1. Frank Cowell: Microeconomics General Equilibrium: price taking MICROECONOMICS Principles and Analysis Frank Cowell Almost essential General Equilibrium: Basics Almost essential General Equilibrium: Basics Prerequisites November 2006

2. Frank Cowell: Microeconomics Puzzles in competitive equilibrium analysis So far we have focused on competitive equilibrium analysis. So far we have focused on competitive equilibrium analysis.  But why?  Why concentrate on equilibrium?  Why assume competitive behaviour? Here we re-examine the basics of market interaction by agents. Here we re-examine the basics of market interaction by agents. Let’s start by having another look at the exchange economy. Let’s start by having another look at the exchange economy. We’ll redraw the Edgeworth box. We’ll redraw the Edgeworth box.

3. Frank Cowell: Microeconomics Overview... An exchange economy The solution concept Prices and the Core General Equilibrium: price taking The offer curve as a tool of analysis

4. Frank Cowell: Microeconomics The Edgeworth Box Remember that the Edgeworth Box is a 2  2 representation of an exchange economy: Remember that the Edgeworth Box is a 2  2 representation of an exchange economy:  Two goods  Two persons, Alf and Bill Represent the equilibrium for each person given: Represent the equilibrium for each person given:  Price-taking behaviour  Ownership of the resources Introduce the materials balance condition… Introduce the materials balance condition… …achieved by inverting one diagram to complete the “box” …achieved by inverting one diagram to complete the “box”

5. Frank Cowell: Microeconomics Behaviour out of equilibrium First let’s see why the CE is of such significance. First let’s see why the CE is of such significance. To do so consider a simple question: To do so consider a simple question:  If Alf and Bill are price takers, what will they do in situations other than equilibrium? To answer this use a familiar tool To answer this use a familiar tool  The offer curve  Introduced in consumer demand To get this re-examine the optimisation problems To get this re-examine the optimisation problems  First Alf  Then Bill Jump to consumer demand

6. Frank Cowell: Microeconomics Alf’s responses to changes in p 1 /p 2 x1ax1a R2aR2a x2ax2a OaOa R a R1aR1a What about Bill?   Alf’s endowment   Alf’s reservation utility   Alf’s preference map   No trade if p 1 is too high   Trades offered as p 1 falls   Alf’s offer curve

7. Frank Cowell: Microeconomics x1bx1b R2bR2b x2bx2b ObOb R1bR1b R b x1bx1b R2bR2b x2bx2b ObOb R1bR1b R b Bill’s responses to changes ObOb   Bill’s situation...  ...as an Australian   No trade if p 1 is too low   Trades offered as p 1 rises   Bill’s offer curve

8. Frank Cowell: Microeconomics Edgeworth Box and CE x2bx2b ObOb x1ax1a x2ax2a OaOa x1bx1b R1bR1b R1aR1a R2aR2a R2bR2b [x*]   Offers are only consistent where curves intersect   By construction this is CE   Price-taking U-maximising Alf   Price-taking U-maximising Bill   Satisfies materials balance   The endowment point (property distribution).   The two offer curves [R]

9. Frank Cowell: Microeconomics The nature of CE Given competitive behaviour, the CE is the only “consistent” allocation. Given competitive behaviour, the CE is the only “consistent” allocation. Clearly the location of the CE depends upon the initial resource endowment [R]. Clearly the location of the CE depends upon the initial resource endowment [R]. But why assume competitive behaviour? But why assume competitive behaviour? Why should Alf and Bill behave as price-takers? Why should Alf and Bill behave as price-takers?

10. Frank Cowell: Microeconomics Where do the prices come from?  p i x i h  y h n i=1 max U h (x h ) subject to  p i x i h  y h The “rules of the game” assume that people act as price takers and that prices are “given” The “rules of the game” assume that people act as price takers and that prices are “given” Then people can solve the standard optimisation problem. Then people can solve the standard optimisation problem. But where do the prices come from? But where do the prices come from? We can’t appeal to invented “shadow” prices We can’t appeal to invented “shadow” prices Nor to “world markets” Nor to “world markets” Nor to some external agency… Nor to some external agency…

11. Frank Cowell: Microeconomics How to make progress It would be convenient to assume there is a big hand…. It would be convenient to assume there is a big hand….  …given the prices the system almost solves itself  But we have to manage without the artificial construct.  How? We need a more general solution concept. We need a more general solution concept. Base this on a broader concept of trading behaviour. Base this on a broader concept of trading behaviour. We will describe the type of equilibrium associated with this concept. We will describe the type of equilibrium associated with this concept. Then we examine how price-taking equilibrium relates to this. Then we examine how price-taking equilibrium relates to this.

