Chapter Equilibrium and Efficiency
Reading Essential reading Hindriks, J and G.D. Myles Intermediate Public Economics. (Cambridge: MIT Press, 005) Chapter. Further reading Duffie, D. and H. Sonnenschein (989) Arrow and General Equilibrium Theory, Journal of Economic Literature, 7, 565 598. Koopmans, T.C. Three Essays on the State of Economic Science. (New York: McGraw-Hill, 957) [ISBN 067803977 hbk]. Ng, Y.-K. Welfare Economics. (Basingstoke: Palgrave Macmillan, 004) [ISBN 0333973 hbk]. Advanced reading Debreu, G. Theory of Value, (New York: Wiley, 959) [ISBN 030005593 pbk]. Mirrlees, J.A. The theory of optimal taxation in K.J. Arrow and M.D. Intrilligator (eds) Handbook of Mathematical Economics (Amsterdam: North-Holland, 986) [ISBN 044486054 hbk].
Introduction Adam Smith s invisible i ibl hand described d the link between competition and efficiency Individually id motivated t decision i produce a socially efficient outcome The competitive model captures this independent decision-making Equilibrium is achieved by the adjustment of prices Prices act as the coordinating mechanism for individual decisions 3
Economic Models A model is a simplification designed d to capture essential aspects of the problem under investigation Most models begin with a specification of objectives for the economic agents Equilibrium is found and the effect of policy choice is determined The art of successful modeling is the choice of the level of detail There is typically a trade-off between generality and clear predictions 4
Competitive Economies The fundamental competitive assumption is that t firms and consumers do not believe they can affect prices This is justified when economic agents are negligible in size It can always be imposed as an assumption on behavior Prices are central to the analysis Prices measure values Act as signals to consumers and firms Adjustment of prices equates supply and demand 5
Competitive Economies Information is central to the functioning i of an economy It is assumed now that t all agents have symmetric information The consequences of relaxing this are investigated in Chapter 9 Symmetric information permits uncertainty as long as all agents are equally uninformed Future profits of firms may be uncertain But the directors must no better informed than the shareholders 6
Competitive Economies The exchange economy models the simplest form of economic trading Assume there are two consumers Each consumer has an initial endowment of the two commodities The consumers exchange quantities of the two commodities to achieve an improved consumption plan Market prices determine the rate of exchange Consumers believe their choices cannot affect prices 7
Competitive Economies The initial iti endowment of consumer h is ω h = ( ω h h ),ω p h = ( h h x x ) When prices are, p a consumption plan x, satisfies the budget of h if p h h h h x + px = pω + pω The consumption plan is chosen to maximize the utility function ( h h ) x x h U, subject to the budget constraint 8
Competitive Economies The Edgeworth box depicts the set of feasible consumption allocations Feasible allocations are represented as points in a rectangle The length of the sides are equal to total endowments Consumptions plans for the two consumers are measured from the corners of the box The south-west corner is the zero consumption point for consumer The north-east corner is the zero consumption point for consumer 9
Competitive Economies A pair of consumption plans is feasible if x x i i i i + x = ω + ω, i =, Feasible plans are shown in the Edgeworth box in Fig.. All feasible plans are located in this box The common endowment point is denoted ω Point x is a feasible consumption allocation ω ω + x x ω ω + x Figure.: Edgeworth Box x ω 0
Competitive Economies The consumers have a common budget constraint that passes through the endowment point x The gradient of the budget constraint is p / p Given this budget x and x are utility-maximising choices Figure.: Preferences and demand x ω
Competitive Economies In equilibrium i supply equals demand Prices adjust to achieve this Plans x and x do not x x form an equilibrium Fig..3 shows the effect of an increase in p relative to p The budget constraint pivots about the endowment point ω Figure.3: Relative price change
Competitive Economies An equilibrium is a position where supply is equated to demand Prices adjust until the indifference curves have a common point of tangency on the budget line When this is achieved demand is equal to supply These are called equilibrium prices Given consumer demand functions the equilibrium prices solve x i ( p, p ) + x ( p, p ) = ω + ω, i =, i i i x h i = x h i ( ) p, p 3
Competitive Economies An equilibrium i is shown in Fig..3 The budget constraint is at a point of common tangency with the indifference curves The sum of demands from the two consumers is equal to the sum of endowments Does such an equilibrium always exist? x x Figure.3: Equilibrium ω 4
Competitive Economies Observe that only relative prices matter in determining the level of demand The budget constraint can be written as h h h p ω + ω h p h h p p = x = ω + ω p p p p p h x x p This is determined by relative prices p /p As a consequence demand is homogenous of degree zero h h x p p = x λp, λp, for λ i ( ) ( ) 0, i > 5
Competitive Economies Consumers are concerned with real purchasing power Nominal income is equal to the value of endowment A change in the level of prices raises nominal income and the cost of purchases equally The implication is price normalization must be adopted to fix the level of prices One normalization is to select a numeraire and set its price equal to The numeraire becomes the unit of account 6
