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Calculate the substitution and income effects for the following utility function:
$$ u(x,y)=\frac{x^a}{a}+\frac{y^a}{a}$$
I know that we are supposed to use the Slutsky equation which accounts for the substitution effect minus the income effect however I am not quite sure how to solve it. The only thing which I know how to do is derive the $x$ and I will get $x^{a-1}$ but after I am not sure what to do.
Solve the "consumer's problem":
$\max U(x,y)$ subject to $I=p_xx+p_yy$
Then you will have $x=x(p_x,p_y,I)$ so x is some function of the parameters. The same goes for y.
Then determine the Hicksian demand functions, either by using some duality result or solving the dual problem:
$\min p_xx+p_yy$ subject to $U(x,y)=\bar u$ and you'll have both x and y in terms of prices and $\bar u$. We denote these demands as $h_x$ and $h_y$.
Now Slutsky's equation for x with respect to $p_x$: $\displaystyle\frac{\partial x}{\partial p_x}=\displaystyle\frac{\partial h_x}{\partial p_x}-\displaystyle\frac{\partial x}{\partial I}x(p_x,p_y,I)$
The substitution effect is the first term on the RHS, and the income effect is the third term.
The general formula for all cases is here https://en.wikipedia.org/wiki/Slutsky_equation
