1. Frank Cowell: General Equilibrium Basics GENERAL EQUILIBRIUM: BASICS MICROECONOMICS Principles and Analysis Frank Cowell Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimisation Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimisation Prerequisites March 2012 1 Note: the detail in slides marked “ * ” can only be seen if you run the slideshow

2. Frank Cowell: General Equilibrium Basics Limitations of Crusoe model  The Crusoe story takes us only part way to a treatment of general equilibrium: there's only one economic actor… …so there can be no interaction  Prices are either exogenous (from the mainland? the world? Mars?) or hypothetical  But there are important lessons we can learn: integration of consumption and production sectors decentralising role of prices When we use something straight from Crusoe we will mark it with this logo March 2012 2

3. Frank Cowell: General Equilibrium Basics Onward from Crusoe…  This is where we generalise the Crusoe model  We need a model that will incorporate: many actors in the economy… …and the possibility of their interaction the endogenisation of prices in the economy  But what do we mean by an “economy”…?  We need this in order to give meaning to “equilibrium” March 2012 3

4. Frank Cowell: General Equilibrium Basics Overview… The economy and allocations Incomes Equilibrium General Equilibrium: Basics The components of the general equilibrium problem March 2012 4

5. Frank Cowell: General Equilibrium Basics The components  At a guess we can model the economy in terms of: Resources People Firms  Specifically the model is based on assumptions about: Resource stocks Preferences Technology  (In addition –for later – we will need a description of the rules of the game) March 2012 5

6. Frank Cowell: General Equilibrium Basics What is an economy?  Resources (stocks) U 1, U 2,…     ,… R 1, R 2,… n h of these n f of these n of these  Households (preferences)  Firms (technologies) March 2012 6

7. Frank Cowell: General Equilibrium Basics An allocation  A collection of bundles (one for each of the n h households)  A collection of net-output vectors (one for each of the n f firms) [x] := [x 1, x 2, x 3,… ] [q] := [q 1, q 2, q 3,… ] p := (p 1, p 2, …, p n ) utility-maximising ⋌ profit-maximising ⋌ A competitive allocation consists of:  A set of prices (used by households and firms) Note the shorthand notation for a collection March 2012 7

8. Frank Cowell: General Equilibrium Basics { } {, h=1,2,…,n h } How a competitive allocation works  Implication of firm f ’s profit maximisation p  q f (p)  Implication of household's utility maximisation  Firms' behavioural responses map prices into net outputs {, f=1,2,…,n f }  Households’ behavioural responses map prices and incomes into demands p, y h  x h (p)  The competitive allocation just a minute! Where do these incomes come from?? An important model component March 2012 8

9. Frank Cowell: General Equilibrium Basics An important missing item  For a consumer in isolation it may be reasonable to assume an exogenous income Derived elsewhere in the economy  Here the model involves all consumers in a closed economy There is no “elsewhere”  Incomes have to be modelled explicitly  We can learn from the “simple economy” presentation March 2012 9

10. Frank Cowell: General Equilibrium Basics Overview… The economy and allocations Incomes Equilibrium General Equilibrium: Basics A key role for the price system March 2012 10

11. Frank Cowell: General Equilibrium Basics Modelling income  What can Crusoe teach us?  Consider where his “income” came from Ownership rights of everything on the island  But here we have many persons and many firms So we need to proceed carefully We need to assume a system of ownership rights March 2012 11

12. Frank Cowell: General Equilibrium Basics What does household h possess?  Resources R 1 h, R 2 h, …  1 h,  2 h, … R i h  0, i =1,…,n R i h  0, i =1,…,n  Shares in firms’ profits 0   f h  1, f =1,…,n f 0   f h  1, f =1,…,n f introduce prices March 2012 12

13. Frank Cowell: General Equilibrium Basics Incomes  Resources Profits Rents The components of h’s income look more closely at the role of prices  Shares in firms  Net outputs  Prices March 2012 13

14. Frank Cowell: General Equilibrium Basics The fundamental role of prices  Net output of i by firm f depends on prices p: q i f = q i f (p)  Supply of net outputs  Thus profits depend on prices: n  f (p):=  p i q i f (p) i=1  So incomes can be written as: n n f y h =  p i R i h +   f h  f (p) i=1 f=1  Again writing profits as price- weighted sum of net outputs directly  Income depends on prices : y h = y h (p)  Income = resource rents + profits  y h () depends on ownership rights that h possesses Holding by h of resource i Holding by h of shares in f indirectly March 2012 14

15. Frank Cowell: General Equilibrium Basics Prices in a competitive allocation  The allocation as a collection of responses  Put the price-income relation into household responses  Gives a simplified relationship for households p  q f (p) {, f=1,2,…,n f } { } {, h=1,2,…,n h } p, y h  x h (p) p y h = y h (p)  Summarise the relationship p  [q(p)][q(p)] [x(p)][x(p)] Let's look at the whole process March 2012 15

