Prerequisites Almost essential A Simple Economy Useful but
Prerequisites Almost essential A Simple Economy Useful, but optional Firm: Optimisation Consumer Optimisation GENERAL EQUILIBRIUM: BASICS MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: General Equilibrium Basics 1
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: • either exogenous (from the mainland? the world? Mars? ) • or hypothetical (shadow prices) § But there are important lessons: • integration of consumption and production sectors • decentralising role of prices When we use something straight from Crusoe we will mark it with this logo April 2018 Frank Cowell: General Equilibrium Basics 2
Onward from Crusoe… § This is where we generalise the Crusoe model § We need a model that will incorporate: • many actors in the economy • the possibility of interaction amongst these actors • the endogenisation of prices in the economy § But what do we mean by an “economy”? • need this in order to give meaning to “equilibrium” April 2018 Frank Cowell: General Equilibrium Basics 3
Overview General Equilibrium: Basics The components of the general equilibrium problem The economy and allocations Incomes Equilibrium April 2018 Frank Cowell: General Equilibrium Basics 4
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 § Also needed: • specification of the rules of the game • (to be discussed in later presentations) April 2018 Frank Cowell: General Equilibrium Basics 5
What is an economy? § Resources (stocks) R , … 1 2 n of these § Households (preferences) U 1, U 2 , … nh of these § Firms (technologies) F 1, F 2 , … nf of these April 2018 Frank Cowell: General Equilibrium Basics 6
An allocation A competitive allocation consists of: shorthand notation for a collection utility-maximising § A collection of⋌ bundles (one for each of the nh households) [x] : = [x 1, x 2, x 3, … ] profit-maximising § A collection of⋌ net-output vectors (one for each of the nf firms) [q] : = [q 1, q 2, q 3, … ] § A set of prices (used by households and firms) p : = (p 1, p 2, …, pn) April 2018 Frank Cowell: General Equilibrium Basics 7
How a competitive allocation works § Implication of firm f’s profit maximisation p { qf(p) , f = 1, 2, …, nf } p, {yh } {xh(p) , h = 1, 2, …, nh } where do these incomes come from? April 2018 § Firms' behavioural responses map prices into net outputs § Implication of household's utility maximisation § Households’ behavioural responses map prices and incomes into demands § The competitive allocation An important model component Frank Cowell: General Equilibrium Basics 8
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 April 2018 Frank Cowell: General Equilibrium Basics 9
Overview General Equilibrium: Basics A key role for the price system The economy and allocations Incomes Equilibrium April 2018 Frank Cowell: General Equilibrium Basics 10
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 April 2018 Frank Cowell: General Equilibrium Basics 11
What does household h possess? § Resources R 1 h, R 2 h, … Rih 0, i =1, …, n 0 Vfh 1, f =1, …, nf § Shares in firms’ profits V 1 h, V 2 h, … introduce prices April 2018 Frank Cowell: General Equilibrium Basics 12
Incomes § Resources § Shares in firms Rents § Net outputs § Prices Profits April 2018 ts n ne e o p m m o inco c e ’s h T fh o look more closely at the role of prices Frank Cowell: General Equilibrium Basics 13
The fundamental role of prices § Net output of i by firm f depends § Supply of net outputs on prices p: qif = qif(p) § Thus profits depend on prices: § Again writing profits as price. P f(p): = n S i=1 weighted sum of net outputs pi qif(p) directly indirectly § So incomes can be written as: n nf i=1 f=1 yh = S pi Rih + S Vfh P f(p) Holding by h of resource i Holding by h of shares in f § Income depends on prices : yh = yh(p) April 2018 § Income = resource rents + profits § yh( • ) depends on ownership rights that h possesses Frank Cowell: General Equilibrium Basics 14
