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General Equilibrium

Part 1. Why GE?

Economics 313

Why General Equilibrium?

Slide 2

Equilibriums are states of rest: sets of endogenous variables such that the forces in the model balance, and there is no tendency to “change”. What about dynamic models??? It’s all good.

“Partial equilibrium” is a modeling approach in which these dependencies are ignored. We study equilibrium in an “isolated” market that is isolated by assumption, made explicit by the saying “everything else held constant.”

Markets do not exist in isolation. Instead every market is embedded in a system of markets. The strength of the connections between any two markets these depends on many factors. For example, consumer preferences and firm technologies depend on the prices of other goods and factors of production.

Why General Equilibrium?

Slide 3

“General Equilibrium” builds on this by asking about equilibrium in the system as a whole.

Why do we do this? 1. Check to make sure that the “feedback” we held

constant doesn’t reverse the prediction of the partial equilibrium model. (Positive)

2. Ask questions about the efficiency of the market system as a whole (Normative)

Why General Equilibrium?

Slide 4

In the simple partial equilibrium model of a market there are a lot of possible “parameters,” some of can change (i.e. are variables). But with only a few equations, just a small set of variables can be “predicted” by the model (i.e. endogenous), with the rest held constant (i.e. exogenous)

=> remember, comparative statics asks what happens to the endogenous variables when an exogenous variable changes.

Why General Equilibrium?

Slide 5

General equilibrium is not a theory of everything. General equilibrium does try to make more of a model endogenous.

Macroeconomists almost always work in some kind of general equilibrium. The workhorse model in much modern macro is the “Dynamic Stochastic General Equilibrium” model (DSGE). What about the government?

General Equilibrium is often called GE, and usually this refers to static models of without uncertainty, designed to explore the properties of finding equilibrium across many competitive markets. Key questions: existence, uniqueness, and efficiency properties of equilibrium. This is our world in 313. Often only one period models.

Arrow-Debreu found an approach to combine uncertainty, many goods, and an infinite time horizon and compute and analyze competitive general equilibrium. Finance applications are significant.

Simplest example

Slide 6

Imagine a world with some consumers and firms that produce two goods, X and Y. One set of variables that might describe this world assumes every has the same preferences, say Cobb-Douglas (tastes)

𝑈𝑈 𝑋𝑋, 𝑌𝑌 = 𝑋𝑋𝛼𝛼𝑌𝑌𝛽𝛽

(Because consumers are identical, we don’t need notation to identify whose preferences there are. More generally we would.)

Similarly, assume that all the firms have the same technology for producing the goods, depending only on labour, with cost function given, respectively, by (technology)

x= 𝑤𝑤𝛾𝛾

y= 𝑤𝑤𝜃𝜃

Finally, there are the prices of the two goods that, in equilibrium, “clear the market” 𝑝𝑝𝑋𝑋 and 𝑝𝑝𝑌𝑌, and the income of consumers 𝑀𝑀.

Simplest example

Slide 7

Even this simple model has many variables: 𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝜃𝜃, 𝑤𝑤, 𝑋𝑋, 𝑌𝑌, , 𝑥𝑥, 𝑦𝑦, 𝑝𝑝𝑋𝑋, 𝑝𝑝𝑌𝑌, 𝑀𝑀

Equilibrium is found by assuming firms and consumers maximize profits and utility, and finding prices where supply equal to demand in all markets:

𝑋𝑋 = 𝑥𝑥 𝑌𝑌 = 𝑦𝑦

In the partial equilibrium approach we do this separately. for each market. In the market for X for example, the endogenous variables are 𝑋𝑋 and 𝑝𝑝𝑋𝑋. The exogenous variables are 𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝜃𝜃, 𝑤𝑤, 𝑦𝑦, 𝑝𝑝𝑌𝑌, 𝑀𝑀. Swapping 𝑋𝑋 for 𝑌𝑌 gives the partition for market Y.

Tastes and technology

Slide 8

These variables are not of equal status: as we saw the prices and quantities are determined by the partial equilibrium models. The taste and technology “variables”

𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝜃𝜃 are not usually “variable” at all, but fixed “parameters”. We don’t usually do comparative statics with them. In fact, markets and institutions do affect these, so we could model that formally. But in this course, we will always take these as given by the setting we are considering. (Usually, by introducing them with the words “Consider a market with…)

What about M and w?

Slide 9

The remaining two variables represent consumer income and the wage paid by firms.

These are clearly set by markets somewhere. The key question where? In the partial equilibrium approach, they are taken as fixed. Which means it can’t be the market for X or Y that determine these, or they would also be endogenous not exogenous.

There must be some other place that Consumers get income and Firms hire Workers. Income could come from wages, and profits.

The approach ahead.

Slide 10

In 313 we build increasingly elaborate models to “endogenize” more variables, where as mentioned tastes and technology are held constant (so can be called “parameters” rather than variables).

We start by thinking only of exchange in a two good, two person economy. Goods are not produced, but instead “endowed.” There is only one period, and the question is what constitutes and equilibrium and what are its efficiency properties.

Exogenous variables are (the parameters) and the endowments. Endogenous are income, prices and quantities consumed.

-- end of part 1 --

General Equilibrium

Part 2: Margins of Efficiency.

