Prerequisites Almost essential General equilibrium Basics Useful but

Prerequisites Almost essential General equilibrium: Basics Useful, but optional General Equilibrium: Price Taking GENERAL EQUILIBRIUM: EXCESS DEMAND THE RÔLE OF PRICES MICROECONOMICS Principles and Analysis Frank Cowell April 2018 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? § Will this determine the price system for us? § We will address these questions using the standard model of a general-equilibrium system § To do this we need just one more new concept April 2018 Frank Cowell: GE Excess Demand & Prices 2

Overview General Equilibrium: Excess Demand+ Definition and properties Excess Demand Functions Equilibrium Issues Prices and Decentralisation April 2018 Frank Cowell: GE Excess Demand & Prices 3

Ingredients of the excess demand function § Three ingredients: 1. Aggregate demands (sum of individual households' demands) 2. Aggregate net-outputs (sum of individual firms' net outputs) 3. Resources § Incomes (in 1 and 2) determined by the price system April 2018 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 resources and shares xih = xih(p) § 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) April 2018 § standard supply functions/ demand for inputs §valid if there are no externalities. As in “Firm and the market”) 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 p 1 p 1 b a Alf April 2018 x 1 Bill x 1 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 q 1 low-cost firm April 2018 q 1+q 2 high-cost firm both firms Frank Cowell: GE Excess Demand & Prices 7

Subtract q and R from x to get E: p 1 Demand Supply q 1 x 1 p 1 Resource stock p 1 R 1 net output of i aggregated over f demand for i aggregated over h Ei(p) : = xi(p) – qi(p) – Ri Exces April 2018 s Dem and 1 Resource stock of i E 1 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: Ei(p*) 0 The materials balance condition (dressed up a bit) If this is violated, then somebody, § For each good i that has a positive price in equilibrium somewhere isn't maximising… (i. e. if pi* > 0): Ei(p*) = 0 April 2018 Can only have excess supply of a good in equilibrium if 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 1. From technology compute firms’ net output functions and profits 2. From property rights compute 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 • is there a solution? • is there just one solution? • will the market mechanism pick this solution? April 2018 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 April 2018 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? Note: these hold for any competitive allocation, not just equilibrium April 2018 Frank Cowell: GE Excess Demand & Prices 12

* detail on slide can only be seen if you run the slideshow Price normalisation § We may need to convert from n numbers p 1, p 2, …pn to n 1 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 Mars. Bar to give a neat oftheory n-1 standard value system “Marxian” Mars bar theory of ofvalue set of set prices thatprices sum to 1 April 2018 § 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 § J: the set of normalised prices p 2 l § The price vector (0, 75, 0. 25) (0, 1) J={p: p 0, p 1 + p 2 = 1} (0, 0. 25) • (0. 75, 0) l (1, 0) April 2018 p 1 Frank Cowell: GE Excess Demand & Prices 14

Normalised prices, n = 3 p 3 § J: the set of normalised prices l § The price vector (0, 5, 0. 25) (0, 0, 1) J={p: p 0, p 1+p 2+p 3 = 1} p 2 (0, 0, 0. 25) • l (0, 1, 0) l (1, 0, 0) (0, 0. 25 , 0) 0 (0. 5, 0, 0) p 1 April 2018 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 April 2018 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 • p J • J is a compact, convex set § We can examine this in the special case n = 2 • in this case normalisation implies that p 2 º 1 p 1 April 2018 Frank Cowell: GE Excess Demand & Prices 17

Why? * detail on slide can only be seen if you run the slideshow Existence of equilibrium? Why boundedness below? As p 2 0, by normalisation, p 1 1 § ED diagram, normalised prices § Case with well-defined eqm price § Case with discontinuous E § Case where E 2 is unbounded below As p 2 0 if E 2 is bounded below then p 2 E 2 0 By Walras’ Law, this implies p 1 E 1 0 as p 1 1 So if E 2 is bounded below then E 1 can’t be everywhere positive 1 p 1 Excess supply good 2 is free here l p 1* good 1 is free here § 0 April 2018 Excess demand § E-functions are: continuous, bounded below § No equilibrium price where E crosses the axis E 1 E never crosses the axis Frank Cowell: GE Excess Demand & Prices 18

