State the rule for optimum input allocation to produce a given level of output at the lowest possible cost -when two inputs are variable and the prices of the inputs are given- and explain why it makes sense
The rule for the optimum input allocation to produce a given level of output at the lowest possible cost when two inputs are variable and the prices of the inputs are given is known as the cost minimization rule.
According to this rule, the optimal input allocation is achieved when the ratio of the marginal products of the two inputs is equal to the ratio of their prices. Mathematically, it can be represented as:
Marginal Product of Input 1 / Price of Input 1 = Marginal Product of Input 2 / Price of Input 2
To understand why this rule makes sense, let's break it down:
1. Marginal Product: Marginal product refers to the additional output produced by using an additional unit of input while keeping all other inputs constant. It measures the increase in output resulting from an increase in the quantity of an input.
2. Prices of Inputs: The prices of inputs represent the cost of acquiring or using each input. In a competitive market, inputs have prices associated with them, and businesses need to consider these prices when determining their input allocation.
Now, let's consider why the cost minimization rule makes sense:
1. Equalizing Marginal Products: The rule aims to allocate inputs in a way that maximizes output for a given level of cost. By equalizing the ratios of the marginal products and prices of the two inputs, the allocation ensures that the additional output gained from each additional unit of input is proportional to its cost. This means that the business is making the most efficient use of its inputs to achieve the desired output level.
2. Cost Minimization: When the rule is followed, it ensures that no input is overused or underused in the production process. Suppose the marginal product of input 1 is higher relative to its price compared to the marginal product of input 2. In that case, it suggests that input 1 provides a greater increase in output per unit cost, indicating that more of input 1 should be used until the ratio of marginal product to price is equal to that of input 2. This reallocation helps to minimize costs while maintaining the desired output level.
Therefore, the cost minimization rule is a rational approach to optimizing input allocation by considering the trade-off between the marginal product of inputs and their prices. It ensures that resources are used efficiently and effectively, resulting in cost-effective production.