Economics 313

We Will Work With Simple Examples

Slide 13

 Mainly the 2x2 production model (2 goods and 2 inputs) with 2 consumers

 We will break this model down into three components:

I. Consumption (who gets what)

II. Production (who produces what)

III. Allocation (how much of what is produced)

Economic Efficiency

Slide 14

 Economic efficiency is about social surplus maximization:  This is our criteria to assess the desirability of economic

policy.

 A situation is efficient if there is no way to reorganize things such that at least one individual is made better off without any individual being made worse off.  Known as “Pareto” efficiency.

 A situation is inefficient if there is a way to reorganize things such that at least one individual is made better off without any individual being made worse off.

Economic Efficiency

Slide 15

 Pareto Improvement (PI) – possible to make at least one person better off without making anyone worse off

 If Pareto improvements are possible an outcome is not efficient

 Potential Pareto Improvement (PPI) - if it is possible for the winners to compensate the losers so that after the compensation, both are made better off.  A movement from an inefficient point to an efficient point is

always a potential Pareto improvement

Question

Slide 16

 The United States experiences economic growth. However only those with incomes at the very top of the distribution benefit. This is an example of

A. A Pareto Improvement B. A potential Pareto Improvement C. An inefficient outcome D. An efficient outcome E. Both A and B

Three Margins of Economic Efficiency

Slide 17

 Analyze efficiency with respect to each component of the model:

1. Consumption relates to Distributive efficiency  Given a fixed quantity of goods, who should get to consume

them?

2. Production relates to Productive efficiency  Given a fixed quantity of goods, who should we have produce

them?

3. Allocation relates to Allocative efficiency  What quantities of goods should be produced given a fixed

quantity of inputs

Economic Efficiency

Slide 18

 First we are going to examine what conditions must hold for each type of economic efficiency and why

 Then we are going to examine how and under what conditions we are lead to efficiency in each case

 Recall that perfect competition leads to efficient outcomes

Distributive Efficiency

Slide 19

Is a distribution like this efficient?

-- end of part 2 --

General Equilibrium

Part 3: Distributive Efficiency.

Economics 313

Where We Are Going

Slide 22

 We will begin with distributional efficiency

 The Edgeworth Box is the tool we used to examine this

 An allocation of goods among consumers satisfies distributive efficiency if we cannot reallocate the goods in such a way as to make at least one consumer better off without making another consumer worse off

Distributive Efficiency

Slide 23

2 goods, 𝑥𝑥 and 𝑦𝑦 (𝑥𝑥, 𝑦𝑦), and 2 consumers, 𝐴𝐴 and 𝐵𝐵 A has bundle (10, 5); B has bundle (5,10)

Their utility functions are respectively 𝑈𝑈𝐴𝐴 = 𝑥𝑥 + 𝑦𝑦 and 𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦

Is this an efficient way to distribute the goods 𝑥𝑥 and 𝑦𝑦?

Distributive Efficiency

Slide 24

 Given A’s Bundle (10,5): 𝑈𝑈𝐴𝐴 = 𝑥𝑥 + 𝑦𝑦 = 15  Given B’s Bundle (5,10): 𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦 = 50

Suppose we reallocate the one unit of x and one unit of y so that A has bundle (9, 6) & B has bundle (6, 9):

– UA = x + y = 9 + 6 = 15

– UB = x y = 6 × 9 = 54

⇒ Original allocation was inefficient.

𝑈𝑈𝐴𝐴 9,6 = 𝑈𝑈𝐴𝐴(10,5) 𝑈𝑈𝐵𝐵 6,9) > 𝑈𝑈𝐵𝐵10,5

Distributive Efficiency

Slide 25

 Not every re-allocation would be an improvement

 Suppose instead we had reallocated so that A had (11,4) and B had (4,11)

– UA = x + y = 11 + 4 = 15

– UB = x y = 4× 11 = 44

 A is neither worse nor better off but B is worse off.

Distributive Efficiency

Slide 26

 How did we know to which way to reallocate to get a PI ?

 From the utility function we can derive Marginal Rate of Substitution (MRS) which tells us how much the consumer values one good relative to the other good

 Recall that the absolute value of the MRS is equal to: 𝜕𝜕𝑈𝑈(𝑥𝑥, 𝑦𝑦)/𝜕𝜕𝑥𝑥 𝜕𝜕𝑈𝑈(𝑥𝑥, 𝑦𝑦)/𝜕𝜕𝑦𝑦

Concept Check

Slide 27

 If you are told that, given his current consumption bundle, a consumer’s MRS equals 4, which of the following statements is TRUE? (Assume good Y is on the vertical axis and X on the horizontal)

I. The consumer is willing to give up 4 y for one additional x II. The consumer is willing to give up ¼ x for one additional y III. The consumer is willing to give up ¼ y for one additional x

A. I only B. II only C. I and II D. III only E. All I, II and III

Concept Check

Slide 28

 If you are told that, given his current consumption bundle, a consumer’s MRS equals 4, which of the following statements is TRUE? (Assume good Y is on the vertical axis and X on the horizontal)

I. The consumer is willing to give up 4 y for one additional x II. The consumer is willing to give up ¼ x for one additional y III. The consumer is willing to give up ¼ y for one additional x

A. I only B. II only C. I and II D. III only E. All I, II and III

Distributive Efficiency

Slide 29

𝑈𝑈𝐴𝐴 = 𝑥𝑥 + 𝑦𝑦 → 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑥𝑥 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑦𝑦

= 1 1

= 1

𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦 → 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑥𝑥 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑦𝑦

= 𝑦𝑦 𝑥𝑥

 MRSA = 1 ⇒ A is willing to pay up to 1 unit of good y in order to receive an extra unit if good x.  Trading 1 y for 1 x will leave A’s utility unchanged. i.e., she is

indifferent between making or not making such a trade.  Trading (say) 0.5y for 1x will increase A’s utility. i.e., she would

strictly prefer making this trade to not making it.

