Quản trị kinh doanh - Chapter 16: General equilibrium theory

General Equilibrium TheoryChapter 16Copyright (c)2014 John Wiley & Sons, Inc.1Chapter Sixteen OverviewGeneral Equilibrium – Analysis IPartial Equilibrium BiasEfficiency and Perfect CompetitionGeneral Equilibrium – Analysis IIThe Efficiency if CompetitionThe Edgeworth BoxAnalysis of Allocation: A Pure Exchange EconomyAnalysis of ProductionChapter SixteenCopyright (c)2014 John Wiley & Sons, Inc.2Chapter SixteenPartial vs. General EquilibriumIf there are spillover effects from one market to another, then the effects of a change in one market on the economy must be analyzed by examining its effect on all marketsCopyright (c)2014 John Wiley & Sons, Inc.3Chapter SixteenPartial vs. General EquilibriumFurther, many exogenous events (or policy changes) affect many markets simultaneously (example: discovery of a major oil deposit that raises the income of all citizens in an economy and so affects equilibrium in all markets).If we do not take into account all markets in our equilibrium calculation, we induce a bias in our analysis.Copyright (c)2014 John Wiley & Sons, Inc.4Chapter SixteenPartial vs. General EquilibriumDefinition: General Equilibrium analysis is the study of how equilibrium is determined in all markets simultaneously (e.g. product markets and labor markets).Definition: Partial Equilibrium analysis is the study of how equilibrium is determined in only a single market (e.g. a single product market).Copyright (c)2014 John Wiley & Sons, Inc.5Chapter SixteenPartial vs. General EquilibriumExample: Equilibrium in two marketsQ1D = 12 – 3p1 + p2 Q1s = 2 + p1Q2D = 4 – 2p2 + p1 Q2s = 1 + p2What is the general equilibrium level of prices and output in this economy?Market 1 equilibrium: 12 – 3p1 + p2 = 2 + p1 p1 = 10/4 + p2/4Market 2 equilibrium: 4 – 2p2 + p1 = 1 + p2 p2 = 1 + p1/3Copyright (c)2014 John Wiley & Sons, Inc.6Chapter SixteenPartial vs. General EquilibriumSubstituting condition 1 into condition 2:4 – 2p2 + 10/4 + p2/4 = 1 + p2 2 = p2e 3 = p1e Q1e = 5 Q2e = 3Copyright (c)2014 John Wiley & Sons, Inc.7Q1P1Market 14.6714P1 = 4 + P2/3 – QD1/3Chapter SixteenEquilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.8Q12 P1 = Q1s - 24.67Chapter SixteenP1Market 114Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.9Q12 5 P1 = 4 + P2/3 - Q1D/3•3e14.67Chapter SixteenP1 = Q1s - 2P1Market 114Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.10Q21P2 = Q2s - 1Chapter SixteenP2Market 2Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.11Q21 11P2 = Q2s - 1P2 = 4 + P1/2 - Q2D/25.5Chapter SixteenP2Market 2Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.12Q21 11P2 = Q2s - 1P2 = 4 + P1/2 - Q2D/2•25.5e24Chapter SixteenP2Market 2Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.13Chapter SixteenEquilibrium in Two Markets 32/11 = p2e 63/11 = p1e Q1e = 85/11 Q2e = 43/11Suppose an exogenous shock increases demand in market 1 to: Q1D = 22 – 3p1 + p2 . What is the new general equilibrium? Market 1 equilibrium: p1 = 22/4 + p2/4 Market 2 equilibrium: p2 = 1 + p1/3Copyright (c)2014 John Wiley & Sons, Inc.14Chapter SixteenSuppose you used the partial equilibrium price and output level in market 2 in order to compute the market 1 equilibrium. What would be the bias in your conclusions for market 1?If we re-solve for market 1 price with the new demand but p2e = 2, we obtain p1e = 11/2 = 5.5 – but in part (b), p1e = 63/11 = 5.72. In other words, we would underestimate the true price for good 1.Equilibrium in Two MarketsCopyright (c)2014 John Wiley & Sons, Inc.15Chapter SixteenConsider an economy with:2 types of households – white-collar households and blue-collar households purchasing 2 goods – energy and food – each