Calculate income and sustitution effect from utility funcion

Question

Utility function $U(x_1 , x_2) = x_1 + 4 * x_2 $

$ p_1 = 3, p_2 = 8, m =120 $

$p_2$ changes from $8$ to $10 $

How can I calculate the income and substitution effect.

I first thought about calculating the MRS making the partial derivatives of the Utility function which resulted in $MRS = 1/4$ which means that the consumer will always prefer at any given moment good $x_2$ thus only buying that good. At first he can only buy $m/p_2 = 120/8 = 15$ so the bundle would be $(0,15)$ and after the price change it would be $(0,12)$ (not sure about that).

I also calculated the utility at the previous point by plugging $(0,15)$ into $U(x_1 , x_2) = x_1 + 4 * x_2 $ which resulted in $60$

$60/p'_2 - 60/p_2 $ = $60/10 - 60/8= - 1.5$ = substitution effect

I am not sure if what I have done is correct and if the substitution effect is 1.5. Is there any way to visualize this graphically ? I thought about doing linear equations with $m/p_2$ and $m/p'_2$ as seen below:

Am I missing something ? Any help is appreciated

Answer 1

First, the fact that $MRS=\frac{1}{4}$ does not tell you by itself that the consumer will only buy $x_2$. We need to go back to the 2nd Gossen's law: $\frac{Um{x_1}}{p_1}=\frac{Um{x_2}}{p_2}$ (this is from where the $MRS$ comes by the way) which is not the case since $\frac{1}{3}<\frac{4}{8}=\frac{1}{2}$.

So you are right the consumer in the first place will not consume any $x_1$. This is also due to the linear form of your function since you do not have a decreasing marginal utility with respect to $x_2$ even if you get more and more $x_2$ (which is quite a common form to model money for instance).

After a change in the price $p_2$ the decision of consuming only $x_2$ remains since $\frac{1}{3}<\frac{4}{10}=\frac{2}{5}$. With such reasoning, you will easily understand that there is no substitution effect for an increase in the price $p_2$. If you want a formal demonstration:

Substitution effect: how much does the consumer substitute a good from another when the price varies with a constant purchasing power? With the demand for the good 2: $x_2=\frac{m}{p_2}$, and denoting $m'$ and $p_2'$ the "new" revenue and the new price we have: $\Delta x_2^s= x_2(p_2',m')-x_2(p_2,m)$. The $m'$ is the "new" revenue such that I can buy the same amount of $x_2$ before the change in the price ie an augmentation of: \begin{aligned} \Delta m&=15(10-8)=30 \\ m'&=m+\Delta m=150 \\ x_2^s&=\frac{150}{10}-\frac{120}{8}=0 \end{aligned}

Nevertheless, it remains the income effect which is the variation of the demand due to the modification of purchasing power.

\begin{aligned} \Delta x_2^i=& x_2(p_2',m)-x_2(p_2',m') \\ =& x_2(10,120)-x_2(10,150) \\ =& 12-15=-3 \end{aligned}

So the total variation of the demand is only related to the income effect and is -3.