I am stuck on a multiple choice question that came up on a Micro university past papers.
The question is:
Marshallian demand curves derived from utility function: $U = log(x) + log (y)$. What is the own price elasticity, cross price elasticity, and income elasticity?
The answers are -1, 0, and 1 respectively, yet I don't understand how.
Many thanks!
as you see that is a Cobb Douglas function(Monotonous transformation), I will assume you already know how to get the X* and Y* Marshallian . "w" is the wealth of the consumer. In case you dont know how to get the marshallians you have to maximize the utility ( "U = log(x) + log (y)") subject to the constrain budget (w = Xpx+Ypy)
$ X(*) = w/2px $
$ Y(*) = w/2py $
So let's start with the income elasticity, we want to know how the consumption of X will change when the income//price of x (own-price) // (cross price )price of y changes, but we want it in terms of percentage/relative ( that's why we want elasticities ) .
So the income elasticity is :
$(\frac{\partial X}{\partial w})*(\frac{w}{X}) =(\frac{1}{2px} )(\frac{w}{\frac{w}{2px}})\ = 1 $
cross price elasticity:
$(\frac{\partial X}{\partial py})*(\frac{py}{X}) =(0)(\frac{py}{\frac{w}{2px}}) = 0 $
own price elasticity :
$ (\frac{\partial X}{\partial px})*(\frac{px}{X}) =(\frac{-w}{2px^{2}} )(\frac{px}{\frac{w}{2px}}) = -1 $
I will let you search what means that numbers! :D
Hope this helps you!
