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\lhead{\textcolor{StatBlue}{\textbf{Statstrive ExamIQ}}}
\rhead{ISI MSQE PEA 2025}
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\begin{document}

\begin{center}
{\Huge \textbf{Statstrive ExamIQ}}\\[6pt]
{\Large ISI MSQE PEA 2025 Question Paper}\\[4pt]
{\small Regenerated from the local solution TeX source. Items missing full source detail are marked Needs Review.}
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\begin{longtable}{c l c}
\toprule
Question & Topic & Website Status \\
\midrule
\endhead
1 & Macroeconomics & Ready \\
2 & Macroeconomics & Ready \\
3 & Macroeconomics & Ready \\
4 & Macroeconomics & Ready \\
5 & Macroeconomics & Ready \\
6 & Macroeconomics & Ready \\
7 & Macroeconomics & Ready \\
8 & Macroeconomics & Ready \\
9 & Statistics & Ready \\
10 & Probability & Ready \\
11 & Econometrics & Ready \\
12 & Probability & Ready \\
13 & Probability & Ready \\
14 & Econometrics & Ready \\
15 & Econometrics & Ready \\
16 & Calculus & Ready \\
17 & Sequences and Series & Ready \\
18 & Optimization & Ready \\
19 & Calculus & Ready \\
20 & Number Theory & Ready \\
21 & Functional Equations & Ready \\
22 & Microeconomics & Ready \\
23 & Microeconomics & Ready \\
24 & Consumer Theory & Ready \\
25 & Public Goods & Ready \\
26 & Consumer Theory & Ready \\
27 & Consumer Theory & Ready \\
28 & Consumer Theory & Ready \\
29 & Consumer Theory & Ready \\
30 & General Equilibrium & Ready \\
\bottomrule
\end{longtable}

\newpage
\tableofcontents
\newpage

\section*{Question 1}

\addcontentsline{toc}{section}{Question 1}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Consider a standard Solow style economy where the aggregate production function is $Y = KN$. Assume no technological progress ($g=0$) and no population growth ($n=0$). Let $\bar K$ and $\bar Y$ be the steady state capital stock and output respectively, $s \in [0,1]$ the saving rate and $\delta \in [0,1]$ the depreciation rate. Find the steady-state capital per worker and output per worker.
\begin{enumerate}[label=(\Alph*)]
\item $s\delta$ and $(s\delta)^2$
\item $s/\delta$ and $s$
\item $s^2/\delta^2$ and $s/\delta$
\item None of the above
\end{enumerate}

\newpage

\section*{Question 2}

\addcontentsline{toc}{section}{Question 2}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Continue with Q1. Assume $\delta = 0.1$. At the steady state, consumption per worker and the saving rate that maximises consumption per worker are given by, respectively:
\begin{enumerate}[label=(\Alph*)]
\item $s(1-s)$ and $s = 0.5$
\item $(s/\delta)^2$ and $s = 0.4$
\item $s/\delta^2$ and $s = 0$
\item $\delta/(1-s)$ and $s = 1$
\end{enumerate}

\newpage

\section*{Question 3}

\addcontentsline{toc}{section}{Question 3}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Wage setting: $W = P(1-u)$. Production: $Y = N$. Markup $\mu = 20\%$. Labour force $L = 60$. Find the natural rate $u_n$ and natural output $Y_n$.
\begin{enumerate}[label=(\Alph*)]
\item $16.667\%$ and $50$
\item $23.5\%$ and $15.5$
\item $20\%$ and $15.5$
\item $20\%$ and $20$
\end{enumerate}

\newpage

\section*{Question 4}

\addcontentsline{toc}{section}{Question 4}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Open economy, fixed exchange rate $=1$. $IM = 0.3Y$, $X = 0.3Y^*$. The foreign country has the same structure. Find the domestic multiplier.

\begin{enumerate}[label=(\Alph*)]
\item $1$ \quad \item $2$ \quad \item $0.4$ \quad \item $0.8$
\end{enumerate}

