and Welfare Lectures 2 and 3, ECON 4240 Spring 2017 University of Oslo 24.01.2017 and 31.01.2017 1/37
Outline General equilibrium: look at many markets at the same time. Here all prices determined in the model. Goal: Study effects that occur when changes in one market have repercussions in other markets Study connections between markets for goods and markets for factors of production Make general welfare statements about how well a market economy performs 2/37
Outline We proceed by steps: A graphical model of with 2 consumption goods and 2 factors of production A mathematical model of exchange with n goods A mathematical model of production and exchange with n goods 3/37
Perfectly Competitive Price System In all the general equilibrium models we borrow many assumptions from the partial equilibrium analysis: All individuals are price-takers, and utility-maximizers All firms are price-takers and profit-maximizers Zero transaction costs Perfect information 4/37
A graphical GE Model with 2 goods Many identical individuals, many identical firms 2 goods (x and y) and 2 inputs (capital and labor) The endowments of capital and labor are fixed For simplicity, all individuals have identical endowments of capital and labor, and they all own equal shares of each firm 5/37
A graphical GE Model with 2 goods We first focus on the supply side. As firms are profit maximizers, in equilibrium inputs will be used in an efficient ways to produce the two goods (efficient = it is not possible to produce more of one good without producing less of the other) We draw an "Edgeworth box", where the horizontal side measures total labor endowment, the vertical side total capital endowment Any point in the box is a full-employment allocation of inputs Each point in the Edgeworth box measures how much of each factor is devoted to the production of x and how much is devoted to the production of y Nothing in the Edgeworth about how much of each factor is used in each single firm (for now it does not matter) 6/37
A graphical GE Model with 2 goods In the Edgeworth box we can draw isoquants for good x and isoquants for good y Definition An isoquant (for good x) shows those combinations of capital and labor that can produce a given level of good x. Note that these are aggregate isoquants, not at single firm level The efficient allocations are the ones where the isoquants for the production of the two goods are tangent 7/37
A graphical GE Model with 2 goods The information about all possible efficient allocations can be used to construct a production possibility frontier in the graph of combinations of good x and good y Definition The production possibility frontier is the curve in the x-y space that delimits all combinations of x and y that can be produced, given the initial endowment of capital and labor 8/37
Efficient Allocations of Inputs Definition The rate of product transformation (RPT) between two outputs is the negative of the slope of the production possibility frontier for those outputs: RPT = dy dx (along the production possibility frontier) Pick any 2 prices for the inputs For any point (x a,y a ) on the production possibility frontier the total cost for producing x a units of good x and y a units of y is the same. Note that RPT = MC x MC y 9/37
Efficient Allocations of Inputs Concave frontier MC x MC y Possible reasons behind concavity of frontier: Diminishing returns Specialized inputs Differing factor intensities increases as x increases and y decreases 10/37
Determination of Equilibrium Prices At any given (good) price ratio p x p y firms altogether produce some amount (x f,y f ) such that (a) (x f,y f ) lies on the production possibility frontier, (b) at (x f,y f ) the production possibility frontier has slope p x p y. In order to find the equilibrium price ratio (denote it p x p y ) we need to consider the demand side Note that the competitive markets only determine equilibrium relative prices, not absolute prices (more on this in a bit) 11/37
Determination of Equilibrium Prices In the x y space, we can draw individuals indifference curves (aggregating individuals indifference curve is not a trivial issue, here for simplicity say that if (x b,y b ) is consumed altogether and there are n individuals, then each individual gets x b /n units of good x and y b /n units of good y) For any pair of prices p x and p y, individuals altogether demand some amount (x c,y c ) such that the indifference curve passing through the point (x c,y c ) has slope p x p y 12/37
Determination of Equilibrium Prices A pair of prices are equilibrium prices if the overall quantities chosen by firms coincide with the overall quantities chosen by individuals: (x f,y f ) = (x c,y c ) = (x,y ) What about the budgets of individuals? How do individuals pay for x and y? In equilibrium total revenues (=sum of revenues of all firms) are x p x + y p y. These revenues cover the costs of inputs and if anything is left they become profits. Whether the revenues are used to cover costs or they are profits they all go to individuals anyway For individuals, the revenue from selling labor and capital, together with the profits, add up to x p x + y p y 13/37
