Final answer:
A homothetic production function maintains constant input ratios regardless of the scale of operation, meaning that the proportions in which a firm combines inputs remains the same for every level of output.
Step-by-step explanation:
To show that when the production function is homothetic, the proportions in which the firm will combine any given pair of inputs is the same for every level of output, we must first understand what homothetic production functions are. A production function is homothetic if it can be represented as a monotonic transformation of a homogeneous function. This implies that as the scale of production changes, the input ratios used in the production process remain constant.
In detail, if a production function is homogeneous of degree k, then when all inputs are scaled by a common factor t, the produced output is scaled by a factor of tk. If the production function is homothetic, the proportionality between the amounts of inputs remains the same regardless of the level of output. Hence, the ratio of any two inputs used in the production process will be the same at any scale of operation. This is an extension of the Law of Variable Proportion, which describes how output changes when adjusting the quantity of one input, keeping others constant.
For example, considering a firm that uses capital and labor to produce goods, if the production function is homothetic, the ratio of capital to labor used when producing 100 units will be the same as when producing 200 units. This characteristic is crucial for firms since it simplifies the decision-making process regarding resource allocation because they can maintain the same input proportions while scaling up production.