12. Frank Cowell: Microeconomics Overview... An exchange economy The solution concept Prices and the Core General Equilibrium: price taking Blocking and the core

13. Frank Cowell: Microeconomics A fresh approach Develop the approach for an exchange economy. Develop the approach for an exchange economy. But it could apply to more interesting economies. But it could apply to more interesting economies.  To do it for production usually involves some strong assumptions. Imagine this as the economics of a PoW camp. Imagine this as the economics of a PoW camp. The rules of the game are very simple: The rules of the game are very simple:  Each person is endowed with a given bundle of goods  Each person has absolute right of disposal over this bundle.  Everyone is free to associate with others to form coalitions. RhRh RhRh No-one is forced to trade/exchange

14. Frank Cowell: Microeconomics Coalitions K2K2 K1K1 K0K0 coalition concept gives us an important clue   Viewed as n h separate individuals   A coalition K...  ...is formed by any subgroup   The population... we've got our endowments with us

15. Frank Cowell: Microeconomics The idea of blocking: a story One day you take your bundle to the “swap shop.” One day you take your bundle to the “swap shop.” Some bossy person there proposes (insists on?) a particular feasible allocation. Some bossy person there proposes (insists on?) a particular feasible allocation. You and some others don’t like the bundle you get under this allocation. You and some others don’t like the bundle you get under this allocation. Your group finds that you could do better Your group finds that you could do better  just by using its own resources  you could all get as much or more utility…  … as that offered under the proposed allocation. You guys therefore refuse to accept the proposal. You guys therefore refuse to accept the proposal. Your coalition has blocked the proposed allocation Your coalition has blocked the proposed allocation

16. Frank Cowell: Microeconomics A formal approach Consider a proposed allocation for the Consider a proposed allocation for the economy ^ [x][x] A coalition A coalition  K  {1,2,...,n h } An allocation [x] preferred by the coalition K: ^ ^  h  K  U h (x h )  U h (x h ), for some h  K  U h (x h )  U h (x h ) The allocation [x] of bundles is feasible for K if:  h  x h   h  R h If there is a feasible, preferred bundle for K then ^ … [x] is blocked by K An allocation is blocked by a coalition if the coalition members can do better for themselves

17. Frank Cowell: Microeconomics Equilibrium concept Use the idea of blocking to introduce a basic solution concept. Use the idea of blocking to introduce a basic solution concept. Surely no blocked allocation could be a solution to the trading game? Surely no blocked allocation could be a solution to the trading game? So we use the following definition of a solution: So we use the following definition of a solution: The Core is the set of unblocked, feasible allocations. The Core is the set of unblocked, feasible allocations. Let’s apply it in the two-trader case. Let’s apply it in the two-trader case.

18. Frank Cowell: Microeconomics Coalitions In a 2-person world there are few coalitions: In a 2-person world there are few coalitions: {Alf } {Bill} {Alf & Bill}

19. Frank Cowell: Microeconomics ObOb OaOa x1x1 b x1x1 a x2x2 a x2x2 b x1x1 a x2x2 a x2x2 b [x b ] x1x1 b bb bb The 2-person core   Draw the contract curve   Alf’s reservation utility   {Bill} blocks these allocations   {Alf, Bill} blocks these allocations   The resulting core   {Alf} blocks these allocations [x a ]   Bill’s reservation utility   The contract curve is the locus of common tangencies aa aa [R]   Bill gets all the advantage from trade at this extreme point.   Alf gets all the advantage from trade at this extreme point.

20. Frank Cowell: Microeconomics The Core: summary Definition of the core follows immediately from: Definition of the core follows immediately from:  The definition of an allocation.  The definition of blocking. It is a general concept. It is a general concept. To find the core you need just: To find the core you need just:  A complete description of the property distribution.  An enumeration of the possible coalitions.  A certain amount of patience. The major insight from the core comes when we examine the relation to CE. The major insight from the core comes when we examine the relation to CE.

