Microeconomics Solutions 13

8/12/2019 Microeconomics Solutions 13

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Chapter 13

Government and the

Individual

Exercise 13.1 A local public good is one that is specic to a particular mu-nicipality: within the municipality good 1 is provided as a public good, but theconsumer has to be resident there to benet from good 1. Residence in a mu-nicipality also determines liability to provide for the public good. Residents canchoose which municipality to live in. Suppose that exactly one unit of privategood 2 is required to produce one unit of the local public good 1 and that, ina given municipality of N people, total output is determined by a productionfunction

q= (N) :

where is a strictly concave function. Each individual in a municipality getsutility U(x1; x2) where x1 is the amount of the local public good, x2 is theindividuals consumption of the private good andUis a utility function with theusual properties.

1. What are the transformation curve and the production-possibility set for asingle municipality of sizeN?

2. If the residents of the municipality could choose both the proportion of totaloutput devoted to the local public good and the size of the municipality,explain how these would be determined. Show that if individuals are paidtheir marginal product then the amount required to nance the public goodis exactly the dierence between the wage bill and the total output.

3. If the size of a municipality N can take any value between 1 and N in-clusive, whereNis the total population in the economy, use your answerto part 1 to explain what the production possibility set is when N is al-lowed to vary. Hence show that the optimal size of municipality could bemultivalued.

4. Assuming that there is an interior solution to the optimal value ofNsketchthe relationship between utility and size of municipality.

5. In an economy there are two municipalities with identical production con-ditions. Individuals can migrate costlessly from one community to another

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Microeconomics CHAPTER 13. GOVERNMENT AND THE INDIVIDUAL

x2

x1

Figure 13.3: Optimal size of municipality

N

(N)

N N

(N)

N

Figure 13.4: Utility and size of municipality

cFrank Cowell 2006 266


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N

(N)(N N )

Utilityinfirstmunicipality

Utilityinsecondmunicipality

?

N

Figure 13.5: Equilibrium with migration

where () is assumed to have the form illustrated on the left-hand sideof Figure13.4. Now consider the migration mechanism: individuals willmove to the municipality where there is the higher utility. Now, on theleft-hand side of Figure13.5we have

NN> (N)

so that N must be falling (migration leads to an increase in the size ofthe second municipalityNN) the migration mechanism continues untilN= 0. On the right-hand side of the gure

(N)>

NN

and so the reverse happens Nis rising and carries on rising until N=N.In this case, therefore, it is clear that there are three equilibrium values

ofN, namely0, 12N andNwhere the casesN= 0and N=N are locallystable and the case N = 12N is unstable. On the other hand if () isassumed to have the form illustrated on the right-hand side of Figure13.4then the migration diagram becomes as in Figure 13.6; clearly there is asingle stable equilibrium where both municipalities have size 12N.

6. From Figure13.5it is clear that12N

> (0) =

N

, establishing the

point that the locally stable equilibria atN = 0 andN=Nare inecient.. .

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Figure 13.6: Equilibrium with migration 2



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Exercise 13.2 (A continuation of Exercise 4.13) There is a single rm pro-

ducing good 1 with costs xed cost

C0+cq (13.6)

where the constant marginal cost c is assumed to be large as in Exercise4.13;all consumers are identical with preferences given by

U(x1; x2) =x1

2

1 +x2 (13.7)

as in Exercise4.12. The government allows the rm to charge what price it likesbut oers to pay the rm a subsidy equal to the consumers surplus generated bythe price that it charges.

