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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 Note: the detail in slides marked “ * ” can only be seen if you run the slideshow

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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

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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

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

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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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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

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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

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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

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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

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

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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

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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

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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

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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 Budget constraint is 2 p i x i b ≤ p i R i b i=1 R b x *b

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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

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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

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