Solution to the Sample Questions for the Mid-term Exam
Analysis of Market and Public Policy (Section 3 and 4), Spring 2011
Created by
Sunwoo Hwang (Teaching Fellow)
Draft: March 22, 2011.
1. (Marshalliand & Hicksian Demand, Indirect Utility Function, Cobb-Douglaus)
(1) A consumer’s Marshallian demand function specifies what the consumer would
buy in each price and wealth (or income), assuming it perfectly solves the utility
maximization problem
.
(1)
The problem has a unique optimal solution when
(2)
since the utility function in (1) represents well-behaved preference. By putting
1 into (2) and then rearranging the equation, we have
the MRS of
(3)
At a consumption bundle on the budget line
that satisfies the
condition in (3), the utility-maximization problem (1) has the solution and we
find Jenny’s Marshallian demand function for each good is
,
(4)
.
(2) The indirect utility function
prices ,
and income I, i.e.
where
and
is the maximum utility attained with
solves the utility maximization problem in (1). Therefore,
1
1
(5)
(3) Given her utility level , Jenny’s Hicksian demand function (or compensated
demand function) for each good solves the expenditure minimization problem
(6)
As in Question 1.(1), we find a unique solution when the rate of exchange
between two goods of
equals the MRS of
, i.e., where the utility
function is tangent to the expenditure function. By rearranging the condition in (3)
with respect to
and putting it into the budget constraint in (6), we have
Safely assuming that Jenny only consumes a positive amount of each good, we
derive a Hicksian demand function for good 1 and good 2 as follows.
(7)
.
(4) By putting the given values ( = $1,
= $2,
= $400) into the equations in (4)
and (5), we find that her maximum utility is 20,000 at
= (200, 100). That is,
Jenny’s utility is maximized when she consumes 200 units of beef and 100 units
of rice with her (ordinal) utility of 20,000.
(5) Excise tax of $1 imposed on beef consumption increases
from $1 to $2. By
putting the changed set of values ( = $2,
= $2,
= $400) into the equation
(4), we find that
= (100, 100). That is, Jenny will buy 100 units of beef and
100 units of rice.
(6) Tax revenue is determined by multiplying the excise tax
consumption of good 1.
Tax Revenue =
= $1
by the increased
100 = $100.
(7) You may use either Slutsky substitution effect or Hicksian substitution effect.
(i) Hicksian substitution effect, which keeps utility constant
By putting the after-tax values of prices and income ( = $2,
= $2,
=
$400) into the equation in (7), we find that
; this
consumption bundle keeps the utility constant. Changes in consumption of
2
each good are decomposed into income effect (the first term) and Hicksian
substitution effect (the second term) as follows.
Specific values are
Income effect on : 100 Substitution effect on :
Income effect on : 100 Substitution effect on :
- 200
– 100.
(ii) Slutsky substitution effect, which keeps purchasing power constant
Left as your exercise.
(8) If the optimal choice under the excise tax is still a feasible solution on the new
budget constraint under the lump sum tax, we can say that the bundle Jenny
chose under the excise tax is still available to her under the lump sum tax. The
form of this budget constraint would be
where
is the lump sum tax that raises the same amount of revenue. For
$1,
= $2, = $400, and
= $100, it becomes
=
The optimal choice under the excise tax
= (100, 100) let the LHS of the
equation above be equal to its RHS. We see here that the optimal choice under
the excise tax is also a feasible solution under the lump sum tax.
(9) While the excise tax imposed on consumption of good 1 increases its price
(Question 1.(5)), the lump sum tax reduces the entire wealth Jenny can consume
(Question 1.(8)). By comparing values the indirect utility functions return with
changed prices and income, therefore, we are able to see which bundle Jenny
would choose. Under the excise tax, price of good 1 is $2 (=$1 + $1), price of good
2 is $2, and income is $400. Under the lump sum tax, price of good 1 is $1, price
of good 2 is $2, and income is $300 (=$400 - $100). With each set of prices and
income, the equation (5) returns utilities as below.
(under the lump sum tax)
(under the excise tax)
3
It turns out that Jenny’s utility is greater under the lump sum tax. We therefore
say that, under the new lump sum tax, Jenny will not choose the bundles she
chose under the excise tax although it is still available to her.
(10)
For
returns
= $1,
= $2, and
= $400 - $100 = $300, the equation (4)
,
.
