<p>Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.</p>

<hr>

<p>Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.</p>

<p>Let the line <span class="math-container">$L$</span> consist of the points <span class="math-container">$((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$</span> where <span class="math-container">$a,b,c \in \mathbb{R}$</span>. I will denote this set of points by <span class="math-container">$L$</span>, the line. It is assumed that we are talking about an actual line, so <span class="math-container">$a,b$</span> are not simultaneously equal to <span class="math-container">$0$</span>.</p>

<p>Let 
<span class="math-container">$$
U(x,y) = \left\{\begin{array}{lc}
\arctan (b \cdot x + a \cdot y) &amp; \text{ if } (x,y) \in L \\
-\pi &amp; \text{ if } (x,y) \notin L. 
\end{array}
\right.
$$</span></p>

<p><strong>Claim</strong> The utility function will always be maximized by a point on <span class="math-container">$L$</span>.</p>

<p>This is easy, because <span class="math-container">$\arctan$</span> maps to <span class="math-container">$(-\frac{\pi}{2},\frac{\pi}{2})$</span>, thus its points always yield greater utility than other points. We specified that the slope of <span class="math-container">$L$</span> is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on <span class="math-container">$L$</span>.</p>

<hr>

<p>We now come back to the question of whether income <span class="math-container">$I$</span> can be negative. If they can be, than every point <span class="math-container">$L$</span> will be a solution of the consumers utility maximization problem for a certain <span class="math-container">$I,p_x,p_y$</span>. The reason for this is that <span class="math-container">$b \cdot x + a \cdot y$</span> creates an ordering of the points of <span class="math-container">$L$</span>. (Lines perpendicular to <span class="math-container">$L$</span> have a slope of <span class="math-container">$-b/a$</span>, these are the level curves of <span class="math-container">$b \cdot x + a \cdot y$</span>.) Select a point <span class="math-container">$(x_0,y_0) \in L$</span>. By setting <span class="math-container">$I$</span> to
<span class="math-container">$$
I = p_x \cdot x_0 + p_y \cdot y_0,
$$</span>
the consumer has barely enough money to purchase this basket. If <span class="math-container">$L$</span> is non-negative sloped and prices are positive then no points with higher utility are attainable.</p>

<hr>

<p>The construction I gave is not unique. One could also use the continuous utility function
<span class="math-container">$$
\hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right)
$$</span>
to achieve the same result. (If <span class="math-container">$a = 0$</span> one may use
<span class="math-container">$$
\hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right)
$$</span>
instead.)</p>

CC BY-SA 4.0
2019-06-09T15:36:25.397Z
Thanks for your help answer! I might start another question with additional restrictions. What are some of the reasonable yet not-too-strong assumptions?
High GPA
CC BY-SA 4.0
2019-06-09T15:41:25.193Z
@HighGPA I have no clue. I don't think this is a much studied problem, the Engel-curve seems to be an intro to Micro tool, and in that context people usually study non-negative good baskets.
Giskard
CC BY-SA 4.0
2019-06-09T09:13:04.713Z
2019-06-09T15:42:21.657Z
2019-06-09T15:42:21.657Z
Giskard
<p>If we restrict the consumption at <span class="math-container">$\mathbb R_+^n$</span>, then it seems like we are implicitly assuming that the Engel curve pass through the origin, so the iff condition would be homothetic preference.</p>

<p>However, if we let <span class="math-container">$x\in\mathbb R^n$</span>, other classes of preference would also give linear Engel curves. For example, those represented by a quasi linear utility in two dimensions: <span class="math-container">$u(x)=x_1+f(x_2)$</span>; in this case, the Engel curves are parallel lines.</p>

<hr>

<p>It could be assumed that, as in the standard consumer theory, the preference is non-satiated and continuous, so the indifference sets are curves.</p>

<microeconomics><consumer-theory>
What is the iff condition for a preference with linear Engel curves (all Engal curves are linear)?
High GPA
# What is the iff condition for a preference with linear Engel curves (all Engal curves are linear)?

(Asked by: username_0 on 2019-06-08)

If we restrict the consumption at $\mathbb R\_+^n$, then it seems like we are implicitly assuming that the Engel curve pass through the origin, so the iff condition would be homothetic preference.

However, if we let $x\in\mathbb R^n$, other classes of preference would also give linear Engel curves. For example, those represented by a quasi linear utility in two dimensions: $u(x)=x\_1+f(x\_2)$; in this case, the Engel curves are parallel lines.

---

It could be assumed that, as in the standard consumer theory, the preference is non-satiated and continuous, so the indifference sets are curves.

**Tags:** microeconomics, consumer-theory


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## 1 answers

### Answer by username_1 on 2019-06-09. Score: 3 | Accepted answer

Are negative incomes (perhaps even negative prices) also allowed? Unless they are, non-horizontal or non-vertical lines are impossible as Engel curves because they have to intersect the negative quadrant of the coordinate system. I will come back to this later.

---

Without additional restrictions (perhaps a lot of additional restrictions) on the preference relations, for any non-negative sloped line I can fabricate a utility function such that the line will be its Engel curve.

Let the line $L$ consist of the points $((x,y)\in\mathbb{R}^2|c = a \cdot x + b \cdot y)$ where $a,b,c \in \mathbb{R}$. I will denote this set of points by $L$, the line. It is assumed that we are talking about an actual line, so $a,b$ are not simultaneously equal to $0$.

Let
$$
U(x,y) = \left\{\begin{array}{lc}
\arctan (b \cdot x + a \cdot y) & \text{ if } (x,y) \in L \\
-\pi & \text{ if } (x,y) \notin L.
\end{array}
\right.
$$

**Claim** The utility function will always be maximized by a point on $L$.

This is easy, because $\arctan$ maps to $(-\frac{\pi}{2},\frac{\pi}{2})$, thus its points always yield greater utility than other points. We specified that the slope of $L$ is non-negative, so for any positive price pair the budget line will cross it, enabling us to chose a point on $L$.

---

We now come back to the question of whether income $I$ can be negative. If they can be, than every point $L$ will be a solution of the consumers utility maximization problem for a certain $I,p\_x,p\_y$. The reason for this is that $b \cdot x + a \cdot y$ creates an ordering of the points of $L$. (Lines perpendicular to $L$ have a slope of $-b/a$, these are the level curves of $b \cdot x + a \cdot y$.) Select a point $(x\_0,y\_0) \in L$. By setting $I$ to
$$
I = p\_x \cdot x\_0 + p\_y \cdot y\_0,
$$
the consumer has barely enough money to purchase this basket. If $L$ is non-negative sloped and prices are positive then no points with higher utility are attainable.

---

The construction I gave is not unique. One could also use the continuous utility function
$$
\hat{U}(x,y) = \min\left(a \cdot \left(x - \frac{c}{a} \right); - b \cdot y\right)
$$
to achieve the same result. (If $a = 0$ one may use
$$
\hat{U}(x,y) = \min\left(a \cdot x; - b \cdot \left(y - \frac{c}{b} \right)\right)
$$
instead.)

> **Comments:**
> **username_0**: Thanks for your help answer! I might start another question with additional restrictions. What are some of the reasonable yet not-too-strong assumptions?  
> **username_1**: @HighGPA I have no clue. I don't think this is a much studied problem, the Engel-curve seems to be an intro to Micro tool, and in that context people usually study non-negative good baskets.  

---

