Int. J. Computational Economics and Econometrics, Vol. 6, No. 1, 2016
Sequential pairwise trading: convergence and welfare implications Maria-Augusta Miceli* Faculty of Economics, Department of Economics and Law, Sapienza University of Rome, 9, via del Castro Laurenziano, Rome 00161, Italy Email:
[email protected] *Corresponding author
Federico Cecconi ISTC-CNR (Institute of Cognitive Sciences and Technologies), 44, via S. Martino della Battaglia, Rome 00185, Italy Email:
[email protected]
Giovanni Cerulli IRCRES-CNR, 19 via dei Taurini, Rome 00185, Italy Email:
[email protected] Abstract: This paper characterises the out-of-equilibrium dynamics of a symmetric, pure-exchange economy with two goods and N agents with uniformly distributed preferences and identical endowments. Relaxing the auctioneer assumption, but maintaining a global price rule, sequential random pairwise trading at out-of-equilibrium prices is allowed. Initial mispricing implies rationing, determining excess demand (supply) fading away only at convergence, when the price of the initially cheaper (more expensive) good becomes more expensive (cheaper) than the walrasian one. The system converges when the sequential price reaches the walrasian price evaluated at current updated endowments. A perfectly symmetric setting, by initial mispricing and consequent rationed trading, creates asymmetric allocations even at convergence, where welfare is just slightly lower than auctioneer Pareto one. This model sketches a basis for price over-reaction micro-foundations and captures endogenous ‘wealth divide’ among the population, induced by whether trading is dominated by preferences over goods or by speculation around their prices. Keywords: pairwise-exchange; tatônnement; rationing; mispricing; walrasianequilibrium; sequential-equilibrium; auctioneer; price-overreaction; wealthdivide; ABM; agent-based-models.
Copyright © 2016 Inderscience Enterprises Ltd.
13
Sequential Pairwise Trading: Convergence and Welfare Implications Maria-Augusta Miceli Sapienza University of Rome
Federico Cecconi ISTC-CNR
Giovanni Cerulli CERIS-CNR
Feb 20th, 2015
Abstract This paper characterizes the out-of-equilibrium dynamics of a symmetric, pure exchange economy with two goods and N agents with uniformly distributed preferences and identical endowments. Relaxing the auctioneer assumption, but maintaining a global price rule, sequentially random pairwise trading at out-of-equilibrium prices is allowed. Initial mispricing implies rationing, determining excess demand (supply) fading away only at convergence, when the price of the initially cheaper (more expensive) good becomes more expensive (cheaper) than the walrasian one. The system converges when the sequential price reaches the walrasian price evaluated at current updated endowments. A perfectly symmetric setting, by initial mispricing and consequent rationed trading, creates asymmetric resource allocations even at convergence, where welfare is lower than a standardized 1% with respect to the auctioneer Pareto one. This model sketches a possible basis for price over-reaction microfoundation and captures endogenous "wealth divide" among the population, induced by whether agent trading is dominated by good preferences or just by speculation around their prices.
Keywords: pairwise exchange, tatônnement, rationing, mispricing, general equilibrium, walrasian equilibrium, sequential equilibrium, auctioneer, price overreaction, wealth divide, agent-based models, agent-based simulations. JEL: B21, C63, C68, D01, D03, D10, D30, D31, D50, D51, D58, D60, D61, D70, G12.
1
1
Introduction
In a totally symmetric pure exchange economy with two goods and N heterogeneous uniformly distributed agents, sequential pairwise exchange, starting at wrong prices affects the distribution of resources at convergence. More precisely, if the initial relative price for one good is cheaper than the walrasian one, the economy ends up with an excess demand for that good even when the price gets more expensive than the walrasian and viceversa. Reaching convergence from a relative price lower than the equilibrium one is not equivalent to starting from a too high price. The surprise is that, although starting from a perfectly symmetric setting, sequential rationed trading starting from mispricing, shifts resouces allocations to an asymmetric allocation even at convergence, neglecting walrasian symmetric prices. We use the simplest general equilibrium setting, keeping all the convex analysis assumptions. We relax the existence of the auctioneer mechanism, in the sense that agents can trade at out-of market clearing prices, but we keep the idea that prices do update following a global pricing rule, partially responding to aggregate excess demand on each market. The simulation explores the sequential pairwise exchange dynamics among heterogeneous agents up to convergence to a stable general equilibrium price. N heterogeneous agents are endowed with an identical quantity of each good. Agent preferences are represented by the classic Cobb-Douglas log linear utility function, where heterogeneity is determined by the coefficient alfa, uniformly distributed on the unit interval. An exogenous relative price is given as initialization. At this price, agents express optimal excess demands, computed by maximizing their utility function under their current budget constraint. Pairs are randomly created between potential buyers and sellers. Only the short side of each trade is satisfied, while the long side is rationed. The sign of the aggregate excess demand changes the price by a tick and "leftovers" are back to be iteratively traded. We did exclude the possibility, used by the game-theory approach on the issue (among many, Debreu and Scarf (1963), Rader (1964), Scarf (1971), Gale (1976), Rubinstein and Wolinski (1985), Shapley and Scarf (1974), Shapley and Shubik (1977), Gale and Sabourian (2005) and Gintis and Mandel (2012a, 2012a)), to have different local prices among pairs, since we doubt coexistence of different local price vectors, one for each pair, in a globalized world. This exercise aims at shedding light on the actual tatônnement mechanism, by simulating its dynamics and analyzing the convergence when out-of-equilibrium trading is allowed. The related literature is vast. The classical existence and stability analysis of the walrasian GE (to mention just a few: Arrow and Debreu (54), Debreu (1959), Hahn (1970), Hahn and Negishi (1962), Arrow and Hahn (1971)), the pairwise tatônnement analysis (Feldman (1973), Goldman, and Starr (1982), Rader (1968) and (1976)) and even the disequilibrium literature [Benassy (1982), Clower and Leijonhufvud (1975), Fisher (1981) and (1983)] didn’t explore the actual agent interactions and the interplay between rationed and non-rationed agents and the consequent allocation dynamics. Differently from the “analytical” approach, the multi-agent simulation allows us to compute, monitor and statistically verify the properties of the allocation evolution during the sequential out-ofequilibrium trading in a setting first used by Foley (1994) and (1999). We focus on the different achievements of satisfied and rationed agents along the convergence path and their consequent different results in terms of utility improvement with respect to the Walrasian allocation that couldn’t be explored by the analytic literature. 2
In this analysis we have voluntarily avoided imitation effects in the utility function (Gintis (2006), (2007), (2010), Ghosal and Morelli (2004), Ghosal and Porter (2010)), because they help convergence. We also neglect the "bargaining" stream of research, because we are actually interested in the pricequantity interaction during trading and the relative implications of it on resource redistribution. Therefore, this study will not consider local bargaining among each pair as, for instance, in Gale (1986a,b), Gale (2000), McLennan and Sonnenshein (1991), and Rubinstein and Wolinski (1985). Progressively the system converges. At which relative price? In this setting we may define many notions of general equilibrium prices: (i) the walrasian static general equilibrium price supposed to clear all market excess demands at initial endowments; (ii) the walrasian dynamic general equilibrium price clearing excess demands at current updated endowments; (iii) the walrasian dynamic general equilibrium price depending on current desired allocations, that, because of rationed trading, does not correspond to actual updated endowments. The system converges when the exogenous price, lead by the aggregate excess demands, converges to (ii): pairwise random trading leads to the same results due to the centralized auctioneer. The combination of the two slowing factors - (i) local and therefore rationed exchange, (ii) "slow" price adjustment preventing the common bang-bang effect (for example Scarf (1960)) occurring when price and quantity perfect flexibility are assumed - drive the system to unexpected convergence. Convergence is realized when, at a certain relative price, all agents with excesses are on the same side of the market, trade is therefore over and eventual infinitesimal price changes move all the unsatisfied agents from one to the other side of the market, making trade impossible. Our main contribution is discovering that, in a perfectly symmetric setting, sequential out-ofequilibrium pairwise exchange starting at wrong prices affects the distribution of resources at convergence, leading to a predictable price “overreaction”. Moreover, due to their interactions, agents cluster in two different classes, satisfied and rationed ones, obtaining different results in terms of utility (and hence welfare) along the convergence process. The design of the proposed model allows us to control for: • which segment of agents win from the mispricing and which lose; • how the measure of initial mispricing induces a well defined minimum measure of rationing in the system, leading to a computationally predictable overpricing. The endogenous clustering of agents between satisfied and rationed ones is a plausible microfoundation for endogenous price overreaction, not strictly induced by "lack" of resources, but by "misallocation" of resources. The paper proceeds as follows. In order to measure the departure of this model from the standard static general equilibrium solution, we start by establishing the general equilibrium analytic results, due to the auctioneer assumption, as benchmark. We then show how agents with different αi preferences react to price changes in terms of different utility maximizing choices. Next, we repeat the same simple example showing how two agents, not symmetric in preferences, exchange in a pair and how consequent and iterative rationing affects endowment updates. Once established the building blocks details, results for the full simulations for N heterogeneous sequentially pairwise interacting agents are presented. To simplify results presentation, we will use initial prices pA /pB = 1/2 and pA /pB = 2, to show how symmetric departures lead to symmetric results in prices and allocations movements, but to 3
asymmetric results in utilities, since trading — as we will show - improves utility anyway.
