Macroeconomics Question

generalequilibrium.pdf

Home >Business & Finance homework help >Economics homework help >Macroeconomics Question

Chapter 11: General equilibrium and welfare economics

Learning outcomes By the end of this chapter, and having completed the Essential reading and activities, you should be able to:

• define the competitive general equilibrium

• analyse the competitive equilibrium in the case of a pure exchange economy

• define Pareto efficiency and apply it to a general equilibrium setting

• find the contract curve

• explain the welfare theorems

• discuss the theory of second best.

Essential reading Morgan, Katz and Rosen Chapter 12.

General equilibrium See MKR Section 12.1 General equilibrium analysis.

In the chapters so far we have always analysed the two sides of the market (consumption and production) separately. Moreover, we have always analysed the consumer’s problem focusing on a single consumer. In partial equilibrium analysis we examine specific aspects of the economic system – for example, the market for a particular good – on the assumption that the neglect of other aspects will not lead us into serious error. In general equilibrium analysis we examine how the interactions of individual economic agents in the economy as a whole determine the allocation of resources and the distribution of income. Your textbook provides a general discussion of the features of general equilibrium analysis. In this section of the subject guide we will focus on understanding in a deeper way one particular case, the pure exchange economy.

Let us start however with a general definition of the equilibrium that we will use. We have a competitive general equilibrium if we have an allocation of resources and a price system where the following conditions hold:

• Each consumer maximises his utility at the equilibrium prices given his endowment.

• Each firm maximises its profit at the equilibrium prices given its technological constraints.

• The firms’ profits are distributed to the consumers.

• Demand equals supply for all goods.

Notice that the first two conditions come directly from the partial equilibrium analysis that we developed in previous chapters. The third and fourth conditions are instead conditions of closure of the economy: the third one says that there is no money inflow or outflow from the economy, and the fourth one asks that all (and only) what is produced is consumed.

66 Microeconomics

For the purposes of this course we will focus on the simplest general equilibrium problem, the pure exchange economy.

Pure exchange economy We have a pure exchange economy when there is no production. We focus on two consumers and two goods in a pure exchange economy. Each consumer is endowed at the beginning with a certain amount of each good, and all that the consumers can do is to trade given the prices of the two goods.

The competitive equilibrium in this case where there is no production requires that:

• Each consumer maximises his utility given prices and his endowment.

• For each good, the sum of the quantities that each consumer demands is equal to the total amount of the good that is available in the economy.

Basically what each consumer does is to solve a maximisation problem like the one that we studied in Chapter 3. The only difference is in the budget constraint; the consumer is not endowed with a certain amount of income as he was before: he is instead endowed with a certain amount of each consumption good. Therefore his income is now endogenous and it is equal to the value of his endowment. Call EA = (eAx,e

A y) the endowment of

consumer A. His budget constraint is therefore

px x A + py y

A = px ex A + py ey

where xA and yA are the amount of goods x and y that A consumes.

One of the reasons why the analysis of the pure exchange economy with two goods and two consumers is particularly interesting is the possibility of representing it in an Edgeworth box. The Edgeworth box puts together two graphs, one for each consumer in the economy. The graph of the second consumer is rotated and composed with the graph of the first consumer in a way that the length of the horizontal sides of the box corresponds to the total amount of good x that is available in the economy, and the length of the vertical sides of the box corresponds to the total amount of good y that is available in the economy. The origin that refers to the first consumer is therefore the bottom left corner of the box, while the origin that corresponds to the second consumer is the top-right corner of the box. You should read the description and look at the graphs contained in the textbook, and practise to understand how to read and use the Edgeworth boxes to analyse a pure exchange economy.

In order to understand how to find the general equilibrium, consider for example the following pure exchange economy with two agents, Ann and Bob, who consume only two goods, x and y. In the economy there are 4 unit of x and 2 units of y. Ann is endowed with 1 unit of x and 2 units of y, and Bob is endowed with 3 units of x and 0 units of y; their utility functions are:

uA(x A, yA) = √x

AyA

uB(x B, yB) = x

B + 2yB.

Let us start by drawing the Edgeworth box, showing the endowment point, E, and the indifference curves.

