---
tags: micro, core, group
---
# Core June 2019 Micro

# S1
## Question
Consider an individual who faces a risky decision and has a Bernoulli utility function over money (x) given by $u(x) = \alpha x + \beta x^2$
a) This utility function is reasonable only for a certain range of values of x. What is that range? Why?

Assume the rest of the problem applies only to this range.
b) Provide sufficient conditions on $\alpha$ and $\beta$ such that the agent will be risk averse. Assuming that these conditions hold, determine whether the agent has increasing, decreasing, or constant absolute risk aversion.

c) Consider a lottery represented by distribution function F that has mean $\mu$ and variance $\sigma^2$.
Show that for an agent with this utility function, it is possible to write $E(u(x)) = g(\mu, \sigma^2)$ for some function $g$.

d) Consider an investor with this utility function deciding between a riskless asset that pays zero interest, and a risky asset. Denote the return of the risky asset by z, so that if the agent invests a in the risky asset, and the realized return is z, she receives a(1+z) dollars from the risky asset (plus whatever she gets from the riskless asset). The random variable z is distributed according to a cdf F that has a strictly positive mean $\mu > 0$ and has variance $\sigma^2$. The agentís initial wealth is w. Find the agent's optimal allocation between the riskless and risky asset as a function of the parameters. Be sure to include the no borrowing and no short-selling constraints. Can you determine whether or not these constraints will be binding? Provide intuition.

## Answer
### a) Reasonable utility function

$$u'(x) > 0 \implies \alpha + 2 \beta x > 0$$ \begin{cases} x > -\frac{\alpha}{2 \beta} &\text{ if } \beta>0 \\
x< -\frac{\alpha}{2 \beta} &\text{ if } \beta<0
\end{cases}

### b)
(i) Sufficient condition for the agent to be risk averse:
Note that risk-averse agent should have $u''(x)<0$ so $u''(x) = 2\beta <0$.

Absolute risk aversion is then
$$-\frac{u''(x)}{u'(x)}= -\frac{2\beta}{\alpha+2\beta x} >0 \implies IARA$$

### c)
$x \sim F(\mu, \sigma^2) \implies E[x] = \mu, V(x) = \sigma^2, E[X^2] = \mu^2 + \sigma^2$
$$\mathbb{E}[u(x)] = \mathbb{E}[ \alpha x + \beta x^2] =\alpha \mu + \beta (\mu^2 + \sigma^2) = g(\mu, \sigma^2)$$

### d)
2 assets: **(safe)** $0$ interest, **(risky)** random return $z \sim F(z)$
$w$: initial wealth, $a$: purchase of risky asset
Realized outcome(asset): $w-a + a(1+z) = w+az$
Agent's utility maximization problem:

$$\max_{a} U = \int u(w+az) dF(z) \\
\text{s.t. } 0<a<w $$

Note that
\begin{aligned}
U'(a) &= \int z u'(w+az) dF(z) \\
U''(a) &= \int z^2 u''(w+az) dF(z) <0 \quad \text{if risk averse} (u''<0)
\end{aligned}

$SOC <0$ so $a^* \text{ such that } U'(a^*)=0$ is sufficient and necessary condition for an optimal allocation in risky asset.

\begin{aligned}
U'(a) &= \int z[ \alpha+2\beta(w+az) ]dF(z) \\
&= \int (\alpha+2\beta w)z dF(z) + \int 2 a \beta z^2 dF(z) \\
&= (\alpha+2\beta w)E[z] + 2\beta E[z^2]a = 0 \\ \\
\Leftrightarrow a^* &= -\frac{\alpha+2\beta w E[z]}{2\beta E[z^2]} = -\frac{(\alpha+2\beta w)\mu}{2\beta(\mu^2 + \sigma^2)}
\end{aligned}

The denominator is greater than zero, the numerator is less than zero so $a^*>0$.

The non short-selling costraint is always satisfied.

# S2
## Question

Theresa May (T) and Parliament (P) are negotiating Brexit and there are $3$ possible outcomes; Soft Brexit $(S)$, Hard Brexit $(H)$, and Very Hard No-Deal Brexit $(\emptyset)$. T's preferences are given by $u$ with $u(S) > u(H) > u(\emptyset)$ = 0. P's preferences are given by $v$ with $v(H) > v(S) > v(\emptyset) = 0$. The European Union will grant exactly one extension so there are two periods of bargaining.