2
Setting and assumptions
In order to monitor the effects of dropping the auctioneer assumption on general equilibrium, we build the simplest general equilibrium setting, keeping all convex assumptions expressed by the following parameters. Assumption 1 Strictly Convex Optimization Context
2.1
Parameters
1. Set of agents: i ∈ I = {1, .., i, ...N } . 2. Two goods economy: j = A, B, from now on "apples" and "bananas". 3. Strictly quasi-concave utility function: log-linear Cobb-Douglas for each agent:
so that u0i > 0, u00i < 0.
¡ B¢ B A ui,t (xA i,t , xi,t ) = αi ln(xi,t ) + (1 − αi ) ln xi,t .
(1)
4. Agents preferences heterogeneity is parametrized by αi ∼ U [0, 1] . o n 2 , where ω A , ωB ⊂ R = ω 5. Endowments: Ωt = {ωi,t }N i,t i,t i,t i=1
2 2 • Symmetric initial endowments: Ω0 = {ωi,0 }N i=1 ⊂ R , where ω i,0 = {ω 0 , ω0 } ⊂ R and ω 0 = 10, ∀agent i, ∀ good j.
6. From now on price of good B ("bananas") is the numeraire1 therefore pB t = 1, ∀t. We will then use notation pA def = p pB intending that we will always have pA at numerator. 7. Initial relative price: pt=0 characterizes every simulation path.
2.2
Implications
´ ³ ∈ R2 , such that 1. Each agent i chooses his desired allocation x∗i = xA , x i,B i,t B maxxA ,xB ui (xA i,t , xi,t ), i,t
i,t
B A B s.t. pA xA i,t + xi,t = pA ω i,t + ωi,t
resulting in the optimal demands ¢ ∗B ¡ ¢ª © ∗A ¡ B A B xi,t pA , ωA i,t , ω i,t , xi,t pA , ω i,t , ω i,t
A B 2. Nominal Income single income : yi,t = pA t ω i,t + ωi,t . Aggregate Income: Yt = 1
PN
i=1 yi,t .
It can very well be money, but we don’t assume anything since its market is redundant by Walras law.
4
3. Excess Demands for agent i and good j = A, B ¡ ¢ B i eji,t αi , pA , ω A i,t , ω i,t = α
³
B pA ωA i,t + ω i,t
pA
´
− ω ji,t
(2)
B 4. Since endowments are symmetric in each good: ω A i,t=0 = ω i,t=0 = 10, ∀i :
• agents utility at initial endowment is the same for all agents ui (ω i ) = ln (10) = 2.302 , P • the total utility at endowment is W (Ω) = N i=1 ui (ω i ) = ln (10) N = 2.302 N,
• Initial income per agent: Yi,t=0 = 10pA t=0 + 10, same for every agent i.
2.3
Simulations Indexes
Simulations are indexed by two elements: • Agent distribution αi ∼ U [0, 1]. • Initially given exogenous relative price: pt=0 , given a fixed pB = 1, ∀t.
2.4
Pairwise Exchange Procedure
We here summarize the procedure. Given the following initial conditions: (i) constant endowments Ω0 = {10, 10}N i=1 for every simulation, (ii) the random draw of the vector α ∼U [0, 1] ,where α ∈RN defining each agent preferences for each good and (iii) any initial pA (t = 0) 6= pW , since pB is the numeraire, ∀ agent i = 1, ..., N and ∀t = 0, ...T, where T is the endogenous date of convergence: 1. we compute: (a) single budget constraint, A B (b) resulting income yi,t = pA t ω i,t + ω i,t as endowment value and P (c) aggregate income Yt = N i=1 yi,t ;
(d) single "realized utilities" evaluated at endowments and their sum Wt (Ω) = (e) optimal demands
© ∗A ¡ ¢ ∗B ¡ ¢ª B A B xi,t pA,t , ω A ; i,t , ω i,t , xi,t pA,t , ω i,t , ω i,t
PN
i=1 ui,t (ω i,t ) ;
(f) excess demands eij , ∀ agent i = 1, ..., N, ∀ good j = A, B;
(g) "potential utilities" evaluated on desired transactions ui (x∗i (pt , ωi,t )) and their sum W (X∗t ) = PN ∗ i=1 ui (xi (pt , ω i,t )) ;
(h) "dynamic walrasian price", reference price able to clear the market,
Result 1 Given preferences and current endowments, the "dynamic walrasian price", able to clear the market at every iteration is µ ¶DW α · ωB pA def t DW pt (Ωt ) = = (3) 0 pB t (1 − α) ·ω A t 5
Proof. There are two markets, apple and bananas. By Walras Law, zero excess demand on one market implies equilbrium on the other. Let’s compute equilibrium on the apple market, j = A to determine the single relative price. ³ ´ B N N N pA ω A + ω X X X ¡ ¢ i,t i,t A i A B i ei,t α , pA , ω i,t , ωi,t ≡ α − ω ji,t = 0 pA i=1 i=1 i=1 ³ ´ A B N N pA ω i,t + ω i,t X X αi − ωji,t = 0 pA i=1
0
pA α
ωA t
0
+α
then = pDW A
ωB t
i=1 − pA 10 ω A t
=0
α0 ω B t (1 − α)0 ωA t
1. Definition of the pair and the exchange procedure: (a) definition of "Net Buyers" and "Net Sellers" sets, given by the excess demands sign, (b) "exchange pairs" are randomly established by one net buyer and one net seller, i. =⇒ the exceeding net buyers or sellers, since they are not able to find a counterpart, they do not exchange in that iteration; ii. =⇒ exchanged quantities will be the minimum quantity between net demand and net supply in the pair. iii. If at the beginning all agents are net sellers or net buyers, there is no trade, and the aggregate excess demand or supply drives the price change. 2. Update phase: (a) endowments update by the traded quantities and leftovers form aggregate net excess demands for each good, (b) the relative price updates as a function of these net aggregate excess demands. Iterations continue up to convergence happening when excesses are only on one side, meaning there are just net buyers or net sellers and there is therefore no exchange. In this situation at a slight increase of the price, all residual net buyers become sellers or viceversa. Neither price, nor quantities change anymore. This procedure will be formalized and described in details in the next sections. In particular, in the next paragraph we describe what the walrasian benchmark in this setting is, we then explore the pairwise trading in a single fixed pair, showing consequences on rationing. Finally results of the complete simulations are shown, while some analytical reasonings explaining the result regularities are exposed in the final section.
