<p>There is already a good answer to this question by VARulle. I am just adding some details.</p>
<p>Given <span class="math-container">$u(x,y)=2xy+y^2$</span>, note that <span class="math-container">$u$</span> is an increasing function and its <span class="math-container">$\text{MRS}=\dfrac{y}{x+y}$</span> which is diminishing. So, <span class="math-container">$u$</span> is quasi-concave.
Given the MRS, it is clear that there is a possibility that the consumer can spend all his income on commodity <span class="math-container">$y$</span>. That will happen if at bundle <span class="math-container">$\left(0,\frac{M}{p_Y}\right)$</span>, MRS <span class="math-container">$= 1\leq  \frac{p_X}{p_Y}$</span>. So, solution to the Utility maximisation problem is as follows:
<span class="math-container">$(x^d,y^d)(p_X,p_Y,M)=\begin{cases}\displaystyle \left(0,\frac{M}{p_Y}\right) &amp; \text{if } p_X\geq p_Y \\  \displaystyle\left(\frac{M(p_Y-p_X)}{(2p_Y-p_X)p_X},\frac{M}{2p_Y-p_X}\right) &amp; \text{if } p_X&lt; p_Y\end{cases} $</span></p>
<p>To determine the price-consumption curve as we vary <span class="math-container">$p_Y$</span> (holding <span class="math-container">$p_X$</span> and <span class="math-container">$M$</span> fixed), we determine the relation between <span class="math-container">$x$</span> and <span class="math-container">$y$</span> using equilibrium conditions and try and remove <span class="math-container">$p_Y$</span>.
Equilibrium conditions for the interior optimum i.e. when <span class="math-container">$p_Y &gt; p_X$</span>:</p>
<ul>
<li><span class="math-container">$\dfrac{y}{x+y}=\dfrac{p_X}{p_Y}$</span> and</li>
<li><span class="math-container">$p_Xx+p_Yy=M$</span></li>
</ul>
<p>So, we get Price-consumption curve as <span class="math-container">$2p_Xx+p_Xy=M$</span>.
For <span class="math-container">$p_Y\leq p_X$</span>, price consumption curve is simply the vertical axis <span class="math-container">$x=0$</span> where <span class="math-container">$y\geq \frac{M}{p_X}$</span>.
So, price-consumption curve as we vary <span class="math-container">$p_Y$</span> (holding <span class="math-container">$p_X$</span> and <span class="math-container">$M$</span> fixed) is:</p>
<p><span class="math-container">\begin{eqnarray*} x= \begin{cases} 0 &amp; \text{if } y\geq \frac{M}{p_X} \\ \frac{M-p_Xy}{2p_X} &amp; \text{if } 0&lt;y&lt; \frac{M}{p_X}\end{cases} \end{eqnarray*}</span></p>
<p>Here is the picture of the price-consumption curve:</p>
<p><a href="https://i.sstatic.net/FC27pbVo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/FC27pbVo.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-07-26T16:00:34.643Z
2024-07-26T16:00:34.643Z
Amit
<p>It's true that given the utility function the <span class="math-container">$y$</span>-good is a normal good, so the question is quite odd. Ignoring this, your calculations are correct, but you could simplify to <span class="math-container">$y(p_x,p_y,b)=\frac{b}{2p_y-p_x}$</span>.</p>

<p>Your curve contains a mistake, since <span class="math-container">$x(1,3,48)=19.2$</span>. Indeed by substituting your expressions for <span class="math-container">$x$</span> and <span class="math-container">$y$</span> you can show that <span class="math-container">$2x+y=\frac{b}{p_x}$</span> as long as <span class="math-container">$p_y\ge p_x$</span>, so the price-consumption curve is piecewise linear.</p>

CC BY-SA 4.0
2020-05-15T15:17:04.637Z
Yes, I think my professor mixed something up. I'll rectify my answer and make it correct. Thank you!
Pedro Cunha
CC BY-SA 4.0
2020-05-15T23:50:31.623Z
It doesn't make much sense to edit your question and correct your own answer, since now Adam Bailey's comment and my answer refer to a version which is no longer visible.
VARulle
CC BY-SA 4.0
2020-05-16T01:28:22.593Z
Sorry. I've rolled it back
Pedro Cunha
CC BY-SA 4.0
2020-05-15T10:42:06.980Z
2020-05-15T10:42:06.980Z
VARulle
<p>Suppose a consumer whose income is <span class="math-container">$b$</span> has a utility function given by <span class="math-container">$U(x,y) = 2xy+y^2$</span> with the price of <span class="math-container">$x$</span> being <span class="math-container">$p_x$</span> and the price of <span class="math-container">$y$</span> being <span class="math-container">$p_y$</span>.</p>

