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Production II: Multi-product firm

Summary

In this Learning Path we introduce three new parameters to our possibilities: we will add a time frame and see how this shapes our choices, we will introduce the ability to produce more than one good or service, and we will also take a first look at prices, production costs and competition as a whole market dynamic.

Period analysis

Short term analysis:

Long term analysis:

Very long term analysis:

Very long run production

Multi-production:

Production possibility frontier
Multiproduct firms

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: MRTS_x=MRTS_Y.

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.

In this second LP on production, we covered everything we need to be able to understand this first aspect of market forces: production decisions. We divided our analysis into the SR, the LR and the VLR, looking at static period analysis in each of the three time frames. What we saw was that in each period, there were additional things we gained control over: such as capital in the LR and technology in the VLR. We also realised that a couple of extra concepts cropped up during our first few years of production, like the concept of returns to scale and the production possibility frontier, which would shape our decisions on what production level to maintain. Finally, we also looked at the effects of introducing a new production line have in multi-product firms.

Production II: Production possibility frontier