Микроэкономика 2 — МИЭФ, 2025 demo midterm

John consumes two goods, $X$ and $Y$. His preferences are represented by

$$U(X,Y)=2\sqrt{X}+Y.$$

Initially,

$$p_X=1,\qquad p_Y=2,\qquad m=6.$$

The government introduces a per-unit tax $t$ on good $X$.

(a) Compute the tax rate $t^*$ that maximizes the government’s tax revenue.

(b) Would the answer to part (a) change if John’s income were

$$\widetilde m=1?$$

(c) Continue to assume

$$(p_X,p_Y,m)=(1,2,6)$$

and suppose the per-unit tax is

$$t=\frac14.$$

The government replaces the per-unit tax by a lump-sum tax $T$ equal to the change in consumer surplus associated with $t$.

By how much does this replacement increase or decrease government revenue?

(d) Would the government obtain more or less revenue if it instead set the lump-sum tax $T$ equal to the equivalent variation associated with $t$?

Would this option be more preferable for the consumer?

Judith lives for two periods, $t=0,1$. Her utility function is

$$U(c_0,c_1)=-\frac{100}{c_0}+\sqrt{c_1},$$

where $c_t$ is consumption in period $t$.

Her current-period income is

$$m_0=\$100.$$

Her future-period income is random:

$$m_1= \begin{cases} 16, & \text{if there is a crisis},\\ 49, & \text{if there is no crisis}. \end{cases}$$

The probability of a crisis is

$$\frac13.$$

Financial markets are absent: Judith can neither borrow nor lend.

(a) The government offers Judith a savings program. She must invest exactly one half of her current income $m_0$ in an asset whose net return is

$$r= \begin{cases} -0.04, & \text{if there is a crisis},\\ 0.02, & \text{if there is no crisis}. \end{cases}$$

Will Judith agree to participate in the program?

(b) The government offers Judith an insurance contract. In period $t=1$:

• she receives an additional $20 if there is a crisis;
• she pays $x if there is no crisis.

What is the maximum amount $x$ that Judith is willing to pay?

By how much does her expected future consumption decrease if she buys this insurance?

Illustrate the answer in a contingent-commodities diagram.

(c) The government offers to pay Judith $y in the current period, while promising to increase her future income by $33 if a crisis occurs.

What is the maximum amount $y$ that Judith is willing to pay?

Illustrate the answer in a wealth-utility diagram.

(a) (10 points) Mixed-strategy equilibrium

Consider the following normal-form game:

The players use a mixed-strategy Nash equilibrium:

$$\left\{ pA_1+(1-p)B_1;\ qA_2+(1-q)B_2 \right\}, \qquad 0<p<1,\quad 0<q<1.$$

It is known that the equilibrium expected payoff of player 1 is 3.

What information about $q$, $p$, or both can be inferred from this fact?

(b) (12 points) Sequential game

Consider the following sequential game with players $A$ and $B$:

(i) Identify all subgames and find all pure- and mixed-strategy subgame-perfect Nash equilibria. Begin from the definition of SPNE.

(ii) Suppose player $A$ chooses $S$ at both of her decision nodes. Player $B$ chooses $S$ at his first decision node and $N$ at his last decision node.

• Is this strategy profile a Nash equilibrium? If yes, prove it. If no, explain what is wrong with it.
• Does the strategy profile contain a non-credible threat? Begin with a definition and explain carefully.

(c) (8 points) Infinitely repeated game

Suppose the following game is repeated infinitely, and both players have a common discount factor

$$\delta\in(0,1).$$

Consider the statement:

For sufficiently high values of $\delta$, there is an SPNE in which $(A_1,A_2)$ is played in the first two periods and $(B_1,B_2)$ is played in every subsequent period.

Is the statement true or false? Explain carefully.