Chapter 11: General equilibrium and welfare economics

E OB

If we want to find the equilibrium we first need to verify that each individual is maximising given prices and endowment. We therefore start our analysis by finding Ann’s demand functions. Ann’s problem is

max √ xAy A (xAy A)

subject to the budget constraint px x A + py y

A = px + 2py. The demand can be found by solving the following system, given as usual by the tangency condition and the budget constraint:

yA xA

px py

pxx A + py y

A = px + 2py

The demands are therefore:

xA (px, py) = px + 2py

2px

yA (px, py) = px + 2py

2py

Now let us look at Bob’s demand functions. Bob’s problem is

max xB + 2yB

(xB, yB)

subject to the budget constraint pxx B + py y

B = 3px . As you have seen in the

VLE exercises related to Chapter 3, a consumer with a utility function characterised by perfect substitutes consumes at an interior point only if the price ratio is equal to his (constant) marginal rate of substitution, and spends all his income (which in this case is 3px) only on the relatively cheaper good otherwise. Given that his MRS is ½, Bob’s demand functions are:

xB( px, py) = 3

px x B( px, py) + py y

B( px, py) = 3px

xB( px, py) = 0

px py

< 1 2y

B( px, py) = 0 px

py =

1 2

px py

> 1 2py

ifyB( px, py) = 3px

In order to find the equilibrium price and allocations, we now equate demand and supply. We require therefore that the sum of the demand for x is equal to the total supply of x. Since Bob’s demand function is defined differently in three different regions, you can equate demand and supply

66 Microeconomics

in each region and see where the equilibrium is. Or you can notice that as long as both prices are different from zero, Ann will want an interior solution; in order to have an interior solution for Bob it has to be the case that

px py

= 1 .2 Therefore

xA( px, py) = px + 2py

2px =

5 2

and the condition such that D = S in the market for x, that is

x A( px , py) + xB( px , py) = 4

gives us xB(px, py) = 3

2 . Given Bob’s demand this implies that yB( px , py)

= 3 4,

and given the equilibrium price ratio we can find that yA( px , py) =

px + 2py 2py

= 5 .4

You can now verify that this price ratio induces equilibrium in the y market as well. Therefore the equilibrium is given by the price ratio

px py

= 1 2 and

the allocation of resources

xA (px, py) = 5 2

y A (px, py) = 5 4

xB (px, py) = 3 2

y B (px, py) = 3 4

Welfare economics See MKR Section 12.2 Welfare economics.

Once we have found the competitive equilibrium, we might want to understand whether the allocation that it proposes is in some sense ‘good’. Welfare economics, as your textbook says, is ‘the branch of economic theory concerned with the social desirability of alternative economic states’.

Welfare economics is essentially interested in the efficiency and the equity of different allocations. In this section we will focus on efficiency issues, while we will discuss equity (and social desirability) issues in Chapter 14.

Pareto efficiency Economists use what is known as the Pareto criterion to evaluate the efficiency of allocative outcomes. The Pareto criterion is a very reasonable way of ranking allocations that says that one allocation is preferred to another one if two conditions are met:

• It does not make any individual worse off.

• It makes at least one individual strictly better off.

The allocation of resources is Pareto optimal (or efficient) if no changes of any sort will make some person better off without making someone else worse off. Clearly, given that it is possible that several allocations are not comparable (consider, for example, one allocation that gives all the available resources to A, and one that gives the available resources to B) we will have in general a large number of Pareto efficient allocations.

You should understand what is meant by a Pareto efficient allocation of resources. You should also note the following two points about this concept of efficiency.

Chapter 11: General equilibrium and welfare economics

• It identifies the welfare of society solely with the welfare of the individuals that make up society.

• There are a number of Pareto efficient allocations that differ primarily in the distribution of well-being between individuals.

The concept of Pareto efficiency is silent on questions concerning the equity of particular distributions of well-being. However, it is the minimal requirement for a desirable equilibrium, given that if we are not in a Pareto efficient allocation there exists another allocation that makes at least one individual better off, without making someone else worse off.

In considering what efficiency means in the economy-wide context of a general equilibrium model it is convenient to start by assuming an economy in which there are two goods (X and Y), two factor inputs (L and K) in fixed supply, and two individuals (A and B). Given well- behaved preference and production functions, it is then possible to show that three conditions are necessary and sufficient for efficiency in general equilibrium. The three conditions are:

• efficiency in consumption (or exchange)

• efficiency in production (in the use of inputs)

• efficiency in product mix (or allocation, or output markets).

Efficiency in consumption means that goods are allocated to individuals so that, on the margin, no one person will be more willing to sacrifice good Y for good X than any other individual or, in other words, the marginal rates of substitution (when defined) in consumption must be equal for all consumers:

MRSAXY = MRS B

XY .

