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Slide 1

General Equilibrium

We will talk about Pareto Efficiency The Free-Market Argument: Competitive

Markets are Efficient Market Failure: An introduction.

Slide 2

The Free-Market Argument

We will discuss the conditions for Pareto efficiency.

We will show that competitive markets are Pareto efficient, given certain assumptions.

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Slide 3

Examples of GE models

We will mostly work with simple examples of general equilibrium models.

You already know the exchange economy in the Edgeworth box.

We will talk about the so-called Robinson Crusoe economy: One input, one consumer.

And the 2x2 production model with two consumers: 2 goods, 2 inputs, 2 consumers

Slide 4

Example of Robinson Crusoe Economy with linear PPF Robinson Crusoe can spend the whole day

letting sun shine on his belly (leisure) or he can gather coconuts. Crusoe can gather 3 coconuts per hour. Crusoes utility function over coconuts and leisure is given by U(c,l)=c1/3l2/3.

Suppose Crusoe has 12 hours a day to spend on leisure and gathering coconuts.

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Slide 5

Problem Crusoe must solve:

How to allocate the total amount of time (resource) between gathering coconuts and leisure so to maximize his utility?

We can do this in two steps: 1. What are the production efficient allocations of

Crusoes time? 2. Once we have production efficiency, what is the

utility maximizing amount of coconuts and leisure?

Slide 6

Production Efficiency Robinson can spend an hour of leisure or he can

spend an hour gathering 3 coconuts. Since he has 12 hours available per day, he could

spend 12 hours of leisure and zero hours of gathering coconuts. This is production efficient, because he spends every hour available to him to produce the maximum amount of leisure.

Similarly he could spend all his time gathering coconuts and then ending up with 36 coconuts per day. Again this would be production efficient.

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Slide 7

Production Efficiency Contd How can Robinson efficiently produce a

combination of leisure and coconuts? As Crusoe increases the production of coconuts by

1 coconut at a time, he sacrifices 1/3 hr or 20 minutes of leisure for each additional coconut. Therefore 1 coconut and 12-1/3 hrs of leisure are another point on the PPF. So are 2 coconuts and 12-2/3hrs of leisure; 3 coconuts and 12- 3/3hrs of leisure and so on and so forth.

Thus, his production possibility frontier would look like this:

Slide 8

Crusoes Production Possibility Frontier

coconuts

leisure

PPF: l = 12 (1/3)*c

1/3

12

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The slope of the PPF is equal to the opportunity cost of coconuts

The slope of the PPF is also called the marginal rate of transformation (MRT leisure for coconut): in order to produce one additional coconut Crusoe sacrifices 1/3 hr of leisure.

Production Possibility Set

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Slide 9

Crusoes Pareto efficient allocation

coconuts

leisure

1/3

12

36

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12

Utility for Robinson increases

Production possibility set in Crusoes problem plays a role similar to the budget set in the consumers problem.

Slide 10

Pareto Efficiency

maxU(c, l) s.t. l = 12 13cIntertiorsolution :(1) MRSleisure for coconut = MRTleisure for coconut==>

l2c =

13

(2) Need to be on PPF : l = 12 13c

12 13c

2c =13

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Slide 11

Pareto Efficient Allocation

Slide 12

Robinson Crusoe Economy with concave PPF

Suppose Crusoe does not care for leisure anymore. Instead he enjoys fish and coconuts: U(f,c)=f1/3c2/3. He can gather coconuts according to the production function c=lc1/2 and he can catch fish according to f =2 lf1/2 where lc denotes the amount of time he devotes gathering coconuts and lf denotes the time he spends catching fish. Overall, he has 12 hours he can use to pursue these activities.

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Slide 13

How can we find the PPF now?

The production possibility frontier tells us which combinations of fish and coconuts are production efficient, that is, how we can use Crusoes total time in such a way as to maximize the amount of fish given that we also want to produce a certain amount of coconuts.

Finding the PPF

Slide 14

max f lf( ) s.t.c lc( ) = c and lf + lc = 12maxl f 2l f1/ 2 + 12 lf( )

1/ 2 c

Interior solution :1lf1/ 2

2 12 lf( )1/ 2= 0

c = 12 l f( )1/ 2

lf = 12 c 2, =2c12 c 2

= dfdc

PPF : f (c) = 2 12 c 2

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Slide 15

Finding the PPF mathematically Intuition

Remember we want to find a relationship between fish and coconuts: If we produce that many coconuts, what is the maximum amount of fish we can produce?

