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Prerequisites Almost essential General equilibrium: Basics Useful, but optional General Equilibrium: Price Taking GENERAL EQUILIBRIUM: EXCESS DEMAND AND THE RÔLE OF PRICES MICROECONOMICS Principles and Analysis Frank Cowell Note: the detail in slides marked “ * ” can only be seen if you run the slideshow March 2012 Frank Cowell: GE Excess Demand & Prices 1 Some unsettled questions Under what circumstances can we be sure that an equilibrium exists? Will the economy somehow “tend” to this equilibrium? And will this determine the price system for us? We will address these using the standard model of a general-equilibrium system To do this we need just one more new concept March 2012 Frank Cowell: GE Excess Demand & Prices 2 Overview… General Equilibrium: Excess Demand+ Definition and properties Excess Demand Functions Equilibrium Issues Prices and Decentralisation March 2012 Frank Cowell: GE Excess Demand & Prices 3 Ingredients of the excess demand function Aggregate demands (the sum of individual households' demands) Aggregate net-outputs (the sum of individual firms' net outputs) Resources Incomes determined by prices check this out March 2012 Frank Cowell: GE Excess Demand & Prices 4 Aggregate consumption, net output From household’s demand function Because incomes depend on prices xih = Dih(p, yh) = Dih(p, yh(p) ) So demands are just functions of p xih(•) depends on holdings of xih = xih(p) resources and shares If all goods are private (rival) then “Rival”: extra consumers require aggregate demands can be written: additional resources. Same as “consumer: aggregation” xi(p) = Sh xih(p) From firm’s supply of net output qif = qif(p) Aggregate: qi = Sf qif(p) March 2012 standard supply functions/ demand for inputs valid if there are no externalities. graphical As in “Firm and the market”) summary Frank Cowell: GE Excess Demand & Prices 5 Derivation of xi(p) Alf’s demand curve for good 1 Bill’s demand curve for good 1 Pick any price Sum of consumers’ demand Repeat to get the market demand curve p1 Alf March 2012 p1 p1 x1a Bill x1b x1 The Market Frank Cowell: GE Excess Demand & Prices 6 Derivation of qi(p) Supply curve firm 1 (from MC) Supply curve firm 2 Pick any price Sum of individual firms’ supply Repeat… The market supply curve p p p q1 low-cost firm March 2012 q1+q2 q2 high-cost firm both firms Frank Cowell: GE Excess Demand & Prices 7 Subtract q and R from x to get E: p1 p1 Demand Supply q1 x1 p1 Resource stock p1 R1 net output of i aggregated over f demand for i aggregated over h Ei(p) := xi(p) – qi(p) – Ri 1 Resource stock of i E1 March 2012 Frank Cowell: GE Excess Demand & Prices 8 Equilibrium in terms of Excess Demand Equilibrium is characterised by a price vector p* 0 such that: For every good i: The materials balance condition (dressed up a bit) Ei(p*) 0 For each good i that has a positive price in equilibrium (i.e. if pi* > 0): Ei(p*) = 0 March 2012 If this is violated, then somebody, somewhere isn't maximising… You can only have excess supply of a good in equilibrium if the price of that good is 0 Frank Cowell: GE Excess Demand & Prices 9 Using E to find the equilibrium Five steps to the equilibrium allocation From technology compute firms’ net output functions and profits 2. From property rights compute household incomes and thus household demands 3. Aggregate the xs and qs and use x, q, R to compute E 4. Find p* as a solution to the system of E functions 5. Plug p* into demand functions and net output functions to get the allocation But this begs some questions about step 4 1. March 2012 Frank Cowell: GE Excess Demand & Prices 10 Issues in equilibrium analysis Existence • Is there any such p*? Uniqueness • Is there only one p*? Stability • Will p “tend to” p*? For answers we use some fundamental properties of E March 2012 Frank Cowell: GE Excess Demand & Prices 11 Two fundamental properties… Walras’ Law. For any price p: You only have to work with n-1 (rather than n) equations n pi Ei(p) = 0 S i=1 Hint #1: think about the "adding-up" property of demand functions… Homogeneity of degree 0. For any price p and any t > 0 : You can normalise the prices by any positive number Ei(tp) = Ei(p) Hint #2: think about the homogeneity property of demand functions… Can you explain why they are true? Reminder: these hold for any competitive allocation, not just equilibrium March 2012 Frank Cowell: GE Excess Demand & Prices 12 Price normalisation* We may