Question Sheet
Question 1 (Optimal size of family)
Suppose a consumer endowed with N hours of labor. Differently from the consumer we studied in
week 2, this one does not value leisure. However, she does value the size of her family such that
the consumer's utility function is
U(C, F),
where C denotes consumption, and F denotes the family size. Assume that the preferences over C
and F are similar to the preferences over consumption and leisure discussed in week 2. The
consumer always prefers more, the consumer likes diversity, and both goods are normal.
a) In a graph with family size F in the x-axis and consumption C in the y-axis, show the
indifference curves for these preferences.
The consumer receives a real wage w for each hour worked and a real dividend r. If the consumer
chooses to have zero children, F = 0, she supplies N hours of labor. However, for each child, the
consumer is required to forgo b hours of labor such that the labor income is w(N - bF). Note that,
since hours worked cannot be negative, F cannot be larger than N/b.
b) Write the budget constraint of the consumer. Draw the budget constraint line in a graph
with family size F in the x-axis and consumption C in the y-axis. (You can draw in the
same graph of item a)
c) Using the graph produced in the items a and b, show the effect of an increase in the non-
labor income on the family size. Recall from the Solow model that richer countries are
associated with smaller families. Is the effect depicted in the graph in accordance with the
Solow model?
d) Using the graph produced in the items a and b, show the effect of an increase in real wages
on the family size. Assume that the substitution effect is stronger than the income effect.
Recall from the Solow model that richer countries are associated with smaller families, but
are also associated with high TFP, which implies higher wages. Is the effect depicted in
the graph in accordance with the Solow model?