Models in International Trade So far we learned that the equilibrium price ratio p x/p y persists as long as technologies or preferences do not change. We can apply this model to questions of International Trade Note that the model can be made more realistic (and more useful) by assuming that there are more inputs (e.g. high and low skilled labor) and individuals differ in the endowments of the inputs 14/37
Models in International Trade The Corn Law Debate: focus on tariffs on grain imports. Grain = x, manufactured goods = y With tariffs high enough to completely prevent trade: equilibrium "E", domestic price ratio: pe x p E y Removing tariffs: pa x, grain imports of x py A B x A, financed with export of manufacturing of y A y B Result of tariff removal: lower (relative) price of grain, less grain production, less farmers, less rent for land owners (different from textbook story) 15/37
Models In International Trade US elections: trade was an important issue Idea: trade affects the relative incomes of various factors of production In the US exports use intensively skilled labor, while imports are products that require low skilled labor More free trade: increasing relative wage of high skilled workers (more inequality (in the US - worldwide not clear)) 16/37
A Mathematical Model of Exchange: Setting Model of exchange = no production, goods already exist n goods, m individuals x i : vector of consumption of individual i = 1..m (vector of size n) x i : vector of endowment of individual i (vector of size n) Here the same individuals can be on both sides of different markets; for simplicity imagine an individual sells all her endowment and uses the revenues to buy whatever she consumes Market value of individual s endowment: px i Budget constraint of individual i: px i px i 17/37
Equilibrium and Walras Law Vector of demand of individual i: x i (p,px i ) These demands are homogeneous of degree 0: x i (p,px i ) = x i (tp,tpx i ) t > 0 Definition Walrasian equilibrium is an allocation of resources and an associated price vector p such that quantity demanded and quantity supplied of each good coincide: Σ m i=1 x i (p,p x i ) = Σ m i=1 x i (1) 18/37
Equilibrium Existence Does an equilibrium exists? Not obvious as: (1) from last slide corresponds to n equations The equations in (1) need not be linear (x i ( ) is a vector of n equations, not necessarily linear) These equations are not independent: they are related by Walras Law 19/37
Walras Law Definition Walras law: Σ m i=1px i = Σ m i=1px i Walras Law is a direct consequence of the individual s budget constraints: for every individual i = 1,..m we have px i = px i : adding over the m individuals we have Walras law 20/37
Existence of Equilibrium Intuitive evidence seems to suggest equilibrium exists Proof of existence: As demands are homogeneous of degree 0, and budget constraints are not affected if all prices are multiplied by a constant t > 0, we can "normalize" prices One way to normalize prices is, for example, to multiply every 1 price by Σ n k=1 p, thus getting p k i = p i Σ n k=1 p k This is called in jargon a normalization as Σ n i=1 p i = 1 (note that relative prices are unaffected by the normalization) 21/37
Existence of Equilibrium Proving existence showing there exists a vector p such that z(p ) := Σ m i=1x i (p,p x i ) Σ m i=1x i = 0 Starting from an arbitrary vector of prices p 0, define p 1 as (textbook expression is wrong) [ p 1 = f (p 0 p1 0 ) := + αz 1(p 0 ) Σ n k=1 (p0 k + αz k(p 0 )),.., pn 0 + αz n (p 0 ] ) Σ n k=1 (p0 k + αz k(p 0 )) where α is some positive constant Function f is continuous and maps normalized prices onto other normalized prices (= if p 0 is a vector of normalized prices, so is p 1 ) 22/37
Brouwer Fixed Point Theorem Function f meets the conditions of Brouwer fixed point theorem for existence of fixed point p (see next slide) Theorem A fixed point of function f is a vector p such that p = f (p ) Brouwer Fixed Point Theorem: Any continuous function from a closed compact set onto itself has a fixed point such that f (x) = x. (No proof) 23/37
Existence of Equilibrium Definition S is a compact subset of R n if S is closed (= contains its boundaries) and bounded (= pick any point in R n : along any "ray" starting from that point and going in any direction there are points outside S) In our case, we consider a subset of R n : [0,1] n 24/37
Welfare Properties of Competitive Equilibria After establishing existence of competitive equilibria, we should ask ourselves whether these equilibria lead to socially desirable divisions of resources (divisions of resources= allocations) Welfare economics studies criteria for choosing among alternative allocations Aside: when considering 1 consumer, any allocation that maximizes his utility is clearly the best one: definition of efficiency is UNIQUE. When considering n consumers, there are many different definitions of efficiency 25/37
Welfare Properties of Competitive Equilibria Definition An allocation of the available goods in an exchange economy is Pareto efficient if it is not possible to devise an alternative allocation in which at least one person is better off and no one is worse off. Other measures of efficiency require to formally define a social welfare function: Definition A social welfare function is a scheme from ranking potential allocations of resources based on the utility they provide to individuals. Social Welfare = SW [U 1 (x 1 ),U 2 (x 2 ),..,U m (x m )] 26/37