E.g. produce 2 coconuts. Since c = lc1/2 , the amount of time required to produce a certain amount of coconuts is given by lc=c2. Thus, 4 hrs are needed to produce 2 coconuts.

Whats the maximum amount of fish Crusoe can produce given that he produces 2 coconuts?

Crusoe has 8 hrs left for catching fish. Since f = 2 lf1/2 , Crusoe can catch 4*21/2 fish when he wants to produce 2 coconuts also.

Repeat with various amounts of coconuts, and you get the whole PPF.

Or, more generally, use amount of coconuts as a parameter to find out how many fish Crusoe can produce for any given number of coconuts.

Slide 16

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Given amount of coconuts requires lc=c2 units of time.

Since lf = 12 lc , time left to produce fish is lf =12 - c2.

Given this amount of time to catch fish, amount of fish caught is f = 2 lf1/2 = 2 (12 - c2)1/2 .

We now can relate any produced amount of coconuts to a maximum amount of fish: PPF is f = 2 (12 - c2)1/2 .

The PPF looks likes this: Slide 17

Slide 18

Concave PPF

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Slide 19

Pareto Efficient Allocation

Robinsons utility increases

Slide 20

Pareto Efficient Allocation In order to find the Pareto efficient

allocation, we need to set

Also need to make sure that we really are on the PPF, hence fish is expressed as a function of coconuts in MRS.

MRT is equal to absolute value of slope of PPF

To find slope of PPF, take derivative of equation describing PPF with respect to c.

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Slide 21

Summing Up

We have looked at a very simple economy with production.

We have learned how to find the PPF if there is only one input.

We have learned how to find the Pareto efficient interior allocation of goods if there is only one consumer: (1) MRS = MRT and (2) need to be on PPF.

In order to determine what is Pareto efficient we need production efficiency. Production efficiency is a necessary condition for Pareto efficiency.

Slide 22

Assumptions for the 2x2 production model Compared to our previous model, we now add one

more production factor (capital) and allow for more than one consumer.

Two inputs: Capital K, Labor L Two consumption goods: good X, good Y Consumers: strictly convex indifference curves

(decreasing MRS between goods). Producers: strictly convex isoquants (decreasing

marginal rate of technical substitution between inputs). We also assume that production of the two goods does not exhibit increasing returns to scale.

Only two people: Person 1 and person 2.

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Slide 23

Problems the economy must solve: How to allocate the existing stock of capital

and labor efficiently between the production of good X and the production of good Y.

How to distribute these goods efficiently among the population once they are produced.

Slide 24

Consumption Efficiency You already know consumption efficiency

from the exchange economy. A distribution of goods is consumption

efficient if it is not possible to reallocate these goods and make at least one person in the economy better off without making someone else worse off.

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Slide 25

Production Efficiency

An allocation of inputs (K and L) is production efficient if it is not possible to reallocate these inputs and produce more of at least one good in the economy without decreasing the amount of some other good that is produced.

Slide 26

Where do the fixed amounts of labor and capital come from? They are determined by past actions of the society:

The amount of labor in the economy is determined by previous amounts of births and deaths in the population,

and the size of the capital stock is determined by previous amounts of investment and depreciation.

In any case, we consider the existing labor force and capital stock as fixed elements that will not be affected by anything we do in our analysis.

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Slide 27

Production Efficiency in the 2x2 production model Use the Edgeworth Box to depict the situation of

the production of two goods with capital and labor.

Instead of consumer 1, we put good X on the lower left hand corner. Instead of consumer 2, we put good Y on the upper right hand corner.

The Edgeworth Box depicts any allocation of labor and capital between the production of the two goods, so that the labor inputs in the production of both goods sum up to always the same total amount. The same is true for capital.

Slide 28

Edgeworth Box of Production

Good X

capital

labor

Good Y

Amount of labor devoted to production of good X

Amount of capital devoted to production of good X

Isoquants of X

Isoquants of Y

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Slide 29

Edgeworth Box of Production

Good X

capital

labor

Good Y

Point A is not production efficient.

Moving to e.g. this point from point A we can produce more of both goods.

A

Slide 30

Edgeworth Box of Production

Good X

capital

labor

Good Y

This point is production efficient: in order to produce more of X we have to reduce production of Y and vice versa.

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Slide 31

How much labor, how much capital should be used in the production of both goods?

Analogy to the consumption efficiency. Noting that the slope of the isoquants is the

marginal rate of technical substitution of capital for labor, interior production efficiency occurs when MRTSXK for L=MRTSYK for L.

