Tatonnement Beyond Gross Substitutes? Gradient Descent to the Rescue

Transcription

1 Tatonnement Beyond Gross Substitutes? Gradient Descent to the Rescue Yun Kuen Cheung Richard Coe Nihi Devanur Abstract Tatonnement is a simpe and natura rue for updating prices in Exchange (Arrow- Debreu) marets In this paper we define a cass of marets for which tatonnement is equivaent to gradient descent This is the cass of marets for which thers a convex potentia function whose gradient is aways equa to the negative of the excess demand We ca this cass the Convex Potentia Function (CPF) marets We show the foowing resuts CPF marets contain the cass of Eisenberg Gae (EG) marets, defined previousy by Jain and Vazirani The subcass of CPF marets for which the demand is a differentiabe function contains exacty those marets whose demand function has a symmetric negative semi-definite Jacobian We define a famiy of continuous versions of tatonnement based on gradient descent using a Bregman divergence As we show, for many CPF marets, every process in this famiy wi converge to an equiibrium and the process based on KL-divergence wi converge for even more of these marets This is anaogous to the cassic resut for marets satisfying the Wea Gross Substitutes property A discrete version of tatonnement converges toward the equiibrium for the foowing marets of compementary goods; its convergence rate for these settings is anayzed using a common potentia function Fisher marets in which a buyers have Leontief utiities; these are a EG marets (Fisher marets are marets in which a agents are either buyers or seers) The tatonnement process reduces the distance to the equiibrium, as measured by the potentia function, to an ɛ fraction of its initia vaun O(1/ɛ) rounds of price updates, and require Ω(1/ ɛ) rounds in the worst case Fisher marets in which a buyers have compementary CES utiities; again, these are a EG marets Here, the distance to the equiibrium is reduced to an ɛ fraction of its initia vaun O(og(1/ɛ)) rounds of price updates This shows that tatonnement converges for the entire range of Fisher marets when buyers have compementary CES utiities, in contrast to prior wor, which coud anayze ony the substitutes range, together with a sma portion of the compementary range Courant Institute, New Yor University Courant Institute, New Yor University Microsoft Research Redmond 1

2 1 Introduction Genera equiibrium theory, the study of marets and their equiibria, has been a core topic in economics for over a century Informay speaing, an equiibrium is a coection of prices at which the suppy and demand of every good in the maret baances Two centra questions in genera equiibrium theory are whether equiibria exist and if so how to compute them The issue of existence was setted for a very genera setting in 1954 by Arrow and Debreu [2] by means of Katutani s fixed point theorem However, the computation of equiibria was aready being wored on in the 1890s by Fisher, who buit a hydrauic apparatus for this tas (see [6] for a description) The forma study of this topic began even earier with thntroduction of an equiibrium mode by Waras in 1874 [30], aong with a natura, simpe, distributed price update process which he named tatonnement Tatonnement is broady defined in terms of the foowing criteria: if the demand for a good is more than the suppy, increase the price of the good, and conversey, decrease the price when the demand is ess than the suppy The price adustment for each good is in the direction of its own excess demand and is independent of the demand for other goods Cassicay, tatonnement has been thought of as a continuous process, with price adustments and demand responses happening continuousy and instantaneousy A computer science approach is to consider updates at discrete timntervas and to bound the number of updates required (though discrete updates were aso considered in the economics iterature as eary as the 60s [29]) Impicit in the question of how to compute equiibria is the assumption that marets do reach these equiibria, which raises the further question of how they reach them Tatonnement is a natura candidate for such a process An eary positive resut, due to Arrow, Boc and Hurwitz [1], showed that a continuous version of tatonnement converges to an equiibrium for marets satisfying the wea gross substitutes (WGS) property, namey that increasing the price of one good does not decrease the demand for any other good 1 However, the hope that tatonnement woud converge for a marets was dashed by Scarf [27], who showed an exampe of a maret where tatonnement does not converge; in fact, it exhibits cycic behavior Thus one can hope to show that tatonnement converges ony for specific casses of marets The question of how the marets might reach equiibria encompasses many issues For exampe, one might as whether strategic behavior of the maret participants is accounted for This has been considered esewhere, and it is nown that in thmit, as participants or agents becomnfinitesima, their strategic power becomes negigibe [26] There have aso been some preiminary attempts to quantify this effect in specific marets [9, 8] One might aso as whether changing suppy in addition to prices is an option This can reay be seen as adding production to the mode, a thoroughy studied topic in genera equiibrium theory Utimatey, one woud ie to obtain a-encompassing resuts, but in this paper we wi focus soey on price adustment, and on tatonnement in particuar We note that tatonnement is often considered to be an agorithmic process and not a maret process; for exampe, [16] states: such a mode of price adustment describes nobody s actua behavior (referring to the cassic auctioneer expanation of tatonnement) The main point of [16], however, is to give an aternate and more pausibe basis for tatonnement More recenty, Coe and Feischer [12] gave another pausibe basis for tatonnement by introducing the Ongoing Maret mode, in which tatonnement and other price update processes can naturay 1 The (strict) gross substitutes property says that increasing the price of one good stricty increases the demand for any other good 2

