Market- General Equilibriun Consider first the case of a pure exchange economy. (One with no production). Suppose there are two consumers, A and B, in a two good economy. A starts with an endowment of wA=(44. wa and B starts with an endowment of wg=(uB,wB) A actually consumes rA of good I and ra of good 2. B actually consumes rB of good 1 and rb of good 2. The consumers cannot purchase more than there is of a particular good. Hence an allocation is feasible whenever 24+x≤4+and+rl≤u2+l Market General Equilibriun Edgeworth Boxes The information on the previous slide can be summarised in an Edgeworth bor. r B IAA The total width of the box is wA+wB and the total height is 4+42 Consumer A consumes nothing at the bottom left consumer B consumes nothing at the top right Label the endowment allocation w and the final allocation r. Remember these represent bundles for all consumers
Market — General Equilibrium 1 Endowments and Allocations • Consider first the case of a pure exchange economy. (One with no production). • Suppose there are two consumers, A and B, in a two good economy. A starts with an endowment of ωA = ¡ ω 1 A, ω 2 A ¢ and B starts with an endowment of ωB = ¡ ω 1 B, ω 2 B ¢ . • A actually consumes x 1 A of good 1 and x 2 A of good 2. B actually consumes x 1 B of good 1 and x 2 B of good 2. • These bundles are called allocations. The initial allocation is (ωA, ωB). The final allocation is (xA, xB). • The consumers cannot purchase more than there is of a particular good. Hence an allocation is feasible whenever: x 1 A + x 1 B ≤ ω 1 A + ω 1 B and x 2 A + x 2 B ≤ ω 2 A + ω 2 B Market — General Equilibrium 2 Edgeworth Boxes • The information on the previous slide can be summarised in an Edgeworth box. ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . ................................................................................................................................................................................................ ...... ...... ... ...... ...... ...... ... . . .................................................................................................................................................................................................. ............... ..................... . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....... . . . . . . . . . . . ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....... . . . . . . . . . . . A B • • ω 1 A ω 1 B ω 2 A ω 2 B x 1 A x 2 A x 1 B x 2 B • The total width of the box is ω 1 A + ω 1 B and the total height is ω 2 A + ω 2 B. • Consumer A consumes nothing at the bottom left corner, consumer B consumes nothing at the top right corner. • Label the endowment allocation ω and the final allocation x. Remember these represent bundles for all consumers
Market- General Equilibriun curves can be drawn in on an Edgeworth box. Consumer A prefers bundles to the north-east diagram and consumer B prefers bundles to the south-west of the box. Hence the diagram looks like Clearly many more indifference be drawn onto the diagram. Hence the Edgeworth box can be used to represent preferences and feasible allocations. These are the economically relevant characteristics of the consumers for the ensuing analysis Market General Equilibriun Pareto Efficiency A Pareto efficient allocation is one where it is not possible to make better off without making another se off. The contract curve is the curve that connects all the Pareto efficient points (the dashed line) B a point like y cannot be Pareto efficient. It is possible to make consumer A better off without making B worse off. by moving to an allocation like r. A has moved onto a strictly higher indifference curve, whilst B is just as well off. Only allocations like r can be Pareto efficient, where the two indifference curves are tangential. It is not possible to make eit better off without making the other worse off. Therefore, the contract curve connects all the indifference curve tangency points
Market — General Equilibrium 3 Preferences • Indifference curves can be drawn in on an Edgeworth box. Consumer A prefers bundles to the north-east of the diagram and consumer B prefers bundles to the south-west of the box. Hence the diagram looks like: ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . ........ ........ ..... . ..................... . .................................................................................................... ...................................................................................................... A B • x • ω • Clearly many more indifference curves can be drawn onto the diagram. • Hence the Edgeworth box can be used to represent preferences and feasible allocations. These are the economically relevant characteristics of the consumers for the ensuing analysis. Market — General Equilibrium 4 Pareto Efficiency • A Pareto efficient allocation is one where it is not possible to make one consumer better off without making another worse off. The contract curve is the curve that connects all the Pareto efficient points (the dashed line). ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . ........................................................................................................ ........................................................................................................ . .................................................................................................................. A B • x • y • A point like y cannot be Pareto efficient. It is possible to make consumer A better off without making B worse off, by moving to an allocation like x. A has moved onto a strictly higher indifference curve, whilst B is just as well off. • Only allocations like x can be Pareto efficient, where the two indifference curves are tangential. It is not possible to make either consumer better off without making the other worse off. • Therefore, the contract curve connects all the indifference curve tangency points
Market- General Equilibriun General Equilibrium a single budget line can be drawn for both consumers. It is a downwardly sloping straight line, which must pass through the endowment and have slope -P/pa(the price ratio) Definition: A general equilibrium is an allocation r and a set of prices p such that, Walrasian equilibrium). What is an equilibrium in this exchange economy? (Often called general competitit 1. Each consumer maximises their utility given their budget constraint. 2. Total demand for each good is no more than the total endowment The second condition is often called market clearing. Alternatively, the allocation must be feasible Market General Equilibriun Walras’Law Equilibrium(allocation z and prices PI and pz)can be illustrated in the Edgeworth box B Budget line Slope=-pi/p2 Both consumers are maximising utility given their budget constraints and the market clears for both goods Define aggregate excess demand for good 1 as z1(p1, p2)=(rA(p1, P2)-4)+(rB(p1, P2)-wB). This is the net amount of good 1 demanded by each consumer. At equilibrium prices, this is Following from the definitions of budget lines. Walras'law states that the value of aggregate excess demand is zero )+p2(1,P2)=0 So if the market clears exactly for one of the goods, it must clear exactly for the other