Welfare Properties of Competitive Equilibria Note: for any "reasonable" SW (.) any allocation that maximizes SW is Pareto Efficient Pareto Efficiency is a VERY limited way to rank allocations: e.g. crazily unequal allocations can be Pareto Efficient (My) view: Pareto efficiency is a necessary but not sufficient requirement for an allocation to be acceptable Utilitarian SW function Maximin SW function SW (U 1,U 2,..,U m ) = U 1 + U 2 +... + U m SW (U 1,U 2,..,U m ) = min[u 1,U 2,...,U m ] 27/37
First Theorem of Welfare Economics Theorem First Welfare Theorem. Every Walrasian equilibrium is Pareto Efficient. Proof by contradiction. (LOTS of typos in the textbook proof) Consider a Walrasian equilibrium with prices p and allocation x k (k = 1,..,m). Suppose there exists an alternative feasible allocation, x k, such that a consumer i is better off and everyone is at least as well off as with x k. Then: p x i > p x i (Suppose instead p x i p x i : then individual i could afford x i, and as we assume individual i is better off with x i then with x i we should conclude that x i is not the optimal choice for consumer i. We reached a contradiction as x i IS optimal. So it must be the case that p x i > p x i ) 28/37
First Theorem of Welfare Economics p x k p x k for k = 1..m (Again, suppose instead that p x k p x k for some k: then consumer k could ensure the same utility obtained from consuming x k at a lower cost by purchasing x k, and with what she saves she could buy more of some good thus increasing overall utility, so x k is not her optimal choice. This is a contradiction as x k is the optimal choice for k = 1..m.) Σ m i=1 x i = Σ m i=1 x i (This is the definition of feasibility.) Multiply both sides of the last equality by p : Σ m i=1 p x i = Σ m i=1 p x i But the previous 2 inequalities, together with Walras Law, imply: Σ m i=1 p x i > Σ m i=1 p x i, hence we reach a contradiction 29/37
Second Welfare Theorem Theorem Second Welfare Theorem. For any Pareto Optimal allocation of resources, there exists a set of initial endowments and a related price vector such that this allocation is also a Walrasian Equilibrium. We provide a graphical proof of the theorem using an Edgeworth box. A model of exchanges is sufficient to talk about welfare properties of different allocation, but if we think of policy instruments, such as taxes and subsidies, used to obtain the desired allocation, then we need to account for the role of these policies on the incentive to produce 30/37
A Mathematical Model of Production and Exchange n goods (including factors of production). Goods consumed and used as factor of production can come from the household endowments or can be produced by some firm r profit-maximizing firms. Production function of firm j: n 1 column vector y j. Positive entries in y j : outputs, negative entries: inputs Profit of firm j: π j (p) = py j Firms can choose y j = 0: exit (long run perspective) (TYPOS in book) Firms are owned by individuals. Simplification: each individual i owns a share s i of every firm (simplification: same share of each firm) 31/37
Budget Constraints and Walras Law Budget constraint of individual i: px i (p) = s i Σ r j=1 py j (p) + px i (p) Note slight inconsistency: we allow for positive profits (like in short run) but we allow for firms exit (like in the long run) Walras law: where: x(p) = Σ m i=1 x i (p), y(p) = Σ r j=1 y j (p) and x = Σ m i=1 x i. px(p) = py(p) + px, 32/37
Walrasian Equilibrium Equilibrium price vector p at which quantity demanded equals quantity supplied in all markets simultaneously: x(p ) = y(p ) + x. (One) use of these models: look at effect on wages of chances in exogenous factors 33/37
Walrasian Equilibrium Going back to the first model of GE considered (goods x and y, inputs K and L) let s look at the relation between returns to scale and the slope of the production possibility frontier x = kx α lx α and y = ky α ly α 2α = 1: constant returns to scale (RTS), 2α < 1: decreasing RTS, 2α > 1: increasing RTS, Full employment of both inputs: l x + l y = l and k x + k y = k 34/37
Contract Curve First we characterize the set of efficient allocations of inputs (that is, the contract curve) Slope isoquant of x. Rate of Technical Substituttion: dx dlx dx dkx = αkα x lx α 1 αkx α 1 lx α = k x lx Slope isoquant of y: k y l y On the contract curve the 2 slopes coincide: k x lx Substituting the "full employment" constraints: k x lx k x lx = k l (and k y l y = k l ) = k k x l l x = k y l y 35/37
Production Possibility Frontier On the conctract curve, both production functions can be expressed as a function of l x only: x(l x ) = ( k l )α lx 2α y(l x ) = ( k l )α ly 2α = ( k l )α (l l x ) 2α Slope of the production possibility frontier: dy dx contract curve = y (l) x (l) = y 1 1 2α d 2 y d 2 x contract curve = (1 1 x 1 1 2α 2α ) y 1 2α x 1 2α 1 < 0 dy dx + (1 1 2α ) y 1 1 2α x 2 2α 1 36/37
Production Possibility Frontier 2α < 1 (decreasing RTS) d 2 y d 2 x contract curve < 0 (concave production possibility frontier (PPF)) 2α = 1 (constant RTS) d 2 y d 2 x contract curve = 0 ("straight" PPF) 2α > 1 (increasing RTS) d 2 y d 2 x contract curve > 0 (convex PPF) 37/37