EC /8 MICHAELMAS TERM SLIDE PACK

Transcription

1 EC /8 MICHAELMAS TERM SLIDE PACK five sections: Firm / Consumer / General Equilibrium / Risk & Uncertainty / Welfare first slide of each lecture has green border full versions at for more, contact f.cowell@lse.ac.uk THE FIRM: Lectures 1-4 Quantities z i amount of input i z = (z 1, z 2,, z m ) input vector Z input requirement set q amount of output (single firm) q f output of firm f Prices and profits w i price of input i w = (w 1, w 2,, w m ) input-price vector p price of output profits Functions production function C cost function H i conditional demand for input i S supply function D i ordinary demand for input i Other Lagrange multiplier (min cost) elasticity of demand 1

2 1-output, 2-input production function q q > (z) q = (z) q < (z) (z 1, z 2 ) Interior: feasible but inefficient Boundary: feasible and efficient Outside : infeasible 0 z 2 In general: q (z 1, z 2,, z m ) Equivalently: q (z) gives maximum output that can be produced from given inputs The input requirement set z 2 q = (z) Z(q) q < (z) Feasible but inefficient Feasible and technically efficient Infeasible points q > (z) Pick a particular output level q Find a feasible input vector z Repeat to find all such vectors for given q Get the input-requirement set: Z(q) := {z: (z) q} z 1 2

3 Case 1: Z smooth, strictly convex z 2 Z(q) Pick two boundary points Draw the line between them Intermediate points lie in the interior of Z z q = (z') q = (z") q< (z) z Important role of convexity Combination of two techniques may produce more output What if we changed some of the assumptions? z 1 Case 2: Z Convex (but not strictly) z 2 Z(q) Pick two boundary points Draw the line between them Intermediate points lie in Z (perhaps on the boundary) z z A combination of feasible techniques is also feasible z 1 3

4 Case 3: Z smooth but not convex z 2 Z(q) Join two points across the dent Take an intermediate point Highlight zone where this can occur in this region there is an indivisibility z 1 Case 4: Z convex but not smooth z 2 q = (z) Slope of the boundary is undefined at this point z 1 4

5 Isoquants Pick a particular output level q Find the input requirement set Z(q) The isoquant is the boundary of Z:{ z : (z) = q } If the function is differentiable at z then the marginal rate of technical substitution is the slope at z: j (z) i (z) Gives rate at which you trade off one input against another along the isoquant, maintaining constant q Think of the isoquant as an integral part of the set Z(q) Where appropriate, use subscript to denote partial derivatives. So (z) i (z) := zi. Isoquant, input ratio, MRTS z 2 MRTS 21 = 1 (z)/ 2 (z) z 2 z z z 2 / z 1 = constant {z: (z)=q} The set Z(q) A contour of the function An efficient point The input ratio Marginal Rate of Technical Substitution Increase the MRTS The isoquant is the boundary of Z Input ratio describes one production technique MRTS 21 : implicit price of input 1 in terms of 2 z 1 z 1 Higher price : smaller relative use of input 1 5

6 MRTS and substitution Responsiveness of input ratio to MRTS z 2 z 2 low high z 1 z 1 Homothetic contours z 2 The isoquants Draw any ray through the origin Get same MRTS as it cuts each isoquant O z 1 6

7 Contours of a homogeneous function The isoquants z 2 Coordinates of input z Coordinates of scaled up input tz tz 2 tz z 2 z t r q (tz) = t r (z) q O z 1 tz 1 z 1 Case 1: IRTS q An increasing returns to scale function Pick an arbitrary point on the surface The expansion path t > 1 implies (tz) > t(z) 0 z 2 Double inputs and you more than double output 7

8 Case 2: DRTS q A decreasing returns to scale function Pick an arbitrary point on the surface The expansion path t > 1 implies (tz) < t(z) 0 z 2 Double inputs and output increases by less than double Case 3: CRTS q A constant returns to scale function Pick a point on the surface The expansion path is a ray (tz) = t(z) 0 z 2 Double inputs and output exactly doubles 8

9 Relationship to isoquants q Take any one of the three cases (here it is CRTS) Take a horizontal slice Project down to get the isoquant Repeat to get isoquant map 0 z 2 Isoquant map is projection of the set of feasible points Marginal products Pick a technically efficient input vector Any z such that q= (z) Keep all but one input constant Measure the marginal change in output w.r.t. this input (z) MP i = i (z) = zi. The marginal product 9

10 CRTS production function again q Now take a vertical slice The resulting path for z 2 = constant 0 z 2 Let s look at its shape MP for the CRTS function q (z) (z) The feasible set Technically efficient points Slope of tangent is the marginal product of input 1 Increase z 1 A section of the production function Input 1 is essential: If z 1 = 0 then q = 0 z 1 1 (z) falls with z 1 (or stays constant) if is concave 10

11 Relationship between q and z 1 q We ve just taken the conventional case q In general this curve depends on shape of z 1 z 1 q Other possibilities for the relation between output and one input q z 1 z 1 Optimisation: the standard approach Choose q and z to maximise := pq m w i z i i=1...subject to the production constraint... q (z) Could also write this as zz(q)..and some obvious constraints: q 0 z 0 You can t have negative output or negative inputs 11

12 A standard optimisation method If is differentiable Set up a Lagrangian to take care of the constraints Write down the First Order Conditions (FOC) Check out second-order conditions Use FOC to characterise solution L (... ) L (... ) = 0 z 2 L (... ) z 2 z * = Stage 1 optimisation Pick a target output level q Take as given the market prices of inputs w Maximise profits......by minimising costs m w i z i i=1 12

13 Iso-cost lines z 2 Draw set of points where cost of input is c, a constant Repeat for a higher value of the constant w 1 z 1 + w 2 z 2 = c" Imposes direction on the diagram... w 1 z 1 + w 2 z 2 = c' w 1 z 1 + w 2 z 2 = c z 1 Use this to derive optimum Cost-minimisation z 2 q The firm minimises cost... Subject to output constraint Defines the stage 1 problem Solution to the problem minimise z* m w i z i i=1 subject to (z) q But solution depends on shape of the input-requirement set Z z 1 What would happen in other cases? 13

14 Convex, but not strictly convex Z z 2 Any z in this set is cost-minimising z 1 An interval of solutions Convex Z, touching axis z 2 z 1 Here MRTS 21 > w 1 / w 2 at the solution. z* Input 2 is too expensive and so isn t used: z 2 * = 0 14

15 Non-convex Z z 2 z* There could be multiple solutions. z** But note that there s no solution point between z* and z** z 1 Non-smooth Z z 2 z* z* is unique costminimising point for q z 1 True for all positive finite values of w 1, w 2 15

16 Cost-minimisation: strictly convex Z Minimise m w i z i i=1 * * 1 (z ) = w 1 * * 2 (z ) = w 2 * * m (z ) = w m q = (z *) + [q q (z)] (z) Because of strict convexity we have an interior solution A set of m+1 First-Order Conditions Use the objective function...and output constraint...to build the Lagrangian Differentiate w.r.t. z 1,..., z m ; set equal to 0... and w.r.t Denote cost minimising values by * From the FOC If both inputs i and j are used and MRTS is defined then... i (z ) w = i j (z ) w j MRTS = input price ratio implicit price = market price If input i could be zero then... i (z ) w i j (z ) w j MRTS ji input price ratio implicit price market price 16

17 The solution Solving the FOC, get a cost-minimising value for each input... z i* = H i (w, q)...for the Lagrange multiplier * = * (w, q)...and for the minimised value of cost itself The cost function is defined as C(w, q) := min w i z i {(z) q} Interpreting the Lagrange multiplier The solution function: C(w, q) = i w i z i * = i w i z i* * [(z * ) q] Differentiate with respect to q: C q (w, q) = i w i H i q(w, q) * [ i i (z * ) H i q(w, q) 1] Rearrange: C q (w, q) = i [w i * i (z * )] H i q(w, q) + * C q (w, q) = * At the optimum, either the constraint binds or the Lagrange multiplier is zero Express demands in terms of (w,q) Lagrange multiplier in (original) stage- 1 problem is just marginal cost Result is just an applications of a general envelope theorem It holds for the untransformed, original version of the problem If we use a transformed version of the constraint (for example log q log (z)) we get a different Lagrange multiplier) 17

18 Properties of C C z 1 * C(w, q+q) C(w, q) Draw cost as function of w 1 Cost is non-decreasing in input prices Increasing in output, if continuous Concave in input prices Shephard s Lemma C(tw+[1 t]w,q) tc(w,q) + [1 t]c(w,q) w 1 C(w,q) = z j * w j What happens to cost if w changes to tw z 2 q Find cost-minimising inputs for w, given q Find cost-minimising inputs for tw, given q So we have: z* C(tw,q) = i tw i z i* = t i w i z i* = tc(w,q) The cost function is homogeneous of degree 1 in prices. z 1 18

19 Average and marginal cost p increasing returns to scale decreasing returns to scale The average cost curve Slope of AC depends on RTS Marginal cost cuts AC at its minimum C q C/q q q Revenue and profits C q C/q A given market price p Revenue if output is q Cost if output is q Profits if output is q Profits vary with q Maximum profits p price = marginal cost q q q q* q q 19