Distributive Efficiency

Slide 30

𝑈𝑈𝐴𝐴 = 𝑥𝑥 + 𝑦𝑦 → 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑥𝑥 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑦𝑦

= 1 1

= 1

𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦 → 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑥𝑥 𝜕𝜕𝑈𝑈𝐴𝐴/𝜕𝜕𝑦𝑦

= 𝑦𝑦 𝑥𝑥

 MRSA always = 1 (a constant) in this case

 MRSB is a function of y and x; specifically, MRSB = y/x.

 So the rate at which B is willing to trade y for x depends on how many units of y and x she currently has.

Distributional Efficiency

Slide 31

 E.g.. If B has bundle (5,10), MRSB = 10/5 = 2, whereas if B has different bundle (10,5), MRSB = 5/10 = 0.5.

 If MRSB = 2 ⇒ B is willing to trade up to 2 units of y in order to receive one more unit of x.

 MRSB declines as x increases.  As B has more x, her willingness to trade y for x diminishes.

Distributional Efficiency

Slide 32

 Person A is willing to trade up to:  1y for 1 extra x; or 1x for 1 extra y

 Person B is willing to trade up to:  2y for 1 extra x; or 1/2 x for 1 extra y

 MRSB > MRSA ⇒ B values extra units of x more highly than A does so we should reallocate so that B has more x (and hence less y); and A has more y (and hence less x).

Trade

Slide 33

- 1 unit of good Y + 1 unit of good X and same off

-2 unit of good Y + 1 units of good X and same off

1 unit of good X

1.5 units of good Y

A B

After trade, both people are better off

Distributional Efficiency

Slide 34

 Given the reallocation of 1 unit more x for B, and one more unit of y for A, we have:  𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 1, given the bundle (9,6) and  𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 3

2 , given the bundle (6,9)

 Still true that 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 > 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴  We want to reallocate further, if feasible

 Tell us that if we reach the point where 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵, then no further reallocation is possible such that at least one person is better off and no person is worse off

 MRSA = MRSB ⇒ each consumer values extra units of good x the same (where that value is measured in units of good y).

-- end of part 3 --

General Equilibrium

Part 4: The Edgeworth Box

Economics 313

Distributional Efficiency

Slide 37

 A graphical tool we use to think about distributional efficiency is known as the Edgeworth box

 Edgeworth exchange box: a diagram used to analyze the general equilibrium of an exchange economy

 It assumes that there is no physical waste. Allocations for consumption exhaust the resources in the economy

 Imagine we have two consumers. Consumer’s A and B. Also, there are two goods, x and y.

Edgeworth Box

A’s possible quantity of x

10 15

There are 15 units of each good available in our example. A has 10 units of x and 5 units of y

A ’s

p os

si bl

e qu

an tit

y of

Edgeworth Box

B’s possible quantity of x

There are 15 units of each good available in our example. Thus B must have 10 y and 5 x, if A has 5 y and 10 x.

B ’s

p os

si bl

e qu

an tit

y of

Edgeworth Box

Note that here we measure good x horizontally and good y vertically.

Edgeworth Box

A has all the y

B has all the x

A has all the x

Edgeworth Box

B has nothing and A has all

A’s has nothing and B has all

Edgeworth Box

B’s quantity of x

A’s quantity of x

So A’s x is increasing as we move from left to right; and A’s y is increasing as we move from lower points to higher points

Edgeworth Box

B’s quantity of x

A’s quantity of x

In contrast, xB↑ as we move from right to left and yB↑ as we move from upper to lower points.

(means that we have to turn ourselves upside down to look at person B)

Edgeworth Box

B’s quantity of x

A’s quantity of x

In our example, at the original allocation A had the bundle (10,5) and B had the bundle (5,10). These two bundles are illustrated by one point in the EB. (point Z). This is known as the endowment point.

Edgeworth Box

B’s quantity of x

A’s quantity of x

We can see that this allocation is inefficient by drawing the indifference curves for each consumer through the allocation.

Edgeworth Box

B’s quantity of x

A’s quantity of x

The MRS is a measure of the slope of the Indifference curves.

We know that at point Z:

𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 1; and 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 2

So we know that A’s IC through point Z is flatter than B’s through point Z.

Edgeworth Box

B’s quantity of x

A’s quantity of x

Begin with A: 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 1

Edgeworth Box

B’s quantity of x

A’s quantity of x

Begin with A: 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 1

And we also know that A’s IC is a straight line (constant MRSA)

ICA

Edgeworth Box

B’s quantity of x

A’s quantity of x

Now Consider B: 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 2

ICA

Edgeworth Box

B’s quantity of x

A’s quantity of x

Now Consider B: 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 2

We also know that B’s 𝑀𝑀𝑀𝑀𝑆𝑆 is not constant and decreases with increasing x

Make sure you can see why B’s MRS is decreasing in this diagram.

ICA

ICB

-- end of part 4 --

General Equilibrium

Part 5: Exchange Equilibrium

Economics 313

The Propensity to Truck and Barter

As we saw before, the allocation of (10, 5) for A and (5,9) for B leaves “surplus on the table”

That is, we constructed a one for one trade that was a PI.

