Suppose we have one consumer (his name is Robinson Crusoe)
with a Cobb-Douglas utility function for consumption C and labor L:u(C,L)=C^2/3(L0-L)^1/3,
Where L0 is his initial endowment of labor. The consumer is endowed with one unit of labor,
that is, L0 = 1. There is one firm with a production technology: C=½L, and the firm is owned by the consumer.
You may assume that the price of consumption good is equal to one.

Find the firm's output supply and input demand functions. What is the maximized profit of the firm?

문제 풀다가 이거 하나때문에 뒤에 문제들을 하나도 못풀고있는데요.
생산함수가 규모에대한 수익불변(constant)일때
등이윤선과 생산함수의 기울기로 푸는걸 언뜻 수업시간에 들었는데 그 때 졸아서.....못풀겠습니다.

혹시 아시는 경제학고수님 없나요?......흑 참고로 과목은 수리경제학입니다.