Calculating the Inverse Function of a Utility Representation

Calculating the Inverse of u(x, y) = √(3x + 3y)

This example demonstrates how to calculate the inverse function of a utility representation.

Understanding the Problem

Maria's preferences are represented by the utility function u(x, y) = √(3x + 3y). This function tells us her level of satisfaction (utility) based on consuming quantities of goods x and y. The inverse function, denoted f^-1(u, y), helps us determine the quantity of good x Maria needs to consume to achieve a specific utility level 'u', given a fixed quantity of good 'y'.

Calculating the Inverse Function

1. Start with the original function: u = √(3x + 3y)

2. Square both sides: u^2 = 3x + 3y

3. Isolate 'x' by subtracting 3y from both sides: u^2 - 3y = 3x

4. Divide both sides by 3 to solve for 'x': x = (u^2 - 3y) / 3

The Inverse Function:

Therefore, the inverse function is:

f^-1(u, y) = ((u^2 - 3y) / 3, y)

Explanation:

This inverse function allows us to input a desired utility level ('u') and a fixed quantity of good 'y' to determine the required quantity of good 'x' for Maria to achieve that specific utility level.