1. Is the regulation mechanism ecient?

2. Does the government as regulator need to know (a) the cost function? (b)the utility function?

3. Show that this mechanism allows the rm to exploit consumers completely.

Outline Answer

1. Given (13.6) the rms prots can be written

= pqC0 cq+B (13.8)

where p is the price that the monopoly charges for good 1 and B is thesubsidy. Given (13.7) the consumer seeks to maximise

x1

2

1 + [y px1]

where y is the consumers income and the price of good 2 has been nor-malised at 1. The FOC for an interior maximum is

1

2x

1

2

1 p= 0 (13.9)

from which we obtain the consumers demand for good 1 as

x1 = 2

4p2 (13.10)

(the condition on the cost function in Exercise 4.12will ensure that thecorner solution, where the person spends everything on good 1, is irrele-vant), From (13.9) the inverse demand function is

p (x1) =1

2x

1

2

1

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So the consumers surplus is

CS =Z x10

p (t) dtp (x1) x1

=

Z x10

1

2t

1

2 dt 12

x1

2

1

= x1

2

11

2x

1

2

1

= 1

2x

1

2

1

= 2

4p (13.11)

From (13.8) rms prots can be written

[p c] q C0+B:

If the size of the market is standardised at 1 for convenience (there is avery large number of small consumers with total mass 1) then q = x1and so, setting B = CV and substituting from (13.10) and (13.11), theexpression for prots becomes

[p c] 2

4p2 C0+

2

4p

Simplifying this, the rms problem can be represented as choosing p tomaximise

= 2

2p c

2

4p2 C0: (13.12)

The FOC is

2

2p2+c

2

2p3 = 0

which on simplifying yields p = c. In other words we have price equalto marginal cost, the necessary condition for an ecient outcome. Thisoutcome will be chosen by the rm if prots are non-negative. Under theabove conditions (13.12)shows that this requires

2

2pp

2

4p2 C0 0; (13.13)

in other wordsCS C0 (13.14)

But this is exactly the condition under which it is ecient to producegood 1 rather than not produce the good.

2. The only information required is the ordinary demand function (13.10).

3. Suppose the monopolist were empowered to charge an entrance feeF forthe right to buy commodity 1. Prots would be

= pqC0 cq+F (13.15)



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and the consumers budget constraint (if he chooses to buy good 1) would

be px1+x2 F y (13.16)Given that the persons utility if he does not consume good 1 (he spendsall of his income on good 2) is simply y then the rm knows that theconsumers participation constraint is

x1

2

1 +x2 y (13.17)

Combining (13.16) and (13.17) we have the constraint

x1

2

1 px1 F 0 (13.18)

Clearly, for any p, the prot-maximising rm will increase F to ensurethat (13.18) is satised with equality. The consumers demand for good 1is given by (13.10) and so (13.18) becomes

F =

2pxp

2

4p2 (13.19)

in other wordsF = CS:

Hence the scheme whereby the rm is subsidised by the amount of theconsumers surplus (and this subsidy is raised from the consumers as alump-sum tax) is equivalent to the scheme where the rm is allowed tocharge an entry fee. The consumer is forced back on to reservation utility

level where (13.16) and (13.18) are binding.



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Exercise 13.3 The government of the tiny island of Mugg is considering whether

it would be a good idea to install piped gas. Once the gas distribution systemis installed each unit of gas (commodity 1) costs a xed amount m of othergoods (commodity 2); there is additionally a xed costF incurred in setting upa distribution system on the island. Before the system is installed Mugg enjoysa total amount R2 of commodity 2. The residents of Mugg are assumed to beidentical in every respect and their tastes are represented by the utility function

1 ex1+x2where (x1; x2) represent the quantities consumed of the two goods and is anon-negative parameter.

1. What is the maximum number of units of gas that could be aorded onMugg?

2. Draw the production-possibility set; draw the indierence curves in twocases: where is large and where is small.

3. Use the diagram to show that whether it is a Pareto improvement to installthe gas system depends on the value of.

4. If the installation of gas on Mugg is Pareto-improving, describe the Pareto-ecient allocation of goods and suggest a scheme by which the publicly-owned corporation MuggGas could implement this allocation if it knowsthe willingness to pay of the residents.