20 What happens if price is low... C q C/q p price < average cost q* = 0 q Profit maximisation Objective: choose q to max pq C (w, q) Revenue minus minimised cost From FOC if q* > 0: p = C q (w, q*) C(w, q*) p q* In general: pq* C(w, q*) Price equals marginal cost Price covers average cost covers both the cases: q* > 0 and q* = 0 20

21 The first response function Review the cost-minimisation problem and its solution Choose z to minimise m w i z i subject to q (z), z 0 i=1 The firm s cost function: C(w, q) := min w i z i {(z) q} Cost-minimising value for each input: z i* = H i (w, q), i=1,2,,m The stage 1 problem The solution function H i is the conditional input demand function Demand for input i, conditional on given output level q Mapping into (z 1,w 1 )-space Conventional case of Z Start with any value of w 1 ( the slope of the tangent to Z) z 2 w 1 Repeat for a lower value of w 1 and again to get the conditional demand curve Constraint set is convex, with smooth boundary Response function is a continuous map: H 1 (w,q) z 1 z 1 21

22 Another map into (z 1,w 1 )-space Now take case of nonconvex Z z 2 w 1 Start with a high value of w 1 Repeat for a very low value of w 1 Points nearby work the same way But what happens in between? A demand correspondence Constraint set is nonconvex Response is discontinuous map: jumps in z* Map is multivalued at the discontinuity z 1 z 1 Use the cost function Recall this relationship? C i (w, q) = z i * Shephard's lemma So we have: C i (w, q) = H i (w, q) Link between conditional input demand and cost functions Differentiate this with respect to w j C ij (w, q) = H ji (w, q) Slope of conditional input demand function derived from second derivative of cost function 22

23 Two simple results Use a standard property 2 () 2 () = w i w j w j w i second derivatives commute So C ij (w, q) = C ji (w,q) The order of differentiation is irrelevant Therefore H ji (w,q) = H ij (w,q) Effect of price of i on conditional demand for j equals effect of the price of j on conditional demand for i We also have: C ii (w, q) = H ii (w, q) We've just put j = i We know C ii (w, q) 0 Because cost function is concave Therefore: H ii (w, q) 0 The relationship of conditional demand for an input with its own price cannot be positive Conditional input demand curve w 1 Consider the demand for input 1 H 1 (w,q) Consequence of result 2? Downward-sloping conditional demand In some cases it is also possible that H i i = 0 z 1 Corresponds to the case where isoquant is kinked: multiple w values consistent with same z* 23

24 The second response function Review the profit-maximisation problem and its solution Choose q to maximise: pq C (w, q) From the FOC: p = C q (w, q * ), if q * > 0 pq * C(w, q * ) Profit-maximising value for output: q * = S (w, p) The stage 2 problem Price equals marginal cost Price covers average cost S is the supply function (again it may actually be a correspondence) Supply of output and output price Use the FOC: C q (w, q) = p Use the supply function for q: C q (w, S(w, p) ) = p marginal cost equals price Gives an equation in w and p Differentiate with respect to p C qq (w, S(w, p) ) S p (w, p) = 1 Use the function of a function rule Rearrange: 1. S p (w, p) = C qq (w, q) Gives slope of supply function 24

25 The firm s supply curve p The firm s AC and MC curves For given p read off optimal q* Continue down to p What happens below p C q C/q Supply response is given by q = S(w,p) p_ q_ q Case illustrated is for with first IRTS, then DRTS Response is a discontinuous map: jumps in q* Map is multivalued at the discontinuity The third response function Recall the first two response functions: z i* = H i (w,q) q * = S (w, p) Now substitute for q* : z i* = H i (w, S(w, p) ) Use this to define a new function: D i (w,p) := H i (w, S(w, p) ) Demand for input i, conditional on output q Supply of output Stages 1 & 2 combined Demand for input i (unconditional ) Use this relationship to analyse further the firm s response to price changes 25

26 Demand for i and the price of output Take the relationship D i (w, p) = H i (w, S(w, p)) Differentiate with respect to p: D pi (w, p) = H qi (w, q * ) S p (w, p) D i increases with p iff H i increases with q. Reason? Supply increases with price ( S p > 0) But we also have, for any q: Shephard s Lemma again H i (w, q) = C i (w, q) H qi (w, q) = C iq (w, q) Substitute in the above: D pi (w, p) = C qi (w, q * )S p (w, p) Demand for input i(d i ) increases with p iff marginal cost (C q ) increases with w i Demand for i and the price of j Again take the relationship D i (w, p) = H i (w, S(w, p)) Differentiate w.r.t. w j : D ji (w, p) = H ji (w, q * )+ H qi (w, q * )S j (w, p) Use Shephard s Lemma again: H qi (w, q)= C iq (w, q) Use this and the previous result on S j (w, p) to give a decomposition into a substitution effect and an output effect : C jq (w, q * ) D ji (w, p) = H ji (w, q * ) C iq (w, q * ) C qq (w, q * ). output effect substitution effect Substitution effect is just slope of conditional input demand curve Output effect is [effect of w j on q][effect of q on demand for i] 26

27 Results from decomposition formula Take the general relationship: C iq (w, q * )C jq (w, q * ) D ji (w, p) = H ji (w, q * ) C qq (w, q * ). The effect w i on demand for input j equals the effect of w j on demand for input i Now take the special case where j = i: C iq (w, q * ) 2 D ii (w, p) = H ii (w, q * ) C qq (w, q * ). If w i increases, the demand for input i cannot rise Input-price fall: substitution effect w 1 conditional demand curve H 1 (w,q) The initial equilibrium price of input falls value to firm of price fall, given a fixed output level price fall Change in cost * z 1 z 1 27

28 Input-price fall: total effect w 1 The initial equilibrium Substitution effect of inputprice of fall Total effect of input-price fall price fall * z 1 ** z 1 z 1 The short-run problem Choose q and z to maximise := pq w m i z i i=1 subject to q (z), q 0, z 0, and z m =z m Letq be the q for which z m =z m would be chosen in the unrestricted problem ~ _ C(w, q, z m ) := min w i z i The solution function with {z m =z m } the side constraint Short-run demand for input i: ~ _ ~ _ H i (w, q, z m ) =C i (w, q, z m ) ~ _ C(w, q) C(w, q, z m ) So, dividing by q: ~ C(w, q) C(w, q, z m ) q q From Shephard s Lemma By definition of cost function Short-run AC long-run AC 28

29 MC, AC and supply in short and long run AC if all inputs are variable p ~ C/q ~ C q C q C/q MC if all inputs are variable Fix an output level AC if input m is now kept fixed MC if input m is now kept fixed Supply curve in long run Supply curve in short run SRAC touches LRAC at the given output SRMC cuts LRMC at the given output q q The supply curve is steeper in the short run Conditional input demand w 1 The original demand curve for input 1 H 1 (w,q) The demand curve from the problem with the side constraint Downward-sloping conditional demand Conditional demand curve is steeper in the short run ~ _ H 1 (w, q, z m ) z 1 29

30 A market with two firms 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 Market supply curve (2) p p p Below p' neither firm is in the market Between p' and p'' only firm 1 is in the market Above p'' both firms are in the market p' p" p" p' q 1 q 2 q 1 +q 2 low-cost firm high-cost firm both firms 30

31 Take two identical firms p p p' p' q 1 q Sum to get aggregate supply p p' q 1 + q 2 31

32 Numbers and average supply p Rescale to get average supply of the firms Compare with S for just one firm Repeat to get average S of 4 firms average S of 8 firms of 16 firms p' average(q f ) The limiting case p The limit: continuous averaged supply curve A solution to the non-existence problem? A well-defined equilibrium Firms outputs in equilibrium average demand average supply p' average(q f )

33 Industry supply: negative externality Each firm s S-curve (MC) shifted by the other s output The result of simple MC at each output level Industry supply allowing for interaction p S 1 (q 2 =5) p S 2 (q 1 =5) p S MC 1 +MC 2 S 1 (q 2 =1) S 2 (q 1 =1) MC 1 +MC 2 q 1 q 2 q 1 + q 2 firm 1 alone firm 2 alone both firms Analysing firms equilibrium price = marginal cost price average cost determines output of any one firm determines number of firms Entry mechanism: if p C/q gap is large then another firm could enter applying this iteratively determines the size of the industry (0) Assume that firm 1 makes a positive profit (1) Is pq C set-up costs of a new firm? if YES then stop. We ve got the eqm # of firms otherwise continue: (2) Number of firms goes up by 1 (3) Industry output goes up (4) Price falls (D-curve) and firms adjust output (individual firm s S-curve) (5) Back to step 1 33

34 Firm equilibrium with entry price marginal cost average cost Draw AC and MC Get supply curve from MC Use price to find output Profits in temporary equilibrium Allow new firms to enter p p p 1 q N q 4 q 3 q 21 output of firm In the limit entry ensures Price-taking profits are competed temporary away equilibrium p = C/q n f = 1234 n f = N Monopoly model structure We are given the inverse demand function p = p(q) Gives the price that rules if the monopolist delivers q to the market For obvious reasons, consider it as the average revenue curve (AR) Total revenue is: p(q)q Differentiate to get monopolist s marginal revenue (MR): p(q) + p q (q)q p q ( ) means dp( )/dq Clearly, if p q (q) is negative (demand curve is downward sloping), then MR < AR 34

