Objective speed. Descriptive depth.

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,

For the linear utility $u(x_1,x_2) = 3x_1 + 2x_2$ with budget $2x_1 + 3x_2 \leq M$, $x_1, x_2 \geq 0$, the Lagrangian is

Determine the equilibrium $(x_1^*, x_2^*, \lambda^*)$.

In month 1: $p_X = 2, p_Y = 3$. Consumer $A$ chose $(3,8)$, consumer $B$ chose $(6,6)$. In month 2: $p_X = 3, p_Y = 2$. Consumer $A$ chose $(8,3)$, consumer $B$ chose $(4,9)$. Which statement is correct?

Production function: $Y(L,K) = \min\{2L, K\}$; cost $C = wL + rK$, $w,r>0$. Find $(L^*, K^*)$ minimizing cost subject to $Y(L,K) \geq \bar Y$.

Inverse demand $P = 1 - q_1 - q_2$; cost $c_i(q_i) = \kappa_i q_i$, $\kappa_i \in (0,1)$. Find firm 2's Cournot equilibrium profit.

Which statement is correct in a two-good world?

A non-transitive preference relation can be represented by a utility function

$U(x,y) = \min\{x,y\}$, income $M = 200$. Old prices $(p_x, p_y) = (2,2)$; new prices $(2,3)$. Find the equivalent variation $A$ and compensating variation $B$.

$U(x,y) = -\big[(10-x)^2 + (10-y)^2\big]$. All prices equal $1$, income $= 40$. Optimal $(x,y)$?

$F(K,L) = \min\{aK, bL\}$, $a,b>0$, $a \neq b$. For fixed $\bar K$, the marginal product of labor is

Let $e_i(p_0)$ be the price elasticity of demand for good $X$ of consumer $i$ at $p_0$, with heterogeneous quantities. What is the elasticity of the aggregate demand at $p_0$?

$m$ identical firms with $C(q) = q^2 + 1$. Industry demand $D(P) = a - bP$, $a,b > 0$. Short-run equilibrium output per firm?

$C(q) = 3q^2 + 800$; $P = 280 - 4q$. Find the price elasticity of demand at the profit-maximizing price.

Solow model with saving $s$, depreciation $\delta$, labor growth $n$, no technological progress. Steady-state capital--output ratio?

Number of arrangements of the word ``PANDEMIC'' (8 distinct letters) such that the vowels appear together.

$f(x) = x^2 - x - 1$, $g(x) = x + 1$. Let $\alpha_1 > 0$ and $\alpha_2 < 0$ be the roots of $f(x)=0$; let $\beta_1 > 0$ and $\beta_2 < 0$ be the roots of $f(g(x))=0$. Identify which statement is incorrect.

$f(x) = \sqrt{1-x}$. Then $f(1 - f(x))$ equals

$f$ is increasing, concave, $C^2$; $g$ is decreasing, convex, $C^2$. Then $G(x) = g(f(x))$ is

$f : \mathbb R \to \mathbb R$ is $C^2$ with $f''(x) > 0$ for all $x$, $f(1)=1$, $f(2)=2$. Then,

$A, B$ non-singular of the same order, $C = B A B^{-1}$. For scalar $\lambda$, $\det(C + \lambda I)$ equals

$\displaystyle \lim_{x \to e} \frac{\ln x - 1}{x - e}$.

$f : [0,\infty) \to \mathbb R$, $f(0) = 0$, $f''(x) > 0$ for $x > 0$. Then $f(x)/x$ is

$f : A \to B$ with $|A|=5$, $|B|=2$. How many onto functions?

$f : \mathbb R \to \mathbb R$, $f(x) = \dfrac{x}{1+x^2}$. Then

$A$ is $3 \times 3$ with rank $3$; $B$ is $3 \times 4$ with rank $3$. Then $\operatorname{rank}(AB)$ is

Let $A$ be a specified $6 \times 8$ (or square, source unclear) matrix [the entries of $A$ were not legibly transcribed in the source]. Which one is an eigenvalue of $A$?

$A$ is a $5\times 5$ non-null singular matrix. Then $Ax = 0$ has

A family has two children. Find the probability that both are boys given that at least one is a boy.