6
3
Static Walrasian Equilibrium Benchmark
We here compute and then visualize the equilibrium analytic solution of this model, i.e. the walrasian equilibrium in this setting. The crucial assumption is that, at given preferences and initial endowments, equilibrium prices are computed as solutions of the (J − 1) market clearing equations and net excess demands are only computed at the equilibrium prices. In other words, agents only exchange at these equilibrium prices. The result is that, tautologically, at equilibrium prices markets clear. Let’s see below what this graphically meansin the 2-goods economy. By assumption, at time t = 0, all agents have the same endowment vector. Each agent evaluates the endowment with his personal utility function parametrized by the coefficient αi , that shows his preference towards apples. Identical endowments for each good and for all agents imply constant initial utility across heterogeneous agents. Proposition 1 Under the assumptions specified in subsection ( 2.1 ), summarized in: ¡ B¢ B A (i) strict convexity of the utility function Âi : ui (xA i , xi ) = αi ln(xi ) + (1 − αi ) ln xi (ii) α e i ∼ U [0, 1] and therefore lim E(α) = lim E (1 − α) = 0.5
N−→∞
N−→∞
(ii) initial endowments ω i = (ω A , ω B ) = (ω, ω) , ∀i, (iii) excess demand functions (2) are homogeneous of degree zero in price levels, (iv) Walras law holds, (v) N −→ ∞,n o © i ªN the economy º , αi n=1 , Ω has a
1. "Population Static walrasian Equilibrium Price" (theoretical-price, p∗ = then =⇒ lim p∗ (Ω0 ) = N−→∞
ωE(α) =1 ωE(1 − α)
³
pA pB
´∗
), (4)
2. "Sample Static Walrasian equilibrium price" (Sample Market Clearing Price, given the initial endowment Ω0 and realized agent distributions) XN XN ω αi ω B αi i α = XN i=1 p∗SW (Ω0 ) = XN i=1 = R1 (5) 1−α (1 − αi ) ω A ω (1 − α ) i i i=1
i=1
Result 2 By the Law of Large Numbers, as the number of agents N −→ ∞ (4) = (5) lim p∗SW =
N→∞
E(α) 1/2 = = p∗ = 1 E(1 − α) 1/2
Proof. In the Appendix. The same result holds if, instead of increasing the number of agents, we increase the number of samples S −→ ∞, ¶ S µ X αs ∗ lim p∗SW = s /S = p = 1 S→∞ 1 − α s=1 where s = 1, ..., S is the number of samples.
7
3.1
Visual representation
©¡ ¢ªN B We can visualize the situation in the xA ∈ R2 plane. Given that all agents are initially i , xi i=1 endowed with symmetric units of each good ωi = (10, 10) , we wonder where would they end up "if" walrasian prices were established by the auctioneer. Knowing that the walrasian equilibrium price would be p∗SW = (pA /pB )∗ = 1, we can draw the budget line passing through the endowment point with that slope. By construction each agent wants to buy a quantity of the first good such that: Definition 1 Static Equilibrium Allocations x∗i = arg max {Ui (xi ) , s.t. p∗ x ≤ p∗ ω i } Using utility function (1) , the solution is ⎧ ¡ A A ¢ αi B B = αi · 20 ⎨ xA∗ i = pA p0 ω i + p0 ωi 0 , i ¡ ¢ ¡ ¢ ) A A B = 1 − αi · 20 ⎩ xB∗ = (1−α p0 ω + pB 0ω i pB
∀i = 1, ..., N
0
Each agent buys each good as a fixed percentage of his income, expressed by his own preference ¡ i ¡ ¢¢ α , 1 − αi . This means that his indifference curve of each agent i would be tangent at the budget line with slope p∗ = 1, in the point such that the marginal rate of substitution is equal to p∗ = 1.E.g. αi xB∗ i =1 (1 − αi ) xA∗ i
(6)
Rearranging, we find the income expansionary path of the indifference curve for each agent i expressend in Fig.1. below. The higher αi , the lower the slope of this curve. ¢ ¡ 1 − αi A∗ B∗ xi xi = αi In Fig.1. for some chosen agents (αi = 0.2, 0.5, 0.8) , we can visualize the indifference curve passing through the endowment ui (10, 10) = αi ln 10 + (1 − αi ) ln(10) = ln (10) (thick curves), and the indifference curve passing through the optimal choice (thin curves). Starting from an endowment B with ωA i = ω i , agents with αi 6= 1 will improve their utility by trading. While for the particular agent αi = 0.5 where the preferred basket has the same quantities of each good, the endowment is his preferred allocation, so he would not trade.
xB 20
15
10
5
0 0
5
10
Fig.1. 8
15
20
xA
Remark 1 At pA /pB = 1, the αi = 0.5 (dashed curves in Fig.1.) would have his optimal allocation equivalent to his endowment x∗i = ω i = (10, 10) , therefore at eventual prices (1, 1) he would not move (dashed indifference curve). In this simple example and following the remark, we notice that at relative price p = 1, the net apple buyers are the agents with αi > 0.5. In the graph they are the ones with optimal choice on the right of the 45◦ line and viceversa. Definition 2 "Net buyers" at walrasian prices are agents such that will buy more than their endowment αi ∈ (0.5, 1] Definition 3 "Net sellers" at walrasian prices are agents such that would have less than their endowment αi ∈ [0, 0.5) How would this walrasian equilibrium situation be implemented? 1. At the market clearing prices pA /pB = 1, the auctioneer gets all the excess demands and perfectly redistributes resources to satisfy all agents. 2. In pairwise trading, each agent could be coupled by his perfectly symmetric counterpart. The latter doesn’t happen by chance and we will study what actually happens in practice. Fig. 2. has αi ∈ [0, 1] on the x-axis and the respective utility functions on the y-axis. The round points express initial utility that by construction is the same for all agents, ui (ω 0 ) = αi ln(10) + (1 − αi ) ln (10) = ln(10) = 2.302,
∀i = 1, ..., N
The blue triangles show the potential utility obtained "if" agents would achieve their optimal allocation at walrasian prices, ui [x∗ (p∗ , ω0 )] . We see that the improvement is zero for the αi = 0.5 agent and max for the αi most distant from the center.
Fig. 2. Single Agent Utilities at t=0 computed on Endowments and WE Allocations at p* =1
3.1 3
u [x (p*, w )] 1
i
0
u (w )
2.9
i
0
2.8 2.7 2.6 2.5 2.4 2.3 2.2
0
0.1
0.2
0.3
0.4
0.5
alfa
9
0.6
0.7
0.8
0.9
1
Remark 2 Utility increase. ∆ui = ui [x∗ (p∗ , ω 0 )] − ui (ω0 ) The farther αi is from 0.5, the more the agent utility can be increased by reaching the walrasian allocation, as shown Fig. 1 and 2. As soon as we drop the two cases above: (1) existence of the auctioneer or (2) the lucky situation, will the agents achieve any improvement with respect to their initial utility. If so, how big? This simplified model can give precise answers. In the next subsection we would refine the definition of "net buyers" and "net sellers" for any level of the relative price.
