Chapter 12: Cost Curves - LUISS Guido Carli

[Pages:13]Chapter 12: Cost Curves

12.1: Introduction

In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level of output. This chapter studies this cheapest cost and explores its properties. In chapter 13 we use these properties to find the profitmaximising output of the firm.

Let us denote by y some level of output of the firm. We denote by C(y) the minimum cost of producing that output. We call this function C(.) the firm's cost function and a graph of it a cost curve.

We should note that this cost function depends upon various things, which we could include in the notation but we leave implicit. The key things that C(.) depend upon are:

1) the technology of the firm 2) the prices of the two inputs.

We should also note that there may be different cost functions ? depending upon the constraints under which the firm is operating. Economists find it useful to distinguish between two scenarios (which determine the constraints under which the firm is operating), which are referred to as the long run and the short run. The long run is the familiar case in which the firm is free to vary the quantities of the two inputs. The short run is defined to be a situation in which just one of the two inputs is freely variable, while the other is fixed; we take input 1 to be freely variable in the short run while input 2 is fixed, its value q2 equal to some fixed level Q2. If it helps, you can imagine input 1 to be labour while input 2 is capital: in the short run the firm can not change the level of its capital ? only its labour; however in the long run it can vary both.

So we can have a long-run cost function and a short-run cost function. Obviously they are different ? in the short run we have this additional constraint that input 2 is fixed. This implies that the shortrun cost function is different from the long-run cost function ? you should be able to work out that the short-run function can never be lower than the long-run function. (We are minimising something ? if we minimise something with a constraint the minimum must be at least as large as the minimum without the constraint.) We will give examples during this chapter.

I should emphasise that C(y) measures the minimum total cost of producing the output y. Later in the chapter we will derive from this total cost function two other cost functions ? the marginal cost function ? which measures the rate at which the total costs are increasing ? and the average cost function ? which measures the average cost of producing a particular output.

12.2: The Long Run Total Cost Curve

We start with the long run ? in which the firm can freely choose both the level of input 1 and the level of input 2. I find it useful to start with a specific example ? which shows the important properties I want to discuss. Let us take the case of Cobb-Douglas technology. This has a

production function given by y = A q1a q2b. In chapter 11 we found the cost-minimising input combinations for any given level of output. These are given in equation (11.3) and are:

q1 = (y/A) 1/(a+b)(aw2/(bw1))(b/(a+b)) and q2 = (y/A) 1/(a+b)(bw1/(aw2))(a/(a+b))

Now the cost of using the combination (q1,q2) is obviously w1q1 + w2q2 so the cheapest cost of producing output y in the long run is found by substituting in the optimal input demands into the production function. This yields:

C(y) = (y/A) 1/(a+b)[w1(aw2/(bw1))(b/(a+b)) + w2 (y/A) 1/(a+b)(bw1/(aw2))(a/(a+b))]

This is the long run total cost function in the Cobb-Douglas case. It can be simplified a little to give:

C(y) = (a+b)(y/A)1/(a+b)(w1/a) a/(a+b)(w2/b)b/(a+b)

(12.1)

Let us take a particular numerical example. Put A =1, a = 0.3 and b = 0.5. This technology exhibits decreasing returns to scale. (Why? Since a + b = 0.8 < 1.) If we substitute these numbers in the expression above we get

C(y) = 1.938 y1.25 w1.375 w2.625

We note that this is an increasing convex function of y and an increasing concave function of the two input prices. If we graph the cost function against y we get the following:

Ensure that you understand the connection between the fact that the technology displays decreasing returns to scale and the fact that the cost function is convex: as output rises the scale has to rise proportionately faster, which makes the cost rise proportionately faster. Note from (12.1) this is a general property. In fact we should write this as a result. It is very important.

The total long run cost function is concave, linear or convex according as the technology displays increasing, constant or decreasing returns to scale.

12.3: The Short Run Total Cost Curve

Before we examine an example, we should think a little about this curve. Let me remind you of the scenario. Input 2 is fixed at the level Q2. The firm cannot change this, It can, however, vary input 1. It still wants to do what it has been doing: for any given level of output it wants to find the cheapest way of producing that output ? and thus it wants to find the cheapest cost of producing each level of

output. Do you think that with this additional constraint (it cannot vary the amount of input 2) it can do better or worse than without it?

The answer I hope is immediate: with this additional constraint it is almost always doomed to do worse than before. Therefore the short run total cost function can never be lower than the long-run total cost function ? if it were then there would be a contradiction ? the long run function could not be minimising the cost in the long run. However, we could be lucky in the sense that the fixed quantity of input 2 might just be the right amount to minimise cost in the long run ? in such a case the long and the short run costs would coincide.

Let us now take an example. Let us continue to use the Cobb-Douglas technology we used in section 12.2 y = A q1a q2b. Now additionally we have q2 = Q2. So we have

y = A q1a Q2b

On the right-hand side the only thing that is variable is q1 ? notice as we vary it then the output y varies. If we want to produce a particular level of output then it is clear that there is a unique value

of q1 for which this is possible ? the one that solves the above equation in terms of y. Solving it we get

q1 = (y/A)1/aQ2-b/a

To produce the output y this and only this quantity of input 1 should be used. (Note that, rather obviously, it depends upon Q2.) It follows that the total cost of producing output y in the short run is given by w1q1 + w2Q2 where q1 is given by the expression above. In this way we get the total short run cost function:

C(y) = w1 (y/A)1/aQ2-b/a + w2 Q2

(12.2)

We note that it is an increasing convex (if 0 ................
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