GENERAL EQUILIBRIUM: EXCESS DEMAND AND THE RÔLE OF PRICES

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MICROECONOMICS Principles and Analysis Frank Cowell Note: the detail in slides marked *

1 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 July 2017 Frank Cowell: GE Excess Demand & Prices 1

2 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 July 2017 Frank Cowell: GE Excess Demand & Prices 2

3 Overview General Equilibrium: Excess Demand+ Definition and properties Excess Demand Functions Equilibrium Issues Prices and Decentralisation July 2017 Frank Cowell: GE Excess Demand & Prices 3

4 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 July 2017 Frank Cowell: GE Excess Demand & Prices 4

5 Aggregate consumption, net output From household s demand function x ih = D ih (p, y h ) = D ih (p, y h (p) ) So demands are just functions of p x ih = x ih (p) If all goods are private (rival) then aggregate demands can be written: x i (p) = Σ h x ih (p) From firm s supply of net output q i f = q if (p) Aggregate: q i = Σ f q if (p) Because incomes depend on prices x ih ( ) depends on holdings of resources and shares Rival : extra consumers require additional resources. Same as consumer: aggregation standard supply functions/ demand for inputs valid if there are no externalities. As in Firm and the market ) July 2017 Frank Cowell: GE Excess Demand & Prices 5

6 Derivation of x i (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 p 1 x 1 a x 1 b x 1 Alf Bill The Market July 2017 Frank Cowell: GE Excess Demand & Prices 6

7 Derivation of q i (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 q 2 q 1 +q 2 low-cost firm high-cost firm both firms July 2017 Frank Cowell: GE Excess Demand & Prices 7

Subtract q and R from x to get E: p 1 p 1 p

i (p) q i (p) R i 1 Resource stock of i E 1

8 Subtract q and R from x to get E: p 1 p 1 p 1 Demand x 1 Supply q 1 Resource stock R 1 p 1 demand for i aggregated over h net output of i aggregated over f E i (p) := x i (p) q i (p) R i 1 Resource stock of i E 1 July 2017 Frank Cowell: GE Excess Demand & Prices 8

9 Equilibrium in terms of Excess Demand Equilibrium is characterised by a price vector p* 0 such that: For every good i: E i (p*) 0 The materials balance condition (dressed up a bit) For each good i that has a positive price in equilibrium (i.e. if p i * > 0): E i (p*) = 0 If this is violated, then somebody, somewhere isn't maximising Can only have excess supply of a good in equilibrium if price of that good is 0 July 2017 Frank Cowell: GE Excess Demand & Prices 9

10 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? July 2017 Frank Cowell: GE Excess Demand & Prices 10

11 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 July 2017 Frank Cowell: GE Excess Demand & Prices 11

Hint #1: think about the "adding-up" property of demand functions Hint #2: think about the homogeneity property of demand functions You

12 Two fundamental properties Walras Law. For any price p: n Σ p i E i (p) = 0 i=1 Homogeneity of degree 0. For any price p and any t > 0 : E i (tp) = E i (p) Can you explain why they are true? Hint #1: think about the "adding-up" property of demand functions Hint #2: think about the homogeneity property of demand functions You only have to work with n 1 (rather than n) equations You can normalise the prices by any positive number Note: these hold for any competitive allocation, not just equilibrium July 2017 Frank Cowell: GE Excess Demand & Prices 12

13 *Price normalisation We may need to convert from n numbers p 1, p 2, p n 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 a numéraire p labour MarsBar p n Σ p i i=1 to give a Marxian Mars set neat standard of set bar prices of theory value n-1 that prices of system sum of value to 1 This method might seem weird But it has a nice property The set of all normalised prices is convex and compact July 2017 Frank Cowell: GE Excess Demand & Prices 13

14 Normalised prices, n=2 p 2 The set of normalised prices The price vector (0,75, 0.25) (0,1) J={p: p 0, p 1 + p 2 = 1} (0, 0.25) (0.75, 0) (1,0) p 1 July 2017 Frank Cowell: GE Excess Demand & Prices 14

15 Normalised prices, n=3 p 3 The set of normalised prices (0,0,1) The price vector (0,5, 0.25, 0.25) J={p: p 0, p 1 +p 2 +p 3 = 1} (0, 0, 0.25) (0, 0.25, 0) (0,1,0) 0 (0.5, 0, 0) (1,0,0) p 1 July 2017 Frank Cowell: GE Excess Demand & Prices 15

16 Overview General Equilibrium: Excess Demand+ Is there any p*? Excess Demand Functions Equilibrium Issues Existence Uniqueness Stability Prices and Decentralisation July 2017 Frank Cowell: GE Excess Demand & Prices 16

17 Approach to the existence problem Imagine a rule that moves prices in the direction of excess demand: if E i >0, increase p i if E i <0 and p i >0, decrease p i 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 1 p 1 July 2017 Frank Cowell: GE Excess Demand & Prices 17

Why? *Existence of equilibrium? Why boundedness below?

