1. Frank Cowell: Microeconomics General Equilibrium: Basics MICROECONOMICS Principles and Analysis Frank Cowell Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimistion Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimistion Prerequisites November 2006

2. Frank Cowell: Microeconomics Limitations of Crusoe model The Crusoe story takes us only part way to a treatment of general equilibrium: 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. Prices are either exogenous (from the mainland? the world? Mars?) or hypothetical. But there are important lessons we can learn: 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

3. Frank Cowell: Microeconomics Onward from Crusoe... This is where we generalise the Crusoe model. This is where we generalise the Crusoe model. We need a model that will incorporate: 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”...? But what do we mean by an “economy”...? We need this in order to give meaning to “equilibrium” We need this in order to give meaning to “equilibrium”

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

5. Frank Cowell: Microeconomics The components At a guess we can model the economy in terms of: At a guess we can model the economy in terms of:  Resources  People  Firms Specifically the model is based on assumptions about: 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) (In addition –for later – we will need a description of the rules of the game)

6. Frank Cowell: Microeconomics What is an economy? Resources (stocks) Resources (stocks) U 1, U 2,...  1,  2,... R 1, R 2,... n h of these n f of these n of these Households (preferences) Households (preferences) Firms (technologies) Firms (technologies)

7. Frank Cowell: Microeconomics An allocation A collection of bundles (one for each of the n h households) 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) 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) A set of prices (used by households and firms) Note the shorthand notation for a collection

8. Frank Cowell: Microeconomics { } {, 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

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

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

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

12. Frank Cowell: Microeconomics What does household h possess? Resources Resources R 1 h, R 2 h,...  1 h,  2 h,... R i h  i =1,…,n. R i h  i =1,…,n. Shares in firms’ profits Shares in firms’ profits  f h  f =1,…,n f.  f h  f =1,…,n f. introduce prices

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

14. Frank Cowell: Microeconomics The fundamental role of prices   Net outputs depend on prices: q i f = q i f (p) firm f, good i. n-vector of prices   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   Incomes depend on prices : y h = y h (p)   Incomes = resource rents + profits   Note that the function y h () depends on the ownership rights that h possesses Holding by h of resource i Holding by h of shares in f indirectly

15. Frank Cowell: Microeconomics 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

16. Frank Cowell: Microeconomics 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

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

18. Frank Cowell: Microeconomics What is an equilibrium? What kind of allocation is an equilibrium? What kind of allocation is an equilibrium? Again we can learn from previous presentations: 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 can do this for the many-person, many- firm case. We just copy and slightly modify our earlier work

19. Frank Cowell: Microeconomics 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 )   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?

20. Frank Cowell: Microeconomics 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   Full additional resources are needed for each additional person consuming a unit of good 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 the q f are feasible will q be feasible?   Yes if there are no externalities   Counterexample: production with congestion... By definition

21. Frank Cowell: Microeconomics To make life simple: Assume incomes are determined privately. Assume incomes are determined privately. All goods are “rival” commodities. All goods are “rival” commodities. There are no externalities. There are no externalities.

22. Frank Cowell: Microeconomics 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

23. Frank Cowell: Microeconomics An example Exchange economy (no production) Exchange economy (no production) Simple, standard structure Simple, standard structure 2 traders (Alf, Bill) 2 traders (Alf, Bill) 2 Goods: 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__ Alf__ Bill__ Bill__ diagrammatic approach

24. Frank Cowell: Microeconomics 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

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

26. Frank Cowell: Microeconomics 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

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