General Equilibrium: Basics

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

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? Rih  0, i =1,…,n. • Resources 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 firm f, good i. n-vector of prices • Net outputs depend on prices: qif = qif(p) • Supply of net outputs • Thus profits depend on prices: • n • Pf(p):=S pi qif(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 pi Rih+SVfh Pf(p) • i=1 f=1 Holding by h of resource i Holding by h of shares in f • Incomes = resource rents + profits • Note that the function yh(•) depends on the ownership rights that h possesses • Incomes depend on prices: yh= yh(p)

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? • 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

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 all goods the materials balance must hold • For each f,maximise n Spi qif, subject toFf(qf ) 0 i=1 aggregate stock of good i. • For each i: what determines these aggregates? xi£ qi + Ri aggregate consumption of good i. 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 • Full additional resources are needed for each additional person consuming a unit of good i. Sum over households • An alternative way to aggregate: • xi=max {xih } • h • Opposite case: a nonrival good • Examples: TV, national defence... • Aggregation of net output: • nf • qi:= S qif • f=1 • if all the 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 S pi xia≤S pi Ria i=1 i=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 S pi xib≤S pi Rib i=1 i=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.