3.2
Net Buyers and Net Sellers Definition
Out of the special case in which pA = pB , not only preferences, but also the relative price, determine whether to be buyer or seller. Definition 4 "Net Buyers" Nb and "Net Sellers" Ns of apples are defined ¡ ¢ ³ ´ b b b∗ b b b b pA ω A + pB ω B Nb : eA α , pA , pB , ω A , ω B = α − ω bA ≥ 0 pA Ns :
s s s s es∗ A (α , pA , pB , ωA , ω B ) = α
(pA ω sA + pB ωsB ) − ωsA < 0 pA
(7) (8)
Remark 3 If and only if pA /pB = 1, the threshold between buyers and sellers is α = 0.5 at pA /pB = 1. Definition 5 In a two goods economy, where all agents have identical endowments, ω0 , and log-linear Cobb-Douglas utility functions and heterogeneity is determined by the coefficient α ∼ U [0, 1] , given b such that at given any relative price pA /pB , the indifferent agent is the one characterized by a α prices, his optimal demand for apples id equal to his optimal demand for bananas α b = (pA , pB ) is equivalent to the relative price vector ¢ ¡ pA ωbA + pB ω bB (pA ωsA + pB ω sB ) − ω 0 = (1 − α b) − ω0 α b: α b pA pA α b 1−α b = (9) pA pB pA =⇒ α b = (pA , pB ) = pA + pB and the two sets will be
Net-Sellers : αi ∈ [0, α b)
Net-Buyers : αi ∈ (b α, 1]
(10)
¡ © ¡ t t¢ £ ¡ ¢¤ª ¢ Corollary 1 At every step t, the vector α b t pA , pB , 1 − α b t ptA , ptB is equivalent to pt = ptA , ptB orthogonal to the budget set pt · (xt −ω t ) = 0.
¡ ¢ £ ¡ ¢¤ Proof. By Definition 5 and equation (9) , the ratio α b t ptA , ptB / 1 − α b t ptA , ptB and the slope ptA /ptB are the same number, and the vector is the geometric separation line distinguishing "net buyers" from "net sellers" at every price update. 10
¡ ¢ Remark 4 By Walras Law.pt = ptA , ptB orthogonal to the budget set pt · (xt −ω t ) = 0 is "not necessarily" an equilibrium price. Definition 6 We define "Speculators" agents whose preference is more motivated by price convenience than by the intrinsec quality of the good. In this article, speculators choose apples over bananas almost essentially if cheaper. Therefore, these agents change the sign of their net demand (or supply) depending on "out-of-equilibrium" prices £ ¡ 0 0¢ ¤ b (p∗A , p∗B ) , if p0A < p∗A b pA , pB ; α Excess-Net-Buyers : αi ∈ α £ ¡ ¢¤ Excess-Net-Sellers : αi ∈ α b (p∗A , p∗B ) , α b p0A , p0B , if p0A > p∗A
Remark 5 The set of agents that, thanks to the wrong price, shift from being net suppliers to being net buyers (or viceversa), will make the difference in sequential equilibrium resource redistributions. Graphically, as in figures 3a and 3b, the "speculator set" is defined by the following separating line: 0.5 • at the initial price pA = 0.5 =⇒ α b = 1+0.5 = 0.33 and therefore the line separating net buyers from net sellers has slope (1 − α b ) /b α = (1 − 0.333) /0.333 = 2.
2 • at the initial price pA = 2 =⇒ α b = 1+2 = 0.667 and therefore the line separating net buyers from net sellers has slope (1 − α b ) /b α = (1 − 0.667) /0.667 = 0.5
Fig. 3.a. Speculators at p(t = 0) = 0.5
At a price for apples that is cheaper than the walrasian, net buyers would increase by the share of "speculators", implying that the share of net buyers would be greater than that of net sellers. The implications of this fact on equilibrium price would be analyzed in Section 6.
11
Fig. 3.b. Speculators at p (t = 0) = 2
At a price for apples that is more expensive than the equilibrium one, for example p = 2, by the same reasoning there would be more net sellers than net buyers. Having specified how net buyers and net sellers sets shift as the relative price move, in the next session, we detail the pairwise exchange and the consequent endowment and price updates.
4
Pairwise Sequential Trading
In the previous section we defined the set of buyers and sellers, here we explore the pairwise trading, i.e. what happens if there is no immediate adjustment to walrasian prices, while agents are many and trade simultaneously and pairwise. We formalize the trading quantities and the endowments and price update rules. We need to define: Definition 7 "Short" part of the trade, the agent trading the minimum quantity in the pairwise trade. Definition 8 "Long" part of the trade, the agent trading the maximum quantity in the pairwise trade. At an initial given price, different from the walrasian one, agents start trading in "trading pairs", randomly coupling one net buyer with one net seller. At given fixed prices agents wish to trade their excesses and therefore only the short side will be satisfied. The "long" side will be rationed and the agents holding the residuals will try to trade them at the next round. While making pairs, at each step, there will be an exceeding number of net buyers or net sellers. Those are supposed to not trade, or, equivalently, they are coupled with satisfied agents (e.g. with zero excess demand). At the end of each step (iteration of the loop), if the aggregate excess demand is positive, it will push the price up by ε and viceversa. We call this the "Price Rule" (see eq.(13)). At the new prices, all agents will reformulate their optimal excess demands/supplies by optimizing their utilities subject to the budget constraint induced by the new reached allocation evaluated at new prices. Note that the rationed agent will be able to update his endowment only by the quantity he will be able to exchange, which is by definition smaller than the desired one. 12
The process continues till aggregate excess demand tends to zero or, most probably, when all unsatisfied agents will be on the same side of the market. In this case, all net buyers (or all net sellers), are just facing satisfied agents that don’t want to trade anymore. Prices still change according to the price rule till the point in which the price change will transform the set of net buyers in a set of net sellers (or viceversa). A bang-bang oscillation settles in, with very small excesses change from "buy" to "sell" side. The relative price oscillation depends on the size of the imposed prices tick change ε around a constant level that is the walrasian price sustained by the reached current endowments. Formally: Definition 9 Trading Pair. We randomly pair one net buyer and one net seller of apples and have them exchange at the given price. Of course their excesses won’t clear. The number of buyers is not necessarily the same as the number of sellers, we therefore define: Definition 10 Number of pairs: k = 1, ..., K, where K = min {#Nb , #Ns } . The exceeding buyers or sellers behave as if they were coupled with an agent wishing to exchange zero quantities. Assumption 2 Rationing. The quantity actually exchanged is the minimum between what each agent is wishing to buy (or sell) at that price and what the couterpart offers at that same price. Only the "short" side is satisfied, while the other is rationed and looks for another counterpart at the next step. o n s∗ , e eA = min eb∗ A A o n s∗ eB = min eb∗ B , eB After trading, there are two consequences: (i) each agent updates his endowments by the realized trade;
Definition 11 Endowments update n o s∗ , |e | ω bA (t + 1) = ωbA (t) + min eb∗ A A n o s∗ ω sA (t + 1) = ωsA (t) − min eb∗ A , |eA |
o n¯ ¯ ¯ ¯ s∗ ω bB (t + 1) = ω bB (t) − min ¯eb∗ , e B¯ B o n¯ ¯ ¯ ¯ s∗ ω sB (t + 1) = ω sB (t) + min ¯eb∗ , e ¯ B B
(11)
(12)
(ii) the external price moves by a tick as a function of the sign of the excess demand. Definition 12 Price Rule: Considering pB as numeraire, pA increases by ε in response of positive aggregate excess demand of apples and viceversa. It will remain constant for zero aggregate excess demand. ⎧ ¤ £ i P i i if EDA (t) ≡ Ii=1 ei∗ ⎪ A α , pA (t) , pB = 1, ω A (t) , ωB (t) > 0 ⎨ pA (t) + ε £ i ¤ P pA (t + 1) = α , pA (t) , pB = 1, ω iA (t) , ωiB (t) = 0 (13) if EDA (t) ≡ Ii=1 ei∗ pA (t) A ⎪ ¤ £ i P ⎩ i i if EDA (t) = Ii=1 ei∗ pA (t) − ε A α , pA (t) , pB = 1, ω A (t) , ωB (t) < 0 where
ε = f [EDA (t)] 13
Established the update rules, in the next section we show the implications of these rules on agent endowment progressive update. To visualize the effects of rationed trading versus non rationed trading we will focus on a single pair.
