Growth, Accumulation, Distribution

From: Robert Vienneau (rvien_at_see.sig.com)
Date: 09/12/04 

Date: Sun, 12 Sep 2004 05:28:36 -0400

1.0 INTRODUCTION

   This post presents a formal model of a steady-state growth path, a
model that is unoriginal except in minor details. Expectations of the
agents in the model are being fulfilled along a steady-state growth
path. Further, no forces in the model are setting up conditions where
current expectations will not be satisfied in the future.

   The model traces out the implications of certain assumptions about
institutions. In particular, I assume that competitive corporations
produce commodities for the market. Capitalists and workers own these
corporations. Stock certificates are used to represent ownership. There
is a pure capitalist class whose entire income comes from ownership. The
workers save part of their income. Consequently, corporate profits are
a source of part of the income of the workers. Savings decisions and the
financing of investment follow certain rules of thumb. These decisions
are embodied in the model through certain parameters.

   The rate of growth is not determined within the model. It is
determined by the corporations. The proportion of profits retained
for financing investment and the proportion of investment financed
by issuing new stock are also under the control of the corporations.
These parameters determine the rate of profit in this model. Assuming
economic profit maximization, the rate of profit determines wages,
prices, and the coefficients of production along a steady-state growth
path. The rate of growth and the coefficients of production determine
how much of the capital good, steel, and how much of the consumption
good, corn, are produced per employed worker. Net investment, the value
of capital, profits, wages, and total income follow. Decisions
within the corporate sector determine all of these variables. The
workers and capitalists' savings propensities help determine the
personal distribution of income, including the proportion of stock
owned by the workers.

2.0 THE MODEL

  Two commodities, steel and corn, are produced in this model
economy. Steel can be combined with labor to produce either more steel
or corn. Corn is the consumption good. Steel is totally used up in the
production processes in the model. Thus, the output of the economy
consists of steel to replace the steel used up in production,
additional steel to support growth, and corn for consumption.

  Let X represent tons of steel used in a production cycle, and let p
represent the price of steel. The value of capital is given by
Equation 1:

             K = p X (1)

   Let g be the steady-state rate of growth. Steel output, corn
output, the employed labor force, and the value of capital all grow at
this rate along a steady-state growth path. Thus, net investment is
related to the value of capital by Equation 2:

            I = g K (2)

   Equation 3 relates accounting profits, P, to the value of
capital:

            P = r K (3)

Equation 3 is a relationship characterizing the return to industrial
capital. The returns to industrial capital and to financial
instruments are distinguished in this model. It is convenient to
follow tradition and refer to r as the rate of profits.

   Let w be the wage and N the number of employed person-years. Total
wages are then given by Equation 4:

           W = w N (4)

I later normalize the work force to unity.

   There is no government and no foreign trade in this model. Net
income, Y, is paid out in the form of wages and profits:

          Y = W + P (5)

Equations 1 through 5 are basic identities in this model.

2.1 FINANCE AND THE RATE OF PROFIT

   The rate of growth is not determined within this model. It is the
result of decisions by corporate managers and depends on their
optimism or pessimism. In short, the rate of growth depends on the
"animal spirits" of the corporate managers.

  Investment has two sources of finance in the model, retained
earnings and the issuing of new stock. This new stock is purchased
by households in the model. Earnings that are not retained are
paid out as dividents to stock owners. Let sc be the proportion
of retained profits. Let f be the proportion of net investment
financed by additional stock. Equation 6 follows:

            I = sc P + f I (6)

Some algebraic manipulation yields equation 7:

          (P/K) = [ (1 - f) / sc ] (I/K) (7)

One obtains Equation 8 from Equations 2, 3, and 7:

          r = [ (1 - f) / sc ] g (8)

Notice that all the parameters on the right-hand side of Equation 8
are controlled by corporate managers. Thus, the rate of profits on
a steady-state growth path is the result of corporate decisions
about the rate of growth and financing.

2.2 PRODUCTION

  Production occurs in the corporate sector. Firms producing steel
face the technology defined by the production function in Equation 9:

       1 = f1( a01, a11 ) (9)

where a01 is the number of person years labor hired per ton produced
steel, and a11 is the number of tons of steel used as input per ton
steel produced. The technology for producing corn is defined by the
production function in Equation 10:

       1 = f2( a02, a12 ) (10)

a02 is the number of person years labor hired per bushel corn produced.
a12 is the number of tons of steel used as input per bushel corn
produced.

   Production functions are assumed to exhibit constant returns to
scale and diminishing marginal returns. The inputs of steel are
purchased at the start of the year, and the outputs of steel and
corn become available at the end of the year. The corporations hire
labor for use throughout the year and pay wages at the end of the
year.

