See W. J. Spillman and E. Lang, The Law of Diminishing Returns (1924).
Malthus and Ricardo, who lived in 19th century England, were worried that land, a factor of production in limited supply, would lead to diminishing returns. In order to increase output from agriculture, farmers would have to farm less fertile land or farm with more intensive production methods. In both cases, the returns from agriculture would diminish over time, causing Malthus and Ricardo to predict population would outstrip the capacity of land to produce, causing a Malthusian catastrophe. (Case & Fair, 1999: 790).
In economics, the term "marginal" is used to mean on the edge of productivity in a production system. The difference in the investment of seed in these three scenarios is one kilogram — "marginal investment in seed is one kilogram." And the difference in output, the crops, is one ton for the first kilogram of seeds, a half ton for the second kilogram, and one quarter of a ton for the third kilogram. Thus, the marginal physical product (MPP) of the seed will fall as the total amount of seed planted rises. In this example, the marginal product (or return) equals the extra amount of crop produced divided by the extra amount of seeds planted.
A consequence of diminishing marginal returns is that as total investment increases, the total return on investment as a proportion of the total investment (the average product or return) also decreases. The return from investing the first kilogram is 1 t/kg. The total return when 2 kg of seed are invested is 1.5/2 = 0.75 t/kg, while the total return when 3 kg are invested is 1.75/3 = 0.58 t/kg.
Another example is a factory that has a fixed stock of capital, or tools and machines, and a variable supply of labor. As the firm increases the number of workers, the total output of the firm grows but at an ever-decreasing rate. This is because after a certain point, the factory becomes overcrowded and workers begin to form lines to use the machines. The long-run solution to this problem is to increase the stock of capital, that is, to buy more machines and to build more factories.
In this example, the marginal cost equals the extra amount of money spent on seed divided by the extra amount of crop produced, while average cost is the total amount of money spent on seeds divided by the total amount of crop produced.
Cost can also be measured in terms of opportunity cost. In this case the law also applies to societies; the opportunity cost of producing a single unit of a good generally increases as a society attempts to produce more of that good. This explains the bowed-out shape of the production possibilities frontier.
Statement: As a firm in the long-run increases the quantities of all factors employed, other things being equal, the output may raise initially at a more rapid rate than the rate of increase in inputs, then output may increase in the same proportion of the input, and ultimately, output increases less proportionately.
There is evidence for possible increasing marginal returns in certain circumstances. A single fax machine is useless and returns nothing, but if two exist, they can exchange messages, increasing the network by 2 exchanges. A third allows each machine to send messages to two points, increasing the network by 4 exchanges (3*2-2). A fourth allows three points of exchange, with a marginal return of 8 exchanges, and so on. This law remains to be proven mathematically.
It's important to note that the "law of diminishing returns" says that there would be a moment when, for example, an increasing number of faxes will not improve productivity nor efficiency. If your company has a fax machine, it's good enough, having 2, would not improve your performance. It would not sense everyone having a fax machine. There is a moment when the efficiency starts to decrease.