of which is produced with2 input services – labor and capitalEquilibrium in Many MarketsCopyright (c)2014 John Wiley & Sons, Inc.16Chapter SixteenEquilibrium in Many MarketsCopyright (c)2014 John Wiley & Sons, Inc.17Chapter SixteenThe law that states that in a general competitive equilibrium with a total of N markets, if supply equals demand in the first N–1 markets, then supply will equal demand in the Nth market as well.Walras’ LawCopyright (c)2014 John Wiley & Sons, Inc.18Chapter SixteenEconomic EfficiencyDefinition: Allocation of goods and inputs is a pattern of consumption and input usage that might arise in a general equilibrium in an economy.Definition: An economic situation is Economically Efficient or Pareto Efficient if there is no other feasible allocation of goods and inputs that would make any person better off without hurting somebody else.Definition: An economic situation is Economically Inefficient or Pareto Inefficient if there is an alternative feasible allocation of goods and inputs that would make all consumers better off than the initial allocation does.Copyright (c)2014 John Wiley & Sons, Inc.19Chapter SixteenEfficiency and Competitive MarketsA competitive equilibrium needs to satisfy three conditions to be efficient:Exchange Efficiency – A characteristic of resource allocation in which a fixed stock of consumption goods cannot be reallocated among consumers in an economy without making at least some consumers worse off.Input Efficiency – A characteristic of resource allocation in which a fixed stock of inputs cannot be reallocated among firms in an economy without reducing the output of at least one of the goods that is produced in the economy.Substitution Efficiency – A characteristic of resource allocation in which, given the total amounts of capital and labor available in the economy, there is no way to make all consumers better off by producing more of one product and less of another.Copyright (c)2014 John Wiley & Sons, Inc.20Chapter SixteenExchange EfficiencySimplifying Assumptions1. Consumers and producers are price takers.2. There are only two individuals and two goods in the economy.3. Individuals have fixed allocations (endowments) of goods that they might trade. No production occurs for now.4. Consumers maximize utility with usually-shaped indifference curves (and non-satiation). Utilities are not interdependent.Copyright (c)2014 John Wiley & Sons, Inc.21Chapter SixteenEdgeworth BoxA graph showing all the possible allocations of goods in a two-good economy, given the total available supply of each good.Copyright (c)2014 John Wiley & Sons, Inc.22Chapter SixteenEdgeworth Box DiagramCopyright (c)2014 John Wiley & Sons, Inc.23Chapter SixteenEdgeworth Box DiagramCopyright (c)2014 John Wiley & Sons, Inc.24Chapter SixteenEdgeworth Box DiagramCopyright (c)2014 John Wiley & Sons, Inc.25Chapter SixteenEdgeworth Box Diagram1. The length of the side of the box measures the total amount of the good available.2. Person A’s consumption choices are measured from the lower left hand corner, Person B’s consumption choices are measured from the upper right hand corner.3. We can represent an initial endowment, (wA1,wA2), (wB1,wB2) as a point in the box. This is the allocation that consumers have before any exchange occurs.Copyright (c)2014 John Wiley & Sons, Inc.26Chapter SixteenEdgeworth Box Diagram4. Any other feasible consumption allocation is a point in the box such that, for each individual:"final demand" < "initial supply" xA1+xB1 < wA1 + wB1 xA2+xB2 < wA2 + wB25. We can represent indifference curves of the individuals between the goods in the standard way measured from the appropriate corners.Copyright (c)2014 John Wiley & Sons, Inc.27Chapter