21. Frank Cowell: Microeconomics Overview... An exchange economy The solution concept Prices and the Core General Equilibrium: price taking Competitive equilibrium, large numbers and a limit theorem

22. Frank Cowell: Microeconomics ObOb OaOa x1x1 b x1x1 a x2x2 a x2x2 b The core and CE   The endowment point   The 2-person core again [R]   Competitive equilibrium again [x*]   A competitive equilibrium must always be a core allocation

23. Frank Cowell: Microeconomics ObOb OaOa x1x1 b x1x1 a x2x2 a x2x2 b The core and CE (2)   Indifference curves that yield multiple equilibria   Endowment point and reservation utility   Equilibrium: low p 1 /p 2   Equilibrium: high p 1 /p 2   The core [x*] [x**] bb bb aa aa [R]

24. Frank Cowell: Microeconomics A simple result Every CE allocation must belong to the core. Every CE allocation must belong to the core. It is possible that no CE exists. It is possible that no CE exists. But what of other core allocations which are not CE? But what of other core allocations which are not CE?  Remember we are dealing with a 2-person model.  But will there always be non-CE points in the core? Let's take a closer look… Let's take a closer look… Core CE Introducing Alf and Bill's twin brothers

25. Frank Cowell: Microeconomics So let’s clone the economy Assume that the economy is replicated by a factor N, so that there are 2N persons. Assume that the economy is replicated by a factor N, so that there are 2N persons. Start with N=2: Start with N=2:  We move from a 2-person economy to a 4- person economy.  Alf and his twin brother Arthur have the same preferences and endowments.  Likewise the twins Bill and Ben. Now of course there are more possibilities of forming coalitions. Now of course there are more possibilities of forming coalitions.

26. Frank Cowell: Microeconomics Coalitions in the N=2 economy All old coalitions are still possible... All old coalitions are still possible... {Alf & Bill} {Alf} {Bill} {Arthur & Ben} {Arthur} {Ben} {Alf & Arthur} {etc, etc} {Bill&Ben} {Alf, Arthur &Bill}... plus some new ones... plus some new ones

27. Frank Cowell: Microeconomics Effect of cloning on the core Ben   This new allocation is not a solution...   But it shows that the core must have become smaller   The core in the 2-person case   {Alf,Arthur,Bill} can block [x a ]...  ...leaving the Ben twin outside the coalition [R]   Are the extremes still core allocations in the 4-person economy? ° Alf, Arthur   The extremes of the two- person core Bill [x a ] [x b ]

28. Frank Cowell: Microeconomics How the blocking coalition works Alf x a = ½[x a +R a ] Arthurx a = ½[x a +R a ] Bill [2R a +R b – 2x a ] ————— 2R a + R b   The consumption within the coalition equals the coalition’s resources.   So the allocation is feasible.   Consumption in the coalition   Sum to get resource requirement Ben R b   Consumption out of coalition

29. Frank Cowell: Microeconomics If N is bigger: more blocking coalitions? numbers of… a-tribe b-tribe   The 2-person core   An arbitrary allocation - can it be blocked? 500 250 310 360 400 450   We’ve found the blocking coalition.   If line is not a tangent this can always be done.   Draw a line to the endowment   Take N=500 of each tribe.   Divide the line for different coalition numbers. [x b ] [R] An a-type would be better off somewhere here [x a ]

30. Frank Cowell: Microeconomics In the limit [x b ] [x*]   If N  a coalition can be found that divides the line to [R] in any proportion you want.   Only if the line is like this will the allocation be impossible to block.   With the large N the core has “shrunk” to the set of CE [R] [x a ]

31. Frank Cowell: Microeconomics A powerful result: the shrinking core As you clone the economy the core becomes smaller. As you clone the economy the core becomes smaller. If you make N large enough you will find some coalition that blocks any non-CE allocation. If you make N large enough you will find some coalition that blocks any non-CE allocation. So in the limit the core contains only CE allocations. So in the limit the core contains only CE allocations. In a suitably large economy the core exactly equals the set of competitive equilibria. In a suitably large economy the core exactly equals the set of competitive equilibria.

32. Frank Cowell: Microeconomics The shrinking core: discussion The result rules out non-price-taking behaviour as a solution. But: The result rules out non-price-taking behaviour as a solution. But: There are some weasel words: “suitably large”. There are some weasel words: “suitably large”.  In principle N should be infinite Process requires balanced replication of the Alf and Bill tribes. Process requires balanced replication of the Alf and Bill tribes.  Problems arise if there is one large b-trader and many a-traders All possible coalitions are assumed relevant to negotiations about blocking. All possible coalitions are assumed relevant to negotiations about blocking.  Only valid if communication and other coalition costs are negligible. The Internet? We have argued only using an exchange economy. We have argued only using an exchange economy.  Can be extended to production economies with CRTS and (with some difficulty) others too.

33. Frank Cowell: Microeconomics Review Basic components of trading equilibrium: Basic components of trading equilibrium:  Coalitions  Blocking  Core as an equilibrium concept Relation to CE Relation to CE  Every CE must lie in the core  In the limit of a replication economy the core consists only of CE Answer to question: why price-taking? Answer to question: why price-taking?  In a large economy with suitably small agents... ....it's the only thing to do. Review