econs 305 - general equilibrium part 2

EconS 305 - General Equilibrium Part 2

Eric Dunaway

Washington State University

[email protected]

October 20, 2015

Eric Dunaway (WSU) EconS 305 - Lecture 22 October 20, 2015 1 / 39

Introduction

Let�s continue our discussion from last time.

Recall that we described the Edgeworth box, Pareto e¢ ciency, and thecontract curve

Today, we will �nally �nd our equilibrium allocation and prices, thenlook at some extensions of the general equilibrium model.


General Equilibrium

Let�s do a quick review of what we have found so far.

We have consumers A and B with initial allocations of pizza and beerof

eA = (1, 5) eB = (7, 3)

and marginal rates of substitution

MRSA = �zAxA

MRSB = �zBxB

From these pieces of information, we were able to derive the contractcurve

zA = xA


General Equilibrium

xA

zA

0 1

5e

0

8

8 7

3

xB

zB


General Equilibrium

xA

zA

0 1

5e

UA

0

8

8 7

3

xB

zB

UB


General Equilibrium

xA

zA

0 1

5e

UA

0

8

8 7

3

xB

zB

UB

Contract Curve


General Equilibrium

Remember that our solution has to lay within the lense where eachconsumer reaches at least their original utility level from the initialendowment.

It also has to lay on the contract curve.

This allows us to restrict our possible solutions to a small segment ofthe contract curve that falls within the lens.


General Equilibrium

xA

zA

0 1

5e

UA

0

8

8 7

3

xB

zB

UB

PossibleSolutions


General Equilibrium

Let�s solve for the equilibrium!

We�ll need to use another trick from consumer theory.

Remember that when consumers optimize, they use their owntangency condition and a budget constraint.

Their tangency condition is exactly the same as before

MRS = MRT

We can substitute consumer A�s values in to obtain

� zAxA= �px

pz


General Equilibrium

zAxA=pxpz

We found the relationship between zA and xA from the contract curve

zA = xA

Substituting this into the tangency condition gives us something quiteuseful

zAz}|{xAxA

= 1 =pxpz

Thus,pxpz= 1 =) px = pz

and the price of good x has to equal the price of good z inequilibrium.


General Equilibrium

px = pz

Right here, we can make things easier by assuming the value of one ofthe prices.

Why? It has to do with the system of equations (which I will show atthe end). Basically, one of the equations is redundant, which makes itso we can�t actually solve for everything.In essence, there are an in�nite amount of price pairs that can solveour market, as long as the px = pz equation is satis�ed.Thus, we will set px = 1, which we call the numeraire. This turns theprice of good z into a relative price. In this case, good z is always asexpensive as good x , and thus pz = px = 1.


General Equilibrium

pz = px = 1

Now we have prices, and just need one more step to �nd ourallocations.

From the consumer choice problem, we have just the equations weneed: the budget constraints! For consumer A (the same thinghappens for consumer B), we have

pxxA + pzzA = YA

They�re a little bit di¤erent, though. We can actually substitute forincome in this case.


General Equilibrium

Rather than just having Income be exogenous (given from outside ofthe model), we can determine both consumers�incomes based ontheir initial endowments.

Basically, whatever their initial endowment is worth is their income.

Thus, consumer A�s income will be his initial endowment of pizza andbeer, multiplied by their respective prices (This converts them intotheir worth in money units). Since consumer A is endowed with oneunit of pizza and �ve units of beer, we have

eA = (1, 5)

YA = px (1) + pz (5)

And for consumer B, whose initial endowment is seven units of pizzaand three units of beer,

eB = (7, 3)

YB = px (7) + pz (3)


General Equilibrium

Updating out budget constraints,

pxxA + pzzA = 1px + 5pzpxxB + pzzB = 7px + 3pz

Just using consumer A�s budget constraint, remember that px = pz .Substituting,

pxxA + px zA = 1px + 5px

and all of the px terms cancel out

xA + zA = 1+ 5 = 6

Lastly, from the contract curve, we know that xA = zA, which we canagain substitute

xA + xA = 6

2xA = 6


General Equilibrium

2xA = 6

Dividing both sides by 2 gives our equilibrium allocation of good x forconsumer A,

x�A = 3

and from the contract curve.