SixteenExchange Any voluntary barter trade (a point that makes at least one consumer better off) must lie in a “lens” formed by the indifference curves that intersect the initial endowment. Allocation through trading that potentially improves utilitybut is infeasible: There is excess demand for good 1 and excess supply of good 2. Neither the market for good 1 nor the market for good 2 is in equilibrium. Copyright (c)2014 John Wiley & Sons, Inc.28Chapter SixteenEdgeworth Box – Infeasible AllocationCopyright (c)2014 John Wiley & Sons, Inc.29Chapter SixteenEdgeworth Box – Infeasible AllocationCopyright (c)2014 John Wiley & Sons, Inc.30Chapter SixteenEdgeworth Box – Infeasible AllocationCopyright (c)2014 John Wiley & Sons, Inc.31Chapter SixteenEdgeworth Box –Allocation Can be ImprovedCopyright (c)2014 John Wiley & Sons, Inc.32Chapter SixteenEdgeworth Box –Allocation Can be ImprovedCopyright (c)2014 John Wiley & Sons, Inc.33Trading will continue until no mutually improving trades are possible. (e.g. at M)Chapter SixteenEdgeworth BoxesCopyright (c)2014 John Wiley & Sons, Inc.34Chapter SixteenEdgeworth Box – Economically Efficient AllocationCopyright (c)2014 John Wiley & Sons, Inc.35Chapter SixteenEdgeworth Box – Economically Efficient AllocationCopyright (c)2014 John Wiley & Sons, Inc.36Chapter SixteenEdgeworth Box – Economically Efficient AllocationCopyright (c)2014 John Wiley & Sons, Inc.37Chapter SixteenEdgeworth Box – Economically Efficient AllocationCopyright (c)2014 John Wiley & Sons, Inc.38Chapter Sixteen1. M is Pareto efficient2. M is at a tangency point of the two individuals’ indifference curvesMRSA1,2 = MRSB1,23. Definition: The set of all Pareto efficientpoints in the Edgeworth box is known as the Pareto set or the Contract Curve.Pareto Set / Contract CurvesThis set typically will stretch from one corner to the other of the box (M not unique). A subset of this set will contain the points that are Pareto efficient with respect to the initial endowment.Copyright (c)2014 John Wiley & Sons, Inc.39Chapter SixteenThe Contract CurveCopyright (c)2014 John Wiley & Sons, Inc.40Chapter SixteenThe Contract CurveCopyright (c)2014 John Wiley & Sons, Inc.41Chapter SixteenThe Contract CurveCopyright (c)2014 John Wiley & Sons, Inc.42Chapter SixteenThe Contract CurveCopyright (c)2014 John Wiley & Sons, Inc.43Chapter SixteenPure Exchange Economy A pure exchange economy in which completely decentralized trading is allowed such that agents have access to each other and each is able to maximize utility subject to a feasibility constraint gets the economy to A Pareto Efficient Allocation. This requires each individual to have information on his endowment and preferences onlyCopyright (c)2014 John Wiley & Sons, Inc.44Chapter SixteenCalculating a Contract CurveTwo individuals, A and B with "Cobb-Douglas" utility functions over 2 goods, X and Y.UA = (XA)(YA)1- UB = (XB)(YB)1-MUXA = XA-1YA1-MUYA = (1-)XAYA-MUXB = XB-1YB1-MUYB = (1-)XBYB-XA + XB = 100 – This gives the size of the Edgeworth BoxYA + YB = 200Therefore: MRSX,YA = MUXA/MUYA = [/(1-)][YA/XA] MRSX,YB = MUXB/MUYB = [/(1-)][YB/XB]Copyright (c)2014 John Wiley & Sons, Inc.45Chapter SixteenCalculating a Contract CurveAndXA = 100 – XB – feasibility constraintsYA = 200 - YBMRSX,YA = MRSX,YB – tangency condition for contract curve[/(1-)][(200 – YB)/(100 – XB)] = [/(1-)][YB/XB]Or(-)YBXB - (1-)(100YB) + (1-)200XB = 0or (-)YAXA + (1-)(100YA) - (1-)200XA = 0Copyright (c)2014 John Wiley & Sons, Inc.46Chapter SixteenDraw the contract curve for  =  = ½The equations for the contract curves simplify to: YA = 2XA and YB = 2XBCalculating a Contract CurveCopyright (c)2014 John Wiley & Sons, Inc.47Chapter SixteenThe Role of PricesSuppose that agents are