In the short run, production activity is analyzed by the Law of Variable Proportions. The Law of Variable Proportions, also known as the Law of Diminishing Marginal Returns, states that as a firm increases the quantity of one input while keeping other inputs fixed, the marginal product of the variable input will eventually decline.

Explanation:
1. Law of Variable Proportions:
The Law of Variable Proportions is a fundamental concept in economics that explains the relationship between inputs and outputs in the short run. It states that as a firm increases the quantity of one input, such as labor or capital, while keeping other inputs fixed, the marginal product of the variable input will initially increase but eventually decline.

2. Short Run Production:
In the short run, a firm is constrained by fixed inputs, such as machinery or facilities, that cannot be easily changed. Only the variable inputs, such as labor or raw materials, can be adjusted to increase or decrease production. This means that the firm can increase its output in the short run by adding more of the variable input, but there is a limit to how much it can increase.

3. Law of Diminishing Marginal Returns:
The Law of Variable Proportions is also referred to as the Law of Diminishing Marginal Returns because it explains how the marginal product of the variable input changes as more of it is added. Initially, when a firm increases the quantity of the variable input, the marginal product also increases. This is because the fixed inputs are being utilized more efficiently and the firm is able to produce more output.

However, as the firm continues to increase the quantity of the variable input, there comes a point where the marginal product starts to decline. This happens because the fixed inputs become a constraint and the additional units of the variable input become less productive. For example, if a firm keeps adding more workers to a fixed amount of machinery, there will be a point where the workers start getting in each other's way, leading to a decrease in productivity.

4. Analysis of Production Activity:
The Law of Variable Proportions allows economists to analyze and understand the production activity in the short run. By examining how the marginal product of the variable input changes as more is added, economists can determine the optimal level of input usage for a given level of output. This analysis helps firms make decisions about resource allocation and production planning in order to maximize efficiency and profits.

In conclusion, the production activity in the short run is analyzed by the Law of Variable Proportions, which explains how the marginal product of the variable input changes as more is added. This analysis helps firms understand the relationship between inputs and outputs and make informed decisions about production planning.