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.