Pareto Efficiency, Social Welfare and Size of Government

Doc. Dr. Sezgin Polat

Topics in Public Economics CourseGalatasaray University

Fall, 2017

Outline

Welfare Theorems

Competitive Equilibrium: Example 1

Walrasian Economy (exchange with production): Example 2Consumer ProblemProducer Problem

Social Welfare

Public Sector GrowthTheories of Public Sector Growth

Allocation and Efficiency

Efficiency: Price mechanism as a means of allocating resources.

I First Fundamental Theorem of Welfare Economics: The competitiveequilibrium, where supply equals demand, maximizes social efficiency.

First Theorem

I Second Fundamental Theorem of Welfare Economics : Society can attainany efficient outcome by suitably redistributing resources amongindividuals and then allowing them to freely trade. Second Theorem

The Edgeworth Box 1

I Graphical tool known as the Edgeworth box 1 can be used to analyze theexchange of two goods between two people.

1The Edgeworth box is named in honor of Francis Ysidro Edgeworth (1845–1926), an English economist who was one of the first to use

this analytical tool.

The Edgeworth Box 2I The exchange of two goods between two people.

The Edgeworth Box 3I The exchange of two goods between two people.

Pareto Efficiency

I A Pareto efficient allocation can be described as an allocation where:1. There is no way to make all the people involved better off2. There is no way to make some individual better off without making someone

else worse off3. All of the gains from trade have been exhausted4. There are no mutually advantageous trades to be made

I The set of all Pareto efficient points in the Edgeworth box is known as thePareto set, or the contract curve. The latter name comes from the ideathat all ”final contracts” for trade must lie on the Pareto set otherwisethey wouldn’t be final because there would be some improvement thatcould be made!

The Edgeworth Box 4 - Market equilibrium

Competitive Equilibrium 1Lest suppose that we have a 2 consumer-economy where endowments ofagents are wa

1,wa2 and wb

1,wb2

The utility function is given as :

MaxUa = xα1 (x2)(1−α) (1)

p1c1 + p2c2 = p1w1 + p2w2 (2)

F.O. C

∂L∂x1

= αx(α−1)1 (x2)(1−α) − λp1 = 0

∂L∂x2

= (1− α)x(α)1 (x2)(1−α−1) − λp2 = 0

x1 =p2

p1

(1− α)

αx2

replace x2 into eq. 2

xa2 =

1− αp2

(p1w1 + p2w2)

xa1 =

α

p1(p1w1 + p2w2)

Competitive Equilibrium 2

Demand for x1 and x2 are conditional on prices (p1, p2) and preferences (α).Lets assume that agent b has a similar utility function but with different tastes.

MaxUb = xβ1 (x2)(1−β)

xb1 =

β

p1(p1w1 + p2w2)

Supply of good 1 is the total amount of endowments of agents a, b

S1 = wa1 + wb

1

Total demand can be written as : //

D1 =α

p1(p1wa

1 + p2wa2) +

β

p1(p1wb

1 + p2wb2)

Competitive Equilibrium 3

p1(wa1 + wb

1) = α(p1wa1 + p2wa

2) + β(p1wb1 + p2wb

2)

p2 = p1(1− α)wa

1 + (1− β)wb1

αwa2 + βwb

2

Endowments and parameters

w1 w2

Agent (a) 12 8Agent (b) 8 12Total 20 20

α = 12

β = 12

p1 = 1

x1 x2

Agent (a) 10 10Agent (b) 10 10Total 20 20p2 = 1

Edgeworth box and Exchange

Figure: Autarkic equilibrium (absence of interaction)

Figure: Market equilibrium through prices

Consumer Problem 1

Let’s suppose that we have a two-consumer economy. Each consumer has 1unit of time endowment. The utility function and the budget constraint aregiven as:

MaxU1 = cα1 (1− l1)(1−α) (3)

subject to constraint

pc1 + w1n = w1 (4)

n1 + l1 = 1 (5)

Consumer Problem 2

Utility maximization w.r.t c1 and l1First order conditions:

∂L∂c1

= αc(α−1)1 (1− l1)(1−α) − λp = 0

∂L∂l1

= −(1− α)c(α)1 (1− l1)(1−α−1) + λw1 = 0

pw1

=αc(α−1)

1 (1− l1)1−α

(1− α)c(α)1 (1− l1)(1−α−1)

pw1

=α(1− l1)

(1− α)c1)

l1 = 1− (1− αα

)pc1

w1

pc1 = w1l1 =⇒ pc1 = w1(1− (1− αα

)pc1

w1)

Consumer Problem 3

First order conditions:

pc1 = w1l1 =⇒ pc1 = w1(1− (1− αα

)pc1

w1)

c1 = αw1

p(6)

l1 = 1− (1− αα

)p

w1

αw1

p

l1 = α (7)

For the second consumer we have the following demand and supply withsimilar preferences:

c2 = αw2

p(8)

l2 = α (9)

Producer Problem 1

Production technology has the functional form of Cobb-Douglas

Y = lβ1 l1−β2 (10)