Market — General Equilibrium 5 General Equilibrium • A single budget line can be drawn for both consumers. It is a downwardly sloping straight line, which must pass through the endowment and have slope −p1/p2 (the price ratio). • What is an equilibrium in this exchange economy? (Often called general, competitive or Walrasian equilibrium). • Definition: A general equilibrium is an allocation x and a set of prices p such that: 1. Each consumer maximises their utility given their budget constraint. 2. Total demand for each good is no more than the total endowment. • The second condition is often called market clearing. Alternatively, the allocation must be feasible. Market — General Equilibrium 6 Walras’ Law • Equilibrium (allocation x and prices p1 and p2) can be illustrated in the Edgeworth box: ....................................................................................................................................................................................................................................................................................................................................................................................................................... ....................................................................................................................................................................................................................................................................................................................................................................................................................... . . .................................................................................................. ...... ...... ... ...... ...... ...... ... . ........................................................................................................ ........................................................................................................ A B • x Budget line Slope = −p1/p2 • ω • Both consumers are maximising utility given their budget constraints and the market clears for both goods. • Define aggregate excess demand for good 1 as z1(p1, p2) = (x 1 A(p1, p2) − ω 1 A) + (x 1 B(p1, p2) − ω 1 B). This is the net amount of good 1 demanded by each consumer. At equilibrium prices, this is zero. • Following from the definitions of budget lines, Walras’ law states that the value of aggregate excess demand is zero: p1z1(p1, p2) + p2z2(p1, p2) = 0 • So if the market clears exactly for one of the goods, it must clear exactly for the other
Market- General Equilibriun Production Pure exchange ave no production. All of the abowe analysis can be extended to the case of production ner, one producer and two goods. (This is the simplest possible case- it can be generalised) are the"same person". This is not unreasonable. most of us are both and producers simultaneously everyone has to work for a living, including Robinson Crusoe. The goods are coconuts, C, and leisure. Recall labour supply (L)can be derived from leisure demand. The price of leisure is u and coconuts are the numeraire. There is a production technology which turns labour into coconuts. Indifference Production function L Market General Equilibriun Robinson Crusoe Economy Economists call this model the Robinson Crusoe economy. The firm maximises profit- pushing the isoprofit line upward until it is tangent to the production function. he c maximises utility -pushing the indifference curve upward until it is tangent to the budget line. Indifference curve roduction function The slope of the budget line(and the isoprofit line)is w. The marginal product is set equal to u at the optimum (as usual) and the MRS is set equal to the price ratio(a)as usual. Putting these two pictures together yields the diagram on the last slide. The budget line and the isoprofit line must coincide- any profits that are made during production are all that is available to be spen
Market — General Equilibrium 7 Production • Pure exchange economies have no production. All of the above analysis can be extended to the case of production. • There is one consumer, one producer and two goods. (This is the simplest possible case — it can be generalised). • In fact, the firm and the consumer are the “same person”. This is not unreasonable, most of us are both consumers and producers simultaneously — everyone has to work for a living, including Robinson Crusoe. • The goods are coconuts, C, and leisure. Recall labour supply (L) can be derived from leisure demand. The price of leisure is w and coconuts are the numeraire. There is a production technology which turns labour into coconuts. ............. ............. ............. ............. ............. ........... . . . . . . ......................................................................................................................................................................................................................................................................................................................................................................................................................... ................................................................................................................................................................................................................................................................................................... ........................................................................................................................ ........................................................................................................................ ........................................................................................................................ 0 L C Production function Indifference curves C ∗ • L ∗ Market — General Equilibrium 8 Robinson Crusoe Economy • Economists call this model the Robinson Crusoe economy. • The firm maximises profit — pushing the isoprofit line upward until it is tangent to the production function. • The consumer maximises utility — pushing the indifference curve upward until it is tangent to the budget line. ............. ............. ............. ............. ............. ........... . . . . . . ............. ............. ............. ............. ............. ........... . . . . . . . . . . . . . . ....................................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................................................................................................................................................................................. ...................................................................................................................................................................................................................................................................................................................................... ...................................................................................................................................................................................................................................................................................................... . ................................................................................................................................................................................................................................................................................................... ........................................................................................................................ 0 L 0 C L C Production function Isoprofit line Indifference curve Budget line C ∗ • • L ∗ C ∗ L ∗ L • The slope of the budget line (and the isoprofit line) is w. The marginal product is set equal to w at the optimum (as usual) and the MRS is set equal to the price ratio (w) as usual. • Putting these two pictures together yields the diagram on the last slide. The budget line and the isoprofit line must coincide — any profits that are made during production are all that is available to be spent
