Lecture9
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This document provides an overview of general equilibrium models in economics. It discusses how all markets are interconnected and how changes in one market can affect others. A general equilibrium model models all markets simultaneously, assuming perfect competition and that consumers maximize utility and firms maximize profits. The document uses Edgeworth boxes and production possibility frontiers to illustrate how resources are allocated across industries and how equilibrium prices are determined. It discusses how technological changes or changes in preferences can shift the general equilibrium.
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Lecture9
- 1. General Equilibrium Dr. Andrew McGee Simon Fraser University
- 2. Interconnectedness of markets • Suppose new research is released showing that wine consumption is good for one’s overall health. • Demand for wine increases. Price of wine increases. Marginal revenue product of wine workers increases. More workers and firms enter this market. • Cloth, used in wine production, also sees its demand increase, and similarly its price will go up and the marginal revenue product of workers in the cloth industry will go up. • The new firms and workers entering these industries, however, come from somewhere—namely other markets. Hence the developments in one market will affect many other markets.
- 3. General equilibrium model • Models all markets at once • Markets include not just markets for consumption goods but factor markets (e.g., capital and labor) • In the equilibrium of a general equilibrium model, Qs=Qd for all markets simultaneously – There is some vector of equilibrium prices such that all markets clear
- 4. Assumptions of general equilibrium model • All markets are perfectly competitive (i.e., selling homogenous products, many buyers, many sellers, zero transaction costs, firms are price-takers) – Law of one price must hold • Consumers maximize their utility • Firms maximize their profits
- 5. General equilibrium model with 2 goods • Assume that all individual have identical preferences concerning goods X and Y. • Assume that there are fixed amount of inputs (capital and labor) available for the production of X and Y. • Edgeworth box: represents all possible allocations of inputs to the production of X and Y with dimensions given by the total amounts of capital and labor available
- 6. Edgeworth box Total capital Total labor 0x 0y Where point A is in the Edgeworth box tells you how much labor and capital are being devoted to the production of X and how much to Y A Labor in X production Capital in X production Labor in Y production Capital in Y production At any point in the Edgeworth box, all of the capital and labor in the economy is fully employed All possible allocations of K & L are depicted in the Edgeworth box
- 7. Efficient allocations • Many allocations of K & L to the production of X and Y are inefficient insofar as more of both X and Y could be produced by allocating K & L to the production of X and Y differently • Assume that perfectly competitive markets result in efficient allocations of inputs • Overlay isoquant maps on top of the Edgeworth box
- 8. Efficient allocations 0x 0y Red: isoquants for Y Blue: isoquants for X A Points such as A are clearly inefficient because more X and more Y can be produced given society’s endowments of K & L. The green line represents the locus of all efficient allocations of K & L—allocations at which there exists no allocation at which more X and more Y can be produced relative to the current outputs. Notice that the isoquants are tangent to each other at these efficient allocations, meaning that MRTSx=MRTSy. Contract curve
- 9. Production possibility frontier (PPF) Y X 0x 0y Note that at each efficient allocation in an Edgeworth box, there are corresponding output levels of X & Y. The graph below depicts those output levels. Note that the bottom left-hand corner of an Edgeworth box corresponds to the case in which no X is produced and only Y is produced. The production possibility frontier shown here depicts the alternation combinations of two outputs that can be produced with fixed quantities of inputs if those inputs are employed efficiently. The output bundle A corresponds to the inefficient input combination A in the Edgeworth box. A The slope of the PPF reflects the rate at which X can be substituted for Y when total resources are held constant. For instance, the slope might be -1/4 near 0x, meaning 1 unit of Y can be sacrificed in order to produce 4 more units of X. Near Oy, this slope might be -5, meaning that 5 units of Y must be sacrificed to produce 1 more unit of X. The negative of the slope is known as the rate of product transformation (RPT).
- 10. Shape of the PPF
- 11. Diminishing Returns • If the production processes of both X and Y exhibit diminishing returns to scale, then increasing X output will increase MCx, while reducing Y output will decrease MCy, which would cause the production function to exhibit increasing RPT. • But economists don’t find DRS theoretically compelling….
- 12. Specialized inputs • Suppose some inputs are better suited for X and others better suited for Y (e.g., marshlands for the production of rice and prairies for the production of wheat). Producing more X requires using more inputs ill-suited for the production of X and better suited for the production of Y and vice versa. • But we have assumed inputs are homogenous (i.e., land is land is land).