Existence: a basic result § An equilibrium price vector must exist if: 1. excess demand functions are continuous and 2. bounded from below • (“continuity” can be weakened to “upper-hemi-continuity”) § 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” April 2018 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 April 2018 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 April 2018 Frank Cowell: GE Excess Demand & Prices 21

* detail on slide can only be seen if you run the slideshow Multiple equilibria § Three equilibrium prices § Suppose there were more of resource 1 § Now take some of resource 1 away 1 p 1 single equilibrium jumps to here!! three equilibria degenerate to one! 0 April 2018 E 1 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 April 2018 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 April 2018 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 April 2018 Frank Cowell: GE Excess Demand & Prices 25

* detail on slide can only be seen if you run the slideshow 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 April 2018 Frank Cowell: GE Excess Demand & Prices 26

“Gravity” in the CE model § Imagine there is an auctioneer • to announce prices • to adjust prices 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 April 2018 Frank Cowell: GE Excess Demand & Prices 27

* detail on slide can only be seen if you run the slideshow “Gravity” in the CE model: the auctioneer using tâtonnement Announce p individual dd & ssdd Adjust p individual & ss dd & ss Adjust p …once we’re at equilibrium we trade Equilibrium? April 2018 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 April 2018 Frank Cowell: GE Excess Demand & Prices 29

Globally stable… 1 p 1(0) Excess supply • p * 1 Excess demand § Start with a very high price § Yields excess supply § Under tâtonnement price falls § Start instead with a low price § Yields excess demand § Under tâtonnement price rises § If E satisfies p 1(0) E 1(0) April 2018 0 WARP then the system must converge E 1(0) Frank Cowell: GE Excess Demand & Prices 30

* detail on slide can only be seen if you run the slideshow Not globally stable… § Start with a very high price § …now try a (slightly) low price 1 § Start again with very low price p 1 • Excess supply Unstable April 2018 Locally Stable § …now try a (slightly) high price § Check the “middle” crossing Excess demand § Here WARP does not hold • • 0 § Two locally stable equilibria Also locally stable § One unstable E 1 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 April 2018 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) 0 } • 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 April 2018 Here “Better-than” is used as shorthand for “Better-than-orjust-as-good-as-” Frank Cowell: GE Excess Demand & Prices 33

* detail on slide can only be seen if you run the slideshow Decentralisation again § The attainable set § The “Better-than-x* ” set § The price line § Decentralisation x 2 § A = {x: x q+R, F(q) 0} x* § B = {Shxh: Uh(xh) Uh(x*h)} B p 1 § x* maximises income over A p 2 § x* minimises expenditure over B A 0 April 2018 x 1 Frank Cowell: GE Excess Demand & Prices 34

Problems with prices § Given (1) non-convex technology for some firms • increasing returns or other indivisibilities § Or (2) non-convexity of B-set for some households • non-concave contoured preferences § Then there may be discontinuous excess demand functions § So there may be no equilibrium § But if there are large numbers of agents everything may be OK two examples April 2018 Frank Cowell: GE Excess Demand & Prices 35

* detail on slide can only be seen if you run the slideshow One unit of input produces exactly one of output A non-convex technology 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 April 2018 Frank Cowell: GE Excess Demand & Prices 36

* detail on slide can only be seen if you run the slideshow Non-convex preferences § The case with 1 person § Rescaled case of 2 persons, x 2 § A continuum of consumers § The attainable set No equilibrium here § “separating” prices and equilibrium x' • x* B x° § In the limit, better-than set is convex § Equilibrium x* is sustained by a mixture of consumers at x° and x' A x 1 April 2018 Frank Cowell: GE Excess Demand & Prices 37

Summary § Excess demand functions are handy tools for getting results § Continuity and boundedness ensure existence of equilibrium § WARP ensures uniqueness and stability § But requirements of continuity may be demanding April 2018 Frank Cowell: GE Excess Demand & Prices 38