MARX MODEL OF REPRODUCTION

Marx's models or 'schemes' of reproduction were inspired by 'Tableaux  economique' of the 18th century economist physiocrat Francois Quesnay.

Notation:
X ... gross value of output
C ... cost of used producer goods - 'constant capital'
V ... wage cost - 'variable capital'
M ... profits - 'surplus value'

subscripts:
t ... time
1 ... sector 1 - production of producer goods
2 ... sector 2 - production of consumer goods

X1t                   ... output of producer goods in period t
X2t                   ... output of consumer goods in period t

(1)        Xt  = X1t  + X2t           

... total output in period t

C1t                   ... use of producer goods in production of X1t

C2t                   ... use of producer goods in production of X2t 

(2)        Ct  = C1t  + C2t            

... total use of producer goods

 V1t                  ... wages paid in sector 1

 V2t                  ... wages paid in sector 2

(3)        Vt  = V1t  + V2t            

... total wages in both sectors

M1t                  ... profits in sector 1

M2t                  ... profits in sector 2

(4)        Mt  = M1t  + M2t            

... total profits

Scheme of reproduction:

(5)       X1t  = C1t  + V1t  +  M1t

(6)       X2t  = C2t  + V2t  +  M2t

          --------------------------------

(7)       X  =  C  +  V  +  Mt

Types of reproduction:

simple..........output remains constant - no growth;

expanded ....growth of output

assumptions: 
   output of each sector is constrained by the available producer goods,
   there is always enough labor to be hired
   output of producer goods produced in period t is used as input in period t+1;
   consumer goods are consumed in the same period they were produced.
   technology remains unchanged
   workers do not save.

 

Simple Reproduction:

To maintain constant level of output in both sectors it is necessary to reproduce the same amount of producer goods that was used in the production of both commodities.1t =  Ct

Because output of producer goods of the period t will be used as input in the period t+1

(9)                    Ct+1  =  X1t

and therefore

(10)                  C t+1  =  Ct

 the total amount of producer goods remains unchanged, allowing for reproduction of the same quantity of both producer and consumer goods in the period t+1. 

Note, that this condition is necessary but not sufficient for simple reproduction. To keep the production in both sectors unchanged, producer goods must be allocated between sectors in the same proportion as in the previous period. 

(11)                  C 1t+1 /Ct+1  =  C1t /Ct

Conditions (10) and (11) guarantee

(12)                C 1t+1    =  C1t

(13)                C 2t+1    =  C2t

To see that both (8) and (12),(13) are necessary conditions for simple reproduction consider the situation in which (8) is valid but more producer goods would be allocated to the production of consumer goods than in the previous period. Obviously there would be less producer goods in the sector 1 and its output would decline. Only if each of the sectors uses exactly the same amount of producer goods as in past, the production of both sectors is reproduced on the same unchanged level.
In the simple reproduction the output of consumer goods must be equal to the sum of total wages and profits.

(14)                  X2t  =  Vt + Mt

As workers do not save by assumption, it implies that capitalist must not save either. They both must spend all their income on purchases of consumer goods. 

period 1       
 

 

   X            C          V         M
Sector 1        90  =   50  +   20  +  20
Sector 2      100  =   40  +   30  +  30
  -------------------------------
Total           190  =   90  +  50  +  50

Producer goods balance:

    X1   =  90  =  50 + 40  =   C1  + C2
Period 2


If 90 units of producer goods that were produced in the period 1 are allocated in the proportion 50 to sector 1 and 40 to sector 2 then the production schema of the period 2 is exactly the same as in the period 1, i.e. output of both sectors remains unchanged.

Let us now investigate what happens when the necessary conditions (8) is satisfied, but producer goods are allocated differently than in the initial period. For example out of 90 units of producer goods produced in period one 50 units would be allocated to sector 2 and only 40 to sector 1. This reallocation of producer goods would require appropriate reallocation of labor. Assuming unchanged technology, the producer goods to labor ratio C/V - or in the Marxian terminology the 'organic composition of capital' - must remain the same in both sectors. In the period 1 this ratio was 50/20 = 2.5 in the sector 1 and 40/30 = 1.333 in the sector 2. It follows that after the reallocation of producer goods period 2 wages in the sector 1 must be  equal to 40/2.5 = 16  and wages in the sector 2 must be equal to 50/1.333 = 37.5. If in addition to that the 'rate of exploitation' M/V remains equal to 1, that is the same as in period 1 then the gross value of output in the sector 1 would change to 40 + 16 + 16 = 72 and in the sector 2 would change to 50 + 37.5 + 37.5 = 125.
After the just described reallocation of producer goods from sector 1 in favor of sector 2 the production schema of the period 2 would look like this 
period 2       
 

 

