David Besanko and Ronald Braeutigam. Prepared by Katharine Rockett Dieter Balkenborg. Microeconomics, 2 nd Edition

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1 Microeconomics, nd Edition David esanko and Ronald raeutigam Chapter : General Equilibrium Theory Prepared by Katharine Rockett Dieter alkenborg 00 John Wiley & Sons, Inc. Trade involves more than one market General equilibrium model replaced labour theory of value, Debreu: Theory of value Main results in general equilibrium theory: Conditions for the eistence of a competitive market equilibrium, i.e., a price system here all markets clear if all agents produce, sell and buy such that they maimize their utility or profits hile taking all prices as given. First elfare theorem: Market equilibria are Pareto efficient in the sense that no one can be made better off by a different plan of production and a different allocation of goods ithout making anybody else orse off. Second elfare theorem: conversely, every Pareto optimum is a market equilibrium for a suitable system of prices and a ellchosen initial allocation of goods Assumptions needed for the theorems to hold: Homo oeconomicus (individuals must be rational + selfish in the sense their utility only depends on their on consumption) Survival assumptions. Roughly speaking If individuals do not on enough by themselves to survive, a market equilibrium might not eist. Conveity assumptions First: Pure barter economy, to goods, to people (e.g. Robinson and Freitag) Edgeorth bo Description of barter Notion of Pareto-efficiency The contract curve Market equilibrium First and second elfare theorem Then: Adding production 3. For simplicity there are only to individuals and to goods in the economy. Otherise there could not be any echange.. There is a fied, total amount of each good in the economy. For no there is no production. Let be the total amount available of commodity one and let be the total amount available of commodity. 3. Once these goods are allocated to the to agents, property rights are assigned and e obtain a barter economy if people are alloed to trade freely (or if a black markets can form). 4 In such a barter economy the agents can voluntarily trade given their initial allocations. Let the initial endoment (or allocation) of Ann (ob) be ( ) units of good and A ( ) units of good. Let (, A ) and (, ) denote the amounts Ann and ob end up ith. A feasible allocation is PARETO EFFICIENT if there is no trade from hich the to can gain, more precisely, there is no other feasible allocation that makes at least one individual better off and no individual orth off. Feasible means hereby "final demand" < "initial supply" + < = + + < = + The notion of Pareto-efficiency is defined for any allocation of goods. Strictly speaking, it does not require markets (that just helps ith the intuition) and certainly not prices. Superficial criticisms often ignore that. Agents are required to have clear preferences. They may have altruistic preferences, although this creates eternalities and therefore the elfare theorems may not hold. 5

2 Assuming selfish preferences and that no commodities are thron aay, e can use the Edgeorth o to see hether an allocation is efficient or not. We also assume the usual shape of the indifference curves. Any point in the Edgeorth bo represents an allocation of the available goods here noting is thron aay, as for instance the initial allocation E in the folloing graph. Good Good E Eample: An Edgeorth o Diagram Good 7 Good 8 In an Edgeorth o. The length of the side of the bo measures the total amount of the good available.. s consumption choices are measured from the loer left hand corner, s consumption choices are measured from the upper right hand corner. Ann s preferred allocations Ann prefers any allocation that is on or above her indifference curve through E. 3. We can represent an initial endoment, (, ), (, ) as a point in the bo. This is the allocation that consumers have before any echange occurs. 9 0 Good Good ob s preferred allocation E IC A Eample: An Edgeorth o Diagram For ob things are reversed. The top right corner is the point here he gets nothing. Moving don and left. ob prefers any allocation that is on or belo his indifference curve through E. Good Good

3 Good Good Gains from trade E IC A IC Eample: An Edgeorth o Diagram Good The lens that is above Ann s and belo ob s indifference curve shos the feasible trades that ould make both agents better off. In such a trade Ann ould give aay apples in echange for oranges. The initial allocation is not Pareto-efficient. Good 3 4 Good Good Good Eample: An Edgeorth o ith an Allocation that could be Improved Upon Good Eample: An Edgeorth o ith an Allocation that could be Improved Upon E E Good Good Good 5 A A Good Good Good Good Eample: Edgeorth o ith Economically Efficient Allocation Good Eample: Edgeorth o ith Economically Efficient Allocation E E Good Good Good 7 Good 8 3

