<p>First let's consider the following problem:
<span class="math-container">\begin{eqnarray*} \max_{(x, y) \in \mathbb{R}^2_+} &amp; x + 4\sqrt{y} \\ \text{s.t.} \ &amp; p_Xx + p_Yy \leq M\end{eqnarray*}</span>
Solution to the above problem is known as demand, and it is given by
<span class="math-container">\begin{eqnarray*} (x^d, y^d) (p_X, p_Y, M) = \begin{cases} \left(0, \frac{M}{p_Y}\right) &amp;\text{if } \frac{M}{p_Y} \leq \frac{4p_X^2}{p_Y^2} \\ \left(\frac{M}{p_X}-\frac{4p_X}{p_Y}, \frac{4p_X^2}{p_Y^2}\right) &amp;\text{if } \frac{M}{p_Y} &gt; \frac{4p_X^2}{p_Y^2}\end{cases} \end{eqnarray*}</span>
To see how to find it, procedure is similar to this one: <a href="https://economics.stackexchange.com/a/16475/11824">https://economics.stackexchange.com/a/16475/11824</a></p>
<p>Now let's consider the following problem:
<span class="math-container">\begin{eqnarray*} \min_{(x, y) \in \mathbb{R}^2_+} &amp; p_Xx + p_Yy \\ \text{s.t.} \ &amp; x + 4\sqrt{y} \geq \mu\end{eqnarray*}</span>
Solution to the above problem is known as Hicksian demand, and it is given by
<span class="math-container">\begin{eqnarray*} (x^h, y^h) (p_X, p_Y, \mu) = \begin{cases} \left(0, \frac{\mu^2}{16}\right) &amp;\text{if } \frac{p_X}{p_Y} \geq \frac{\mu}{8} \\ \left(\mu-\frac{8p_X}{p_Y}, \frac{4p_X^2}{p_Y^2}\right) &amp;\text{if } \frac{p_X}{p_Y} &lt; \frac{\mu}{8}\end{cases} \end{eqnarray*}</span></p>
<p>So, total price effect <span class="math-container">$ = (x^d, y^d)(0.25,1,1) - (x^d, y^d)(1,1,1) = \left(3,\frac{1}{4}\right) -(0,1) = \left(3, -\frac{3}{4}\right) $</span></p>
<p>and the initial satisfaction level is <span class="math-container">$\mu = u(0,1) = 4$</span>.</p>
<p>Now we can find the Income and Substitution effect using Hicksian method:</p>
<p>Hicksian SE <span class="math-container">$=(x^h, y^h)(0.25, 1, 4) - (x^d, y^d)(1,1,1) = \left(2,\frac{1}{4}\right)-(0,1) = \left(2, -\frac{3}{4}\right)$</span></p>
<p>Hicksian IE <span class="math-container">$= (x^d, y^d)(0.25,1,1)-(x^h, y^h)(0.25, 1, 4)  = \left(3,\frac{1}{4}\right)-\left(2, \frac{1}{4}\right) = (1,0)$</span></p>

CC BY-SA 4.0
2022-06-15T07:09:16.767Z
i found the hicksian demand by simply solving the new budget with new prices and i did get the same answer, i.e. (3, 1/4). But i prefer your method more, could you please tell how you got μ/8 while solving for the hicksian demand?
petearGriffin
CC BY-SA 4.0
2022-06-15T07:09:48.953Z
i think that is MRS
petearGriffin
CC BY-SA 4.0
2022-06-15T02:00:17.930Z
2022-06-15T02:00:17.930Z
Amit
<p>utility function given to me is as follows, <span class="math-container">$$u(x,y) = x+4 \sqrt{y}$$</span> which is a simple quasilinear function. There is a change in the given price vector, <span class="math-container">$(p_x,p_y) = (1,1) \to (0.25,1)$</span>, and the income, M=1. the question asks about the absolute values of <strong>substitution and Income Effect</strong> using Hicks method.</p>
<p>approaching this in any way results in the demand <span class="math-container">$(x^* , y^*) = (0,1)$</span>, consumer would have prefered to consume <span class="math-container">$y^* = 4$</span> but as it is not affordable for the given budget so consumer would choose to consume all of y and nothing of x initially.</p>
<p>upon further solving i am getting <span class="math-container">$SE= 3$</span> and <span class="math-container">$ME=15$</span> but the income effect in quasilinear case is always zero.</p>
<p>can anyone please assist me with this?</p>

CC BY-SA 4.0
2022-06-12T11:30:31.867Z
You forgot to specify the income of the consumer.
Michael Greinecker
CC BY-SA 4.0
2022-06-12T11:34:49.900Z
Income effect in case of quasi-linear is not always $0$.
Amit
CC BY-SA 4.0
2022-06-12T11:39:01.600Z
@Amit it just occurred to me that ME = 0 only for the non linear good of the function, in this case it is y.
petearGriffin
CC BY-SA 4.0
2022-06-12T11:39:59.680Z
Yes but even for the non-linear one, it may not be $0$. This is because of the non-negativity constraints.
Amit
CC BY-SA 4.0
2022-06-12T11:48:10.063Z
Yes, but in the above case the demand for $Y$ is not independent of income. Demand for $Y$ is 1 when income is 1 and both prices are also 1, and demand for $Y$ is 2 when income is 2 and both prices are still 1. So demand for $Y$ increased with income.
Amit
CC BY-SA 4.0
2022-06-12T11:59:15.947Z
@Amit can you tell me if my answer to the above questions are correct or wrong?
petearGriffin
CC BY-SA 4.0
2022-06-12T12:08:55.333Z
I haven't solved it, but they don't look correct. Reason is that with $M=1$, and $0.25\leq p_X \leq 1$, such a high income effect is impossible because even if you spend all your income on $X$ when price of $X$ is $p_X =0.25$, the maximum possible demand for $X$ is not more than $4$ which is a strict upper bound on these effects because you'll always spend some amount on $Y$.
Amit
<slutsky-equation>
is income effect in the given quasilinear function equal to 0?
petearGriffin
# is income effect in the given quasilinear function equal to 0?