5
Rationed Pairwise Trading in a Fixed Pair: Example
In this section we focus the attention to what happens in a fixed pair to better understand simulation results. In the actual simulations the pair is not fixed, e.g. the next time every rationed agent will exchange his excesses in a new pair, where the new counterpart will presumably have another α`ı preference. Here, we stick with the same two counterparts to simplify the graphs. In a pair, excess demands depending on preferences, ´ current prices and endowments, are typically ³ αi 1−αj non symmetric for the two agents typically pA 6= pB . We are interested in observing where desired and realized allocation, as functions of prices and updated endowments, would go. As an example let’s consider a non symmetric pair where the two agents have preferences α = 0.3 and α = 0.8 respectively. The α = 0.8 agent starts with an endowment that is 0.5 of his revenue, therefore he wishes to buy 0.3 more of apples to reach his preference of 0.8 of his revenue in apples. On the other hand the α = 0.3 agent wants only 0.3 of his revenue in apples while he has 0.5 of his endowment in apples. He would like to sell just 0.2 of his revenue from apples. So the buyer would not be able to buy 30% of his revenue in apples, because the other only sells 20% of the same revenue (In the beginning, revenues are the same for everyone, given the assumptions). So he will not be able to get all the apples he would buy, nor sell all the bananas: he is rationed. He can’t reach any optimal allocation (tangency position), but only the position at the intersection between the indifference curve and the budget line, e.g. marginal rate of substitution MRS (t) = p (t) . On the contrary, the agent α = 0.3 would be able to sell all the apples he intends to and buy from him all the bananas he wants: the α = 0.3 is not rationed. By the assumed price rule (13), if excess demands will dominate at aggregate level, relative price will move up. Just for the sake of exposition, we photograph the trading positions at fixed instants expressed by def exogenously raising prices for apples p = pA /pB = 0.5, 0.75, 1. At these changing prices the "short" side trader with α = 0.3 is never rationed. In Fig. 4. we observe the evolution of his allocations. Starting at endowment ω, he will reach position 1 (at prices p = 0.5), position 2 (at prices p = 0.75), we then consider two different situations for the relative equilibrium price p = 1 : (i) allocation called W D (walrasian dynamic) as the tangency position at the budget line defined by equilibrium prices p∗ = 1 and current updated endowments, WD :
ui [x∗ (p∗ = 1, ωi,t )]
(ii) allocation called W S (walrasian static equilibrium ) following the strict definition of walrasian as tangency position at the budget line passing through initial endowments ωi = (10, 10) at equilibrium prices p∗ = 1. WS : ui [x∗ (p∗ = 1, ω i,0 )] Notice that at each price level, the non rationed agent is able to reach the position where his indifference curve is tangent to the current budget line (e.g. MRS (t) = p (t)) . Fig. 4. Short and Satisfied Side of the Trade: αi = 0.3 14
On the contrary the rationed trader α = 0.8 will never reach his preferred positions, at any price, not even at the theoretical equilibrium p∗ = 1. The reason is that his counterpart is always offering to trade "too small" quantities. The rationed agent therefore "never" reaches tangency positions, but he is forced to stop at intersection points between the budget line and the utility function (MRS (t) > p (t)) . Fig. 5. Long and Rationed Side of the Trade: αi = 0.8
What would it happen in a potential non-rationed behaviour? The situation is "potential" because it can only happen with symmetric counterparts. We draw it in Fig. 6. In the no-rationing environment, the agent-type α = 0.8, sinceby assumption will be able toi find his counterpart, he will immediately buy all apples he wants at the cheap price to reach position 1 and then, as the price would eventually move up, he will start selling apples, and become richer with higher utility. In general, when finding a compatible counterpart, he will systematically reach his optimal choices at the eventually moving prices p = 0.5, 0.75, 1. 15
Fig. 6. αi = 0.8 if NO Rationing
There are some remarks about the comparison between rationed and non-rationed situation for agent αi = 0.8. 1. The movement of the αi = 0.8 agent allocation in the rationed situation Fig. 5. is roughly mirroring the non rationed situation of the counterpart agent αi = 0.3 in Fig. 4. This is normal because within the pair, agent αi = 0.8 is buying just what αi = 0.3 is selling. On the other hand if αi = 0.8 would not be rationed, as mentioned above, he would immediately buy the big quantity of apples he wants, to reach position 1. Then as price would eventually move up, because of aggregate excess demand in the market, he will have a budget line pivoting in the updated endowment and find optimal to sell some of the apples, improving income and utility. Technically, his allocation moves north-west in the R2 plane. We will therefore see these two very different behaviours emerging in the full simulation: (a) the non-rationed one, where agents are "lucky" in finding a compatible counterpart. This behaviour will be characterized by reaching optimal allocations and therefore use price change to increase wealth and utility. This situation is quite likely for agents with intermediate preferences, e.g. agents trading more for price convenience than for tastes (αi ). This same situation is very unlikely for "extreme αi ” for two reasons: i. the probability to meet an agent with "extreme" αi of the opposite sign has a low probability2 ; ii. after the first run, most of the excesses are traded and only leftovers are back to be iteratively traded, and leftovers are, by construction, "small" and decrease in size as t increases. (b) the rationed one, when the counterpart trades less than desired quantities, and that is very likely for the same reasons (i) and (ii) above. 2
The formal probability calculus is developed in further work.
16
2. The "Static walrasian Equilibrium" depending on initial endowments and equilibrium prices attains: (a) a lower wealth and utility for the agent that has a strong preference for the initially cheap good, because buying his preferred good at a "cheaper than equilibrium" price, increases his wealth and utility. (b) a higher wealth and utility for the agent that has a low preference for the initially cheap good, because he actually wants to sell the good that is cheaper than at equilibrium. For the agent α = 0.8, the optimal position feasible from the initial endowment at equilibrium prices p∗ = 1 (static walrasian equilibrium, W S) (cyan colour) attains a lower wealth and a lower utility than at W D since he is a net buyer of the initially cheap good. The static equilibrium notion doesn’t allow the opportunity to buy apples at the initial "cheaper than equilibrium" price p < 1, but only at the equilibrium price, p∗ = 1. The satisfied side of the trade α = 0.3 improves utility during trading, but he would have been best in the static equilibrium since he is a net seller of the cheap good. Selling at the initial price p = 0.5 instead than at equilibrium price p∗ = 1 damages his wealth and utility. These two behaviours will emerge in the "complex" interaction shown in the simulations.
6
Sequential Pairwise Trading: Results
We finally present the simulations results for the N interacting agents. Price results can be classified in two symmetric groups, depending on whether the initial price is higher or lower than the sample walrasian price (p∗SW ≈ 1). Therefore, without loss of generality and for simplicity, we run two sets of simulations starting from two symmetric initially wrong prices pA /pB = 0.5 and pA /pB = 2.
