Multi-product firms are firms that produce multiple goods, and therefore have to deal with allocating inputs more properly in order to attain higher production levels. This is a greater problem than the one single-product firms face, the production maximisation problem, since multiproduct firms must allocate their factors not only to produce one good, but multiple goods.
Indeed, production analysis can be extended to include multiproduct firms, as it can be derived from F. Y. Edgeworth‘s “Mathematical Psychics”, 1881. In this case, a company produces two goods, X and Y, using the same two factors, L and K, and subject to the following restrictions:
In this first figure, which is known as an Edgeworth box, if we take A as a starting point, we can see that it is far from an optimum, as we could increase our production of X whilst maintaining production of Y constant by moving from A to B, and maintain production of X constant whilst increasing production of Y by moving to point C. These optimum points pertain to the contract curve, along which production is efficient. Along all these points, the marginal rate of technical substitution for both products is equal: MRTSx=MRTSY.
From this Edgeworth box and, more precisely, the contract curve that takes all efficient production levels into account, we can derive the production possibility frontier (PPF), shown in the figurebelow. This is just another way to look at the same problem: how to better allocate our resources in order to obtain the maximum output possible. However, by using the PPF, we concentrate more on the amounts being produced, rather than the inputs being used.
Of course, what is missing here to complete the analysis would be the cost of both inputs and how much is required to produce each type of good. In this case, by factoring in the selling price of each, we would have a closed optimisation problem that we would be able to resolve analytically, determining an efficient production function that could optimise the ratio of production of one good to another.