Pure Exchange

1. Pure Exchange Chapter 6

2. Introduction • 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 • 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

3. Introduction • 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

4. Introduction • 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

5. Gains from Trade • 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 • MRS1 = MRS2 = … = MRSn • 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

6. Two-Commodity and Two-Household Economy • Consider an economy with two commodities, bread and fish, and two individuals (households), Robinson (R) and Friday (F ) • 50 units of bread and 100 units of fish are to be allocated • Can be allocated in various ways • Could be all 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

7. Two-Commodity and Two-Household Economy • 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 • Friday’s MRSF (bread for fish) = 1/1 • Allocation does not result in an efficient allocation because • MRSR = 2 ≠ 1 = MRSF • 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 • 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

8. Table 6.1 Gains from trade

9. Two-Commodity and Two-Household Economy • As long as MRSs between Robinson and Friday are unequal • Gains from trade are possible • Many possible trades that will result in gains • Use Robinson’s and Friday’s initial endowments and indifference curves to determine all possible trades leading to gains • Shown in Figure 6.1

10. Figure 6.1 Preferences and endowments for the two-commodity…

11. Edgeworth Box • 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 • 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

12. Figure 6.2 Edgeworth box in a pure-exchange economy

13. Edgeworth Box • An allocation is feasible if total quantity consumed of each commodity is equal to total available from endowments • xR1 + xF1 = eR1 +eF1, commodity 1, fish • xR2 + xF2 = eR2 +eF2, commodity 2, bread • An allocation where Robinson receives nothing and Friday receives all is a feasible allocation • Represented by 0R in Figure 6.2 • Robinson’s utility is minimized and Friday’s utility is maximized • At 0F, allocation is nothing for Friday and everything for Robinson • Friday’s utility is minimized and Robinson’s maximized • 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 0R, Friday receives more of either fish or bread • Increases her utility • Robinson receives less, which decreases his utility • Reverse occurs for a movement toward 0F

14. Pareto-Efficient Allocation • 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

15. Figure 6.3 Efficiency in a pure-exchange economy

16. Pareto-Efficient Allocation • 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

17. Contract Curve (Pareto-Efficient Allocation) • 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 Pareto-efficient allocation

18. Contract Curve (Pareto-Efficient Allocation) • 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

19. Figure 6.4 Contract curve in a pure-exchange economy

20. Contract Curve (Pareto-Efficient Allocation) • 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 • 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

21. Contract Curve (Pareto-Efficient Allocation) • 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 • 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 • Some analytical results can be obtained with a Pareto-welfare criterion • For example, point rationing is ordinarily better than fixed-ration quantities

22. Efficiency of a Price System • 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

23. Efficiency of a Price System • 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 • 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

24. Efficiency of a Price System • 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 • 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

25. Efficiency of a Price System • Recall that Pareto efficiency in exchange requires • MRS1 = MRS2 = … = 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(x2 for x1) = p1/p2 • 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

26. Offer Curve • 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

27. Figure 6.5 Offer curve

28. Offer Curve • 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

29. Figure 6.6 Pareto efficiency and competitive pricing

30. Offer Curve • 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 p1*/p2* • Aggregate supply equals aggregate demand for each of the commodities • Households are maximizing their utility

31. Walras’s Law • 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

32. Walras’s Law • For example, if at p1'/p2' 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 • 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

33. Walras’s Law • Friday’s demand functions for commodities x1 and x2, respectively • x1F(p1, p2, e1F, e2F) and x2F( p1, p2, e1F, e2F) • Robinson’s demand functions • x1R(p1,p2,e1R,e2R) and x2R( p1, p2,e1R,e2R) • Walrasian equilibrium set of prices (p1*, p2*) is where aggregate demand equals aggregate supply

34. Walras’s Law • 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

35. Walras’s Law • 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 (p1*, p2*) but for all possible prices • In two-commodity economy, if z1 > 0 (excess demand), then, given positive prices, z2 < 0 (excess supply) for Walras’s Law to hold

36. Walras’s Law • Proof of Walras’s Law • Involves adding up households’ budget constraints and rearranging terms • Consider Friday’s budget constraint, where the right-hand side is Friday’s income, represented as value of Friday’s endowments of x1 and x2 • Total expenditures on x1 and x2 will equal this value of endowments • Rearranging terms, we get

37. Walras’s Law • Define Friday’s excess demands for commodities 1 and 2 as • For z1F > 0 Friday has excess demand for fish and z1F < 0 Friday has excess supply • Represent Friday’s budget constraint as • Value of Friday’s excess demand for the two commodities is zero

38. Walras’s Law • 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

39. Relative Prices • 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

40. Relative Prices • By Walras’s Law, if p2* > 0, then • A set of prices where aggregate demand for one market equals aggregate supply will result in other market clearing • Mathematically, this implies (k - 1) independent equations in a k-commodity 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 and supply for commodities

41. Social Welfare • 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 • 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 • 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

42. Equitable Distribution of Endowments and Fair Allocations • 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 • 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

43. Equitable Distribution of Endowments and Fair Allocations • 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

44. Equitable Distribution of Endowments and Fair Allocations • Impossible for Robinson to envy Friday at Pareto-efficient allocation • So a competitive equilibrium from equitable initial endowments is a fair allocation • 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 • One problem with achieving an equal distribution of initial endowment values is • Incentives to work and invest are reduced • 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

45. Equitable Distribution of Endowments and Fair Allocations • An alternative to an equal distribution of value of endowments • Providing equal opportunities for enriching a household’s endowments • Equal opportunity was one of driving forces for large migration of households to United States in 19th 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

46. Equitable Distribution of Endowments and Fair Allocations • 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 1980s 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