markets in developing world lecture

8/13/2019 Markets in Developing world Lecture

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Lecture Note 1: Introduction to Welfare

EconomicsPart 2B: Paper 1.

Dr. T.S. AidtUniversity of Cambridge

Michaelmas 2012

1 Introduction

This lecture note provides a brisk reminder of the basic ideas, results andconcepts of welfare economics. Most of this should be familiar from last

year. The note sets out the details of a baseline model. We shall makeuse of various versions of this model to illustrate and discuss many of the

j lt b t ti l t li d i l t b t l i i


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j lt b t ti l t li d i l t b t l i i


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8. Consumers have preferences dened over the Ncommodities and laboursupply (leisure forgone) described by the direct utility function

Uh =Uh(lh0 ; xh1 ;:::;x

hN): (1)

Utility is increasing in consumption of goods and decreasing in labour

supply. Utility is often assumed to be cardinal.

9. The budget constraint of consumerhis

NX

i=1

qixhi =m

h + q0lh0 ; (2)

wheremh are other sources of (exogenous) income (in particular prots

from ownership of rms) than labour income.

10. For many purposes, we need to use the indirect utility function and theexpenditure function (which should be familiar from last year). Theindirect utility function, Vh(q; mh) is simply linking the maximizedutility to consumer prices and income, i.e., is dened as

Vh(q; mh) = maxflh0 ;xh1 ;:::;xhNg

Uh(lh0 ; xh1 ;:::;x

hN)

subject to the budget constraint. The expenditure function is dened


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and the demand for labour and capital are given by

@j(p)

@p0= lj0(p) (4)

@j(p)

@pk= kj(p): (5)

14. An important consideration for many policy questions is whether rmsmake prots or not. This partly depends on technology. If the pro-duction technology has constant returns to scale, then prots must bezero in a competitive market. However, if, in the short run, there isdecreasing returns to scale or if there is imperfect competition, protsare positive. In this case, we assume that the prots are distributed

to consumers as income. If we denote the share of prots from rmj

that consumer h gets by hj , then we can write mh as

JPj=1

hj j(p). The

income of the consumer now depends on the whole vector of producerprices and, as we shall see, this will play a role for some of the taxresults that we shall discuss.

15. If the government needs to raise revenueR, then it must levy taxes on

commodities (including labour and other inputs). We need to make adistinction between two cases. Firstly, if producer prices are xed, thenth t b th b dl f i t it d t d i


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Exercise 1 What is the dierence between cardinal and ordinal utility?

Exercise 2 Why is it true that@j(p)

@p0=lj0(p)?

Exercise 3 If the production technology has constant returns to scale, whymust prots be zero in a competitive market?

3 The Fundamental Welfare Theorems

In the absence of any (distortionary) taxes and market or information im-perfections, we expect that a competitive equilibrium will naturally emergein which a unique set of prices will clear all markets. This is the startingpoint for thinking about economic policy. In particular, the two Fundamen-tal Welfare Theorems are applicable and can be used as a guide to policy. Todiscuss the two Theorems, it is useful to consider a special case of the generalmodel with two commodities, two factors of production and two (types of)consumers.

3.1 The 2 2 2 model

Consider the following economy:

1. Two outputs or commodities x1 and x2 produced with two inputs,


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3. There are two consumers,h 2 fA,Bg, with preferences dened over thetwo commodities described by the direct utility functions2

Uh =Uh(xh1 ; xh2): (9)

The consumers budget constraints are

q1xh1+ q2x

h2 =m

h; (10)

where mh is the (given) income of consumer h andqi is the consumerprice of commodity i.