- 14. Why is the PPF bowed outward? Y X 0x 0y y4 y2 x1 x3 If we assume CRS, then this output combination is possible (because ½ the inputs used to produce x1 will produce ½ of x1 and likewise for the other output levels). We also know that all K& L will be used at this new allocation (though we can’t see that in this graph). B We know that bundle B uses ½ of the K used in x1 and ½ the K used in x3 to produce the new amount of X and similarly ½ the L used in x1 and ½ the L used in x3 to produce the new amount of X. Similarly for the amounts of K & L used to produce Y.
- 15. Why is the PPF bowed outward? 0x 0y x1 y4 x3 y2 This input bundle, C, involves using ½ the K used to produce output combination P1 and ½ the K used to produce the output combination P2. P1 P2 Notice that C is not on the contract curve; it is not efficient. It is possible to produce more of both X and Y. Thus there must exist output bundles to the northeast of B on the PPF in the preceding slide. If the K/L mix (factor intensity) does not vary between production of X and Y, the contract curve and PPFs will be linear C Total capital Total labor
- 16. PPF & Opportunity cost • For a society with finite resources, producing more X requires producing less Y if all resources are fully employed. • The slope of the PPF gives the opportunity cost of producing an additional unit of X in terms of units of Y foregone and vice versa
- 17. Equilibrium prices • Given that society faces tradeoffs in the production and consumption of X and Y, how does it decide how much X & Y to produce (and consume)? • The prices of X and Y must adjust so that the quantity demanded of each equals the amount produced by the society.
- 18. Determination of equilibrium prices Y X 0x 0y U4 U3U2 U1 PPF At price ratio px/py, firms produce (x1, y1), but consumers demand (x1’, y1’)—supply does not equal demand and markets do not clear. x1 y1 x1’ y1’ px/py x* y* px*/py* Why is the tangency between the PPF and the price ratio the profit maximizing choice for firms? Because at this bundle the price ratio equals the ratio of marginal costs (all firms are earning zero profit by assumption). Note that the red lines here are budget constraints. X and Y sell for pxX+pyY, but in this economy the producers are also the consumers, so whatever they earn in revenues is returned to consumers through wages and rents from capital. Given that these red lines are budget constraints, the point of tangency with the indifference curves is the utility maximizing consumption bundle (from 1st ½ of course).
- 19. Determination of equilibrium prices Y X 0x 0y U4 U3U2 U1 PPF x1 y1 x1’ y1’ px/py x* y* px*/py* At px/py, there will be excess demand for X and excess supply of Y. Markets do not clear. In order for the markets to clear, the price of X must go up, and the price of Y must go down (or some combination) such that the price ratio gets steeper. A higher price for X both reduces demand for X and increases its supply. At px*/py*, the supply of X equals the demand for X and similarly for Y. This is a general equilibrium: all markets clear simultaneously.
- 20. Persistence of general equilibria • Consistent with the definition of an equilibrium, a GE will tend to persist unchanged without a change in the underlying economic fundamentals • The economic fundamentals that could lead to a change in the equilibrium prices and quantities are preferences and technologies
- 21. Technological Change Y X 0x 0y U2 PPF1 PPF2 U3 Here, producers get better at producing X as evidenced by the fact that they can produce more X along with any given amount of Y still using the same fixed quantities of K & L. This is what technological progress looks like! Notice that the market clearing price ratio changes following this technological change. Society gets better at producing X; its relative price falls in equilibrium.
- 22. Change in preferences Y X 0x 0y U2 Original preferences Suppose people decide that they like X less. This would lead to “flatter” indifference curves. Notice that the price of X declines relative to that of Y in equilibrium (the price ratio gets flatter) and less X is consumed. New equilibrium price ratio Old equilibrium price ratio xoldxnew yold ynew New preferences, U2’
- 23. A simple general equilibrium model: Production
- 24. A simple general equilibrium model: Consumption
- 25. A simple general equilibrium model: Equilibrium
- 26. A simple general equilibrium model: Consumers’ budget constraint
- 27. Corn Laws • 19th century British laws placing high tariffs on grain imports (1815-1846) • Served the interests of landowners, who benefits from higher grain prices • Disliked by manufacturers • Assuming Corn Laws prevented trade (which they didn’t), repealing the Corn Laws would have an unambiguously positive effect on consumers, but what about the effects on factor owners (landowners and capitalists)?
- 28. An argument for trade Y X 0x 0y PPF1 U1 U2 A E B yA yE yB xA xE xB In the absence of trade, a society is limited to consuming only those bundles which it can produce. In this economy under autarky (i.e., a no trade regime), the GE would occur at point E. If trade with other countries/economies is allowed, then this country could consume bundle B while producing bundle A. It would be a net exporter of good Y while being a net importer of good X.
