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```									 Topic #2                Household Production, the Family, and the Life
Cycle
Suppose a 65-year old can earn \$30,000 per year working and that this individual dies
upon his 70th birthday. Also, suppose initially that no social security retirement benefits
are provided. Calculate remaining lifetime income for retirement at ages 65 through 69
under plan 1.

Now, consider the following change. Let social security retirement benefits of \$20,000
be paid for each year of retirement, where the individual is eligible to receive social
security retirement benefits at age 65. Calculate remaining lifetime earnings for ages 65
through 69 under plan 2.

Next, suppose that the minimum age at which social security retirement benefits can be
received is raised to age 67. Calculate remaining lifetime income under plan 3.

Finally, suppose that annual social security retirement benefits are reduced from \$20,000
per year to \$10,000 per year. Calculate remaining lifetime income under plan 4.

Retirement Age          Plan 1        Plan 2         Plan 3              Plan 4
65
66
67
68
69

When remaining lifetime income increases, individuals retire earlier. Conversely, when
remaining lifetime income decreases, individuals retire later. This is the income effect.

When the opportunity cost of retiring increases, then retirement decisions are postponed.
When the opportunity cost of retiring decreases, then individuals retire sooner. This is
the substitution effect.

Consider how each of the plans described above affect retirement decisions.

Suppose that social security retirement benefits are \$20,000 a year beginning at age 65.
Also, suppose that a 62-year old can earn \$30,000 per year working and that this
individual dies upon his 70th birthday. Also, suppose that social security retirement
benefits of \$20,000 per year are provided at age 65. Remaining lifetime income and total
retirement benefits for retirement at ages 62 through 69 are presented in the table below.

Now, suppose that the individual can begin drawing early social security retirement
benefits (begin receiving benefits before age 65). However, if one begins drawing social

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security retirement benefits early, a penalty is incurred: 5/9 of 1 percent of benefits for
each month of retirement before age 65.

Annual retirement benefits for retirement at age 62 will be 20,000 – 20,000*5/9(0.01)(36)
= \$16,000. Such benefits for retirement at age 63 will be 20,000 – 20,000*5/9(0.01)(24)
= \$17,333 and at age 64 will be 20,000 – 20,000*5/9(0.01)(12) = \$18,666. The chart
below gives the remaining lifetime income for retirement at ages 62 through 69 and
social security benefits.

Plan 1 - No Early Retirement   Plan 2 - Early Retirement Benefits
Age    Retirement Benefits      Income    Retirement Benefits     Income
62          100,000            100,000         128,000          128,000
63          100,000            130,000         121,333          151,333
64          100,000            160,000         112,000          172,000
65          100,000            190,000         100,000          190,000
66           80,000            200,000          80,000          200,000
67           60,000            210,000          60,000          210,000
68           40,000            220,000          40,000          220,000
69           20,000            230,000          20,000          230,000

Remaining lifetime income is depicted in the figure below. Providing early social
security retirement benefits increases remaining lifetime income, which prompts a
negative income effect. That is, individuals should retire earlier because they have
greater remaining lifetime income. Further, a negative substitution effect is created
because the opportunity cost of retiring earlier has decreased from \$30,000 (age ages 62
through 65) to less than \$30,000. So, senior citizens will unambiguously retire earlier.

2

260,000
240,000

220,000
200,000
180,000
160,000
140,000                               No Early Benefits
120,000                               Early Benefits
100,000
80,000
60,000
40,000
20,000
0
70     69      68     67      66         65     64    63        62
Retirement Age

Across the lifecycle, wages gradually rise and then fall as retirement age approaches.
Though wages change, we might assume that household production remains relatively
constant.

Wages

Age      45                                 70

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Perhaps this explains why time at work mirrors wages and time at home is inversely
related to wages.