4. The resource constraint of the economy is

l = l1

0+ l2

0 (11)k = k1 + k2: (12)

3.2 The Pareto conditions

We are interested in the conditions that characterize an ecient allocationof resources in this economy. Loosely speaking, by ecient we mean anallocation of resources such that it is not possible to nd another allocationwhat would make some consumer better o without making someone elseworse o. We are interested in this for two reasons. Firstly, we want to

d h k ll d


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to achieve something else. However, it is more convenient and intuitive torefer to their absolute value, so that MRS, MRTS and MRT are positivenumbers, and I shall do so throughout the lectures and in the notes. Morespecically, the three conditions are:

1. Production eciency: The marginal rate of technical substitution (MRTS)

between any two factors of production must be equal in the productionof all commodities:

MRTSx1k;l0 =MRTSx2k;l0

: (13)

2. Consumption eciency: The marginal rate of substitution (MRS) be-tween any two commodities must be equal for all consumers, i.e.,

M RSAx1;x2 =M RSBx1;x2

: (14)

3. Product mix eciency: The (equal) marginal rate of substitution be-tweenx1 andx2 in consumption must be equal to the marginal rate oftransformation (MRT) between x1 andx2 in production:


=M RTx1;x2 : (15)

Let us start with a discussion of production eciency. Production e-


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that goes through point A. The output associated with this allocation isthenx02 andx

01. It is clear, however, that this is not production ecient: we

can move capital out of and labour into the of production of x1 in such away to keep production at x01 (move down the isoquant labelled x

01) but in

the process of doing so, we increase the amount of commodity x2 that canbe produced. This process can be continued until we get to point B wherethe allocation x01 and x12 is production ecient. At point B, we see thatthe slopes of the two isoquants are the same. The (absolute values of the)slopes represent the marginal rate of technical substitution (MRTS), i.e., therate at which it is technically possible to replace labour with capital whilekeeping output constant in each sector. Accordingly, we see that productioneciency requires that the marginal rates of technical substitution betweenany to inputs must be the same across all outputs, i.e.,

MRTSx1k;l0 =MRTSx2k;l0

: (16)

Of course, there are many such allocations dened by the tangency points inthe box diagram. We can plot these in(x1; x2)space to get what is called theproduction eciency frontier. This is shown in Figure3. The productioneciency frontier shows the combinations of the maximumamounts of thetwo commodities that can be produced with the economys scarce resources.

At any point on the frontier, the marginal rates of technical substitution areequal. Since production eciency will play an important role later on in ourdiscussion of optimal taxation it is worthwhile pausing and thinking through


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is called the marginal rate of transformation (MRT)

dx2

dx1=M RTx1;x2 (17)

and is illustrated in Figure 3.

Let us now consider consumption eciency. The requirement here is sim-ply that the marginal rate of substitution between any two commodities mustbe the same for all consumers. This can be illustrated in the consumptionbox diagram shown in Figure 4. This consumption box is superimposed onthe production frontier diagram from Figure 3. Suppose that the economyis producing a point E on the production eciency frontier. This denesthe quantities of x1 and x2 available for consumption. We can draw theindierence curves for the two consumers in the box (for consumer B with

reference to OB and for consumer A with reference to OA). Consumptioneciency requires that the available amounts of the two commodities aredivided between the two consumers along the contract curve, where


: (18)

The reason this is required for eciency is simply that if the condition fails,then it is possible to reallocate the two commodities between the two con-sumers in such a way that would make at least one of them better o.

Product mix eciency requires, in addition, that this common marginal


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It should be clear that there exist many allocations of resources whichsatisfy the three necessary conditions for an ecient allocation. We cansummarize these by the utility possibility frontierof the economy. Thisis constructed from Figure5 by, rst, xing a point on the production fron-tier, say, E, and then recording the utility levels of the two consumers atthe corresponding consumption ecient allocation and plotting them in adiagram where we have the (ordinal) utility of consumerA on thex-axis andthe (ordinal) utility of consumer B on they-axis. Pick another point on theproduction eciency frontier and repeat the exercise. In this way, we cantrace out the utility possibility frontier shown in Figure 6. It is downwardssloping, reecting the fact that allocations on the frontier are Pareto ecient.

