Answer Key to Problem Set 3
Problem Solving
1) Calculate the provincial tax liability and the average tax rate of a person who makes a)
20,000, b) 40,000, c) 80,000, d) 120,000 in Alberta and in Ontario using the tax tables
below, given that the first $15,435 (the personal amount) are taxed at a zero tax rate in
Alberta and the first $8,553 (the personal amount) are taxed at a zero tax rate in Ontario.
Provinces / Territories
Rate(s)
Ontario
6.05% on the first $35,488 of taxable income, +9.15% on the next $35,488,
+11.16% on the amount over $70,976
Alberta
10% of taxable income
The provincial tax liability for each income group in Alberta is given in the table below.
Tax
Liability =
Column 3: column 3- Average Tax
Column 1:
Column 2:
Tax Rate column2* Rate = Tax
Taxable Income Personal Amount
=10% .1
liability/column 1
20,000
15,435
2,000
456.5
2.28%
40,000
15,435
4,000
2,456.5
6.14%
80,000
15,435
8,000
6,456.5
8.07%
120,000
15,435
12,000 10,456.5
8.71%
In order to calculate the provincial tax liability for each income group in Ontario I use
the Ontario tax form given in the table below.
Ontario
Taxable Income Personal Amount Tax
Tax Liability=Ontario Tax Average Tax
– pers. amt.*.0605
Rate
20,000
8,553 1,210.00
692.54
3.46%
40,000
8,553 2,559.85
2,042.39
5.11%
80,000
8,553 6,401.08
5,883.62
7.36%
120,000
8,553 10,865.08
10,347.62
8.62%
2) Consider the following five families. All spouses are in good health and can engage in
household production. To make comparison easier assume none of the couples have
children. Assume that no matter how much a spouse receives as his or her own income,
family income is shared equally among family members.
Family 1: Wife earns 80,000 working full-time, husband earns 30,000 working half-time.
Family 2: Husband earns 80,000 working full-time, wife earns 30,000 working full-time.
Family 3: Wife earns 60,000 working full-time, husband earns 50,000 working full-time.
Family 4: Both spouses are retired. Wife’s pension is 30,000, husband’s pension is
80,000.
Family 5: Both spouses are retired. Wife’s pension is 60,000, husband’s pension is
50,000.
a) Applying the principle of ability to pay, which families have equal ability to pay?
Which families have unequal ability to pay? Rank families in decreasing ability to
pay.
The ability to pay should not only take into account how much income a family has but
also the time a household has left for household production. Based on that criterion,
families 2 and 3 have equal ability to pay. Families 4 and 5 have also equal ability to
pay. Family 1 has a higher ability to pay than families 2 and 3. Families 4 and 5 have a
higher ability to pay than families 1,2 and 3. Thus the ranking of families in decreasing
ability to pay: Families 4,5
Family 1
Families 2,3
b) Calculate each family’s federal tax liability as given by the tax schedule in the
lecture notes, slide 15 (i.e. individual taxation and tax credit of 9,600*.15=1,440).
Rank families according to decreasing tax liability. How does this ranking
compare to your ranking in a)?
Family 1: Wife’s tax liability = 13,783.18; husband’s tax liability = 3060; family 1’s tax
liability = 16,843.18.
Family 2: Wife’s tax liability = 3060; husband’s tax liability = 13,783.18; family 1’s tax
liability = 16,843.18.
Family 3: Wife’s tax liability = 9,157.84; husband’s tax liability = 6,957.84; family 1’s
tax liability = 16,115.68.
Family 4: Wife’s tax liability = 3060; husband’s tax liability = 13,783.18; family 1’s tax
liability = 16,843.18.
Family 5: Wife’s tax liability = 9,157.84; husband’s tax liability = 6,957.84; family 1’s
tax liability = 16,115.68.
Ranking families in terms of their tax liability:
Families 1,2,4
Families 3,5
The ranking based on individual taxation is ad odds with the ranking based on ability to
pay. It only gets the ranking of family 1 versus 3 right and family 4 versus 3 right.
c) Calculate each family’s federal tax liability assuming income splitting. Rank
families according to decreasing tax liability. How does this ranking compare to
your ranking in a)?
As all families have the same family income, with income splitting all families pay the
same amount of taxes. Tax liability is equal tax liability under individual taxation if each
spouse had earned half of the family’s income. Half the family’s income is 55,000; tax
liability on 55,000 is 8,057.84 and hence tax liability of each family is 16,115.68.