If this condition did not hold, the individuals could mutually gain by trade. You should be able to show this in terms of an Edgeworth (consumption) box diagram. You should also be able to use this diagram to describe a contract curve, the locus of all points of tangency between two indifference maps, that is, the locus of all points in the Edgeworth box where the marginal rates of substitution between two goods are the same for both individuals.

To better understand what a contract curve is let us consider the following example. Consider a pure exchange economy where there are two consumers A and B, and two goods x and y. The utility functions are:

uB(x B, yB) = √x

ByB.

uA(x A, yA) = 2x

A + yA,

The two marginal rates of substitution are:

MRSA = 1 ,2

MRSA = yB . xB

The contract curve, therefore, is the straight line coming out of the top right corner (B’s origin) where xB = 2yB.

Activity 11.1

Notice that in some cases the MRS is not defined; however we may still be able to determine the set of Pareto efficient points. Consider two consumers A and B in a two- good pure exchange economy where the total endowment of good x is 2, and the total endowment of good y is 2 (hence the Edgeworth box is a square).

66 Microeconomics

Find the set of consumption efficient allocations when the consumers’ utility functions are:

uA(x A, yA) = min{xA, yA},

uB(x B, yB) = √xB yB.

Efficiency in production means that factor inputs are allocated to the production of goods so that the marginal rates of technical substitution between inputs in all lines of production are equal:

MRTSXLK = MRTS Y

LK.

If this condition did not hold it would be possible to reallocate inputs and thereby increase the output of goods (without using additional inputs).

The production contract curve shows the locus of tangencies between isoquants in a production box diagram. The information implicit in this curve can be used to derive a production possibilities frontier or transformation curve. This describes the locus of the maximum attainable production combinations: the maximum output of X attainable for any given output of Y, for given factor availability and technology. Together with individuals’ indifference mappings, the transformation curve can be used to explain the third condition for efficiency in general equilibrium, efficiency in product mix, which concerns the interface between consumption and production decisions.

Efficiency in product mix (or allocative efficiency) means that the rate at which producers can convert Y into X is equal to the rate at which consumers are willing to sacrifice Y for X in their consumption:

MRTXY = MRSXY.

Suppose MRSXY > MRTXY. This situation is inefficient because, in principle, it is possible to make at least one person better off without making anyone else worse off if resources are reallocated so that more X is produced and less Y is produced. Each of the three conditions discussed is necessary for an efficient allocation of resources in general equilibrium.

Welfare theorems Let us now introduce the two results that characterise the efficiency of the competitive general equilibrium, the two welfare theorems.

The first welfare theorem gives the conditions under which the competitive general equilibrium allocation is efficient. The theorem states:

First welfare theorem. If producers and consumers are price-takers and markets are complete the equilibrium allocation of resources is Pareto efficient.

Notice that the assumptions of the first welfare theorem are mild. Essentially we rule out markets in which firms (or consumers) are powerful enough to create distortions (monopolies, monopsonies, oligopolies, and so on), and situations where markets are not well-defined (for example, situations where we have public goods or externalities).

The textbook provides a sketch of the proof of this theorem. The mechanism through which the proof works is to recognise that in order to have optimality, consumers and producers observe the prices and adjust their consumption and production to them. Therefore, for example, each consumer will choose consumption such that his marginal rate of substitution is equal to the price ratio. But, since the price ratio that every consumer observes is the same, this induces consumption efficiency, because it implies that all the marginal rates of substitution are the same.

Chapter 11: General equilibrium and welfare economics

Prices here work, therefore, as a tool for decentralisation. They make all the agents in the economy converge to an efficient outcome without any need for a coordinated intervention. The power of this concept of equilibrium is that it simply needs each agent to maximise his own utility or profit individually given the observed price, and despite this full decentralisation it still leads to an efficient outcome (under the mentioned hypotheses).

As we said in the introductory part of this section, however, efficiency and social desirability may be very different concepts. Suppose that we are interested in a particular Pareto efficient allocation, and that we want to know whether it is implementable as a competitive equilibrium, through a redistribution of the initial endowments. The second welfare theorem gives us the conditions under which it is possible to implement every Pareto efficient allocation as a competitive general equilibrium.

Second welfare theorem. If all indifference curves and isoquants are convex to the origin (i.e. they satisfy the condition that the set of all weakly or strictly preferred points is a convex set), each Pareto efficient allocation can be implemented as a competitive equilibrium for a given initial distribution of resources.

