Prerequisites Almost essential A Simple Economy Frank Cowell

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

Limitations of Crusoe model Frank Cowell: Microeconomics n The Crusoe story takes us only part way to a treatment of general equilibrium: u u n n 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: u u integration of consumption and production sectors decentralising role of prices. When we use something straight from Crusoe we will mark it with this logo

Onward from Crusoe. . . Frank Cowell: Microeconomics n n This is where we generalise the Crusoe model. We need a model that will incorporate: u u u n n 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”

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

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

What is an economy? Frank Cowell: Microeconomics n Resources (stocks)R 1 , R 2 , . . . n of these n Households (preferences) U 1, U 2 , . . . nh of these n Firms (technologies) F 1, F 2 , . . . nf of these

An allocation Frank Cowell: Microeconomics A competitive allocation consists of: Note the shorthand notation for a collection utility-maximising n A collection of⋌bundles (one for each of the nh households) [x] : = [x 1, x 2, x 3, . . . ] profit-maximising n A collection of net-output vectors ⋌ 1, q 2, q 3, . . . ] [q] : = [q (one for each of the nf firms) n A set of prices (used by households and firms) p : = (p 1, p 2, . . . , pn)

Frank Cowell: Microeconomics 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 } just a minute! Where do these incomes come from? ? § 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

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

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

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

What does household h possess? Frank Cowell: Microeconomics n Resources R 1 h, R 2 h, . . . Rih 0, i =1, …, n. 0 Vfh 1, f =1, …, nf. h, V h, . . . V Shares in firms’ n 1 2 profits introduce prices

Incomes Frank Cowell: Microeconomics Resources n Shares in firms n Rents Net outputs n Prices n Profits 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

The fundamental role of prices Frank Cowell: Microeconomics firm f, n-vector i. ofdepend prices h. Net good outputs on prices: qif = qif(p) h. Thus profits depend onindirectly prices: n directly P f(p): = S pi qif(p) § Supply of net outputs § Again writing profits as priceweighted sum of net outputs i=1 Holding by hcan be written. Holding h. So incomes as: by§h. Incomes = resource rents + of resource n i nf of shares in f yh = S pi Rih + S Vfh P f(p) i=1 profits f=1 h. Incomes depend on prices : yh = yh(p) § Note that the function yh( • ) depends on the ownership rights that h possesses

Prices in a competitive allocation Frank Cowell: Microeconomics § The allocation as a collection of responses p {qf(p) , f=1, 2, . . . , nf } 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 [q(p)] [x(p)] Let's look at the whole process

The price mechanism Frank Cowell: Microeconomics resource distribution share a ownership a d distribution 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] allocation [q(p)] [x(p)]

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

What is an equilibrium? Frank Cowell: Microeconomics n n What kind of allocation is an equilibrium? Again we can learn from previous presentations: u u u n 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

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

Consumption and net output Frank Cowell: Microeconomics h“Obvious” way to aggregate § Appropriate if i is a rival good overi? § Full additional resources are neede consumption of Sum good households nh x i = S x ih for each additional person consuming a unit of good i. h=1 h. An alternative way to aggregate: § Opposite case: a nonrival good § Examples: TV, national defence. . . xi = max {xih } h By definition h. 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. . .

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

Competitive equilibrium: summary Frank Cowell: Microeconomics h. It must be a competitive allocation § A set of prices p § Everyone maximises at h. The materials balance condition must hold § Demand cannot exceed those prices p supply: x£q+R

An example Frank Cowell: Microeconomics n n Exchange economy (no production) Simple, standard structure 2 traders (Alf, Bill) 2 Goods: Alf__ Bill__ l resource endowment (R 1 a, R 2 a) (R 1 b, R 2 b) 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) diagrammati c approach

Alf’s optimisation problem Frank Cowell: Microeconomics § 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 Oa R 1 a x 1 a § Alf sells some endowment of 2 for good 1 by trading with Bill

Bill’s optimisation problem Frank Cowell: Microeconomics § 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 R 1 b x 1 b

Combine the two problems Frank Cowell: Microeconomics x 1 b R 1 b Incomes from the distribution… x 2 a R 2 a Ob ·l [R] R 2 b …match expenditures in the allocation ·l [x*] § Bill’s problem (flipped) § Superimpose Alf’s problem. § Price-taking trade moves agents from endowment point… §. . . to the competitive equilibrium allocation § The role of prices § This is the Edgeworth box. § Width: R 1 a + R 1 b § Height: R 2 a + R 2 b. x 2 b Oa R 1 a x 1 a

Alf and Bill as a microcosm Frank Cowell: Microeconomics n n n The Crusoe equilibrium story translates to a manyperson 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.