Time

At Home

At Work

Age           45               70
What happens as wages change? Expected wage changes create an intertemporal
substitution effect. An expected wage increase in period t creates a positive intertemporal
substitution effect of more work in period t. No income effect is created because the
wage change was expected. Therefore, expected lifetime income has not changed.
However, unexpected wage changes create both an intertemporal substitution effect and
an income effect. The unexpected wage change creates a positive intertemporal
substitution effect of more work in period t. It also creates a negative income effect of
less work in all periods because expected lifetime wealth is (unexpectedly) higher.

Now consider household productivity. Suppose time is allocated among three activities:
marketplace work, leisure, and household work. Let there be a couple, where H indicates
husband, W indicates wife, and MPHH indicates the marginal product of household
production, with WH = \$20, MPHHH = \$12, WW = \$15.00, and MPHHW = \$15.00. Suppose
the couple decides that one should work for pay and one should stay home. The spouse
with the highest net gain from working should optimally work. The wife should
optimally stay home (WH + MPHHW = 35 > WW + MPHHH = 27). Why is WH>WW?
Perhaps it is due to discrimination. Or, perhaps MPHHW > MPHHH due to socialization.

Now, suppose that the wife works and that WW increases. Also assume that an increase
in WW increases the wife’s marketplace work hours. What happens to the husband’s
labor supply decision? For the husband, real income has increased because his wife is
earning more. Thus, a negative income effect is created where the husband works less.
If the husband and wife are complements in household production, then if WW and wife
marketplace work hours increase, MPHHH decreases and hours of husband marketplace
work increase (though this increase may be partially or completely offset by the negative
income effect described above). Why be at home at dinner if your wife isn’t home until
8:00? On the other hand, if the husband and wife are substitutes in household production,
then if WW and wife marketplace work hours increase, MPHHH increases and hours of
husband marketplace work decrease (which would be augmented by the negative income
effect described above). Both husband and wife may not be needed to clean the toilettes.

What if a baby comes along? MPHHW may increase and real income may decrease.

4
Problem Set#2 Household Production, the Family, and the Life
Cycle
1.     Consider a household where the husband can earn a market wage of \$15.00 per
hour and has a marginal productivity in the household that is the equivalent of \$12.00.
The wife can earn a market wage of \$20.00 and has a marginal productivity in the
household that is the equivalent of \$15.00. Suppose initially that their marginal
household productivities do not depend on the time that the other spouse spends at home.

a) If the couple decides to have only one of them work for pay outside the home,
who should it be?
b) Would it make sense for both the husband and wife to work for pay outside of the
home?
c) Assuming both members of the couple are working, how would an increase in the
wife’s wage affect her work hours? For this part of the question, assume there are
no cross effects on the husband’s household marginal product. Would an increase
in the wife’s wage cause the husband to increase or decrease his work hours?
d) Now let the wife’s labor supply decisions have cross effects on the husband’s
household marginal product. Also, assume that if WW increases, the wife’s
marketplace labor supply increases. How would your answer to part c be
different if you knew that the couple were substitutes in household production?
e) Now let the wife’s labor supply decisions have cross effects on the husband’s
household marginal product. Also, assume that if WW increases, the wife’s
marketplace labor supply increases. How would your answer to part c be
different if you know that the couple were complementary in household
production?

2. The year is 1972. Katherine and Chad are married and both work. However,
Katherine is pregnant.

a) How will Katherine’s labor supply change after she gives childbirth? Why? Use
b) Will Chad’s labor supply be affected?

Now consider the effects of child care tax credits. The federal government subsidizes
childcare through a myriad of programs. One of the largest of these in dollar terms is
the Child Care Tax Credit (CCTC), which has been part of the U.S. income tax
system since 1976. The CCTC directly reduces the tax liability of a family. It is
worth 30% of annual childcare expenditures for a family with an adjusted gross
income of \$10,000 or less. Assume that the CCTC only applies to formal childcare
expenditures.

c) Suppose passage of the 1976 CCTC was unexpected. How will the CCTC affect
Katherine’s current and lifetime labor supply?