need to convert from n numbers p1, p2,…pn to n1 relative prices The precise method is essentially arbitrary The choice of method depends on the purpose of your model It can be done in a variety of ways: You could divide by n pi S i=1 ppapnuméraire n labour MarsBar to give a neat oftheory n-1 standard value system “Marxian” Mars bar theory of ofvalue value set of set prices thatprices sum to 1 March 2012 This method might seem weird But it has a nice property The set of all normalised prices is convex and compact Frank Cowell: GE Excess Demand & Prices 13 Normalised prices, n=2 The set of normalised prices p2 The price vector (0,75, 0.25) (0,1) J={p: p0, p1+p2 = 1} (0, 0.25) • (0.75, 0) (1,0) March 2012 p1 Frank Cowell: GE Excess Demand & Prices 14 Normalised prices, n=3 p3 The set of normalised prices The price vector (0,5, 0.25, 0.25) (0,0,1) J={p: p0, p1+p2+p3 = 1} (0, 0, 0.25) • (0,1,0) (1,0,0) (0, 0.25 , 0) 0 (0.5, 0, 0) p1 March 2012 Frank Cowell: GE Excess Demand & Prices 15 Overview… General Equilibrium: Excess Demand+ Is there any p*? Excess Demand Functions Equilibrium Issues •Existence •Uniqueness •Stability Prices and Decentralisation March 2012 Frank Cowell: GE Excess Demand & Prices 16 Approach to the existence problem Imagine a rule that moves prices in the direction of excess demand: • “if Ei >0, increase pi” • “if Ei <0 and pi >0, decrease pi” • An example of this under “stability” below This rule uses the E-functions to map the set of prices into itself An equilibrium exists if this map has a “fixed point” • a p* that is mapped into itself? To find the conditions for this, use normalised prices • pJ • J is a compact, convex set We can examine this in the special case n = 2 • In this case normalisation implies that p2 1 p1 March 2012 Frank Cowell: GE Excess Demand & Prices 17 Why? Existence of equilibrium? * Why boundedness below? As p2 0, by normalisation, p11 ED diagram, normalised prices Excess demand function with well-defined equilibrium price Case with discontinuous E Case where excess demand for good2 is unbounded below As p2 0 if E2 is bounded below then p2E2 0 By Walras’ Law, this implies p1E1 0 as p11 So if E2 is bounded below then E1 can’t be everywhere positive 1 p1 Excess supply good 2 is free here p1* good 1 is free here 0 March 2012 Excess demand E-functions are: continuous, bounded below No equilibrium price where E crosses the axis E1 E never crosses the axis Frank Cowell: GE Excess Demand & Prices 18 Existence: a basic result An equilibrium price vector must exist if: excess demand functions are continuous and 2. bounded from below • (“continuity” can be weakened to “upper-hemi-continuity”) 1. Boundedness is no big deal • Can you have infinite excess supply…? However continuity might be tricky • Let's put it on hold • We examine it under “the rôle of prices” March 2012 Frank Cowell: GE Excess Demand & Prices 19 Overview… General Equilibrium: Excess Demand+ Is there just one p*? Excess Demand Functions Equilibrium Issues •Existence •Uniqueness •Stability Prices and Decentralisation March 2012 Frank Cowell: GE Excess Demand & Prices 20 The uniqueness problem Multiple equilibria imply multiple allocations, at normalised prices… …with reference to a given property distribution Will not arise if the E-functions satisfy WARP If WARP is not satisfied this can lead to some startling behaviour… let's see March 2012 Frank Cowell: GE Excess Demand & Prices 21 Multiple equilibria Three equilibrium prices Suppose there were more of resource 1 Now take some of resource 1 away 1 p1 single equilibrium jumps to here!! three equilibria degenerate to one! 0 March 2012 E1 Frank Cowell: GE Excess Demand & Prices 22 Overview… General Equilibrium: Excess Demand+ Will the system tend to p*? Excess Demand Functions Equilibrium Issues •Existence •Uniqueness •Stability Prices and Decentralisation March 2012 Frank Cowell: GE Excess Demand & Prices 23 Stability analysis We can model stability similar to physical sciences We need… • A definition of equilibrium • A process • Initial conditions Main question is to identify these in economic terms Simple example March 2012 Frank Cowell: GE Excess Demand & Prices 24 A stable equilibrium Stable: Equilibrium: If we apply a small shock Status quo isadjustment left the built-in undisturbed by gravity process (gravity) restores the status quo March 2012 Frank Cowell: GE Excess Demand & Prices 25 An unstable equilibrium * Equilibrium: Unstable: This If weactually