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Utility function
$$ U=4{q_1}^{0.5}+q_2 \\ Y=10 $$
$$ p_1=p_2=1 $$
Then $p_1$ rises to 2. I would like to ask how to calculate the substitution effect on the demand for q1?
What I have done is calculate the optimal bundle initially which is (4,6) then subs into the utility function and get $U=14$. However, I am stuck here for calculating the subs effect on $q1$.
You made the right start by calculating the utiltiy at $p_1=p_2=1$.
When we change a price, two things happen. Firstly, one of the goods becomes relatively more expensive, so people substitute away from that good.
Secondly, since the total amount of goods someone can afford is lower when a price increases, it is as if their income went down.
To find the substitution effect, we need to shut down the second of these effects and focus on the first.
It sounds like you are after what is more properly known as the Hicksian substitution effect. To calculate that, we need to compensate the consumer for the aparent loss of income. In other words, we need to answer the following question: "Given $p_1=2$ and $p_2=1$, what income would be needed to achieve the same utility as before (i.e. $U=14$)?" Call the income level that answers this question $\widetilde{Y}$.
Then, all we need to do is calculate the optimal consumption bundle when $p_1=2$, $p_2=1$, and income is $\widetilde{Y}$. The substitution effect is the the difference in $q_1,q_2$ between this new bundle and the first one you calculated.
Hopefully, this is enough to help you figure out the solution.
