What is marginal analysis? How does it relate to the Law of Diminishing Returns for a company?

Answer:
Let's define some key terms. Marginal means the effect of adding 1 extra unit to, or subtracting 1 unit from, a particular variable or factor of production.

Marginal product is the additional output that results from a firm which adds one more additional unit of labour or other input of production to the production process.

Marginal Cost is the increase in total cost that results from producing one more unit of output to a production process. Marginal costs reflect changes in variable costs, which are any cost that a firm bears that depends on a level of production chosen.

(For all definitions in this section, I have relied on The Concise Dictionary of Economic Terminology, a valuable dictionary of economic terms which you can find in Karl E. Case et al., Principles of Microeconomics, Scarborough: Prentice-Hall Canada Inc., 1998, pp. 605-611.)

Marginal analysis is concerned with questions of allocative efficiency. For example, how well inputs of production are allocated to various production levels by management.
The Law of Diminishing Returns states that when additional units of a variable unit are added to fixed inputs, after a certain point, the marginal product of the variable input declines.

Let's consider this production schedule:

No. Workers Total Product Average Product Marginal Product
1 10.0 10.0 10
2 22.0 11.0 12
3 42.0 14.0 20
4 72.0 18.0 30
5 100.0 20.0 28

When hiring additional labour, any company will seek to increase its marginal product, and thereby lessen its cost. Increasing marginal product occurs only to a certain level of hiring, in fact, to that point where marginal product is greatest and, though we have not shown it here, where marginal cost is lowest. In our example, the hiring of workers 1 through 4 allows labour to specialize in various tasks, as long as their efforts are properly unified through efficient management. The marginal product of the fourth worker is greater than that of the third, the third greater than the second, and the second is greater than having just one worker. Marginal product starts to fall due to the Law of Diminishing Returns. At that point, the marginal product of successive units of labour starts to decline. Marginal cost starts to rise and, if unchecked, they can kill a firm's profits.

What managers need to do, given a surging marginal cost that is greater than marginal product, is to require more accountability and results from the labour force rather than just hiring more labour. If management just hires more workers and does not have a clear results plan, then new employees become disinterested in their work, and this leads to diminishing returns for the firm. This issue has often been identified as a principal problem of large bureaucracies, where management does not have a clear plan for goals and workflow. Note that I have provided here only an example of a small firm, but the shape of the data and the product curves will be the same for large and small firms.
Marginal calculations can help a manager plan new work, decide on how many employees are required, and decide on the quality of results that is expected by management.