2. (Marshalliand & Hicksian Demand, Indirect Utility Function, Perfect Substitutes)
(1) James’ Marshallian demand function solves the utility maximization problem
.
When
and
,
are perfect substitutes, we have three possible cases to consider:
, and
. The function for each good for each case is
,
(2) Indirect utility function is given by the sum of demand of each good. That is,
(3) Given the utility level , James’ Hicksian demand function (or compensated
demand function) for each good solves the expenditure minimization problem
.
When
,
. If we substitute it into the budget constraint we have
. In a similar fashion, we develop the entire set of demand function for
each case
4
,
(4) Since
,
(5) Since
,
,
(6) Tax Revenue (R) =
, and
,
= $1
1200 = $1200.
(7) Since
and
are perfect substitutes, only income effect exists; i.e. total effects
equal income effects.
Income effect on
Income effect on
=
,
=
.
(8) The budget constraint under the lump sum tax is
where
is the lump sum tax that raises the same amount of revenue. For
$2,
= $4, = $3600, and
= $1200, it becomes
5
=
The optimal choice under the excise tax
= (1200, 0) let the LHS of the
equation above be equal to its RHS. We see here that the optimal choice under
the excise tax is also a feasible solution under the lump sum tax.
(9) It turns out that utilities are the same under the excise tax and under the lump
sum tax.
(under the lump sum tax)
(under the excise tax)
James may choose the bundle he chose under the excise tax since he is indifferent
between the two alternatives.
(10)
3.
,
.
(Perfect Complements)
(1)
(2) This consumer purchases the same amount of good 1 and good 2 no matter what
the prices. Let this amount be denoted by . Then, we have to satisfy the budget
constraint
.
Solving for
gives us the optimal choices of goods 1 and 2:
.
(3)
(4)
(5)
Since optimal choices are
from the Question 3.(2),
.
Therefore, Mike’s Hicksian demand functions for each good are
.
6
(6)
,
No substitution effect.
Income effect on : 25-100/3
Income effect on : 25-100/3
4. (Market demand)
(1) We know Jenny’s Marshallian demand function obtained in Question 1.
Considering it as a demand function of each consumer, we compute the market
demand, which is the sum of these individual demands over all consumers, using
the given information:
(2) As discussed in Question 1, the excise tax imposed on consumption of good 1
increases its price as much as the amount of tax. Therefore, the market demand as
a function of
and is
5. Based on given information and Jenny’s Marshallian demand function, we know
that
,
.
Since we have two unknowns and two equations, we can solve this problem:
= 800,000, and
= 40.
6. (Inter-temporal Choice, Cob-Dauglas)
(1) If he can borrow or save at an interest rate of 10%, his budget constraint is given
by
7
.
(2) Neil’s Marshallian demand solves the following utility-maximization problem
when MRS equals the rate of exchange between
and
, i.e.,
2.
By rearranging this condition with respect to
budget constraint, we have
and substituting it into Neil’s
.
It turns out that Neil’s utility-maximizing consumption in period 1 is $200 while
his income is only $100 in that period. Thus, he needs to borrow $100 in order to
consume optimally $200.
(3) .
(4) Two cases need to be considered.
and
2
8
.
(i) Borrow (100 < ) – when consumption is greater than income in period 1.
(We already have covered this case in Question 6.(2) and (3).)
(ii) Save (100 > ) – when consumption is smaller than income in period 1.
The slope of Neil’s budget constraint is changed from -1.1 to -1.23 since the
interest rate is increased from 10% to 20%. Therefore, the optimal condition
where MRS equals the rate of exchange between
and
also changes to
By substituting this new condition into Neil’s budget constraint after
rearranging it with respect to , we have
But, since
> 100, we do not have an optimal solution in case of “Save.”
 Neil’s choice does not change.
7. (Inter-temporal Choice, Perfect Complements)
(1) The budget constraint is the same as that in Question 6.(1).
3
9
(2) Neil’s Marshallian demand solves the following utility-maximization problem
when
=
. Then, the optimal consumption in each period is
.
 Borrow $65.
(3) .
(4) BetweenTwo cases need to be considered.
(i) Borrow (100 < ) – when consumption is greater than income in period 1.
(We already have covered this case in Question 7.(2) and (3).)
(ii) Save (100 > ) – when consumption is smaller than income in period 1.