35 Average and marginal revenue p AR curve is just the market demand curve Total revenue: area in the rectangle underneath Differentiate total revenue to get marginal revenue AR MR q Monopoly optimisation problem Introduce the firm s cost function C(q) Same basic properties as for the competitive firm From C we derive marginal and average cost: MC: C q (q) AC: C(q) / q Given C(q) and total revenue p(q)q, profits are: (q) = p(q)q C(q) The shape of is important: We assume it to be differentiable Whether it is concave depends on both C() and p() Of course (0) = 0 Firm maximises (q) subject to q 0 35

36 Monopoly solving the problem Problem is max (q) s.t. q 0, where: (q) = p(q)q C(q) First- and second-order conditions for interior maximum: q (q)= 0 qq (q)< 0 Evaluating the FOC: p(q)+ p q (q)q C q (q) = 0 p(q)+ p q (q)q = C q (q) MR = MC This condition gives the solution from above get optimal output q * put q * in p( ) to get monopolist s price p * = p(q * ) Check this diagrammatically Monopolist s optimum p AR and MR Marginal and average cost Optimum where MC=MR Monopolist s optimum price MC Monopolist s profit AC p* q* MR AR q 36

37 Monopoly pricing rule Introduce the elasticity of demand : := d(log q) / d(log p) = p(q)/ qp q (q) < 0 First-order condition for a maximum: p(q)+ p q (q)q = C q (q) can be rewritten as: p(q) [1+1/] = C q (q) This gives the monopolist s pricing rule: C q (q) p(q) = 1 + Monopoly analysing the optimum Take the basic pricing rule p(q) = C q (q) 1 + 1/ Use the definition of demand elasticity p(q) C q (q) p(q) > C q (q) if < price > marginal cost Clearly as decreases: output decreases gap between price and marginal cost increases What happens if 1 ( -1)? 37

38 Monopolistic competition: 1 MC AC Take linear demand curve (AR) The derived MR curve Marginal and average costs Optimal output for single firm Price and profits p q 1 MR AR output of firm outcome is effectively the same as for monopoly Monopolistic competition: 2 p q 1 output of firm 38

39 THE CONSUMER: Lectures 5-8 Quantities x i consumption of good i x = (, x 2,, x n ) consumption vector X consumption set R i resource stock of good i R = (R 1,R 2,,R n ) resource endowment Prices and income p i price of good i p = (p 1, p 2,, p n ) price vector y money income Functions U utility function C cost (expenditure) function H i compensated demand for good i D i ordinary demand for good i V indirect utility function Other Lagrange multiplier (min cost) Lagrange multiplier (max utility) utility level The budget constraint x 2 A typical budget constraint Slope determined by price ratio Distance out of budget line fixed by income or resources Two important cases determined by 1. amount of money income y 2. vector of resources R p 1 p 2 39

40 Case 1: fixed nominal income x 2 Budget constraint determined by the two end-points Examine the effect of changing p 1 by swinging the boundary thus y.. p1 Budget constraint is n p i x i i=1 y Case 2: fixed resource endowment x 2 Budget constraint determined by location of resources endowment R Examine the effect of changing p 1 by swinging the boundary thus R Budget constraint is n p i x i n p i R i i=1 i=1 40

41 Revealed Preference x 2 Let market prices determine a person's budget constraint Suppose the person chooses bundle x Use this to introduce Revealed Preference x x Axiom of Rational Choice Consumer always makes a choice and selects the most preferred bundle that is available Essential if observations are to have meaning Weak Axiom of Revealed Preference Weak Axiom of Revealed Preference (WARP) If x RP x' then x' not-rp x If x was chosen when x' was available then x' can never be chosen whenever x is available Suppose that x is chosen when prices are p If x' is also affordable at p then: Now suppose x' is chosen at prices p' This must mean that x is not affordable at p': 41

42 WARP in action x 2 Take the original equilibrium Now let the prices change WARP rules out some points as possible solutions x x x Clearly WARP induces a kind of negative substitution effect But could we extend this idea? Trying to extend WARP x 2 Take basic idea of revealed preference Invoke revealed preference again Invoke revealed preference yet again Draw the envelope x'' x' x Is this an indifference curve? No. WARP does not rule out cycles of preference You need an extra axiom to progress further on this 42

43 The (weak) preference relation The basic weak-preference relation: x x' From this we can derive the indifference relation "Basket x is regarded as at least as good as basket x' " x x' and x' x x~x' and the strict preference relation x x' x x' and not x' x Fundamental preference axioms Completeness Transitivity Continuity Greed For every x, x' X either x x' is true, or x' x is true, or both statements are true For all x, x', x" X if x x' and x' x" then x x" For all x' X the not-better-than-x' set and the not-worse-than-x' set are closed in X (Strict) Quasi-concavity Smoothness 43

44 Continuity: an example x 2 Take consumption bundle x Construct two other bundles, x L with Less than x, x M with More There is a set of points like x L and a set like x M Draw a path joining x L, x M x L x x M If there s no jump But what about the boundary points between the two? Do we jump straight from a point marked better to one marked worse"? Utility function Representation Theorem: given completeness, transitivity, continuity preference ordering can be represented by a continuous utility function In other words there exists some function U such that x x implies U(x) U(x') and vice versa U is purely ordinal defined up to a monotonic transformation So we could, for example, replace U( ) by any of the following log( U( ) ) ( U( ) ) φ( U( ) ) where φ is increasing All these transformed functions have the same shaped contours 44

45 A utility function Take a slice at given utility level Project down to get contours 0 x 2 Another utility function By construction U* = φ(u) Again take a slice Project down 0 x 2 45

46 The greed axiom x 2 Bliss! B Pick any consumption bundle in X Greed implies that these bundles are preferred to x' Gives a clear North-East direction of preference What can happen if consumers are not greedy x' Greed: utility function is monotonic Conventionally shaped indifference curves x 2 A Slope well-defined everywhere Pick two points on the same indifference curve Draw the line joining them Any interior point must line on a higher indifference curve C B ICs are smooth ICs strictly concaved-contoured I.e. strictly quasiconcave (-) Slope is the Marginal Rate of Substitution U 1 (x). U 2 (x). 46

47 Other types of IC: Kinks x 2 Strictly quasiconcave But not everywhere smooth A C B Other types of IC: not strictly quasiconcave x 2 Slope well-defined everywhere Not quasiconcave Quasiconcave but not strictly quasiconcave A C B Indifference curves with flat sections make sense But may be a little harder to work with 47

48 The problem Maximise consumer s utility U(x) Subject to feasibility constraint x X and to the budget constraint n p i x i y i=1 U assumed to satisfy the standard shape axioms Assume consumption set X is the non-negative orthant The version with fixed money income The primal problem x 2 The consumer aims to maximise utility Subject to budget constraint Defines the primal problem Solution to primal problem x* max U(x) subject to n p i x i y i=1 But there's another way of looking at this 48

49 The dual problem x 2 Constraint set Alternatively the consumer could aim to minimise cost Subject to utility constraint Defines the dual problem Solution to the problem Cost minimisation by the firm x* minimise n p i x i i=1 subject to U(x) But where have we seen the dual problem before? A lesson from the firm Compare cost-minimisation for the firm and for the consumer z 2 q x 2 The difference is only in notation So their solution functions and response functions must be the same z* x* z 1 49

50 Cost-minimisation: strictly quasiconcave U Minimise n p i x i i=1 + λ[ U(x)] Because of strict quasiconcavity we have an interior solution A set of n + 1 First-Order Conditions U 1 (x ) = p 1 U 2 (x ) = p 2 U n (x ) = p n = U(x ) Use the objective function and utility constraint to build the Lagrangian Differentiate w.r.t.,, x n and set equal to 0 and w.r.t Denote cost-min values with a * From the FOC If both goods i and j are purchased and MRS is defined then U i (x ) p = i U j (x ) p j MRS = price ratio implicit price = market price If good i could be zero then U i (x ) p i U j (x ) p j MRS ji price ratio implicit price market price 50

51 The solution Solve FOC to get a cost-minimising value for each good x i * = H i (p, ) for the Lagrange multiplier * = *(p, ) and for the minimised value of cost itself The consumer s cost function or expenditure function is defined as C(p, ) := min p i x i {U(x) } Main results are immediate The cost function has same properties as for the firm Shephard's Lemma gives demand as a function of prices and utility H i (p, ) = C i (p, ) Properties of the solution function determine behaviour of response functions Same problem as for firm; so results are the same H is the compensated or conditional demand function Downward-sloping with respect to its own price, etc Short-run results can be used to model side constraints For example rationing 51

52 Comparing firm and consumer Cost-minimisation by the firm and expenditure-minimisation by the consumer are effectively identical problems So the solution and response functions are the same: Problem: Firm m min w i z i z i=1 Solution: C(w, q) + [q (z)] Consumer n min p i x i x i=1 C(p, ) + [ U(x)] Response: z i * = H i (w, q) x i * = H i (p, ) The Primal and the Dual There s an attractive symmetry about the two approaches to the problem In both cases the ps are given and you choose the xs. But constraint in the primal becomes objective in the dual n p i x i + [ U(x)] i=1 n U(x) + [ y p i x i ] i=1 and vice versa 52