Box 1 contains $\{1$ black, $1$ white$\}$; Box 2 contains $\{1$ black, $2$ white$\}$. Select a box at random and then a ball at random. Probability that the drawn ball is black?

$f : \mathbb R \to \mathbb R$, $f(x) = (x^2 + 1)^{2022}$.

A consumer's budgetary allocation for two commodities $x$ and $y$ is $m$. Her demand for $x$ is

Initially $m=1000$, $p_y=20$, $p_x=5$. The price of $x$ falls from $5$ to $4$. The substitution effect of the price change is:

\begin{center} \begin{tabular}{lcc|lcc} \toprule \multicolumn{3}{c|}{Year 1} & \multicolumn{3}{c}{Year 2} \ & Qty & Price & & Qty & Price \ \midrule Good 1 & 100 & 100 & Good 1 & 120 & 100 \ Good 2 & 100 & 100 & Good 2 & $x$ & 80 \ \bottomrule \end{tabular} \end{center} For which range of $x$ is the consumer's behaviour inconsistent with WARP?

$p=100-q$, $q=\sum_{i=1}^{23}q_i$, $c_i(q_i)=\dfrac{q_i^{2}}{2}$. Cournot--Nash equilibrium:

$p=100-q$, $n=10$ firms, $c_i(q_i)=q_i$. Total deadweight loss is

(The intended options correspond to $\frac{9^2}{2},\frac{9^2}{2},\frac{10^2}{2},\frac{100^2}{2}$, the first being $\frac{9^2}{2}$.)

$p=100-q$; large number of identical firms with

The competitive equilibrium price is:

$p=100-q$. Two firms with cost

i.e.\ each firm has capacity $10$. Demands: undercut wins the market, ties split. Bertrand--Nash equilibrium:

$500$ consumers uniform on $[0,1]$. Two hospital locations $a<b$. Travel cost = distance. Each individual values service at $4$. Fixed cost per hospital = $300$; marginal cost of treatment = $2$. Optimal locations?

Unit mass of consumers, valuation $\theta\sim F$ on $[\underline\theta,\overline\theta]$. Each buys $1$ unit iff $\theta\ge p$. Inverse demand $p(q)$. Find $p'(q)$.

Demand-determined output. Workers receive share $\lambda Y$, capitalists $(1-\lambda)Y$. Saving rates $s_w$ (workers) and $s_c$ (capitalists), with $s_w<s_c$. Investment $\bar I$ autonomous. If both $s_c$ and $s_w$ rise, in the new equilibrium:

With saving rates unchanged but $\lambda$ rising (more income to workers), the new equilibrium has:

$Y=\bar K^\alpha L^{1-\alpha}$, $0<\alpha<1$. Perfectly competitive. Find $L^d$ as a function of real wage $W/P$.

Household has endowment $\bar L$, supplies $L^s$, enjoys $\bar L-L^s$ leisure. $u=C^\beta+(\bar L-L^s)^\beta$, $0<\beta<1$. Budget: $PC=WL^s$. Find $L^s$ as a function of $W/P$.

Given the labour demand and supply derived above, the aggregate supply curve is:

Goods market: $S(Y,r)=I(r)$ with $0<S_Y<1,\;S_r>0,\;I_r<0$. Money: $M/P=L(Y,r)$ with $L_Y>0,\;L_r<0$. Find $dY/dP$.

$g_i=\alpha+\beta_0\log y_{i,0}+\beta_1\log n_i+\beta_2\log s_i+\gamma X_i+\varepsilon_i$. The Solow model predicts the sign of $\beta_0$:

Rank of $A$?

Bowl A: 2 red. Bowl B: 2 white. Bowl C: 1 white, 1 red. A bowl is selected at random and a coin drawn at random. Given the drawn coin is white, what is the probability that the other coin in the bowl is red?

Choose $n\in\{1,2,\dots,6\}$ uniformly, then choose $m\in\{1,\dots,n\}$ uniformly. Probability that $m=5$?

$F(1.4)=0.92,\;F(0.14)=0.555,\;F(-0.2)=0.42,\;F(-1.6)=0.055$. Diameter $\sim N(\mu,\sigma^2)$. $P(\text{diam}<1.8)=0.08$ and $P(\text{diam}>2.4)=0.055$. Find $\mu$.