   X            C          V            M
Sector 1        72  =   40  +   16    +  16
Sector 2      125  =   50  +   37.5 +  37.5
  ----------------------------------
Total           197  =   90  +  53.5  +  53.5
Compared to the period 1 the output of consumer goods increased by 25 percent and the output of producer goods decreased from 90 to 72, i.e. by 20 percent. From the schema we see quite clearly that the necessary condition for simple reproduction in period 2 is violated. To be able to reproduce the output in the period 3 at the same level we would need 90 units of producer goods but only 72 were produced. Notice also that to keep the consumer market in equilibrium capitalists would have to spend 125 - 53.5 = 71.5 on purchases of consumer goods, which is more than their profits. In such a situation capitalist would have to 'dissave', i.e. 'eat up' part of their capital. what would happen in future depends again on the way 72 units of producer goods are allocated in the period 3.

 

Expanded Reproduction (economic growth):

To achieve economic growth, it is necessary to produce  more producer goods than is needed for simple reproduction
(15)                     X1t >  Ct

Because output of producer goods of the period t will be used as input in the period t+1

(16)                    Ct+1  =  X1t

and therefore

(17)                     Ct+1  >  Ct

 the total amount of producer goods increased, allowing  production of larger quantity of both producer and consumer goods in the period t+1. Again this is a necessary but not sufficient condition. The actual result depends on the allocation of produced producer goods. For economic growth to continue it is necessary that the quantity allocated to sector 1 is greater than in the preceding year.

(18)                      C1t+1  >  C1t

CONTINUATION OF NUMERICAL  EXAMPLE:

We need to start in period 1 with larger production of producer

goods.                                              

                     

    X         C        V        M
Sector 1      99  =   55  +   22  +  22
Sector 2  100  =   40  +   30  +  30
  -------------------------
Total 199  =   95  +  52  +  52

                  

Availability of producer goods for the next period is slightly greater.
                        

X1   =  99  >  95 = C

It implies that quantity of consumer goods is smaller than total wages and profits.

                             

X2   = 100  <  104  =  52 + 52  =  V + M

 

Obviously somebody must spend on consumer goods less than total income, or in other words somebody must save  4 units and invest them (use them to buy producer goods).          

Now several alternative investment policies may be chosen. If we want at least the same amounts of both goods as in the previous year we must put at least 55 into production of producer goods and 40 in the production of consumer goods. What to do with remaining 4 units? Let us investigate the following 3 policies:

A  invest all 4 in sector 1

B  invest 2.3 in sector 1 and 1.7 in sector 2.

C  invest all 4 in sector 2

Second year output under policy A:            

 
  X            C         V           M

C1 = 55 + 4 = 59   

Sector 1 106.2  =   59  +   23.6  +  23.6 V1 = 59/2.5 = 23.6
Sector 2 100     =   40  +   30     +  30  
  -------------------------------------- Test the reproduction condition
Total 206.2  =   99  +   53.6  +  53.6 106.2 - 99 = 7.2 > 0; expanded

 

Second year output under policy B:

 

   X               C            V           M C1 = 55 + 2.3 = 57.3; C2 =40 + 1.7 = 41.7
Sector 1 103.19  =   57.33  +   22.93  +  22.93 V1 = 57.3/2.5 = 22.9 V2 = 41.7/1.333 = 31.3
Sector 2

104.17  =   41.67  +   31.25  +  31.25

   
  --------------------------------------

Test the reproduction condition

Total 207.36  =   99       +   54.18  +  54.18

103.1 - 99 = 4.1 > 0;   expanded

 

Second year output under policy C:

 

    X         C       V        M C2 = 40 + 4 = 44
Sector 1 

  99  =   55  +   22  +  22

V2 = 44/1.333 = 33
Sector 2 110  =   44  +   33  +  33  
  -------------------------------- Test the reproduction condition
Total  209  =   99  +   55  +  55 99 - 99 =  0;     simple

 

Judging by total output policy C seems to be the best: total output increased by 10 units (5%). With policy B the total output increased only by 8.4 units (4.2%) and with policy A  by 7.2 units (3.6%).

However, under policy C the number of units of producer goods is only 99 that is the same as in preceding year. Obviously there will be no growth in the third year.

Under policy B there is more producer goods produced: 103.19 instead of 99.  The allocation of extra units was made proportional to the existing ratio of producer goods in the two sectors, and as a result both sectors grew by the same rate 4.2%. That also means that enough producer goods will be available in the third year to continue by the same rate 4.2%.

Under the policy A the increment of the output of producer goods is 7.2 which means that the  growth in the third year can be considerably accelerated.

We can conclude that allocation of the increment of producer goods in proportion to the existing quantities (policy B) results in maintaining the same rate of growth. The preferential allocation to sector 1 (policy A) results in the acceleration of the rate of growth. With  preferential allocation to the production of consumer goods (sector 2)  the growth slows down.

 

 

 

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