In period $1$, T makes an offer $(S; H, \text{ or } \emptyset )$ and P accepts or rejects the offer. If T's offer is accepted, that outcome is implemented. If T's o§er is rejected, the game moves to period $2$. In period $2$, T makes another offer $(S; H, \text{ or } \emptyset )$ and P accepts or rejects the offer. If T's offer is accepted, that outcome is implemented and payoffs are discounted by $\delta \in (0, 1)$. If T's offer is rejected, $\emptyset$ is implemented.

(a) What is the subgame perfect equilibrium?

Now suppose for the rest of the problem, that there are two possible types of P. The "regular" type has the preferences above. The "hard-or-bust" type will always accept $H$ and reject $S$ (even if it means $\emptyset$ will be implemented). P's type is privately known by P, and T believes P is the regular type with probability $\gamma$.

Assume $v(S) < \delta v(H)$ and $\gamma u(S) > u(H)$ and consider the solution concept perfect Bayesian equilibrium.

(b) Consider the subgame after T's offer is $S$ in period 1. Characterize an equilibrium where the regular type of P mixes in period 1 and then, conditional on P rejecting, T mixes between $S$ and $H$ in period 2.

(c ) Show that there is a generically unique equilibrium in the whole game (generically only because there could be one value of $\delta$ where there are multiple equilibria). For what values of $\delta$ is there sometimes delayed agreement, even if P is regular?
## Answer
### a) Subgame perfect equilibrium
Solve by backward induction.

In the final period, $P$ would accept any offer (since $v(S)>0, v(H)>0, \text{ and } v(\phi)\ge 0$) so $T$ would offer $S$ which gives her the highest payoff.

Then, in the first period, $P$ would accept any offer such that $v(\cdot) \ge \delta v(S)$ since $v(H)>v(S)>\delta v(S)$

Given this, $T$ would offer $H$ in $T=1$ if $u(H)>\delta u(S)$ and $S$ if $u(H) < \delta u(S)$

Hence, the subgame perfect equilibrium in this game is:
$T=1$
$T$ offers \begin{cases} S &\text{ if } & u(H) < \delta u(S) \\ H &\text{ if } & u(H) > \delta u(S)\end{cases}
$P$ accepts $H$ or $S$.

$T=2$
$T$ offers $S$, $P$ accepts any offer.

### b) A subgame after T offers S in period 1.
incomplete extensive form of the game:


Non-trivial information set is $\{P \text{ rejects}\}$
Denote $T$'s beliefs about $P$ being regular and HoB by $(\pi, 1-\pi)$. Regular type $P$ mixes (accept, reject) with probability $(\alpha, 1-\alpha)$, and $T$ mixes $(S,H)$ w.p. $(\beta, 1-\beta)$ in $T=2$.

In $T=2$, expected payoff for $T$ given beliefs $(\pi, 1-\pi)$:
$EU(S) = \pi \delta u(S), \quad EU(H) = \delta u(H)$

If $T$ was mixing the two strategies, it is the case that $\pi = \frac{u(H)}{u(S)}$

Also, $T$ would mix the two strategies which would make $P$ indifferent between accepting and rejecting in period $1$;
$$\underbrace{v(S)}_{EV(A)} = \underbrace{\beta \delta v(S) + (1-\beta) \delta v(H)}_{EV(R)}$$

$$\beta =\frac{\delta v(H) - v(S)}{\delta v(H) - \delta v(S)}$$

In $T=1$, $P$ is mixing the strategies which makes $T$ indifferent between offering $S$ and $H$.

$$EU(H) = EU(S) \Leftrightarrow u(H) = \gamma \alpha u(S) \\
\therefore \alpha = \frac{\gamma u(S)}{u(H)}$$

### c

If $v(S) > \delta v(H)$, i.e. $\frac{v(s)}{v(h)} > \delta$ , regular P will choose $(S, S)$, and regular P will use $( \text{always accept at } T=1, \text{always accept at } T=2)$, irregular P will choose $(\text{ accept if H}, \text{ accept if H} )$.

