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

$\backslash ($ x$($y$+1)+\backslash$lambda$\backslash$left$($B$-$P$\_\{$x$\}$ x$-$P$\_\{$y$\}$ y$\backslash$right$)\backslash )$

$($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

$\backslash ($ U$^\{*\}=$U$\backslash$left$($P$\_\{$x$\},$ P$\_\{$y$\},$ B$\backslash$right$)\backslash )$and derive an expression for the expenditure function

$\backslash ($ E$=$E$\backslash$left$($P$\_\{$x$\},$ P$\_\{$y$\},$ U$^\{*\}\backslash$right$)\backslash )$

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

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

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,$\backslash (\backslash$partial E $/\backslash$partial P$\_\{$x$\}\backslash )$ and $\backslash (\backslash$partial E $/\backslash$partial P$\_\{$y$\}\backslash ),$ respectively.