C(w1,w2) = w1l1 + w2l2 (11)

Cost minimization

MinC(w1,w2)l1,l2 = w1l1 + w2l2

subject to constraint

Y = lβ1 l1−β2

l1(w1,w2, Y) = Y(w2

w1

1− ββ

)1−β

l1(w1,w2, Y) = Y(w1

w2

β

1− β )β

Producer Problem 2

Profit maximization:

Π = pY − (w1l1 + w2l2) (12)

Π = pY − Y[w1(w2

w1

1− ββ

)1−β + w2(w1

w2

β

1− β )β ]

∂Π

∂Y= 0

p = wβ1 w1−β2 K Pricing equation (13)

where K = ( 1−ββ

)1−β + ( β1−β )β

General EquilibriumGoods demand equals goods supply:Adding individual demands eq. 6 and 8 to get aggregate demand Y:

Yd = αw2

p+ α

w1

p

Y s = lβ1 l1−β2 = αβ + α1−β = α

Lets choose p = 1 as a numeraire and find goods market equilibrium

α

p(w1 + w2) = α

(w1 + w2) = 1

and we have the pricing equation:

1 = wβ1 w1−β2 K

Solution to the equation below in terms of parameters (β, α)

1 = (1− wβ2 )w1−β2 (

1− ββ

)1−β + (β

1− β )β

Social Welfare Functions 1

Equity: Contradictory objectives, preferences and values ?I The traditional means for representing the values of the community in

economics is to use a social welfare function (SWF). SWF requires a socialplanner maximizes in order to determine the socially optimal policy.

I Utilitarian social welfare function: The seminal paper on SWFs is by Bergson(1938), with the most significant further explication by Samuelson (1947, ch.8). The SWF can be written as follows:SWF = U1 + U2 + U3 + ...+ Unwhere W is a real valued function of all variables, and the Ui s and SWF arechosen to represent the ethical values of the society or of the individuals in it(Samuelson, 1947, p. 221).

Social Welfare Functions 2

I The traditional means for representing the values of the community ineconomics is to use a social welfare function (SWF)

I Rawlsian Social Welfare Function: Another popular form of social welfarefunction is the Rawlsian SWF, named for the philosopher John Rawls. Hesuggested that society’s goal should be to maximize the well -being of itsworst -off member.5 The Rawlsian SWF has the form:SWF = min(U1,U2,U3, ...Un) Rawls- Theory of Justice

Since social welfare is determined by the minimum utility in society, socialwelfare is maximized by maximizing the well -being of the worst -off personin society.(The maximin criterion.) Human Rights

John Rawls (Theory of Justice): The Difference Principle permits divergingfrom strict equality so long as the inequalities in question would make theleast advantaged in society materially better off than they would be understrict equality.

I First: Each person is to have an equal right to the most extensive scheme ofequal basic liberties compatible with a similar scheme of liberties for others.

I Second: Social and economic inequalities are to be arranged so that they areboth

I to the greatest expected benefit of the least advantagedI attached to offices and positions open to all under conditions of fair equality of

opportunity.In Rawls’s theory, life is a game of chance in which Nature deals out attributes and social positions ina random or accidental way. Now this natural distribution of attributes and chance determination ofsocial position is neither just nor unjust. But it is unjust for society simply to accept these randomoutcomes, or to adopt institutions that perpetuate and exaggerate them. Thus, a set of justinstitutions is one that mitigates the effects of chance on the positions of individuals in the social

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I (The maximin criterion: Declaration des droits de l’homme et du citoyen,1789, article 1) Les hommes naissent et demeurent libres et egaux endroits. Les distinctions sociales ne peuvent etre fondees que sur l’utilitecommune - Men are born and remain free and equal in rights. Socialdistinctions can only be based upon common utility

I Amartya Sen develops a more concrete approach to rights and”capabilities”: A. Sen defines capabilities as ”the freedom that a personhas in terms of the choice of functionings, given his personal features(conversion of characteristics into functionings) and his command overcommodities.”

I The functioning of a person is an achievement; it iswhat the person succeeds in doing with the commodities and characteristics at his or hercommand.For example, bicycling has to be distinguished from possessing a bike. It has to bedistinguished also from the happiness generated by [bicycling].

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Public Sector Growth

I The historical development of the public sector over the past century canbe summarized as one of significant growth.

I For the typical industrially developed economy, government expenditurewas only a small proportion of gross domestic product at the start of thetwentieth century.

I Expenditure then rose steadily over the next sixty years, leveling outtoward the end of the century.

Government Expenditure / GDP

Types of Government Expenditures

Theories of Public Sector Growth

I Development Models : The basis of the development models of publicsector growth is that the economy experiences changes in its structureand needs as it develops.

I infrastructural expenditure in the development of citiesI Increasing proportion of public expenditure is diverted away from spending

on infrastructure to urbanization and its externalitiesI Transfer payments, such as social security, health, and education, becoming

the main items of expenditure

Size of Government

I Wagner’s Law : Adolph Wagner was a nineteenth-century economist whoanalyzed data on public sector expenditure for several Europeancountries, Japan, and the United States.