The ranking based on income splitting is ad odds with the ranking based on ability to
pay. It only gets the ranking of family 2 versus 3 right and family 4 versus 5 right.
d) Now assume that in families 4 and 5 one of the spouses is taking care of the ailing
partner. Given this information, would this change your answer to a)?
Yes it would. This reduces the time that households 4 and 5 have available for household
production. The household loses the time the ailing partner could spend in household
production would he or she be healthy. The healthy spouse now spends most of the time
taking care of the ailing partner and there will be also be other costs due the medical
condition of the ailing spouse. These are expenditures that families 1, 2, and 3 do not
have and they would decrease families 4 and 5’s ability to pay. I would now rank families
4 and 5 about the same as families 2 and 3.
e) Canada currently taxes families 1 to 3 based on individual taxation, but families 4
and 5 based on income-splitting. Rank the families according to decreasing tax
liability. How does this ranking compare to your ranking in d)?
The current tax system yields the following ranking:
Families 1 and 2, tax liability: 16,843.18
Families 3, 4 and 5, tax liability: 16,115.68.
Compared to the ranking in d) the tax system gets only the ranking of family 2 wrong.
3) Consider the effect of a wage tax on hours worked, using drawings to illustrate your
answers. Use the leisure-consumption model discussed in class. Assume that consumers
are endowed with a certain number of hours of leisure time each week that they can sell
to the market in the form of work. Assume the price of consumption is equal to 1.
a) Start with no wage tax, and draw a picture showing the effect of the introduction of the
wage tax. Assume that leisure is a normal good. What can you predict theoretically
about the effect of the tax on hours worked?
There is not a clear theoretical prediction, unless we make an assumption about the
relative size of the income and substitution effects. In the diagram below, I assume that
the substitution effect dominates, so the wage tax causes hours worked to fall.
Income/
consumption
A
C
L L’
T leisure
The consumer’s original choice is denoted by A, with consumption of L hours of leisure.
Therefore the hours worked on the original (dotted) budget constraint are T-L. After the
tax is imposed the after-tax wage falls. The consumer’s optimal choice on the new (solid)
budget constraint is at C, with leisure of L’. This means hours worked fall to T-L’. Had
the income effect dominated, then the hours worked would have risen.
b) Decompose the overall effect of the tax into income and substitution effects.
Income/
consumption
A
B
C
L L’’
T leisure
L’
The movement from A to B is the substitution effect (from L to L’’). The movement from
B to C is the income effect (from L’’ to L’). The substitution effect has the person work
fewer hours due to the lower wage. This means that leisure has become relatively less
expensive compared to consumption and thus the person substitutes away from earning
income. The income effect makes the person feel poorer and causes her to reduce both
consumption and leisure. That is the income effect has the person work more hours due to
the reduced income. In this way I have drawn it, the substitution effect dominates the
income effect.
c) What can you predict about the effect of this tax on the choice of consumption
(income)?
Income/consumption is shown on the y-axis. Income and substitution effects on
consumption can be seen by measuring the moves from A to B to C along the y-axis. The
income effect of the wage decline reduces consumption. The substitution effect (which
argues for more leisure, and therefore less consumption) also reduces consumption.
Therefore consumption falls unambiguously.
d) What assumption is required for the tax to cause a decrease in hours worked?
As noted above, we need to assume that the substitution effect dominates the income
effect.
e) Using the same leisure-consumption framework, illustrate the deadweight loss
associated with the tax.
Government revenue is the income paid by the individual to the government at their new
consumption choice, C. It can be measured as the vertical distance between the original
(solid) budget constraint and the new (solid) budget constraint at point C.
Notice that a lump-sum tax equal to distance EV would generate the same utility loss for
the individual as the wage tax, and yet would generate more government revenue. The
fact that the wage tax does not generate as much revenue for a given amount of utility
loss as some other tax suggests that there is deadweight loss associated with the wage
tax. The deadweight loss is equal to EV-GR, and can be indicated by the vertical
distance between point C and the dotted line below.
Note that if the only effect of the tax were to move the person from point A to point B,
there would be no deadweight loss. It’s in the move from B to C (subs effect) that the
deadweight loss occurs.
Income/
consumption
•A
EV
GR
•B
•C
T leisure