Production Efficiency

Slide 32

maxL1 ,K1 ,

f1 L1,K1( ) + f2 L L1,K K1( ) y2( )Interiorsolution :f1L1

f1L2

= 0

f1K1

f1K2

= 0

y2 = f2 L L1,K K1( )

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Slide 33

Condition for Interior Production Efficiency A given set of inputs available in an

economy should be allocated across a set of producers until the marginal rate of technical substitution for each pair of inputs is equal for each producer.

Slide 34

Product Mix Efficiency

Which combination of goods will give us Pareto-efficiency?

We can have efficiency in production and in consumption, and yet there is still room for a Pareto improvement, because we are producing too much of one good and not enough of the other.

Product mix efficiency puts together both sides, consumers and producers.

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Slide 35

The Production Possibility Frontier with 2 Inputs

While the set of Pareto efficient allocations in the Edgeworth box of production makes all the efficient input combinations visible, the PPF gives us all the combinations of goods that are production efficient.

The PPF is the value function of the problem that gives us production efficient allocations as its solution.

Slide 36

From Edgeworth Box of Production to PPF

Good X Good Y

capital

labor

X=30, Y=10

X=24, Y=20 X=12,

Y=28

Good X

PPF

Good Y

28

20

10

12 24 30

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Slide 37

The Slope of the PPF

Marginal rate of transformation (MRT) of good Y into good X, indicates how many units of good Y the economy would have to sacrifice (by transferring inputs from the production of good Y to the production of good X) in order to produce 1 more unit of good X.

Slide 38

Putting Production and Consumption Together Graphically

We can draw an Edgeworth box from each of the points on the PPF and identify the consumption efficient allocations for each combination of goods.

The MRS typically changes along the set of Pareto efficient allocations. The selected point on the PPF is product mix efficient if MRS=MRT.

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Slide 39

Pareto Efficiency in Graph

PPF

Keep person 1 at this utility level.

Person 2s highest utility given X and Y

X,Y

Good X

Good Y

The green point gives us an allocation of goods that satisfies production efficiency and consumption efficiency, but does it satisfy product mix efficiency?

Slide 40

Pareto Efficiency in Graph

PPF

Keep person 1 at this utility level.

Person 2s highest utility given X and Y

X,Y

Good X

Good Y With X and Y, we have found a point on the PPF that allows us to increase person 2s utility without making person 1 worse off as compared to X and Y. Do we have production efficiency now?

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Slide 41

Pareto Efficiency in Graph

PPF

Keep person 1 at this utility level.

Person 2s highest utility given any X and any Y.

X,Y

X,Y

Good X

Good Y By shifting to yet another point on the PPF, we can again increase person 2s utility without making person 1 worse off.

Slide 42

Pareto Efficiency in Graph

Keep person 1 at this utility level.

Person 2s highest utility given any X and any Y.

Slope of PPF less steep than slope of indifference curves

Slope of PPF steeper than slope of indifference curves

Slope of PPF same as slope of indifference curves

Good X

Good Y

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Slide 43

Suppose MRSY for X>MRTY for X. To be more specific assume that MRSY for X=2 and

MRTY for X=1. If we want to produce one more unit of good X,

we have to give up one unit of good Y. Both consumers are the same off if they give up

two units of good Y and receive one more unit of good X.

But this means that we will make both of them better off by producing two more units of good X and sacrificing 2 units of good Y!

Only if MRS=MRT is such a reallocation not possible.

Solving for Pareto Efficient Allocation

Slide 44

maxu1 x1,y1( )s.t. u2 x2,y2( ) = u2s.t. x1 + x2 = fx Lx,Kx( )s.t. y1 + y2 = fy Ly,Ky( )s.t. Lx + Ly = Ls.t. Kx + Ky = K

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Using PPF, the problem is

Slide 45

maxu1 x1,y1( )s.t. u2 x2,y2( ) = u2s.t. x1 + x2 = xs.t. y1 + y2 = y x( )maxx1 ,y1 ,x,

u1 x1,y1( ) + u2 x x1,y x( ) y1( ) u2( )

Slide 46

Exercise Suppose there are 2 consumers, Ara and Bahar,

and two goods, good X and good Y. Aras utility function is given by UA=XY and Bahars utility function is given by UB=X1/4Y3/4. Both goods are produced with labor and capital and good Xs production function is fX=(KXLX)1/2 while good Ys production function is fY=(KYLY)1/2. Suppose there are 8 units of labor and 8 units of capital in this economy.