This means that in principle efficiency and equity are issues that can be separated in the analysis of the equilibrium.

Notice that these two theorems are very important; you should know their assumptions and be able to understand when they can be applied and when the environment does not satisfy the conditions that they require.

Market failure and the theory of second best There are four reasons why markets may fail to work efficiently. First, firms or consumers may have market power in input or output markets. Second, consumers or producers may have incomplete information. Third, externalities may be present. Fourth, public goods may not be produced in sufficient quantities. The last three reasons are the subjects of the next two chapters and so explanations of these sources of market failure are deferred until then.

If a firm or firms that supply a product have market power then price exceeds marginal cost. Thus, if one good, say x, in a two-good general equilibrium model is monopolised, the equilibrium can be characterised as follows:

MCx MCy

MRTxy= = MRx py

< px py

= MRSxy .

Therefore, MRTXY < MRSXY and the condition for efficiency in the product mix is violated. The problem here is related to the fact that the monopolist’s marginal revenues are different from the price of good x, and therefore we lose the allocative efficiency. This implies that if the monopoly could somehow be abolished, the gain would be great enough to permit compensating the monopolist while still leaving something over for the rest of society.

The idea that there is an efficiency loss associated with monopoly has already been examined in a partial equilibrium context in Chapter 8. Various policy options were also considered. It was suggested, implicitly, that if policy intervention in any given market reduced price towards marginal cost, efficiency loss would be reduced. General equilibrium analysis, however, shows that the efficiency consequences of any

66 Microeconomics

measure that reduces the gap between price and marginal cost cannot be considered in isolation. To the extent that there are other markets in which price is not equal to marginal cost, and the goods in these markets are related, the overall efficiency impact depends on what is going on in all markets. Reducing the gap between price and marginal cost in just one market may (not necessarily will) increase efficiency loss not reduce it. This insight is called the theory of second best. It indicates that if at least one of the first best, that is, Pareto-efficient conditions are not met, then a second-best allocation can only be achieved by departing from all other first-best conditions. The implications for policies intended to correct market failures are unclear. Practitioners may simply assume that the amount of inter-relatedness between the market of their concern and other markets is sufficiently small that cross-effects can safely be ignored, but the reasonableness of this assumption cannot be taken for granted.

A reminder of your learning outcomes Having completed this chapter, and the Essential reading and activities, you should be able to:

• define the competitive general equilibrium

• analyse the competitive equilibrium in the case of a pure exchange economy

• define Pareto efficiency and apply it to a general equilibrium setting

• find the contract curve

• explain the welfare theorems

• discuss the theory of second best.

Sample examination questions

Section A

1. Consider a pure exchange economy where there are two consumption goods, pasta and bread, and two consumers, Alice and Bart. In the economy there are 2 units of bread and 2 units of pasta. Alice’s utility function is:

uA(p,b) = √pb and Bart’s utility function is:

uB(p,b) = p + b. Find the contract curve and draw it in the Edgeworth box that

corresponds to this economy.

2. Consider a given allocation of resources that is not Pareto efficient. Then, not all possible Pareto-efficient allocations in the economy are necessarily socially preferred to it. True, false or uncertain? Explain your answer.

3. Is the allocation of resources in a competitive equilibrium necessarily efficient?

Chapter 11: General equilibrium and welfare economics

Section B

1. Consider the following pure exchange economy with two agents, Karl and Jasmine, who consume only two goods, x and y. In the economy there are 4 units of x and 2 units of y. Karl is endowed with 1 unit of x and 1 unit of y, and Jasmine is endowed with 3 units of x and 1 unit of y; their utility functions are:

uJ(x J, yJ) = √x

Jy J

uK(x K, yK) = (x

K) 1/3( yK)

2/3,

a. Draw the Edgeworth box, showing the endowment point, and the indifference curves.

b. Find the contract curve.

c. Find Karl’s demand functions.

d. Find Jasmine’s demand functions.

e. Find the equilibrium price and allocations.

2. In an exchange economy, Grace has endowment (4,2) of two goods, and preferences uG( x

G , y

G) = xG(yG)2, while Madhav has endowment (4,4) and preferences uM(x

M, yM) = (x M)

5/7( yM) 2/7,

a. Find Grace’s demand for the two goods.

b. Find Madhav’s demand for the two goods.

c. Find the Walrasian-equilibrium prices and allocations.