General Equilibrium: Basics

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

2. 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

3. 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”

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

5. 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)

6. What is an economy? • Resources (stocks) R1 , R2 ,… n of these • Households (preferences) nh of these U1, U2 ,… • Firms (technologies) nfof these F1, F2,…

7. An allocation A competitive allocation consists of: Note the shorthand notation for a collection • A collection of bundles (one for each of the nh households) • A collection of net-output vectors (one for each of the nf firms) utility-maximising ⋌ [x] := [x1, x2, x3,… ] profit-maximising ⋌ [q] := [q1, q2, q3,… ] • A set of prices (used by households and firms) p := (p1, p2, …, pn)

8. How a competitive allocation works • Implication of firm f’s profit maximisation • Firms' behavioural responses map prices into net outputs p qf(p) { , f=1,2,…,nf} • Implication of household's utility maximisation • Households’ behavioural responses map prices and incomes into demands p, yhxh(p) { } { , h=1,2,…,nh} • The competitive allocation just a minute! Where do these incomes come from?? An important model component

9. 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

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

11. 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

12. What does household h possess? • Resources Rih 0, i=1,…,n R1h, R2h, … 0  Vfh 1, f =1,…,nf V1h, V2h, … • Shares in firms’ profits introduce prices

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

14. The fundamental role of prices • Net output of i by firm f depends on prices p: • qif = qif(p) • Supply of net outputs • Thus profits depend on prices: • n • P f(p):= S piqif(p) • i=1 • Again writing profits as price-weighted sum of net outputs indirectly directly • So incomes can be written as: • n nf • yh= S piRih+SVfhPf(p) • i=1 f=1 • Income = resource rents + profits Holding by h of shares in f Holding by h of resource i • Income depends on prices: • yh= yh(p) • yh(•) depends on ownership rights thathpossesses

15. Prices in a competitive allocation • The allocation as a collection of responses p qf(p) { , f=1,2,…,nf} • Put the price-income relation into household responses • Gives a simplified relationship for households p, yhxh(p) { } { , h=1,2,…,nh} p • Summarise the relationship yh= yh(p) [q(p)] Let's look at the whole process p [x(p)]

16. resource distribution share ownership R1a, R2a, … V1a, V2a, … R1b, R2b, … V1b, V2b, … … … The price mechanism* • System takes as given the property distribution • Property distribution consists of two collections • Prices then determine incomes • Prices and incomes determine net outputs and consumptions a d • Brief summary… [y] [q(p)] [x(p)] prices allocation distribution

17. Overview… General Equilibrium: Basics The economy and allocations Specification and examples Incomes Equilibrium

18. What is an equilibrium? We just copy and slightly modify our earlier work • 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

19. Competitive equilibrium: basics • For each h,maximise • Households maximise utility, given prices and incomes Uh(xh), subject to n Spi xih  yh i=1 • Firms maximise profits, given prices • For each f,maximise • For all goods the materials balance must hold n S pi qif, subject toFf(qf ) 0 i=1 • For each i: xi£ qi + Ri what determines these aggregates? aggregate consumption of good i aggregate stock of goodi aggregate net output of good i

20. Consumption and net output • “Obvious” way to aggregate consumption of good i? • nh • xi=S xih • h=1 • 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… • An alternative way to aggregate: • xi= max {xih} • h • Aggregation of net output: • nf • qi:= Sqif • f=1 • if all qf are feasible will q be feasible? • Yes if there are no externalities • Counterexample: production with congestion… By definition

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

22. Competitive equilibrium: summary • It must be a competitive allocation • A set of prices p • Everyone maximises at those prices p • The materials balance condition must hold • Demand cannot exceed supply: • x ≤q + R

23. An example • Exchange economy (no production) • Simple, standard structure • 2 traders (Alf, Bill) • 2 Goods: Alf__ Bill__ • resource endowment (R1a, R2a) (R1b, R2b) • consumption (x1a, x2a) (x1b, x2b) • utility Ua(x1a, x2a) Ub(x1b, x2b) diagrammatic approach

24. Alf’s optimisation problem • Resource endowment Increasing preference • Prices and budget constraint x2a • Preferences • Equilibrium • Ra R2a • x*a • Budget constraint is • 22 • Spi xia≤Spi Ria • i=1i=1 • Alf sells some endowment of 2 for good 1 by trading with Bill Oa x1a R1a

25. Bill’s optimisation problem • Resource endowment Increasing preference • Prices and budget constraint x2b • Preferences • Equilibrium • x*b • Budget constraint is • 22 • Spi xib≤Spi Rib • i=1i=1 • Rb R2b • Bill, of course, sells good 1 in exchange for 2 Ob R1b x1b

26. R1b x1b Ob x2a R2b R2a x2b Oa x1a R1a Combine the two problems • Bill’s problem (flipped) • Superimpose Alf’s problem Incomes from the distribution… • Price-taking trade moves agents from endowment point… • …to the competitive equilibrium allocation • [R] …match expenditures in the allocation • The role of prices • [x*] • This is the Edgeworth box • Width: R1a + R1b • Height: R2a + R2b

27. 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