Example of production without consumption:

Let  the  economy  produce  n  final  goods  y₁,  y₂,  ..., y_n,  These  goods  are produced using  m  factors of production, X₁,  ..,  x_m.  The prices of final goods are p₁,  ..., p_n,  respectively, assumed to be fixed  in  the world market. This is called Small country assumption.

Let, x_ij = amount of factor i  used  in the production of good j  i=l, .., m, j=l,  .., n.
The production function for good  j  is written as

y_j  = P  (x_ij, ..  .,  x_mj).

The total resources of the economy are fixed at the levels x₁, .  .  ..,x_m i.e., the quantities are not choice variables, but parameters. In  a competitive  set up,  the exhaustion of gains from  trade implies that  the value of total (final) output, or GNP is maximised. The above statement  can be written mathematically as  

We,  for the simplicity of analysis, will assume that there are only two goods and two factors. Let the factors are labor L and capital K. Let  the  labor and  capital  allocated  to  industry  j  be  denoted  by  L,  and K, respectively.