Use of Slutsky equation
Utility function $u(x_1, x_2) = x_1^2x_2$.


  Q. Derive the demand for $x_1$ and $x_2$ as a function of $p_1$, $p_2$ and $m$.


Here are the demand functions for $x_1$ and $x_2$:
$$x_1(p_1, p_2, m) = \frac{2m}{3p_1}$$
$$x_2(p_1, p_2, m) = \frac{m}{3p_2}$$


  Q. Illustrate the equilibrium in a diagram when $p_1=1$, $p_2=1$ and $m=12$.





  Q. Suppose $p_1$ rises to 3. Calculate the substitution effect and income effect.


If $p_1$ rises to 3, the new equilibrium choice is $\left(\frac{8}{3}, 4\right)$. To find the Substitution effect and Income effect using Slutsky approach, we will find the equilibrium at new set of prices when the consumer has just enough money to buy the old equilibrium bundle i.e. we will find the demand at prices $(3,1)$ when income is $m' = 3(8) + 1(4) = 28$. Substituting this data in the demand functions, we get the equilibrium choice as: $\left(\frac{56}{9}, \frac{28}{3}\right)$. Here is how the situation looks in graph:


Substitution effect = $\displaystyle\frac{56}{9} - 8 = -\frac{16}{9}$

Income effect = $\displaystyle\frac{8}{3} - \frac{56}{9} = - \frac{32}{9}$


  Q. Explain your exact results using the appropriate Slutsky equation.


Slutsky equation:
Change in Demand =  Change in Demand due to substitution effect + Change in Demand due to income effect

$$\Delta x_1 = \Delta^s x_1 + \Delta^i x_1 = -\frac{16}{9} - \frac{32}{9} 
 = -\frac{16}{3}$$