There is nothing (yet) in our model that restricts the ability to trade, no information problems, no cost of trade, no laws against trade, so it seems unlikely that this initial “endowment” allocation would survive the desire for gain.

What must be true about an equilibrium?

Edgeworth Box

Slide 55

B’s quantity of x

A’s quantity of x

ICA

ICB

Consider point H.

Would A trade her initial allocation for H?

Would B?

Results:  Result 1: Equilibrium allocation must be individual

rational allocation  Individually Rational (IR) as they rely only on the choice for

the individual: these are trades an agent would agree to because they do not reduce utility.

 Allocations that are Individually Rational for both agents are in the lens

 These are all the allocations that is as good or better than the initial allocation for both agents

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

In this “lens”, both are better off

On border of the lens, one consumer is better off while the other is no worse off.

The region is a function of the initial allocation of the goods.

Edgeworth Box

Slide 58

B’s quantity of x

A’s quantity of x

ICA

ICB

Region of individually rational trades and Pareto improvements

Are all the allocations in the lens Pareto Efficient?

Edgeworth Box

Slide 59

B’s quantity of x

A’s quantity of x

ICA

ICB

Efficient allocations occur when no more Pareto improvements are possible so, in this example, 𝐼𝐼𝐶𝐶𝐵𝐵 must be tangent to 𝐼𝐼𝐶𝐶𝐴𝐴

for an allocation to be efficient.

𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵

Results  Result 1: Equilibrium allocation must be individual rational

allocation

 Result 2: Equilibrium allocations must be (Pareto) efficient  (Pareto) efficient allocation or Pareto-optimal allocation:

allocation such that it is not possible to reallocate goods such that all agents are better off

 It is easy to see that not all IR points are Pareto Efficient. Even when a trade is IR for both agents (i.e. is in the lens).

 What about the reverse? Can allocations be Pareto efficient but not IR for both agents?

Results  Result 1: Equilibrium allocation must be individual rational allocation

 Result 2: Equilibrium allocations must be (Pareto) efficient  (Pareto) efficient allocation or Pareto-optimal allocation: allocation

such that it is not possible to reallocate goods such that all agents are better off

 Yes: Spotting Pareto efficient allocations…  For ‘nice’ utility functions  Indifference curves (IC’s) tangent  IC’s tangent  MRSAmy = MRSBekayla

 Not just one possible efficient allocation but a set of efficient points: this set is called the contract curve (careful about definitions: this term is sometimes defined differently).

Edgeworth Box

Slide 62

 Recall we found that the points the obey 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵, are efficient

 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 → 𝑦𝑦𝐵𝐵 𝑥𝑥𝐵𝐵

= 1

 𝑦𝑦𝐵𝐵 = 𝑥𝑥𝐵𝐵

 Tells us that the “contract curve (CC)” – the set of Pareto efficient allocations - is the set of all points such that B’s consumption of x = B’s consumption of y.

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

Tells us that the CC is a straight line and runs from B’s origin to A’s origin.

Contract Curve

Edgeworth Box

A’s quantity of x

ICA

ICB

Core if initial allocation z

Core if initial allocation G

Results  Result 1: Equilibrium allocation must be individual rational

allocation

 Result 2: Equilibrium allocations must be (Pareto) efficient

 Result 3: Equilibrium allocation must be in core (of the economy)  Core: set of allocations that are IR and on the contract

curve  The equilibrium is not necessarily unique

Result 3 is all we can say with two agents  Actual equilibrium allocation depends on relative

“bargaining power” of A and B

-- end of part 5 --

General Equilibrium

Part 6: Potential Pareto Improvements

Economics 313

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

Diagram makes a very important point.

Every movement from a point not on the CC to a point on the CC is a move from an inefficient point to an efficient point. .

L ICA

ICB

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

BUT, not every movement from an inefficient point to an efficient point is a PI.

L ICA

ICB

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

We call the move Z to L a Potential Pareto Improvement (PPI) A gains x and y at B’s expense ⇒ NOT a PI

L ICA

ICB

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

But, in principle, A could compensate B for losses such that both of them are better off than at the starting point Z

L ICA

ICB

Edgeworth Box

B’s quantity of x

A’s quantity of x

ICA

ICB

Where there exists the possibility for such compensation, we say that the move from Z to L represents a PPI. Compensation does NOT need to take place for move from Z to L to be a PPI. Just need the “potential” for such a transfer.

L ICA

ICB

Distributive Efficiency: Policy Analysis

Slide 73

 At this point you might wonder why are we even talking about point L at all. Surely the compensation will happen, right?  Why not cut straight to the chase and only talk about

movements from Z to M?