16. Frank Cowell: General Equilibrium Basics The price mechanism* d resource distribution R 1 b, R 2 b, … R 1 a, R 2 a, … … share ownership  1 b,  2 b, …  1 a,  2 a, … …  System takes as given the property distribution  Property distribution consists of two collections  Prices then determine incomes [y][y]  Prices and incomes determine net outputs and consumptions [q(p)] [x(p)]  Brief summary… a distribution prices allocation March 2012 16

17. Frank Cowell: General Equilibrium Basics Overview… The economy and allocations Incomes Equilibrium General Equilibrium: Basics Specification and examples March 2012 17

18. Frank Cowell: General Equilibrium Basics What is an equilibrium?  What kind of allocation is an equilibrium?  Again we can learn from previous presentations: Must be utility-maximising (consumption)… …profit-maximising (production)… …and satisfy materials balance (the facts of life)  We can do this for the many-person, many-firm case We just copy and slightly modify our earlier work March 2012 18

19. Frank Cowell: General Equilibrium Basics U h (x h ), subject to n  p i x i h  y h i=1 Competitive equilibrium: basics  Households maximise utility, given prices and incomes  Firms maximise profits, given prices  For each h, maximise  For all goods the materials balance must hold n  p i q i f, subject to  f (q f )  0 i=1  For each f, maximise  For each i: x i  q i + R i aggregate consumption of good i aggregate net output of good i aggregate stock of good i what determines these aggregates? March 2012 19

20. Frank Cowell: General Equilibrium Basics Consumption and net output  “Obvious” way to aggregate consumption of good i? n h x i =  x i h h=1  An alternative way to aggregate: x i = max {x i h } h  Appropriate if i is a rival good  Additional resources needed for each additional person consuming a unit of i Sum over households  Opposite case: a nonrival good  Examples: TV, national defence…  Aggregation of net output: n f q i :=  q i f f=1  if all q f are feasible will q be feasible?  Yes if there are no externalities  Counterexample: production with congestion… By definition March 2012 20

21. Frank Cowell: General Equilibrium Basics To make life simple:  Assume incomes are determined privately  All goods are “rival” commodities  There are no externalities March 2012 21

22. Frank Cowell: General Equilibrium Basics Competitive equilibrium: summary  A set of prices p  Everyone maximises at those prices p  It must be a competitive allocation  Demand cannot exceed supply: x ≤q + R  The materials balance condition must hold March 2012 22

23. Frank Cowell: General Equilibrium Basics An example  Exchange economy (no production)  Simple, standard structure  2 traders (Alf, Bill)  2 Goods: resource endowment (R 1 a, R 2 a ) consumption (x 1 a, x 2 a ) utility U a (x 1 a, x 2 a ) (R 1 b, R 2 b ) (x 1 b, x 2 b ) U b (x 1 b, x 2 b ) Alf__ Bill__ diagrammatic approach March 2012 23

24. Frank Cowell: General Equilibrium Basics Alf’s optimisation problem Increasing preference x1ax1a R1aR1a R2aR2a x2ax2a OaOa  Resource endowment  Preferences  Prices and budget constraint R a x *a  Equilibrium  Budget constraint is 2   p i x i a ≤   p i R i a i=1  Alf sells some endowment of 2 for good 1 by trading with Bill March 2012 24

25. Frank Cowell: General Equilibrium Basics Bill’s optimisation problem Increasing preference x1bx1b R1bR1b R2bR2b x2bx2b ObOb  Resource endowment  Preferences  Prices and budget constraint  Equilibrium  Bill, of course, sells good 1 in exchange for 2 March 2012 25  Budget constraint is 2   p i x i b ≤   p i R i b i=1 R b x *b

26. Frank Cowell: General Equilibrium Basics Combine the two problems  Bill’s problem (flipped)  Price-taking trade moves agents from endowment point…  Superimpose Alf’s problem x1bx1b R1bR1b R2bR2b x2bx2b ObOb  …to the competitive equilibrium allocation  This is the Edgeworth box  Width: R 1 a + R 1 b  Height: R 2 a + R 2 b x1ax1a R1aR1a R2aR2a x2ax2a OaOa [R][R]  [x * ]  The role of prices Incomes from the distribution… …match expenditures in the allocation March 2012 26

27. Frank Cowell: General Equilibrium Basics Alf and Bill as a microcosm  The Crusoe equilibrium story translates to a many- person economy  Role of prices in allocations and equilibrium is crucial  Equilibrium depends on distribution of endowments  Main features are in the model of Alf and Bill  But, why do these guys just accept the going prices…?  See General Equilibrium: Price-Taking March 2012 27