# Linear Homothetic Utility

A Homothetic Utility is where $$\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax,ay)=au(x,y)$$ (or its monotonic transformation).

A linear Homothetic utility is defined as $$\forall x,y, \forall a \in \mathbb{R}_+: \ u(ax+b,ay+c)=au(x+b,y+c)$$ where $$b,c$$ are constants.

This preference has very similar property as the homothetic preference. In fact, if we simply translate the coordinate system in the direction of (b,c), then the preference becomes homothetic.

Are there any works covering this property? I've checked a lot of theory papers in homothetic preference but found no luck.

Homothetic Preferences by James DOW· and Sergio Ribeiro da Costa WERLANG

Homothetic and weakly homothetic preferences by J.C. Candeal, E. Indurain

Linear-homothetic preferences, by B Datta, H Dixon

• in the first line, should $au(x +y)$ be $au(x,y)$, as $u$ seems to take 2 arguments? – 201p May 27 '19 at 23:53
• @201p You are right, that was a typo – High GPA May 28 '19 at 0:03
• @Giskard You are right about a,x,y. b,c are constants – High GPA May 30 '19 at 14:26
• Can you give an example of a function $u \neq 0$ satisfying this identity? – Bertrand May 30 '19 at 17:13
• If you want to allow for $(b,c)$-translations, your then the preference should instead satisfy $$\forall x,y, \forall a \in \mathbb{R}_+: \ u(a(x+b),a(y+c))=au(x+b,y+c),$$ but this is equivalent to homotheticity. – Bertrand Jun 2 '19 at 10:44

## 1 Answer

The only utility function that comes to mind is the Stone-Geary utility function. For 2 goods, $$x$$ and $$y$$, this takes the form: $$u(x,y) = (x - a)^\alpha (y- b)^{1- \alpha}.$$ This is a Cobb-Douglas type of utility function where $$a$$ and $$b$$ are subsistence levels, i.e. you need to consume at least $$a$$ from $$x$$ and $$b$$ from $$y$$ to survive. It is the utility function that leads to the Linear expenditure system.

To see that it is linear homothetic notice that: \begin{align*} u(\beta \tilde x + a, \beta \tilde y + b) &= (\beta \tilde x + a - a)^\alpha (\beta \tilde y + b - b)^{1-\alpha},\\ &=\beta (\tilde x)^\alpha (\tilde y)^{1-\alpha},\\ &=\beta ((\tilde x + a) - a)^\alpha ((\tilde y + b) - b)^{1-\alpha},\\ &= \beta u(\tilde x + a, \tilde y + b). \end{align*} It could be that there is more work on utility functions with subsistence levels that lead to other preferences that are also linear homothetic.

For example you could define a CES utility function with subsistence levels: $$u(x,y) = (\alpha_x(x -a)^\sigma + \alpha_y(y-b)^\sigma)^{1/\sigma}$$ This will also satisfy linear homogeneity. This paper of Baumgärtner, Drupp & Quaas does something like this.

In general, if you take any homothetic utility function $$u(x,y)$$ then the modified 'subsistence-augmented' function: $$\tilde u(x, y) \equiv u(x - a, y - b),$$ will be linear homothetic.