53 A useful connection Compare the primal problem of the consumer with the dual problem x 2 x 2 Two aspects of the same problem So we can link up their solution functions and response functions x* x* Utility maximisation Maximise U(x) n y p i x i + μ[ y p i x i ] i=1 If U is strictly quasiconcave we have an interior solution A set of n+1 First-Order Conditions U 1 (x ) = p 1 U 2 (x ) = p 2 U n (x ) = p n n y = p i x i i=1 Use the objective function and budget constraint to build the Lagrangean Differentiate w.r.t.,, x n and set equal to 0 and w.r.t Denote utility maximising values with a * 53

54 From the FOC If both goods i and j are purchased and MRS is defined then U i (x ) p = i (same as before) U j (x ) p j MRS = price ratio implicit price = market price If good i could be zero then U i (x ) p i U j (x ) p j MRS ji price ratio implicit price market price The solution Get U-maximising value for each good and Lagrange multiplier x i * = D i (p, y), i = 1,,n * = *(p, y) Also for the maximised value of utility itself The indirect utility function is defined as V(p, y) := max U(x) p i x i y} V is non-increasing in every price, decreasing in at least one price is increasing in income y is quasi-convex in prices p is homogeneous of degree zero in (p, y) satisfies Roy's Identity 54

55 Another useful connection Indirect utility function maps prices and budget into maximal utility: = V(p, y) Cost function maps prices and utility into minimal budget: y = C(p, ) Therefore we have: = V(p, C(p, )) The indirect utility function works like an "inverse" to the cost function The two solution functions have to be consistent with each other. Odd-looking identities can be useful 0 = V i (p,c(p,))+v y (p,c(p,)) C i (p,) Function-of-a-function rule 0 = V i (p, y) + V y (p, y) x * i Shephard s Lemma Rearrange to get Roy s identity x i* = V i (p, y)/v y (p, y) The right-hand side is just D i (p, y) Utility and expenditure Utility maximisation and expenditure-minimisation by the consumer are effectively two aspects of the same problem So their solution and response functions are closely connected: Primal Dual Problem: Solution: V(p, y) n max U(x) + μ[ y p i x i ] x i=1 n min p i x i x i=1 C(p, ) + [ U(x)] Response: x i * = D i (p, y) x i * = H i (p, ) 55

56 Solving the max-utility problem The primal problem and its solution n Lagrangean for the max U problem max U(x) + [ y p i x i ] i=1 U 1 (x * ) = p 1 U 2 (x * ) = p 2 U n (x * ) = p n n p i x i* = y i=1 Solve this set of equations: * = D 1 (p, y) x 2* = D 2 (p, y) x n* = D n (p, y) n p i D i (p, y) = y i=1 The n + 1 first-order conditions, assuming all goods purchased Gives a set of demand functions, one for each good: functions of prices and incomes A restriction on the n equations. Follows from the budget constraint The response function Primal problem response function is demand for good i: x i* = D i (p,y) Should be treated as just one of a set of n equations The system has an adding-up property:, Each equation is homogeneous of degree 0 in prices and income. For any t > 0: x i* = D i (p, y )= D i (tp, ty) Follows from budget constraint: LHS is total expenditure Again follows from the budget constraint 56

57 Effect of a change in income x 2 Take the basic equilibrium Suppose income rises The effect of the income increase Demand for each good does not fall if it is normal x * x ** But could the opposite happen? An inferior good x 2 Take same original prices, but different preferences Again suppose income rises The effect of the income increase Demand for good 1 rises, but Demand for inferior good 2 falls a little x * x ** Can you think of any goods like this? How might it depend on the categorisation of goods? 57

58 Effect of a change in price x 2 Again take the basic equilibrium Allow price of good 1 to fall The effect of the price fall The journey from x* to x** broken into two parts x * x ** A fundamental decomposition Take the two methods of writing x i* : H i (p,) = D i (p,y) Use cost function to substitute for y: H i (p,) = D i (p, C(p,)) Differentiate with respect to p j : H ji (p,) = D ji (p,y) + D yi (p,y)c j (p,) Simplify : H ji (p,) = D ji (p,y) + D yi (p,y) H j (p,) Remember: they are two ways of representing the same thing Gives us an implicit relation in prices and utility Uses y = C(p,) and function-of-afunction rule again Using cost function and Shephard s Lemma = D ji (p,y) + D yi (p,y) x j * From the comp. demand function And so we get: D ji (p,y) = H ji (p,) x j* D yi (p,y) This is the Slutsky equation 58

59 The Slutsky equation D ji (p,y) = H ji (p,) x j * D yi (p,y) Gives fundamental breakdown of effects of a price change x * x ** Income effect: I'm better off if the price of jelly falls; I m worse off if the price of jelly rises. The size of the effect depends on how much jelly I am buying if the price change makes me better off then I buy more normal goods, such as icecream Substitution effect: When the price of jelly falls and I m kept on the same utility level, I prefer to switch from icecream for dessert The Slutsky equation: own-price Set j = i to get the effect of the price of ice-cream on the demand for ice-cream D ii (p,y) = H ii (p,) x i * D yi (p,y) Own-price substitution effect must be negative This is non-negative for normal goods So the income effect of a price rise must be non-positive for normal goods Important special case Follows from the results on the firm Price increase means less disposable income Theorem: if the demand for i does not decrease when y rises, then it must decrease when p i rises 59

60 Price fall: normal good p 1 ordinary demand curve D 1 (p,y) compensated (Hicksian) demand curve H 1 (p,) The initial equilibrium price fall: substitution effect total effect: normal good income effect: normal good For normal good income effect must be positive or zero price fall * ** Consumer equilibrium: another view x 2 Type 2 budget constraint: fixed resource endowment Budget constraint with endogenous income Consumer's equilibrium Its interpretation x * Equilibrium is familiar: same FOCs as before R 60

61 The offer curve x 2 x ** x *** x * Take the consumer's equilibrium Let the price of good 1 rise Let the price of good 1 rise a bit more Draw the locus of points This path is the offer curve Amount of good 1 that household supplies to the market R Household supply x 2 p 1 Flip horizontally, to make supply clearer Rescale the vertical axis to measure price of good 1 Plot p 1 against x *** x ** This path is the household s supply curve of good 1 R x * supply of good 1 supply of good 1 The curve bends back on itself Why? 61

62 Decomposition another look Take ordinary demand for good i: x i* = D i (p,y) Substitute in for y : x i* = D i (p, j p j R j ) Differentiate with respect to p j : dx * i dy = D ji (p, y) + D yi (p, y) dp j dp j = D ji (p, y) + D yi (p, y) R j Now recall the Slutsky relation: D ji (p,y) = H ji (p,) x j* D yi (p,y) Use this to substitute for D ji : dx * i = H ji (p,) + [R j x j* ] D yi (p,y) dp j Function of prices and income Income itself now depends on prices The indirect effect uses function-of-a-function rule again Just the same as on earlier slide This is the modified Slutsky equation The modified Slutsky equation: dx i * = H ji (p, ) + [R j x j* ] D yi (p,y) dp j Substitution effect has same interpretation as before Two terms to consider when interpreting the income effect This term is just the same as before This term makes all the difference: Negative if the person is a net demander Positive if the person is a net supplier 62

63 The savings problem x 2 Resource endowment is noninterest income profile Slope of budget constraint increases with interest rate, r Consumer's equilibrium Its interpretation x *,x 2 are consumption today and tomorrow Determines time-profile of consumption What happens to saving when the interest rate changes? R Labour supply x 2 Endowment: total time & non-labour income Slope of budget constraint is wage rate Consumer's equilibrium,x 2 are leisure and consumption Determines labour supply x * Will people work harder if their wage rate goes up? R 63

64 The two aspects of the problem Primal: Max utility subject to the budget constraint Dual: Min cost subject to a utility constraint x 2 V(p, y) x 2 C(p,) What effect on max-utility of an increase in budget? What effect on min-cost of an increase in target utility? V C x* x* Interpreting the Lagrange multiplier The solution function for the primal: At the optimum, either the V(p, y) = U(x * ) = U(x * ) + * constraint binds or Lagrange [y i p i x i* ] multiplier is zero Differentiate with respect to y: Use the ordinary demand V y (p, y) = i U i (x * )D i y(p, y) + * [1 i p i D i y(p, y)] functions Rearrange: Lagrange multiplier in the V y (p, y) = i [U i (x * ) * p i ]D i y(p,y)+ * = * primal is MU of income The solution function for the dual: C(p, ) = i p i x i* = i p i x i* * [U(x * ) ] Differentiate with respect to and rearrange: C (p, ) = i [p i * U i (x * )] H i (p, )+ * = * Same argument as above Lagrange multiplier in the dual is MC of utility We can also show:. * = 1/ * A useful connection between C and V 64

65 The problem of valuing utility change x 2 ' Take the consumer's equilibrium and allow a price to fall... Obviously the person is better off....but how much better off? x* x** Story number 1 (CV) Price of good 1 changes p: original price vector p': vector after price change This causes utility to change = V(p, y) ' = V(p', y) Value this utility change in money terms: what change in income would bring a person back to the starting point? Define the Compensating Variation: = V(p', y CV) Amount CV is just sufficient to undo effect of going from p to p' original utility level at prices p new utility level at prices p' original utility level restored at new prices p' 65

