There are 10 families in a slum in india, each of which
has one adult and one child. the adults would prefer not to have their children work, but the
family needs to eat, and children’s wages contribute to the food available. adults, however,
will always work. children are exact substitutes for adults—a child earns the same wage as
an adult. each family immediately consumes all its income, whether from adults or children.
if the going wage is w the income of a family is
i = w + wc
where c is 1 if the family sends it child to work, and 0 if not. the utility of the each family
is given by
u(i, c) = −1/i − c = −1/w + wc − c.
a. what is the expression for the utility of a family if it does not send its child to work so
c=0? what is the expression for the utility of the family if it does send its child to work
so c=1? sketch the two curves as w changes (put w on the x-axis, utility on the y axis;
your sketch does not need to be from a computer, but should get how the curves relate
to each other; negative numbers for utility are fine, utility is ordinal not cardinal).
not sending children to work gives u(w, c = 0) = −1/w, while sending
them to work gives u(w, c = 0) = −1/(2w) − 1. these curves are shown in figure
1. the important point is that the curves cross. at a high wage not sending children to
work is better. but the value of the extra income becomes more and more important at
low wages, when it is better to send children to work..