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Production Function


A producer uses raw materials, capital, and labor to produce goods and services. Here, we will present a simple model for how they decide how much to produce and which technology to use for the production. A large part of producer theory is very similar to consumer theory. Basic assumptions for consumer theory are that consumers have a goal to maximize their utility, but that they have restrictions due to limited income and prices. Producers also have a goal. They wish to maximize their profit. They also have restrictions. These are, for instance, the costs of labor and capital; but they also have restrictions regarding the technology of production. An aspect that will also prove important for a firm is the amount of competition they face: Do they have one, a couple, or many competitors? Alternatively, do they not face any competition at all?

  • The producers have certain restrictions. Primarily because different combinations of inputs (labor and capital) have different associated costs.
  • The firm operates in a market that, in turn, has certain structures that the firm cannot influence.
  • Technology: Different combinations of input produce different quantities of goods.
  • We will distinguish between production in the short run and in the long run. In the short run, the quantity of available capital is fixed; in the long run, both labor and capital are variable.
  • Given production and restrictions, the producer maximizes her profit.
  • Another important question is how large a firm should be. The important concept here is returns to scale. How firm size affects how efficiently it can transform input to output.

The Profit Function

We will use a very simple model of a firm. It produces a single good, and the most important input factors are labor and capital (for instance machines). The producer has a certain cost, C, and a certain revenue, R. Her profit, π, can then be written as the difference between revenue and cost:


The revenue, in turn, depends on the price of the good and how large quantity she sells,


The costs are, of course, also dependent on how large quantity she produces, but usually in a more complicated way. The profit can therefore be written as


where C(q) means that the cost, C, is a function of the quantity, q. We will analyze each of the three variables, p, q, and C, in the profit function. The price is often set by the market. C depends both on the costs of the input factors and the quantity produced. The firm therefore chooses the quantity that maximizes profit.

The Production Function

The quantity the producer will produce of the single good, depends on the number of working hours, L (for Labor), and the amount of capital, K, that she uses. q is consequently a function of L and K:


The letter f in the expression means that we have a function of L and K. That could mean 'L + K', 'L*K' or 'L2 + 9*ln(K)' to just mention a few arbitrary examples. Which function that is appropriate depends on the technology, which good one produces, etc. Note that we, of course, assume that the production of q units is done in the most efficient way possible. If it, for instance, is possible to produce 10 units of the good using a certain combination of L and K, it is also possible to produce only 9 units with the same combination. However, that production cannot be efficient, since one is wasting resources that could be used for the production of one more unit.

Average and Marginal Product

Before we begin the analysis, we need to define a few concepts that will be important later on. The average product is how much of the good that on average is produced by a certain input, L or K. We therefore have that the average products, AP, for labor and capital, respectively, are


The marginal product, MP, is how much extra quantity that can be produced if one increases the amount of either labor or capital with one unit, keeping the other one constant:


The expression for MP is, just as in the case of MRS, only approximate. It will also become more and more exact the smaller one chooses ΔK or ΔL.

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