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

Question

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 $y$).

Answer 1

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.

Answer 2

$$D(p,I) = \arg\max {U(x):px≤I}.$$ The problem $$\arg\max_x {U(x/I):px/I≤1}$$ has solution $$x/I = D(p,1)$$ hence $$D(p,I)/I = D(p,1)$$ or $$D(p,I) = ID(p,1)$$

Answer 3

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.