 Throughout the term we are going to identify market outcomes that are not efficient  That is, points such as Z

 We will see that in such cases it is often relatively easy to design policies that move us from inefficiency to efficiency

 In the vast majority of cases policies will NOT be PIs, but PPIs  They will be moves such as the one from K to M

Concept Check

Slide 74

 Which of the following statements is false A. Point G is efficient B. Point F is efficient C. Moving from H to G is a PI D. Moving from H to F is a PPI E. Moving from H to F is a PI

Concept Check

Slide 75

 Which of the following statements is false A. Point G is efficient B. Point F is efficient C. Moving from H to G is a PI D. Moving from H to F is a PPI E. Moving from H to F is a PI

-- end of part 6 --

General Equilibrium

Part 7: MRS in a Pareto Efficient Allocation

Economics 313

Question

Slide 78

 Will there be efficient outcomes where 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 ≠ 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵? A. Yes B. No C. Maybe

Distributive Efficiency

Slide 79

 Note that efficiency is not always where 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵

 It is possible for efficient allocations where 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 ≠ 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵

 In these cases, simple calculus needs some help us to find the CC

 Recall conceptually what the math and pictures represent

Distributive Efficiency

Slide 80

 Recall that we want to reallocate such that the consumer with the higher 𝑀𝑀𝑀𝑀𝑆𝑆𝑦𝑦 𝑓𝑓𝑓𝑓𝑓𝑓 𝑥𝑥 has more x

 This consumer values 𝑥𝑥 more on the margin than the other consumer (willing to give up more 𝑦𝑦 than 𝑥𝑥).

 Note that this involves the consumers swapping units of x for units of y.

 This is only feasible if, in fact each consumer has something to swap (what if the consumers with the higher 𝑀𝑀𝑀𝑀𝑆𝑆 has no 𝑦𝑦 left to swap?)

Distributive Efficiency

Slide 81

 For instance, preferences are the same as in our previous example is the same, but we have 30 units of good x and 15 units of good y. The endowment is now E  EB is a rectangle, not a square.

 Exercise: Draw a carefully labeled diagram illustrating the contract curve in this case

 Hint: it is still true that 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 → 𝑦𝑦𝐵𝐵 = 𝑥𝑥𝐵𝐵. But now this line does not run from B’s origin to A’s origin

Edgeworth Box

Slide 82

15 15

B’s x increasing

A’s 𝑥𝑥 increasing

ICA

ICA ICB

ICB E

Edgeworth Box

Slide 83

15 15

B’s x increasing

A’s 𝑥𝑥 increasing

ICA

ICA ICB

ICB E

Edgeworth Box

Slide 84

15 15

B’s x increasing

A’s 𝑥𝑥 increasing

ICA

ICA ICB

ICB L

Trade to PO on the Boundary

Summary

Slide 85

 An allocation satisfies distributive efficiency if we cannot reallocate such that at least one person is made better off without making another individual worse off.

 Graphically: allocation is efficient we can’t put one consumer on a higher IC without putting the other on a lower IC.

 If 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 at an allocation, then that allocation is always efficient but it is also possible to have efficient allocations where 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 ≠ 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵

 General rule: If 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 > 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵, allocate more x to person A and more y to person B, if that reallocation is possible.

-- end of part 7 --

General Equilibrium

Part 8: Competitive Markets

Economics 313

Competitive Markets

Slide 88

 Typically assume lots of buyers and sellers. In exchange economies, that would be lots of “traders.”

 The are justifications for the important assumption: price taking behaviour. We can assume this directly, abnd look at competitive markets in our 2x2 exchange economy, even though there are only two traders.

 The method is old: a “Walrasian Auctioneer” calls out a price for every good, and agents propose trades. These are just the familiar “demand curves” from 203, with a twist, as will become clear.

 If supply is greater than demand for any market, a new price is proposed. Only when the prices called out clear all of the markets at once do the trades take place. This, then is a Competitive Equilibrium, where price taking is assumed directly.

Competitive Markets

Slide 89

 The “twist” is that now we are computing a general equilibrium. There is no labour (production is a future topic) but income is well defined. The income of each trader is just the value of their endowment at the prices called out by the auctioneer.

 By definition, when they demand, they supply: these are trades after all!

 So it is really a “net demand” that matters, and with two goods, the value of supply must equal the value of demand: otherwise they are “violating their budget constraint.” And that can’t happen for the usual reasons.

Competitive Markets

Slide 90

 You will see that this equilibrium price vector has already been introduced. We didn’t call it a price vector of course, but in all the examples above we did implicitly describe a “rate of exchange” between X and Y. But this is exactly what a set of prices gives you.

 In 203 we talked about prices and income as if money existed, but it was only used as a “unit of account.” Here the “price of X” is how much Y it takes to “buy” a unit, and the price of Y is how much X is takes to buy a unit of Y.

 We will act as if there is a 𝑝𝑝𝑋𝑋 and a 𝑝𝑝𝑌𝑌, but you will see what really matters is the ratio, and we could arbitrarily “normalize” everything by setting one of these to be one.

Competitive Markets

Slide 91

 In the end this is going to take us a long way the the goal we talked about in Part 1.

 There is no production, so no “technology” and no labour market, wages or profit. But we do have “tastes,” as always treated as a fixed parameter, prices for each good, and income.

 That is, we are going to solve for the variables in red, given the parameters in green, and for the moment setting aside the rest (in grey).

 The relationship between supply and demand, that is X and x, Y and y, is something you should look for.

𝛼𝛼, 𝛽𝛽, 𝛾𝛾, 𝜃𝜃, 𝑤𝑤, 𝑋𝑋, 𝑌𝑌, , 𝑥𝑥, 𝑦𝑦, 𝑝𝑝𝑋𝑋, 𝑝𝑝𝑌𝑌, 𝑀𝑀

Competitive Markets

Slide 92

 How does this all work….

 First, how can “consumer” behavior result in distributive efficiency?