6.1
Results: Prices
Let’s remind that different notions of prices are considered in this work: 1. the "sample static walrasian price" p∗SW computed at initial sample endowments, is not exactly equal to 1 because it depends on the sample mean α as in eq. (5) p∗SW (Ω0 ) =
α (1 − α)
2. the "price of the simulation"updated at every t by the Price Rule (13) , p (t) = pA (t)/pB (t), ∀t : (a) Initial price def
p0 = p (t = 0) =
(
0.5 2
(b) Price of the simulation induced by the aggregate excess demand, e.g. the price rule p (t) , as in eq. (13) , (c) Price of the simulation at convergence p (T ) .
17
3. "Dynamic walrasian price" at evolving endowments p∗DW (Ωt ) =
α · ωB t (1 − α) · ω A t
(14)
The sequential path of the different prices are described in figures 8, where the left figure shows paths starting from p(t = 0) = 0.5 and the right figure shows the symmetric paths. 1. Fig.8. Price Dynamics Price Evolutions from p(0) = 0.5
Price Evolutions from p(0) = 2
2
1.3 1.2
1.8 p(t)
1.1
pDW(t)
1.6 1
pSW(t) 0.9
1.4 p(t)
0.8
1.2
pDW(t) 0.7
pSW(t) 1
0.6
0.5
0
50
100
0.8
150
0
50
100
150
t
t
We observe the sequential simulation price updated by the excess demands of the economy by the price rule p(t) in eq. (13) : Result 3 It moves monotonically: excess demand maintains the same sign up to convergence. Result 4 It converges at the level of the walrasian price computed on updated endowments. I.e. if the initial relative price is lower than the static walrasian, it converges to a price higher than the static walrasian one and viceversa. If
p0 < p∗SW
=⇒
If
p0 = p∗SW
=⇒
If
p0 > p∗SW
=⇒
pT −→ pDW > p∗SW T pT −→ pDW = p∗SW T
(15)
pT −→ pDW < p∗SW T
Despite the apparently smooth convergence of excess demands, price results are not conventional. Something much more complex is happening in the agent interaction process. To understand these results we examine the "quantity" results.
6.2
Results: Allocation Redistribution
We here describe how agents allocations evolve during iterated pairwise trading. At initial price, net buyers, as defined in equations (7) will try to reach immediately their desired position on the budget line represented by the initial relative price. In just one round of trading they are able to get the majority of their desired allocation jumping from the fixed endowment point (10, 10) to a point on the budget line. It’s difficult to reach extreme positions though. 18
Trading with pA(t=0) = 0.5: Evolution of Agents Allocations during iterations 1:150
Fig. 9.2. pA = 0.5 - iter =1:3 20 18 16 14 12 10 8 6 4 2 0
18
16
16
14
14
12
12 xB
18
10 8
8
6
6
4
4
2
2
5
10
15 xA
20
10
15
20
25
10
25
0
30
0
5
10
15 xA
20
25
30
Fig. 9.5. pA(0)=0.5, pA(50)=1, iter1:50 20
Fig.9.6. pA(0)=0.5, pA(150)=1.26, iter 1:126 20
18 18
16 16
14 14
12 12
10
xB
xB
xB
20
0
5
Fig. 9.4. pA(0)=0.5 - pA(20)=0.7 - iter 1:20
Fig. 9.3. pA=0.5 - iter 1:6 20
0
0
8
10
8
6
6
4
4
2
2
0
0
5
10
15 xA
20
25
0
0
5
10
15 xA
20
25
Trading with p(t=0) = 2: Evolution of Agent Allocations during iterations 1:150
Fig. 10.2. p(0) = 2 Iter = 1:3 30
25
25
20
20
xB
xB
Fig. 10.1. p(0) = 2 Iter = 1 30
15
15
10
10
5
5
0
0
2
4
6
8
10 xA
12
14
16
18
0
20
0
2
4
6
8
10
12
14
16
18
20
16
18
20
xA
Fig. 10.3. p(0) = 2 Iter = 1:12
Fig. 10.4. p(0) = 2 Iter = 1:40
30
25
25 20
20
xB
xB
15 15
10 10
5 5
0
0
2
4
6
8
10 xA
12
14
16
18
20
0
0
2
4
6
Fig. 10.5. p(0) = 2 Iter = 1:100
8
10 xA
12
14
Fig. 10.6. p(0) = 2 Iter = 1:150
20
20
15
15
xB
25
xB
25
10
10
5
5
0
0
2
4
6
8
10
xA
12
14
16
18
20
0
0
2
4
6
8
10
xA
12
14
16
18
20
What we write is showed in Fig. 9 for initial price p = 0.5 and in Fig. 10. for initial price p = 2. We describe the situation for initial price p = 0.5. Agent αi = 0.9 needs the luck to meet agents with α`ı ≈ 0.1 able to sell him many apples to reach the desired position. Not many are so lucky. But in a few steps some succeed to attain extreme positions. The ideal line passing through initial endowment Ω with slope (−p(t)) becomes well drawn by the points representing agents endowments. As pA increases, progressively, only those that attain "optimal" allocations at current price are able to change side of the market (net buyers will become net sellers and viceversa). Since the satisfied agents are more likely to be those with weak preferences, e.g. "the speculators", we notice them to become richer and attain higher levels of income and utility. We start to see a decoupling of the budget line. The upper part gathers the successful traders, the below one gathers the rationed ones. Satisfied agents, rich in the good that appreciates, are better off as price increases. On the contrary, rationed agents, are unable to sell the depreciating good to buy the appreciating one and they just observe their income worsening and can’t trade to increase utility. Since agents are utility maximizers, none of them will trade, while the income, as result of price change, worsens. We summarize the results in these few points. 1. Market expresses excess demand for the cheaper good. Because of pairwise rationing, all the excess demand is not satisfied and it sticks in the market even when the price reaches walrasian prices. 2. The most persistent rationed agents are those with strong preferences for one of the goods (αi ≈ 1 for apples and αi ≈ 0 for bananas). They remain net buyers even at and over equilibrium price p∗SW and they are the responsible for price overreactions. 3. Satisfied net buyers, at further increasing prices will express net supply for the good they own, enjoying income effects and becoming richer in nominal wealth. Result 5 "Speculators are winners". The moral is that the most succeeding agents, i.e., those reaching more easily the optimal allocations, are those with weak preferences for the goods, being net buyers essentially for price reasons and not for preference reasons: we defined those as "speculators". Contrarily to common wisdom, speculators are here strongly rational, loving goods only for their price and not for "preferences", while price overreaction is caused by buyers with extreme preferences for a good. An easy interpretation for an "extreme" preference is the absolute need for housing, even when prices are irrationally high. In these cases an extreme preference, in the presence of very inconvenient prices may look obstinate. This obstination drives the irrational bubble.
6.3
Results: Welfare
We reconsider Fig. 2, representing utilities for each agent in the (α, u) plane. In Fig. 10 below we add agent "actual" utilities computed at final allocations and compare them with initial and final "potential" utilities. We remind the definitions of the utilities: 1. utilities at initial endowments represented by the flat line {ui [ω i,0 ]}N i=1 = 2.302,
19
∀i = 1, ..., N;
2. utilities evaluated at walrasian optimal allocation {ui [xi (p∗SW , ωi,0 )]}N i=1 represented by the regular parables already shown in section 3, having its lowest point at α = 0.5, because at p∗ = 1, as already said, the αi = 0.5 agent would have the endowment as optimal allocation; 3. utilities evaluated on the final endowments at convergence {ui [ω iT ]}N i=1
(16)
4. agents’ potential utilities at wrong initial prices, or "desired utilities" {ui [xi (ωi,0 , p0 = 0, 5 or 2)]}N i=1 .
