amppsolution to the sample questions midterm

8/11/2019 AMPPSolution to the Sample Questions Midterm

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Solution to the Sample Questions for the Mid-term ExamAnalysis of Market and Public Policy (Section 3 and 4), Spring 2011

Created bySunwoo Hwang (Teaching Fellow)

Draft: March 22, 2011.

1.

(Marshalliand & Hicksian Demand, Indirect Utility Function, Cobb-Douglaus)(1)A consumers Marshallian demand function specifies what the consumer would

buy in each price and wealth (or income), assuming it perfectly solves the utilitymaximization problem

. (1)

The problem has a unique optimal solution when

(2)

since the utility function in (1) represents well-behaved preference. By puttingthe MRS of 1 into (2) and then rearranging the equation, we have

(3)

At a consumption bundle on the budget line that satisfies thecondition in (3), the utility-maximization problem (1) has the solution and wefind Jennys Marshallian demand function for each good is

, (4)

.

(2)The indirect utility function is the maximum utility attained with

prices , and income I, i.e.

where and solves the utility maximization problem in (1). Therefore,

1


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(5)

(3)

Given her utility level , Jennys 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 exchangebetween two goods of equals the MRS of , i.e., where the utilityfunction 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, wederive 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,

Jennys utility is maximized when she consumes 200 units of beef and 100 unitsof rice with her (ordinal) utility of 20,000.

(5)Excise tax of $1 imposed on beef consumption increases from $1 to $2. Byputting 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 by the increasedconsumption of good 1.

Tax Revenue = = $1 100 = $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 ; thisconsumption bundle keeps the utility constant. Changes in consumption of


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each good are decomposed into income effect (the first term) and Hicksiansubstitution effect (the second term) as follows.

Specific values are

Income effect on : 100 -

Substitution effect on : - 200

Income effect on : 100 -

Substitution effect on : 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 newbudget constraint under the lump sum tax, we can say that the bundle Jennychose under the excise tax is still available to her under the lump sum tax. Theform 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 underthe 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 withchanged prices and income, therefore, we are able to see which bundle Jennywould choose. Under the excise tax, price of good 1 is $2 (=$1 + $1), price of good2 is $2, and income is $400. Under the lump sum tax, price of good 1 is $1, priceof good 2 is $2, and income is $300 (=$400 - $100). With each set of prices andincome, the equation (5) returns utilities as below.

(under the lump sum tax) (under the excise tax)


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It turns out that Jennys utility is greater under the lump sum tax. We thereforesay that, under the new lump sum tax, Jenny will not choose the bundles shechose under the excise tax although it is still available to her.

(10) For = $1, = $2, and = $400 - $100 = $300, the equation (4)returns

,

.

2. (Marshalliand & Hicksian Demand, Indirect Utility Function, Perfect Substitutes)(1)JamesMarshallian 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 compensateddemand 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


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,

(4)

Since , , , and

(5)Since , ,

(6)

Tax Revenue (R) = = $1 1200 = $1200.

(7)

Since and are perfect substitutes, only income effect exists; i.e. total effectsequal 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


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The optimal choice under the excise tax = (1200, 0) let the LHS of theequation above be equal to its RHS. We see here that the optimal choice underthe 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 lumpsum tax.

(under the lump sum tax) (under the excise tax)

James may choose the bundle he chose under the excise tax since he is indifferentbetween the two alternatives.

(10)

, .

3. (Perfect Complements)

(1)

(2)

This consumer purchases the same amount of good 1 and good 2 no matter whatthe prices. Let this amount be denoted by . Then, we have to satisfy the budgetconstraint

.

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, Mikes Hicksian demand functions for each good are

.


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(6)

,

No substitution effect.Income effect on : 25-100/3Income effect on : 25-100/3

4. (Market demand)

(1)We know Jennys Marshallian demand function obtained in Question 1.Considering it as a demand function of each consumer, we compute the marketdemand, which is the sum of these individual demands over all consumers, usingthe given information:

(2)As discussed in Question 1, the excise tax imposed on consumption of good 1increases its price as much as the amount of tax. Therefore, the market demand asa function of and is

5. Based on given information and Jennys Marshallian demand function, we knowthat

,

.

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


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.

(2)

Neils 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 and substituting it into Neils

budget constraint, we have

.

It turns out that Neils utility-maximizing consumption in period 1 is $200 whilehis income is only $100 in that period. Thus, he needs to borrow $100 in order toconsume optimally $200.

(3)

.

(4)Two cases need to be considered.

2 and .


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(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 Neils budget constraint is changed from -1.1 to -1.23 since theinterest rate is increased from 10% to 20%. Therefore, the optimal conditionwhere MRS equals the rate of exchange between and also changes to

By substituting this new condition into Neils budget constraint afterrearranging it with respect to , we have

But, since > 100, we do not have an optimal solution in case of Save.

Neils choice does not change.

7. (Inter-temporal Choice, Perfect Complements)

(1)The budget constraint is the same as that in Question 6.(1).

3


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(2)Neils 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%, Neils budgetconstraint is changed to

.

Satisfying the optimal condition , . But, since > 100,

we do not have an optimal solution in case of Save.

Neils choice does not change.


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8. (Inter-temporal Choice, Perfect Substitutes)

(1)The budget constraint is the same as that in Question 6.(1).

(2)Neils Marshallian demand solves the following utility-maximization problem

Since , (See Question 2.(1))

, .

Save(or lend) $100.

(3).


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(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 Neils budget constraint is changed from -1.1 to -1.2,4 isstill greater than ; i.e., . Therefore,

, .

Neils choice does not change, and he enjoys greater amount of interest.

4


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9. (Food Stamp, Cobb Douglas)

(1)Jacks budget lines (a) when there is no subsidy of any kind, (b) when there isfood 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 tosee what he prefers between them. As illustrated in the figure above, Jack preferscash subsidy to food stamps if . On the other hand, Jack is indifferentbetween 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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is 5 and the slope is -1, the optimal choice is a solution that satisfies thefollowing condition

.

By substituting it into the budget line, we find that . Jacks

optimal choice is illustrated in the figure below. Since , he is

indifferent between the two alternatives.

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


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11.(Food Stamp, Perfect Substitutes)

(1)The budget lines are the same as those in Question 9.(1).

(2)Jacks optimal choice solve the following utility-maximization problem

As illustrated in the figure below, Jack is indifferent between cash subsidy andfood stamps for if . On the other hand, Jack prefers cash

subsidy to food stamps for if

12.

(Labor Supply)


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(1)Marys 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 Marys budget constraint and rearranging the equationwith 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 anew 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, wedevelop a new utility-maximizing condition

.

Also, utilities before and after the change in Marys wage must be the same: i.e.,

which gives another equation

.7

Now we have two equations that have two unknowns and .

6 and .

7 , .


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(5)Reflecting the change made in the amount of leisure time to 14 hours, Marysutility-maximization problem becomes

Satisfying the same optimal condition of and the new budgetconstraint, Marys choice for leisure time is . Then,

.

(6)

A change is made on her wage. Thus, we have a utility maximization problemwith 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 ofcomputation 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 ofpositively-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 normalgoods.


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(2)(Marshallian demand curve)

(3)(Hicksian demand curve) Using the Hicksian (or uncompensated) demandfunction that we have developed in Question 1, let us first compute the utility

level using given information:

.

(4)

(Hicksian demand curve)

amppsolution to the sample questions midterm

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