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

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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.

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    $\begingroup$ (+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 $u$. $\endgroup$ – Alecos Papadopoulos Oct 9 '15 at 13:38
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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, $w$.

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

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    $\begingroup$ 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. $\endgroup$ – ramazan Oct 9 '15 at 7:15
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    $\begingroup$ note: $u(x,y)=x+y+5$ is not homogenous but it is homothetic and it represents nomothetic preferences. $\endgroup$ – ramazan Oct 9 '15 at 7:17
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    $\begingroup$ Indeed the definition given is for homogeneous functions, which is a subset of homothetic functions. $\endgroup$ – Alecos Papadopoulos Oct 9 '15 at 13:40
  • $\begingroup$ You guys are correct, I am making a stronger assumption than homotheticity. $\endgroup$ – Kitsune Cavalry Oct 9 '15 at 14:40

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