66 The compensating variation x 2 A fall in price of good 1 Reference point is original utility level CV measured in terms of good 2 x* x** Story number 2 (EV) Price of good 1 changes p: original price vector p': vector after price change This causes utility to change = V(p, y) ' = V(p', y) Value this utility change in money terms: what income change would have been needed to bring the person to the new utility level? Define the Equivalent Variation: ' = V(p, y + EV) Amount EV is just sufficient to mimic effect of going from p to p' original utility level at prices p new utility level at prices p' new utility level reached at original prices p 66

67 The equivalent variation x 2 ' Price fall as before Reference point is the new utility level EV measured in terms of good 2 x* x** Welfare change as (cost) Compensating Variation as (cost): CV(pp') = C(p, ) C(p', ) ( ) change in cost of hitting utility level. If positive we have a welfare increase Equivalent Variation as (cost): EV(pp') = C(p, ') C(p', ') ( ) change in cost of hitting utility level '. If positive we have a welfare increase Using these definitions we also have CV(p'p) = C(p', ') C(p, ') = EV(pp') Looking at welfare change in the reverse direction, starting at p' and moving to p 67

68 Cost-of-living indices An index based on CV: C(p', ) I CV = C(p, ) An approximation: i p' i x i IL = i p i x i I CV. An index based on EV: C(p', ') I EV = C(p, ') An approximation: i p' i x' i IP = i p i x' i I EV. What's the change in cost of hitting the base utility level? What's the change in cost of buying the base consumption bundle x? This is the Laspeyres index (the basis for the Consumer Price Index) What's the change in cost of hitting the new utility level '? What's the change in cost of buying the new consumption bundle x'? This is the Paasche index. Another (equivalent) form for CV Use the cost-difference definition: CV(pp') = C(p, ) C(p', ) Assume that the price of good 1 changes from p 1 to p 1 ' while other prices remain unchanged. Then we can rewrite the above as: CV(pp') = C 1 (p, ) dp 1 p 1 p 1 ' Further rewrite as: p 1 ( ) change in cost of hitting utility level. If positive we have a welfare increase Using definition of a definite integral CV(pp') = H 1 (p, ) dp 1 Using Shephard s lemma again p 1 ' CV is an area under the compensated demand curve 68

69 Compensated demand and the value of a price fall (CV) p 1 compensated (Hicksian) demand curve H 1 (p, ) The initial equilibrium price fall: (welfare increase) value of price fall, relative to original utility level The CV provides an exact welfare measure price fall Compensating Variation But it s not the only approach * Compensated demand and the value of a price fall (EV) price fall p 1 Equivalent Variation compensated (Hicksian) demand curve H 1 (p, ) As before but use new utility level as a reference point price fall: (welfare increase) value of price fall, relative to new utility level The EV provides another exact welfare measure But based on a different reference point Other possibilities ** 69

70 Ordinary demand and the value of a price fall p 1 ordinary (Marshallian) demand curve The initial equilibrium price fall: (welfare increase) An alternative method of valuing the price fall? D 1 (p, y) CS provides an approximate welfare measure price fall Consumer's surplus x* 1 x** 1 Three ways of measuring the benefits of a price fall p 1 D 1 (p, y) H 1 (p,) H 1 (p, ) Summary of the three approaches. Illustrated for normal goods price fall For normal goods: CV CS EV For inferior goods: CV > CS > EV x* 1 x** 1 70

71 GENERAL EQUILIBRIUM: Lectures 9-12 Quantities q i aggregate net output of good i x i aggregate consumption of good i R i resource stock of good i R h i resource holding by h of i q f i net output by f of i x h i consumption by h of i [,x 2, ] allocation across households [q 1,q 2, ] allocation across firms Prices and incomes p i market price of good i i shadow price of good i f profits of firm f y h money income of h Functions U h utility function of h f production function of firm f excess demand for good i E i Other N replication factor h reservation utiity for h h f share of h in the profits of f Q technology set A Attainable set B better than set K Coalition Approaches to outputs and inputs NET OUTPUTS q 1 q 2... OUTPUT INPUTS z 1 z 2... A standard accounting approach An approach using net outputs How the two are related A simple sign convention q n-1 q n q z m q 1 q 2... q n-1 q n = z 1 z 2... z m +q Outputs: Inputs: Intermediate goods: + 0 net additions to the stock of a good reductions in the stock of a good your output and my input cancel each other out 71

72 The technology set Q q 1 Q 0 Tradeoff in inputs q 3 q 4 (given q 1 = 500) (given q 1 = 750) 72

73 Tradeoff between outputs q 2 Again take slices through Q For low level of inputs For high level of inputs CRTS high input low input q 1 The technology set Q and the production function q 2 (q) > 0 (q) = 0 A view of set Q: production possibilities of two outputs. The frontier is smooth (many basic techniques) Feasible but inefficient points in the interior Feasible and efficient points on the boundary Infeasible points outside the boundary q Q (q) 0 (q) < 0 (q 1, q 2,,q n ) nondecreasing in each q i Boundary is the transformation curve Slope: marginal rate of transformation MRT ij := j (q) / i (q) q 1 73

74 The Crusoe problem (1) max U(x) by choosing x and q subject to... x X (q) 0 x q + R a joint consumption and production decision logically feasible consumption technical feasibility: equivalent to q Q materials balance: can t consume more of than is available from net output + resources Crusoe s problem and solution x 2 Attainable set with R 1 = R 2 = 0 Positive stock of resource 1 More of resources 3,,n Crusoe s preferences The optimum The FOC x * Attainable set derived from technology and materials balance condition MRS = MRT: U 1 (x) 1 (q) = U 2 (x) 2 (q) 0 74

75 Profits and income at shadow prices We know that there is no system of prices Invent some shadow prices for accounting purposes Use these to value national income n profits 1 q q n q n 1 R R n R n 1 [q 1 +R 1 ] n [q n +R n ] value of resource stocks value of national income National income contours q 2 +R 2 1 [q 1 + R 1 ] + 2 [q 2 + R 2 ] = const q 1 +R 1 75

76 National income of the Island x 2 Attainable set Iso-profit income maximisaton The Island s budget set Use this to maximise utility 1 (x) 1 = 2 (x) 2 x * U 1 (x) 1 = U 2 (x) 2 Using shadow prices we ve broken down the Crusoe problem into a two-step process: 1.Profit maximisation 2.Utility maximisation 0 A separation result By using shadow prices a global maximisation problem is separated into sub-problems: max U(x) subject to x q + R (q) 0 1. An income-maximisation problem 2. A utility maximisation problem n max i [ q i R i subj. to i=1 (q) 0 max U(x) subject to n i x i y i=1 Maximised income from 1 is used in problem 2 76

77 Crusoe problem: another view x 2 The attainable set The Better-than-x * set The price line Decentralisation A = {x: x q + R, (q) 0} x * B 1 2 B = {x: U(x) U(x * )} x * maximises income over A A x * minimises expenditure over B 0 Optimum cannot be decentralised x 2 x A nonconvex attainable set The consumer optimum Implied prices: MRT=MRS Maximise profits at these prices A x * Production responses do not support the consumer optimum In this case the price system fails 77

78 Crusoe's island trades x 2 q 2 ** q ** Equilibrium on the island The possibility of trade Max national income at world prices Trade enlarges the attainable set Equilibrium with trade x 2 ** x * Domestic prices x ** World prices x * is the Autarkic equilibrium: * = q 1* ; x 2* =q 2 * World prices imply a revaluation of national income In this equilibrium the gap between x ** and q ** is bridged by imports & exports q 1 ** ** The nonconvex case with world trade q 2 ** x 2 q ** Equilibrium on the island World prices Again maximise income at world prices The equilibrium with trade Attainable set before and after trade x 2 ** x ** Trade convexifies the attainable set x * A A q 1 ** ** 78

79 What is an economy? Resources (stocks) Households (preferences) R 1, R 2, U 1, U 2, n of these n h of these Firms (technologies), n f of these An allocation A competitive allocation consists of: utility-maximising A collection of bundles (one for ^ each of the n h households) profit-maximising A collection of net-output vectors ^ (one for each of the n f firms) [x] := [, x 2, x 3, ] [q] := [q 1, q 2, q 3, ] A set of prices (used by households and firms) p := (p 1, p 2,, p n ) 79

80 How a competitive allocation works p { q f (p), f=1,2,,n f } p, { y h } { x h (p), h=1,2,,n h } Implication of firm f s profit maximisation 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 What does household h possess? Resources R 1h, R 2h, R ih 0, i =1,,n Shares in firms profits 1h, 2h, 0 fh 1, f =1,,n f 80

81 Incomes Resources Shares in firms Net outputs Prices Rents Profits The fundamental role of prices Net output of i by firm f depends on prices p: q if = q if (p) Thus profits depend on prices: n f (p):= p i q if (p) i=1 So incomes can be written as: n n f y h = p i R ih + fh f (p) i=1 f=1 Supply of net outputs Again writing profits as priceweighted sum of net outputs Income = resource rents + profits Income depends on prices : y h = y h (p) y h ( ) depends on ownership rights that h possesses 81