4 Good Good Good Eample: Edgeorth o ith Economically Efficient Allocation Good Eample: Edgeorth o ith Economically Efficient Allocation M M E E Good Good Good 9 Good 0 A Pareto-efficient allocation Repeat: Is a distribution of the commodities (i.e., an allocation) of the available commodities in such a ay that is not possible to change the allocation such that at least one individual is made better off an no individual is made orth off. In other ords, there is no reason to trade. M is Pareto efficient implies in the interior:. M is at a tangency point of the to individuals indifference curves MRS A, = MRS, 3. Definition: The set of all Pareto efficient points in the Edgeorth bo is knon as the Pareto set or the Contract Curve. Good This set typically ill stretch from one corner to the other of the bo (M not unique) Good Eample: The Contract Curve A subset of this set ill contain the points that are Pareto efficient ith respect to the initial endoment. What is the set of Pareto efficient allocations if there is only one commodity? E Good 3 Good 4 4

5 Good Good Good Eample: The Contract Curve Good Eample: The Contract Curve E E Good Good Good 5 Good Good Good Eample: The Contract Curve Terminology and findings E A A Good Good 7 The contract curve consists of the set of all Pareto efficient allocations. At Paretoefficient allocations inside the Edgeorth bo the agents have identical marginal rates of substitution. An initial endoment E defines a lens of mutually beneficially trades. The intersection of this lens ith the contract curve as called the CORE by Edgeorth. This is ere efficient trade ould end. 8 Eample: Calculating a Contract Curve individuals, A and ith "Cobb-Douglas" utility functions over goods, and. U A = ( A ) α ( A ) -α U = ( ) β ( ) -β MU A = (-α) Aα -α MU = β A β- -αβ MU = (-β) Aβ -β A + = 00 A + = 00 This gives the size of the Edgeorth o

6 Therefore, MRS,A = MU A /MU A = [α/(-α)][ A / A ] MRS, = MU /MU = [β/(-β)][ / ] And A = 00 - A = 00 - feasibility constraints MRS,A = MRS, tangency condition for contract curve [α/(-α)][(00 )/(00 )] = [β/(-β)][ / ] or 3 3 (β-α) - (-α)β(00 ) + α(-β)00 = 0 or equivalently (β-α) A A + α(-β)(00 A ) - (-α)β00 A = 0. Dra the contract curve for α = β = ½ The equations for the contract curves simplify to: A = A and = Eample: A Contract Curve The market equilibrium Contract Curve Suppose both agents are price takers. (This is very artificial and ould be satisfactory only for many agents.) Suppose there are prices p and p. uy selling his initial endoments each agent can guarantee the income y A = pa+ p A or y = p + p A Slope = This money can then be used to buy the commodity bundle hich maimizes utility. 3

7 If markets then clear, e have a market equilibrium If Ann s optimal demand is (, A ) and (, ) denote the commodity bundles that Ann and ob ill buy at the given prices. Then markets clear if the optimal demands define a feasible allocation: Good Good A Eample: Prices hich do not yield a market equilibrium. "final demand" < "initial supply" E + < = + + < = + Good 37 A A Good 38 Good Good Eample: A Market equilibrium Good Good Eample: A Market equilibrium E E Good Good Good 39 Good 40 Good Good Eample: A Market equilibrium Good Good Eample: A Market equilibrium E E Good Good -P /P Good 4 -P /P Good 4 7

8 Market equilibrium We see: A market equilibrium must be a point in the core and hence Paretoefficient. This means that the indifference curves of the to agents have the same tangent (i.e., both have the same MRS). Moreover, this tangent must go through the initial endoment and hence define the budget line for the agents. The latter implies that the price ratio equals the marginal rate of substitution in equilibrium. 43 We have a system of to simultanoeus equations ith to unknons. The eistence of a solution is not obvious. General conditions have been orked out by Arro, Debreu and McKensey. Moreover, it follos: 44 The first elfare theorem The second elfare theorem Every market equilibrium is Paretoefficient. 45 Every Pareto efficient allocation is a market equilibrium for a suitable initial allocation of commodities. Take a point on the contract curve. Set the relative price such that it is equal to the MRS of the to consumers. Choose as the initial endoment E a point on the tangent to the indifference curves. Relative to this initial endoment the prices and the point form a market equilibrium. Take =E. Then no consumer ants to trade at these prices and a Market equilibrium is achieved. We obtain a no-trade equilibrium. 4 Good Good Eample: A Pareto optimum Good Good Eample: A Market equilibrium E Good Good -P /P Good 47 -P /P Good 48 8

