\subsection*{Problem 7}
\begin{enumerate}
\item The risky bundle is described by the lottery $L:\{5999,\frac{4}{5};0,\frac{1}{5}\}$. \\
Therefore the two possible outputs are $V_1=5999$ with probability $p_1=\frac{4}{5}$ and corresponding utility $U_1=9.39$,
and $V_2=0$ with probability $p_2=\frac{1}{5}$ and corresponding utility $U_2=0.69$
\item $EV=p_1V_1+p_2V_2=\frac{4}{5}*5999=4799.2$
\item $EU=p_1U_1+p_2U_2=\frac{4}{5}*9.39+\frac{1}{5}*0.69=7.65$
\item $U(EV)$ is the utility value Stephen would get from a value corresponding to the expected value $EV$. \\
By substitution $U(EV)=U(4799.2)=ln(9600.4)=9.17$
\item Certainty equivalent $CE$ is the amount of wealth that would yield the expected utility, in a certainty condition. Analytically this means: $$U(CE)=EU$$ $$ln(2CE+2)=7.65 \ \ \ \ 2CE+2=e^{7.65} \ \ \ \ CE=\frac{e^{7.65}}{2}-1=1049.32$$
\item Risk premium is the extra compensation that Stephen would require in order to take the risk: Analytically it is defined as:
$$RP=EV-CE=4799.2-1049.32=3749.87$$
\item \includegraphics[scale=0.5]{Q77.png}
\item Stephen is risk averse because its RP is positive. This means that he would not take the risk without an adequate compensation. This is also demonstrated by his Utility function, which is a concave function.
\item A risk averse is only willing to buy a policy if it is fair, which means that the premium paid for every dollar covered is lower or equal to the probability of the event. Therefore Stephen is willing to buy the policy if $\delta \leq 0.2$. But $\delta=\frac{3000}{5999}=0.5$ which is greater than 0.2. So Stephen won't buy the policy.
\item Stephen will continue avoiding the policy because, although the new probability of theft is $p=0.05$, $\delta=0.5$ which is higher than the probability.
\item The maximum premium per unit that Stephen is willing to pay for a full coverage is $\delta_1=p_1=0.2$ in the first case and $\delta_2=p_2=0.05$ in the second case.

\end{enumerate}

\subsection*{Problem 9}
\begin{enumerate}
\item The risky bundle is described by the lottery $L:\{ 0,\frac{1}{10};5000,\frac{9}{10}\}$. \\
Therefore the two possible outcomes are $V_1=0$ with probability $p_1=\frac{1}{10}$ and corresponding utility $U_1=10$, and $V_2=5000$ with probability $p_2=\frac{9}{10}$ and corresponding utility $U_2=1424.21$.
\item $EV=p_1V_1+p_2V_2=\frac{9}{10}*5000=4500$
\item $EU=p_1U_1+p_2U_2=\frac{1}{10}*10+\frac{9}{10}*1424.21=1282.78$
\item By substitution $U(EV)=U(4500)=20*\sqrt{4500}+10=1351.64$
\item $$U(CE)=EU=1282.78$$ Therefore
$$20\sqrt{CE}+10=1282.78 \ \ \ \ \ \sqrt{CE}=\frac{1272.78}{20} \ \ \ \ \ CE=(\frac{1272.78}{20})^2=4049.92$$
\item $RP=EV-CE=4500-4049.92=450.07$
\item Niccolò is risk averse because its RP is positive. This means that he would not take the risk without an adequate compensation. This is also demonstrated in point 8. by his Utility function, which is a concave function.
\item \includegraphics[scale=0.6]{Q88.png}
\item The options A and B can be summarized through the lotteries: \\ $A=\{4900,1\}$ and $B=\{0,0.05;4925,0.95\}$ \\
First of all, we calculate the premium per unit for option A, which is $\delta=\frac{100}{5000}=0.02$. $\delta<0.1$, so Niccolò is willing to ensure the entire value and he prefers option A to the the previous situation. \\
Now we evaluate $EU_A$ and $EU_B$ to understand which option grants Niccolò an higher Expected Utility.
$$EU_A=U(4900)=1410$$ because he is sure he will be getting a value of 4900.
$$EU_B=0.05*10+0.95*1413.56=1343.38$$
Therefore $EU_A>EU_B$ and Niccolò prefers option A.
\item The policy of option A is now unfair because $\delta=\frac{560}{5000}=0.112>0.1$. Therefore he is not willing anymore to take option A, which would grant him $EU_A=1342.66$. In this situation, option B grants him the same expected utility as before $(EU_B=1343.38)$, which is higher than his utility without any coverage ($EU=1282.78$). \\
His final choice will be option B.
\item Knowing that Lorenzo is risk averse, we can state that he will choose option A because with option B he will face a probability of losing the entire barrel. On the other hand, with option A the barrel will surely reach the destination and Lorenzo won't face any risk.


\end{enumerate}