What if the factors of production are not xed? This is an important casethat plays a key role in much of the discussion about taxation so it is worth

considering briey what happens if labour is variable and its supply is a choicemade by the two consumers. To this end, suppose that there is an endowmentof labor time, T0, which is the same for the two consumers and that T0 =lh0 +L

h0 where L

h0 is the amount of leisure enjoyed by consumer h. Leisure

generates utility and should thus be counted as a good in the utility functionUh(xh1 ; x

h2 ; l

h0) and the budget constraint should be q1x

h1 + q2x

h2 + q0L

h0 =

q0T0+ mh orq1x

h1 + q2x

h2 =q0l

h0 + m

h. This extension aects the conditionsfor consumption and product mix eciency but has no implications for theconditions for production eciency.


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M RSAx1;L0 = M RSBx1;L0

=M Px1l0 (21)

M RSAx1;L0 = M RSBx1;L0

=M Px2l0 : (22)

3.3 The First Welfare TheoremThe First Welfare Theorem says that a competitive equilibrium is necessarilyecient, i.e., that the allocation of resources satises the three Pareto con-ditions. The reason for this is simply that consumers and producers face thesame relative prices and that they optimize by equating these relative pricesto the relevant marginal rates. Firstly, rms minimize costs by employingthe two factors of production till the marginal rate of technical substitution

is equal to the relative wage (p0) rental rate (pk), so

MRTSx1l0;k= MRTSx2l0;k

= p0

pk. (23)

Secondly, rms maximize prots by equating the marginal cost of productionto the relevant producer prices, i.e.,

M Cx1 = p1 (24)

M Cx2 = p2: (25)

Notice that the total cost of employing labour and capital in the production


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the markets depend on the location of the endowment point. Suppose westart with the endowment corresponding to point D0 in Figure 7. This in-duces a particular competitive equilibrium and a corresponding allocation onthe utility possibility frontier (at point D). Suppose society prefers an equaldistribution of utility at point C. This can be achieved by redistributing theendowments to point C0 and then let the market do the rest!

Exercise 5 What if labour supply is endogenous? How does the Second Wel-fare Theorem work in that case?

4 Social Welfare Functions

The Pareto criterion is clearly insucient to allow the government to decidebetween dierent allocations on the utility possibility frontier. By denitionthey are all Pareto ecient and any move in one direction or the other froma given starting point will make some individuals better o at the expenseof others. In order to rank dierent allocations (and thus to determine thedegree to which is it desirable to redistribute endowments through lump sumtaxes and subsidies) the government must trade o the utility gains andlosses associated with any given policy proposal.

The Bergson-Samuelson approach to this is to specify a social welfarefunction:

SW F = W (U1 UH)


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In fact, a Bergson-Samuelson SWF can only produce a unique socialchoice if 1) individual utility functions are cardinal and 2) interpersonalcomparable. To prove this consider the following little example. Two in-dividuals,h2 fA; Bg, live in a society that needs to allocate 4 beers amongthem. Individuals prefer more beer to less. Social welfare is maximized atthe allocation {2,2}. Can we nd ordinalindividual utility functions such

that social welfare is uniquely maximized at {2,2}? The answer is no. To seethis, let the social welfare function be the sum of individual utilities, i.e.,

SW F =UA(xA) + UB(xB) (35)

wherexh is the number of beer given to individual h. If social social welfareis really maximized at f2; 2g, then it must (among other things) be the casethat

UA(2) + UB(2)> UA(1) + UB(3): (36)

Rewrite this to get

UA(2) UA(1)> UB(3) UB(2): (37)

Now suppose that utility is ordinal. In that case, we can transform theutility function in any way we like as long as the transformation is positivemonotonic. So, let us scale up the utility function of individual B by aconstant (we are free to do so without scale the utility of individual A).Then we get

