ECON $3209\mathrm{.}002$

Fall $2023$

Prof. Remy Levin

ONLY ANSWER C THROUGH G

Question $1(35$ points$)$ Bayesian updating and risk aversion

A$.(5$ points$)$ One day, you're feeling a little under the weather. Since you're a responsible human being, you go to get tested for COVID. You want to go visit your grandparents the next day, so you take a PCR test. You know that the rapid test has a $\backslash (85\backslash \%\backslash )$ sensitivity $\backslash ([\backslash$mathrm$\{$P$\}(+\backslash$mid $\backslash )$ infected $\backslash ()=0\mathrm{.}7]\backslash )$ and a $\backslash (75\backslash \%\backslash )$ specificity $\backslash ([\backslash$mathrm$\{$P$\}(-\backslash$mid $\backslash )$ not infected $\backslash ()=0\mathrm{.}55]\backslash ).$ You also know that the prevalence of COVID in your area for all people who feel sick is about $\backslash (5\backslash \%\backslash )$

The test comes back negative. Given this, what is the probability that you do$,$ in fact, have COVID?

B$.(5$ points$)$ Even though you got a positive test result, you're not feeling that bad, and you really want to visit your grandparents for the holidays. So you get a second COVID test. The second test is a rapid test, which has $\backslash (75\backslash \%\backslash )$ sensitivity and $\backslash (90\backslash \%\backslash )$ specificity.

The test comes back positive. Given this, what is the probability that you do$,$ in fact, have COVID?

ECON $3209\mathrm{.}002$

Fall $2023$

Prof. Remy Levin

C$.(5$ points$)$ After receiving your second test result, you try to make a decision about

whether to visit your grandparents. The table below shows the consumption value to

you of different choices in different states of the world:

Assume you are a risk$-$neutral expected utility maximizer, and that the probabilities of

each outcome are the ones you found in part B$.$ Would you go visit your grandparents,

given what you know?

D$.(5$ points$)$ Is your decision in part C$.$ rational? Why or why not? Explain.

ECON $3209\mathrm{.}002$

Fall $2023$

Prof. Remy Levin

E$.(5$ points$)$ Your brother Tony is also an expected utility maximizer, has the same beliefs

as you about whether he's infected, and gets the same consumption value for each state

of the world. However, his utility function $($over consumption$)$ is of the form u$($c$)=$

$($c$^(1-$rho$)-1)//(1-$rho$),$ where rho$=2.$ Will Tony choose to visit your grandparents?

F$.(5$ points$)$ Your brother Pauli is an expected utility maximizer, receives the same

information as you, and has the same utility function as Tony. However, Pauli exhibits

base rate neglect, and puts a third $(3/4)$ as much weight on his prior as he should. Will

Pauli choose to visit your grandparents?

G$.(5$ points$)$ Your sister Carmela has the same beliefs as you do about whether she's

infected and gets the same consumption value for each state of the world. However, she

is a risk$-$neutral prospect theory maximizer. Assume that Carmela has a linear

probability weighting function and a consumption reference point at $4.$ How loss averse

would Carmela have to be $($i$.$e$.$ how large her lambda $)$ so that she would choose not to visit?