$1.$ John consumes two goods, good and good $.$ His utility function $is$ given $byXY(,),$ where $is$ the amount $of$ good and $is$ the amount $of$ good $.uQQQQQXQYXYXYXYIn$ this case, John's marginal rate $of$ substitution $of$ good for good $()at$ a point $XYMRS(,)onan$ indifference curve $is$: $at$ a point $(,).QQMRSQQQQXYXYYX$ The price $of$ good $is$ $$2$ per unit, the price $of$ good $is$ $$4$ per unit, and his money $XY$ income $is$ $$200$: $$2,$ $$4,$ and $$200.PPMXY(1)$ Express John's budget set mathematically. $(5$ points$)(Thatis,$ write the inequality which represents John's budget set.$)(2)$ Write the equation for his budget line $(also$ called his budget equation$).(5$ points$)(3)$ What $is$ his real income $in$ terms $of$ good $?(2$ points$)Y(4)$ Graph the budget set and the budget line that $he$ faces. $(5$ points$)(The$ horizontal axis measures the quantity $of$ good $,$ and the vertical axis $QXX$ measures the quantity $of$ good $.)QYY(5)$ Draw the indifference curve associated with a utility $(indicator)of50.$ That $is,$ draw the following indifference curve: $50.QQXY$ You are required $to$ show $at$ least $on$ the indifference curve. four points $(5$ points$)(6)$ Calculate John's $at$ a combination $of20$ units $of$ good and $40$ units $ofMRSX$ good $.(5$ points$)Y(7)$ Find the optimal choice. $(5$ points$)(Recall$ that John's marginal rate $of$ substitution $of$ good for good $()XYMRSat$ a point $(,)onan$ indifference curve $is$: $at(,).)QQMRSQQQQXYXYYX(8)$ Suppose that the price $of$ good falls $to$ $$1$ per unit, $.X$ ceteris paribus Find the optimal choice. $(5$ points$)(9)$ Suppose that the price $of$ good $is$ $$4$ per unit, $.X$ ceteris paribus Find the optimal choice. $(5$ points$)(10)$ Using the answers $to$ parts $(7),(8),$ and $(9),$ plot three points $on$ John's demand curve for good $.(5$ points$)X($ : The horizontal axis measures the quantity $of$ good Hint $QXXPX$ and the vertical axis measures the price $of$ good $.)X(11)$ Now suppose that $$4,$ $$8,$ and $$400.PPMXY$ That $is,$ the prices $of$ good and good and his money income are doubled. $XY$ How does his optimal choice change? $(3$ points$)$

John consumes two goods, good $x$ and good $Y.$ His utility function $is$ given $by$

$u(Q_x,Q_Y)=Q_x{Q}_Y,$ where $Q_{x}is$ the amount $of$ good $x$ and $Q_{Y}is$ the amount $of$ good $Y.$

$In$ this case, John's marginal rate $of$ substitution $of$ good $x$ for good $Y(MRS)at$ a point

$(Q_x,Q_Y)onan$ indifference curve $is$: $MRSat$ a point $(Q_x,Q_Y)=\frac{Q_Y}{Q_x}.$

The price $of$ good $xis$ $$2$ per unit, the price $of$ good $Yis$ $$4$ per unit, and his money

income $is$ $$200$:$P_x=$$$2,P_Y=$$$4,$ and $M=$$$200.$

$(1)$ Express John's budget set mathematically. $(5$ points$)$

$(Thatis,$ write the inequality which represents John's budget set.$)$

$(2)$ Write the equation for his budget line $(also$ called his budget equation$).$

$(5$ points$)$

$(3)$ What $is$ his real income $in$ terms $of$ good $Y{Q}_{x}ofgdx,$ and the vertical axis

measures the quantity $Q_{Y}of$ good $Y_{.}50=Q_x{Q}_Y.$

You are required $to$ show $at$ least four points $on$ the indifference curve.

$(5$ points$)$

$(6)$ Calculate John's $MRSat$ a combination $of20$ units $of$ good $x$ and $40$ units $of$

good $Yx$ for good $Y{Q}_x,Q_{Y}MRSat(Q_x,Q_Y)=\frac{Q_Y}{Q_x}.x$ falls $to$ $$1$ per unit, ceteris paribus.

Find the optimal choice. $(5$ points$)$

$(9)$ Suppose that the price $of$ good $xis$ $$4$ per unit, ceteris paribus.

Find the optimal choice. $(5$ points$)$

$(10)$ Using the answers $to$ parts $(7),(8),$ and $(9),$

plot three points $on$ John's demand curve for $g$