A A B B


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The choice of social welfare function involves an ethical judgement aboutthe extent to which welfare gains and losses of dierent individuals can betraded o. In other words, the choice embodies a statement about the soci-etys aversion to inequality, about its inequality aversion. The two mostcommonly used social welfare functions are the utilitarian and the RawlsianSWF. The utilitarian SWF takes the form

SW F =P

h hUh: (39)

It is simply the sum of the utilities of all individuals where individuals aregiven a social welfare weight h > 0. A special case is when all individualsare given the same weight h = 1. In this case, the utility of one individual isa perfect substitute for that of another and the social indierence curves arestraight lines with slope1. A Rawlsian SWF is concerned with the welfare

of the worst o individual in society. We can express this as follows:

SW F= min

U1;:::;UH

(40)

and we note that the utility of one individual is a perfect complement tothat of another. Consequently, the social indierence curves are Lshapedwith the "kink" located at an allocation with equal utility to everybody. Itis clear that the Rawlsian approach supposes an extremely high degree of

inequality aversion. The utilitarian approach, on the other hand, is muchless equalitarian. Some examples of social choices on the utility possibilityfrontier are shown in Figure 8.


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government. The rst best is dened as an allocation of resources suchthat allPareto conditions are satised and in which the only constraintson the economy are those irreducible ones associated with technology andendowments. The utility frontier shown in Figure6 is for this reason calledtherst best utility frontier.

The only eciency-related task of the government in a perfect world is to

ensure that markets are competitive and property rights etc. are protected.Moreover, any redistribution that might be desired can be done withoutinterfering with eciency simply by an appropriate lump sum reallocation ofendowments. We have perfect separation between eciency and equity.

What is needed for the world to be perfect? Well, we need the conditionsfor the two Fundamental Welfare Theorems to apply, which are:3

1. Perfect competition and price taking behavior in all markets.

2. Complete set of markets (or the absence of externalities and publicgoods).

3. Complete and symmetric information.

Whenever one or more of these conditions fail, the world in no longerperfect and the potential role of government widens and becomes more com-

plicated. Firstly, the government can take it upon itself to (try to) restoreeciency whenever one or more of the three Pareto conditions fail withoutgo ernment inter ention This ma in ol e regulating monopolies pro iding


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world.

1. A rst best world is one in which the government has the capacityto correct allmarket failures and therefore ensure that all the Paretoconditions are satised. Example: Think about a polluting monopoly.Here there are two market failures, an externality and a monopoly. If

the government can regulate both pollutionandthe monopoly, then allPareto conditions can be satised and the rst best restored.

2. A second best world is one in which the government only has the ca-pacity to correct some market failures and, therefore, cannot ensurethatallthe Pareto conditions are satised. Another way to put this isthat the economy is characterized by some important failures that thegovernment simply cannot remove. Example: Think, again, about thepolluting monopoly. If the government cannot regulate pollution, andonly got the power to deal with the monopoly, it will not be possibleto restore all Pareto conditions, and the design of government policybecomes second best.

6 Distortions

It is useful to make a distinction between two types or classes of distortionsthat might cause one or more of the Pareto conditions to fail. The two main


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(b) Production ineciencies may arise from wage dierentials for afactor between two sectors, e.g., the public and the private sector,and be associated with dual labour markets or unionization. Thisimplies that the MRTS is not the same in the two sectors becauseproducers do not face the same prices.

(c) Consumption ineciencies may arise because of a consumptionexternality, such a envy between consumers. This will put a wedgebetween market prices and the MRS for dierent consumers.

2. Policy-imposed distortions. Policy-imposed distortions refer to distor-tions originating from policy choices made by the government thatcause one or more of the Pareto conditions (which would otherwisehave be satised) to fail. Examples include taxes and subsidies, taris,

regulation of entry, public production of goods etc. Again, we can listsome examples of how the Pareto conditions might break down due topolicy-imposed distortions:

(a) Product mix eciency breaks down as soon as the governmentintroduces a commodity tax since producers and consumers arenot facing the same prices anyone and so MRS and MRT will bedierent.

