# applications of the slutsky equation

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 for the demands for each good, meaning get each good in terms of wealth and price. Then use the Slutsky equation by taking partial derivatives for each good with respect to income and price. – VCG Aug 31 '16 at 3:19
• so i took the partial derivatives of the utility function and obtained x^(a-1) and y^(a-1). not quite sure what to do from here, would appreciate a bit more help. – Irina Aug 31 '16 at 5:33

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

• I computed the demand functions of the maximization that is x and y at the optimum. I also found the Hicksian demands for x and y. However, I am not sure how to proceed from here. – Irina Sep 2 '16 at 23:27
• @Irina Take the derivatives of those functions with respect to the price of x. The equation I have there defines that relationship.Show some work in your question and I will give you a full solution. – VCG Sep 2 '16 at 23:44