presented with prices, (p1,p2) that they take as given and can use to value their initial endowment of goods p1w1 + p2w2 = I Hence, these prices define a budget constraint for each individualtangency with the budget constraint determines where the individuals will desire to consume:Copyright (c)2014 John Wiley & Sons, Inc.48Chapter SixteenPrices & EquilibriumDefinition: If, at the announced prices, the amount that A wants to buy (sell) of good 1 exactly equals the amount B wants to sell (buy) of good 1 and if the same holds for good 2 as well, the market is in equilibrium.MRS = p1/p2So that in the General Equilibrium – MRSA1,2 = MRSB1,2 = p1/p2Copyright (c)2014 John Wiley & Sons, Inc.49Chapter SixteenEconomically Efficient Price AllocationCopyright (c)2014 John Wiley & Sons, Inc.50Chapter SixteenEconomically Efficient Price AllocationCopyright (c)2014 John Wiley & Sons, Inc.51Chapter SixteenEconomically Efficient Price AllocationCopyright (c)2014 John Wiley & Sons, Inc.52Chapter SixteenEconomically Efficient Price AllocationCopyright (c)2014 John Wiley & Sons, Inc.53Chapter SixteenEconomically Efficient Price Allocation In other words, income is determined by the value of the endowment and equilibrium holds when, in every market, demand equals supply. Further, since the market equilibrium holds where the marginal rates of substitution are equal and the preferred bundles of each agent lie above the budget set, the market equilibrium is Pareto Efficient.Copyright (c)2014 John Wiley & Sons, Inc.54Chapter SixteenEconomically Efficient Price AllocationThis means that society can achieve efficiency by allowing competitionThis equilibrium requires very little information (prices only) or co-ordination.In fact, any Pareto-efficient equilibrium can be obtained by competition, given an appropriate endowment. For example, any Pareto efficient allocation, x, can be obtained as a competitive equilibrium if the initial endowment is x.Copyright (c)2014 John Wiley & Sons, Inc.55Chapter SixteenEconomically Efficient Price Allocation This means that society can obtain a particular efficient allocation by appropriately redistributing endowments (income). This can be achieved through taxes/subsidies to endowments (lump sum taxes) that do not affect choice (prices) In fact, this redistribution could be viewed as the main role of government in the perfectly competitive modelCopyright (c)2014 John Wiley & Sons, Inc.56Chapter SixteenAt Equilibrium Prices:These Equilibriums Must be Present 1. Allocative Efficiency: MRSX,Y for all the individuals must be equal2. Private Utility Maximization: MRSX,Y for each and every individual must equal pX/pY3. Market Equilibrium: Qd = QS must hold for each and every good.4. Feasibility: Total supply must equal the original endowment for each and every good.Copyright (c)2014 John Wiley & Sons, Inc.57Chapter SixteenInput Efficiency Edgeworth Box for Inputs – A graph showing all the possible allocations of fixed quantities of labor and capital between the producers of two different goods. Input Contract Curve – A curve that shows all the input allocations in an Edgeworth box for inputs that are input efficient. MRTS XL,K = MRTSYL,K = w/rCopyright (c)2014 John Wiley & Sons, Inc.58Chapter SixteenInput EfficiencyCopyright (c)2014 John Wiley & Sons, Inc.59Chapter SixteenSubstitution EfficiencySuppose that all individuals in the economy have a dual role: they are consumers, but they also are the producers. In other words, the individual's role as a producer will determine their income. Definition: The production possibility frontier (PPF) of an individual is the maximum combinations of goods A and B that can be produced with the individual’s input (e.g., labor) per