$f$ differentiable, $f'$ strictly increasing. $f(1/2)=1/2,\;f(1)=1$. Compare $f'(1/2),\;1,\;f'(1)$.

$f(x)=\lfloor x\rfloor$. Evaluate $\displaystyle\int_0^{\sqrt 5}f(x^2)\,dx$.

Find the constant term in $\bigl(x+\tfrac{1}{x^2}\bigr)^{19}$.

Arjun and Gukesh each toss 3 fair coins. $p_1=P(\text{Arjun's heads}>\text{Gukesh's heads})$. Find $p_1$.

Find the number of real solutions of $x|x|+1=3|x|$.

Positive integers $k_1,\dots,k_n$ (not necessarily distinct) with

Maximum $n$?

Monkey at $(0,0)$ in $\mathbb R^2$ jumps a distance of $5$ to an integer point. Number of possible end positions?

Strip of $n+2$ squares; squares $1$ and $n+2$ are black; squares $2,\dots,n+1$ are white. Girl picks a white square uniformly, then picks one of its two neighbours uniformly. Probability the neighbour is white?

$f(x)=\max(|x|,x^2)$ for all $x\in\mathbb R$. Which is true?

With $f$ as in Q28, $D=\{(x,y)\in\mathbb R^2:y\ge f(x)\}$ (epigraph of $f$). Which is true?

$f:[-1,1]\to\mathbb R$ with

(Standard interpretation: $f(x)=f\!\Bigl(\dfrac{2x^2}{2-x^2}\Bigr)$ or similar.) Find $f(-1)$.

If $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function at $a\in\mathbb{R}$ such that $f'(a)=af(a)$, then what is

Suppose $S_n$ is defined as follows for every positive integer $n\ge 2$:

The value of $\displaystyle\lim_{n\to\infty}S_n$ is

Suppose

where $a_1,a_2,a_3$ are given real numbers. Then it is necessarily true that

Let $f:\mathbb{R}\to\mathbb{R}$ be defined as

where $a,b,c$ are positive real numbers. Which of the following statements is correct, under the assumption that $f$ is continuous?

Consider $f:[0,1]\to[0,1]$ such that $f(x)=\dfrac{x}{2-x}$. Which of the following statements is incorrect?

Let $f:\mathbb{R}\to\mathbb{R}$ be the function $f(x)=|x|+x^{2}\ \forall x\in\mathbb{R}$. Which of the following statements about $f$ is correct?

$\displaystyle\int x^{3}e^{x^{2}}\,dx$ equals

Let $A$ be a $3\times 3$ matrix having eigenvalues $2,7,5$. What is the determinant of $A+2I$?

Let $x$ and $y$ be two column vectors of length $3$ such that $\sum_{i}x_{i}y_{i}=1$. What is the rank of $xy^{T}$?

Let $A$ be a $3\times 3$ matrix such that $Ax=x$ for all column vectors $x$ of length $3$. Which of the following statements is correct?

Let $S=\{(x_1,0):x_1\in\mathbb{R}\}$ and $T=\{(0,x_2):x_2\in\mathbb{R}\}$. Which of the following is correct?

In a chess tournament, there are both boys and girls. Each player plays with another player exactly once. If there are $45$ games in total and exactly $15$ of them feature only boys, how many games feature a boy and a girl?

What is the number of possible arrangements of the letters of the word madam' such that the two a's never appear in consecutive positions?

A monkey starts at $(0,0)$ on the $xy$-plane in period $1$. From any position $(x,y)$ in a period, the monkey can only jump to $(a,b)$ in the next period, where $a\in\{x+1,x,x-1\}$ and $b\in\{y+1,y,y-1\}$. How many possible positions can the monkey be in period $2$?

In the previous question, suppose in every period the monkey can go to any of the possible positions in the next period with equal probability. Then what is the probability that the monkey is at a distance of more than $1$ from $(0,0)$ in period $2$?

For a given data set, let the least squares regression line be $y=10+2x$. It is given that variance of $x$ is $9$ and variance of $y$ is $81$. What is the correlation coefficient between $x$ and $y$?

An urn contains $4$ white, $6$ red, and $5$ black balls. $5$ balls are randomly selected from the urn. Let $X$ and $Y$ denote respectively the number of white and black balls selected. Suppose $Y=2$, i.e., $2$ of the $5$ selected balls are black. What is the probability that $X$ takes the value $2$?