- 29. Winners & losers from trade? 0x 0y x1 y4 x3 y2 P1 P2 The movement from E to A corresponds to a move from P2 to P1;it causes the K to L ratio to increase in both industries. This means that the relative price of capital must fall (or the relative price of labor increases). Total labor Thus in this simple, stylized model, trade benefits laborers while hurting capital owners. Total capital Total labor
- 30. Walrasian Equilibrium • Suppose there is no production in the economy. The economy is just endowed with a certain amount of goods X and Y. • Individuals hold endowments of X & Y and then trade among themselves. • Once trade among individuals occurs, a Walrasian equilibrium is an allocation of resources and an associated price vector (i.e., set of prices for all goods in the economy) such that the demand for each good equals the economy’s total endowment for that good (i.e., markets clear).
- 31. Trading: how do we get from an initial allocation to an equilibrium allocation 0B 0A (4,4) (8,2) Price vector px/py has slope -1/2: you can trade on unit of x for 2 units of y Initial allocation Equilibrium allocation Here, A gives B 4 units of X in exchange for 2 units of Y. In this sense the price vector defines the trades that can be made and the set of possible final allocations
- 32. Mathematical model of exchange (no production) Vector notation
- 33. Mathematical model of exchange (no production)
- 34. Exchange
- 37. Existence of a general equilibrium
- 39. Fixed point theorem 1 1 x f(x) 0 45-degree line x=f(x)f(x*) x* Fixed point Any continuous function that maps from a closed compact set (here the interval [0,1]) onto that same set ([0,1]) will have a fixed point where x=f(x).
- 40. Existence of GE proof • Existence proof relied on a number of non- trivial assumptions: 1. Price-taking by all individuals 2. Homogeneity of degree 0 demand functions 3. Continuity of demand functions 4. Budget constraints 5. Walras’ law: value of all quantities demanded equals value of all endowments
- 41. First Fundamental Theorem of Welfare Economics • FFWT: Every Walrasian equilibrium is Pareto efficient • Pareto efficient allocation: An allocation of the available goods in an exchange economy is Pareto efficient if it is not possible to devise an alternation allocation in which at least one person is better off and no one is worse off. • FFWT does not imply that every Walrasian equilibrium is socially desirable.
- 42. FFWT • Given initial endowments, individuals trade. They move from their initial endowment to some final allocation of goods. The equilibrium that emerges (at the equilibrium prices) will be Pareto efficient insofar as none of the traders can be made better off without making another trader worse off. • Pareto efficient equilibria can be highly inequitable (points A or C, for instance) • The location of the WE depended substantially on the initial endowment, E • The initial endowment constrains the set of possible WE (between A & C). There may exist other far more socially desirable but not attainable Pareto efficient allocations.
- 43. FFWT in a 2 person, 2 good exchange economy 0B 0A Initial endowment Px/Py Walrasian equilibrium (WE): after trade, A and B end up here given prices Px and Py Notice that the WE is Pareto efficient (FFWT) A C Region between A & C is the Pareto- improving set.
- 44. Proof of FFWT
- 45. Second Fundamental Theorem of Welfare Economics • SFWT: For any Pareto optimal allocation of resources, there exists a set of initial endowments and a related price vector (Px/Py in the 2-good case) such that this allocation is also a Walrasian equilibrium • Implication: if you can adjust initial allocations (i.e., through taxes and other forms of coercion), then any Pareto efficient allocation can be sustained as a Walrasian equilibrium
- 46. SFWT in a 2-good, 2-person exchange economy 0B 0A Initial endowment Px/Py A C Q Q is thought to be more socially desirable than any WE between A and C If one can re-allocate goods from the initial endowment to any point such as Z on the price line passing through the tangency between UA and UB, then Q can be sustained through trade as a WE Z UA UB
- 47. Example of SFWT in a 2-person (A & B) exchange economy
- 48. Example of SFWT in a 2-person (A & B) exchange economy
- 49. Example of SFWT in a 2-person (A & B) exchange economy
- 50. Example of SFWT in a 2-person (A & B) exchange economy
- 52. Insights from GE models • In economic models, we take preferences and production technologies as given. With these in hand, all prices are necessarily determined. • Firms and productive inputs are owned by households. All income (profits and returns to inputs) accrues to households. • Bottom line in policy evaluations is the utility of households. Firms and governments are just intermediaries. • All taxes distort economic decisions along some dimensions. The welfare costs of such distortions must be weighed against the benefits of such taxes (whether from public good provision or equity-enhancing transfers)