Homogenous of degree one in utility function.

Question

Question

My solution is as follows. Please check my solution. If I make a mistake, please tell. I am really not sure about my solution. Thank you

U(x) is homogenous of degree one i.e. u(tx)=tu(x)

Firstly I show that the indirect utility function is homogenous of degree one in m.

By the utility maximization,

V(p,m)=max u(x) subject to px$\le$ m

tv(p,m)=max tu(x) subject to px$\le$ m

Since u(tx)=tu(x), tv(p,m)=max u(tx) subject to px$\le$ m

Then v(p,tm)=tv(p,m)

That is the indirect utility function is homogenous of degree one.

I show that the expenditure function is homogenous of degree one in u by using previous result.

I know that

v(p,m)=v(p, e(p,u))=u(x)

Since u(x) is homogenous of degree one and v(p,m) is homogenous of degree one in m, v(p, e(p,u)) have to be homogenous of degree one in e(p,u).

In other words, v(p, e(p,u(tx)))=v(p, e(p,tu(x)))=tv(p, e(p,u)) holds iff e(p,tu(x))=te(p,u(x))

i.e. The expensive function e(p,u) is homogenous of degree one in u.

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Now I will show that the marshallian demand x(p,m) is homogenous of degree one in m.

By Roy's identity,

$$\frac{\partial v(p,m)/\partial p}{\partial v(p,m)/\partial m}=x(p,m)$$

By the first result, since v(p,m) is homogenous of degree one in m, then x(p,m) is homogenous of degree one in m.

now lets show that the hicksian demand is homogenous of degree one in u.

I know that

x(p,m)= x(p,e(p,u))=h(p,u) ........(1)

x(p,tm)=tx(p,m)=tx(p,e(p,u))=x(p,te(p,u))

Since e(p,u) is homogenous of degree one by the second part,

x(p,te(p,u))=x(p,e(p,u(tx))=h(p,u(tx))=h(p,tu(x))=th(p,u(x)) must hold since the equality(1) exist.

That is the hicksian demand is homogenous of degree one in u.

Answer 1

The way you show that $v(p,m)$ is homogeneous of degree one in $m$ is correct, but the reason why this implies that, $e(p,u)$ is homogeneous of degree one in $u$, is not very precise in your argument. For example, duality tells us $$v(p,e(p,u))=u,$$ where $u$ is just a target utility level, but should not be $u(x)$ as in your proof.

Here is one possible way to proceed: Since $v(p,m)$ is homogeneous of degree one in $m$, it can be written as $$v(p,m)=mv(p,1)=m\tilde v(p).$$ Applying the equality $v(p,e(p,u))=u$ gives $$e(p,u)=\frac{u}{\tilde v(p)},$$ which clearly implies that $e(p,u)$ is homogeneous of degree one in $u$. You can use a similar argument to prove homogeneity of the Hicksian demand.

With all that said, I would suggest you prove the original statement directly using the definitions of expenditure function and Hicksian demand. For instance, \begin{align*} e(p,\lambda u)&= \min p\cdot x ~~\text{ s.t. } u(x)\geq \lambda u\\ &=\lambda\min p\cdot\frac{1}{\lambda}x ~~\text{ s.t. }\frac{1}{\lambda}u(x)\geq u\\ &=\cdots \end{align*}