 From Econ 203, we know that consumers will choose the consumption bundle where 𝑀𝑀𝑀𝑀𝑆𝑆 = 𝑃𝑃𝑥𝑥

𝑃𝑃𝑦𝑦

 Tells us that if two consumers - A and B - face the same prices, then we will have:

 MRSA = px/py,

 MRSB = px/py.

We end up with MRSA = MRSB i.e., we have distributive efficiency

⇒

Distributional Efficiency & Behaviour

Slide 94

 But where does this price ratio come from?

 The Walrasian Auctioneer!

Towards a Competitive Equilibrium  Result 4: If there are very many agents the core often reduces

to single point, i.e. the competitive equilibrium allocation

 Definition: Competitive equilibrium [in an exchange economy]: set of prices, and the corresponding allocation, such that:

1. Consumers each solve their consumer’s problem  they choose optimal bundles, i.e. bundles which maximize utility given

prices and initial endowment  No consumer wishes to change her consumption choice

2. These optimum bundles form feasible allocation  In other words, (net) demand = (net) supply for all goods  In other words, markets are in equilibrium and Prices have no tendency to

change

 Note: here General equilibrium = competitive equilibrium

The Invisible Hand  The First Welfare Theorem of Economics  Equilibrium in Competitive Markets is Pareto Optimal

 The Second Theorem of Welfare Economics  Any allocation on the contract curve can be sustained as a

competitive equilibrium

1. Each consumer is maximizing her utility.

i. 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑃𝑃𝑥𝑥 𝑃𝑃𝑦𝑦

ii. 𝑝𝑝𝑥𝑥𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝑦𝑦𝐴𝐴 = 𝑝𝑝𝑥𝑥𝜔𝜔𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝜔𝜔𝑦𝑦𝐴𝐴

iii. 𝑀𝑀𝑀𝑀𝑆𝑆𝐵𝐵 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

iv. 𝑝𝑝𝑥𝑥𝑥𝑥𝐵𝐵 + 𝑝𝑝𝑦𝑦𝑦𝑦𝐵𝐵 = 𝑝𝑝𝑥𝑥𝜔𝜔𝑥𝑥𝐵𝐵 + 𝑝𝑝𝑦𝑦𝜔𝜔𝑦𝑦𝐵𝐵

2. Supply and demand are equal for each good, given each consumer’s choice.

v. 𝑥𝑥𝐴𝐴 + 𝑥𝑥𝐵𝐵 = 𝑥𝑥𝑇𝑇 = 𝜔𝜔𝑥𝑥𝐴𝐴 + 𝜔𝜔𝑥𝑥 𝐵𝐵 (total available x)

vi. 𝑦𝑦𝐴𝐴 + 𝑦𝑦𝐵𝐵 = 𝑦𝑦𝑇𝑇 = 𝜔𝜔𝑦𝑦𝐴𝐴 + 𝜔𝜔𝑦𝑦 𝐵𝐵 (total available y)

 We can solve this system of equations for five variables:

 𝑥𝑥𝐴𝐴, 𝑥𝑥𝐵𝐵, 𝑦𝑦𝐴𝐴, 𝑦𝑦𝐵𝐵, 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

We use symbol ω to denote endowments.

Distributional Efficiency & Behaviour

-- end of part 8 --

General Equilibrium

Part 9: Examples

Economics 313

An example: test for an equilibrium.

 Let the preferences and endowments be given by

Utility: Elvis: UE(xE,yE) = xE 2yE; Costello: U

C(xC,yC) = xCyC Endowments: Elvis: (xEE,yEE)=(7,2); Costello: (xEC,yEC)=(2,4))

 Do the prices px=1 and py=1 and the bundles Elvis: (xE,yE)=(6,3) and Costello: (xC,yC)= (3,3) constitute a competitive equilibrium?

The Competitive Equilibrium  Perform check by going carefully through the definition of the

competitive equilibrium:  “Consumers choose bundles so as to maximize their utility taking

these prices as given.”  In math:

 “These optimum bundles are such that net demand equals net supply for each good.” In math

 Check whether these 6 formula’s all hold and you’re done!  Note: Recall that we assume ICs are ‘enough’ bowed inward

E C

E ECE

E C

E ECE

yyyy xxxx

+=+

and

= + = +

E Ex E x E y E x E y E

E Ex C x C y C x C y C

p MRS p x p y p x p y

p p

MRS p x p y p x p y p

The Competitive Equilibrium  “Consumers choose bundles so as to maximize their

utility taking these prices as given.”  𝑀𝑀𝑀𝑀𝑆𝑆𝐸𝐸 =

2𝑥𝑥𝐸𝐸𝑦𝑦𝐸𝐸 𝑥𝑥𝐸𝐸 2 =

2𝑦𝑦𝐸𝐸 𝑥𝑥𝐸𝐸

= 2 3 6

= 1 = 𝑃𝑃𝑥𝑥 𝑃𝑃𝑦𝑦

= 1

 𝑀𝑀𝑀𝑀𝑆𝑆𝐶𝐶 = 𝑥𝑥𝐶𝐶𝑦𝑦𝐶𝐶 𝑥𝑥𝐶𝐶

= 𝑦𝑦𝐶𝐶 𝑥𝑥𝐶𝐶

= 3 3

= 1 = 𝑃𝑃𝑥𝑥 𝑃𝑃𝑦𝑦

= 1

 “These optimum bundles are such that net demand equals net supply for each good.”