5
3.      Suppose a 65-year old can earn \$30,000 per year working and that this individual
dies upon his 70th birthday. Also, suppose that social security retirement benefits of
\$20,000 per year are provided for each year in which the individual is retired. However,
the individual cannot work (earning \$30,000) and draw social security benefits (of
\$20,000) in the same year. That is, the individual must either work or be retired.
Calculate remaining lifetime income for retirement at ages 65 through 69 under plan 1.

In 2000, the federal government passed the Senior Citizens’ Freedom to Work Act, which
allows senior citizens to begin drawing full social security retirement benefits upon their
65th birthday regardless of whether they work. That is, the legislation allows senior
citizens to draw full social security retirement benefits and work full time simultaneously.

Let social security retirement benefits of \$20,000 be paid for each year (starting at age
65) regardless of whether the individual works. Thus, 65-year olds who work (earning
\$30,000) now simultaneously drawing social security retirement benefits (of \$20,000 per
year). Calculate remaining lifetime earnings for ages 65 through 69 under plan 2.

Retirement Age                 Plan 1                     Plan 2
65
66
67
68
69

a)     Calculate remaining lifetime income for retirement at ages 65 through 69 under
plan 1.
b)     Calculate remaining lifetime earnings for ages 65 through 69 under plan 2.
c)     Draft a lifetime income – retirement age indifference map showing remaining
lifetime income for retirement at ages 65 through 69 under plan 1 and under plan
2.
d)     How does the Senior Citizens’ Freedom to Work Act affect the retirement
decision of a worker who is 65 years of age? That is, will the individual retire
sooner or later? Use the intuition behind the substitution and income effects in

4. Examine the effect of federal “continuation of coverage” mandates on retirement
decisions. These mandates grant individuals the right to continue purchasing health
insurance through a previous employer after retiring from a firm. Although
individuals must pay the average employer cost of their group insurance, the price to
the early retiree is typically well below that of a policy purchased in the individual
market. The importance of health insurance coverage for the potential early retiree is
underscored by the high medical costs of older individuals.
In 1986, the federal government mandated continuation of coverage at the nation
level under the 1985 Consolidated Omnibus Budget Reconciliation Act (COBRA).

6
Continuation of coverage mandates offer retiring individuals the right to continue
their health-insurance coverage through the employer’s plan for a specified period of
time. For the following questions, assume the 1985 passage of the COBRA was
unexpected. Therefore, upon retirement, workers expected to pay the higher
insurance premium characteristic in the market for health insurance rather than the
lower-cost premium offered by employers under the COBRA.
a) Frank, a 62 year old man, is considering retiring at age 65. How will the passage
of the COBRA legislation affect Frank’s retirement decision? Use an optimal
b) Andrew is Frank’s grandson. Andrew is only 22 years old but has taken note of
the new legislation. How will the federal “continuation of coverage” legislation
change Andrew’s life-cycle labor supply?
c) Assume now that Andrew expected the passage of continuation of coverage
legislation. How will such legislation affect Andrew’s lifetime labor supply?

5. One of the hallmarks of President Franklin D. Roosevelt’s administration was the
creation of Social Security, a government fund into which individuals pay while they
work that will offer a stipend to those individuals upon retirement. Social Security
was created under the Social Security Act of 1935 and initially applied to retired
workers aged 65 and older. The act required employers and employees to each pay a
1 percent tax on the first \$3,000 of the employee’s wage. This fund would be used to
provide a base of financial subsistence to retired workers, ensuring a bare minimum
standard of living. This was thought to be necessary because in absence of this
government provision, people often failed to provide for the financial consequences
of old age.
a) According to the theory of labor supply, how did the creation of Social Security
change life-cycle labor supply for individuals during the pre-retirement period of
life?
b) Now examine how the Social Security Act effected retirement decisions: how
would the Social Security Act have effected the retirement decision of a worker
who was 65 years of age in 1935? Draft a lifetime income – retirement age
the discount rate is 0 (zero), yearly income when working is \$3,000, retirement
benefits are \$2,000 per year, the life expectancy of the worker is 70 years of age,
and any 65 year old is eligible for retirement benefits immediately.
c) The Social Security entitlement program is one of the most expensive the
government offers. Over the years this program has been expanded to provide
coverage for others in need. By 1982, expenditures on Social Security for old age
benefits reached 141.9 billion dollars. Evaluate the overall merits of this
government program given your results from part a and b.

