth.*i*is the interest rate.*j*is the proportion of stock owned by workers.*r*is the rate of profits earned by the corporations on the book value of their capital stock.*s*_{c}is the proportion of profits retained by corporations.*s*_{r}is the (average and marginal) to save of the capitalists.*s*_{w}is the (average and marginal) to save of the workers.*v*is the valuation ratio, that is, the ratio of the value of the stocks of the corporations to their book value.

**Reference**

- Scott J. Moss (Dec. 1978). The Post-Keynesian Theory of Income Distribution in the Corporate Economy,
*Australian Economic Papers*, V. 17, N. 31: pp. 302-322.

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