Competitive Economies Define excess demand d by Zi = xi + xi ωi ωi Then p [ ] [ ] x + x ω ω + p x + x ω 0 Z + pz = p ω = This is Walras law: the value of excess demand is zero If demand is equal to supply for good then demand must also equal supply for good Equilibrium in one market necessarily implies equilibrium in other second market 7
Competitive Economies The existence of an equilibrium i can be demonstrated as follows: Choose good as the numeraire (so p = ) Plot the excess demand for good as a function of p An equilibrium is occurs where the graph of excess demand crosses the horizontal axis If the graph starts s above the axis and finishes below there must be a crossing point When the excess demand for good is zero by Walras law excess demand d must also be zero for good 8
Competitive Economies Fig..5 illustrates t the argument that an Z equilibrium exists (, p ) Excess demand for good is positive when p is low Excess demand is negative when p is high If the excess demand function is continuous it must cross the axis Generally there will be an odd number of equilibria Figure.5: Equilibrium and excess demand p 9
Competitive Economies Adding production to the exchange economy provides a complete model of economic activity Production is undertaken by firms which aim to maximize profit Some goods can be initial endowments (e.g. labor), some intermediate inputs, and some final consumption goods The fully-developed model is called the Arrow- Debreu economy 0
Competitive Economies Firms use input to produce outputs aiming to maximize profit Each firm has production technology The technology describes feasible input-output p combinations The profits of the firms are distributed as dividend to consumers Consumers hold shares in firms The dividend received is proportional to shareholding
Competitive Economies Good Afi firm s technology is described by a production set, denoted Y j A typical production set is shown in Fig..6 Y j 3 Inputs are negative (a subtraction from economy s stock) and - outputs are positive (an addition) The production plan (-, 3) is feasible Figure.6: Typical production set Good
Competitive Economies The profit of firm j from production plan ( j j y y ) is, j j j π = p y + p y Isoprofit curves show all production plans giving a constant level of profit The curve for π = 0 always goes through the origin Profit is maximized on the highest attainable isoprofit curve This is at y * in Fig..7 π < 0 π = 0 π > 0 * y Good p Figure.7: Profit maximization p Good 3
Competitive Economies Profit maximization i by firm j implies a supply of good i given by y j p and a level of profit i π ( ) j n i = = ( p) With m firms the aggregate g supply for good i is The shareholding of consumer h in firm j is p i y j i j ( p) y ( p) m j Y = = i h j H h θ, i h θ j = = Including dividends the budget constraint of consumer h is n h n h m i= pi xi = i= piωi + j = θ h j π j 4
Competitive Economies Summing over consumers gives the level of aggregate demand h ( p) x ( p) X = = i Excess demand including the supply from firms is Z i ( p) X ( p) Y ( p) H h = i i h H = Equilibrium occurs when p = for all goods i i ( ) 0 Equilibrium can be shown to exist under fairly weak conditions Z i ω h i 5
Efficiency of Competition Efficiency i is about the best use of resources Efficiency is achieved when more cannot be obtained with existing resources If there is a single decision-maker an efficient allocation will maximize utility This is illustrated by the single-consumer economy Characterizing efficiency is more complex when there are multiple decision makers Competing preferences must be resolved Pareto-efficiency is one solution to this difficulty Chapter 3 considers other solutions 6
Efficiency of Competition Fig..8 shows utility Good maximization in a single consumer economy ~ x * Good is supplied by the consumer to the firm Good is supplied by the π firm to the consumer The consumer receives dividend p Given the budget constraint utility is maximized at ~ x * p Figure.8: Utility maximization Good 7
Efficiency of Competition The equilibrium of the economy is shown in Fig..9 * The production plan y is profit maximizing given the prices Consumption ~ x * is equal to production y * The equilibrium is efficient It achieves the highest attainable indifference curve given the production set Good ~ x * = y * π p Figure.9: Efficient equilibrium Good 8
Efficiency of Competition The gradient of the indifference curves is the marginal rate of substitution MRS, The rate at which hgood d can be traded dfor good d holding utility constant The gradient of the production frontier is the marginal rate of transformation MRT, The rate at which good can be transformed into good At the efficient equilibrium the two gradients are equal so MRS, = MRT, 9
Efficiency of Competition Utility maximization i equate the gradient of the indifference curve and budget constraint MRS, = p p Profit maximization equates the gradient of the production frontier to an isoprofit curve MRT = The consumer and the firm react to the same prices so MRS, = p p = MRT, Prices succeed in coordinating decisions, p p 30
Efficiency of Competition Constant returns to scale provide some simplification In equilibrium the firm must earn zero profit so the price vector is orthogonal to the production frontier The budget constraint goes through the origin and is coincident with the π = 0 ~ x * = y * Good p Good production frontier This is illustrated in Fig..0 Figure.0: Constant returns to scale 3