Irregular P's strategy is just based on its preference.

Since $v(S) > \delta v(H)$, if T use $S$ with positive probability at first stage, always accept at first stage dominates any other strategy for regular P. Given regular P will accept any offer at first stage and $\gamma u(S) > u(H)$, T will choose $S$ at first stage.

Since $\gamma u(S) > u(H)$, T will choose $S$ at second stage and regular P will also choose accept. This is an unique NE.

If $v(S) < \delta v(H)$, the NE is what we did in part b, which is unique.

If $v(S) = \delta v(H)$, the above two NEs hold, so we have equilibria.

# S3
https://open.lib.umn.edu/principleseconomics/chapter/14-2-monopsony-and-the-minimum-wage/

# S4
See Micro2 DQ1

# L1
## Question
Suppose $n$ risk-neutral agents compete for a prize worth $V$. If agent $i$ exerts effort $x_i$; this costs her $x_i^\gamma/\gamma$ and the probability she wins is $x_i/X$ where $\gamma> 0$ and $X = \sum_j x_j$ (where $j = 1 \cdots , n$, and includes $i$).

Agent $i$'s expected payoff is thus

$$
\pi_i = \frac{V x_i}{X} - \frac{x_i^\gamma}{\gamma}, i=1, \cdots, n
$$

(a) Find $i$'s first-order condition for the choice of $x_i$.

(b) Evaluate this at a symmetric (pure-strategy) equilibrium candidate and hence find the candidate-equilibrium common level of $x$.

(c ) Find the corresponding candidate-equilibrium level of each agentís expected payoff, $\pi$.

(d) Under what conditions is the expected payoff in (c ) positive?

(e ) Evaluate the second-order condition at the candidate symmetric equilibrium.

(f) Explain what your answers in (d) and (e) say about the candidate equilibrium.

(g) Under what conditions on the parameters does a symmetric pure-strategy equilibrium exist? Are the conditions necessary as well as sufficient?
(h) Assuming a symmetric equilibrium exists, determine whether an increase in $n$ increases or lowers
(i) each agentís expected payoff, $\pi$.
(ii) the sum of all agentsípayoffs.

Explain the intuition in parts (i) and (ii).
(i) Explain intuitively the issues regarding the existence of a symmetric equilibrium and, if a symmetric equilibrium doesnít exist, what an equilibrium might look like.

## Answer
### a)
$$FOC:\ \frac{\partial \pi_i}{\partial x_i} = \frac{\partial}{\partial x_i}\left( \frac{Vx_i}{x_i + \sum_{j \neq i}x_j}- \frac{x_i^{\gamma}}{\gamma} \right)
= \frac{VX-Vx_i}{X^2} - x_i^{\gamma-1} = 0$$
$$x_i^{\gamma -1} = \frac{VX_{-i}}{X^2}, \text{ where } X_{-i}= \sum_{j \neq i}x_j$$

### b)
Assuming a symmetric (pure strategy) equilibrium, let $x_i = x_j = x$ and $X = nx$

Then the first order condition becomes:
\begin{aligned}
x^{\gamma-1} &= \frac{V(n-1)x}{(nx)^2} \\
x^{\gamma} &= \frac{V(n-1)}{n^2} \\
x &= \left( \frac{V(n-1)}{n^2}\right)^{1/\gamma}
\end{aligned}

### c)
Candidate equilibrium payoff is then
$$\pi = \frac{V}{n} - \frac{V(n-1)}{\gamma n^2} = \frac{\gamma n V - nV + V}{\gamma n^2}$$

### d)
Since $\gamma, n, V>0$, the expected payoff in part c) is positive if $$ n(\gamma - 1)+ 1 >0 $$

### e) SOC
\begin{aligned}
&-2VX_{-i} (x_i + X_{-i})^{-3} - (\gamma-1)x_i^{\gamma-2} \\
=& 2V(n-1)x(nx)^{-3} - (\gamma-1)x^{\gamma-2} \quad \text{ imposing symmetry}\\
=& -\frac{2V(n-1)}{n^3x^2}-(\gamma-1)x^{\gamma-2} \\
=& -\frac{x^{\gamma}}{x^2}\left( \frac{2}{ n } \underbrace{\frac{V(n-1)}{x^{\gamma}n^2}}_{=1, \text{ if symmetric}} + \gamma -1\right)
\end{aligned}