Outline Answer

1. Because setting up the gas supply costs F and each unit costs m, thesupply of x1 units will cost F +mx1 units of commodity 2. Given thatthe total amount of commodity 2 available is R2, the maximum supply ofgas is

x1 =R2 F

m

2. In Figure13.7the feasible set is represented by the solid triangle with thespike on top: the height of the spike is F and the slope of the boundaryism. The MRS for the specied family of preferences is given by

dx2

dx1=const =e

x1

:

which means that they are vertically displaced in (x1; x2)-space, theyget steeper as the parameter increases and they can touch the x2-axis .Clearly picks up the strength of the willingness-to-pay for gas.

3. If < mex1then there can be no solution on the x1-axis. There remaintwo dierent types of solution, depending on the values of:

The high-valuation tastes (high-indierence curves) have steep in-tercepts on the vertical axis: clearly the optimum in this case wouldbe as in Figure13.8.

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x1

x2

R2

F

m

[R2F]/m

Figure 13.7: Gas: production possibilities

The low-valuation tastes are given by the at indierence curves andthe optimum is as in Figure 13.9 no gas should be supplied andeveryone consumes (0; R2). The no-gas utility level is obviously just

R2.

Clearly it is Pareto-optimal to provide gas if satises

mm log

m

> F:

4. If the installation is undertaken, the maximum utility that can then beobtained is at the point where the indierence curve touches the produc-tion possibility set. Since the slope of the production possibility frontierismthis means that

x1 = log

m

;

x2 = R2 Fm logm :At(x1; x

2)utility is

m+R2 F m log

m

:

Evidently this yields greater utility than the no-installation case. The gas-supply corporation can induce the(x1; x

2)outcome by making consumerspayFas a xed charge and then a (marginal) price ofm per unit of gas.

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x1

x2

R2

Figure 13.8: High valuation: optimum production is positive

x1

x2

R2

Figure 13.9: Low valuation: optimum production is zero



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Exercise 13.4 Take the model of the Island of Mugg (Exercise 13.3) again.

Suppose that the government of Mugg has a horror of public enterprise and de-cides to delegate the decision on installation and supply by selling o MuggGas.

1. Will this generate an improvement on the no-gas situation?

2. Will it generate an ecient allocation?

3. How would your answers be aected if MuggGas were split into a numberof private companies, or if consumers were allowed to resell gas to eachother?

Outline Answer

1. The answer depends on whether the rm is allowed to make a xed chargeto cover the xed cost. If the rm is restricted to charging a uniform priceto all customers then privatisation here could be a disaster. There are twopossibilities:

x1

x2

R2

Figure 13.10: Gas: simple prot maximisation

If the rm (or rms) were to set price equal to marginal cost at theecient outcome it would make a loss.

If there were a monopoly with a uniform price for all units of gasand no xed charge then the result is still bound to be inecient.First note that given gas-price p a Mugg consumer will choose x1 tomaximise

1 ex1+ [R2 px1]The rst-order condition for an interior maximum is

ex1 p= 0

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which implies

x

1(p) = log p as the consumers demand for commodity 1, given the (uniform) pricep. Second, note that maximum prots for the rm are found bychoosingp to maximise

(p) := px1 Fmx1:

This becomes

(p) = px1(p) F mx1(p)

= [p

m]log

pF (13.20)

First-order conditions for an interior maximum are

@(p)

@p = log

p

p m

p = 0 (13.21)

This implies that prots at the optimum are given by

(p) = [pm]2

p F

This may imply a Pareto-improvement over the no-gas case: theconsumer will be better o than at B if:

h

1 ex1(p)ipx1(p)> 0:

in other words:

pp log

p

> 0

which implies

) log

p

m Fm log

m

which implies

mh

1 pm

+ logp

m

i> [pm]log

p

F (13.23)

Given that p > m, the left-hand side of (13.23) must be negative Cf (13.22); but the expression for prots (13.20) implies that theright-hand side of (13.23) must be nonnegative a contradiction.

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Exercise 13.5 In an economy there are two rms each producing a single out-

put from a single non-produced resource according to

q1 =p

z

q2 = maxp

R z q1; 0

whereqi is the amount produced of goodi, z is the amount of the resource usedin the production of good 1, R is the total stock of the resource and is aparameter.

