Suppose Your Tastes Are Defined by the Utility Function $u \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$

Suppose your tastes are defined by the utility function $u \left( x _ { 1 } , x _ { 2 } \right) = x _ { 1 } x _ { 2 }$ .
a.Suppose your income is $1,000,the price of
$x _ { 2 }$ is 1 and the price of
$x _ { 1 }$ is
$p _ { 1 }$ .Set up your utility maximization problem.
b.Derive the quantity of
$x _ { 2 }$ you will consume.
c.What happens to your consumption of
$x _ { 2 }$ if
$p _ { 1 }$ increases?
d.Now suppose that your and my tastes are captured by the utility function
$u \left( x _ { 1 } , x _ { 2 } \right) = \left( x _ { 1 } ^ { - p } + x _ { 2 } ^ { - p } \right) ^ { - 1 / \rho }$ ,with the parameter
$\rho$ different for you than it is for me.When
$p _ { 1 }$ increases,you consume more
$x _ { 2 }$ than before and I consume less.What range of
$\rho$ is consistent with your behavior,and what range is consistent with mine? Use your answer to (b)to explain.