Micro question

[RichaS]

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 rupees 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 Rs. 10 per hour. Write
down the consumer's utility function and budget con-
straint with c and h as the choice variables. How
many hours will he choose to work, and what will be
the resulting utility?
b)Suppose the wage rate is Rs. 10 per hour for the
first 8 hours of work each day, and Rs. 30 per hour
for each daily hour of work beyond the first 8. Write
down the consumer's utility function and budget con-
straint with c and h as the choice variables. How
many hours will he choose to work, and what will be
the resulting utility?

[Abhitesh]

(a) Budget constraint : h<=24  and c<=240
u(h) = ln(10h) + 3ln(24-h) ; u(c) = ln(c) + 3ln(24-c/10)
Now the consumer problem is : max u(h) s.t h<=24
So u'(h) = 0 => h=6 and u(6) = (ln10+3)ln6

(b) Budget constraint B(h) : h<=24
B(c): c<=80 if h<=8
         80<c<=560 if h>8
u(h) = ln(10h) + 3ln(24-h) if h<=80
          ln(30(h-8) + 80) + 3ln(24-h) = ln(30h-160) + 3ln(24-h) if h>8
Now we have to maximise u(h).
So u'(h)=0 => h=6 if h<=8
                       h=10 if h>8
Substituting h = 6 and 10 in u(h) we get
u(6) = (ln10+3) ln6
u(10) = (ln10+3)ln14
Since u(10)>u(6). So the consumer will work for 10 hours in this case.