2. Equlibrium and Efficiency

Transcription

1 2. Equlibrium and Efficiency 1

2 2.1 Introduction competition and efficiency Smith s invisible hand model of competitive economy combine independent decision-making of consumers and firms into a complete model of the economy existence of competitive equilibrium competitive equilibrium =efficient coordinating role played by prices 2

3 2.2 Economic Models Why employ models to make predictions? experiments --> difficult economic system is too large and too complex formal model = logical consistency Many different models <-- policies important common features objectives of the individual agents in the economy constraints they face aggregate individual decisions to markets equilibrium --> partial or general trade-off between specified and general 3

4 2.3 Competitive Economies competitive individual agents do not consider their actions to have any effect on prices prices = fixed or parametric negligible in size relative to the market prices values --> relative values of different goods signals --> coordinating the decisions of agents economic efficiency 4

5 secondary feature in the model symmetric information uncertainty = all agents are equally uninformed any agent has no informational advantage two forms of competitive model exchange economy production economy 5

6 2.3.1 Exchange Economy trade of commodities between two parties for mutual advantage two-consumer, two-good exchange economy all results extend to large numbers endowment = initial stock of the goods (constant) exchange (market prices = exchange rates ) competitive = price-taking behavior desirable consumption plans 6

7 consumer h endowment : h s initial stock of good i prices p 1, p 2 budget constraint : h s consumption plan preference = utility function marginal utility is positive for both goods indifferent curve has convex shape 7

8 Edgeworth box feasible plan rectangle length southwest corner: consumer 1 s origin northeast corner: consumer 2 s origin 8

9 2 1 9

10 Equilibrium budget constraints consumer 1: relative to the southwest corner consumer 2: relative to the northeast corner same endowment point gradient equilibrium prices utility maximization for both consumers supply = demand = 10

11 2 1 11

12 2 good 1: excess demand p 1 goes up good 2: excess supply p 2 goes down 1 12

13 2 good 1: excess demand p 1 goes down good 2: excess supply p 2 goes up 1 13

14 2 equilibrium 1 14

15 consumers demand functions max of utility subject to budget constraint demand demand = supply (market clearing ) equilibrium the prices lead to a budget line on which the indifference curves has a point of common tangency number of prices, number of independent equations 15

16 only relative prices matter budget constraint always passes through the endowment point gradient is determined by the price ratio demand and supply depend on the relative pricebut not on the absolute values homogeneous of degree 0 any scaling up or down will also be equilibrium prices price normalization numeraire = unit of account 16

17 Walras s law interdependence between the two equilibrium equations excess demand total value of excess demands, for any prices 17

18 Walras s law (continued) if Z 1 =0, then Z 2 =0 if Z 2 =0, then Z 1 =0 since p 1 Z 1 +p 2 Z 2 =0 equilibrium in one market implies equilibrium in the other choose good 1 as the numeraire (p 1 =1) and plot the excess demand for good 2 as a function of p 2 if the ED of 2 is zero, then the ED of 1 is zero ---> equilibrium extension of many consumers and goods number of equilibrium conditions 18

19 0 19

20 2.3.2 Production and Exchange addition of production to exchange economy initial endowments (e.g. labor) consumption goods produced from ini. end. intermediates : produced from ini. end. and input for another Arrow-Debreu economy model of competitive economy producer (firm): production technology consumer: preference, initial endowments shareholdings of the firms 20

21 firm (producer) production set available production technology = knowledge inputs ---> outputs Y j : firm j s production set input: negative 0 = subtraction from the stock of that good available for consumption output: positive 0 production plan: prices: profit: 21

22 Good 2 Y j 3-2 Good 1 22

23 Process of profit maximization isoprofit curve <--- all production plans that give a specified level of profit π : given π =0 <---> inner product of p and y = 0 (price vector is perpendicular to isoprofit line) π >0 <---> inner product > 0 (acute angle) π <0 <---> inner product < 0 (obtuse angle) 23

24 Process of profit maximization profit maximization y* : tangency point between the highest attainable isoprofit curve and production set the higher the isoprofit curve, the bigger the profit (northeast shift gives more profit) 24

25 π = 0 π < 0 π > 0 Good 2 p 2 p 1 Good 1 25

26 economy with n goods m firms production plan chosen to maximize profits s.t. production set supply function level of profit aggregate supply negative or positive 26

27 private ownership economy H consumers : initial endowments ω h shareholdings in firms budget constraint change in prices ---> endowment dividend income 27

28 demand function aggregate demand equilibrium <---> demand = supply (not fixed) excess demand=0 demand (supply) is determined by relative prices ---> Z i is homogeneous of degree zero normalization, numeraire 28

29 2.4 Efficiency of Competition efficiency if more cannot be achieved, then the outcome is efficient <--- individual decision-maker different decision-makers aggregation of preferences first-best only the production technology and the limited endowments restrict the choice of DM second-best other constraints could be limits 29

