Adam$$s preferences for opera tickets $($good $1)$ and ballet tickets $($good $2)$ can be

rep$-$ resented by the utility function uA$($xA

$1,$ xA

$2)=$ xA

$1$ x

A

$2.$ Eve$$s utility

function is given by uE $($xE

$1,$ x

E

$2)=$ xE

$1+$ xE

$2.$ Here, the subscripts denote goods

whereas the superscripts denote the individuals. Adam$$s initial endowment

consists of $4$ opera tickets and $2$ ballet tickets per year whereas Eve$$s initial

endowment is given by $6$ opera tickets and $3$ ballet ticket per year. Assume that

Adam and Eve are the only potential traders in this exchange economy.

$($a$)(6$ points$)$ Draw an Edgeworth box for this pure exchange economy with two

consumers. Indicate the point of initial endowments and sketch several

indifference curves for both consumers. Which allocations are attainable in this

economy? Indicate graphically the set of mutually beneficial exchanges $($lens of

exchange$).$

$($b$)(6$ points$)$ Is the initial endowment Pareto$-$efficient? Determine the set of all

Pareto$-$efficient allocations and draw the contract curve. $($Hint: in this economy,

some Pareto$-$efficient allocations may be located on the boundary of the

Edgeworth box. I.e$.$ it may be efficient for one of the consumers to consume

positive amounts of both goods and, for the other one, only one of two goods.$)$

$($c$)(6$ points$)$ Consider the utility maximization problem of Adam. Derive Adam$$s

demand for opera tickets and ballet tickets as functions of prices p$1,$ p$2.$ Is there a

price ratio such that Adam demands a positive quantity of only one of both

goods? Now consider Eve$$s preferences and endowments. At which price ratio

p$1/$p$2$ would Eve demand positive quantities of both goods?

$($d$)(6$ points$)$ Using your results from $($c$),$ find the equilibrium price ratio between

opera tickets and ballet tickets. Clearly explain how you achieve your result.

$($e$)(6$ points$)$ Find the equilibrium allocations. Is this market equilibrium Pareto$-$

efficient? Why or why not? Illustrate your answer graphically.