2.2.1 PRICE EQUATIONS

   The same rate of profits is earned along a steady-state growth path
in each industry:

       a11 p (1 + r) + a01 w = p (11)

       a12 p (1 + r) + a02 w = 1 (12)

Notice that corn is the numeraire.

   One can solve Equations 11 and 12 for the wage and the price of
steel. Equation 13 gives the wage-rate of profits curve for the
technique defined by a choice of coefficients of production.

               1 - a11 (1 + r)
    w = -------------------------------- (13)
          (a01 a12 - a02 a11)(1 + r) + a02

Equation 14 gives the price of steel, given coefficients of production
and the rate of profits:

                     a01
    p = -------------------------------- (14)
          (a01 a12 - a02 a11)(1 + r) + a02

   The choice of technique is determined by appending marginal
productivity equations to either the system of equations 11 and
12 or to the equivalent set of equations 13 and 14. Marginal
productivity equations follow from assuming that firms minimize
cost, given technology. Alternatively, one can assume that firms
maximize economic profits. Equation 15 shows that the wage is
equal to the value of the marginal product of labor in producing
steel:

           del f1
    w = p ------- (15)
           del a01

The wage is also equal to the value of the marginal product of
labor in producing corn:

        del f2
    w = ------- (16)
        del a02

The price of steel, discounted to the end of the year when output
becomes available, is equal to the value of the marginal product
of steel in producing steel:

                    del f1
    p (1 + r) = p ------- (17)
                    del a11

Similarly, the discounted price of steel is equal to the value of
the marginal product of steel in producing corn:

                  del f2
    p (1 + r) = ------- (18)
                  del a12

   This analysis of the price equations shows that, given the rate
of profits, coefficients of production, the wage and prices of
commodities are determined by cost-minimization. Notice no equation
exists equating the marginal product of the value of capital, K, and
the rate of profits. Marginal productivity conditions are part of
the determination of the choice of technique. They do not determine
distribution.

2.2.2 QUANTITY EQUATIONS

   There are X tons of steel used as inputs in production in a single
year. By assumption, [ ( 1 + g ) X ] tons are produced at the end of
the year. Let c be the bushels of corn produced and available at the
end of the year. Equation 19 relates the steel used in production
to the quantities of final outputs:

     X = a11 (1 + g) X + a12 c (19)

The size of the employed labor force, normalized to unity, is related
to final outputs by Equation 20:

     1 = a01 (1 + g) X + a02 c (20)

Notice that c is the amount of corn produced per worker, and X is tons
steel used per worker.

   One can solve Equations 19 and 20 for c and X as functions of the
rate of growth and the coefficients of production. Equation 21 shows
the trade-off between per-capita consumption and the rate of growth:

               1 - a11 (1 + g)
    c = -------------------------------- (21)
          (a01 a12 - a02 a11)(1 + g) + a02

Notice this trade-off is of the same form as the wage-rate of profits
curve (Equation 13). Steel per worker is given by Equation 22:

                     a12
    X = -------------------------------- (22)
          (a01 a12 - a02 a11)(1 + g) + a02

   This completes the analysis of the corporate sector. I have shown
how the rate of profit, the wage, prices, coefficients of production,
and quantities produced per worker are determined in this model. The
accounting identities can be used to calculate the value of capital,
net investment, net income, and the distribution between profits and
wages. The distribution of income between persons requires an
analysis of the financial sector and of households.

2.3 THE VALUATION RATIO, DIVIDENDS, CAPITAL GAINS, AND THE INTEREST RATE

   The value of the capital goods owned by corporations is K.
Households do not own these capital goods. Rather, they own stock.
The market value of stock is (v K). The ratio of the market value
of stocks to the value of the capital goods used by corporations
(the "book value") is known as the valuation ratio, v. I assume
that the valuation ratio is constant along a steady-state growth
path; variations in the valuation ratio reflect short-term
speculation.

   The steady-state valuation ratio cannot be below unity.
Otherwise, corporations would expand not by purchasing capital
goods. Rather, they would purchase financial assets. Since
households do not typically buy blast-furnaces and other real
capital goods, the valuation ratio can exceed unity.

   Profits not retained by the corporate sector are paid out to
households in the form of dividends. By assumption, dividends
are [ (1 - sc) P ].

   The increase in the value at the end of a year of the capital
goods owned by corporations is net investment, I. The increase
in the value of all shares is (v I), while the value of new
shares sold is (f I). This latter quantity is needed to finance
that portion of net investment not covered by retained profits.
The difference in these quantities, [ (v - f) I ], is the increase
in the value of old shares. In other words, [ (v - f) I ] is the
value of capital gains.

