Second Income

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					Endowment vs Income

A. Up until now, people have only had money to exchange
for goods. But in reality, people sell things they own
(e.g., labor) to acquire goods. Want to model this idea.

In several types of applications of consumer theory, we will deal with
    consumer endowments rather than incomes.
An endowment is a list of the items the consumer owns at the start of the
  choice problem. In our analysis we will take the endowment to be of the
  same form as the consumption bundles, i.e., as amounts of the two
  goods. This approach might be familiar to those of you who have taken
  a course in international trade. It is used to generate offer curves for
  trade. Two major topics for which we will use this approach are:
i) labor supply/leisure choice
ii) intertemporal choice.

Labor supply/leisure choice problem

For the labor supply/leisure choice problem, the two goods we will analyze
leisure time and a composite good, “dollars spent on all other goods at
   fixed prices.”
The consumer/worker owns her own leisure time (and perhaps some non-
  labor income) at the start. She sells some of her leisure time to an
  employer for a wage, w (i.e., she uses some of her time as labor time,
  working for an employer for w per unit of time).
Her remaining leisure time is one of the goods she consumes. For each
  unit of leisure time she consumes, she must forgo $w of the other good
  (“dollars spent on all other goods”), so the price of leisure is w. The other
  good has price 1 since it is measured in dollars.

In our previous analysis, as a good becomes more expensive, the consumer is
made worse off, because the budget set shrinks, as in the first figure below.
When the consumer owns the good whose price increases, and will sell it, the
consumer is made better off, since the budget set expands as in the second
figure below (which is drawn for an endowment (L, 0)).

If the endowment consists of positive amounts of both goods, then when
prices change the budget rotates through the endowment bundle, as in the
figure below.

This is the typical situation for an intertemporal choice problem, where x is
   the number of dollars available to purchase goods in the first period and
   y is the number of dollars available to purchase goods in the second
The endowment represents the consumer’s income in each of the two
  periods (perhaps the paychecks of a salaried worker).
The consumer can turn first period income into second period dollars by
  saving, with $1 in the first period returning $(1 + r) in the second period,
  where r is the per period interest rate paid on savings.
The consumer can turn second period income into first period dollars by
  borrowing, with $1 available in the first period for a payment of $(1 + R)
  in the second period, where R is the per period interest rate for loans.
  For simplicity, we will usually assume r = R, so the budgets are straight

The effect of a price change now depends on whether the consumer is a
  net seller or net buyer of the good whose price changes. She may even
  switch between being a net seller and a net buyer.
[For the labor/leisure problem, even with a strictly positive endowment of all
   goods, the points to the right of the endowment are irrelevant since the
   consumer is a net seller of leisure. She cannot consume more than her
   endowment of 24 hours per day of leisure.]

                            Unusual budgets

   In many applications, the budget set will not be a straight line.
For example, in a labor/leisure problem, if the job pays an overtime wage of
   (1.5)w for each hour worked beyond say eight hours per day, then the
   budget line will change slope from - (1.5)w to - w at the leisure amount
   corresponding to 8 hours of work per day, i.e., at L= 8 hours of leisure
   per day where L is her endowment of leisure per day.
As another example, if the job is only available as a full time job of forty
   hours per week, then the consumer/worker’s choice is “work or don’t
   work,” but the number of hours worked must be either 0 or 40 per week.
As a final example, consider an intertemporal choice problem in which the
   interest rate the consumer faces for borrowing is higher than the interest
   rate the consumer can obtain when lending money (saving). In this case
   the budget will change slope at the endowment point.

                  Labor/leisure choice example

If the consumer is able to choose the number of hours worked, how does
    the introduction of overtime pay at a rate 50% above the base wage for
    every hour worked beyond 40 hours per week affect the number of
    hours worked?
Let x be the number of hours of leisure consumed per week and let y be the
   number of dollars spent on all other goods per week.
The following figures are drawn for an endowment of (L, 0), but an
  endowment which included non-wage income would have no effect on
  the results.

The first two figures are for individuals
who initially work less than 40 hours per
There is no effect in the first figure since
the individual does not find it worthwhile
to take advantage of the overtime pay.
She consumes L* hours of leisure and
works L- L* hours each week.
In the second figure the individual does
find it worthwhile to take advantage of
the overtime pay, and increases the
number of hours she works each week
from L- L* to L-- L** (or equivalently,
decreases the number of hours of
leisure she consumes each week from
L* to L**).

The third figure shows an individual who originally works 40 hours per
week, and thus has L-40 hours of leisure per week.
 By the Slutsky version of the substitution effect, she will decrease her
leisure hours (they have become more expensive) and increase her
work hours from 40 to L - L**.

For an individual who initially worked more than 40 hours per week,
depending on her preferences, she might increase or decrease the number
of hours per week she works. For example, the individual in the fourth
figure has perfect complements preferences. She will work less and
consume more leisure.

The fifth figure shows an individual with smooth indifference curves who
  initially worked more than 40 hours per week. Depending on her
  preferences, she might increase or decrease the number of hours per
  week she works.

For example, if we knew leisure were an
inferior good for her, then we could
determine that she would work more.
The reasoning is as follows.
First consider the artificial, dashed
budget with slope - (1.5)w. As in the
third figure, we know she would
consume less than L* hours of leisure
(i.e., she would work more) with that
The relevant portion of the true budget
she faces has the same slope but is out
further to the right.
This corresponds to an increase in
income, so the fact that leisure is an
inferior good means that she would
consume even less leisure (i.e., she
would work even more) when facing the
true budget.

If leisure were a normal good, but
not strongly so, in the last step she
would increase leisure, but not by
The net effect of the two steps could
still be a decrease in leisure (i.e., an
increase in work).
If leisure were strongly normal, then
the net effect would be an increase
in leisure and a decrease in work.


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