# Calculate income and sustitution effect from utility funcion

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

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.

• thanks a lot for your detailed response, it really helped a lot and I know understand this topic much more. Just to see if I grasp everything correctly, in the case that p2 increases to 18, I should calculate the substitution effect by doing 270/18 - 120/8 = 0, but I feel that there should be a substitution effect as 1/3 > 4/18. Income effect with this price is -8.33. Am I on the right track to solving this ? Is there something I am missing ? Again, thanks a ton for your comment it really helped a lot. Sep 9 at 19:00
• I managed to do it again, which resulted in a substitution effect of - 25. for the price change of p2 to 18. I also did another method and got -15, any idea on which one could be the correct one ? Sep 10 at 2:20
• I do not clearly understand what you are doing, but if the inequality is reversed as you see then the consumer will only buy $x_1$ this time. Just remind you that it is a very particular case where goods are perfect substitutes (see the link of the previous comment). Sep 10 at 11:31
• I am unable to see the previous comment, could you please send the link again. So sorry for this and thanks for your help Sep 10 at 19:31