Consumer's Problem

[abhinav619]

Denote a consumer’s daily hours of work by H, and hours of non-work by N = 24− H.
Consider a consumer who has no other source of income than wages for hours worked, and
no debt or other obligations. He consumes what he earns each day. Writing C for the dollar
amount of his consumption, suppose his utility function is
U(C, N) = ln(C) + 3 ln(N).
All these quantities are to be treated as continuous variables

(a)  Suppose the wage rate is $10 per hour for the first 8 hours of work each day, and $30
per hour for each daily hour of work beyond the first 8. Write down the consumer’s utility
function and budget constraint with C and H as the choice variables. How many hours will
he choose to work, and what will be the resulting utility?

[ashutosh]

You can convert the utility function into simpler one.

 To maximize -
V= C*N^3      ( Given U is just the monotonic transformation V)

Budget constraint will be

C= { 10(24-N)   when N=>16
    {  80 + 30( 16-N)    N <= 16



You can solve for optimal points in both the cases now.

[abhinav619]

Where did log disappear ?? Isn't it the other way
log(mn) = log(m) + log(n).

If sub-divided the budget constraint but then how to do i proceed to find the optimal condition. I get two different optimal condition.