Market- General Equilibriun General Equilibrium Revisited General equilibrium when there is production in the is a natural extension of the earlier definition Definition: A general equilibrium is an allocation r a production plan y and a set of prices p such that: 1. Each consumer maximises utility subject to their budget constraint. 2. Each firm maximises profit given the market prices. 3. The market for each good(whether input or output)clears. at they only take an interest in their This is a static concept. There is no mention of how a general equilibrium might arise, whether it would aris indeed whether it even exists in a given economy (is there a set of prices and an allocation such that all three conditions hold? ). So far it is just a definition Market General Equilibriun Production Possibility Sets A production possibility set is the set of bundles of goods that it is possible to produce. roduction possibility set Production possibility fron Slope= MRT The above diagram illustrates the production possibility set. The outermost line represents the production possibility frontier. A profit maximising firm will produce at the point at which the isoprofit line is tangential to the frontier The frontier slope is the marginal rate of transformation- the rate at which one good can be turned into another. The slope of the isoprofit line is the price ratio between the two goods(C is equilibrium cost P2
Market — General Equilibrium 9 General Equilibrium Revisited • General equilibrium when there is production in the economy is a natural extension of the earlier definition. • Definition: A general equilibrium is an allocation x, a production plan y and a set of prices p such that: 1. Each consumer maximises utility subject to their budget constraint. 2. Each firm maximises profit given the market prices. 3. The market for each good (whether input or output) clears. • Each consumer and firm acts “selfishly” in the sense that they only take an interest in their own well-being. • This is a static concept. There is no mention of how a general equilibrium might arise, whether it would arise, indeed whether it even exists in a given economy (is there a set of prices and an allocation such that all three conditions hold?). So far it is just a definition. Market — General Equilibrium 10 Production Possibility Sets • A production possibility set is the set of bundles of goods that it is possible to produce. ............. ............. ............. ............. ............. ........... . . . . . . ................................................................................................................................................................................................................................................................................ ............................................................................................................................................................................................................................................................................................... . . . ........ ......... .... .................................................................................................. .................................................................................................. 0 x1 0 x2 x1 x2 Production possibility set Isoprofit lines Slope = −p1/p2 Production possibility frontier Slope = MRT x • ∗ 2 x ∗ 1 • The above diagram illustrates the production possibility set. The outermost line represents the production possibility frontier. A profit maximising firm will produce at the point at which the isoprofit line is tangential to the frontier. • The frontier slope is the marginal rate of transformation — the rate at which one good can be turned into another. • The slope of the isoprofit line is the price ratio between the two goods (C is equilibrium cost): π = p1x1 + p2x2 − C =⇒ x2 = π + C p2 − p1 p2 x1
Market- General Equilibriun Why is the production possibility set this shape? Suppose firm 1 was particularly good at making good 1. For every good 2 it makes it could make 2 of good 1. On the other hand firm 2 is an expert in good 2. For every good I it makes it could make 2 of good 2. Firm I has a comparatine aduantage in making good 1. Firm 2 has a comparative advantage in good 2. The above picture shows their production possibility sets and the joint production possibility set notice the shape. Market General Equilibriun MRS and MRT In the below picture, an Edgeworth box is drawn under the production possibility set The general equilibrium is illustrated. The production plan is y=(I1, I2). The allocation is r* and the set of prices is given by the slope of the budget line that separates the two indifference curves. Notice that the prices are also given by the slope of the isoprofit line tangential to the production frontier Hence, in a general equilibrium the mrS of each er and the MRT of the technology are equal and given by the price ratio. This is true of a general equilibrium no matter how many firms
Market — General Equilibrium 11 Comparative Advantage • Why is the production possibility set this shape? ................................................................................................................................................................................................................................................................................ .................................................................................................................................................................................................................................................................................. .................................................................................................................................................................................................................................................................................. ...................................................................................................................................................................................... . ...................................................................................................................................................................................... 0 x1 0 0 x2 x1 x2 x1 x2 • Suppose firm 1 was particularly good at making good 1. For every good 2 it makes it could make 2 of good 1. On the other hand firm 2 is an expert in good 2. For every good 1 it makes it could make 2 of good 2. • Firm 1 has a comparative advantage in making good 1. Firm 2 has a comparative advantage in good 2. • The above picture shows their production possibility sets and the joint production possibility set — notice the shape. Market — General Equilibrium 12 MRS and MRT • In the below picture, an Edgeworth box is drawn under the production possibility set. ............. ............. ............. ............. ............. ............. ............. ............. ............. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................................................................................................................................................................................................................................. . . . ............................................................................................................................................................... .................................................................................. .......... ........ ....... ...... .... ... ... 0 x1 x2 • • x ∗ x1 x2 • The general equilibrium is illustrated. The production plan is y = (x1, x2). The allocation is x ∗ and the set of prices is given by the slope of the budget line that separates the two indifference curves. • Notice that the prices are also given by the slope of the isoprofit line tangential to the production frontier. • Hence, in a general equilibrium the MRS of each consumer and the MRT of the technology are equal and given by the price ratio. This is true of a general equilibrium no matter how many firms, consumers and goods there are
