PEA 2026 PEA Q30 | ISI MSQE

PEAMCQModerate

2026 PEA Q30

A two-period consumer maximises

U(C_1, C_2) = \log C_1 + \tfrac{1}{1+\rho}\log C_2, \qquad \rho > 0,

subject to $C_1 + S = w$ and $C_2 = (1+r) S$ , with $w, r > 0$ . If $w$ rises by $1\%$ and $r$ falls by $1\%$ , then $S$

Reveal answer and solution

Answer

Solution

1
Substitute $C_2 = (1+r)(w - C_1)$ into $U$ and differentiate w.r.t.\ $C_1$ :
2
$\frac{1}{C_1} - \frac{1}{1+\rho}\cdot \frac{1+r}{C_2}\cdot(1+r)\cdot\frac{1}{1+r} = \frac{1}{C_1} - \frac{1}{(1+\rho) C_2}\,(1+r)\cdot\frac{1}{1+r} \cdot (1+r).$
3
A cleaner derivation: the Euler equation $\dfrac{C_2}{C_1} = \dfrac{1+r}{1+\rho}$ , combined with the lifetime budget constraint $C_1 + \dfrac{C_2}{1+r} = w$ , gives
4
$C_1 + \frac{1}{1+r}\cdot \frac{1+r}{1+\rho}\,C_1 \;=\; w \quad\Longrightarrow\quad C_1\!\left(1 + \frac{1}{1+\rho}\right) = w \quad\Longrightarrow\quad C_1 = \frac{(1+\rho)\,w}{2+\rho}.$
5
Therefore
6
$S = w - C_1 = w\!\left(1 - \frac{1+\rho}{2+\rho}\right) = \frac{w}{2+\rho}.$
7
This is the well-known log-utility result: *saving is proportional to wage
8
and independent of $r$ * (income and substitution effects of an interest-rate
9
change cancel exactly under log preferences).
10
Since $S$ depends only on $w$ (not on $r$ ):
11
\begin{itemize}
12
- A $1\%$ rise in $w$ raises $S$ by exactly $1\%$ .
13
- A $1\%$ fall in $r$ leaves $S$ unchanged.
14
\end{itemize}
15
Therefore $S$ rises by exactly $1\%$ .

Answer structure / marking notes

Under log utility the income and substitution effects of a change in $r$ on saving cancel out exactly; this is a special property of log preferences.

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Review Flags

Question 18. As transcribed (with $u_A = x_A + y$ , $u_B = 2x_B + y$ and production constraint $y + m(x_A + x_B) = 1$ ), the Samuelson condition yields $m = 2/3$ , which is not among the printed options $\{4/5,\,4/7,\,4/9,\,4/11\}$ . The problem statement (likely the utility functions or the production technology in the underlying source) should be verified before finalising the option choice.

All other questions are flagged as either Verified or Draft (see the Answer-Key Summary at the front of this booklet).

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\begin{center} \textit{End of PEA Solutions --- ISI MSQE 2026, prepared for Statstrive.} \end{center}

Content note

Imported from public/resources/isi/msqe/solutions/pea/2026/ISI_MSQE_PEA_2026_Solutions.tex. Question wording is retained from the available local TeX source; incomplete option blocks or ambiguous source status are flagged for review.