Economics 201B–Second HalfLecture 11, 4/20/1017.I and Handout: Nonconvex Preferences and Indivisibilities• Why do we care?– General story that justifies convexity (diminishing MRS along each indifference curve) worksfor “most” goods, but if there is a single pair of goods for which preferences are nonconvex, orif there is a single good which is indivisible, Existence of Walrasian Equilibrium and the SecondWelfare Theorem fail.∗12house in Berkeley and12house in SF∗ Two trips to Winnemucca are not preferable to one trip to Salt Lake if you like to ski.∗ Painting a room with orange and green stripes is not preferable to solid orange or solidgreen.• Theorem 1 (Shapley-Folkman) Suppose x ∈ con (A1+ ···+ AI),whereAi⊂ RL. Then we maywrite x = a1+ ···+ aI,whereai∈ con Aifor all i and ai∈ Aifor all but L values of i.Proof: The proof is in the handout, it just uses the fact that m ≥ L + 1 vectors in RLmust belinearly dependent.Theorem 2 Suppose we are given a pure exchange economy, where for each i =1,...,I, isatisfies1. continuity: {(x, y) ∈ RL+× RL+: x iy} is relatively open in RL+× RL+;12. for each individual i, the consumption set is RL+, i.e. each good is perfectly divisible, and ea chagent is capable of surviving on zero consumption;3. acyclicity: there is no collection x1,x2,...,xmsuch that x1ix2i···ixmix1;4. str ong monotonicity:x>y⇒ x iyThen there exists p∗∈ Δ0with 0 ∈ con E(p∗) and x∗i∈ Di(p∗) such that1IL=1p∗ Ii=1x∗i−Ii=1ωi ≤2LImax{ωi∞: i =1,...,I}where x∞=max{|x1|,...,|xL|}.The inequality bounds the market value of the surpluses and shortages in the economy. There mightbe a large surplus of a good whose price is nearly zero; although the statement seems to allow a largeshortage of a good whose price is nearly zero, in practice goods which are in short supply tend notto be inexpensive. Dividing by I on the left side of the inequality expresses the market value of thesurplus and shortages in per capita terms. On the right side, we typically assume that the number ofindividuals (I) is much larger than the number of goods (L); this will certainly be true if we considera model in which goods are somewhat aggregated (food, clothing, housing, transportation, ...,; orhamburger, steak, milk, ...); it will be false if we consider each dwelling unit to be an entirely separatecommodity from every other dwelling unit. Remember that we are using an exchange economy tostudy the allocation of consumption,2taking the production decisions as exogenous; thus, max{ωi} should be thought of as the max-imum resources devoted to consumption by any individual, which is typically much less than thatindividual’s total wealth.Thefactthat0∈ con E(p∗) has its own, separate interpretation. Imagine that not everyone tradesat exactly the same time, but people come to the market at different times, and choose consumptionvectors out of their demand sets. If there is a little inventory in the market, then the demands canbe accommodated for a while. Once the inventory starts running out of some goods, the price can bechanged very slightly to shift the demand. Very small shifts in price can move the demand aroundE(p∗); by spending various amounts of time at different points in E(p∗), the market can effectivelyproduce any excess demand in con E(p∗), in particular zero excess demand.Theorem 3 Suppose we are given a pure exchange economy, where for each i =1,...,I, isatisfies1. continuity: {(x, y) ∈ RL+× RL+: x iy} is relatively open in RL+× RL+;2. for each individual i, the consumption set is RL+, i.e. each good is perfectly divisible, and ea chagent is capable of surviving on zero consumption;3. acyclicity: there is no col lection x1,x2,...,xmsuch that x1ix2i···ixmix1;3Then there exists p∗ 0 and x∗i∈ Di(p∗) such that1IL=1maxIi=1x∗i−Ii=1ωi, 0≤ 2LImax{ωi1: i =1,...,I} (1)where x1=L=1|x|.Notice that the theorem does not assume any monotonicity, or even local nonsatiation. It boundsonly the shortages in the economy; there may be large surpluses of some goods.Proof Outline:– LetΔ=⎧⎨⎩p ∈ RL:LI≤ p≤ 1( =1,...,L)⎫⎬⎭Why?∗ Convenient to normalize prices by p∞= 1, would have to prove Kakutani’s Theoremholds on that set, which is not convex. Δis convex and compact.∗ In the proof of Debreu-Gale-Kuhn-Nikaido, we define the correspondence on Δ0and extendit to Δ. Get a fixed point ˆp∗∈ Δ0, but we get no control over the price of the cheapestgood. This gives us no control over the diameter of the budget set, which bounds the sizeof the nonconvexity in the demand set.∗ By trimming Δso all prices bounded below byL/I, we give up the fact that the Kakutanifixed point is in the interior of Δ, but we get control over the diameter of the budget set.4– Consider the correspondence z :Δ→ RLdefined byz(p)=Ii=1con Di(p)− ¯ω– Acyclicity implies that z(p) = ∅.– Look at the “Offer Curve”{x : ∃p∈Δx ∈ z(p)}The picture is almost the same as the picture in the convex case.– Choose a compact set X ⊂ RLsuch thatp ∈ Δ⇒ z(p) ⊆ XDefine a correspondence f :Δ× X → Δ× X byf(p, x)={(q, y):y ∈ z(p), ∀q∈Δq · x ≥ q· x}By Kakutani’s Theorem, there exists a fixed point (p∗, ¯x∗)¯x∗=Ii=1¯x∗i¯x∗i∈ con Ei(p∗)(i =1,...,I)(2)∀q∈Δq · ¯x∗≤ p · ¯x∗=0 (3)– In the proof of Debreu-Gale-Kuhn-Nikaido, we showed that∀q∈Δq · ¯x∗≤ 0 ⇒ ¯x∗≤ 0Use a similar argument and Equation (3) to put an upper bound on the positive components of¯x∗.– From Equation (2) and the Shapley-Folkman Theorem, we can assume that¯x∗i∈ con Ei(p∗)( =1,...,L)¯x∗i∈ Ei(p∗)fori ∈{i1,...,iL}5– Choose arbitrarilyx∗i1∈ Ei1(p∗),...,x∗iL∈ EiL(p∗)and letx∗i=¯x∗ifor i ∈{i1,...,iL}sox∗i∈ Ei(p∗) for all i–Ii=1x∗i=¯x∗+L=1x∗i− ¯x∗iError Term– The diameters of the budget sets are bounded above by the endowments and the lower boundon prices in Δ, which bounds the Error Term.Indivisibilities:Theorem 3 applies verbatim to the case of indivisibilites, except that one must substitute Qifor Di.Withindivisibilities, Qihas closed graph but Digenerally does