Final Answer:
a) The given production function,
, represents a Cobb-Douglas functional form.
b) The Cobb-Douglas functional form assumes constant returns to scale, implying that the exponents on capital
sum to 1. This assumption is economically sensible as it allows for flexibility in adjusting inputs while maintaining a consistent output scale.
c) Setting
.
Explanation:
a) The Cobb-Douglas functional form is characterized by a production function of the form
is a positive constant and
is the output elasticity of capital. In the given production function
, confirming it as a Cobb-Douglas form.
b) The assumption of constant returns to scale is integral to the Cobb-Douglas model. It means that if both inputs, capital
, are increased by a certain factor, output
will increase by the same factor. This assumption aligns with economic intuition, as it allows for easy interpretation of the impact of input changes on output.
c) Setting
. This expression represents a linear relationship between capital
.
In conclusion, the Cobb-Douglas functional form provides a useful framework for analyzing production relationships, and the constant returns to scale assumption simplifies the interpretation of input-output dynamics in economic production models. The rearranged function
expresses the proportional relationship between capital and labor in the given production context.