Black Diamond Snowboards (BDS) is a startup snowboard producing enterprise. Its founder has invented a new lamination process that gives extra strength to his boards. He has set up a production line in his garage that has four workstations: Laminating, attaching the steel edge, waxing, and packing.
With this process in place, he must examine how productive his firm can be. After extensive testing, he has determined exactly how his productivity depends upon the number of workers. If he employs only one worker, then that worker must perform several tasks, and will encounter 'down time' between workstations. Extra workers would therefore not only increase the total output; they could, in addition, increase output per worker. He also realizes that once he has employed a critical number of workers, additional workers may not be so productive: Because they will have to share the fixed amount of machinery in his garage, they may have to wait for another worker to finish using a machine. At such a point, the productivity of his plant will begin to fall off, and he may want to consider capital expansion. But for the moment he is constrained to using this particular assembly plant. Testing leads him to formulate the relationship between workers and output that is described in Table 8.1.
Table 8.1 Snowboard production and productivity
1 
2 
3 
4 
5 
Workers 
Output 
Marginal 
Average 
Stages of 

(TP) 
product 
product 
production 


(MP_{L}) 
(AP_{L}) 

0 
0 


MP_{L} increasing 
1 
15 
15 
15 
2 
40 
25 
20 
3 
70 
30 
23.3 
4 
110 
40 
27.5 
5 
145 
35 
29 
MP_{L} positive and declining 
6 
175 
30 
29.2 
7 
200 
25 
28.6 
8 
220 
20 
27.5 
9 
235 
15 
26.1 
10 
240 
5 
24.0 
11 
235 
5 
21.4 
MP_{L} negative 
By increasing the number of workers in the plant, BDS produces more boards. The relationship between these two variables in columns 1 and 2 in the table is plotted in Figure 8.1. This is called the total product function (TP), and it defines the output produced with different amounts of labour in a plant of fixed size.
Total product is the relationship between total output produced and the number of workers employed, for a given amount of capital.
This relationship is positive, indicating that more workers produce more boards. But the curve has an interesting pattern. In the initial expansion of employment it becomes progressively steeper – its curvature is slightly convex; following this phase the function's increase becomes progressively less steep – its curvature is concave. These different stages in the TP curve tell us a great deal about productivity in BDS. To see this, consider the additional number of boards produced by each worker. The first worker produces 15. When a second worker is hired, the total product rises to 40, so the additional product attributable to the second worker is 25. A third worker increases output by 30 units, and so on. We refer to this additional output as the marginal product (MP) of an additional worker, because it defines the incremental, or marginal, contribution of the worker. These values are entered in column 3.
More generally the MP of labour is defined as the change in output divided by the change in the number of units of labour employed. Using, as before, the Greek capital delta () to denote a change, we can define
In this example the change in labour is one unit at each stage and hence the marginal product of labour is simply the corresponding change in output. It is also the case that the MP_{L} is the slope of the TP curve – the change in the value on the vertical axis due to a change in the value of the variable on the horizontal axis.
Marginal product of labour is the addition to output produced by each additional worker. It is also the slope of the total product curve.
During the initial stage of production expansion, the marginal product of each worker is increasing. It increases from 15 to 40 as BDS moves from having one employee to four employees. This increasing MP is made possible by the fact that each worker is able to spend more time at his workstation, and less time moving between tasks. But, at a certain point in the employment expansion, the MP reaches a maximum and then begins to tail off. At this stage – in the concave region of the TP curve – additional workers continue to produce additional output, but at a diminishing rate. For example, while the fourth worker adds 40 units to output, the fifth worker adds 35, the sixth worker 30, and so on. This declining MP is due to the constraint of a fixed number of machines: All workers must share the same capital. The MP function is plotted in Figure 8.2.
The phenomenon we have just described has the status of a law in economics: The law of diminishing returns states that, in the face of a fixed amount of capital, the contribution of additional units of a variable factor must eventually decline.
Law of diminishing returns: when increments of a variable factor (labour) are added to a fixed amount of another factor (capital), the marginal product of the variable factor must eventually decline.
The relationship between Figures 8.1 and 8.2 should be noted. First, the MP_{L} reaches a maximum at an output of 4 units – where the slope of the TP curve is greatest. The MP_{L} curve remains positive beyond this output, but declines: The TP curve reaches a maximum when the tenth unit of labour is employed. An eleventh unit actually reduces total output; therefore, the MP of this eleventh worker is negative! In Figure 8.2, the MP curve becomes negative at this point. The garage is now so crowded with workers that they are beginning to obstruct the operation of the production process. Thus the producer would never employ an eleventh unit of labour.
Next, consider the information in the fourth column of the table. It defines the average product of labour (AP_{L})—the amount of output produced, on average, by workers at different employment levels:
This function is also plotted in Figure 8.2. Referring to the table: The AP column indicates, for example, that when two units of labour are employed and forty units of output are produced, the average production level of each worker is 20 units (=40/2). When three workers produce 70 units, their average production is 23.3 (=70/3), and so forth. Like the MP function, this one also increases and subsequently decreases, reflecting exactly the same productivity forces that are at work on the MP curve.
Average product of labour is the number of units of output produced per unit of labour at different levels of employment.
The AP and MP functions intersect at the point where the AP is at its peak. This is no accident, and has a simple explanation. Imagine a softball player who is batting .280 coming into today's game—she has been hitting her way onto base 28 percent of the time when batting, so far this season. This is her average product, AP.
In today's game, if she bats .500 (hits her way to base on half of her atbats), then she will improve her average. Today's batting (MP) at .500 therefore pulls up the season's AP. Accordingly, whenever the MP exceeds the AP, the AP is pulled up. By the same reasoning, if her MP is less than the season average, her average will be pulled down. It follows that the two functions must intersect at the peak of the AP curve. To summarize:
If the MP exceeds the AP, then the AP increases;
If the MP is less than the AP, then the AP declines.
While the owner of BDS may understand his productivity relations, his ultimate goal is to make profit, and for this he must figure out how productivity translates into cost.