<p>Draw the price-consumption curve assuming <span class="math-container">$y$</span> is an inferior good (keep the price of <span class="math-container">$x$</span> constant).</p>

<p>If we set up the Lagrangian <span class="math-container">$$ L = 2xy+y^2 + \lambda \cdot (b - p_xx-p_yy) $$</span></p>

<p>and solve the maximization problem, we'll end up with</p>

<p><span class="math-container">$$x(p_x,p_y,b) = \left( \frac{p_y-p_x}{2p_xp_y - p_x^2} \right) \cdot b, \,\,\,\, y(p_x,p_y,b) = \left( \frac{p_x}{2p_xp_y - p_x^2} \right) \cdot b $$</span></p>

<p>If <span class="math-container">$y$</span> is to be inferior, then we must have (considering income and prices to be strictly positive)</p>

<p><span class="math-container">$$ \frac{\partial y(p_x,p_y,b)}{\partial b} &lt; 0 \implies \left( \frac{p_x}{2p_xp_y - p_x^2} \right)  &lt; 0 \Leftrightarrow 2p_xp_y - p_x^2 &lt; 0  $$</span></p>

<p>However, in order for the amount consumed of <span class="math-container">$y$</span> to be non-negative, we must have</p>

<p><span class="math-container">$$ \left( \frac{p_x}{2p_xp_y - p_x^2} \right) \cdot b \geq 0 \Leftrightarrow 2p_xp_y - p_x^2 \geq 0  $$</span></p>

<p>Thus, we can conclude that if the demand for <span class="math-container">$y$</span> is to be non-negative at strictly positive prices and income, then <span class="math-container">$y$</span> can't be an inferior good.</p>

<p>However, if we respect the condition given by the non-negativity of <span class="math-container">$y$</span>, we can still plot the price-consumption curve. Taking the following pairs of prices </p>

<p><span class="math-container">$$ (p_x,p_y) = \{ (1,1),(1,3),(1,5) \}$$</span></p>

<p>and letting <span class="math-container">$b = 48$</span>, we'll have:</p>

<p><span class="math-container">$$\begin{cases}
x(1,1,48) = 0, y(1,1,48) = 48 \\ x(1,3,48) = 9.60, y(1,3,48) = 9.60 \\ x(1,5,48) \approx 21.34, y(1,5,48) \approx 5.34
\end{cases}$$</span></p>

<p>Plotting these values on a graph, we have:</p>

<p><a href="https://i.sstatic.net/65JVU.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/65JVU.png" alt="Price-consumption curve for y"></a></p>

<p>Is my answer correct?</p>

<p>Thanks!</p>

CC BY-SA 4.0
2020-05-15T11:23:59.463Z
You seem to have inferred the shape of the curve from just 3 points.  It would have been safer to test whether the U-shape is correct by also calculating $x,y$ at $(1,4,48)$.
Adam Bailey
CC BY-SA 4.0
2020-05-15T15:16:23.200Z
You are completely right! If had I done that I'd have noticed that the price-consumption is piece-wise linear!
Pedro Cunha
<utility><consumer-theory><demand>
Price-consumption curve
Pedro Cunha
# Price-consumption curve

(Asked by: username_0 on 2020-05-15)

Suppose a consumer whose income is $b$ has a utility function given by $U(x,y) = 2xy+y^2$ with the price of $x$ being $p\_x$ and the price of $y$ being $p\_y$.

Draw the price-consumption curve assuming $y$ is an inferior good (keep the price of $x$ constant).

If we set up the Lagrangian $$ L = 2xy+y^2 + \lambda \cdot (b - p\_xx-p\_yy) $$

and solve the maximization problem, we'll end up with

$$x(p\_x,p\_y,b) = \left( \frac{p\_y-p\_x}{2p\_xp\_y - p\_x^2} \right) \cdot b, \,\,\,\, y(p\_x,p\_y,b) = \left( \frac{p\_x}{2p\_xp\_y - p\_x^2} \right) \cdot b $$

If $y$ is to be inferior, then we must have (considering income and prices to be strictly positive)

$$ \frac{\partial y(p\_x,p\_y,b)}{\partial b} < 0 \implies \left( \frac{p\_x}{2p\_xp\_y - p\_x^2} \right) < 0 \Leftrightarrow 2p\_xp\_y - p\_x^2 < 0 $$

However, in order for the amount consumed of $y$ to be non-negative, we must have

$$ \left( \frac{p\_x}{2p\_xp\_y - p\_x^2} \right) \cdot b \geq 0 \Leftrightarrow 2p\_xp\_y - p\_x^2 \geq 0 $$

Thus, we can conclude that if the demand for $y$ is to be non-negative at strictly positive prices and income, then $y$ can't be an inferior good.

