PEA 2022 PEA Q1 | ISI MSQE

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2022 PEA Q1

Consider an economy with two goods $X$ and $Y$ . Let the utility function be given by $u(x,y) = A\sqrt{xy}$ where $A > 0$ , $x \geq 0$ and $y \geq 0$ . The budget constraint is $P_X x + P_Y y \leq M$ , with $M > 0$ . Let $P_X = P_Y > 1$ and let $(x^*, y^*)$ denote the utility maximizing bundle. Then,

Reveal answer and solution

Answer

Solution

1
The function $u(x,y) = A\sqrt{xy}$ is a symmetric Cobb--Douglas with equal exponents.
2
Maximizing $\tfrac{1}{2}\ln x + \tfrac{1}{2}\ln y$ subject to $P_X x + P_Y y = M$ gives
3
$x^* = \frac{M}{2P_X}, \qquad y^* = \frac{M}{2P_Y}.$
4
Since $P_X = P_Y$ , we obtain $x^* = y^*$ .

Answer structure / marking notes

Option (D) is a budget identity only when $P_X = P_Y = 1$ ; here prices exceed $1$ .

Content note

Imported from public/resources/isi/msqe/solutions/pea/2022/ISI_MSQE_PEA_2022_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.