Solutions to assigned questions (Chs2-5).
Ch2: Questions 4 and 5
4.
We are given the formula for the marginal rate of substitution of leisure for
income (or consumption, since there is a one-for-one trade-off between the two of them).
The MRS is the slope of the indifference curve.
The formula that appears is actually the
absolute value of the MRS, since that expression must be positive (A(x) must always be
positive, and C and l must always be non-negative), and the indifference curves have a
negative slope.
a)
As we move from left to right, the amount of leisure increases, while A(x)
remains constant, and the value of C falls.
This implies that the value of
the MRS decreases.
The interpretation is that as the worker consumes
more and more leisure, he/she values the marginal hour of leisure less and
less, and is willing to trade off less and less income.
The variables
(labelled X) which might affect the MRS are the number of children that
the woman has, as well as their ages, her level of education, and her
marital status (tied with her husband’s income).
b)
Recall that the reservation wage is equal to the slope of the indifference
curve at the lower right-hand corner solution, which corresponds to the
situation in which no hours are worked.
If h = 0, the number of hours
worked, then all of the time endowment T goes to leisure.
We can thus
write:
MRS = A (x) C / T = w*.
In order for this woman to participate in
the labour market, the market wage has to exceed the reservation wage, so
we can write: w > A(x) C / T.
Taking the natural logarithm of both sides
of the equation yields:
ln w > ln (A(x)) + ln C - ln T.
Since the
logarithmic operation is the inverse of the exponential operation, and at
the corner solution, C = non-labour income (y), and we obtain the desired
result.
The log of the time endowment T can probably be interpreted as a
constant across almost all women, and so it can probably be ignored at this
stage of the problem.
c)
We treat Z as a random variable which is distributed normally.
That
means that it has mean zero and a variance of unity.
The graph has a bell
shape on a diagram with the probability density of the vertical axis and the
values of Z on the horizontal axis. Any factor which raises Z makes
labour force participation more likely.
The form of that distribution,
however, is not really the focus of this question.
An increase (decrease) in
non-labour income ln Y would shift the income constraint upward
(downward), making participation less (more) likely.
An increase
(decrease) in ln W would rotate the income constraint upward
(downward), making participation more (less) likely.
The impact of the
taste shifters depends on whether the sign of the beta coefficient is positive
or negative.
They have the effect of changing the slope of the indifference