However, if we respect the condition given by the non-negativity of $y$, we can still plot the price-consumption curve. Taking the following pairs of prices

$$ (p\_x,p\_y) = \{ (1,1),(1,3),(1,5) \}$$

and letting $b = 48$, we'll have:

$$\begin{cases}
x(1,1,48) = 0, y(1,1,48) = 48 \\ x(1,3,48) = 9.60, y(1,3,48) = 9.60 \\ x(1,5,48) \approx 21.34, y(1,5,48) \approx 5.34
\end{cases}$$

Plotting these values on a graph, we have:

[![Price-consumption curve for y](https://i.sstatic.net/65JVU.png)](https://i.sstatic.net/65JVU.png)

Is my answer correct?

Thanks!

**Tags:** utility, consumer-theory, demand

> **Comments:**
> **username_1**: You seem to have inferred the shape of the curve from just 3 points.  It would have been safer to test whether the U-shape is correct by also calculating $x,y$ at $(1,4,48)$.  
> **username_0**: You are completely right! If had I done that I'd have noticed that the price-consumption is piece-wise linear!  

---

## 2 answers

### Answer by username_3 on 2020-05-15. Score: 4 | Accepted answer

It's true that given the utility function the $y$-good is a normal good, so the question is quite odd. Ignoring this, your calculations are correct, but you could simplify to $y(p\_x,p\_y,b)=\frac{b}{2p\_y-p\_x}$.

Your curve contains a mistake, since $x(1,3,48)=19.2$. Indeed by substituting your expressions for $x$ and $y$ you can show that $2x+y=\frac{b}{p\_x}$ as long as $p\_y\ge p\_x$, so the price-consumption curve is piecewise linear.

> **Comments:**
> **username_0**: Yes, I think my professor mixed something up. I'll rectify my answer and make it correct. Thank you!  
> **username_3**: It doesn't make much sense to edit your question and correct your own answer, since now username_1 username_1's comment and my answer refer to a version which is no longer visible.  
> **username_0**: Sorry. I've rolled it back  

---

### Answer by username_2 on 2024-07-26. Score: 1

There is already a good answer to this question by username_3. I am just adding some details.

Given $u(x,y)=2xy+y^2$, note that $u$ is an increasing function and its $\text{MRS}=\dfrac{y}{x+y}$ which is diminishing. So, $u$ is quasi-concave.
Given the MRS, it is clear that there is a possibility that the consumer can spend all his income on commodity $y$. That will happen if at bundle $\left(0,\frac{M}{p\_Y}\right)$, MRS $= 1\leq \frac{p\_X}{p\_Y}$. So, solution to the Utility maximisation problem is as follows:
$(x^d,y^d)(p\_X,p\_Y,M)=\begin{cases}\displaystyle \left(0,\frac{M}{p\_Y}\right) & \text{if } p\_X\geq p\_Y \\ \displaystyle\left(\frac{M(p\_Y-p\_X)}{(2p\_Y-p\_X)p\_X},\frac{M}{2p\_Y-p\_X}\right) & \text{if } p\_X< p\_Y\end{cases} $

To determine the price-consumption curve as we vary $p\_Y$ (holding $p\_X$ and $M$ fixed), we determine the relation between $x$ and $y$ using equilibrium conditions and try and remove $p\_Y$.
Equilibrium conditions for the interior optimum i.e. when $p\_Y > p\_X$:

* $\dfrac{y}{x+y}=\dfrac{p\_X}{p\_Y}$ and
* $p\_Xx+p\_Yy=M$

So, we get Price-consumption curve as $2p\_Xx+p\_Xy=M$.
For $p\_Y\leq p\_X$, price consumption curve is simply the vertical axis $x=0$ where $y\geq \frac{M}{p\_X}$.
So, price-consumption curve as we vary $p\_Y$ (holding $p\_X$ and $M$ fixed) is:

\begin{eqnarray\*} x= \begin{cases} 0 & \text{if } y\geq \frac{M}{p\_X} \\ \frac{M-p\_Xy}{2p\_X} & \text{if } 0<y< \frac{M}{p\_X}\end{cases} \end{eqnarray\*}

Here is the picture of the price-consumption curve:

[![enter image description here](https://i.sstatic.net/FC27pbVo.png)](https://i.sstatic.net/FC27pbVo.png)


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