Pure Exchange Chapter 6 Slides by Pamela L

Introduction n n Economists have different treatments for addressing economic problems If problem of allocation is generally localized in one market § Partial-equilibrium analysis would provide correct solution • Only one segment of an economy is analyzed ¨ n Without consideration for possible interactions with other segments If it is a general problem infecting numerous markets or whole economy § Use general equilibrium analysis • Study of interaction among agents across markets within an economy 2

Introduction n A (general) equilibrium model of all markets, where supply and demand for each commodity are equated § Will result in necessary conditions for economic efficiency Achieved by agents trading commodities to increase their utility § Agents will trade until all gains are exhausted Efficiency gains from agents’ trading are most apparent when households are the only agents § Initially endowed with some quantities of commodities, and there is no production • Called a pure-exchange economy ¨ Supply of each commodity is sum of each household’s endowment of that commodity 3

Introduction n This chapter § Explores gains from trade in a pure exchange economy • Edgeworth box is a method of illustrating these gains for two traders § Investigates efficiency of a free-market price system for allocating commodities • Relate this to Pareto-efficient allocation using First and Second Fundamental Theorems of Welfare Economics § Develops trial-and-error process of establishing an § equilibrium set of prices • Relates this to optimal social-welfare allocation Discusses a fair allocation of initial resources • Yielding an optimal social-welfare allocation 4

Gains from Trade n n In pure-exchange economies, we assume a certain amount of various commodities exist Problem is to efficiently allocate these commodities among households § An allocation of existing commodities is efficient if no one household can be § made better off without making some other household worse off Necessary condition for such an efficient allocation of commodities is • MRS 1 = MRS 2 = … = MRSn ¨ ¨ ¨ n Subscripts denote households n represents number of households MRS measures how much a household is willing to trade one commodity for another When how much each household is willing to trade one commodity for another are equal § Gains from trade are exhausted • Any reallocation of commodities will not increase utility of one household without decreasing utility of another 5

Two-Commodity and Two. Household Economy Consider an economy with two commodities, bread and fish, and two individuals (households), Robinson (R) and Friday (F ) n 50 units of bread and 100 units of fish are to be allocated n § Can be allocated in various ways • Could be allocated to Robinson, all to Friday, or • some combination in between Egalitarian allocation would divide commodities equally between Robinson and Friday ¨ Such an allocation may not be efficient if Robinson’s and Friday’s MRSs are not equal at this equal allocation 6

Two-Commodity and Two. Household Economy n Suppose Robinson’s MRSR (bread for fish) = 2/1 at an equal allocation § where Robinson obtains 25 units of bread and 50 units of fish n n Friday’s MRSF (bread for fish) = 1/1 Allocation does not result in an efficient allocation because § MRSR = 2 ≠ 1 = MRSF n If 2 units of bread are taken from Robinson with 1 unit traded to Friday for 1 fish § Level of utility remains the same with 1 unit of bread leftover n • Indicated in Table 6. 1 By trading, utilities of Robinson and Friday remain unchanged, with 1 unit of bread left over § Represents gains from trade • Could then be divided between Robinson and Friday ¨ Resulting in their utility increasing 7

Table 6. 1 Gains from trade 8

Two-Commodity and Two. Household Economy n As long as MRSs between Robinson and Friday are unequal § Gains from trade are possible Many possible trades that will result in gains n Use Robinson’s and Friday’s initial endowments and indifference curves to determine all possible trades leading to gains n § Shown in Figure 6. 1 9

Figure 6. 1 Preferences and endowments for the two-commodity… 10

Edgeworth Box n n Provides a convenient method for representing the two households’ preferences and endowments in one diagram § See Figure 6. 2 Construct box by turning preference space for Friday 180 § Place it on top of Robinson’s preference space at point where their endowments are together • Point C in Figure 6. 2 n n Horizontal width represents total quantity of fish available Vertical height represents total quantity of bread available § Size of box depends on total amount of fish and bread available in § economy Every point inside box represents a feasible allocation of fish and bread 11

Figure 6. 2 Edgeworth box in a pure-exchange economy 12

Edgeworth Box n An allocation is feasible if total quantity consumed of each commodity is equal to total available from endowments § x. R 1 + x. F 1 = e. R 1 +e. F 1, commodity 1, fish § x. R 2 + x. F 2 = e. R 2 +e. F 2, commodity 2, bread n An allocation where Robinson receives nothing and Friday receives all is a feasible allocation § Represented by 0 R in Figure 6. 2 • Robinson’s utility is minimized and Friday’s utility is maximized n At 0 F, allocation is nothing for Friday and everything for Robinson § Friday’s utility is minimized and Robinson’s maximized n Feasible allocations between these two extreme points represent combinations of commodities with varying levels of satisfaction for both Robinson and Friday § For a movement toward 0 R, Friday receives more of either fish or bread • Increases her utility § Robinson receives less, which decreases his utility • Reverse occurs for a movement toward 0 F 13