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Slide 47

Exercise Contd

Suppose Ara receives one unit of good X. How many units of good Y must Ara receive and how many units of good X and good Y must Bahar receive for the allocation to be Pareto efficient?

Slide 48

Answer

Production efficiency: We have 8 units of labor and eight units of capital. If for both goods the capital labor ratio is equal to one, then KX/LX=1= MPLX/MPKX and KY/LY=1= MPLY/MPKY so

MRTSX capital for labor = MRTSY capital for labor Moreover PPF: Y= 8 X.

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Slide 49

Answer Contd

Next we need to ensure that consumption efficiency and product mix efficiency are satisfied.

For consumption efficiency MRSAra Y for X = MRSBahar Y for X We know that XB =X 1, YA + YB = 8-X YA /XA=YB /3XB

Slide 50

Answer Contd

For product mix efficiency MRS = MRT, YA /XA=YB /3XB = 1. The allocation KX=LX=2.5, KY=LY=5.5,

XA=1 , XB=1.5 , YA=1, YB= 4.5 is Pareto efficient.

We have shown that with this allocation of resources and consumption goods all three conditions of Pareto efficiency are satisfied.

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Slide 51

Competitive Equilibrium

Now that we understand Pareto efficiency in a general equilibrium model with production, we want to find out if the competitive equilibrium in this model is Pareto efficient.

First discuss what happens in the competitive equilibrium, then determine whether its Pareto efficient.

Slide 52

Perfectly Competitive Markets Satisfy the Conditions for Pareto Efficiency

Consumers maximize utility taking prices as given. From utility max MRS=price ratio. If consumers are price takers, they all face the

same prices and therefore all their marginal rates of substitution have to be equal.

The competitive equilibrium satisfies consumption efficiency.

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Slide 53

Perfectly Competitive Markets Satisfy the Conditions for Pareto Efficiency

Firms maximize profits and hence minimize costs. From profit maximization, MRTS=factor price ratio (because cost minimization is a necessary condition for profit maximization). If all firms are price takers in the factor

markets, then all firms have equal MRTS. The competitive equilibrium satisfies

production efficiency.

Slide 54

Perfectly Competitive Markets Satisfy the Conditions for Pareto Efficiency

Firms maximize profits. From profit maximization of competitive firms (if the number of firms is sufficiently large) P=MC. For any two goods, consumers set MRS=pX/pY. From profit max a firm produces an amount of X where

pX=MCX, and a firm produces an amount of Y where pY=MCY.

This implies pX/pY=MCX/MCY, but MCX/MCY=MRT and therefore MRS=MRT.

The competitive equilibrium satisfies product mix efficiency.

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Slide 55

The Two Fundamental Theorems of Welfare Economics

The First (FFTW): The competitive equilibrium is Pareto efficient.

The Second (SFTW): Any Pareto efficient allocation can be achieved by the competitive equilibrium with the appropriate redistribution of initial endowments.

Slide 56

Market Failure and Redistribution

Market failure addresses the issue when the competitive equilibrium is not Pareto efficient. Takes FFTW as starting point.

Redistribution (interventionist argument) is unhappy with the distribution of goods when economy is left to market forces alone. Takes SFTW as starting point.

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Slide 57

The Interventionist Argument

The Second Fundamental Theorem of Welfare Economics is not very useful. It requires that income be redistributed in such a way, that people and firms cannot change their behavior in order to receive more transfers/pay less taxes (only lump-sum taxes and transfers allowed). This is next to impossible in the real world.

We then have to trade off equity for efficiency.

Slide 58

Reading Suggestion

Amarty Sen, one of the Nobel Prize winners in Economics, has an article that examines the first and second fundamental theorems of welfare economics. The title is The Moral Standing of the Market, in Ellen Frankel Paul, Fred D. Miller, Jr., and Jeffrey Paul (eds) (1985), Ethics & Economics, pp.1-19, Basil Blackwell Publisher Limited.

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Slide 59

FFTW and Market Failure

Rather than equity considerations, this approach questions whether the competitive equilibrium is Pareto efficient.

We made several assumptions in order to show that the competitive equilibrium is Pareto efficient. If these assumptions do not hold in the real world, competitive markets are not efficient.

Slide 60

Assumptions in FFTW Consumers and Producers are price takers. If not,

people have market power (e.g. oligopoly, monopoly, monopsony).

Producers face DRS or CRS technology. If not, natural monopoly.

Symmetric information. If not, people can take advantage of their superior knowledge. Markets may not exist, or may not be efficient.

Private goods. If not, markets provide too little of public goods and goods with positive externalities and too much of goods with negative externalities.