1. What phenomenon does this model represent?

2. Draw the production-possibility set.

3. Assuming that all consumers are identical, sketch a set of indierencecurves for which (a) an ecient allocation may be supported by a pseudomarket in externalities; (b) a pseudo market is not possible.

4. What role does the parameter play in the answer to the previous parts?

Outline Answer

1. There is a simple production externality: production of good 1 reducesthe amount of good 2 that can be produced for a given amount of inputdevoted to the production of good 2.

Figure 13.11: Externality: two cases

2. Note that production of good 2 is positive ifz < z0 where

z0 := R

1 +2 (13.24)



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The production-possibility set is as illustrated in the two panels of Figure

13.11.Note that the line segment L from pz0; 0 to pR; 0 belongs tothe set. The set is non-convex.

3. In panel (a) the solution is at the point of tangency and can be decen-tralised using prices shown by the broken line. However in panel (b) thisdecentralisation is not possible. If the prices given by the broken line areused then production takes place, not at the point of tangency, but at

pointp

R; 0

.

4. From equation (13.24) shows that as gets larger so too does the line Land the problem raised by the non-convexity becomes more salient.

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Exercise 13.6 In a large economy all agents have a utility function of the form

(x1) +xh2

wherex1is the amount provided of a public good andxh2 is agenths consumption

of a private good. All agents are endowed with the same amount of private goodRh2 = 1. Each individual can choose whether to contribute to the public good:

zh2 =

1 contribute0 not contribute

The unit contribution costs an amount ch to agent h; the individual costs areunobservable but the distribution functionF()of costs is known. The productionof the public good is given by

x1= ()

where is the proportion of contributing individuals.

1. Show that an ecient outcome implies that there is cost levelc0 such that

zh2 =

1 ch c00 ch > c0

2. The government introduces a tax-subsidy scheme based on individual ac-tions as follows. Each contributor receives a subsidy s and each non-contributor has to payt. Givenc0 and the distribution of costsF what isthe condition for a balanced budget if agents behave as in part 1?

3. Under the conditions of part 2 what is the utility of someone withch < c0?

Of someone withch > c0?

4. By requiring that someone with ch = c0 be indierent between contribut-ing and non-contributing show that this tax-subsidy scheme induces anecient equilibrium.

5. How much of the public good is provided and what is the tax rate andsubsidy rate?

Outline Answer

1. Pick a given level of provision of the public good x1. This then determinesa given number representing the proportion of the population that must

contribute. Consider the situation where there is somec0 such thatzh2 = 0if and only ifch > c0 . Under this scheme

= F

c0

x1 = (F

c0

):

Take a high-cost type individual that does not contribute (ch > c0) anda low-cost type individual who does contribute (c` < c0). Now change thesituation described above so thathcontributes and` does not. The utilityloss toh is ch which is strictly greater than the utility gainc` to individual`. Clearly no such perturbation can lead to a Pareto improvement.



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2. Under the above scheme there is a proportionF

c0

who are contributing,

so that the total subsidy required is given by sFc0. Given that thereis a proportion 1F c0 who are not contributing, the tax raised ist

1 Fc0. So the budget constraint issF

c0

= t

1 Fc0 (13.25)3. Consider someone with exactly costc0. This person is indierent between

contributing and not contributing if

(x1) + 1 c0 +s| {z }utility if contribute

= (x1) + 1 t| {z }utility if dont contribute

(13.26)

wherex1 = F(c0). Clearly if (13.26) holds then

(x1) + 1 c` +s > (x1) + 1 t

forc` < c0, and

(x1) + 1 ch +s < (x1) + 1 t

for ch > c0. It is also clear that c0 is publicly available informationgiven that F is known. So this scheme is indeed an equilibrium and isimplementable

4. Condition (13.26) impliess+t = c0 (13.27)

Combining the budget constraint (13.25) with(13.27) we get

c0 tFc0= t 1 Fc0

and so the required tax rate is

t= c0F(c0); (13.28)

the subsidy rate iss= c0

1 F(c0) ; (13.29)

and the amount of public good provided is

F

c0

. It remains todeterminec0. This follows from the standard rst-order condition PMRS=MRT. In the present case this gives

F

c0

1

F

c0

= 1

F

c0

(13.30)

where 1 and 1 denote the rst derivatives of and respectively.Given particular functional forms for F, and the value ofc0 can bedetermined from (13.30).



