Let there be n industries producing n products and m resources. Let the input requirement of the various industries for the resources be given by the set of coefficients b_i,k, where b_i,j is the input of the k-th resource required per unit of production of the i-th industry. Let x_i be the output of the i-th industry. The amount of the k-th resource used up is

Σ_ibi,k

If the economy only has R_k units of the k-th resource then

Σ_{ibi,k ≤ Rk}

When there are only two products the levels of production satisfying the above constraint are easily shown, as below:

All the points satisfying the condition that no more of the resource is used than is available are shown in red. These are the set of feasible production levels.

When there are two constraints the situation is more interesting.

The points satisfying the condition that no more of either resource is being used than is available are shown in violet. The points shown in violet are the feasible production levels. Those satisfying the first constraint but not the second are shown in red. Those satisfying the second constraint but not the first are shown in blue.

Now consider maximizing the value of production which can be achieved with the limited amounts of two resources. The prices are depicted by the green iso-value lines in the diagram.

The value of production will be maximized at some corner point of the set of feasible production levels.

(To be continued.)