\newpage

\section*{Question 5}

\addcontentsline{toc}{section}{Question 5}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Continue with Q4. Target $Y = 250$, foreign $G^*$ unchanged. Required increase in $G$?
\begin{enumerate}[label=(\Alph*)]
\item $112.4$ \quad \item $96$ \quad \item $38.8$ \quad \item $86$
\end{enumerate}

\newpage

\section*{Question 6}

\addcontentsline{toc}{section}{Question 6}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$Y = A\min\{K,L\}$, $A > 0$. With $W=\partial Y/\partial L$ and $R = \partial Y/\partial K$:
\begin{enumerate}[label=(\Alph*)]
\item $R$ and $W$ same for every $k$.
\item Same for $k\le 1$ but not for $k>1$.
\item Same for $k>1$ but not for $k\le 1$.
\item None of the above.
\end{enumerate}

\newpage

\section*{Question 7}

\addcontentsline{toc}{section}{Question 7}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$\dfrac{d}{dt}\bigl[e^{-\rho t}U'(C)\bigr] = e^{-\rho t}U'(C) r$. When is $\dot c > 0$?
\begin{enumerate}[label=(\Alph*)]
\item $r < \rho$ \quad \item $r > \rho$ \quad \item $r = \rho$ \quad \item Never.
\end{enumerate}

\newpage

\section*{Question 8}

\addcontentsline{toc}{section}{Question 8}

\noindent\textbf{Topic:} Macroeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Income stream $\{y_h, y_l, y_h, y_l, \ldots\}$, discount factor $\beta\in(0,1)$. Find constant $\bar c$ with equal present value.
\begin{enumerate}[label=(\Alph*)]
\item $\beta(y_h + y_l)/(1-\beta)$
\item $\beta^2(y_h+y_l)/(1-\beta)$
\item $\beta^2(y_h+y_l)(1-\beta)$
\item $(y_h + \beta y_l)/(1+\beta)$
\end{enumerate}

\newpage

\section*{Question 9}

\addcontentsline{toc}{section}{Question 9}

\noindent\textbf{Topic:} Statistics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Out of 20 questions, average correct $= 11.7$, average wrong $= 5.3$. Has the TA made a mistake?
\begin{enumerate}[label=(\Alph*)]
\item Yes \item No \item Maybe \item Insufficient information
\end{enumerate}

\newpage

\section*{Question 10}

\addcontentsline{toc}{section}{Question 10}

\noindent\textbf{Topic:} Probability \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Under the null $p = 1/2$, compare $P(\text{15 of 20 smokers die first})$ vs.\ $P(\text{12 of 20})$.
\begin{enumerate}[label=(\Alph*)]
\item $=1$ \item $<1$ \item $>1$ \item Insufficient info.
\end{enumerate}

\newpage

\section*{Question 11}

\addcontentsline{toc}{section}{Question 11}

\noindent\textbf{Topic:} Econometrics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$\hat\beta_0 = 5$, $\bar y = 65$, $\bar x = 90$. Predict $y$ for $x = 75$.
\begin{enumerate}[label=(\Alph*)]
\item $55$ \item $75$ \item $40$ \item $65$
\end{enumerate}

\newpage

\section*{Question 12}

\addcontentsline{toc}{section}{Question 12}

\noindent\textbf{Topic:} Probability \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Two fair coins. $A$: head on first coin. $C$: head on second coin. $D$: coins match. $G$: two heads. Which statement is false?
\begin{enumerate}[label=(\Alph*)]
\item $C$ and $D$ independent.
\item $A$ and $G$ independent.
\item $A$ and $D$ independent.
\item $A$ and $C$ independent.
\end{enumerate}

\newpage

\section*{Question 13}

\addcontentsline{toc}{section}{Question 13}

\noindent\textbf{Topic:} Probability \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Bowl: 3 chips $=$ Re.\ 1, 2 chips $=$ Rs.\ 4. Draw 2 without replacement. Expected sum?
\begin{enumerate}[label=(\Alph*)]
\item $<2$ \item $3$ \item $(3,4)$ \item $(4,5)$
\end{enumerate}