Along with the change in interest rate from 10% to 20%, Neil’s budget
constraint is changed to
.
Satisfying the optimal condition
,
we do not have an optimal solution in case of “Save.”
 Neil’s choice does not change.
10
. But, since
> 100,
8. (Inter-temporal Choice, Perfect Substitutes)
(1) The budget constraint is the same as that in Question 6.(1).
(2) Neil’s Marshallian demand solves the following utility-maximization problem
Since
, (See Question 2.(1))
,
 Save(or lend) $100.
(3) .
11
.
(4) BetweenTwo cases need to be considered.
(i) Borrow (100 < ) – when consumption is greater than income in period 1.
(We already have covered this case in Question 8.(2) and (3).)
(ii) Save (100 > ) – when consumption is smaller than income in period 1.
After the slope of Neil’s budget constraint is changed from -1.1 to -1.2,4
still greater than ; i.e.,
. Therefore,
,
.
 Neil’s choice does not change, and he enjoys greater amount of interest.
4
12
is
9. (Food Stamp, Cobb Douglas)
(1) Jack’s budget lines (a) when there is no subsidy of any kind, (b) when there is
food stamp worth $100, and (c) when the government gives Jack $100 cash, are
(2) We may compare what Jack chooses between cash subsidy and food stamps to
see what he prefers between them. As illustrated in the figure above, Jack prefers
cash subsidy to food stamps if
. On the other hand, Jack is indifferent
between two alternatives if
. The utility-maximization problem
has an optimal solution when MRS equals the slope of the budget line. Since MRS
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5 and the slope is -1, the optimal choice is a solution that satisfies the
is
following condition
.
By substituting it into the budget line, we find that
optimal choice is illustrated in the figure below. Since
indifferent between the two alternatives.
. Jack’s
, he is
10. (Food Stamp, Perfect Complements)
(1) The budget lines are the same as those in Question 9.(1).
(2) The following utility-maximization problem
has an optimal solution when
. Therefore,
. Again,
Since
, Jack is indifferent between the two alternatives.
5
14
11. (Food Stamp, Perfect Substitutes)
(1) The budget lines are the same as those in Question 9.(1).
(2) Jack’s optimal choice solve the following utility-maximization problem
As illustrated in the figure below, Jack is indifferent between cash subsidy and
food stamps for
if
. On the other hand, Jack prefers cash
subsidy to food stamps for
if
12. (Labor Supply)
15
(1) Mary’s utility is maximized at the solution of the problem
where MRS equals the slope of the budget line; i.e.,
.6
Substituting it into the Mary’s budget constraint and rearranging the equation
with respect to gives us her demand function for leisure
.
(2)
.
(3)
,
(4)
,
For decomposition of change in demand, we first need to find a bundle (a) with a
new budget line that has the same relative prices as the final budget line and (b)
with preferences that is sufficient to purchase a bundle that is just indifferent to
his original bundle. Using the relative price of the final budget line of 20, we
develop a new utility-maximizing condition
.
Also, utilities before and after the change in Mary’s wage must be the same: i.e.,
which gives another equation
.7
Now we have two equations that have two unknowns
and
and
6
,
7
16
.
.
.
(5) Reflecting the change made in the amount of leisure time to 14 hours, Mary’s
utility-maximization problem becomes
Satisfying the same optimal condition of
constraint, Mary’s choice for leisure time is
and the new budget
. Then,
.
(6) A change is made on her wage. Thus, we have a utility maximization problem
with a new budget line.
Reflecting the change in , a new optimal condition is
. After
substituting this condition into the budget constraint, a few more steps of
computation gives us
.
She work 32/2 hours long. Also, the tax revenue is
Tax Revenue =
=
= 0.25 x $20
32/2 = $80/3.
13.
(1) (Normal good vs. Inferior good) Given the same utility function as in Question 1,
we know that the corresponding Marshallian demand function is also the same:
,
.
Let us use the income elasticity of demand for our judgment.
Assume that the consumer has a positive income and consume a positive amount of
positively-priced good 1 and good 2. Income elasticity of demand is then positive
both for good 1 and good 2. Therefore we can tell both good 1 and good 2 are normal
goods.
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(2) (Marshallian demand curve)
(3) (Hicksian demand curve) Using the Hicksian (or uncompensated) demand
function that we have developed in Question 1, let us first compute the utility
level using given information:
.
(4) (Hicksian demand curve)
18