z�A = x�A = 3

There are two ways to get consumer B�s values. We could either usetheir budget constraint or the resource constraints. I prefer using theresource constraints

xA + xB = 8

zA + zB = 8


General Equilibrium

x�A = 3 z�A = 3

xA + xB = 8

zA + zB = 8

Solving the resource constraints for xB and zB , respectively, yield

x�B = 8� x�A = 8� 3 = 5z�B = 8� z�A = 8� 3 = 5

And we can de�ne our equilibrium allocation as the following:

x�A = 3 z�A = 3

x�B = 5 z�B = 5

p�x = 1 (numeraire) p�z = 1


General Equilibrium

xA

zA

0 1

5e

UA

0

8

8 7

3

xB

zB

UB

3

3

5

5a*


General Equilibrium

As we can see in the �gure, consumer A traded two units of beer toconsumer B in exchange for two units of pizza.

They both consumed beer and pizza in equal amounts (This won�talways happen, but was a result of their utility functions).

In equilibrium, since the relative prices were equal, the goods weretraded one for one.


General Equilibrium

Pretty simple, right?

This is a lot of steps, and again, I don�t expect you to reproduce thisfor a quiz or exam.Know how to solve it graphically, though.

Did you know that you just solved a system of six equations and sixunknowns? Those equations were

MRSA = MRT

MRSB = MRT

pxxA + pzzA = YApxxB + pzzB = YB

xA + xB = 8

zA + zB = 8


General Equilibrium

Here is a step by step guide to solving these problems mathematically:

1. Use MRSA = MRSB and the resource constraints to �nd the contractcurve (like we did last time)

2. Use the contract curve and MRSA = MRT to solve for relative prices(remember to make one of them the numeraire).

3. Use the relative prices, contract curve, and consumer A�s budgetconstraint to solve for consumer A�s equilibrium allocation.

4. Use the resource constraints and consumer A�s equilibrium allocation tosolve for consumer B�s allocation.


General Equilibrium

Let�s set up one more that�s not symmetric.

Consider a market where consumers A and B both enjoy theconsumption of cheeseburgers (good x) and fries (good z).

Consumer A is endowed with 1 unit of cheeseburgers and 6 units offries while consumer B is endowed with 3 units of cheeseburgers and 2units of fries. Mathematically,

eA = (1, 6) eB = (3, 2)

Let�s draw the Edgeworth Box.


General Equilibrium

eA = (1, 6) eB = (3, 2)

For the width (horizontal axis) of the box, we sum up the totalendowment of good x .

x = 1+ 3 = 4

Likewise, for the height (vertical axis) of the box, we sum up the totalendowment of good z .

z = 6+ 2 = 8

Thus, our Edgeworth box should have a width of 4 and a height of 8.


General Equilibrium

xA

zA

xB

zB

0

04

4

8

8


General Equilibrium

xA

zA

xB

zB

0

04

4

8

8

e


General Equilibrium

From here, as long as we specify either utility functions or marginalrates of substitution, we can follow the steps from before to solve forthe equilibrium values.

You�ll be solving this one for homework.

For a quiz or exam, I would expect you to be able to draw theprevious Edgeworth box, put a few general utility functions on it aswell as a contract curve, then tell me a few things about theequilibrium allocation.

I might ask you to derive the contract curve. That�s not so bad.


General Equilibrium

What we just solved is known as a pure exchange model.

This is because the consumers were endowed with the goods, then theytrade among themselves to �nd an e¢ cient allocation.

There are many ways to extend the general equilibrium model.

The more factors we include in the model, the closer it gets tomodelling reality.Let�s look at two extensions.


General Equilibrium

We can add perfectly competitive �rms into our model. This changesseveral factors in the model.

First of all, instead of being endowed with the consumption goods, theconsumers can either be endowed with an input good, or labor andcapital.The budget constraints will now include wages paid to the consumers,rents from their capital, and even pro�ts if they own the �rms and the�rms manage to make economic pro�ts.The resource constraints now include the output from the �rms, as theconsumption goods are created by the �rms.Finally, we have to add more equations for the �rm�s choice problem,since they will choose capital and labor to maximize their pro�ts. Thisallows us to �nd equilibrium values for wages and rents, too.