(b) Production ineciencies can be created by a set of sector-specicfactor taxes or subsidies These implies that producers in dierent


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7 The Principle of Targeting

In a rst best world, the policy advise to the government is clear: since it canrestoreallthe failing Pareto conditions, this is exactly what it should do. Endof story. Yet, the question remains, how should it do it? Bhagwati (1971)suggests a simple principle which we shall call the Principle of Targeting.

It runs like this: if you want to correct a distortion (whether endogenousor policy-imposed) in a rst best world where you (well, your government)can, in fact, restore all the (failing) Pareto conditions, you should use theinstrument that directly osets the source of the distortion. You shouldtarget your instrument at the source of the problem.

The point in that there often are many ways to eliminate a particulardistortion (regulate a monopoly, internalize an externality, provide a public

good) and you want to make sure you do it in a way that does not create anynew distortions which, in turn, will create new deviations from the Paretoconditions somewhere else in the economy. A good example to illustrate thislogic is to consider a simple production externality and the choice between anexternality tax and a green tax. An externality tax (also called a Pigouviantax) is levied directly on the externality. A green tax, in contrast, is leviedon the output of the good the production of which is causing the externality.

To be specic, suppose that production of a good yi sold in a competitive

market at price pi generates emission of pollution ei. This emission causesdamage to consumers and we denote the social marginal damage thus createdb SMD d h i i i i i h f i i I i


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left-hand panel an increase in the output (green) tax shifts the M Bi curvedown because the reduction in output reduces the marginal protability ofemission. As a consequences, emission is reduced belowei . In the right-handpanel, the demand curve shifts down (if the tax is collected from consumers).The reduction in output creates a deadweight loss. For a small output taxthe (marginal) deadweight cost is small while the (net) environmental benet

is large, so welfare can be improved by increasing the tax. At some point,the marginal deadweight cost in the product market is equal to the marginal(net) environmental benet and it does not pay to increase the output taxmore. This happens at eSBi 2 (e

i ; e

i ). The point to notice, however, is

that the green tax can be used to x the pollution problem but in doingso, it distorts the choice between yi and other goods thus imposing a newineciency on the economy. The Pigouvian tax avoids this by aiming directly

at the source of the problem and is, therefore, the preferred instrument.

8 Literature

References

[1] Stiglitz, chapters 3 to 5.

[2] Cullis and Jones, chapter 1;


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labour (l0)

Outputofx1

Production frontier

Production set

Figure 1: The production frontier and production set.


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1xO

2x

O

Labour

Capit

al

0

1x

A

0

2x12x

B

Figure 2: Production Efficiency

Contract

curve

Isoquants


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x1

x2

Production inefficient

allocations

Production efficient allocations

Figure 3: The production frontier

Production efficiency

frontier

MRT


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x1

x2


frontier

E

OB

Contract curve

Indifference curves for consumer A

Indifference curves

for consumer B

Figure 4: Consumption efficiency.

OA


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x1

x2


frontier

E

OB

Contract curve

Figure 5: Product mix efficiency.

OA

same slope

CD

)(CUB

)(CUA


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x1

x2


frontier

E

OB

Contract curve

Figure 7: The Second Welfare Theorem.

OA

Budget

line

CD

)(CU

B

)(CUA

Endowmentpoint after

lump sum tax

Initial endowment

CD

UA

UB

D

C


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RawlsB

U

A

U

Utilitarian

Figure 8: Social choices with Utilitarian and Rawlsian SWFs


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'iMB

'ie*

ie

'iSMD

)0(' tMBi

'iSMD

ip

*

p

iy

Dead weight

loss

**

ie

),( SBii epy

*

iy**

iy

A

t => y => e

tp*

SB

ie

)0(' tMBi

Figure 9: The principle of targeting

markets in developing world lecture

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