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Exercise 13.7 An economy consists of individuals whose income is only from

labour: each person is endowed with a specic level of ability which is reectedin his or her market wage w, and chooses `, the amount of time he or sheworks 0 ` 1. The minimum value of w in the population is w0, andthe mean value is w0, where > 1. The government imposes a tax-transferscheme such that a person with pretax cash incomey has after-tax cash incomeofx= [1 t][y y0] +y0.

1. Interpret the parameterst andy0.

2. Assume that every individuals preferences can be represented by the utilityfunctionx [1 `]1. What is optimal labour supply as a function ofw,t andy0?

3. If the government wants to ensure that everyone works what constraint willthis impose on the values oft andy0? Assuming that everyone does work,and that the tax raised is used solely for the purposes of redistribution,show that this implies thatt andy0 must satisfy the constraint:

y0 = w01 t

1 t

4. If the government seeks to maximise the after-tax cash income of the poor-est person (the welfare function with = 1in Exercise9.5) subject to theabove constraints show that the optimal tax rate is

t

= 1

1r 1

Interpret this result.

Outline Answer

1. The operation of the tax and transfer system is illustrated in Figure13.12:t is the marginal tax rate, y0 is the break-even income and ty0 gives aminimum guaranteed income.

2. The consumers problem is exactly the same as that of Exercise 5.7 with

the eective wage rate [1 t] w and nonwage income ty0. So we can justadapt the result from there to get

` =

[1 ] t1t y0w , ifw w1

0 , otherwise

: (13.31)

where

w1:= y0t

1 t1

(13.32)

is the reservation wage. Equation (13.31) gives labour supply as a func-tion ofw , t and y0.

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disposableincome

x

y0

1y0

pre-tax incomey

Figure 13.12: The tax-transfer system and disposable income

3. To ensure that everyone works the government must set the tax parame-ters so that w0 > w1 in (13.31). From (13.32) this requires y0 < y1 :=1tt

1w0: Net revenue raised in this system is given byZ

t[w` y0]dF(w) (13.33)

and if the tax is purely redistributive, then (13.33) should be zero. Ifeveryone works then (13.31) and (13.33) imply

Z t

w [1 ] t

1 t y0 y0

dF(w) = 0 (13.34)

Simplifying we get Z wdF(w) =

Z 1 t

1 t y0dF(w) (13.35)

Using the denition of the mean of the unknown distribution F we get

immediatelyy0 = w0

1 t1 t : (13.36)

Choosing the tax rate t then automatically xes the guaranteed income(ty0) that can be aorded.

4. The after-tax cash income of the poorest person is given by[1t]w0`+ty0which, if the person works, becomes[1 t]w0 + ty0 using the expressionfor` in (13.31); in view of the net-revenue constraint (13.36) this in turnbecomes

[1 t]w0+2t 1 t1 t w0: (13.37)



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Expression (13.37) then becomes the objective function for a government

with Rawls type objectives. To nd the optimum we dierentiate (13.37)with respect tot so as to get

w0+21 2t1 t w0+

3t 1 t

[1 t]2w0 = 0 (13.38)

which implies

1 = 1 2t1 t +

2t 1 t

[1 t]2: (13.39)

Simplifying this rst-order condition yields a quadratic in t:

2 [ 1] t2 2t [ 1] + 1 = 0: (13.40)

Applying the standard solution algorithm we get:

t = [ 1]

q2 [ 1]2 +2 [ 1][1 ]

2 [ 1] (13.41)

If we rearrange (13.41) and ignore the irrelevant root we get the result.Note that the optimal tax rate increases with : the larger is the meanwage (relative to the lowest wage) the more well-o people there are topay for transfers.

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Microeconomics Solutions 13

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