 𝑥𝑥𝐸𝐸 + 𝑥𝑥𝑐𝑐 = 𝑥𝑥𝐸𝐸 𝐸𝐸 + 𝑥𝑥𝑐𝑐𝐸𝐸 ⇒ 7 + 2 = 6 + 3

 𝑦𝑦𝐸𝐸 + 𝑦𝑦𝑐𝑐 = 𝑦𝑦𝑐𝑐𝐸𝐸 + 𝑦𝑦𝑐𝑐𝐸𝐸 ⇒ 2 + 4 = 3 + 3

IT IS A COMPETITVE EQUILIBRIUM!! We can then use the budget constrains to find the “consumption bundles”.

Example of Prices In Edgeworth Box

𝑂𝑂𝐴𝐴

𝑂𝑂𝐵𝐵

𝑥𝑥𝐵𝐵

𝑦𝑦 𝐵𝐵𝑦𝑦 𝐴𝐴

𝑥𝑥𝐴𝐴

15 30

7 7

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = −1

𝐸𝐸0

 Example of Disequilibrium

Equilibrium

𝐼𝐼𝐶𝐶𝐵𝐵

𝐼𝐼𝐶𝐶𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = −1

𝐸𝐸0

𝑂𝑂𝐴𝐴

𝑂𝑂𝐵𝐵

𝑥𝑥𝐵𝐵

𝑥𝑥𝐴𝐴

15 30

7 7

𝑦𝑦 𝐴𝐴

Aside: Look Familiar?

𝐼𝐼𝐶𝐶𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = −1

𝐸𝐸0

𝑂𝑂𝐴𝐴 𝑥𝑥𝐴𝐴

15 30

𝑦𝑦 𝐴𝐴

Aside: A change in relative prices

𝐼𝐼𝐶𝐶𝐴𝐴

𝐸𝐸0

𝑂𝑂𝐴𝐴 𝑥𝑥𝐴𝐴

15 30

𝑦𝑦 𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = − 6

 Example of Disequilibrium

Equilibrium

𝐼𝐼𝐶𝐶𝐵𝐵

𝐼𝐼𝐶𝐶𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = −1

𝐸𝐸0

𝑂𝑂𝐴𝐴

𝑂𝑂𝐵𝐵

𝑥𝑥𝐵𝐵

𝑥𝑥𝐴𝐴

15 30

7 7

5 more X demanded by A

3 y supplied A

𝑦𝑦 𝐴𝐴

 Example of Disequilibrium

Equilibrium

𝐼𝐼𝐶𝐶𝐵𝐵

𝐼𝐼𝐶𝐶𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = −1

𝐸𝐸0

𝑂𝑂𝐴𝐴

𝑂𝑂𝐵𝐵

𝑥𝑥𝐵𝐵

𝑥𝑥𝐴𝐴

15 30

7 7

5 more X demanded by A

3 y supplied A5 more x demanded by B

5 y supplied by B

𝒚𝒚 𝑨𝑨

Equilibrium

𝐸𝐸0

 Example of Equilibrium

𝑂𝑂𝐵𝐵

𝑥𝑥𝐵𝐵

15 30

7 7

𝑦𝑦 𝐴𝐴

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = − 6

Equilibrium

𝑂𝑂𝑎𝑎

𝑂𝑂𝑏𝑏

𝐼𝐼𝐶𝐶𝐵𝐵

𝐼𝐼𝐶𝐶𝐴𝐴

Y oram

’s Q of Sour Jujubes

𝐸𝐸0

 Example of Equilibrium

15 30

7 7

Y supplied by B

X supplied by A

Y demanded A

X demanded by B

Slope=− 𝑝𝑝𝑥𝑥 𝑃𝑃𝑌𝑌

∗ = − 6

-- end of part 9 --

General Equilibrium

Part 10: Another example.

Economics 313

113

 Example: Suppose we have consumers A and B with:

 𝑈𝑈𝐴𝐴 = 𝑥𝑥𝑦𝑦, 𝜔𝜔𝑥𝑥𝐴𝐴 = 50, 𝜔𝜔𝑦𝑦𝐴𝐴 = 60

 𝑈𝑈𝐵𝐵 = 𝑥𝑥𝑦𝑦, 𝜔𝜔𝑥𝑥𝐵𝐵 = 10, 𝜔𝜔𝑦𝑦𝐵𝐵 = 60

 We want to solve for the equilibrium xA, xB, yA, yB & px/py.

1. Consumer A is maximizing utility:

i. 𝑀𝑀𝑀𝑀𝑆𝑆𝐴𝐴 = 𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦 → 𝑦𝑦𝐴𝐴

𝑥𝑥𝐴𝐴 = 𝑝𝑝𝑥𝑥

𝑝𝑝𝑦𝑦 → 𝑦𝑦𝐴𝐴 =

𝑝𝑝𝑥𝑥 𝑝𝑝𝑦𝑦

𝑥𝑥𝐴𝐴 ii. 𝑝𝑝𝑥𝑥𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝑦𝑦𝐴𝐴 = 𝑝𝑝𝑥𝑥𝜔𝜔𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝜔𝜔𝑦𝑦𝐴𝐴 iii.

