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Intermediate Microeconomics W3211
Lecture 7: Introduction The Endowment Economy
Columbia University, Spring 2016
Mark Dean: [email protected] 2
3 4 The Story So Far…. Today’s Aims
• Remember: the course had two basic aims: We are now going to take our first stab in thinking about what happens when economic agents interact 1. Introduce you to models of how people make choices Constrained optimization In particular, we are going to think about how prices and 2. Introduce you to models of what happens when people interact income are determined in an economy Equilibrium So far we have treated these as exogenous parameters But they have to come from somewhere! We will assume that they come about through the interaction of economic agents buying and selling • We have now done a fairly thorough job of modelling the behavior of one type of economic agent – the consumer They move in order to balance supply and demand This is the study of equilibrium Set up the consumer’s problem Solved the consumer’s problem We will start with the simplest possible setting Derived demand functions Two consumers Thought about how demand changes with income and prices Two goods No firms
This is the study of an endowment economy Varian Ch. 9, 15, 16 Feldman and Serrano Ch 15
6 An Endowment Economy
We are going to think about the simplest possible example of an economy
There are two people: 1 and 2
There are two goods: apples and bananas
A Simple Endowment Each person starts off with an endowment of apples and bananas
Economy , : amount of apples and bananas that person 1 has 1: Revisiting the consumer’s problem , : amount of apples and bananas that person 2 has
People are allowed to buy and sell at market prices and
Questions we want to answer: What should prices be in this economy? 5 Who ends up with what stuff?
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7 8 An Endowment Economy Updating the Consumer’s Problem
Note: Apples and bananas only exist in the endowment of the Remember the ‘standard’ consumer problem consumers 1. CHOOSE a consumption bundle They are not produced 2. IN ORDER TO MAXIMIZE preferences There are no firms 3. SUBJECT TO the budget constraint This is a simplification, makes it easier to see what is going on The budget constraint (for consumer 1) was given by Don’t worry, firms will appear soon enough
First order of business: think what the consumer’s problem looks Now, rather than getting mysterious income I, the income the consumer like in an endowment economy gets comes from their endowment
Or, equivalently 0
is the net demand for good I Budget constraint says that the cost of net demand has to be less than zero
9 10 Updating the Consumer’s Problem Updating the Consumer’s Problem
What happens to the budget constraint if the price of good a What happens to the budget constraint if the price of good a changes? changes?
Good b implies wa, wb Could buy units of good b (increasing in )
Could buy units of good a (decreasing in )
Always feasible to consume , ) Increase in pa
Budget constraint pivots round the endowment Good a
11 12 Updating the Consumer’s Problem Updating the Consumer’s Problem
Another question: what happens when the prices of both Now proportionally increase prices of both a and b goods change in proportion? A quick example A quick example 2 2 3 3 4 2 2 1 Budget constraint is Budget constraint is 2∗2 3 2 2∗2 3 2 7 2 7 2
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13 14 Updating the Consumer’s Problem Updating the Consumer’s Problem
The only thing that matters for the consumers problem is the We have now changed the parameters of the consumer’s ratio of the price of good a to that of good b problem
This is obvious from the budget constraint Rather than the parameters being , , they are now , , We can similarly think of the demand function in terms of these This means that we can normalize the price of good b to 1 parameters: i.e. assume that the price of good b is 1 , , ) is the demand of person 1 for good a, given prices Think only about changes in the price of good a and endowment , ) What we really mean by the price of good a is , , ) = , , ) - is the net demand of person 1 for good a This will make our life a lot easier
But is quite a subtle point. Make sure you understand it!
15 16 A Worked Example A Worked Example
A worked example: , ,
Choose , to Maximize , = Then take derivatives 0 Subject to 0 First set up the Lagrange Function , , 0 Then take derivatives First two equations gives
Last equation gives the
17 A Worked Example
Substituting in gives
which implies 1 , , 2 1 A Simple Endowment , , 2 Economy And so 2: The Edgeworth Box 1 , , 2 1 , , 2
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19 20 The Edgeworth Box The Edgeworth Box
For the consumer’s problem we found it handy to draw graphs First step: how can we represent all the stuff that there is in the which allowed us to see what is going on economy?