7
Answer Key #2 Household Production, the Family, and the Life
Cycle

a)     A household’s total production will be maximized when the spouse with the
highest net gain from market work is the one that works for pay. Even though
the wife is more productive at home than the husband, the net gain to the
household of having the wife work is \$5.00, while the net gain to having the
husband work is \$3.00. For every hour the wife works and the husband stays
home, the household produces an equivalent of \$32.00, which having the
husband work and the wife stay home could result in only \$30.00.
b)     Yes, since an extra hour of market work allows each to buy at least enough
goods and services to compensate for the hour of lost productivity at home.
c)     An increase in the wife’s wage would decrease her work hours if the income
effect of the wage increase dominated the substitution effect. Vice versa if the
substitution effect dominated the income effect. Empirical studies suggest
that the latter is more common for women. The income effect associated with
the wife’s wage increase reduces the husband’s incentive to work for pay
provided there are no cross effects on the husband’s household marginal
productivity or the marginal utility that the husband derives from household
consumption.
d)     If the husband and wife are substitutes in household production, an increase in
the wife’s work hours will increase his household marginal product, creating
an incentive for the husband to spend more time in household production and
less time in marketplace work.
e)     If the husband and wife are complements in the consumption of household
commodities, an increase in the wife’s work hours will decrease the utility the
husband derives from the additional consumption of household commodities
and create an incentive to spend more time in marketplace work.

a)     When young children are present in the household, Katherine’s household
marginal productivity will rise, creating an incentive for Katherine to work
less in the market and more in the household.
b)     When Katherine works less due to a new baby, household income will
decline. This income effect will provide incentive for Chad to work more.
Also, if Katherine and Chad are substitutes in household production, then
when Katherine’s household marginal product increases and she works less in
the marketplace, Chad’s household marginal product will decrease and he will
unambiguously work more. However, if Chad and Katherine are
complements in household production, then when Katherine works less in the
marketplace and more at home, Chad’s household marginal product increases,
providing incentive for him to work less in the marketplace. If Chad and
Katherine are complements in household production, then we can’t tell if
Chad will work more or less because the income effect from Katherine’s

8
fewer hours of marketplace work will induce Chad to work more but the
less.
c)      The tax credit is similar to an increase in the wage since the opportunity cost
of working falls with childcare becoming more affordable. Each hour of work
brings in more take-home earnings now than before. This will cause the
standard income and substitution effects where the increase in real income
leads Katherine to work less (the income effect) but the increase in real wages
induces Katherine to work more (the substitution effect). It is not clear which
effect dominates. If the change is unexpected, then the increase in real
income induces Katherine to work less in all time periods (the income effect)
and the substitution effect induces Katherine to work more in years when the
wage rate is higher (when the child needs childcare and the tax credit is
received) and work less in years when the wage rate is not higher (when the
child needs no childcare and Katherine therefore gets no tax credit).

Retirement Age                Plan 1                     Plan 2
65                     100,000                    100,000
66                     110,000                    130,000
67                     120,000                    160,000
68                     130.000                    190,000
69                     140,000                    220,000

The retirement age indifference map follows from this chart. The Senior Citizens
Freedom to Work Act (the Act) will have an ambiguous effect on retirement age. The
Act generates a substitution effect that prompts later retirement because the opportunity
cost of retiring (one year earlier) has increased (from \$10,000 per year to \$30,000 per
year). The Act also generates an income effect that prompts earlier retirement: the Act
increases remaining lifetime income which encourages less work.

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a)     The COBRA legislation makes retirement less costly because the cost of
health insurance that must be purchased by Frank upon retirement is less
expensive now than before. We are thus assuming that a fixed amount was
unexpectedly added to lifetime income at each retirement age. This shifts the
budget constraint out in a parallel fashion, creating an income effect but no
substitution effect. The optimal age of retirement would unambiguously be
reduced because real income has increased.