In answering an MCQ with $4$ choices, a student either knows the answer (with probability $\tfrac14$) or guesses (with probability $\tfrac34$). A guess is correct with probability $\tfrac14$. What is the probability that the student knew the answer, given that he answered correctly?

Let $X_1,X_2,\dots,X_5$ be i.i.d.\ random variables with $E(X_i)=0$ and $V(X_i)=1$. What is the value of

Consider $X\sim\text{Bin}(n,p)$. Which of the following is incorrect?

Consider an economy with $y=Ak$ (per-capita form), saving rate $s\in(0,1)$, population growth rate $n>0$ and depreciation rate $\delta\in(0,1)$. Assume parameters ensure positive long-run growth of $y$. Which of the following is incorrect?

Solow economy with $Y=K^{1/3}L^{2/3}$, no population growth or technological change, $\delta=0.05$, $s=0.2$. What is the steady-state capital--labour ratio?

With the same Solow economy (no population growth, $\delta=0.05$, $Y=K^{1/3}L^{2/3}$), a social planner wishes to maximize steady-state per-capita consumption. What savings rate $s$ achieves this?

A Solow economy has $y=k^{1/2}$, $\delta=0$, labour grows at $n>0$. If the steady-state value of $k$ is $50$ and the current value of $k$ is $2$, what is the current growth rate of per-capita output?

Two countries $A$ and $B$ produce wheat and television respectively, with $Y_A=200$ and $Y_B=250$. The TV price is normalised to $1$, the relative price of wheat is $p$. Let $E_A$ (in wheat units) and $E_B$ (in TV units) denote total expenditures. The income identity is $pY_A+Y_B=pE_A+E_B$. Initially $E_A=100$; $A$ reduces $E_A$ to $70$. Suppose $1/3$ of each country's expenditure is on wheat. What happens to the equilibrium $p$?

Same setup as Q25, except now each country spends $2/3$ of its expenditure on its own good and $1/3$ on the foreign good. What is the impact on $p$ when $E_A$ falls from $100$ to $70$?

A consumer consumes two goods $X$ and $Y$. It is observed that her consumption of $X$ always falls when the price of $X$ falls, ceteris paribus. Suppose now income rises, with prices of $X$ and $Y$ held constant. What happens to consumption of $X$?

$A$ has a pond which yields $f$ fish at $t=1$ and nothing at $t=2$. He consumes only fish. Storage is costless and undepreciated; no discounting. Utility in any period $t$ is $u(c_t)=c_t^{\,n}$ with $0<n<1$. Optimal consumption?

Country $C$ has $10{,}000$ identical farmers with cost $C(q_i)=0.5q_i^{2}+4q_i+100$, aggregate demand $Q=-10{,}000p+400{,}000$. Current price support $p=30$; government buys all unsold output at $30$. The government proposes removing the support and giving each farmer an equal share of the saved expenditure as a lump-sum transfer. Each farmer must, however, pay $300$ to open a bank account to receive the transfer. By how much does a farmer gain (or lose)?

A consumer consumes three goods $X,Y,Z$ over three periods. Prices and bundles:

Which of the following statements about her preferences is correct?

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.

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:

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 $Y_n$.

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

Continue with Q4. Target $Y = 250$, foreign $G^*$ unchanged. Required increase in $G$?

$Y = A\min\{K,L\}$, $A > 0$. With $W=\partial Y/\partial L$ and $R = \partial Y/\partial K$:

$\dfrac{d}{dt}\bigl[e^{-\rho t}U'(C)\bigr] = e^{-\rho t}U'(C) r$. When is $\dot c > 0$?

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.

Out of 20 questions, average correct $= 11.7$, average wrong $= 5.3$. Has the TA made a mistake?

Under the null $p = 1/2$, compare $P(\text{15 of 20 smokers die first})$ vs.\ $P(\text{12 of 20})$.

$\hat\beta_0 = 5$, $\bar y = 65$, $\bar x = 90$. Predict $y$ for $x = 75$.

Two fair coins. $A$: head on first coin. $C$: head on second coin. $D$: coins match. $G$: two heads. Which statement is false?