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'

X_1t                   ... output of producer goods in period t
X_2t                   ... output of consumer goods in period t

(1)        X_t  = X_1t  + X_2t            ... total output in period t
 

C_1t                   ... use of producer goods in production of X_1t

C_2t                   ... use of producer goods in production of X_2t 

(2)        C_t  = C_1t  + C_2t     ... total use of producer goods
 

  V_1t                  ... wages paid in sector 1

  V_2t                  ... wages paid in sector 2

(3)        V_t  = V_1t  + V_2t      .. total wages in both sectors
 

M_1t                  ... profits in sector 1

M_2t                  ... profits in sector 2

(4)        M_t  = M_1t  + M_2t             ... total profits
 

Scheme of reproduction:

(5)       X_1t  = C_1t  + V_1t  +  M_1t

(6)       X_2t  = C_2t  + V_2t  +  M_2t

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

(7)       X_t   =  C_t   +  V_t   +  M_t
 

Types of reproduction:

simple reproduction

output remains constant - no growth;

expanded reproduction

 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.

(8)                    X_1t =  C_t
 

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

(9)                    C_t+1  =  X_1t
 

and therefore

 
(10)                  C _t+1  =  C_t
 

 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  /C_t+1  =  C_1t /C_t
 

Conditions (10) and (11) guarantee
 

 (12)                  C _1t+1  =  C_1t

   (13)                 C _2t+1  =  C_2t
 

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)                  X_2t  =  V_t + M_t
 

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. 

NUMERICAL  EXAMPLE:
 

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
 

Consumer goods balance:
 

                         X₂   = 100  =  50 + 50  =  V + M          
      

   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 

                     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)                    X_1t >  C_t
 

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

(16)                    C_t+1  =  X_1t
 

and therefore

(17)                  C_t+1  >  C_t
 

 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)                  C_1t+1  >  C_1t
 

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

  Sector 1   106.2  =   59  +   23.6  +  23.6

Sector 2   100     =   40  +   30     +  30

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

 Total        206.2  =   99  +   53.6  +  53.6
 

C₁ = 55 + 4 = 59
V₁ = 59/2.5 = 23.6

Test the reproduction condition

106.2 - 99 = 7.2 > 0; expanded

Second year output under policy B:
 

                   X            C        V           M

Sector 1   103.19  =   57.33  +   22.93  + 22.93

Sector 2   104.17  =   41.67  +   31.25  +  31.25

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

Total        207.36  =   99       +   54.18  +  54.18

C₁ = 55 + 2.3 = 57.3; C₂ =40 + 1.7 =41.7
V₁ = 57.3/2.5 = 22.9
V₂ = 41.7/1.333 = 31.3

Test the reproduction condition

103.1 - 99 = 4.1 > 0;    expanded

Second year output under policy C:
 

                  X         C       V        M

Sector 1    99  =   55  +   22  +  22

Sector 2  110  =   44  +   33  +  33

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

Total       209  =   99  +   55  +  55
 

C₂ = 40 + 4 = 44
V₂ = 44/1.333 = 33

Test the reproduction condition

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.