Hart Notes Matthew Basilico April, 2013 Part I General Equilibrium Chapter 15 - General Equilibrium Theory: Examples • Pure exchange economy with Edgeworth Box • Production with One-Firm, One-Consumer • [Small Open Economy] 15B. Pure Exchange: The Edgeworth Box Denitions and Set Up Pure Exchange Economy • An economy in which there are no production opportunities. Agents possess endowments, eco- nomic activity consists of trading and consumption Edgeworth Box Economy • Preliminaries Assume consumers act as price takers Two consumers i = 1; 2; Two commodities l = 1; 2 • Consumption and Endowment 0 Consumer i s consumption vector is xi = (x1i; x2i) Consumer 0s consumption set is 2 ∗ i R+ Consumer has preference relation over consumption vectors in this set ∗ i %i 0 Consumer i s endowment vector is !i = (!1i;!2i) ∗ Total endowment of good l is !¯l = !l1 + !l2 • Allocation 1 An allocation 4 is an assignment of a nonnegative consumption vector to each con- x 2 R+ sumer ∗ x = (x1; x2) = ((x11; x21) ; (x12; x22)) A feasible allocation is ∗ xl1 + xl2 ≤ !¯l for l = 1; 2 A nonwasteful allocation is ∗ xl1 + xl2 =! ¯l for l = 1; 2 ∗ These can be depicted in Edgeworth Box • Wealth and Budget Sets Wealth: Not given exogenously, only endowments are given. Wealth is determined by by prices. ∗ p · !i = p1!1i + p2!2i Budget Set: Given endowment, budget set is a function of prices 2 ∗ Bi (p) = xi 2 R+ : p · xi ≤ p · !i ∗ Graphically, draw budget line [slope = − p1 ]. Consumer 1's budget set consists of p2 all nonnegative vectors below and to the left; consumer 2's is above and to the right • Graph: Axes: horizontal is good 1, vertial is good 2 Origins: consumer 1 in SW corner (as usual), consumer 2 in NE corner (unique to Edgeworth boxes) • Oer Curve = Demand (as a function of p) (A depiction of preferences of each consumer) %i ∗ Asume strictly convex, continuous, strongly monotone 2 As p varies, budget line pivots around !. Given this line, consumer demands most preferred point in B (p). This set of demanded consumptions makes up oer curve ∗ Just like demand function: x1 (p; p · !1) ∗ Curve passes through the endowment point. Also, since at each point endowment is aordable, every point on oer curve must be at least as good as !i. • Walrasian Equilibrium (a.k.a. Competitive Equilibrium), for an Edgeworth Box: Denition: Price vector ∗ and an allocation ∗ ∗ ∗ in the Edgeworth box such that p x = (x1; x2) for i = 1; 2 ∗ for all 0 ∗ ∗ xi %i xi xi 2 Bi (p ) At equilibrium, the oer curves of the two consumers intersect ∗ Any intersection of the oer curves outside of ! corresponds to a WE p∗ Only relative prices 1 are determined in equilibrium, since each consumer's demand is p2∗ homogenous of degree zero ∗ ∗ ∗ is a WE ∗ ∗ ∗ is a WE ∗ p = (p1; p2) =) αp = (αp1; αp2) • Solving for WE strategy: 1. Optimization: Max utility, using budget constraint to reduce number of variables 2. Demand: Solve for Demand (OC) as a function of price xli (p) 3. Market Clearing: Set total demand for a good equal to endowment: xl1 + xl2 = !l (a) AKA solve for excess demand function zl (p) = xl1 + xl2 − !l, and then set it equal to 0. ∗ p1 4. Solve for price ratio ∗ p2 (a) To determine p∗, only need to nd that for which one market clears. Other will neces- sarily clear at these prices. See PS1 Q6 3 • Multple WE Can certainly have multiple WE, as you will have a WE any time the Oer Curves intersect more than once ∗ i.e. quasilinear utility with respect to dierent numeraires (doctored a bit, MWG 521) • Non-existance of WE Two classic examples 1. Non-monotonic Preferences (a) Endowment on the boundary (b) Consumer 2 only desires good 1 and has all good 1. (c) Price of good 2 is zero. Consumer 2 strictly prefers receiving more of good 1 (but won't). But also prefers to receive more of good 2 at price 0. (d) No p∗ at which demands are compatible i. [Since Consumer 1's demand for good 2 is innite at p2 = 0] ii. Can also say: there is an 0 ∗ 0 ∗ , hence not a WE x1 1 x1; x1 2 B (p ) 2. Nonconvex Preferences (a) In example, consumer 1's oer curve is disconnected, so it does not intersect consumer 2's 4 • Pareto Optimum Denition: An allocation x in the EWB is Pareto Optimal if there is no other allocation x0 in the Edgeworth box with 0 and and 0 for some . xi %i xi i = 1; 2 xi i xi i ∗ An economic outcome is Pareto optimal if there is no alternative feasible outcome at which every individual in the economy is at