Growth Strategies 



based on


The two sector model of growth which Marx formulated in his 'theory of reproduction' became a basis for development strategies in the Soviet Union and Eastern Europe. The following presentation of this theory builds on the models of simple and expanded reproduction presented earlier but changes the notation and terminology to facilitate understanding of results. Simple algebraic solutions are used and the whole process is illustrated by numerical simulations and graphs. 

As before the economy is assumed to consist of two sectors:

1. production of producer goods

                        2. production of consumer goods


X1 be the gross output of producer goods (at constant prices)

       X2 be the gross output of consumer goods (at constant prices)

X   be the aggregate gross value of output of the whole economy.

Obviously it must hold

                                                   X = X1 + X2

There are two factors of production: capital and labor. 

 K1 be the capital used in the production of producer goods 

be the
capital used in the production of consumer goods 

be the aggregate capital of the whole economy.

Clearly it must also hold

                                                   K = K1 + K2


It is assumed that capital is the only constraining factor of production. Labor is assumed to be abundant and can be increased without limits. We do not need therefore to bring labor into the model explicitly.

Production technology is characterized by fixed coefficients, i.e. fixed amounts of capital are needed to produce one unit of output in each sector.

Let  a1, a2  
be   the requirements of capital per unit of output

According to these definition we can write

(1)                    K1 = a1 X1,    K2 = a2 X2,


Further it is assumed that there is no distinction between capital goods and intermediate goods, i.e. capital lasts just one unit of time and it is fully worn out and depreciated at the end of the period. Each period's investment must replace the whole stock of capital worn out during that period and add some additional capital if growth is to occur. The gestation period is one unit of time, i.e. investment of the period t began to produce in the period t+1.

Let I  be  the total net investment in the economy.

It must be equal to the excess of the output of producer goods over the need to replace the worn out capital

  (2)                              I = X1 - K

The total net investment must be allocated in some way to the two sectors. The following identity must always hold

(3)                              I = I1 + I2


Net investment add to the existing stock of capital in each sector.

(4a)                              K1 t+1 = K1 t + I1 t

(4b)                              K2 t+1 = K2 t + I2 t

The main implication of the Marxian 'theory of reproduction' for a centrally planned economy is that planners decide how the net investment are allocated  between the producer goods and consumer goods sectors and by doing so control the rate of economic growth. According to this model the preferential allocation of investment to the first sector would speed up the overall rate of economic growth. In the short run some consumption would be sacrificed but the faster growth would increase the consumption in the long run. This can be seen from the following analysis:


This is an animated diagram of the two sector model. Notice that the output of each sector is dependent on labor and capital. The crucial parts of the model are allocations of labor and capital represented by red and green circles. However, in this section we assume that labor is not a limiting factor, so that the green loops are less important for determination of rates of growth. Allocation of investment (product of the sector 1) between two sectors remains the sole most important decision


 g be the ratio of I1 to K1.
g   =   ----

Investment create an increment of capital as indicated by (4a). It follows that capital of the first sector will grow by the rate g and because of fixed coefficients in (1a) the output of producer goods will grow also by the same rate g. According to (2) the growth of X1 will  result in increased total net investment and that will - with a lag - lead also to the growth of production of consumer goods. The policy advice to planners is "if you want to improve the standard of living of population in the long run, you must allocate investment preferentially to the production of producer goods and have the sector 1 grow in the short run faster than sector 2. The rate of growth of sector 2 will later catch up with the rate of growth of sector 1". This is the theoretical basis of the quest for industrialization with priorities given to heavy industry.


Let us now derive the growth equations of the model. 
As stated above :
                                               I1t = gt K1t

g is the desired ratio of investment to capital in sector 1.

Substituting this into (4a)
                                                   K1 t+1 = K1 t + gt K1t

        (5a)                              K1 t+1 =  (1  + gt )K1t

Dividing both sides of equation by a1 and using (1a) we get

    (5b)                                X1t+1  =  (1  + gt )X1t

is also the rate of growth of output in sector 1.


From (4b) we have

K2 t+1 = K2 t + I2 t 

from (3): I2 t  = I t - I1 t   so                

                                         K2 t+1 = K2 t + I t - I1 t

from  (2):
I X1 t - K1 t - K2 t so

  K2 t+1 = K2 t + X1 t - K1 t - K2 t   - I1 t 

I1t = gt K1t    and therefore       

                                           K2 t+1 = X1t  - (1 + gt )K1t

from (1)  K1 = a1 X1,  

                                           K2 t+1 = X1t - a1(1  + gt )X1t

which yields  

                                           K2 t+1 = [1 - a1(1 + gt )]X1t

Dividing the resulting equation by a2 we get the growth equation for output of sector 2

                                                [1 - a1(1 -gt)]X1t
(6)                         X2t+1 =   -------------------

Notice that output of consumer goods depends exclusively on technical coefficients, the decision parameter g and the output of producer goods in the preceding period. It explicitly does not depend on the past history of the production of consumer goods at all.



Now divide equation (5) by equation (6) to get

                                         X1 t+1
t+1=   ---------

                                         X2 t+1

This gave us the relation between the rate of growth of the sector 1 and economic structure (denoted as f) in the following time period. Again notice the peculiarity of this model: The ratio between the output of producer and consumer goods depends solely on technical coefficients and
the decision parameter g of the previous period. The past history of the production of producers and consumer goods is irrelevant.

Now if g were kept constant the proportion between outputs of the two sectors would also remain constant. In such a case sector 2 would grow by the same rate as sector 1. From that we can derive an expression for a steady (constant) rate of growth in both sectors.

                                          (1 - a1)ft - a2
(8)            g(
ft,a1,a2)    =   -----------------
                                            (a1a2 + a2)

This is to be understood in the following way: If the structural parameter 
ft  reaches certain value at time t than choosing the decision parameter g as indicated by (8) will make both sectors to grow by the same rate g. That will, naturally keep structure ft unchanged.


Equation (8) shows that a steady state growth rate, i.e. the growth rate that is equal for both sectors and constant in time depends only on technological coefficients a1, a2 and on the proportion of outputs ft

How the steady state growth rate depends on the proportion
ft and on values of technological parameters is shown in the following table and graph. Note that some of these rates are unrealistic as .5 means 50% rate. The two numbers in the heading of the table are the technical parameters a1 and a2. Obviously: more capital demanding technology slower the steady rate of growth is. Increased proportion of producer goods speeds the economic growth, but the marginal effect is small at very high values of ft.


    Steady State Rates of Growth 

 Technical coefficients  a1 , a2



We shall now explore three alternative
strategies of growth using numerical examples:

Technology: Initial conditions:
a1 = .8, X1, 0 = 151.879f0 = 3.16
a2 = .6, X2, 0 = 48.121






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