(17)
Naturally at a cheap pA /pB net buyers are the majority. If they actually could buy so much of the cheap good, they would be sky happy. These points plot the upper parable. Unfortunately this allocation is unattainable since there wouldn’t be a compatible supply at that price. So these utilities could only be realized in a partial equilibrium setting and that’s why we call these just "potential utilities"; 5. agents’ potential utilities at convergence, or utilities evaluated at desired allocation at convergence (18) {ui [xi (pT , ωiT , )]}N i=1 . Result 6 At convergence, utility measured at updated endowment (eq.(16)), equals utility at desired optimal allocations, that is function of current prices and endowments (eq. (18)) : N {ui [ω iT ]}N i=1 = {ui [xi (pT , ω iT )]}i=1
Fig. 10. Utilities Comparisons Agents' Utilities: p = 0.5 0
3.4 ui (w0)
3.2
ui (w0)
3.2
ui [ x* ( p0*=1, w 0)] Walras Eq. ui pot [x*( p0=0.5, w0 )]
3
Agents' Utilities: p 0 = 2
3.4
ui [x*(p0= 0.5, w0)] Walr. Eq. ui pot [x*(p0=0.5, w0 )]
3
ui ( wT)
ui (wT)
u [ x* ( p , w ) ] i
2.8
T
T
2.6
2.6
2.4
2.4
2.2 0
0.1
0.2
0.3
0.4
0.5 alfa
ui [x* (pT, w T) ]
2.8
0.6
0.7
0.8
0.9
2.2 0
1
0.1
0.2
0.3
0.4
0.5
alfa
0.6
0.7
0.8
0.9
1
Let’s observe Fig. 10 and we only comment the case p0 = 0.5, because the two cases are perfectly symmetric. We know the flat line of utilities at initial endowments and the symmetric parable expressing utilities at the static walrasian equilibrium allocations. The asymmetric well signed parable expresses the "potential utility" at initial price. It is the set of points where agents would be, if desired excess 20
demand at initial mispricing is realized, as in eq. (17). Finally there is a cloud of points between the two parables, area delimited by the walrasian equilibrium and the potential unrealizable equilibrium, representing utilities at convergence that are both (16) and (18) induced by the pairwise sequential trading. • It is very clear that, in the case of p0 = 0.5, net sellers of apples, on the left hand side of the xaxis, are essentially all worse off than in the walrasian equilibrium: their utilities are all below the walrasian parable. In fact, they are selling part of the apple excess at a cheaper-than-equilibrium apple price. • Net buyers of apples, in the case of p0 = 0.5, have more dispersed utilities. All are better than in walrasian equilibrium, because they bought apple at a cheaper than walrasian equilibrium price. Some perfectly satisfied ones even attain the highest threshold of potential utility, but some rationed ones improve only partially their utility. The issue is: are the satisfied agents a sufficient number, able to compensate the rationed ones and the net sellers that are worse off? In Fig. 11 we show the results of computing this "gap measure" ∆ui (T ) = It shows quite a lot of evidence.
ui [ω iT ] − ui [xi (p∗SW , ωi,0 )] £ ¡ ¢¤ ui xi p∗SW , ω i,0
Result 7 Utility Divide: 1. In Fig.11.a., successful right hand side agents are not able to compensate the utility losses of the agents located on the left hand side, because they are not all located on the positive sloped line. The ones on the positive sloped line are the successful ones. We see that there are more successful agents on the central segment where our defined "speculators" are (symmetric reasoning holds for Fig. 11.b). 2. Buyers with strong preference for the mispriced cheap good, if not rationed, can achieve maximum utility improvement, since they may mostly profit of the income effects. Fig. 11. Sequential Trading Utilities at T : Improvements w.r.t. Walrasian Utilities Single Utilities %, p0 = 0.5
0.15
0.1
0.1
0.05
0.05
0
0
-0.05
-0.05
-0.1
Single Utilities %, p0 = 2
0.15
-0.1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
alfa
alfa
21
0.6
0.7
0.8
0.9
1
Let α∗ ∈ [0, 1] be defined by the intersection of the two utility parables such that: ui [xi (p∗SW , ω i,0 )] = ui [ωiT ] . In Fig. 11 we show how much final utility at convergence is larger than the optimal utility obtained if prices wouldbe the walrasian prices, ui [ω iT ] > ui [xi (p∗SW , ω i,0 )] in percentage of the latter. In the LHS graph, when initial price is lower for apples and higher for bananas, we see that consumers with low α, preferring bananas, the expensive good, are damaged with certainty in terms of utility. In the same graph we see that not all the consuners with high α, preferring the cheap apples, are able to enjoy the utility advantage. The reason is that "many" of them are rationed, in the sense that they are unable to reach their optimal allocation and therefore cannot enjoy the income effect. As a result, the advantaged consumers do not completely compensate for the disadvantaged ones. But, surprisingly enough, the aggregate loss is in the order of 0.5%, as proved in Miceli (2015).
7
Conclusions
This paper characterizes the out-of-equilibrium trading dynamics of a stylized, perfectly symmetric, walrasian pure exchange two-goods3 economy with N agents with uniformly distributed preferences and identical endowments for each good. Initial mispricing induces initial excess demand for the cheaper-than-equilibrium good and excess supply for the more-expensive-than-equilibrium good that are only partially satisfied by the sequential pairwise trading. This trading mechanism impedes full adjustment even at walrasian prices, because, within the pairs, excesses remain unsatisfied for rationed unlucky agents that iteratively do not meet counterparts with adequate quantities to exchange. These asymmetric excesses install in the economy, and some do not disappear even while the simulation passes through the theoretical walrasian equilibrium price and they instead push a price overreaction of opposite sign with respect to the initial mispricing, up to convergence. The relatively slower adjustment of quantities with respect to prices - pairwise trading is stickier than the auctioneer redistribution - induces more persistent price effects, leading to the “price overreactions”. For any set of initial prices, allowing sequential out-of-equilibrium pairwise exchange, the good having an initial relative price cheaper (more expensive) than the walrasian price will have at convergence a relative price higher (lower) than the walrasian (price) benchmark. The convergent "price and quantity" allocation is characterized by the classic orthogonality condition between the price vector and the budget line that interpolates the cloud central mass at these prices, but at redistributed quantities with respect to the walrasian allocation. The agents based simulation and the explicit decision to keep the model in two dimensions allows us to keep progressive endowment redistribution, single income, and single and aggregate utility under control. We can actually monitor the system dynamics and understand the interplay of substitution and income effects along the way. The model shows how pairwise out-of-equilibrium trading clusters agents into satisfied and unsatisfied ones, and how the unsatisfied ones push price overreaction The main results are: 1. Static Walrasian prices evaluated at initial endowment become meaningless, since the endowments evolve along the trading process. 3
General results for K goods confirming the thesis are exposed in a forthcoming paper by the same authors.
22
2. The winners are those able to buy their desired allocations at the cheaper-than-equilibrium price, so that as the cheap price increases, they will enjoy positive income effects on wealth and utility. Those are the "satisfied" agents, more likely among the ones desiring "small trades". Typically indifferent between the two goods (αi → 0.5) they choose to move just for price advantages ("speculators"). They take the largest improvements with respect to static walrasian allocations in terms of utility and income. The losers are consumers with strong preferences for ¢ ¡ the characteristic of the good αi → 0 or αi → 1 . When unable to get their desired choice because unlucky in finding the "right" counterpart, they persist demanding the good even at increasing price, pushing the price over walrasian levels. 3. Pairwise trading at non equilibrium prices, through rationing, (i) slows down price instability, (ii) allows convergence at updated endowments and (iii) gains more than 95% of Pareto optimal welfare, defined as total utility achievable at walrasian prices. 4. The more the agents and/or the more heterogeneity they represent, the faster convergence is, because finding a counterparts is easier. 5. With quite few ingredients, the present model builds a benchmark able to analyze the mispricing effects on resource allocations. Starting from perfect symmetry, the model is able to measure the asymmetry induced by mispricing. Concluding, in such a simple setting, this model is able to endogenously capture two important features of our economies. It is able to disclose and account for the size of price over-reaction due to an initial mispricing, thus sketching a possible basis for a price overreaction microfoundation. Moreover, the model is able to capture endogenous effects of "wealth divide" among a population, induced by whether agent trading is dominated by strict preference for the good or just by speculation around the price. The model proves that not being dominated by a strong preference (for example, necessity) and trading just for price reasons, gives the agent a strong advantage in fully exploiting income effects. To notice how misleading aggregate or average indicators can be, welfare negligible losses, in terms of aggregate or average figures, hide the fact that winners from out-of-equilibrium trading are less than one third of the population, but they win so much, to be able to almost counterbalance losses pertaining to more than two thirds of the population.
