Problem $1($Nicholson & Snyder, $2017,$ Ex $2\mathrm{.}3,$ p$.77)$

Suppose that $f(x,y)=xy.$ Find the maximum value for $f$ if $x$ and $y$ are constrained to sum to $1.$ That is$,x+y=1.$

Solve this problem in two ways: i$)$ by substitution; and, ii$)$ by using the Lagrange multiplier method.

Problem $2$

Let $x$ and $y$ represent consumption bundles of goods $1$ and $2($non$-$negative consumptions of each one of the goods$).$ That is$,$ each one of these bundles are represented by $($qnt of good $1,$ qut of good $2).$ In a mathematical form, we can indicate that $x,yin{R}_{+}^2.$ For each of the preference relations below:

i$.x>-=y>x_1+x_2{y}_1+y_2$

ii$.x>-=y>$min$(x_1,2x_2)$min$(y_1,2y_2)$

iii. $x>-=y>x_1{x}_2{y}_1{y}_2$

Answer the following:

$($a$)$ Are these preference relations complete, transitive, and continuous? Briefly justify each one of your answers.

$($b$)$ We know that, if a preference relation on a choice set is rational $($i$.$e$.,$ complete and transitive$)$ and continuous, then it can be represented by utility functions. For each preference relation that is rational and continuous $($according to your answer in part $($a$)),$ draw their indifference curves.

$1$

Problem $3$

Consider the following utility functions:

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

ii$.U(x,y)=x+y-3$

iii. $U(x,y)=$min$(x,y)$

iv$.U(x,y)=x+y^{0\mathrm{.}5}$

For each of them:

$($a$)$ Indicate the type of the underlying preference relation $($i$.$e$.,$ whether they are: complete, transitive, continuous, monotonic, convex$).$

$($b$)$ Represent the indifference map.

$($c$)$ Calculate the marginal utilities.

$($d$)$ Determine the Marginal Rate of Substitution of $y$ for $x.$

Problem $4$

Which of the following functions are monotonic transformations of $xy?$ For those that are, what is the monotonic transformation that gives this equivalence?

$($a$)7x^2{y}^2+2$

$($b$)ln(x)+ln(y)+1$

$($c$)x^{2}y$