82 Prices in a competitive allocation p q f (p) {, f=1,2,,n f } p, p{ y h } { x h (p), h=1,2,,n h } The allocation as a collection of responses Put the price-income relation into household responses Gives a simplified relationship for households Summarise the relationship p [q(p)] [x(p)] The price mechanism ddistribution resource distribution share ownership R 1a, R 2a, 1a, 2a, R 1b, R 2b, 1b, 2b, 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 Brief summary a [y] allocation [q(p)] [x(p)] 82

83 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 Competitive equilibrium: basics For each h, maximise n U h (x h ), subject to p i x ih y h i=1 For each f, maximise n p i q if, subject to f (q f ) 0 i=1 Households maximise utility, given prices and incomes Firms maximise profits, given prices For all goods the materials balance must hold For each i: x i q i + R i 83

84 Consumption and net output Obvious way to aggregate consumption of good i? n h x i = x i h h=1 Appropriate if i is a rival good Additional resources needed for each additional person consuming a unit of i An alternative way to aggregate: x i = max {x ih } h Aggregation of net output: n f q i := q i f f=1 Opposite case: a nonrival good Examples: TV, national defence if all q f are feasible will q be feasible? Yes if there are no externalities Counterexample: production with congestion 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 84

85 Alf s optimisation problem x 2 a R 2 a R a Resource endowment Prices and budget constraint Preferences Equilibrium x *a Budget constraint is 2 2 p i x ia i=1 i=1 p i R i a Alf sells some endowment of 2 for good 1 by trading with Bill O a R 1 a a Bill s optimisation problem x 2 b Resource endowment Prices and budget constraint Preferences Equilibrium R 2 b x *b R b Budget constraint is 2 2 p i x ib i=1 i=1 p i R i b Bill, of course, sells good 1 in exchange for 2 O b R 1 b b 85

86 Combine the two problems x 2 a R 2 a b R 1 b [R] O b R 2 b 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 [x * ] x 2 b This is the Edgeworth box Width: R 1 a + R 1 b Height: R 2 a + R 2 b O a R 1 a a Response to changes in prices x 2 a R 2 a R a Alf s endowment Alf s reservation utility Alf s preference map No trade if p 1 is too high Trades offered as p 1 falls Alf s offer curve R 1 a a O a 86

87 Response to changes in prices (2) x b x b 1 2 R b 1 R b R b 2 O b R 2 b R b 1 x b 2 O b Bill s situation as an Australian No trade if p 1 is too low Trades offered as p 1 rises Bill s offer curve R b b Edgeworth Box and CE x b 1 x a 2 R b 1 R 2 a [R] Endowment point O b (property distribution) The two offer curves Offers consistent at intersection By construction this is CE Price-taking R bu-maximising Alf 2 Price-taking U-maximising Bill Satisfies materials balance [x*] O a R 1 a a x 2 b 87

88 Coalitions The population Viewed as n h separate individuals A coalition K is formed by any subgroup K 1 K 2 K 0 A formal approach An allocation is blocked by a coalition if the coalition members can do better for themselves 88

89 Equilibrium concept Use the idea of blocking to introduce a solution concept if allocation is blocked a coalition could stop it happening such an allocation could not be a solution to the trading game So we use the following definition of a solution: the Core is the set of unblocked, feasible allocations Let s apply it in the two-trader case In a 2-person world there are few coalitions: {Alf } {Bill} {Alf & Bill} let s see what allocations are blocked by them and what remains unblocked The 2-person core x 2 a b [R] a b [x b ] Alf s reservation O b utility b x {Alf} blocks 2 these allocations Bill s reservation utility {Bill} blocks these allocations Draw the contract curve {Alf, Bill} blocks these allocations The resulting core x 2 a [x a ] b Bill gets all the advantage from trade at this extreme point Alf gets all the advantage from trade at this extreme point The contract curve is the locus of common tangencies O a a a x 2 b 89

90 The core and CE x 2 a b [R] The endowment O b point The 2-person core again Competitive equilibrium again [x*] A competitive equilibrium must always be a core allocation O a a x 2 b The core and CE (2) x 2 a b [R] Indifference O b curves that yield multiple equilibria Endowment point and reservation utility Equilibrium: low p 1 /p 2 a [x*] [x**] b Equilibrium: high p 1 /p 2 The core O a a x 2 b 90

91 A simple result and a question Every CE allocation must belong to the core Core It is possible that no CE exists What about core allocations which are not CE? CE Remember we are dealing with a 2-person model Will there always be non-ce points in the core? To find out, let's clone the economy economy replicated by a factor N, so there are 2N persons start with N = 2 Alf and twin brother Arthur have same preferences and endowments likewise the twins Bill and Ben Now there are more possibilities of forming coalitions so more blocking! {Alf} {Bill} {Arthur} {Ben} {Alf & Arthur} {Alf, Arthur &Bill} {Alf & Bill} {Arthur & Ben} {Alf, Arthur &Ben} {Bill&Ben} {Bill, Ben &Alf} {etc, etc} Effect of cloning on the core [R] The core in the 2-person case The extremes of the two-person core {Alf,Arthur,Bill} can block [x a ] leaving the Ben twin outside the coalition [x a ] [x b ] Are the extremes still core allocations in the 4-person economy? This new allocation is not a solution But it shows that the core must have become smaller 91

92 How the blocking coalition works Consumption in the coalition Sum to get resource requirement Consumption out of coalition Alf x a = ½[x a +R a ] Arthur x a = ½[x a +R a ] Bill [2R a +R b 2x a ] 2R a + R b The consumption within the coalition equals the coalition s resources So the allocation is feasible Ben R b If N is bigger: more blocking coalitions? [R] [x a ] [x b ] The 2-person core An arbitrary allocation - can it be blocked? Draw a line to the endowment Take N=500 of each tribe Divide the line for different coalition numbers We ve found the blocking coalition If line is not a tangent this can always be done numbers of a-tribe b-tribe

93 In the limit [R] [x b ] [x*] [x a ] If N a coalition can be found that divides the line to [R] in any proportion you want Only if the line is like this will the allocation be impossible to block With the large N the core has shrunk to the set of CE Review Basic components of trading equilibrium: Coalitions Blocking Core as an equilibrium concept Relation to CE Every CE must lie in the core In the limit of a replication economy the core consists only of CE Answer to question: why price-taking? In a large economy with suitably small agents it's the only thing to do 93

94 Aggregates 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) = h x ih (p) From firm s supply of net output q f i = q if (p) Aggregate: q i = 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 ) 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 b x Alf Bill 1 x The Market 1 a 94

95 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 Subtract q and R from x to get E: p 1 p 1 p 1 Demand Supply q 1 Resource stock R 1 p 1 E i (p) := x i (p) q i (p) R i E

96 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 You can only have excess supply of a good in equilibrium if the price of that good is 0 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 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 raises some questions about step 4 96

97 Two fundamental properties Walras Law. For any price p: n p i E i (p) = 0 i = 1 You only have to work with n-1 (rather than n) equations Homogeneity of degree 0. For any price p and any t > 0 : E i (tp) = E i (p) You can normalise the prices by any positive number Reminder: these hold for any competitive allocation, not just equilibrium Price normalisation We may need to convert from n numbers p 1, p 2, p n 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 a numéraire p labour MarsBar p n p i i=1 to give a Marxian set Mars neat standard of set prices bar of theory value n-1 that prices of system sum value to 1 This method might seem weird But it has a nice property The set of all normalised prices is convex and compact 97

98 Normalised prices, n = 2 p 2 The set of normalised prices The price vector (0,75, 0.25) (0,1) J={p: p0, p 1 +p 2 = 1} (0, 0.25) (0.75, 0) (1,0) p 1 The existence problem Imagine a rule that moves prices in 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 So the mapping has a fixed point We can examine this in the special case n = 2 In this case normalisation implies that p 2 1 p 1 98

99 Existence of equilibrium? Excess supply 1 p 1 1. E-functions are: continuous, bounded below p 1 * 0 Excess demand 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 2. No equilibrium price where E crosses the axis 3. E never crosses the axis E 1 Multiple equilibria Three equilibrium prices Suppose there were more of resource 1 Now take some of resource 1 away p E 1 99

100 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 Globally stable Excess supply 1 p 1 (0) 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 100

101 Not globally stable p 1 1 Start with a very high price now try a (slightly) low price Start again with very low price now try a (slightly) high price Excess supply 0 Excess demand Check the middle crossing E 1 Here WARP does not hold Two locally stable equilibria One unstable Decentralisation again x 2 The attainable set The Better-than-x * set The price line Decentralisation A = {x: x q+r, (q) 0} A x * B p 1 p 2 B = { h x h : U h (x h ) U h (x *h )} x* maximises income over A x* minimises expenditure over B 0 101

102 A non-convex technology q' B output The case with 1 firm Rescaled case of 2 firms, 4, 8, 16 Limit of the averaging process q* separating prices and equilibrium The Better-than set A q input Limiting attainable set is convex Equilibrium q * is sustained by a mixture of firms at q and q' Non-convex preferences x 2 The case with 1 person Rescaled case of 2 persons, A continuum of consumers The attainable set separating prices and equilibrium x' A x* x B Limiting better-than set is convex Equilibrium x * is sustained by a mixture of consumers at x and x' 102

