<p>It is quite simple. The engels curve is the change in demand for a good as a function of income, keeping prices fixed</p>

<p>Solve the constrained maximisation problem
$$
\max B^{0.67}Z^{0.33}-\lambda(P_bB+P_zZ-Y) \Leftrightarrow \\
B=\frac{0.67Y}{P_b} \\
Z=\frac{0.33Y}{P_z} \\
\lambda = \frac{0.5303709372}{P_z^{33/100}P_b^{67/100}}
$$</p>

<p>Fix prices at some level. I choose 1 for simplicity. Then the engels is simply
$$
Y=\frac{1}{0.67}B \\
Y=\frac{1}{0.33}Z
$$
<img src="https://i.sstatic.net/sNDBq.png" alt="enter image description here"></p>

CC BY-SA 3.0
2015-03-05T09:25:51.223Z
2015-03-05T09:25:51.223Z
Rud Faden
<p>To derive the Engel curve follow the following steps:</p>
<ol>
<li><p>Derive the demand functions by maximizing the Utility function subject to the budget constraint.<br />
For this you can use either:</p>
<ul>
<li>The Lagrange method like in @Rud-Faden solutions;</li>
<li>Or set <span class="math-container">$MRS = P_b/P_z$</span>;</li>
<li>Or even better since we have a Coub Douglas Utility let's use a common knowledge, for any utility function of type <span class="math-container">$$U = B^{\alpha}Z^{\beta}$$</span> <span class="math-container">${\alpha}$</span> and <span class="math-container">${\beta}$</span> shows the share of income spent on each good. Thus the demand functions will be <span class="math-container">$$B = \frac{{\alpha}Y}{P_B} $$</span> <span class="math-container">$$Z = \frac{{\beta}Y}{P_Z} $$</span> In your utility function <span class="math-container">${\alpha}=0.67;{\beta} = 0.33$</span> so <span class="math-container">$$B = \frac{0.67Y}{P_B} $$</span> <span class="math-container">$$Z = \frac{0.33Y}{P_Z} $$</span></li>
</ul>
</li>
<li><p>On the demand functions derived 1, Treat the price as constant.
The demand function itself is the Engel Curve but you need to treat it as a function of income not price because price is constant. <span class="math-container">$B(Y), Z(Y)$</span>.</p>
</li>
<li><p>To plot the Engel curve we usually set the income on the <span class="math-container">$y$</span> axis and the quantity in the <span class="math-container">$x$</span> axis. Just arrange the demand functions so that you will have Income in the left and quantity of goods in the right. and fix prices to a certain constant.</p>
</li>
</ol>
<p><span class="math-container">$$Y(B) = \frac{B*P_B}{0.67} $$</span>  <span class="math-container">$$Y(Z) = \frac{Z*P_Z}{0.33} $$</span></p>
<p><a href="https://i.sstatic.net/0OZNv.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/0OZNv.png" alt="Engel curve for good B &amp; Z" /></a></p>
<p>The price for both goods were fixed to 1.</p>

CC BY-SA 4.0
2021-12-07T14:51:14.680Z
2021-12-07T14:56:54.870Z
2021-12-07T14:56:54.870Z
Macosso
<p>I have a utility function of $U = B^{.67}Z^{.33}$ with Income $Y$, Price of Good B is $P_b$ and price of Z is $P_z$ I now need to derive an engel curve for this. I have no idea where to start.</p>

CC BY-SA 3.0
2015-03-05T08:25:03.233Z
This might help http://www.udel.edu/johnmack/frec343/engel.html
Rud Faden
<microeconomics>
Deriving Equation for Engel Curve
All About Groups
# Deriving Equation for Engel Curve

(Asked by: username_0 on 2015-03-05)

I have a utility function of $U = B^{.67}Z^{.33}$ with Income $Y$, Price of Good B is $P\_b$ and price of Z is $P\_z$ I now need to derive an engel curve for this. I have no idea where to start.

**Tags:** microeconomics
> **Comments:**
> **username_1**: This might help http://www.udel.edu/johnmack/frec343/engel.html  

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## 2 answers

### Answer by username_1 on 2015-03-05. Score: 8

It is quite simple. The engels curve is the change in demand for a good as a function of income, keeping prices fixed

Solve the constrained maximisation problem
$$
\max B^{0.67}Z^{0.33}-\lambda(P\_bB+P\_zZ-Y) \Leftrightarrow \\
B=\frac{0.67Y}{P\_b} \\
Z=\frac{0.33Y}{P\_z} \\
\lambda = \frac{0.5303709372}{P\_z^{33/100}P\_b^{67/100}}
$$

Fix prices at some level. I choose 1 for simplicity. Then the engels is simply
$$
Y=\frac{1}{0.67}B \\
Y=\frac{1}{0.33}Z
$$
![enter image description here](https://i.sstatic.net/sNDBq.png)


---

### Answer by username_2 on 2021-12-07. Score: 0

To derive the Engel curve follow the following steps:

1. Derive the demand functions by maximizing the Utility function subject to the budget constraint.  
   For this you can use either:

   * The Lagrange method like in @Rud-Faden solutions;
   * Or set $MRS = P\_b/P\_z$;
   * Or even better since we have a Coub Douglas Utility let's use a common knowledge, for any utility function of type $$U = B^{\alpha}Z^{\beta}$$ ${\alpha}$ and ${\beta}$ shows the share of income spent on each good. Thus the demand functions will be $$B = \frac{{\alpha}Y}{P\_B} $$ $$Z = \frac{{\beta}Y}{P\_Z} $$ In your utility function ${\alpha}=0.67;{\beta} = 0.33$ so $$B = \frac{0.67Y}{P\_B} $$ $$Z = \frac{0.33Y}{P\_Z} $$
2. On the demand functions derived 1, Treat the price as constant.
   The demand function itself is the Engel Curve but you need to treat it as a function of income not price because price is constant. $B(Y), Z(Y)$.
3. To plot the Engel curve we usually set the income on the $y$ axis and the quantity in the $x$ axis. Just arrange the demand functions so that you will have Income in the left and quantity of goods in the right. and fix prices to a certain constant.

$$Y(B) = \frac{B\*P\_B}{0.67} $$ $$Y(Z) = \frac{Z\*P\_Z}{0.33} $$

[![Engel curve for good B & Z](https://i.sstatic.net/0OZNv.png)](https://i.sstatic.net/0OZNv.png)

The price for both goods were fixed to 1.


---