least as well o, and some individual is strictly better o. Interior solution: consumers' indierence curves through x must be tangent Corner solution: tangency need not hold • Pareto Set Set of all Pareto Optimal allocations • Contract Curve The part of the Pareto Set where both consumers do at least as well as at their initial endowments • 1st Welfare Theorem Any Walrasian equilibirum allocation x∗ necessarily belongs to the Pareto Set • 2nd Welfare Theorem Verbally: Under convexity assumptions (not required for 1st WT), a planner can achieve any desired Pareto optimal allocation by appropriately redistributing wealth in a lump-sum fashion and then letting the market work Denition: An allocation x∗ in the EWB is supportable as an equilibrium with transfers if ∗ there is a price system p and wealth transfers T1 and T2 satisfying T1 + T2 = 0 such that for each consumer we have: ∗ 0 0 2 such that ∗ 0 ∗ i xi % xi 8xi 2 R+ p · xi ≤ p · !i + Ti ∗ Note that p∗ doesn't change; it is essentially the endowment ! that is moved by the transfers to get to the new x∗ Failures ∗ Nonconvexities: May fail when preferences aren't convex (given budget line, consumer with nonconvex preference may prefer dierent point to intended x∗) ∗ Monotonicity: (Figure 10(a) above). ! is PO, but cannot be supported as an equilib- rium with transfers 5 15C: The One-Consumer, One-Producer Economy • Set up Single producer, single rm; both are price taking. x1 = leisure (and labor = z = L¯ − x1); x2 = good produced by rm Endowment: L¯ of leisure, 0 of x2 (although consumer owns rm and gets prots, which are positive) Continuous, convex and strongly monotone dened over and % x1 (leisure) x2 • Production max pf (z) − wz z≥0 Given prices (p; w),[p is price of x2, w is wage] rm has optimal labor demand z (p; w), output q (p; w) and π (p; w) • Consumption max u (x1; x2) 2 (x1;x2)2R+ s.t. px2 ≤ w L¯ − x1 + π (p; w) Budget constraint is from wage earning and prot from rm • Walrasian Equilibrium Price vector (p∗; w∗) at which consumption and labor markets clear: ∗ ∗ ∗ ∗ ∗ x2 (p ; w ) = q (p ; w ) ∗ ∗ ∗ ∗ L¯ − x1 (p ; w ) = z (p ; w ) 6 Graphically: 15.C.1(b) is not an equilibrium; 15.C.2 is A particular consumption-leisure combination is an WE () it maximizes the consumer's utility subject to the economy's technological and endowment constraints • Graph Of (producer's origin) in SE corner; takes x−axis as −z Oc (consumer's origin) in SW corner as usual Budget constraint line has slope w and intercept π(p;w) . This is the isoprot line of the − p p rm's prot maximization problem: (−z; q): pq − wz = π (p; w) 16: Equilibrium and Its Basic Welfare Properties Basic Model Denitions • I consumers, J consumers, L goods Consumer consumption set L • i Xi ⊂ R Preferences dened on , rational (complete & transitive) • %i Xi Endowment vector L • !¯ = (!1;:::;!L) 2 R Firm production set (or technology) L. Every (nonempty), closed. • Yj ⊂ R Yj 6= ; Basic data on preferences, technologies and resources for this economy are summarized by I J f(Xi; %i)gi=1 ; fYjgj=1 ; !¯ Feasible Allocation (x; y) • Denition: A feasible allocation is a vector (x1; : : : ; xI ; y1; : : : ; yI ) s.t. 1. xi 2 Xi 8i 7 2. yj 2 Yj 8j 3. P P i xi =! ¯ + j yj n o The set of feasible allocations is denoted by P P A = (x; y) 2 X1 × · · · × XI × Y1 × · · · × YI : i xi =! ¯ + j yj ⊂ RL(I+J) Pareto Optimum Denition: A feasible allocation is Pareto Optimal if 0 0 s.t. 0 • (x; y) 2 A @ (x ; y ) 2 A xi %i xi 8i and 0 for some . xi xi i Private Ownership Economies Endowment: Each consumer has initial endowment L • i !i 2 R • Shareholding: Each consumer i has shareholding θij in rm j θij 2 [0; 1] 8i; j P i θij = 1 8j Data on preferences, technology, resources and ownership are summarized by I J I f(Xi; %i)gi=1 ; fYjgj=1 ; f!i; θi1; : : : θiJ gi=1 Walrasian Equilibrium n o Given a private ownership economy specied by I J I , f(Xi; %i)gi=1 ; fYjgj=1 ; (!i; θi1; : : : ; θiJ )i=1 ∗ ∗ An allocation (x ; y ) and a price vector p = (p1; : : : pL) constitute a Walrasian (or competitive) Equilibrium if: 1. ∗ p · yj ≤ p · yj 8yj 2 Yj (a) [ max: That is, for every ∗ maximizes prots in ] π j; yj yj n o 2. For every , ∗ is maximal for in the budget set P ∗ i xi %i xi 2 Xi : p · xi ≤ p · !i + j θijp · yj (a) [Hence ∗ ∗ ∗ P ∗] xi i xi =) p xi > p !i + j θijp · yj (b) [Consumers maximize utility subject to budet constraint, and anything strictly better is unattainable] 3.