103 UNCERTAINTY AND RISK: Lectures Quantities x scalar payoff under state x vector payoff under state R i resource stock of good i Prices and income p i price of good i y money income certainty-equivalent income L loss insurance premium Functions U utility function u felicity function Other state of the world set of all states of the world P prospect probability of state proportionate bond holding r rate of return E expectation absolute risk aversion relative risk aversion Concepts state-of-the-world pay-off (outcome) x X prospects {x : } ex ante before the realisation ex post after the realisation time The time line The "moment of truth" The ex-ante view The ex-post view 103

104 The state-space diagram: # x BLUE The consumption space under uncertainty: 2 states A prospect in the 1-good 2-state case The components of a prospect in the 2-state case But this has no equivalent in choice under certainty payoff if RED occurs P 0 O 45 x RED The state-space diagram: #=3 x BLUE The idea generalises: here we have 3 states = {RED,BLUE,GREEN} A prospect in the 1-good 3- state case P 0 O 104

105 Another look at preference axioms Completeness Transitivity Continuity Greed (Strict) Quasi-concavity Smoothness to ensure existence of indifference curves to give shape of indifference curves Ranking prospects x BLUE Greed: Prospect P 1 is preferred to P 0 Contours of the preference map P 1 P 0 O x RED 105

106 Implications of Continuity x BLUE Pathological preference for certainty (violates continuity) Impose continuity An arbitrary prospect P 0 Find point E by continuity Income is the certainty equivalent of P 0 E P 0 O x RED Reinterpret quasiconcavity x BLUE Take an arbitrary prospect P 0 Given continuous indifference curves find the certainty-equivalent prospect E Points in the interior of the line P 0 E represent mixtures of P 0 and E If U strictly quasiconcave P 1 is preferred to P 0 E P 1 P 0 O x RED 106

107 A change in perception x BLUE The prospect P 0 and certaintyequivalent prospect E (as before) Suppose RED begins to seem less likely Now prospect P 1 (not P 0 ) appears equivalent to E Indifference curves after the change This alters the slope of the ICs E O P 0 P1 x RED The independence axiom Let P(z) and P (z) be any two distinct prospects such that the payoff in state-of-the-world is z x = x = z If U(P(z)) U(P (z)) for some z then U(P(z)) U(P (z)) for all z One and only one state-of-the-world can occur So, assume that the payoff in one state is fixed for all prospects Level at which payoff is fixed has no bearing on the orderings over prospects where payoffs differ in other states of the world We can see this by partitioning the state space for > 2 107

108 Independence axiom: illustration x BLUE What if we compare all of these points? Or all of these points? A case with 3 states-of-theworld Compare prospects with the same payoff under GREEN Ordering of these prospects should not depend on the size of the payoff under GREEN Or all of these? O The revealed likelihood axiom Let x and x be two payoffs such that x is weakly preferred to x Let 0 and 1 be any two subsets of Define two prospects: P 0 := {x if 0 and x if 0 } P 1 := {x if 1 and x if 1 } If U(P 1 ) U(P 0 ) for some such x and x then U(P 1 ) U(P 0 ) for all such x and x Induces a consistent pattern over subsets of states-of-the-world 108

109 A key result A result that is central to the analysis of uncertainty Introducing the three new axioms: State irrelevance Independence Revealed likelihood implies that preferences must be representable in the form of a von Neumann-Morgenstern utility function: ux Alternatively, write as Eux expectation uses the numbers to weight the payoff evaluations ux Implications of vnm structure (1) x BLUE A typical IC Slope where it crosses the 45º ray? From the vnm structure So all ICs have same slope on 45º ray RED BLUE O x RED 109

110 Implications of vnm structure (2) x BLUE A given income prospect From the vnm structure Mean income Extend line through P 0 and P to P 1 P 1 P _ By quasiconcavity U(P) U(P 0 ) P 0 O Ex x RED Risk aversion and concavity of u Use the interpretation of risk aversion as quasiconcavity If individual is risk averse then U() U(P 0 ) Given the vnm structure u(ex) RED u(x RED ) + BLUE u(x BLUE ) u( RED x RED + BLUE x BLUE ) RED u(x RED ) + BLUE u(x BLUE ) So the function u is concave 110

111 The felicity function u u of the average of x BLUE and xand RED x RED equals higher the expected than the u of xexpected BLUE and uof of xx RED BLUE and of x RED Diagram plots utility level (u) against payoffs (x) Payoffs in states BLUE and RED If u is strictly concave then person is risk averse If u is a straight line then person is risk-neutral If u is strictly convex then person is a risk lover x BLUE x RED x Attitudes to risk Risk-neutral u(x) Shape of u associated with risk attitude Neutrality: will just accept a fair gamble Aversion: will reject some better-than-fair gambles x BLUE Ex x RED x Loving: will accept some unfair gambles Risk-averse u(x) Risk-loving u(x) x BLUE Ex x RED x x BLUE Ex x RED x 111

112 Risk premium and risk aversion x BLUE A given income prospect The certainty equivalent income Slope gives probability ratio Mean income The risk premium P P 0 RED BLUE Risk premium: Amount that amount you would sacrifice to eliminate the risk Useful additional way of characterising risk attitude O Ex x RED Risk premium: an example u u(x RED ) u(ex) Eu(x ) u(x) Utility values of two payoffs Expected payoff and the utility of expected payoff Expected utility and the certainty-equivalent The risk premium again u(x BLUE ) x BLUE Ex x RED x 112

113 Change the u-function u The utility function and risk premium as before Now let the utility function become flatter u(x RED ) u(x BLUE ) u(x BLUE ) Making the u-function less curved reduces the risk premium and vice versa More of this later x BLUE Ex x RED x Absolute and relative risk aversion Define absolute and relative risk aversion for scalar payoffs u xx (x) u xx (x) (x) := ; (x) := x u x (x) u x (x) For risk-averse individuals For risk-neutral individuals independent of scale and origin of u can show this from the definitions are two different ways of capturing curvature of u The definitions are linked: (x) = x (x) d(x) d(x) = (x) + x dx dx 113

114 Special cases: CARA and CRRA 1. Constant Absolute Risk Aversion Assume that (x) = for all x Felicity function must take the form 1 u(x) = e x 2. Constant Relative Risk Aversion Assume that (x) = for all Felicity function must take the form 1 u(x) = 1 Each induces a distinctive pattern of indifference curves Constant Absolute Risk Aversion x BLUE Case where = ½ Slope of IC is same along 45 ray (standard vnm) For CARA slope of IC is same along any 45 line O x RED 114

115 Constant Relative Risk Aversion x BLUE Case where = 2 Slope of IC is same along 45 ray (standard vnm) For CRRA slope of IC is same along any ray ICs are homothetic O x RED Lotteries Consider lottery as a particular type of uncertain prospect Take an explicit probability model Assume a finite number of states-of-the-world Associated with each state are: A known payoff x, A known probability 0 Lottery is probability distribution over the prizes x, =1,2,, The probability distribution is just the vector := (,,, ) Of course, = 1 What about preferences? 115

116 The probability diagram: #=2 Probability of state RED BLUE (0,1) Probability of state BLUE Cases of perfect certainty Cases where 0 < < 1 The case (0.75, 0.25) (0, 0.25) Only points on the purple line make sense This is an 1-dimensional example of a simplex (0.75, 0) (1,0) RED The probability diagram: #=3 BLUE (0,0,1) Third axis corresponds to probability of state GREEN There are now three cases of perfect certainty Cases where 0 < < 1 The case (0.5, 0.25, 0.25) (0, 0, 0.25) (0, 0.25, 0) (0,1,0) Only points on the purple triangle make sense, This is a 2-dimensional example of a simplex 0 (0.5, 0, 0) (1,0,0) RED 116

117 Preferences over lotteries Take probability distributions as objects of choice lotteries, ', ", Each lottery has same payoff structure state-of-the-world has payoff x probability or ' or " depending on which lottery Axioms of preference over lotteries Transitivity over lotteries If ' and ' " then " Independence of lotteries If ' and (0,1) then ]" ' ] " Continuity over lotteries If ' " then there are numbers and such that ]" ' and ' ]" Basic result Take the axioms transitivity, independence, continuity Imply that preferences must be representable in the form of a von Neumann-Morgenstern utility function: ux or equivalently: where ux So we can also see the EU model as a weighted sum of s 117

118 -indifference curves (0,0,1) Indifference curves over probabilities Effect of an increase in the size of BLUE. (1,0,0) (0,1,0) Trade 118

119 Contingent goods: equilibrium trade a x BLUE b x RED Certainty line for Alf Alf's indifference O b curves Certainty line for Bill Bill's indifference curves Endowment point Equilibrium prices & allocation O a a x RED b x BLUE Trade: problems Do all these markets exist? If there are states-of-the-world there are n of contingent goods Could be a huge number Consider introduction of financial assets Take a particularly simple form of asset: a contingent security pays $1 if state occurs Can we use this to simplify the problem? 119

120 Attainable set: buying a risky asset x BLUE Endowment If all resources put into bonds All these points belong to A Can you sell bonds to others? Can you borrow to buy bonds? If loan shark willing to finance you _ y _ P y+r, y+r [1+rº]y _ [1+r ]y, [1+r]y P 0 A _ y [1+r' ]y _ x RED Attainable set: insurance x BLUE Endowment Full insurance at premium All these points belong to A Can you overinsure? Can you bet on your loss? _ y _ P L y 0 L A _ y P 0 y 0 x RED 120