I Economic growth requires continual introduction of new laws and thedevelopment of the legal structure. Law and order imply continuingincreases in public sector expenditure.

I The process of urbanization and the increased externalities associated with.I The goods supplied by the public sector have a high income elasticity of

demand.I Demand increases more than proportionally with respect to income.

Size of Government

I Baumol’s Law: The basic hypothesis is that the technology of the publicsector is labor-intensive relative to that of the private sector.

I The public sector cannot substitute capital for labor, the wage increases in theprivate sector feed through into cost increases in the public sector.

I Technological advances in the private sector lead to increases in productivity.

Size of Government

I The government as re-distributor of income and wealthI The Meltzer and Richard model. Meltzer and Richard (1978, 1981, 1983)

r = tymean inequalityI Kuznets’ (1955) famous inverted-U curve. Piketty inequality curve

I Redistribution is limited through deadweight loss in taxation.I Politics is majoritarian, equal (one person, one vote) and with full participation

(all economic agents vote).I Cusack (1997) : Left of-center governments are assumed to favor more

redistribution and larger budgets than right-of-center governments.I Kristov, Lindert, and McClelland (1992) : Redistribution as a function of the

social affinity between different groups in the income distribution.I Peltzman (1980): Increasing equality of income among potential coalition

members drive the growth of government

Size of Government

I Interest groups and the growth of governmentI Tullock (1959) : discussion of majority rule. more is spent than would be

spent under the unanimity rule. Second, if the unanimity rule were in use,there would be no incentive to have the government.

I Rice (1986) presented evidence suggesting that labor unions and otherinterest groups were able to induce governments to introduce programs tooffset economic hardships

I North and Wallis (1982): Growth of government and the growth ofwhite-collar and managerial employment in the private sector. Response tothe greater transaction costs from organizing a market economy withincreasing specialization.

Size of Government

I Bureaucracy and the growth of governmentI Budget-maximizing bureaucrats: Niskanen (1971) postulated that a

bureaucrat’s ”salary, perquisites of the office, public reputation, power [and]patronage” are all positively related to the size of the bureau.

I Government Agency :The lack of information available to voters. Theimperfect information of voters enables the government to grow larger byincreasing the tax burden. Mill (1861) felt that direct taxes were more visibleand, by implication, that excessive government growth would have to rely onindirect taxes.The issue of what sources of revenue are less visible to citizens,as well as the magnitude of any fiscal illusion caused, must be regarded aslargely empirical.

I Corruption: Predatory regulation. The government intentionally createsregulations that entrepreneurs have to pay bribes to get around. Goel andNelson (1998) use convictions for public abuse of office as an index ofcorruption, and find that corruption at the state level in the United Statesincreases with the size of state governments.

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Meltzer-Richard Model (1981) 1

Meltzer-Richard (1981) ” A Rational Theory of the Size of Government”Example Taken from D. Mueller ”Public Choice III” (p. 512-514)Standart model with Stone-Geary utility function.

MaxU = ln(c + γ) + aln(l + λ) (14)

The parameters γ, λ are thought of as ”subsistence” levels of consumption,below which utility is not defined subject to constraint.

c1 = (1−t)y+r each worker receives a lump-sum transfer r and pays a proportional taxe t

y = xn each worker works n and has a different productivty level x

l = 1− n total endowment is unity

Meltzer-Richard Model (1981) 2

MaxU = ln((1− t)nx + r︸ ︷︷ ︸c

+γ) + aln(l + λ) (15)

n =(1− λ)(1− t)x + a(r + γ)

x(1− t)(1− a)

n cannot be negative; thus there is a critical level of ability, x0, at whichoptimal n = 0so c = r

x0 =a(r + y)

(1 + λ)(1− t)(16)

take the derivative eq. 16 with respect to t

∂x0

∂t=

a(r + γ)

(1 + λ)> 0

as t increases, the level of x0 increases thus more people will choose not towork, on the other hand, people might not be willing to work if t increases

∂n∂t

= −a(r + γ)

(1 + a)x(1− t)−2 < 0

Figure: Meltzer-Richard(1981) page 922

Preferences for Distribution - Alesina and Giuliano(2011) Handbook of Social Economics, Volume

1A

One unit of labor is inelastically supplied and the individual productivity is ai.Assume that the government uses a linear income tax t on income to financelump sum transfers and that there is a wastage equal to wt2 per person whichcapture the distortionary cost of taxation. aA is the average productivity of thepopulation.

ci = yi = ai(1− t) + aAt− wt2

Take the derivative with respect to t

t =aA − ai

2w

if the voter i is the median or decisive voter aM then she/he will choose thetaxe rate

t =aA − aM

2w

Figure: Alesina and Giuliano(2011) Handbook of Social Economics, Volume 1A

Figure: Alesina and Giuliano(2011) Handbook of Social Economics, Volume 1A