It will be equally handy for us to do so for our simple 2 person, 2 With a box! good economy Width of the box is total amount of good a However, the graph we need to draw is a bit more complicated Height of the box is the total amount of good b
The Edgeworth Box!
21 22 Fig 2: An Edgeworth Box The Edgeworth Box
1 2 w b+w b Second step, how do we represent a feasible allocation in the Edgeworth Box? A feasible allocation is an amount of each good for person 1 and 2 that uses up the total allocation , , , such that + = + Good b + = +
Any point in the box is a feasible allocation
1 2 w a+w a Good a
23 24 A Feasible Allocation The Edgeworth Box 2 x a
1 2 w b+w b Third step, we can represent the consumer’s problem for person 1 2 x b
1 x a
1 2 w a+w a 1 x a
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25 26 Consumer 1 in the Edgeworth Box Fig 5: Consumer 1’s Optimal Bundle
1 2 1 2 w b+w b w b+w b
Budget Line
Good b Slope: -pa Good b
Optimal 1 x b Consumption Bundle for Consumer 1’s Indifference Curves Allocation Consumer 1 Allocation
1 2 1 2 w a+w a w a+w a Good a 1 Good a x a
27 28 The Edgeworth Box Consumer 2’ Budget Set in the Edgeworth Box
1 2 w b+w b Fourth step (and this is the clever bit), we can represent the consumer’s problem for person 2 Consumer 2’s Budget Set
Good b
1 2 w a+w a Good a
29 30 Consumer 2’s Preferences in the Consumer 2’s Optimal Bundle 2 Edgeworth Box x a
1 2 1 2 w b+w b w b+w b
2 x b
Good b Good b
Optimal Better Bundles Consumption Bundle for Consumer 2
1 2 1 2 w a+w a w a+w a Good a Good a
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32 Equilibrium
We are now ready to calculate the equilibrium for our nice, simple, two person, two good exchange economy
An important first step: we better define what we mean by an equilibrium A Simple Endowment Economy 3: Equilibrium
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33 34 Equilibrium Equilibrium
An equilibrium is a description of what we think will happen in So in our simple, two person two good economy, an equilibrium an economy consists of
A price of good a: (remember we don’t need a price for good b) We can think of it as a prediction of where we might expect the The amount of stuff that person 1 gets: and economy to end up The amount of stuff that person 2 gets: and Consists of two types of thing Prices of the various goods Quantities of each good that each person receives.
A handy tip: If I ask you, in an exam or homework, to calculate an equilibrium, you have to tell me what the prices and quantities are If you haven’t, you haven’t answered the question!
35 36 Equilibrium The Definition of an Equilibrium
What properties do you think an equilibrium should have? We define an equilibrium in the following way: , , , and form an equilibrium if two things are true i.e., if I told you that , , , and were the equilibrium for the economy, what would you expect to be true? 1. Optimality: The amount that each person gets is what they want, given the prices and their endowments It may be useful to think of the equilibrium as the resting point of the economy , , ) , , ) , , ) , , )
2. Market Clearing: Supply equals demand for each good + + + +
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37 38 Finding an Equilibrium A Worked Example
So now we know what an equilibrium is Let’s work through an example
How do we find one? First, we need to describe the economy
Luckily we have another recipe! 1. The endowment of each agent 1. Calculate the demand for each consumer and each good =3 as a function of prices =2 i.e. calculate , , ), , , ), , , ), , , ) =1 =5 2. Find the price at which markets clear ∗ i.e. find the price such that 2. The preferences of each agent ∗ ∗ , , )+ , , )= + , = ∗ ∗ , , )+ , , )= + , = 3. Use this price to calculate what each person gets
39 40 A Worked Example A Worked Example
Stage 1: Calculate the demand for each consumer and each Stage 2: Find the price at which markets clear good as a function of prices Let’s first think about the market for apples Luckily we have already done this for these preferences (earlier ∗ ∗ in the lecture) , , )+ , , ) + 3 1 1 5 1 3 1 , , ∗ ∗ 3 1 2 2 2 2 2 7 1 3 , , 1 4 ∗ 8 2 2 1 1 5 ∗ 7 , , 4 2 2 2 1 5 , , 2 2 2
41 42 A Worked Example A Worked Example
Great! We have the price that clears the market for apples! The same price clears the market for apples and bananas!
But there is also the market for bananas! A miracle!