Pareto-Efficient Allocation n n Allocation where there is no way to make all households better off No way to make some households better off without making someone else worse off § All gains from trade are exhausted • Illustrated in Figure 6. 3 • Point C is not Pareto efficient ¨ Possible to reallocate commodities in such a manner that one household can be made better off without making another worse off • Any point within shaded lens represents a gain (Pareto • improvement) At points A, B, and all points on the cord ¨ MRSR = MRSF and are Pareto-efficient allocations 14

Figure 6. 3 Efficiency in a pureexchange economy 15

Pareto-Efficient Allocation n n Cord from points A to B is called the core solution § All gains from trade are exhausted Exact solution point within this core depends on bargaining strength of two agents § If Friday is a relatively strong bargainer • She will receive more of gains from trade ¨ Solution will be closer to point B § Alternatively, if Robinson has the upper hand • He will receive a larger portion of gains ¨ Solution will approach point A 16

Contract Curve (Pareto-Efficient Allocation) n Optimal allocation resulting from allocating commodities between Robinson and Friday § Will depend on how initial total endowment of bread and fish is divided between them • If Robinson initially has most of the bread and fish ¨ Optimal allocation near point D in Figure 6. 4 may result § The more fish and bread a household initially has, the higher the level of utility it can achieve • Distribution of income determines resulting Paretoefficient allocation 17

Contract Curve (Pareto-Efficient Allocation) n By varying allocation of initial endowments § Can trace out complete set of Pareto-efficient allocations • Called a contract curve ¨ Illustrated in Figure 6. 4 • Contract curve represents a curve in interior of Edgeworth box ¨ Intersecting tangencies between indifference curves for two agents § MRSs are equal • If efficient allocations exist, where an agent will not consume a positive amount of all commodities ¨ Contract curve will correspond with a segment of an axis (corner solution) § MRSs will not equate 18

Figure 6. 4 Contract curve in a pure-exchange economy 19

Contract Curve (Pareto-Efficient Allocation) n n n Any point not on contract curve is inefficient Agents will adjust terms of trade until a contract is made Represents all Pareto-efficient allocations for a given set of initial endowments § Every point on contract curve results in economic efficiency • Social welfare is not maximized at every point n n Movement along a contract curve will increase one agent’s utility at expense of reduced utility for other agent Maximum social welfare depends on § Economic efficiency § Optimal distribution of income 20

Contract Curve (Pareto-Efficient Allocation) n Pareto optimality does provide a necessary condition for an allocation to maximize social welfare § No inefficiencies in resource allocation exist • Necessary condition for maximum social welfare n Major inadequacy of Pareto-welfare criterion § Does not lead to a complete social ranking of alternative allocations for an economy • Useless criterion for many policy propositions n Some analytical results can be obtained with a Paretowelfare criterion § For example, point rationing is ordinarily better than fixed-ration quantities 21

Efficiency of a Price System n n n A medium of exchange is particularly useful as number of households increases § Process of bartering becomes cumbersome All commodities are valued by this medium § Medium is accepted in exchange for commodities § Medium of exchange is money Money is accepted not for direct utility it provides § But for indirect utility via commodities it can purchase 22

Efficiency of a Price System n Allocation device that has received by far the greatest attention by economists is price system § Assumes all commodities are valued in market by their money equivalence n Permits decentralization of allocation decisions § Provides a method for relating household preferences with supply at a reduced cost for society • Prices act as signals to economic agents in guiding their supply and demand decisions § Under a perfectly competitive price system, households have no control over market prices • Take prices as given ¨ Yields a Pareto-efficient market system 23

Efficiency of a Price System n In a perfectly competitive price system § Correspondence between Pareto-efficient allocation of resources and perfectly competitive price system is exact • Called First Fundamental Theorem of Welfare Economics ¨ n Provides formal and general confirmation of Adam Smith’s invisible hand Second Fundamental Theorem of Welfare Economics states § Every Pareto-efficient allocation has an associated perfectly § competitive set of prices States possibility of achieving any desired Pareto-efficient allocation as a market-based equilibrium using an appropriate distribution of income • Not every Pareto-efficient allocation is a social-welfare optimum 24

