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

$12$

$2$

$2$

$1$

$1(,)$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

$$

p

y

p

y

v p y

where y is income, pi 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