And we only have one price (it’s not a miracle. I will explain why in a minute)
This sounds like trouble First, is it the case that ∗ is an equilibrium? Let’s see what happens in the market for bananas NO! ∗ ∗ , , )+ , , ) + An equilibrium consists of ∗ , , , and 3 5 1 2 5 2 2 2 ∗ 7 4 14 7 ∗ 4
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43 44 A Worked Example Walras’ Law
Stage 3: Use this price to calculate what each person gets So let’s go back to the miracle 3 4 29 ∗ , , 2 7 14 We had two markets (apples and bananas) and only one price 21 29 ( ∗ ) ∗ , , 1 8 8 1 20 27 How did the one price clear the two markets? ∗ , , 2 14 14 7 5 27 It turns out this is always true ∗ , , 8 2 8 Walras law: The price that clears the market for good a will also The equilibrium of this economy is these allocations, plus the clear the market for good b ∗ price Why (strap in for some algebra)
45 Walras’ Law
Both agents obey their budget constraint
So A Simple Endowment + - - + - - 0 Economy 4: Equilibrium in Pictures If the market for good a clears then + - - 0
Implies + - - 0
Market for good b clears 46
47 48 Equilibrium in the Edgeworth Box Equilibrium in the Edgeworth Box
1 2 w b+w b Having introduced the Edgeworth box, it might be useful to think about what equilibrium looks like in this diagram. Offer curve for consumer 1 First lets think what happens to the demand of consumer 1 as ∗ we change the price
Good b Increase
in pa
1 2 w 1+w 1 Good a
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49 50 Equilibrium in the Edgeworth Box Equilibrium in the Edgeworth Box
1 2 w b+w b Having introduced the Edgeworth box, it might be useful to think about what equilibrium looks like in this diagram. Offer curve for consumer 2 First lets think what happens to the demand of consumer 1 as we change the price
Offer curve of consumer a traces out their optimal bundle as Good b changes Increase
in pa Similarly, we can trace out the offer curve of consumer 2
1 2 w a+w a Good a
51 52 Equilibrium in the Edgeworth Box Equilibrium in the Edgeworth Box 2 x a
1 2 w b+w b
Having introduced the Edgeworth box, it might be useful to x2 think about what equilibrium looks like in this diagram. b
First lets think what happens to the demand of consumer 1 as we change the price
Offer curve of consumer a traces out their optimal bundle as Good b changes 1 x b Similarly, we can trace out the offer curve of consumer 2
Where is equilibrium going to occur?
1 2 w a+w a 1 Good a x a
+ 53 54 Equilibrium in the Edgeworth Box Equilibrium in the Edgeworth Box 2 x a
1 2 w b+w b
In the Edgeworth box, equilibrium occurs where the offer curves x2 intersect b
Good b
1 x b
1 2 w a+w a 1 Good a x a
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55 56 Equilibrium in the Edgeworth Box Equilibrium
Change In Equilibrium 1 2 w 2+w 2 In the Edgeworth box, equilibrium occurs where the offer curves intersect
This is also the point at which three things are tangent The price line The indifference curve of person 1 Good 2 The indifference curve of person 2
This will be important for the next lecture!
Notice: different endowments will give rise to different equilibria
1 2 Change In Endowment w 1+w 1 Good 1
57 58 Equilibrium, Supply and Demand Equilibrium, Supply and Demand
Good 1 Demand Here is a final way of thinking about equilibrium 1 1 1 2 2 2 Curve= xa ( pa , wa , wb ) xa ( pa , wa , wb )
Remember the market clearing condition: Supply ∗ ∗ curve , , )+ , , )= + 1 2 w 1+w 1 Left hand side is the total (or market) demand for good a Typically (though not always) downward sloping
Right hand side is the total supply of good a
Equilibrium occurs when prices make supply equal to demand Equilibrium P1 Price
60 Summary
Today we have done the following
1. Defined a simple endowment economy Rather than income, consumers have endowments of each good We have defined the consumer’s problem in this case Also defined the demand function , , )
2. Introduced the Edgeworth Box as a way of representing a simple two person, two good endowment economy Shown how it represents feasible allocations Summary Shown how it can represent the consumer’s problem for both consumer 1 and 2
Defined the concept of an equilibrium
Shown how to solve for the equilibrium of a simple endowment economy
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