Efficiency of Competition If there is more than consumer a definition of efficiency must accommodate competing preferences Pareto efficiency is a test of efficiency that can be applied with many consumers Pareto efficiency searches for unexploited gains Is there a feasible reallocation of resources that can benefit at least one consumer without harming any other? If no improving reallocation can be found the initial position is Pareto efficient. 3
Efficiency of Competition This can be defined formally An allocation x is feasible if x = y + ω A feasible consumption allocation xˆ is Pareto- efficient if there is does not exist an alternative feasible allocation x such that: Allocation gives all consumers at least as much utility as xˆ x ; Allocation x gives at least one consumer more utility that. xˆ Note how Pareto-efficiency efficienc is defined by a negative property and sidesteps distribution 33
Efficiency of Competition The Two Theorems of Welfare Economics are the basis for claims concerning the desirability of the competitive outcome The First Theorem states that a competitive equilibrium is Pareto-efficientefficient The Second Theorem states that any Pareto- efficient allocation can be decentralised as a competitive equilibrium These theorems are based on applying Paretoefficiency to the competitive economy 34
Efficiency of Competition In Fig.. allocation a is not Pareto efficient The move to b raises the utility of both consumers Allocation c is Pareto- efficient Any change in allocation must make at least one consumer worse-off At c MRS, = MRS, There are many Pareto- efficient allocations d illustrates a second c d b a Figure.: Pareto-efficiency 35
Efficiency of Competition Pareto-efficient efficient allocations occur at the tangency of two indifference curves The locus of tangencies is the contract curve In Fig.. the competitive equilibrium is at point e The budget line is a common tangent The competitive equilibrium is Paretoefficient e Contract Curve Figure.: First Theorem ω 36
Efficiency of Competition Each consumer maximizes i utility given the prices so, = p p MRS, MRS = This equality of the MRSs is the Pareto-efficiency condition Individual decisions are coordinated via prices This is achieved through h individual id optimization i Theorem (First Theorem of Welfare Economics) The allocation of commodities at a competitive equilibrium is Pareto-efficient 37
Efficiency of Competition The Second Theorem asks: can a given Pareto efficient allocation be made into a competitive equilibrium? This is the process of decentralisation Decentralisation is possible if the consumers indifference curves are convex. The common tangent at a Pareto-efficient allocation provides the budget constraint Convexity ensures that given this budget line the Pareto-efficient allocation will also be the optimal choice of the consumers Decentralization is completed by choosing a point on this budget line as the initial endowment point 38
Efficiency of Competition Decentralization is illustrated in Fig..3 The Pareto-efficient i t allocation e is made an equilibrium by: Selecting ω as the endowment point Allowing trade to move the allocation from ω to e If the initial endowment point is ω a transfer of endowment is necessary ω e' ω' Figure.3: Second Theorem 39
Efficiency of Competition The formal description of this construction ti is the Second Theorem Theorem (Second Theorem of Welfare Economics) With convex preferences, any Pareto-efficient efficient allocation can be made a competitive equilibrium The important step is placing the economy at the correct starting point This requires a process for redistributing initial wealth 40
Efficiency of Competition The extension of the Second Theorem to production is shown in Fig..4 W is the set of feasible production plans Z describes quantities of the two goods that would permit a Paretoimprovement over xˆ, xˆ The price line separates W and Z { } ˆx W Feasible Set ˆx xˆ Z Figure.4: Proof of the Second Theorem Price Line 4
Lump-Sum Taxation It is implicit it in the Second Theorem that t wealth is reallocated The practical value of the Second Theorem depends on the possibility of redistribution The Theorem sees this as done is by making lump-sum transfers between consumers Quantities of endowments are transferred between consumers A transfer is lump-sum if no change in a consumer s behaviour can affect the size of the transfer The transfer is optimal if the resulting equilibrium is the policy maker s most preferred outcome 4
Lump-Sum Taxation A lump-sum transfer is shown in Fig..5 At the initial iti point the income level of h at the prices pˆ is p ˆpω ω h e The value of the required transfer to consumer h is h h pˆ ω ' pˆ ω Achieved by transferring a quantity ~ x of good Figure.5: Lump-sum transfer ω' ~ x ω 43
Lump-Sum Taxation Lump-sum transfers can be rephrased in terms of lump-sum taxes T = p ˆx ~ Consumer pays tax of amount The tax revenue is given to consumer This pair of taxes moves the budget constraint in exactly the same way as the lump-sum transfer The taxes are also lump-sum The value of the taxes cannot be affected by any change in behaviour 44
Lump-Sum Taxation Lump-sum taxes have a central role in public economics They achieve distributional objectives without reducing the total endowment Redistribution is achieved with no efficiency cost If they can be employed in the manner described they are the perfect taxes But in practice information limits the use of lump-sum taxation This is explored in Chap. 45
Discussion of Assumptions The competitive economy was based on a strong set of assumptions Competitive behavior rules out monopoly power but does not explain who sets prices Asymmetry of information can develop through product experience Market failure occurs when one or more assumption is not met and efficiency is not achieved 46