\newpage

\section*{Question 14}

\addcontentsline{toc}{section}{Question 14}

\noindent\textbf{Topic:} Econometrics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Two samples; $R^2$ of first $>$ $R^2$ of second. Which follows?
\begin{enumerate}[label=(\Alph*)]
\item Coefficient on $X$ greater in first.
\item $|$coefficient$|$ on $X$ greater in first.
\item Variance of $Y$ smaller in first.
\item None of the above.
\end{enumerate}

\newpage

\section*{Question 15}

\addcontentsline{toc}{section}{Question 15}

\noindent\textbf{Topic:} Econometrics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$\mathrm{sd}(x) = 2$, $\mathrm{sd}(y) = 4$. The OLS slope of $Y$ on $X$ equals:
\begin{enumerate}[label=(\Alph*)]
\item $8 r$ \item $2r$ \item $r/2$ \item None.
\end{enumerate}

\newpage

\section*{Question 16}

\addcontentsline{toc}{section}{Question 16}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$f(x) = 4x^3 - 6x^2 - 72x$. Where is $f$ increasing / decreasing?
\begin{enumerate}[label=(\Alph*)]
\item Increasing everywhere.
\item Decreasing everywhere.
\item Increasing in $(-\infty,-2)$, decreasing in $(-2,\infty)$.
\item Increasing in $(-\infty,-2)$ and $(3,\infty)$, decreasing in $(-2,3)$.
\end{enumerate}

\newpage

\section*{Question 17}

\addcontentsline{toc}{section}{Question 17}

\noindent\textbf{Topic:} Sequences and Series \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Find $\sum_{k=1}^{2024}\dfrac{1}{k(k+1)}$.
\begin{enumerate}[label=(\Alph*)]
\item $1$ \item $2025/2026$ \item $2024/2025$ \item $2025/2024$
\end{enumerate}

\newpage

\section*{Question 18}

\addcontentsline{toc}{section}{Question 18}

\noindent\textbf{Topic:} Optimization \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Find non-negative $x,y$ with $x + y = 16$ minimising $x^3 + y^3$.
\begin{enumerate}[label=(\Alph*)]
\item $x=y=8$ \item $x=15,y=1$ \item $x=12,y=4$ \item None.
\end{enumerate}

\newpage

\section*{Question 19}

\addcontentsline{toc}{section}{Question 19}

\noindent\textbf{Topic:} Calculus \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



$\displaystyle \int_1^2 \log x\, dx$.
\begin{enumerate}[label=(\Alph*)]
\item $\log 4$ \item $\log 4 - 1$ \item $\log 2 - 1$ \item None.
\end{enumerate}

\newpage

\section*{Question 20}

\addcontentsline{toc}{section}{Question 20}

\noindent\textbf{Topic:} Number Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Find $6^{2025} \mod 25$.
\begin{enumerate}[label=(\Alph*)]
\item $0$ \item $1$ \item $24$ \item $6$
\end{enumerate}

\newpage

\section*{Question 21}

\addcontentsline{toc}{section}{Question 21}

\noindent\textbf{Topic:} Functional Equations \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$f : \mathbb{N}_+ \to \mathbb{N}_+$, $f(x+y) = f(x) f(y)$, $f(1) = 2$. Find $2 + \sum_{n=1}^{2025} f(n)$.
\begin{enumerate}[label=(\Alph*)]
\item $2^{2025}$ \item $2^{2026}$ \item $3^{2024}$ \item None.
\end{enumerate}

\newpage

\section*{Question 22}

\addcontentsline{toc}{section}{Question 22}

\noindent\textbf{Topic:} Microeconomics \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Two demand groups. Both stop at $p = 5$. Per Re.\ price decrease: student demand $+5$, professor demand $+10$. $MC = 1$. Who is charged more?

\begin{enumerate}[label=(\Alph*)]
\item Professors \item Students \item Same \item Insufficient info.
\end{enumerate}

\newpage

\section*{Question 23}

\addcontentsline{toc}{section}{Question 23}

\noindent\textbf{Topic:} Microeconomics \quad \textbf{Difficulty:} Easy \quad \textbf{Status:} Ready



Bharat: linear demand, zero above $p=5$. Chinmayi: linear demand, zero above $p=10$. Market demand kink and flatness?

