<p>You made the right start by calculating the utiltiy at $p_1=p_2=1$.</p>

<p>When we change a price, two things happen. Firstly, one of the goods becomes relatively more expensive, so people <strong>substitute</strong> away from that good.</p>

<p>Secondly, since the total amount of goods someone can afford is lower when a price increases, it is as if their <strong>income</strong> went down.</p>

<p>To find the substitution effect, we need to shut down the second of these effects and focus on the first.</p>

<p>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 $\widetilde{Y}$.</p>

<p>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.</p>

<p>Hopefully, this is enough to help you figure out the solution.</p>

CC BY-SA 3.0
2015-04-30T13:31:48.213Z
You mean find the Y such that it passes through the original bundle (4,6),which is Y=14? But it seems like when Y =14, I got $$q_1=1$$ and $$q_2=12$$ which is not true, to maintain the same utility, I must have $$q_1=1$$ and $$q_2=10$$ I know this because sub back into the utility function.
UnusualSkill
CC BY-SA 3.0
2015-04-30T12:22:27.303Z
2015-04-30T12:22:27.303Z
Ubiquitous
<p>Utility function</p>

<p>$$
U=4{q_1}^{0.5}+q_2 \\
Y=10
$$</p>

<p>$$
p_1=p_2=1
$$</p>

<p>Then $p_1$ rises to 2. I would like to ask how to calculate the substitution effect on the demand for q1?</p>

<p>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 $q1$.</p>

CC BY-SA 3.0
2015-04-30T09:49:49.100Z
Which substitution effect?

Here are some links that may help you find out what you need to do:

http://wikieducator.org/SUBSTITUTION_EFFECT/Next

http://en.wikipedia.org/wiki/Substitution_effect
Giskard
CC BY-SA 3.0
2015-04-30T09:52:32.083Z
@denesp what i want to find is the quantity of A reduced as a result of the substitution effect. Now I have U=14 But I am not sure how to find that new $$q_1$$ with the utility held constant.
UnusualSkill
CC BY-SA 3.0
2015-04-30T11:26:42.423Z
@UnusualSkill did you follow the link that denesp posted? There is more than one substitution effect.
FooBar
<consumer-theory>
Calculate the substitution effect
UnusualSkill
# Calculate the substitution effect

(Asked by: username_0 on 2015-04-30)

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 $q1$.

**Tags:** consumer-theory
> **Comments:**
> **username_1**: Which substitution effect?  Here are some links that may help you find out what you need to do:  http://wikieducator.org/SUBSTITUTION_EFFECT/Next  http://en.wikipedia.org/wiki/Substitution_effect  
> **username_0**: @denesp what i want to find is the quantity of A reduced as a result of the substitution effect. Now I have U=14 But I am not sure how to find that new $$q_1$$ with the utility held constant.  
> **username_2**: @UnusualSkill did you follow the link that denesp posted? There is more than one substitution effect.  

---

## 1 answers

### Answer by username_3 on 2015-04-30. Score: 2

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.

> **Comments:**
> **username_0**: You mean find the Y such that it passes through the original bundle (4,6),which is Y=14? But it seems like when Y =14, I got $$q_1=1$$ and $$q_2=12$$ which is not true, to maintain the same utility, I must have $$q_1=1$$ and $$q_2=10$$ I know this because sub back into the utility function.  

---

