Answer:
To find the indirect utility function, we need to solve the utility maximization problem subject to the budget constraint and express the maximum utility achieved as a function of the prices and income.
Given the utility function U = max(X, Y) and the budget constraint P₁X + P₂Y = M, we can solve for X and Y in terms of prices (P₁, P₂) and income (M).
First, let's consider the different cases:
If P₁ ≤ P₂:
In this case, the individual would choose to consume only good X. Therefore, X = M / P₁ and Y = 0.
If P₂ < P₁:
In this case, the individual would choose to consume only good Y. Therefore, X = 0 and Y = M / P₂.
Now, we can express the indirect utility function in terms of the prices (P₁, P₂) and income (M) for each case:
a) If P₁ ≤ P₂:
In this case, the individual maximizes utility by consuming only good X.
Therefore, the indirect utility function is V(P₁, P₂, M) = U(X, Y) = U(M / P₁, 0) = M / P₁.
b) If P₂ < P₁:
In this case, the individual maximizes utility by consuming only good Y.
Therefore, the indirect utility function is V(P₁, P₂, M) = U(X, Y) = U(0, M / P₂) = M / P₂.
c) and d) do not match any of the cases above.
Therefore, among the given options, the correct answer is:
a) M / max(P₁, P₂).