Efficiency of a Price System n Recall that Pareto efficiency in exchange requires § MRS 1 = MRS 2 = … = MRSn • For n households in economy § For utility maximization subject to a wealth or income constraint, each household equates its MRS with price ratio • MRS(x 2 for x 1) = p 1/p 2 § Every household faces same price ratio § Market in equilibrium creates a societal trade-off rate that is a correct reflection of every household’s trade-off rate • Information on this trade-off rate (if it could be gathered) would require large expenditures by a government ¨ Instead, this trade-off can be generated by perfectly competitive interaction of supply and demand § At zero governmental cost 25

Offer Curve n n n Traces out points where household maximizes utility for a given level of income across various ratios of prices § See Figure 6. 5 At each point household’s indifference curve is tangent to a budget constraint for a given price ratio Represents how much a household is willing to offer one commodity in exchange for the other at a given price ratio Analogous to price consumption curve with focal point at initial endowment e At alternative price ratios, endowment is affordable § Every point on offer curve is at least as good as agent’s endowment point e 26

Figure 6. 5 Offer curve 27

Offer Curve n Represents a set of demanded bundles § Each demanded bundle associated with a price ratio • As price ratio continues to increase, new demanded • • • bundles unfold Locus of all these demanded bundles is offer curve Each household has an offer curve and initial endowment of commodities Relating offer curves and endowment of two households in an Edgeworth box ¨ Walrasian equilibrium is illustrated in Figure 6. 6 28

Figure 6. 6 Pareto efficiency and competitive pricing 29

Offer Curve n Where two offer curves intersect, point A, price ratio is same for both Robinson and Friday § Demanded bundles exactly match supply § MRSs for Robinson and Friday are equated • Yields one-to-one correspondence between Pareto efficiency and perfectly competitive markets MRSs are both equal to p 1*/p 2* • § Aggregate supply equals aggregate demand for each of the commodities • Households are maximizing their utility 30

Walras’s Law n n n Static equilibrium conditions are generally theoretical results in economics Economy does not operate on a set of natural laws that describe its evolution Process of how an economy in disequilibrium reaches an equilibrium state (called tâtonnement stability) can be described only in limited detail § First described by Walras § Tâtonnement is French • Means groping or trial and error 31

Walras’s Law n For example, if at p 1'/p 2' quantity demanded for fish is greater than quantity supplied and quantity demanded for bread is less than quantity supplied § Price ratio would rise • Quantity demanded for fish would decline and that for bread would increase ¨ n Adjustment would continue until prices converge to competitive equilibrium levels Adjustment assumes prices will respond to market shortages and surpluses § If prices are rigid then tâtonnement process will not work 32

Walras’s Law n n n Friday’s demand functions for commodities x 1 and x 2, respectively § x 1 F(p 1, p 2, e 1 F, e 2 F) and x 2 F( p 1, p 2, e 1 F, e 2 F) Robinson’s demand functions § x 1 R(p 1, p 2, e 1 R, e 2 R) and x 2 R( p 1, p 2, e 1 R, e 2 R) Walrasian equilibrium set of prices (p 1*, p 2*) is where aggregate demand equals aggregate supply 33

Walras’s Law n Alternatively, this Walrasian equilibrium may be represented in terms of aggregate excess demand functions § For zj > 0 commodity j is in excess demand § For zj < 0 commodity j is in excess supply • For j = 1, 2 34

Walras’s Law n Walrasian equilibrium exists when these aggregate excess demand functions are zero § Result of specifying markets for commodities in terms of excess demand is Walras’s Law • Value of aggregate excess demand is zero for not only the equilibrium set of prices (p 1*, p 2*) but for all possible prices n In two-commodity economy, if z 1 > 0 (excess demand), then, given positive prices, z 2 < 0 (excess supply) for Walras’s Law to hold 35

Walras’s Law n Proof of Walras’s Law § Involves adding up households’ budget constraints and rearranging terms n Consider Friday’s budget constraint, where the right-hand side is Friday’s income, represented as value of Friday’s endowments of x 1 and x 2 • Total expenditures on x 1 and x 2 will equal this value of endowments § Rearranging terms, we get 36

Walras’s Law n Define Friday’s excess demands for commodities 1 and 2 as § For z 1 F > 0 Friday has excess demand for fish and z 1 F < 0 Friday has excess supply n Represent Friday’s budget constraint as § Value of Friday’s excess demand for the two commodities is zero 37

