<p>I think this statement is untrue. Gorman form is a sufficient condition for linear Engle curves, but it is not necessary. Consider the following example:
<span class="math-container">$$v(p,a_i)=\beta(p)a_i^2$$</span>
Applying Roy's lemma yield demands which are still linear in <span class="math-container">$a$</span>.</p>

CC BY-SA 4.0
2023-03-08T02:39:57.200Z
The reasoning seems reasonable. I would check it. Thanks a lot!
DRM
CC BY-SA 4.0
2023-03-07T20:25:34.953Z
2023-03-07T20:25:34.953Z
Fića
<p>How to prove that if the Engel curves (expenditures as a function of wealth) are linear in wealth, then the indirect utility function has the form <span class="math-container">$v_{i}(p,a_{i})=\alpha_{i}(p)+\beta(p)a_{i}$</span> for an agent with utility function <span class="math-container">$u_{i}$</span>? I think the converse statement is easy by using Roy's Identity, but how about this one? Thanks!</p>

CC BY-SA 4.0
2023-03-07T15:49:44.280Z
Could you please define the variables in your notation?
Giskard
CC BY-SA 4.0
2023-03-08T02:31:46.797Z
Sorry. I forgot to give the notation. p is the price and a is the wealth. alpha and beta is two coefficients depend on p only.
DRM
<utility><preferences><demand><wealth>
Linear Engel Curve
DRM
# Linear Engel Curve

(Asked by: username_0 on 2023-03-07)

How to prove that if the Engel curves (expenditures as a function of wealth) are linear in wealth, then the indirect utility function has the form $v\_{i}(p,a\_{i})=\alpha\_{i}(p)+\beta(p)a\_{i}$ for an agent with utility function $u\_{i}$? I think the converse statement is easy by using Roy's Identity, but how about this one? Thanks!

**Tags:** utility, preferences, demand, wealth
> **Comments:**
> **username_1**: Could you please define the variables in your notation?  
> **username_0**: Sorry. I forgot to give the notation. p is the price and a is the wealth. alpha and beta is two coefficients depend on p only.  

---

## 1 answers

### Answer by username_2 on 2023-03-07. Score: 2

I think this statement is untrue. Gorman form is a sufficient condition for linear Engle curves, but it is not necessary. Consider the following example:
$$v(p,a\_i)=\beta(p)a\_i^2$$
Applying Roy's lemma yield demands which are still linear in $a$.

> **Comments:**
> **username_0**: The reasoning seems reasonable. I would check it. Thanks a lot!  

---