apply afulfils small the shock definition the built-in adjustment process (gravity) moves us But… away from the status quo March 2012 Frank Cowell: GE Excess Demand & Prices 26 “Gravity” in the CE model Imagine there is an auctioneer to announce prices, and to adjust if necessary If good i is in excess demand, increase its price If good i is in excess supply, decrease its price (if it hasn't already reached zero) Nobody trades till the auctioneer has finished March 2012 Frank Cowell: GE Excess Demand & Prices 27 “Gravity” in the CE model: the auctioneer using tâtonnement* Announce p individual dd & ss individual individual dd & ssdd Adjust p Adjust p individual & ss dd & ss Adjust p …once we’re at equilibrium we trade Equilibrium? Equilibrium? Equilibrium ? Equilibrium? March 2012 Evaluate Evaluate excess dd Evaluate excess dd excess dd Evaluate excess dd Frank Cowell: GE Excess Demand & Prices 28 Adjustment and stability Adjust prices according to sign of Ei: • If Ei > 0 then increase pi • If Ei < 0 and pi > 0 then decrease pi A linear tâtonnement adjustment mechanism: Define distance d between p(t) and equilibrium p* Given WARP, d falls with t under tâtonnement Two examples: with/without WARP March 2012 Frank Cowell: GE Excess Demand & Prices 29 Globally stable… Start with a very high price 1 Yields excess supply Under tâtonnement price falls p1(0) Excess supply •p * 1 Start instead with a low price Yields excess demand Excess demand Under tâtonnement price rises If E satisfies p1 WARP then the system must converge… p1(0) E1 0 E1(0) March 2012 E1(0) Frank Cowell: GE Excess Demand & Prices 30 Not globally stable… * Start with a very high price 1 Start again with very low price p1 • Excess supply …now try a (slightly) low price …now try a (slightly) high price Locally Stable Check the “middle” crossing Excess demand Here WARP does not hold Unstable March 2012 • •0 Two locally stable equilibria Also locally stable One unstable E1 Frank Cowell: GE Excess Demand & Prices 31 Overview… General Equilibrium: Excess Demand+ The separation theorem and the role of large numbers Excess Demand Functions Equilibrium Issues Prices and Decentralisation March 2012 Frank Cowell: GE Excess Demand & Prices 32 Decentralisation Recall the important result on decentralisation • discussed in the case of Crusoe’s island The counterpart is true for this multi-person world Requires assumptions about convexity of two sets, defined at the aggregate level: A := {x: x q+R, F(q) • the “better-than” set: B(x*) := {Shxh: Uh(xh )Uh(x*h ) } • the “attainable set”: To see the power of the result here… • use an “averaging” argument • previously used in lectures on the firm March 2012 Frank Cowell: GE Excess Demand & Prices 33 Decentralisation again * The attainable set The “Better-than-x* ” set The price line Decentralisation x2 A = {x: x q+R, F(q)0} x* B = {Shxh: Uh(xh) Uh(x*h)} B x* maximises income over A p1 p2 x* minimises expenditure over B A 0 March 2012 x1 Frank Cowell: GE Excess Demand & Prices 34 Problems with prices Either non-convex technology (increasing returns or other indivisibilities) for some firms, or… …non-convexity of B-set (non-concave- contoured preferences) for some households… …may imply discontinuous excess demand function and so… …absence of equilibrium But if there are large numbers of agents everything may be OK two examples March 2012 Frank Cowell: GE Excess Demand & Prices 35 A non-convex technology* output One unit of input produces exactly one of output B q' The case with 1 firm Rescaled case of 2 firms, … 4 , 8 , 16 Limit of the averaging process The “Better-than” set • q* “separating” prices and equilibrium Limiting attainable set is convex q° input Equilibrium q* is sustained by a mixture of firms at q° and q' A March 2012 Frank Cowell: GE Excess Demand & Prices 36 Non-convex preferences* The case with 1 person Rescaled case of 2 persons, x2 A continuum of consumers The attainable set No equilibrium here “separating” prices and equilibrium x' • x* Limiting better-than set is convex B Equilibrium x* is sustained by a mixture of consumers at x° and x' x° A x1 March 2012 Frank Cowell: GE Excess Demand & Prices 37 Summary Review Excess demand functions are handy tools for getting results Review Continuity and boundedness ensure existence of equilibrium Review WARP ensures uniqueness and stability Review But requirements of continuity may be demanding March 2012 Frank Cowell: GE Excess Demand & Prices 38