(Asked by: username_0 on 2022-06-12)

utility function given to me is as follows, $$u(x,y) = x+4 \sqrt{y}$$ which is a simple quasilinear function. There is a change in the given price vector, $(p\_x,p\_y) = (1,1) \to (0.25,1)$, and the income, M=1. the question asks about the absolute values of **substitution and Income Effect** using Hicks method.

approaching this in any way results in the demand $(x^\* , y^\*) = (0,1)$, consumer would have prefered to consume $y^\* = 4$ but as it is not affordable for the given budget so consumer would choose to consume all of y and nothing of x initially.

upon further solving i am getting $SE= 3$ and $ME=15$ but the income effect in quasilinear case is always zero.

can anyone please assist me with this?

**Tags:** slutsky-equation

> **Comments:**
> **username_1**: You forgot to specify the income of the consumer.  
> **username_2**: Income effect in case of quasi-linear is not always $0$.  
> **username_0**: @username_2 it just occurred to me that ME = 0 only for the non linear good of the function, in this case it is y.  
> **username_2**: Yes but even for the non-linear one, it may not be $0$. This is because of the non-negativity constraints.  
> **username_2**: Yes, but in the above case the demand for $Y$ is not independent of income. Demand for $Y$ is 1 when income is 1 and both prices are also 1, and demand for $Y$ is 2 when income is 2 and both prices are still 1. So demand for $Y$ increased with income.  
> **username_0**: @username_2 can you tell me if my answer to the above questions are correct or wrong?  
> **username_2**: I haven't solved it, but they don't look correct. Reason is that with $M=1$, and $0.25\leq p_X \leq 1$, such a high income effect is impossible because even if you spend all your income on $X$ when price of $X$ is $p_X =0.25$, the maximum possible demand for $X$ is not more than $4$ which is a strict upper bound on these effects because you'll always spend some amount on $Y$.  

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## 1 answers

### Answer by username_2 on 2022-06-15. Score: 3 | Accepted answer

First let's consider the following problem:
\begin{eqnarray\*} \max\_{(x, y) \in \mathbb{R}^2\_+} & x + 4\sqrt{y} \\ \text{s.t.} \ & p\_Xx + p\_Yy \leq M\end{eqnarray\*}
Solution to the above problem is known as demand, and it is given by
\begin{eqnarray\*} (x^d, y^d) (p\_X, p\_Y, M) = \begin{cases} \left(0, \frac{M}{p\_Y}\right) &\text{if } \frac{M}{p\_Y} \leq \frac{4p\_X^2}{p\_Y^2} \\ \left(\frac{M}{p\_X}-\frac{4p\_X}{p\_Y}, \frac{4p\_X^2}{p\_Y^2}\right) &\text{if } \frac{M}{p\_Y} > \frac{4p\_X^2}{p\_Y^2}\end{cases} \end{eqnarray\*}
To see how to find it, procedure is similar to this one: <https://economics.stackexchange.com/a/16475/11824>

Now let's consider the following problem:
\begin{eqnarray\*} \min\_{(x, y) \in \mathbb{R}^2\_+} & p\_Xx + p\_Yy \\ \text{s.t.} \ & x + 4\sqrt{y} \geq \mu\end{eqnarray\*}
Solution to the above problem is known as Hicksian demand, and it is given by
\begin{eqnarray\*} (x^h, y^h) (p\_X, p\_Y, \mu) = \begin{cases} \left(0, \frac{\mu^2}{16}\right) &\text{if } \frac{p\_X}{p\_Y} \geq \frac{\mu}{8} \\ \left(\mu-\frac{8p\_X}{p\_Y}, \frac{4p\_X^2}{p\_Y^2}\right) &\text{if } \frac{p\_X}{p\_Y} < \frac{\mu}{8}\end{cases} \end{eqnarray\*}

So, total price effect $ = (x^d, y^d)(0.25,1,1) - (x^d, y^d)(1,1,1) = \left(3,\frac{1}{4}\right) -(0,1) = \left(3, -\frac{3}{4}\right) $

and the initial satisfaction level is $\mu = u(0,1) = 4$.

Now we can find the Income and Substitution effect using Hicksian method:

Hicksian SE $=(x^h, y^h)(0.25, 1, 4) - (x^d, y^d)(1,1,1) = \left(2,\frac{1}{4}\right)-(0,1) = \left(2, -\frac{3}{4}\right)$

Hicksian IE $= (x^d, y^d)(0.25,1,1)-(x^h, y^h)(0.25, 1, 4) = \left(3,\frac{1}{4}\right)-\left(2, \frac{1}{4}\right) = (1,0)$

> **Comments:**
> **username_0**: i found the hicksian demand by simply solving the new budget with new prices and i did get the same answer, i.e. (3, 1/4). But i prefer your method more, could you please tell how you got μ/8 while solving for the hicksian demand?  
> **username_0**: i think that is MRS  

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