Marshallian Demand and Indirect Utility Function Derivation
How to derive Marshallian demand and indirect utility function for u(x, y) = (0.3√x + 0.7√y)^2?
What steps are involved in deriving the Marshallian demand and indirect utility function for the given utility function?
Marshallian Demand and Indirect Utility Function Derivation:
To derive the Marshallian demand and indirect utility function, we need to follow a series of steps:
1. Take the derivative of the utility function with respect to each good (x and y).
2. Set the derivatives equal to zero to find the optimum quantity of goods that maximize utility.
3. Solve for the optimal quantities of x and y.
4. Substitute the optimal quantities back into the utility function to find the indirect utility function.
Step-by-Step Explanation:
1. Begin by calculating the derivative of the utility function u(x, y) = (0.3√x + 0.7√y)^2 with respect to x and y.
2. Set the derivatives equal to zero: d(u)/dx = 0 and d(u)/dy = 0.
3. Solve the two equations simultaneously to find the optimal quantities of x and y: x* and y*.
4. Substitute x* and y* back into the utility function to obtain the indirect utility function V(p, w): V(p, w) = (0.3√x* + 0.7√y*)^2.
5. Now, you have derived the Marshallian demand and indirect utility function for the given utility function u(x, y).
By following these steps, you can determine the optimal quantities of goods x and y that maximize utility and calculate the corresponding indirect utility function.