Marshallian Derivation and Indirect Utility

Indirect utility is a concept in economics that represents the maximum utility an individual can achieve given their budget constraint and the prices of goods. The Marshallian derivation is a method used to derive the demand function for goods based on the consumer's utility maximization problem.

In this question, we are given the prices of two goods, Px1 and Px2, and the consumer's income, M. We need to solve for the consumer's indirect utility function given these parameters.

Step 1: Set up the Utility Maximization Problem
The first step in the Marshallian derivation is to set up the consumer's utility maximization problem. The consumer's objective is to maximize their utility subject to their budget constraint.

Let's assume that the consumer's utility function is given by U(x1, x2), where x1 and x2 are the quantities of goods 1 and 2 consumed, respectively. The consumer's budget constraint is represented by the equation:

Px1*x1 + Px2*x2 = M

where Px1 and Px2 are the prices of goods 1 and 2, and M is the consumer's income.

Step 2: Solve for the Consumer's Demand Functions
To solve for the consumer's demand functions, we need to find the values of x1 and x2 that maximize the utility function U(x1, x2) subject to the budget constraint.

Given the utility function and the budget constraint, we can set up the Lagrangian function:

L(x1, x2, λ) = U(x1, x2) - λ(Px1*x1 + Px2*x2 - M)

where λ is the Lagrange multiplier.

Step 3: Take Partial Derivatives
To find the consumer's demand functions, we need to take partial derivatives of the Lagrangian function with respect to x1, x2, and λ, and set them equal to zero.

The first-order conditions are:

∂L/∂x1 = ∂U/∂x1 - λ*Px1 = 0
∂L/∂x2 = ∂U/∂x2 - λ*Px2 = 0
∂L/∂λ = Px1*x1 + Px2*x2 - M = 0

Step 4: Solve the System of Equations
Solving the system of equations will give us the values of x1, x2, and λ.

Using the first equation, ∂U/∂x1 = λ*Px1, we can solve for x1 in terms of λ:

x1 = (1/λ)*∂U/∂x1

Similarly, from the second equation, we can solve for x2 in terms of λ:

x2 = (1/λ)*∂U/∂x2

Step 5: Calculate the Indirect Utility
Now that we have the demand functions for goods 1 and 2, we can substitute these values back into the utility function to calculate the indirect utility.

The indirect utility function is given by V(Px1, Px2, M) = U(x1*, x2*)

where x1* and x2* are the optimal quantities of goods 1 and