30 2.4.1 Single Consumer individual preferences = social preferences best outcome = first-best no constraint on policy choice single firm & single consumer diagram = Robinson Crusoe economy Robinson acts as a firm carrying out production and as a consumer of the product of the firm 1) Robinson = social planner for coordination 2) profit maximization & utility maximization 30

31 Good 2 good 2 =numeraire 1 π p 1 Good 1 31

32 Good 2 good 2 =numeraire 1 π p 1 Good 1 32

33 budget constraint or where is net consumption utility maximization ---> equilibrium for the economy ---> net consumption = supply profit maximization = utility maximization efficient 33

34 Good 2 1 π p 1 Good 1 34

35 Good 2 1 π p 1 Good 1 35

36 Good 2 1 π p 1 Good 1 36

37 characterization of first-best allocation tangency point between the two curves marginal rate of substitution MRS 12 =U 1 /U 2 marginal rate of transformation MRT 12 (=MC 1 /MC 2 ) MRS 12 =MRT 12 marginal value = marginal cost at the first-best 37

38 the market achieves efficiency utility maximization ---> MRS 12 =p 1 /p 2 profit maximization ---> MRT 12 =p 1 /p 2 competitive equilibrium MRS 12 =p 1 /p 2 = MRT 12 38

39 constant return to scale efficient production frontier =straight line through the origin only equilibrium : zero profit positive ---> infinitely large profit isoprofit curve with π =0 equal to efficient PF two further implications equilibrium price ratio is determined by the zeroprofit condition and is independent of demand since the profit income of the consumer is zero, the budget constraint passes through the origin 39

40 Good 2 1 p 1 Good 1 40

41 2.4.2 Pareto-Efficiency more than one consumer Pareto-efficiency no unexploited economic gains consider whether it is possible to undertake a reallocation of resources that can benefit at least one consumer without harming any other if possible, there would exist unexploited gains if not, the initial position is Pareto-efficient efficient allocation of resources 41

42 feasibility : consumption, production, ini. endowment x = y + ω Definition 1: A feasible consp. allocation Pareto-efficient if there does not exist an alternative feasible allocation such that i. Allocation gives all consumers at least as much utility as. ii. Allocation gives at least one consumer more utility than. is Pareto-efficient if there is no that can make someone better off without making anyone worse off is 42

43 2.4.3 Efficiency in an Exchange Economy Two Theorems of Welfare Economics First Theorem 1 a competitive equilibrium is Pareto efficient Second Theorem 2 any Pareto-efficient allocation can be decentralized as a competitive equilibrium encouragement of competition Edgeworth box diagram a is not a Pareto-efficient allocation b is Pareto-preferred to a 43

44 2 1 44

45 Pareto-efficient allocation c beginning at c, any change in the allocation must lower the utility of at least one of the consumers tangency between the two indifference curves MRS 121 =MRS 12 2 Pareto-efficient allocation is not unique ---> contract curve 45

46 2 1 contract curve 46

47 competitive equilibrium a price line through the initial endowment point ω that is tangential to both indifference curves at the same point competitive equilibrium = Pareto-efficient utility maximization ---> MRS 12h =p 1 /p 2 competitive equilibrium MRS 121 =p 1 /p 2 = MRS 12 2 Theorem 1 (First Theorem of Welfare Economics) The allocation of commodities at a competitive equilibrium is Pareto-efficient. 47

48 2 1 48

49 Second Theorem whether any chosen Pareto-efficient allocation can be made into CE by choosing suitable location for the initial endowment possible if indifference curves are convex a budget line through the Pareto-ef.allo. exists consumers potimal choice decentralization Theorem 2 (Second Theorem of Welfare Economics) With convex preferences, any Pareto-efficient allocation can be made a competitive equilibrium. 49

50 2.4.4 Extension to Production production ---> supply is variable consumer s income differs Robinson Crusoe economy: same more than one consumer utility maximization ---> MRS ijh =p i /p j profit maximization ---> MRT ijm =p i /p j competitive equilibrium efficiency in consumption MRS ijh =p i /p j = MRS ijh efficiency in production MRT ijm =p i /p j = MRT ij m MRS ijh = MRT ij m --->First Theorem 50

51 Second Theorem W: feasible output plans w=y+ω Z: quantities for Pareto-improvement if both are convex, a common tangent exists total consumption 51

52 W Z Price line Feasible set 52

53 2.5 Lump-Sum Taxation decentralization mechanism Second Theorem lump-sum transfers lump sum no change in a consumer s behavior can affect the size of the transfer differentiated from income or commodity taxes redistributive instrument 53

54 2 1 54

55 initial endowment Pareto efficient e equilibrium prices income at the initial point budget constraint lump-sum taxes consumer 1 pays a tax consumer 2 pays a negative tax pair of taxes = transfer of endowment efficiency in achieving distributional objectives 55