Which law states that the marginal product of a factor initially rises with employment but, after a certain level of employment, begins to fall as more units of the factor are used?

Introduction / Context:
The behaviour of marginal product as more units of a variable factor are employed with a fixed factor is a key topic in production theory. The question describes a situation where marginal product rises at first and then falls after a certain level of employment. This pattern is explained by a well known law of production that is studied in the context of the short run.

Given Data / Assumptions:

• We are in the short run, where at least one factor is fixed.
• The variable factor is increased while the fixed factor remains constant.
• Marginal product initially increases with employment.
• After some level of employment, marginal product begins to decline.
• We use standard production theory terminology.

Concept / Approach:
The law described is commonly known as the law of variable proportions, also referred to as the law of diminishing marginal returns. It states that as more units of a variable factor (like labour) are employed with a fixed factor (like land or machinery), total product at first increases at an increasing rate, then at a diminishing rate, and eventually marginal product may become zero or negative. This behaviour comes from the fact that the fixed factor becomes increasingly crowded as more variable input is added.

Step-by-Step Solution:
1. Identify that the question focuses on marginal product behaviour as employment increases.2. Note that marginal product initially rises, indicating increasing returns to the variable factor.3. The question then states that marginal product starts falling after a certain point, indicating diminishing returns.4. This pattern of rising and then falling marginal product is exactly what the law of variable proportions explains.5. Therefore, the correct name for the law described in the question is the law of variable proportions.

Verification / Alternative check:
To verify, imagine a small farm with fixed land area. Initially, adding more workers allows better division of labour and use of land, so marginal product increases. After some point, however, the land becomes crowded and each extra worker has less land or equipment to work with, so the additional output contributed by each new worker falls. This pattern is observed in many production processes and is the essence of the law of variable proportions.

Why Other Options Are Wrong:
Option A: The phrase law of diminishing marginal product describes only the declining part and does not emphasise the initial rise in marginal product. The widely used textbook term that covers the complete pattern is law of variable proportions.
Option C: The short run law of supply relates price and quantity supplied, not marginal product behaviour in production.
Option D: The law of the long run is not a standard term in basic production theory and does not refer to marginal product behaviour.
Option E: The law of increasing opportunity cost is related to the shape of the production possibility curve, not specifically to marginal product of a single factor in a firm.

Common Pitfalls:
Students sometimes think the law starts only when marginal product begins to fall, but the law of variable proportions actually covers the entire process of increasing returns, constant returns, and diminishing returns to the variable factor. Another pitfall is mixing this law with economies of scale, which are a long run concept involving changes in all factors. Remember that law of variable proportions is a short run phenomenon, holding some inputs fixed.

Final Answer:
The law described is the law of variable proportions.