   Interest payments on financial capital are such that one can
consume them entirely and leave one's capital value unaltered. Thus,
interest payments in this model are equivalent to the sum of
dividends and capital gains. The rate of interest is then defined
by Equation 23:

           (1 - sc) P + ( v - f ) I
     i = ------------------------ (23)
                     v K

Some algebraic manipulations and the definitions of the rate of
profits and the rate of growth given by Equations 3 and 2,
respectively, yield Equation 24:

     i = g + [ r - ( sc r + f g ) ]/v (24)

Substituting from Equation 8, I obtain

     i = g + ( r - g )/v (25)

Or,

     v = ( r - g ) / ( i - g ) (26)

Assume that the rate of interest is greater than the rate of growth.
Then, a valuation ratio greater than unity is equivalent to
the rate of profits on industrial capital exceeding the rate of
interest on financial capital along a steady state growth path.

2.4 SAVINGS DECISIONS AND INCREASES IN WEALTH

   I assume the existence of a class of workers and a class of
capitalists (rentiers) in the steady state. The workers save
and therefore obtain a share of profits. The rentiers obtain
all their income from profits.

   If both classes are to persist, the wealth of each class
must grow at the steady-state growth rate, g. Equation 27
equates the rate of growth, g, and the rate of growth of
rentier wealth:

         ( 1 - j ) sr [ ( 1 - sc ) P + ( v - f ) I ]
   g = ------------------------------------------- (27)
                   ( 1 - j ) v K

where sr is the capitalists' (marginal and average) savings
propensity and j is the proportion of stock owned by the
workers. Equation 28 relates the steady-state rate of growth
and the rate of growth of workers' wealth:

         sw W + j sw [ ( 1 - sc ) P + ( v - f ) I ]
   g = -------------------------------------------- (28)
                        j v K

where sw is the workers' (marginal and average) savings propensity.

   Equations 27 and 28 determine the valuation ratio and the
proportion of stock owned by the workers by the already determined
variables the wage, profits, net investment, the value of capital
and the parameters the rate of growth, the proportion of net
investment financed by additional stock, the retention ratio, and
the workers' and capitalists' savings propensities.

2.5 PARAMETER RANGES

   Certain parameters must lie within certain ranges for a
steady-state growth path to exist in the above model. Given
a positive price of steel, Equation 29 follows from Equation
11:

       a01 w / p = 1 - a11 ( 1 + r ) (29)

Then the wage is positive if and only if:

           1 - a11 ( 1 + r ) > 0 (30)

Or:

          r < ( 1 - a11 ) / a11 (31)

By similar reasoning based on Equation 19, outputs of corn
and steel are positive if and only if Inequality 32 holds:

          g < ( 1 - a11 ) / a11 (32)

Displays 31 and 32 show a limit on the rate of profit and
on the rate of growth. This limit is imposed by a parameter of
the chosen technique. Inequality 33 must hold for the rate
of profit and the rate of growth to be positive:

            0 < a11 < 1 (33)

   The model imposes a necessary condition on savings propensities.
Define sh as a weighted savings propensity:

           sh = sw j + sr ( 1 - j ) (34)

Total savings, S, is the sum of savings from profits and savings
from wages:

                  Savings From Profits

                      Savings from Consumption Savings
      Savings From Retained From From
        Dividends Earnings Capital Gains Wages

 S = sh ( 1 - sc ) P + sc P - ( 1 - sh ) ( v - f ) I + sw (Y - P)
                                                                  (35)

Since intended savings equals intended investment in a steady state,
Equation 36 must hold:

  I = [ sc + sh ( 1 - sc ) - sw ] P - ( 1 - sh ) ( v - f ) I
                                                         + sw Y (36)

Or

P [ 1 + ( 1 - sh ) ( v - f ) ] I sw
- = ---------------------------- - - --------------------------- (37)
Y [ sc + sh ( 1 - sc ) - sw ] Y [ sc + sh ( 1 - sc ) - sw ]

I assume both profit and wage shares are positive:

          0 < ( P / Y ) < 1 (38)

Display 39 gives a necessary condition to ensure positive shares:

          sw < sc + sh ( 1 - sc ) (39)

This condition is that the savings rate out of wages is less than
the sum of the proportion of retained earnings and the savings rate
out of profits distributed as dividends.

3.0 SUMMARY

   The structure of the above model is summarized in the figure below.
The arrows indicate how the variables are determined or, in some
cases, indicate accounting identities. Notice that the major
macroeconomic variables are determined by decisions within the
corporations, while savings decisions can affect only the overall
value of stocks and the personal distribution of income. Control
over the means of production determines the functional distribution
of income.

