A consumer has the following utility function: U$($x$,$y$)=$x$($y$+1),$ where x and y are quantities of two consumption goods whose prices are Px and Py$,$ respectively. The consumer also has a budget of B$.$ Therefore, the Lagrangian for this consumer is

LaTeX: x$($y$+1)+\backslash$lambda$($B$-$P$\_$xx$-$P$\_$yy$)$

$$

$($

$$

$+$

$1$

$)$

$+$

$$

$($

$$

$$

$$

$$

$$

$$

$$

$$

$$

$)$

$($a$)$ Verify that this is a maximum by checking the second$-$order conditions. By substituting x$*$ and y$*$ into the utility function, find an expression for the indirect utility function

LaTeX: U$^*=$U$($P$\_$x$,$P$\_$y$,$B$)$

$$

$*$

$=$

$$

$($

$$

$$

$,$

$$

$$

$,$

$$

$)$

and derive an expression for the expenditure function

LaTeX: E$=$E$($P$\_$x$,$P$\_$y$,$U$^*)$

$$

$=$

$$

$($

$$

$$

$,$

$$

$$

$,$

$$

$*$

$)$

$($b$)$ This problem could be recast as the following dual problem

LaTeX: Min $\backslash \backslash \backslash$ P$\_$xx$+$P$\_$yy$\backslash \backslash$

$\backslash$text$\{$Subject to$\}\backslash \backslash$ x$($y$+1)=$U$^*$

Find the values of x and y that solve this minimization problem and show that the values of x and y are equal to the partial derivatives of the expenditure function, LaTeX: $\backslash$partial E$/\backslash$partial P$\_$x $\backslash \backslash$text$\{$and$\}\backslash \backslash$partial E$/\backslash$partial P$\_$y

&partial;

$$

$/$

&partial;

$$

$$

and

&partial;

$$

$/$

&partial;

$$

$$

$,$ respectively.