$SOC<0$ if $\frac{2}{n}+\gamma-1 >0$

### f,g) Conditions for the equilibrium to exist

FOC tells that if a symmetric equilibrium exists (necessary condition),
$$x = \left( \frac{V(n-1)}{n^2}\right)^{1/\gamma}$$

SOC should be negative to have a symmetric equilibrium (sufficient condition): $$\gamma > \frac{n-2}{n}$$

However, by part d), we can see that even if the second-order condition is satisfied, we might have a negative payoff at the equilibrium.

In order to ensure that a symmetric pure-strategy equilibrium exist, we'll need to have
$$x = \left( \frac{V(n-1)}{n^2}\right)^{1/\gamma}$$
as a necessary condition and
$$\gamma > \frac{n-1}{n}$$ as a sufficient condition.

### h) Assuming a symmetric equilibrium
(i)

(ii)

### g)

# L2
## Question
## Answer
### a
A's problem:

$$
\max_{c,t} c^{1/2} t^{1/2} \\
\text{s.t. }\\
p_cc+p_tt \leq 4w \\
or, c+pt \leq 4w
$$

we get, $p=\frac{c}{t}$

Ford's problem:
$$
\max_{L} 1L-wL
$$

we get, w=1

GM's problem:
$$
\max_{L} p*2L-wL
$$

we get, w =2p

therefore, combining agent A's problem and ford's problem-

$$
p=\frac{1}{2} \\
c=2t
$$

from market clearing,

$$
c+pt=4w \\
c=2 \\
t=4
$$

### b
B's problem:
$$
\max_{c,t} c^{1/2} t^{1/2} \\
\text{s.t. }\\
p_cc+p_tt \leq 4w \\
or, c+pt \leq 4w
$$

we get, $p=\frac{c}{t}$

Honda's problem:
$$
\max_{L} \frac{3}{4}L-wL
$$

we get, $w=\frac{3}{4}$

Toyota's problem:
$$
\max_{L} pL-wL
$$

we get, $w=p=\frac{3}{4}$

from market clearing,

$$
c+pt=4w \\
t=2 \\
c=\frac{3}{2}
$$

### c
Let $W_A$ and $W_J$ be the wage rate in the US and Japan, respectively. Let $P_C$ adn $P_T$ be the price of car and truck. By the competitive market, the price should equal to the cost, which is labor input times wage rate.

The price/cost of producing vehicles in the US:

$$
P_C= W_A, P_T= \frac{1}{2} W_A
$$

The price/cost of producing vehicles in the Japan:

$$
P_C= \frac{4}{3} W_J, P_T= W_J
$$

By the free trade, the price of the same good should be equal.

If all firms are producing, we have

$$
\frac{3}{4} W_A= W_J \\
\frac{1}{2} W_A= W_J,
$$

which is impossible.

For the Japanese, making cars can exchange more American labors, in other words, more American goods. Hence, the inefficient Toyota will shut.

### d
Normalize $P_T=1, P_C=P$. Since only Honda works, Japanese's output is $3$ cars and their income is $3P$.

Because the production function is linear, when $P>2$, American will only produce cars; when $P<2$, American will only produce trucks.

By the C-D utility function, everyone should consume both goods. Hence, US cannot specialize on car. (Otherwise, no one produce trucks) And $P<=2$.

Suppose $P<2$, and the US specialize on trucks, then American's output is $8$ trucks, and their income is $8$.

By the C-D utility, American will consume $4$ trucks and $4/P$ cars; Japanese will consume $1.5P$ trucks and $1.5$ cars. By the resource constraint,

$$
4 + 1.5P =8, \frac{4}{P} + 1.5 = 3.
$$

We can solve $P=8/3>2$, which is a contradiction. Hence, the only equilibrium price is $P=2$.

American's income is $4P=8$, Japanese's income is $6$. By C-D utility, US consumes $2$ cars and $4$ trucks; Japan consumes $1.5$ cars and $3$ trucks.

US produces $0.5$ cars and $7$ trucks; Japan produces $3$ cars.

### e

### f