→ 𝑝𝑝𝑥𝑥𝑥𝑥𝐴𝐴 + 𝑝𝑝𝑦𝑦𝑦𝑦𝐴𝐴 = 𝑝𝑝𝑥𝑥50 + 𝑝𝑝𝑦𝑦60 → (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦)𝑥𝑥𝐴𝐴+𝑦𝑦𝐴𝐴 = (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦)50 + 60 → (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦)𝑥𝑥𝐴𝐴+ (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦)𝑥𝑥𝐴𝐴= (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦)50 + 60 → (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) 2𝑥𝑥𝐴𝐴 − 50 = 60 → (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) = 60/ 2𝑥𝑥𝐴𝐴 − 50

Given (px/py), we can figure out xA & yA.

Solving For the Price Ratio and Demand

114

1. Now we know that if consumer A is maximizing utility, then:

⇒ (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) = 60/ 2𝑥𝑥𝐴𝐴 − 50 2. Can now solve the same problem for consumer B:

iii. MRSB = px/py ⇒ yB/xB = px/py

iv. pxxB + py yB = px ωxB + py ωyB

Solving this in the same way as the previous slide…. ⇒ (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) = 60/ 2𝑥𝑥𝐵𝐵 − 10

 Now we know what each consumer would like to do, as a function of prices (a demand curve)

 Just need to equate the demand side with the supply side to solve for the set of equilibrium values.

Solving For the Price Ratio and Demand

115

1. Demand side for consumer A: (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) = 60/ 2𝑥𝑥𝐴𝐴 − 50 2. Demand side for consumer B: (𝑝𝑝𝑥𝑥/𝑝𝑝𝑦𝑦) = 60/ 2𝑥𝑥𝐵𝐵 − 10 3. Supply side: ωxA + ωxB = 60.

• i.e., there are 60 units of good x in total. ⇒ xA + xB = 60

⇒ xB = 60 - xA

 Set (px/py) = (px/py) on the demand side (since A and B face the same prices.

⇒ 2xA - 50 = 2xB - 10

⇒ xA - 25 = xB - 5

⇒ xA - 25 = 60 - xA - 5

⇒ 2xA = 80

⇒ xA = 40

Solving For the Price Ratio and Demand

116

• i.e., there are 60 units of good x in total. ⇒ xA + xB = 60

⇒ xB = 60 - xA

 Set (px/py) = (px/py) on the demand side (since A and B face the same prices.

⇒ 2xA - 50 = 2xB - 10

⇒ xA - 25 = xB - 5

⇒ xA - 25 = 60 - xA - 5

⇒ 2xA = 80

⇒ xA = 40

Calculating (px/py)? from demand side for A we know: (px/py) = 60/{2xA-50}

= 60/{80 - 50} = 2

Solving For the Price Ratio and Demand

117

• i.e., there are 60 units of good x in total. ⇒ xA + xB = 60

⇒ xB = 60 - xA

 Set (px/py) = (px/py) on the demand side (since A and B face the same prices.

⇒ 2xA - 50 = 2xB - 10

⇒ xA - 25 = xB - 5

⇒ xA - 25 = 60 - xA - 5

⇒ 2xA = 80

⇒ xA = 40

Calculating (px/py)? from demand side for A we know: (px/py) = 60/{2xA-50}

= 60/{80 - 50} = 2

Solving For the Price Ratio and Demand

And for B we know (px/py) = 60/{2xB-10} ⇒ xB = 20

118

 Recall: we wanted to solve for 5 unknowns:

xA, xB, yA, yB & px/py.

 Solving for yA & yB:

 MRSA = px/py ⇒ yA/xA = px/py ⇒ yA/40 = 2

⇒ yA = 80

 MRSB = px/py ⇒ yB/xB = px/py ⇒ yB/20 = 2

⇒ yB = 40

 Can also see the equilibrium graphically, in the Edgeworth Box.

we have already found three of them

Solving For the Price Ratio and Demand

119

= ωxA

ωyA =

= ωxB

= ωyB

At the endowment: MRSA = y/x = 6/5 MRSB = y/x = 6

MRSB > MRSA ⇒ B values x more highly

than A does ⇒ there exists scope for

trade such that both are better off.

Solving For the Price Ratio and Demand

120

= ωxA

ωyA =

= ωxB

= ωyB

80 40

20 At the endowment:

MRSA = y/x = 6/5 MRSB = y/x = 6

MRSB > MRSA ⇒ B values x more highly

than A does ⇒ there exists scope for

trade such that both are better off.

Only when MRSA = MRSB are the gains from trade exhausted.

Solving For the Price Ratio and Demand

121

= ωxA

ωyA =

= ωxB

= ωyB

BL: slope = px/py = 2

80 40

20 At the endowment:

MRSA = y/x = 6/5 MRSB = y/x = 6

MRSB > MRSA ⇒ B values x more highly

than A does ⇒ there exists scope for

trade such that both are better off.

Only when MRSA = MRSB are the gains from trade exhausted.

Price ensures that - in equilibrium - MRSA = MRSB

Solving For the Price Ratio and Demand

122

 We have seen that, in a pure exchange economy:

1. Consumer sets MRS = price ratio.

2. Price ratio equates demand and (fixed) supply.

 Market behavior results in distributive efficiency if each consumer faces the same price ratio.

Solving For the Price Ratio and Demand

-- end of part 10 --

General Equilibrium

Part 11: Conclusion

Economics 313

What I Expect You to Know

Slide 125

 What Pareto efficiency, Pareto improvements and potential Pareto improvements are and be able to identify when they occur

 Work with and draw an Edgeworth box given utility functions and endowments

 Be able to derive a contract curve  Solve a simple competitive exchange economy model  Define the first and second welfare theorem.