<p>A homothetic preference means that for some utility function representing the preferences,</p>

<p>$$u(\alpha x, \ \alpha y) = \alpha u(x, \ y)$$ for any bundle $(x, \ y)$.</p>

<p>So now consider the sum of two different homothetic utility functions, $w$.</p>

<p>$$u(x, \ y), v(x, \ y)$$
$$w(x, \ y) = u(x, \ y) + v(x, \ y)$$
$$\alpha w(x, \ y) = \alpha u(x, \ y) + \alpha v(x, \ y)$$
$$= u(\alpha x, \ \alpha y) + v(\alpha x, \ \alpha y)$$
$$= w(\alpha x, \ \alpha y)$$</p>

CC BY-SA 3.0
2015-10-09T07:15:52.317Z
It is highly likely that the definition of homotheticity of a function is more general than what is defined here. What you are defining here is more of the definition of homogeneity of degree 1.
ramazan
CC BY-SA 3.0
2015-10-09T07:17:16.313Z
note: $u(x,y)=x+y+5$ is not homogenous but it is homothetic and it represents nomothetic preferences.
ramazan
CC BY-SA 3.0
2015-10-09T13:40:17.313Z
Indeed the definition given is for homogeneous functions, which is a subset of homothetic functions.
Alecos Papadopoulos
CC BY-SA 3.0
2015-10-09T14:40:45.060Z
You guys are correct, I am making a stronger assumption than homotheticity.
Kitsune Cavalry
CC BY-SA 3.0
2015-10-09T04:34:05.063Z
2015-10-09T04:34:05.063Z
Kitsune Cavalry
<p>Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$.</p>

<p>Defn: A function is homothetic if it is a monotonic transformation of a homogenous function.</p>

<p>Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then 
$$
\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} = \frac{g'(u(x,y))\frac{\partial u}{\partial x}}{g'(u(x,y))\frac{\partial u}{\partial y}}=\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}
$$
is homogenous of degree zero. </p>

<p>Let</p>

<p>i) $u_1(x,y)=x+y$, </p>

<p>ii) $u_2(x,y)=\log(2x+y)$</p>

<p>Then, $u_1$ and $u_2$ are homothetic functions since they are monotonic transformations of homogenous functions (of degree 1). Let</p>

<p>iii) $u_3(x,y)=x+y+log(2x+y)$. Then</p>

<p>$$
MRS_{u_3}=\frac{\frac{\partial u_3}{\partial x}}{\frac{\partial u_3}{\partial y}}=\frac{1+\frac{2}{2x+y}}{1+\frac{1}{2x+y}}=\frac{2x+y+2}{2x+y+1}
$$
which is not homogenous of degree 0. </p>

<p>Hence, sum of homothetic functions is not necessarily homothetic. </p>

CC BY-SA 3.0
2015-10-09T13:38:57.283Z
(+1) for including the (not so widely known) lemma. Perhaps you could consider including also its simple proof, so that it is clear that the MRS is homogeneous of degree zero, irrespective of the degree of homogeneity of $u$.
Alecos Papadopoulos
CC BY-SA 3.0
2015-10-09T08:08:00.447Z
2015-10-09T08:13:46.033Z
2015-10-09T08:13:46.033Z
ramazan
<p>If two utility functions represent homothetic preferences, will their sum also be homothetic?</p>

<utility><preferences>
Sum of Homothetic Functions
Sher Afghan
# Sum of Homothetic Functions

(Asked by: username_0 on 2015-10-09)

If two utility functions represent homothetic preferences, will their sum also be homothetic?

**Tags:** utility, preferences


---

## 2 answers

### Answer by username_2 on 2015-10-09. Score: 9 | Accepted answer

Defn: A function $h:\mathbb{R}^2\rightarrow \mathbb{R}$ is homogenous of degree $k$ if for every nonzero $\alpha$, $h(\alpha x, \alpha y)=\alpha^k h(x,y)$.

Defn: A function is homothetic if it is a monotonic transformation of a homogenous function.

Lemma: If $f$ is homothetic, i.e. $f=g\circ u$ for some strictly increasing $g$ and homogenous $u$ then
$$
\frac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} = \frac{g'(u(x,y))\frac{\partial u}{\partial x}}{g'(u(x,y))\frac{\partial u}{\partial y}}=\frac{\frac{\partial u}{\partial x}}{\frac{\partial u}{\partial y}}
$$
is homogenous of degree zero.

Let

i) $u\_1(x,y)=x+y$,

ii) $u\_2(x,y)=\log(2x+y)$

Then, $u\_1$ and $u\_2$ are homothetic functions since they are monotonic transformations of homogenous functions (of degree 1). Let

iii) $u\_3(x,y)=x+y+log(2x+y)$. Then

$$
MRS\_{u\_3}=\frac{\frac{\partial u\_3}{\partial x}}{\frac{\partial u\_3}{\partial y}}=\frac{1+\frac{2}{2x+y}}{1+\frac{1}{2x+y}}=\frac{2x+y+2}{2x+y+1}
$$
which is not homogenous of degree 0.

Hence, sum of homothetic functions is not necessarily homothetic.

> **Comments:**
> **username_3**: (+1) for including the (not so widely known) lemma. Perhaps you could consider including also its simple proof, so that it is clear that the MRS is homogeneous of degree zero, irrespective of the degree of homogeneity of $u$.  

---

### Answer by username_1 on 2015-10-09. Score: 1

A homothetic preference means that for some utility function representing the preferences,

$$u(\alpha x, \ \alpha y) = \alpha u(x, \ y)$$ for any bundle $(x, \ y)$.

So now consider the sum of two different homothetic utility functions, $w$.

$$u(x, \ y), v(x, \ y)$$
$$w(x, \ y) = u(x, \ y) + v(x, \ y)$$
$$\alpha w(x, \ y) = \alpha u(x, \ y) + \alpha v(x, \ y)$$
$$= u(\alpha x, \ \alpha y) + v(\alpha x, \ \alpha y)$$
$$= w(\alpha x, \ \alpha y)$$

> **Comments:**
> **username_2**: It is highly likely that the definition of homotheticity of a function is more general than what is defined here. What you are defining here is more of the definition of homogeneity of degree 1.  
> **username_2**: note: $u(x,y)=x+y+5$ is not homogenous but it is homothetic and it represents nomothetic preferences.  
> **username_3**: Indeed the definition given is for homogeneous functions, which is a subset of homothetic functions.  
> **username_1**: You guys are correct, I am making a stronger assumption than homotheticity.  

---

