Production function

Now we get to answering the question we asked at the end of Section 4.1: what explains the shape of the typical cost function? Throughout the discussion here, we call the variable input labour (L) and the fixed input capital (K). It is obvious from Table 4.1 that the fixed cost is $500, which means that to hire capital K the firm has paid $500. It is fixed, so this firm has to pay $500 regardless of how much they produce. To change the level of output in the short run, it can only vary the level of the other input, labour, which costs w per unit. Our interest is to investigate the relationship between the level of output and the level of labour input.

Note that this firm’s labour hiring cost corresponds exactly to its variable cost (V C), because labour is the only input that is variable in the short run. Look at Table 4.4. It is identical to Table 4.1 except for the final column where labour input is shown. Suppose that it costs $10 to hire one hour of labour (w= 10). Then we can deduce the hours of work (L) that are needed for each level of production (q). For example, in order to produce one unit, the variable cost is $200. It means 20 (200 divided by 10) hours of labour are

Table 4.4. Costs and labour input.

q T C F C V C L

0 500 500 0 0

1 700 500 200 20

2 820 500 320 32

3 920 500 420 42

4 1040 500 540 54

5 1190 500 690 69

6 1390 500 890 89

7 1670 500 1170 117

8 2070 500 1570 157

9 2640 500 2140 214

10 3440 500 2940 294

q q(L)

0 L

Figure 4.5 A rough sketch of the production function.

required. For 10-unit production, the variable cost is $2940, indicating that 294 (2940 divided by 10) hours have to be put in. Following the same steps, you should be able to figure out how the final column of the table is constructed. The information in the first and the last columns in the table gives us the production function: it shows how much output can be produced for each amount of labour input:

q = f (L). (4.2)

Let us plot the information on a diagram; what does the production function look like?

Figure 4.5 shows the plot taking hours of work on the horizontal axis and the level of output on the vertical axis.

We now introduce a new notion called the marginal product of labour. The marginal product of labour is the change in the level of output caused by a one-unit increase in labour. We cannot see the marginal product of labour from the table, but can roughly observe on the diagram how it behaves. You should observe the following.

97 4.3 Production function

(1) At low levels of production, the marginal product of labour increases.

(2) At high levels of production, the marginal product of labour starts to decline.

The increasing marginal product of labour is also referred to as increasing returns to labour whereas the decreasing marginal product of labour is often called diminishing returns to labour.

The pattern of the marginal product of labour increasing up to a certain unit of production then declining explains the pattern of the marginal cost decreasing up to a certain unit of production then increasing (which is observed in Section 4.2: remember, the MC curve is U-shaped). If the marginal product of labour is increasing, then the marginal cost is decreasing, and vice versa.

In Section 4.1 we deferred answering the question: what sort of production technology are we looking at? Let us think about the above pattern of the marginal product of labour.

For instance, consider working in a restaurant and, to start with, suppose you are the only worker. As a sole worker, you have to do everything – taking orders, all the cooking, serving, etc. – by yourself, yet still you are able to serve a certain number of customers in a day. The marginal product of labour is equivalent to your output (i.e. an increase in the restaurant’s total output). Now, if the restaurant hires another worker who has the same skills, then you can imagine that the restaurant’s output in a day more than doubles. Why?

The new worker is as skillful as you are, so if both of you worked independently, then the output should double. That is, the marginal product of the new worker (an increase in the total output caused by the new worker) is the same as yours (i.e. your output when you were the sole worker). But you and the new worker may be able to do better than working independently; both of you can coordinate – e.g. one specialises in cooking and the other in all the other things – and work more effectively as a team. In this case, the marginal product (let me remind you again, it is an increase in the total output caused by the new worker) is greater than yours (an increase in the total output caused by you when you were the sole worker, i.e. your output before the new worker joined). A third worker may allow making the kitchen (and hence the whole restaurant) work more efficiently; two workers may be able to specialise in particular processes in the kitchen while another worker does all the other things in the dining area. The marginal product of the third worker, in this case, is greater than that of the second one.

You may be able to envisage the increasing marginal product described as above up to some level of output but, as you might already have guessed, there may be an end to it;

i.e. the diminishing returns to labour may kick in at some stage. Any restaurant has its capacity (it presumably depends on the size of the restaurant, which has to do with the fixed costs) – both in the kitchen and in the dining area – and so the room for further coordination between the workers becomes less and less. At some stage, an additional worker may increase the total output (so the marginal product is positive) but this increase in the total output may be less than that when the previous worker was hired. You may even be able to imagine the case under which the marginal product is negative (i.e. the total output decreases when the new worker is hired), although numbers provided in Table 4.4 do not consider this extreme situation.

Table 4.5. The marginal revenue and marginal cost.

q MR MC

3 210 100

4 210 120

5 210 150

6 210 200

7 210 280

8 210 400

9 210 570

10 210 800

The numbers provided in the tables in this chapter (and hence the shapes of the produc-tion funcproduc-tion and the cost funcproduc-tion) represent the producproduc-tion technology described as in the story above. In the following section, we will deduce an upward sloping supply curve for a firm supposing that the firm’s production technology is of the above sort; but here is a word of warning before moving on. If you encounter other tables that summarise some production technology, they may not necessarily replicate the story described above. For example, the table may represent the case where the marginal product of a particular input is constant (see Question 2 in Section 4.8).