Income



Retirement Age                  62

b)      Andrew’s lifetime wealth is now higher, producing an income effect that says
work less in each time period. If wages do not change, then no substitution
effect is created. If wages fall to pay for the cost of retirees’ health premiums,
then Andrew will work less in the periods when the wage rate is lower. If
Andrew expects the passage of the continuation of coverage mandates, then
his expected lifetime wealth will not change and no income effect will be
created. If the cost of the continuation of coverage mandates induces
employers to pay lower wages, then Andrew will work relatively more in time
periods before 1985 (when the wage rate is at its original level) and relatively
less in time periods after 1985 (when the wage rate is lower), whether he
expects the passage of COBRA or not.

10
a)     First, assume the changes caused by the Social Security Act were expected.
The Social Security Act will decrease all wages earned after 1935 by 1%.
According to the intertemporal substitution effect, workers will respond by
working relatively more in pre-1935 work years (before the wage is affected)
and relatively less after 1935 (when the effective wage rate is reduced). If the
act was expected, then no income effect is created by its passage. Now
assume the passage of the Social Security Act was unexpected and people first
learn of its provisions in 1935 as it takes effect. If the reduction in wages
caused by the 1% pay cut are offset by income after retirement such that
lifetime wealth is left unchanged, then no income effect will be created. If
workers get more out of the program then they pay into it, resulting in an
increase in lifetime wealth, then the income effect will predict workers will
respond by working less. The opposite will occur if workers pay more into
the fund than they get out because lifetime wealth would have been reduced.
According to the intertemporal substitution effect, workers will work less
when their wage rate goes down. Wages have fallen by 1% in all post-1935
periods, motivating an intertemporal substitution effect that says work less in
each period for which this is the case.
b)     Retirement decisions will also be affected. A workers will unambiguously
retire sooner. Consider the numbers offered in the question. Before, if a
worker retired at age 65, they would earn no wage and retirement benefits
were not in existence (income would have been zero). If they delayed
retirement until age 70 (which is also death), remaining lifetime income at age
65 would be \$15,000 (\$3,000 per year for 5 years of work). With the Social
Security Act in effect, a 65 year old who retires at age 65 would receive
\$2,000 in retirement benefits for each of the remaining 5 years of life
(\$10,000). If the worker delayed retirement until age 70, earnings would be
\$3,000*5 = \$15,000 (the same as before since the Social Security Act only
changes things if you retire). One final example: before, if a 65 year old
retired at age 68, remaining lifetime income was \$3,000* 3 years of work:
\$9,000. With Social Security, the worker gets the \$9,000 from working plus
\$2,000 in retirement benefits for each of the remaining two years of life for a
total remaining lifetime income of \$13,000. The answer to the question is as
follows: greater remaining lifetime wealth has been created and with it an
income effect that moves the worker in the direction of earlier retirement (we
consume more leisure when real income increases); a lower effective wage
rate creates a substitution effect that also pushes for earlier retirement (the
opportunity cost of working has increases since you only get the retirement
benefit if you retire – as leisure becomes relatively less expensive, the worker
substitutes toward it).
c)     Any thoughtful answer to part c is fine, though I would note that the Social
Security Act seems to decrease work. From part a, we see that if the SSA was
unexpected, an intertemporal substitution effect is created, which says work
less in each post-1935 period. We don’t know how the income effect works

11
because we can’t tell from the problem whether lifetime wealth increases or
decreases. But, my guess is that people who were near the age of retirement
in 1935 will not have worked enough before retirement to pay into the
program what they get out of it. If this guess is right, lifetime wealth will
increase and work will decrease. Further, from part B we learn that people
will unambiguously retire earlier. Therefore, the SSA seems to decrease work
in general, though no one retires into poverty.

\$15,000
\$13,000

\$12,000
\$10,000

\$9,000

\$6,000

\$3,000

70     69            68          67        66       65
Retirement Age

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