Walras’s Law n n Robinson’s value of excess demand for the two commodities is also zero Adding Friday’s and Robinson’s value of excess demand functions yields Walras’s Law 38

Relative Prices n Important result of Walras’s Law § If aggregate demand equals aggregate supply in one market, then, for a two-market economy • Demand must equal supply in the other market § Generalizing to k commodities, if aggregate demand equals aggregate supply in (k - 1) markets • Demand must equal supply in remaining excluded market 39

Relative Prices n By Walras’s Law, if p 2* > 0, then § A set of prices where aggregate demand for one market equals aggregate supply will result in other market clearing n Mathematically, this implies (k - 1) independent equations in a kcommodity model § With one less equation than k number of market clearing prices, cannot solve system for a set of k independent prices • Can determine only relative prices § Specifically, in general equilibrium, each household’s income is the value of endowment at given prices • Each household’s budget constraint is homogeneous of degree zero in prices • In general equilibrium, only relative prices are determined, given that all • households’ budget constraints are homogeneous of degree zero in prices Multiplying all prices by some positive constant does not change households’ demand supply for commodities 40

Social Welfare n Walrasian equilibrium may not be an optimal social-welfare point § However, does result in a Pareto-efficient allocation § Assumes a given distribution of initial endowments n If endowments are distributed in such a way that one agent receives a relatively small share of initial endowments and other agent receives a relatively large share § Social welfare may not be maximized • To accomplish optimal distribution of initial endowments along with competitive prices is required n One criticism of perfect competition (capitalist markets in general) involves § Restriction that perfectly competitive markets take this distribution of initial § endowments as given With any initial distribution of endowments that society deems as “unfair” • Perfect competition will not maximize social welfare 41

Equitable Distribution of Endowments and Fair Allocations n Problem of determining an optimal distribution of initial endowments is normative in nature § Involves value judgments concerning satisfaction households receive from their endowments • Possible normative solution is an optimal distribution of initial endowments that is equitable ¨ n May be defined as an allocation of endowments where no household prefers any other household’s initial endowment One equitable allocation of initial endowments is an equal division of commodities § Each household has same initial commodity bundle • Equal division will probably not be Pareto efficient ¨ A competitive market, given this initial equal division of commodities, will yield a Walrasian equilibrium that is Pareto efficient § Such a market allocation is called a fair allocation § Both equitable and Pareto efficient 42

Equitable Distribution of Endowments and Fair Allocations n Can show a fair allocation resulting from a competitive reallocation of equitable initial endowments by contradiction § Assume allocation is not fair and Robinson is envious • Prefers Friday’s allocation to his own • Robinson cannot afford Friday’s allocation • However, equal distribution of initial endowments implies value of initial endowment must be the same • Thus, Friday also cannot afford her optimal allocation ¨ Results in a contradiction 43

Equitable Distribution of Endowments and Fair Allocations n Impossible for Robinson to envy Friday at Pareto-efficient allocation § So a competitive equilibrium from equitable initial endowments is a fair allocation n Some societies have achieved this equal division in value of initial endowments by reducing size of Edgeworth box § Through war, pestilence, and famine, current and future generations of households within these societies are or will be at a near subsistence level • Social welfare is not maximized by achieving an equitable distribution of endowments n One problem with achieving an equal distribution of initial endowment values is § Incentives to work and invest are reduced n • No incentive to provide next generation with additional endowments In a communist society, working for the common good is meant to replace these individual incentives § Results in an enlargement of Edgeworth box for all comrades • This degree of altruism may be too much to ask of individual households 44

Equitable Distribution of Endowments and Fair Allocations n An alternative to an equal distribution of value of endowments § Providing equal opportunities for enriching a household’s endowments n n Equal opportunity was one of driving forces for large migration of households to United States in 19 th century Providing an initial endowment consisting of equal opportunities is an equitable allocation § Provides an underlying justification for equal opportunity legislation • Ranging from minority rights to funding for public education 45

Equitable Distribution of Endowments and Fair Allocations n U. S. history indicates that combining equal opportunity with free markets can greatly enlarge Edgeworth box § Results in increasing all households’ utilities and in fair allocations leading to an optimal social-welfare allocation of commodities • However, a great deal of poverty still exists within United States • The reversal in 1980 s of a more equal distribution of wealth • indicates that United States has not reached local bliss (maximum social welfare) Other societies are generally more socialistic than United States ¨ Unwilling to have such a large inequality in wealth § Generally strive for a more equal distribution of endowments • Which system is preferred is a value judgment 46