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 Excess

Excess demand ED diagram, normalised prices Case with well-defined eqm price Case with discontinuous E

18 Why? *Existence of equilibrium? Why boundedness below? As p 2 0, by normalisation, p 1 1 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 Excess supply good 1 is free here 1 E-functions are: p 1 good 2 is continuous, free here bounded below p 1 * 0 Excess demand ED diagram, normalised prices Case with well-defined eqm price Case with discontinuous E Case where E 2 is unbounded below No equilibrium price where E crosses the axis E 1 E never crosses the axis July 2017 Frank Cowell: GE Excess Demand & Prices 18

19 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 July 2017 Frank Cowell: GE Excess Demand & Prices 19

20 Overview General Equilibrium: Excess Demand+ Is there just one p*? Excess Demand Functions Equilibrium Issues Existence Uniqueness Stability Prices and Decentralisation July 2017 Frank Cowell: GE Excess Demand & Prices 20

21 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 July 2017 Frank Cowell: GE Excess Demand & Prices 21

! p 1 three equilibria degenerate to one!

22 *Multiple equilibria Three equilibrium prices Suppose there were more of resource 1 Now take some of resource 1 away 1 single equilibrium jumps to here!! p 1 three equilibria degenerate to one! 0 E 1 July 2017 Frank Cowell: GE Excess Demand & Prices 22

23 Overview General Equilibrium: Excess Demand+ Will the system tend to p*? Excess Demand Functions Equilibrium Issues Existence Uniqueness Stability Prices and Decentralisation July 2017 Frank Cowell: GE Excess Demand & Prices 23

24 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 July 2017 Frank Cowell: GE Excess Demand & Prices 24

undisturbed process (gravity) by gravity restores the

25 A stable equilibrium Stable: Equilibrium: If we apply a small shock Status the built-in quo is adjustment left undisturbed process (gravity) by gravity restores the status quo July 2017 Frank Cowell: GE Excess Demand & Prices 25

26 *An unstable equilibrium Equilibrium: Unstable: This If we actually apply a fulfils small the shock definition the built-in adjustment process (gravity) moves us But away from the status quo July 2017 Frank Cowell: GE Excess Demand & Prices 26

27 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 July 2017 Frank Cowell: GE Excess Demand & Prices 27

28 * Gravity in the CE model: the auctioneer using tâtonnement Announce p Adjust p Adjust p Equilibrium? Equilibrium? Adjust p Equilibrium? Equilibrium? once we re at equilibrium we trade individual dd & ss individual dd & ss individual dd & ss individual dd & ss Evaluate excess dd Evaluate excess dd Evaluate excess dd Evaluate excess dd July 2017 Frank Cowell: GE Excess Demand & Prices 28

mechanism: Define distance d between p(t) and equilibrium p * Given WARP, d falls with t

29 Adjustment and stability Adjust prices according to sign of E i : If E i > 0 then increase p i If E i < 0 and p i > 0 then decrease p i 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 July 2017 Frank Cowell: GE Excess Demand & Prices 29

30 Globally stable Excess supply 1 p 1 (0) p 1 * 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 E 1 (0) p 1 p 1 (0) 0 E 1 (0) E 1 If E satisfies WARP then the system must converge July 2017 Frank Cowell: GE Excess Demand & Prices 30

price now try a (slightly) high price Check the middle crossing E 1 Here WARP does not hold

31 *Not globally stable Excess supply Unstable p 1 1 Locally Stable Also locally stable 0 Excess demand Start with a very high price now try a (slightly) low price Start again with very low price now try a (slightly) high price Check the middle crossing E 1 Here WARP does not hold Two locally stable equilibria One unstable July 2017 Frank Cowell: GE Excess Demand & Prices 31

32 Overview General Equilibrium: Excess Demand+ The separation theorem and the role of large numbers Excess Demand Functions Equilibrium Issues Prices and Decentralisation July 2017 Frank Cowell: GE Excess Demand & Prices 32

33 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: the attainable set : A := {x: x q + R, Φ(q) 0 } the better-than set: B(x*) := {Σ h x h : U h (x h ) U h (x* h ) } To see the power of the result here use an averaging argument previously used in lectures on the firm Here Better-than is used as shorthand for Better-than-orjust-as-good-as- July 2017 Frank Cowell: GE Excess Demand & Prices 33

34 *Decentralisation again x 2 The attainable set The Better-than-x * set The price line Decentralisation A = {x: x q+r, Φ(q) 0} A x * B p 1 p 2 B = {Σ h x h : U h (x h ) U h (x *h )} x* maximises income over A x* minimises expenditure over B 0 x 1 July 2017 Frank Cowell: GE Excess Demand & Prices 34

35 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 July 2017 Frank Cowell: GE Excess Demand & Prices 35

q* separating prices and equilibrium Limiting attainable set is convex q input Equilibrium q * is

36 *A non-convex technology One unit of input produces exactly one of output q' B output 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 July 2017 Frank Cowell: GE Excess Demand & Prices 36

equilibrium A x* x B Limiting better-than set is convex Equilibrium x * is sustained

37 *Non-convex preferences x 2 The case with 1 person Rescaled case of 2 persons, A continuum of consumers x' No equilibrium here The attainable set separating prices and equilibrium A x* x B Limiting better-than set is convex Equilibrium x * is sustained by a mixture of consumers at x and x' x 1 July 2017 Frank Cowell: GE Excess Demand & Prices 37

38 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 July 2017 Frank Cowell: GE Excess Demand & Prices 38

The Multi-Output Firm

Prerequisites Almost essential Firm: Optimisation Useful, but optional Firm: Demand and Supply The Multi-Output Firm MICROECONOMICS Principles and Analysis Frank Cowell October 2006 Introduction This presentation