     Rate of growth,
     Retention ratio, -----------------------> Rate of Profit
 +- Proportion of investment | |
 | externally financed | |
 | | | | |
 | | | | |
 | | \|/ \|/ |
 | | Corn produced per worker, Wage, Price of Steel, |
 | | Steel produced per worker <------- Coefficients of |
 | | | production |
 | | | | | |
 | | | +------------------------+ +-------+ |
 | | | | | |
 | | | | +------------------------ | ----+
 | | | | | |
 | \|/ \|/ \|/ | |
 | Net Value of \|/ Net \|/
 | investment <--- capital ---> Profits ----> income <---- Wages
 | | | | |
 | | | | +--------------------+
 | | | | |
 | \|/ \|/ \|/ \|/
 +-> Ratio of value of stocks to book value, Saving propensities
     Distribution of stocks between rentiers <---- of rentiers and
       and workers workers

   The above model is one of a family of models. The Pasinetti variant
results from removing the corporate sector (sc = 0, f = 1, v = 1). The
Kaldor variant contains no class of pure capitalists/rentiers (sr = 0,
j = 1). A generalization would assume saving propensities vary by source
of income, as well as class.

   A number of research programs can be organized around these models.
One might generalize the production model to more goods. One might
assume industries exhibit various barriers to entry. This is Paolo
Sylos Labini's (old) Industrial Organization. The model shows the rate
of industrial profits as depending on corporate decisions about growth
and finance. Related issues are explored in managerial theories of the
firm, as developed by Edith Penrose, Robin Marris, Adrian Wood, Alfred
Eichner, and Edward Nell. I don't think this family of models is
particularly strong on environmental issues; nevertheless, the extension
of the production model to include joint production provides an essential
tool for environmental economics. Some work has been done on including
government and international trade in this family of models. One might
see this theory of steady-state growth as being a long period extension
of Keynes' General Theory; those who emphasize the role of money and
uncertainty in Keynes' theory have questioned whether such elements have
yet been adequately incorporated into this family of models. These
are just a few indications of the research that has been or can be
associated with this model.

4.0 REFERENCES

Nicholas Kaldor, "Alternative Theories of Distribution," _Review of
Economic Studies_, V. XXIII, pp. 83-100, 1956.

Nicholas Kaldor, "Marginal Productivity and the Macro-Economic
Theories of Distribution: Comment on Samuelson and Modigliani,"
_Review of Economic Studies_, V. XXXIII, pp. 309-319, 1966.

Scott J. Moss, "The Post-Keynesian Theory of Income Distribution in
the Corporate Economy," _Australian Economic Papers_, V. 17, pp.
303-322, 1978.

Luigi Pasinetti, "Rate of Profit and Income Distribution to the
Rate of Economic Growth," _Review of Economic Studies_, XXIV, pp.
267-279, 1962.

Luigi Pasinetti, _Growth and Income Distribution: Essays in Economic
Theory_, Cambridge University Press, 1974. 

-- 
Try http://csf.colorado.edu/pkt/pktauthors/Vienneau.Robert/Bukharin.html
                            To solve Linear Programs:  .../LPSolver.html
r           c                                 A game:  .../Keynes.html
 v         s a           Whether strength of body or of mind, or wisdom, or
  i       m   p          virtue, are found in proportion to the power or wealth
   e     a     e         of a man is a question fit perhaps to be discussed by
    n   e       .        slaves in the hearing of their masters, but highly
     @ r         c m     unbecoming to reasonable and free men in search of
      d           o      the truth.    -- Rousseau

Relevant Pages

  • Was Q3 GDP growth manipulated upwards because of the coming elections or is the US government cluele
    ... The first estimate of US Q3 GDP growth came out at a dismal 1.6% but, ... Oct. 27 -- An unexpected increase in auto production last ... Without the gain, the economy would ...
    (soc.culture.vietnamese)
  • Re: The writing is on the wall
    ... Reserves divided by current production rate is useLESS. ... Russia's oil ... Industrialism can work just fine without growth. ...
    (rec.martial-arts)
  • Re: Space-based power makes CNN.com - comments may be entertaining
    ... energy consumption equals the revised amounts of production capacity ... Rather than expand the production of synthetic ... it is also true that unlimited growth in human numbers and ... systems - telerobotics and automation - to transform life on Earth, ...
    (sci.space.policy)
  • Singapores Full-year growth to Beat Forecast
    ... Singapore's economy in the third quarter grew by an estimated 6 per ... would benefit from increased pharmaceutical production. ... The government also revised upward its growth figure for gross domestic ... dependence on electronics, its biggest export sector. ...
    (soc.culture.singapore)
  • Re: Fine work by our new Nobelist
    ... >>profits). ... Wrong it is the valuation of assets not how much is currently spent. ... Acquiring useless intangibles do not add to the intangible stock. ... In context of growth models which Presscot loosely bases the anaylsis on ...
    (sci.econ)