This problem set $($PS$)$ is due at $11$:$59$pm on Wednesday, November $13.$ The maximum score is $103$ points. Be sure to review the syllabus for details about homework assignments and their grading! Feel free to contact me via e$-$mail if you have specific questions about the HW assignment.

Note that some Exercises have several parts, and each part may conceal more than one task for you. Be sure to answer every question thoroughly for full credit!

Exercise $1(20$ points$)$ Suppose that the demand curve is given by the following: $\backslash ($ P$=10-$Q $\backslash ).$

$($a$)$ Calculate the own price elasticity of demand from $\backslash ($ Q$=4\backslash )$ to $\backslash ($ Q$=5\backslash ).$ as well as from $\backslash ($ Q$=5\backslash )$ to $\backslash ($ Q$=6\backslash )$

$($b$)$ Find out the elastic and inelastic regions of this demand curve.

Exercise $\backslash (2(20\backslash )$ points$)$

Draw some indifference curves for the preferences that are represented by the following utility functions. Show the direction of increasing utility by putting an arrow between indifference curves.

$($a$)\backslash ($ U$($x$,$ y$)=3$ x$^\{0\mathrm{.}5\}$ y$^\{0\mathrm{.}5\}\backslash )$

$($b$)\backslash ($ U$($x$,$ y$)=5$ x$+4$ y $\backslash )$

Exercise $3(35$ points$)$

Ms$.$ Harris is planning an around$-$the$-$world trip on which she plans to spend $\backslash (\backslash$$ $10,000\backslash ).$ The utility from the trip is a function of how much she actually spends on it $\backslash ((\backslash$Upsilon$)\backslash ),$ given by

$\backslash [$

U$(\backslash$Upsilon$)=\backslash$ln $(\backslash$Upsilon$)$

$\backslash ]$

$($a$)$ If there is a $25$ percent probability that Ms$.$ Harris will loose $\backslash (\backslash$$ $1,000\backslash )$ of her cash on trip, what is the trip's expected utility?

$($b$)$ Suppose that Ms$.$ Harris can buy insurance against loosing $\backslash (\backslash$$ $1,000\backslash )$ at an actually fair premium of $\backslash (\backslash$$ $250\backslash ).$ Show that her expected utility is higher if she purchases this insurance than if she faces the chance of loosing the $\backslash (\backslash$$ $1,000\backslash )$ without insurance.

$($c$)$ What is the maximum amount that Ms$.$ Harris would be willing to pay to insure her $\backslash (\backslash$$ $1,000\backslash ).$

$($d$)$ What is the maximum amount that Ms$.$ Harris would be willing to pay to insure her $\backslash (\backslash$$ $1,000\backslash ),$ if we change her utility function to $\backslash ($ U$(\backslash$Upsilon$)=\backslash$Upsilon $\backslash ).$

$($e$)$ What is the maximum amount that Ms$.$ Harris would be willing to pay to insure her $\backslash (\backslash$$ $1,000\backslash ),$ if we change her utility function to $\backslash ($ U$(\backslash$Upsilon$)=(\backslash$Upsilon$)^\{2\}\backslash ).$ Compare your answer with part $\backslash ($ c $\backslash )$ and $\backslash ($ d $\backslash )$

Exercise $4(25$ points$)$

Suppose that the demand curve is given by the following: $\backslash ($ P$=10-$Q $\backslash ).$ Moreover the supply curve is given by $\backslash ($ P$=$Q $\backslash ).$

$($a$)$ Calculate the market equilibrium as well as the consumer, producer and total surpluses.

$($b$)$ Suppose that the government now imposes a consumption tax equal to $\backslash (\backslash$$ $1\backslash )$ per unit being bought and sold in the market. Calculate the new equilibrium as well as the consumer, producer and the total surpluses. What happens to the total surplus after the introduction of tax? Explain.