\begin{enumerate}[label=(\Alph*)]
\item Kink at Rs.~5, flatter to right.
\item Kink at Rs.~5, flatter to left.
\item Kink at Rs.~15, flatter to right.
\item Kink at Rs.~15, flatter to left.
\end{enumerate}

\newpage

\section*{Question 24}

\addcontentsline{toc}{section}{Question 24}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Income $= 200$. $p_B = 5$, $p_Y = 1$. $U = 4B + 2Y$. Plus a Rs.\ 50 gift card usable only on books.
\begin{enumerate}[label=(\Alph*)]
\item $10$ books, $200$ other goods.
\item $0$ books, $200$ other goods.
\item $20$ books, $150$ other goods.
\item $40$ books, $0$ other goods.
\end{enumerate}

\newpage

\section*{Question 25}

\addcontentsline{toc}{section}{Question 25}

\noindent\textbf{Topic:} Public Goods \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$MC = 25$. $5$ ``high'' people have $MB_H = 100 - 0.5Q$, $10$ ``low'' people have $MB_L = 50 - 0.25Q$. Max number of trees voluntarily provided with cost-sharing?

\begin{enumerate}[label=(\Alph*)]
\item $0$ \item $100$ \item $150$ \item $195$
\end{enumerate}

\newpage

\section*{Question 26}

\addcontentsline{toc}{section}{Question 26}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$x \succ y$ iff (i) $x_a + x_b > y_a + y_b$, or (ii) $x_a + x_b = y_a + y_b$ and $x_a > y_a$. Which statements are true?

\textbf{I.} $d_a = w/p_a$ if $p_a < p_b$. \quad
\textbf{II.} $d_a = w/p_a$ if $p_a \le p_b$. \quad
\textbf{III.} $d_a = 0$ if $p_b \le p_a$. \quad
\textbf{IV.} $d_b = w/p_b$ if $p_b < p_a$. \quad
\textbf{V.} $d_b = 0$ if $p_b < p_a$.

\begin{enumerate}[label=(\Alph*)]
\item I and III \item II and III \item II and IV \item I and V
\end{enumerate}

\newpage

\section*{Question 27}

\addcontentsline{toc}{section}{Question 27}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



Same setting as Q26. Are $d_a, d_b$ continuous in $(p_a, p_b)$?
\begin{enumerate}[label=(\Alph*)]
\item Both continuous.
\item $d_a$ continuous, $d_b$ not.
\item $d_a$ not continuous, $d_b$ continuous.
\item Neither continuous.
\end{enumerate}

\newpage

\section*{Question 28}

\addcontentsline{toc}{section}{Question 28}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Hard \quad \textbf{Status:} Ready



$U(x_1,x_2) = 4 x_1^2 + x_2^2$. Solve $\max U$ subject to $p_1 x_1 + p_2 x_2 = w$. Let $X(p_1, p_2, w)$ be the solution set. Which is true?

\begin{enumerate}[label=(\Alph*)]
\item $X(2,1,10)$ not convex.
\item $X(6,3,30)$ convex.
\item $X(1,1,10)$ not convex.
\item $X(4,2,20)$ convex.
\end{enumerate}

\newpage

\section*{Question 29}

\addcontentsline{toc}{section}{Question 29}

\noindent\textbf{Topic:} Consumer Theory \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



$U(x,y) = \dfrac{x}{1+x} + y$, $p_x = p_y$. Then:

\begin{enumerate}[label=(\Alph*)]
\item $x > 0$, $y = 0$.
\item $x = 0$, $y > 0$.
\item $x > 0$, $y > 0$.
\item $x = y = 0$.
\end{enumerate}

\newpage

\section*{Question 30}

\addcontentsline{toc}{section}{Question 30}

\noindent\textbf{Topic:} General Equilibrium \quad \textbf{Difficulty:} Moderate \quad \textbf{Status:} Ready



Two consumers, same preferences, same endowments. Strictly convex preferences. Then the economy has:
\begin{enumerate}[label=(\Alph*)]
\item No Pareto-efficient allocation.
\item Multiple competitive equilibrium allocations.
\item Exactly two Pareto-efficient allocations.
\item Only one competitive equilibrium allocation.
\end{enumerate}

\end{document}