Prices in a competitive allocation § The allocation as a collection of responses {qf(p) , f = 1, 2, …, nf } p p, p{ } { yh xh(p) , h = 1, 2, …, nh } § Put the price-income relation into household responses § Gives a simplified relationship for households § Summarise the relationship yh = yh(p) p April 2018 [q(p)] [x(p)] Let's look at the whole process Frank Cowell: General Equilibrium Basics 15
* detail on slide can only be seen if you run the slideshow The price mechanism resource distribution share a ownership a d distribution April 2018 R 1 , R 2 , … V 1 a, V 2 a, … R 1 b, R 2 b, … V 1 b, V 2 b, … … … prices § 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 § Brief summary… [y] [q(p)] [x(p)] allocation Frank Cowell: General Equilibrium Basics 16
Overview General Equilibrium: Basics Specification and examples The economy and allocations Incomes Equilibrium April 2018 Frank Cowell: General Equilibrium Basics 17
What is an equilibrium? § What kind of allocation is an equilibrium? § Again we can learn from previous presentations: • must be utility-maximising (consumption) • must be profit-maximising (production) • must 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 April 2018 Frank Cowell: General Equilibrium Basics 18
Competitive equilibrium: basics § For each h, maximise n Uh(xh), subject to S pi xih yh i=1 § Households maximise utility, given prices and incomes § Firms maximise profits, given prices § For all goods the materials balance must hold § For each f, maximise n S pi qif, subject to Ff(qf ) 0 i=1 § For each i: xi qi + Ri aggregate consumption of good i aggregate stock of good i aggregate net output of good i April 2018 what determines these aggregates? Frank Cowell: General Equilibrium Basics 19
Consumption and net output § “Obvious” way to aggregate consumption of good i? nh x i = S x ih h=1 § Appropriate if i is a rival good § Additional resources needed for each additional person consuming a unit of i Sum over households § An alternative way to aggregate: § Opposite case: a nonrival good § Examples: TV, national defence… xi = max {xih } h § Aggregation of net output By definition nf qi : = S qif f=1 April 2018 § if all qf are feasible will q be feasible? § Yes if there are no externalities § Counterexample: production with congestion… Frank Cowell: General Equilibrium Basics 20
To make life simple: § Assume incomes are determined privately § All goods are “rival” commodities § There are no externalities April 2018 Frank Cowell: General Equilibrium Basics 21
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 April 2018 Frank Cowell: General Equilibrium Basics 22
An example § Exchange economy (no production) § Simple, standard structure § 2 traders (Alf, Bill) § 2 Goods: Alf__ Bill__ l resource endowment (R 1 a, R 2 a) l consumption (x 1 a, x 2 a) (x 1 b, x 2 b) l utility Ua(x 1 a, x 2 a) Ub(x 1 b, x 2 b) (R 1 b, R 2 b) diagrammatic approach April 2018 Frank Cowell: General Equilibrium Basics 23
Alf’s optimisation problem § Resource endowment x 2 R 2 a a g sin e a e c r Inc feren pre l § Prices and budget constraint § Preferences § Equilibrium Ra l x*a § Budget constraint is 2 2 S pi xia ≤ S pi Ria i=1 § Alf sells some endowment of 2 for good 1 by trading with Bill Oa April 2018 R 1 a x 1 a Frank Cowell: General Equilibrium Basics 24
Bill’s optimisation problem § Resource endowment x 2 b g sin e a e c r Inc feren pre § Prices and budget constraint § Preferences § Equilibrium l x*b § Budget constraint is 2 R 2 b l 2 S pi xib ≤ S pi Rib i=1 Rb § Bill, of course, sells good 1 in exchange for 2 Ob April 2018 R 1 b x 1 b Frank Cowell: General Equilibrium Basics 25
Combine the two problems x 1 b R 1 b Incomes from the distribution… x 2 a R 2 a Ob ·l [R] § Bill’s problem (flipped) § Superimpose Alf’s problem b R 2 …match expenditures in § Price-taking trade moves agents from endowment point… §…to the competitive equilibrium allocation the allocation § The role of prices ·l [x*] § This is the Edgeworth box § Width: R 1 a + R 1 b § Height: R 2 a + R 2 b x 2 b Oa April 2018 R 1 a x 1 a Frank Cowell: General Equilibrium Basics 26
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 April 2018 Frank Cowell: General Equilibrium Basics 27
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