9 This means that society can achieve efficiency by alloing competition This equilibrium requires very little information (prices only) or co-ordination. In fact, any Pareto-efficient equilibrium can be obtained by competition, given an appropriate endoment. For eample, any Pareto efficient allocation,, can be obtained as a competitive equilibrium if the initial endoment is. This means that society can obtain a particular efficient allocation by appropriately redistributing endoments (income). This can be achieved through taes/subsidies to endoments (lump sum taes) that do not affect choice (prices) In fact, this redistribution could be vieed as the main role of government in the perfectly competitive model Suppose that all individuals in the economy have a dual role: they are consumers, but they also are the producers. In other ords, the individual's role as a producer ill determine their income Definition: The production possibility frontier (PPF) of an individual is the maimum combinations of goods A and that can be produced ith the individual s input (e.g., labor) per unit of time. Definition: The slope of the production possibility frontier is the marginal rate of transformation (MRT). The MRT tells us ho much more of good can be produced if the production of good is reduced by a small amount. Or the MRT tells us ho much it costs to produce one good in terms of foregone production of the other good (opportunity cost). Definition: An individual achieves efficiency in production if s/he produces combinations of goods on the PPF (so that there is no "slacking off"). 5 5 Eample: General Equilibrium Eample: The Production Possibility Frontier s e Slope = -p e /p e Kate s production e A e A s e PPF 53 PPF K MRT K =

10 Eample: The Production Possibility Frontier Eample: The Production Possibility Frontier Kate s production Pierre s production PPF K MRT K = 3 PPF P MRT P = / Kate s production Pierre s production Joint Production MRT J = / PPF K MRT J = MRT K = 3 PPF P MRT P = / Definition: The joint PPF for all possible technologies and all producers in the economy depicts the maimum amount of each good that could be produced in total by all producers. Definition: A producer ho, hen 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. If the MRT of to 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 No, let s look at the efficient product mi At hich point along the joint PPF ould society operate? Any individual consumer ould prefer production to occur at a point here the consumer's indifference curve is just tangent to the PPF. Eample: The Efficient Product Mi Preference direction

11 Eample: The Efficient Product Mi Eample: The Efficient Product Mi Preference direction Preference direction IC MRS, = MRT, IC At this point, the consumer s illingness to give up good in order to get good just equals the rate at hich a producer has to give up good in order to produce more of good. Can the competitive market help us to achieve this optimality? At the Pareto efficient allocations, it is true for all consumers that: MRT, = MRS, MRS, = p /p ut This must be true for all consumers if the economy is to produce optimally for each consumer. 3 4 No, consider the producers problem. Suppose that the producers produce goods and and choose the product mi so as to maimize profits given the prices p and p : Definition: an isoprofit line shos the output combinations that result in a given level of profit, Π 0 or Q = (Π 0 + C*)/p p Q /p Ma Π= p Q + p Q C* Q,Q Where: e ill suppose that the cost of production is fied hatever the optimal output mi (e.g., e just ant to kno ho to employ the labor e have contracted) 5

12 Eample: The Profit Maimizing Product Mi Eample: The Profit Maimizing Product Mi Isoprofit lines (Π 0 +C*)/P -p /p 7 8 (Π 0 +C*)/P Eample: The Profit Maimizing Product Mi Isoprofit lines Direction of increasing profits Hence, If the firm maimizes profits, then, it chooses the product mi that shifts out the isoprofit line as much as possible hile remaining feasible. This is a tangency point such that for all producers: Profit maimising product mi MRT, = p /p -p /p 9 70 In other ords, in equilibrium, the price ratio ill measure the opportunity cost of production of one good in terms of production of the other good. Therefore ecause 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 mi for all producers and all consumers Eample: General Equilibrium Our earlier allocative efficiency results still hold ith production PPF 7 7

13 Eample: General Equilibrium Eample: General Equilibrium s e s e Slope = -p e /p e s e PPF 73 e A s e PPF 74 Eample: General Equilibrium Eample: General Equilibrium s e Slope = -p e /p e s e Slope = -p e /p e e A e A s e PPF 75 e A e A s e PPF 7 here: s and s are the amounts of produced in the economy; ( e A, e A) is the amount of and consumed by person A and ( e,e ) is the amount of and consumed by person. Efficiency in echange (on contract curve) Efficiency in use of inputs (on PPF) Efficiency in product mi(tangency ith PPF) 77 3

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