8
Appendix
h ³ ´ i Proof. of Result 2. Given that α ∼ U (a + b)/2; (b − a)2 /2 , the LLN says that ∀ε > 0 lim Pr ob (|αn − α| ≥ ε) = 0
N →∞
because by Bienaymé-Tchébichev Inequality Pr (|αn − E (α)| ≥ ε) ≤ Pr (|αn − E (α)| ≥ ε) ≤
(b − a)2 1 2 Nε2
Therefore 0 ≤ Pr (|αn − E (α)| ≥ ε) ≤ 23
V (αn ) nε2
(b − a)2 1 2 Nε2
then lim Pr (|αn − E (α)| ≥ ε) = 0
On the other way
N →∞
Pr (|αn − E (α)| < ε) = 1 − Pr (|αn − E (α)| ≥ ε) therefore lim Pr (|αn − E (α)| < ε) = 1
In particular
N →∞
E (α) = so that lim p∗SW =
N→∞
a+b 1 = 2 2
E (α) 1/2 = = p∗ = 1 E(1 − α) 1/2
The same result holds if, instead of increasing the number of agents, we increase the number of samples S −→ ∞, ¶ S µ X αs ∗ ∗ lim pSW = s /S = p = 1 S→∞ 1 − α s=1
References [1] Arrow, K. J.and G. Debreu, (1954) “Existence of an Equilibrium for a Competitive Economy", Econometrica 22(3):265—290. [2] Arrow, K. J., & Hahn, F. H. (1971). General Competitive Analysis, Oliver and Boyd, Edinburgh. [3] Benassy, J.P. (1982) "The Economics of Market Disequilibrium", (Vol. 6). New York, Academic Press. [4] Clower, R.W. and A. Leijonhufvud (1975) "The Coordination of Economic Activities: A Keynesian Perspective," American Economic Review. 65(2), pp. 182-188. [5] Debreu, G. (1959) “Theory of Value: an Axiomatic Analysis of Economic Equilibrium", New Haven. [6] Debreu, G and H. Scarf (1963) "A Limit Theorem on the Core of an Economy", International Economic Review, Vol. 4, No. 3 (Sep., 1963), pp. 235-246 [7] Feldman, A (1973) "Bilateral Trading Processe?, Pairwise Optimality, and Pareto Optimality," Review of Economic Studies, 40(1973), 463-473. [8] Fisher F. M. (1981). "Stability, Disequilibrium Awareness, and the Perception of New Opportunities",. Econometrica 49 (2): 279-317. [9] Fisher, F. M.(1989), "Disequilibrium Foundations of Equilibrium Economics", Cambridge, Cambridge University Press. [10] Foley, D. K. (1994) "A Statistical Equilibrium Theory of Markets" Journal of Economic Theory, 62(2), 321—345. 24
[11] Foley, D. K. (1999) “Statistical Equilibrium In Economics: Method, Interpretation, and an Example”. Paper presented at the XII Workshop on “General Equilibrium: Problems. Prospects and Alternatives” at the Certosa di Pontignano, Italy. [12] Gale D. (1976)."The Linear Exchange Model," Journal of Mathematical Economics, 3(1976), 205-209. [13] Gale D. (1986a). ‘Bargaining and Competition Part I: Characterization’, Econometrica 54:785806. [14] Gale, D. (1986b). ‘Bargaining and Competition Part II: Existence’, Econometrica 54:807-818. [15] Gale, D. (2000). Strategic Foundations of General Equilibrium,. Cambridge University Press. [16] Gale, D. and H. Sabourian (2005) "Complexity and Competition", Econometrica, Vol. 73(3):739769. [17] Gintis, H., (2006).“The Emergence of a Price System from Decentralized Bilateral Exchange", Contributions to Theoretical Economics 6,1,13. [18] Gintis, H. (2007).“The Dynamics of General Equilibrium”, Economic Journal 117:1289—1309. [19] Gintis, H. (2010). "The Dynamics of Generalized Market Exchange". Santa Fe Institute Working Paper. [20] Gintis, S. and A. Mandel (2012a). “The Stability of Walrasian General Equilibrium", WP, Paris I. [21] Gintis, S. and A. Mandel (2012b). “The Dynamics of Walrasian General Equilibrium: Theory and Application", WP, U Mass. [22] Ghosal, S. and M. Morelli (2004). “Retrading in Market Games”, Journal of Economic Theory 115:151-181. [23] Ghosal, S. and J. Porter (2010) "Decentralized Exchange, Out of Equilibrium Dynamics and Convergence to Efficiency", Warwick Economic Research Papers n. 928. [24] Goldman, S. M. and R. M. Starr (1982) "Pairwise, t-wise and Pareto Optimalities", Econometrica 50(3): 593-606. [25] Hahn, F.H (1970) "Some Adjustment Problems", Econometrica, Vol. 38, No. 1 (Jan), pp. 1-17. [26] Hahn, F.H. and T. Negishi (1962) "A Theorem on Non-Tâtonnement Stability" Econometrica, Vol. 30, No. 3 (July), pp. 463-469. [27] Kirman, A.P. (1992) “Whom or What does the Representative Individual Represent?”, Journal of Economic Perspectives 6: 117—136. [28] McLennan, A. and H. Sonnenschein (1991) "Sequential Bargaining as a Noncooperative Foundation for Walrasian Equilibrium". Econometrica 59(5): 1395-1424.
25
[29] Rader, T. (1964). "Edgeworth Exchange and General Economic Equilibrium", Yale Economic Essays, 4(1), 132-180. [30] Rader, T. (1968) "Pairwise Optimality and Non-Competitive Behavior," in Papers in Quantitative Economics, Ed. by J. P. Quirk and A. M. Zarley. Lawrence, Kansas: University Press of Kansas, 1968. [31] Rader, T. (1976). Pairwise Optimality, Multilateral Efficiency and Optimality, with and without Externalities. Theory and Measurement of Economic Externalities, Academic Press, New York. [32] Rubinstein, A. and A. Wolinsky (1985) "Equilibrium in a Market with Sequential Bargaining", Econometrica, Vol. 53(5):1133-1150. [33] Scarf, H. (1960) "Some Examples of Global Instability of the Competitive Equilibrium", International Economic Review, Vol. 1, No. 3 (Sep., 1960), pp. 157-172 [34] Scarf, H. (1971) "On the existence of a cooperative solution for a general class of n-person games", Journal of Economic Theory, vol 3(2), June, pp. 169—181. [35] Shapley, L.and H.Scarf (1974) "On cores and indivisibility" Journal of Mathematical Economics, Volume 1, Issue 1, March, 23-37. [36] Shapley, L. and M. Shubik (1977) "Trade Using One Commodity as a Means of Payment", Journal of Political Economy, Vol. 85, No. 5 (Oct.), pp. 937-968. [37] Walras, L. (1874) Elements of Pure Economics, London.
26
