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.
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.