121 Consumer choice with a variety of financial assets x BLUE Payoff if all in cash Payoff if all in bond 2 Payoff if all in bond 3, 4, 5, Possibilities from mixtures Attainable set The optimum 1 2 only bonds 4 and 5 used at the optimum A 3 4 P * x RED Problem and its solution But corner solutions may also make sense 121

122 Consumer choice: safe and risky assets x BLUE Attainable set, portfolio problem Equilibrium -- playing safe Equilibrium - "plunging" Equilibrium - mixed portfolio _ y _ P P * A P 0 _ y x RED Results (1) Will the agent take a risk? Can we rule out playing safe? Consider utility in the neighbourhood of = 0 Eu( + r) = u y () E r u y is positive So, if expected return on bonds is positive, agent will increase utility by moving away from = 0 122

123 Results (2) Take the FOC for an interior solution Examine the effect on * of changing a parameter For example differentiate E (ru y ( + * r)) = 0 w.r.t. E (ru yy ( + * r)) + E (r 2 u yy ( + * r)) * / = 0 * E (ru yy ( + * r)) = E (r 2 u yy ( + * r)) Denominator is unambiguously negative To sign the numerator we need to impose more structure Assume Decreasing ARA Theorem: If an individual has a vnm utility function with DARA and holds a positive amount of the risky asset then the amount invested in the risky asset will increase as initial wealth increases An increase in endowment x BLUE Attainable set, portfolio problem DARA Preferences Equilibrium Increase in endowment Locus of constant _ y+ _ y P * o P** New equilibrium A _ y _ y+ x RED 123

124 A rightward shift in the distribution Attainable set, portfolio problem x BLUE DARA Preferences Equilibrium Change in distribution Locus of constant New equilibrium _ y A _ P P * o P ** P 0 _ y x RED An increase in spread x BLUE Attainable set, portfolio problem Preferences and equilibrium Increase r, reduce r P * stays put So must have reduced You don t need DARA for this _ y _ P P * y+ * r, y+ * r A P 0 _ y x RED 124

125 WELFARE: Lectures Quantities q f i net output by f of i x h i consumption by h of i R h i ownership by h of resource i Prices and income p i price of good i y h money income of h T h tax revenue raised from h loss Functions constitution C h cost function of h U h utility function of h V h indirect utility function of h v h utility of h as function of f production function of firm f W social welfare function social evaluation function Other social state set of all social states utility possibility set Social objectives Two dimensions of social objectives Set of feasible social states A social preference map? Assume we know the set of all social states How can we draw a social preference map? Can it be related to individual preferences? objective 1 125

126 Elements of a constitution Social states can incorporate all sorts of information: economic allocations, political rights, etc Individual (extended) preferences over ' means that person h thinks state is at least as good as state ' An aggregation rule for the preferences so as to underpin the constitution A function defined on individual (extended) preferences The social ordering and the constitution Where does this ordering come from? Presumably from individuals' orderings over assumes that social values are individualistic Define a profile of preferences as a list of orderings, one for each member of society (,,, ) The constitution is an aggregation function defined on a set of profiles yields an ordering So the social ordering is = (,,, ) 126

127 Axioms and a result Universality should be defined for all profiles of preferences Pareto Unanimity if all consider that is better than ', then the social ordering should rank as better than ' Independence of Irrelevant Alternatives if two profiles are identical over a subset of then the derived social orderings should also be identical over this subset Non-Dictatorship no one person alone can determine the social ordering Kenneth Arrow s Theorem: There is no constitution satisfying these axioms Relaxing universality Could it be that the universal domain criterion is just too demanding? Should we insist on coping with any and every set of preferences, no matter how bizarre? Perhaps imposing restrictions on admissible preferences might avoid the Arrow impossibility result However, we run into trouble even with very simple versions of social states 127

128 Alf, Bill, Charlie decide preference Charlie 1-dimensional social states Scaling of axes is arbitrary Three possible states Views about defence spending Each individual has dramatically different views But all three sets of preferences are single peaked Alf Bill ' " defence spending How do they decide???? Alf Bill Charlie Verdict Alf, Bill, Charlie decide (2) Same states as before preference Charlie Bill Same preferences as before Now Bill changes his mind Now one set of preferences is no longer single peaked How do they decide? Alf Bill ' " defence spending??? Alf Bill Charlie Verdict 128

129 Alternative voting systems Relaxing IIA involves an approach that modifies the type of aggregation rule Simple majority voting may make too little use of information about individual orderings or preferences Here are some alternatives: de Borda (weighted voting) Single transferable vote Elimination voting None of these is intrinsically ideal Consider the results produced by third example The IOC Decision Process An elimination process 1997: Appears to give an orderly convergence Athens is preferred to Rome irrespective of the presence of other alternatives 1993: Violates IIA Ordering of Sydney, Peking depends on whether other alternatives are present 129

130 A definition of efficiency The basis for evaluating social states: v h () A social state is Pareto superior to state ' if: 1. For all h: v h () v h (') 2. For some h: v h ()>v h (') A social state is Pareto efficient if: 1. It is feasible 2. No other feasible state is Pareto superior to the utility level enjoyed by person h under social state Note the similarity with the concept of blocking by a coalition feasibility could be determined in terms of the usual economic criteria Derive the utility possibility set b (a, x 2a ) (b, x 2b ) From the attainable set take an allocation Evaluate utility for each agent Repeat to get utility possibility set A A a =U a (a, x 2a ) b =U b (b, x 2b ) a ) 2 ) a 130

131 Finding an efficient allocation max L( [x ], [q], ) := U 1 ( ) + h h [U h (x h ) h ] f f f (q f ) + i i [q i + R i x i ] where x i = h x ih, q i = f q f i Differentiate L w.r.t x ih. If x ih, x jh positive at the optimum: h U ih (x h ) = i h U jh (x h ) = j Differentiate L w.r.t q if. If q if, q jf nonzero at the optimum: f if (q f ) = i f jf (q f ) = j Interpreting the FOC From the FOCs for any household h and goods i and j: U ih (x h ) i = U jh (x h ) j for every household: MRS = shadow price ratio From the FOCs for any firm f and goods i and j: if (q f ) i = jf (q f ) j for every firm: MRT = shadow price ratio 131

132 Efficiency in an Exchange Economy x 2 a b O b Alf s indifference curves Bill s indifference curves The contract curve Allocations where MRS 12 a = MRS 12 b Set of efficient allocations is the contract curve Includes cases where Alf or Bill is very poor O a a x 2 b Efficiency with production q 2 f x 2 h Household h s indifference curves h s consumption in the efficient allocation MRS Firm f s technology set f s net output in the efficient allocation q ^ f x h MRS = MRT at efficient point 0 h q 1 f 132

133 Two welfare theorems Welfare theorem 1 Assume a competitive equilibrium What is its efficiency property? THEOREM: if all consumers are greedy and there are no externalities then a competitive equilibrium is efficient Welfare theorem 2 Pick any Pareto-efficient allocation Can we find a property distribution d so that this allocation is a CE for d? THEOREM: if, in addition to conditions for theorem 1, there are no non-convexities then an arbitrary PE allocation be supported by a competitive equilibrium Supporting a PE allocation b a x [R] 2 O b The contract curve An efficient allocation Supporting price ratio = MRS The property distribution A lump-sum transfer Allocations where MRS 12 a = MRS 12 b [x] ^ p 1 p 2 Support allocation by a CE This needs adjustment of the initial endowment Lump-sum transfers may be tricky to implement O a a x 2 b 133

134 Individual household behaviour x 2 h Household h s indifference curves h s consumption in the efficient allocation Supporting price ratio = MRS ^ x h p 1 p2 h s consumption in the allocation is utility-maximising for h h s consumption in the allocation is cost-minimising for h 0 h March 2012 Supporting a PE allocation (production) q 2 f Firm f s technology set f s net output in the efficient allocation Supporting price ratio = MRT ^ q f f s net output in the allocation is profit-maximising for f p 1 p 2 0 q 1 f 134

135 Firm f makes wrong choice q 2 f ~ q f Firm f s production function violates second theorem Suppose we want to allocate this net output to f Introduce prices f's choice at those prices ^ q f p 1 p2 A twist on the previous example Big fixed-cost component to producing good 1 0 March 2012 q 1 f market failure once again PE allocations two issues good 2 Same production function Is PE here? or here? Implicit prices for MRS=MRT Competitive outcome 0 March 2012 good 1 Issue 1 what characterises the PE? Issue 2 how to implement the PE 135

136 Indecisiveness of PE b Construct utility-possibility set as previously Two efficient points Points superior to Points superior to ' v v a Boundary points cannot be compared on efficiency grounds and ' cannot be compared on efficiency grounds Potential Pareto superiority Define to be potentially superior to ' if : there is a * which is actually Pareto superior to ' * is accessible from To make use of this concept we need to define accessibility use a tool from the theory of consumer welfare CV h (' ): the monetary value the welfare change for person h of a change from state ' to state valued in terms of the prices at CV h > 0 means a welfare gain; CV h < 0 a welfare loss THEOREM: a necessary and sufficient condition for to be potentially superior to ' is h CV h (' ) > 0 136