General Equilibrium

It gets a lot more complicated.

The bottom line is that while we have added a couple of newequations, and changed a few others, it�s still just algebra once we havespeci�ed all of our equations.It might take us a lot of work to �gure out our equilibrium allocations,but it�s very doable.


General Equilibrium

We can also add a time factor to the model, letting consumers tradeover numerous periods.

These are commonly used in macroeconomics.Interestingly enough, we have to make assumptions on population sizeand how long consumers live.Sometimes, we let consumers live forever (it makes the math easier).Sometimes, we let them live for 2-3 periods, then die, getting replacedby new consumers.


General Equilibrium

We can even do both �rms and numerous periods at the same time.

This is actually how modern economic growth models were developed.When economists do work for the government, they usually choose amodel similar to this, add some policy changes (like taxes), thenanalyze what happens to the model in the long run as a kind ofsimulation.Interestingly enough, the di¤erence in opinions of the two main schoolsof macroeconomics all come down to assumptions that they makeabout these models.

We�ll save discussion of those for a macro class, though.


General Equilibrium

We can keep adding more and more factors to our models to getthem closer and closer to reality.

Things like human capital, labor/leisure decisions, random shocks,technology, etc.

The bottom line is that even though we add more and more layers ofcomplexity, the same rules as before govern how we �nd our solutions.


The Welfare Theorems

All of the work in General Equilibrium led to two very importantresults, which we call the Fundamental Theorems of WelfareEconomics.

I want to discuss them brie�y, because they both get applied to policyall the time!



The First Welfare Theorem states that any equilibrium allocationwe �nd in our model is Pareto e¢ cient.

This should be pretty obvious, since we got out equilibrium point byusing the contract curve, which comprises all of the Pareto e¢ cientequilibria.

This theorem is where people get the idea that "the free marketworks."

It�s also used as an argument against regulation, as it states that if themarket it left to its own methods, an e¢ cient equilibrium will emerge.



Why regulate at all then? Two reasons:

The First Welfare Theorem has all of the assumptions of a perfectlycompetitive market. We talked about these assumptions already andknow that they may not hold.Both welfare theorems break down in the presence of externalities.

Externalities are things that a¤ect you, but are controlled by otherpeople, like garbage, pollution, vaccinations, etc.When externalities are present, individual actions may not lead to asocially optimal outcome. This factoid is conveniently forgotten bymany free market disciples.



The Second Welfare Theorem states that we can move from onePareto e¢ cient allocation to whichever other one we want to bytransfering wealth from one consumer to another.

This one is a bit more complicated.Basically, if we have an equilibrium allocation that isn�t seen as "fair,"a regulator (government, etc.) can cause us to move to a more "fair"equilibrium allocation by taking money from one consumer and givingit to another.



Where do we see the second welfare theorem?

Social Security: Old people receive money from young people in orderto let them consume more than they normally would.Food Stamps: Poor people receive food vouchers from rich people.Basically, any type of government transfer is an application of thesecond welfare theorem.


Summary

Economists love general equilibrium.

It allows us to model complex economic behavior in a verymathematical context.

Several modern macroeconomic models are derived from the generalequilibrium framework.


Preview of Friday

Monopoly!

Not the game!Perlo¤, Chapter 11

We are done with the core microeconomic theory. Now, we�re movingon to small units that we are more likely to see in the real world.


Assignment 5-1 and 5-2 (1 of 1)

1. Consider a market where consumers A and B both enjoy theconsumption of cheeseburgers (good x) and fries (good z).

Consumer A is endowed with 1 unit of cheeseburgers and 6 units offries while consumer B is endowed with 3 units of cheeseburgers and 2units of fries. Mathematically,

eA = (1, 6) eB = (3, 2)

The marginal rates of substitution are as follows:

MRSA = �2zAxA

MRSB = �zBxB

a. Draw the Edgeworth Box. Choose and draw a general utility function(you know the shape) for each consumer, and draw a guess for thecontract curve (a straight line is �ne). Indicate all possible equilibra.

b. Derive the contract curve. Show your work, but you should �nd

zA =8xA8� xA


econs 305 - general equilibrium part 2

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