unit of time. Definition: An individual achieves efficiency in production if s/he produces combinations of goods on the PPF (so that there is no "slacking off").Copyright (c)2014 John Wiley & Sons, Inc.60Chapter SixteenMarginal Rate of TransformationDefinition: The slope of the production possibility frontier is the marginal rate of transformation (MRT). The MRT tells us how much more of good Y can be produced if the production of good X is reduced by a small amount.Orthe MRT tells us how much it costs to produce one good in terms of foregone production of the other good (opportunity cost).Copyright (c)2014 John Wiley & Sons, Inc.61Kate’s Production YX362PPFKMRTK= 2Chapter SixteenProduction Possibility Frontier (PPF)Copyright (c)2014 John Wiley & Sons, Inc.62Kate’s Production Pierre’s Production YXYX633622PPFKPPFPMRTK= 2MRTP= 1/2Chapter SixteenProduction Possibility FrontierCopyright (c)2014 John Wiley & Sons, Inc.63YX66MRTJ= 1/2MRTJ= 2PPFJChapter SixteenProduction Possibility FrontierKate’s Production Pierre’s Production Joint Production YXYX633622PPFKPPFPMRTK= 2MRTP= 1/2Copyright (c)2014 John Wiley & Sons, Inc.64Chapter SixteenJoint PPFDefinition: The joint PPF for all possible technologies and all producers in the economy depicts the maximum amount of each good that could be produced in total by all producers.Definition: A producer who, when producing one good, reduces production of a second good less compared to another producer is said to have a comparative advantage in producing the first good. Copyright (c)2014 John Wiley & Sons, Inc.65Chapter SixteenJoint PPF If the MRT of two different producers (and consumers) differs, then the individuals can potentially gain from trade If many production methods are available, the joint PPF takes a typically “rounded” shape, representing the various MRT’s available to the economy.Copyright (c)2014 John Wiley & Sons, Inc.66Chapter SixteenThe Efficient Product Mix Now, let’s look at the efficient product mix. At which point along the joint PPF would society operate? Any individual consumer would prefer production to occur at a point where the consumer's indifference curve is just tangent to the PPF. With Joint PPFCopyright (c)2014 John Wiley & Sons, Inc.67XYPPFJPreferenceDirectionChapter SixteenThe Efficient Product MixCopyright (c)2014 John Wiley & Sons, Inc.68XICPPFJChapter SixteenYPreferenceDirectionThe Efficient Product MixCopyright (c)2014 John Wiley & Sons, Inc.69X•ICPPFJMRSX,Y = MRTX,YChapter SixteenYPreferenceDirectionThe Efficient Product MixCopyright (c)2014 John Wiley & Sons, Inc.70Chapter SixteenThe Efficient Product MixAt this point, the consumer’s willingness to give up good X in order to get good Y just equals the rate at which a producer has to give up good X in order to produce more of good Y. MRTX,Y = MRSX,YBut this must be true for all consumers if the economy is to produce optimally for each consumer.Copyright (c)2014 John Wiley & Sons, Inc.71Chapter SixteenComparative Markets & OptimalityCan the competitive market help us to achieve this optimality?At the Pareto efficient allocations, it is true for all consumers that:MRSX,Y = pX/pYCopyright (c)2014 John Wiley & Sons, Inc.72Chapter SixteenThe Producers’ ProblemSuppose that the producers produce goods X and Y and choose the product mix so as to maximize profits given the prices pX and pY:Max  = pXQX + pYQY – C*QX,QYWhere: we will suppose that the cost of production is fixed whatever the optimal output mix (e.g., we just want to know how to employ the labor we have contracted)Copyright (c)2014 John Wiley & Sons, Inc.73Chapter SixteenIsoprofitDefinition: an isoprofit line shows the output combinations that result in a given level of profit, 0 orQY = (0 + C*)/pY – pXQX/pYCopyright (c)2014 John Wiley & Sons, Inc.74XYPPFJChapter SixteenThe Profit Maximizing Product MixCopyright (c)2014 John Wiley & Sons, Inc.75X•PPFJ(0+C*)/PY-pX/pYChapter SixteenYThe Profit Maximizing Product MixIsoprofit LinesCopyright (c)2014 John Wiley & Sons, Inc.76X•PPFJDirection of increasing profits(0+C*)/PYProfit maximising product mix-pX/pYChapter SixteenYThe Profit Maximizing Product MixIsoprofit LinesCopyright (c)2014 John Wiley & Sons, Inc.77Chapter SixteenThe Profit Maximizing Product MixHence, If the firm maximizes profits, then, it chooses the product mix that shifts out the isoprofit line as much as possible while remaining feasible. This is a tangency point such that for all producers: MRTX,Y = pX/pYCopyright (c)2014 John Wiley & Sons, Inc.78Chapter SixteenIn other words, in equilibrium, the price ratio will measure the opportunity cost of production of one good in terms of production of the other good.ThereforeBecause competition ensures that both the MRS and the MRT equal the (same) price ratio for all producers and all consumers, a competitive equilibrium achieves an efficient product mix for all producers and all consumersEarlier allocative efficiency results still hold with productionThe Profit Maximizing Product MixCopyright (c)2014 John Wiley & Sons, Inc.79YXPPFChapter SixteenGeneral EquilibriumCopyright (c)2014 John Wiley & Sons, Inc.80XPPF•XsXeBYeBYsChapter SixteenYGeneral EquilibriumCopyright (c)2014 John Wiley & Sons, Inc.81XPPF•XsSlope = -p1e/p2e•XeAXeBYeBYsChapter SixteenYGeneral EquilibriumCopyright (c)2014 John Wiley & Sons, Inc.82XPPF•XsSlope = -p1e/p2e•YeAXeAXeBYeBYsChapter SixteenYGeneral EquilibriumCopyright (c)2014 John Wiley & Sons, Inc.83Chapter SixteenYs and Xs are the amounts of X produced in the economy; (XeA,YeA) is the amount of X and Y consumed by person A and (XeB,YeB) is the amount of X and Y consumed by person B. Efficiency in exchange (on contract curve) Efficiency in use of inputs (on PPF) Efficiency in product mix (tangency with PPF)General EquilibriumWhere:ConsiderCopyright (c)2014 John Wiley & Sons, Inc.84Chapter SixteenFundamental Theorems of Welfare Economics1. The allocation of goods and inputs that arises in a general competitive equilibrium is economically efficient. That is, given the resources available to the economy, there is no other feasible allocation of goods and inputs that could simultaneously make all consumers better off.2. Any economically efficient allocation of goods and inputs can be attained as a general competitive equilibrium through a judicious allocation of the economy’s scarce supplies of resources.Copyright (c)2014 John Wiley & Sons, Inc.85Chapter SixteenSummary1. If there are spillover effects among markets in the economy, we need to calculate equilibrium by determining equilibrium in all markets simultaneously. Otherwise, our results for equilibrium prices and quantities will be biased. This bias can be large.2. In partial equilibrium analysis, a perfectly competitive and allocates goods in a way market produces a Pareto efficient amount of output that is Pareto efficient.Copyright (c)2014 John Wiley & Sons, Inc.86Chapter Sixteen3. We can make a similar statement about perfect competition in a general equilibrium analysis. In other words, taking into account that income is determined endogenously and costs are determined endogenously was well, we can still state that perfect competition produces a Pareto efficient amount of output and allocates it in a Pareto efficient way.4. More specifically, competitive allocations are efficient in exchange, efficient in the use of inputs in production, and efficient in the mix of outputs.SummaryCopyright (c)2014 John Wiley & Sons, Inc.87