Consider the following indirect utility function derived from a consumer's utility maximization:

$v(p,y)={}_1(\frac{y}{p_1}{)}^{{}_1}+{}_2(\frac{y}{p_2}{)}^{{}_2}$

where $y$ is income, $p_i$ is price of commodity $i,$ for $i=1,2.$ and ${}_i$ and ${}_i$ are parameters.

$($a$)$ Determine the Lagrange multiplier of the underlying consumer's utility maximization

problem.

$[8$ Marks$]$

$($b$)$ Determine the Marshallian demand functions.

$[12$ Marks$]$

$($c$)$ Explain the concept of duality in consumer theory.

$[10$ Marks$]$