# How to show that a homothetic utility function has demand functions which are linear in income

A homothetic utility function is one which is a monotonic transformation of a homogeneous utility function.

I am asked to show that if a utility function is homothetic then the associated demand functions are linear in income.

In general if $H$ is monotonic and we compose it with a function $g$ which is homogeneous we get $H(g(x,y))$ which is homothetic and so the ratio of derivatives of $H$ with respect to $x$ and $y$ is the same as the ratio of the derivatives of $g$ with respect to $x$ and $y$ since the outer function cancels out. This means all income expansion paths are rays from the origin and the slope of indifference curves (level sets) have the same slope along the income expansion path.

My question is how does one show generally that the demand functions are linear in income? If you don't have the functional form of the utility function then what can you substitute into the budget constraint to solve for $x$ or $y$ as functions of prices and income only?

All you know is that the negative ratio of derivatives is equal to $p_1/p_2$ but you can't substitute anything into the budget constraint for $x$ or $y$ since you only have general partial derivatives.

The only way I can think to do that is take the inverse of the derivative of the inner (homogenous) function g in order to solve for $x$ (or $y$).

• Hi, welcome to Economics SE. Usually people here don't like straight up homework questions without any indication of work done on it, though it seems you are quite lost. Is this supposed to be a more general proof question? If I may ask, what context are you trying to answer this question in? Are you in a graduate masters or doctorate program? – Kitsune Cavalry Oct 4 '15 at 2:05
• I voted to leave this homework question open, because the OP does present his thoughts on the matter. – Alecos Papadopoulos Oct 13 '15 at 22:42

I think what you need is that if $U(x,y)$ is homothetic then $$\forall \alpha \in \mathbb{R}_{++}, \forall (x,y) : \hskip 6pt \frac{\frac{\partial U(x,y)}{\partial x}}{\frac{\partial U(x,y)}{\partial y}} = \frac{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial x}}{\frac{\partial U(\alpha \cdot x,\alpha \cdot y)}{\partial y}}$$ and love.

• Then it should be "all you need", I guess. – Alecos Papadopoulos Oct 5 '15 at 12:30
• @AlecosPapadopoulos that was my first draft but since the proof is not complete I decided to do away with the "all". – denesp Oct 5 '15 at 13:21

$$D(p,I) = \text{argmax} {U(x):px≤I} = \text{argmax}{U(x/I):px/I≤1}$$ has solution $$x/I = D(p,1)$$ hence $$D(p,I)/I = D(p,1)$$ or $$D(p,I) = ID(p,1)$$

1.Since there exists an assumption that "the preference is homothetic", according to the exact definition, U(tx)=tU(x), where U(x) is the direct utility of goods x and U(x) is homogeneous of degree one. under the previous assumption. (https://en.wikipedia.org/wiki/Homothetic_preferences)

2.I need to prove x(p,m) is also homogeneous of degree one in m. We can prove it by contradiction.

Suppose x(p,tm)≠ tx(p,m) and x(p,m) is the optimal choice which maximizes utility based on the budget constraint m.

It's noticed that both consumptions x(p,tm)* and tx(p,m)* are feasible based on the budget tm*. But x(p,tm)* is the optimal choice, we can get U(x(p,tm)) ＞ U(tx(p,m)).

We can multiply by 1/t both sides, which turns out to be U(x(p,tm))/ t＞U(tx(p,m))/ t. Since U(x) is homogeneous of degree one in x, we can put 1/t inside U(x), which is U(X(p,tm)/t)＞U(tx(p,m)/t)=U(x(p,m)).

Notice that x(p,m) is already the optimal choice based on the budget constraint m, but here we find another optimal consumption x(p,tm)/t . Contradicts.

So x(p,tm) ＝ tx(p,m). And x(p,m) is homogeneous of degree one in m.

3.Because the preference is homothetic, x(p,tm)=tx(p,m), which means that mx(p)=mx(p,1)=x(p,m).

So x(p,m)=mx(p) where the demand is the linear function of the income.

I apologize for my rudeness.

• This answer does not use the fact that the utility function is homothetic. What it does say is that if demand is linear in income, then demand is linear in income. – denesp Nov 10 '16 at 7:29
• If you guys are not able to understand my proof, I won't blame you for it. But still you guys need to prove this question in your ways and convince everyone with your proof, not just a well-known but useless formula and nothing else, which can't prove anything. – Li.shen Nov 12 '16 at 1:24
• BTW, recommend an advanced microeconomics textbook called Microeconomics Analysis by Hal.Varian. That's what you guys and your professors really need. – Li.shen Nov 12 '16 at 1:32
• Thank you for your kind words wise Li.shen. Your new proof, which is as I can still see in the edit history very different from your old proof, is mostly correct. Now if you only had the wisdom not to blame us for not writing this one in the first place. Btw. not all utility functions representing homothetic preferences are homogeneous of degree one. All homothetic preferences have a utility representation which is homogeneous of degree one, but other representations exist as well. – denesp Nov 12 '16 at 8:19