Hi- 90% sure this is correct . . .
Firstly, consider the following equation;
U=I^(1-ε)/(1-ε)×L^(1-l)/(1-l)
L and I are functions of one another, regulated by this condition;
L+I/w = H. H and w are constants.
We need to solve for ε; expressed as a function of h, when U is maximised.
I think I have it solved.
Firstly express the function above as
f(I,L,ε,l)=f(I,ε)×f(L,l)
differentiate to
(du/u)/dh f(I,L,ε,l)=(du/u)/dh f(I,ε)+(du/u)/dh f(L,l)
(du/u)/dh =0 due to maxisation constraint, hence
(du/u)/dh f(I,ε)=-(du/u)/dh f(L,l)
we then decompose the differtial terms;
(du/u)/dh = (dI/I)/dh * (dU/U)/(DI/DI)
(dL/L)/dh = (dL/L)/dh * (dU/U)/(dL/L)
substituting and solving gives;
(1/h ×1/(ε+1))=(1/(H-h)×1/(l+1))
ε=((H-h).(l+1)/h))-1
pretty sure that is right, but not sure about the clumsy proof.
Thanks in advance, sorry if it is not clear.