3 be viewed as in-maret processes The continued interest in the pausibiity of tatonnement is aso refected in some experiments by Hirota [19] which show the predictive accuracy of tatonnement in a non-equiibrium trade setting [19] Returning to the question of convergence, Coe and Feischer [12] showed that a discrete version of tatonnement converges quite quicy, for a cass of marets satisfying the wea gross substitutes property [12] The current paper is motivated by the quest for other broad casses of marets for which the same hods true Of particuar interest are marets that exhibit compementarity, such as the Constant Easticity of Substitution (CES) utiities and Leontief utiities This wor wi argey focus on Fisher marets, marets in which the participants or agents can be partitioned into buyers and seers 2 (See Section 2 for forma definitions) For the WGS case, the existing resuts a rey on very strong properties of WGS marets that are naturay hepfu in showing that tatonnement converges One exampe of such a property is that for Fisher marets the extreme prices aways movn, ie, the bound on the ratio of the current price to the equiibrium prics guaranteed to shrin [12] Another exampe, again for Fisher marets, is that the equiibrium can be reached by starting with very sma prices and monotonicay increasing them It is easy to show that such strong properties cease to hod in the compementary regime Therefore new techniques are needed to hande such marets In this paper we reate the tatonnement process to another simpe and natura process: gradient descent Gradient descent is a famiy of agorithms used to minimize convex functions It wors by starting at some point and moving in the direction of the negative of the gradient We consider the cass of marets for which the tatonnement process is formay equivaent to performing gradient descent on a convex function In particuar, we define the cass of Convex Potentia Function (CPF) marets to be those marets for which thers a convex potentia function whose gradient 3 is aways equa to the negative of the excess demand We show that this cass contains the cass of Eisenberg-Gae (EG) marets introduced by Jain and Vazirani [20] The subcass of CPF marets for which the demand is differentiabe can be characterized in terms of the Jacobian 4 of the demand function These are exacty the marets for which the Jacobian of the demand function is aways symmetric and negative semi-definite 5 We ca this cass the Convex Conservative Vector Fied (CCVF) marets, since functions that have a symmetric Jacobian are caed conservative vector fieds The aforementioned CES and Leontief utiities aong with many other interesting marets (in the Fisher maret mode) are contained in thntersection of EG marets and CCVF marets The equivaence with gradient descent opens up the entire too box deveoped to anayze gradient descent and provides a principed approach to show convergence of the tatonnement process For a arge cass of CPF marets, we show that a continuous version of tatonnement converges to an equiibrium For the specia cases of CES and Leontief utiities, we show stronger con- 2 The term Fisher maret was coined by the computer science community to refer to this cass of marets, which were the marets defined by Fisher and experimented with in his aready mentioned computationa wor in the 1890s 3 More generay, the potentia function need not be differentiabe and the demand need not be unique, in which case the equivaencs between the subgradient of the potentia function and the set of excess demand vectors 4 Reca that the Jacobian of a differentiabe function from R n to R n is the matrix whose (i, ) entry is the rate of change of th th component of the function with respect to a changn the th coordinate 5 By contrast, if the off-diagona entries of the Jacobian are a positive, then the maret satisfies the wea gross substitutes property 3

4 vergence resuts by proving certain structura properties of the corresponding convex functions for these marets We now summarize the main contributions of the paper The cass of Eisenberg-Gae (EG) marets contains a Fisher marets for which the equiibrium aocation is captured by a certain type of convex program caed the Eisenberg- Gae-type (EG-type) convex program We show that EG marets are CPF marets by expicity constructing a convex potentia function (Theorem 4) In fact, the potentia function is the obective function of the dua of the corresponding EG-type convex program We show that a famiy of continuous versions of the tatonnement process converges to the equiibrium for a arge cass of CPF marets This famiy is derived by considering gradient descent with respect to any Bregman divergence and taing thmit as the step size goes to zero (Theorem 22) In addition, the process based on KL-divergence (a particuar Bregman divergence) converges for an even arger cass of CPF marets This mirrors the cassic resut of [1] which shows a simiar resut for gross substitutes marets For Leontief utiities, we show a a fairy fast rate of convergence for a discrete version of the process, namey, the number of time steps required to reduce the distance from the equiibrium to an ɛ fraction of its initia vaue, as measured by the potentia function, is O(1/ɛ) (Theorem 25) 6 This foows from a sma modification of a genera resut of [7] that shows convergence of gradient descent with Bregman divergences at this rate whenever the convex function satisfies a certain sandwiching property 7 We show that the potentia function in this case satisfies this sandwiching property for an appropriate choice of parameters with respect to the KL-divergence We aso show that, in the worst case, tatonnement uses Ω(1/ ɛ) iterations with Leontief utiities Consequenty, thnear bounds achieved for CES utiities (see beow) cannot extend to Leontief utiities For CES utiities we show a inear convergence, that is, the number of time steps required to reduce the distance from the equiibrium to an ɛ fraction of its initia vaue, again as measured by the potentia function, is O(og(1/ɛ)) (Theorem 35) This is obtained by showing that the potentia function in this case satisfies a stronger sandwiching property This stronger property is reminiscent of strong convexity but to the best of our nowedge, this particuar property has not been used before We aso note that when reasonaby near to equiibrium, the potentia function has vaue Θ( z2 p ), where z is the excess demand for good and p is its price (Lemmas 32 and 33) In addition, we show that this anaysis handes CES utiities that are substitutes, thereby providing an aternate anaysis for the resuts in [12] Reated wor The stabiity of the tatonnement process has been considered to be one of the most fundamenta issues in genera equiibrium theory, and the textboo of Mas-Coe, Whinston and Green [24] contains a good summary of the cassic resuts 6 The O() hides maret dependent parameters 7 Actuay we observe that a sighty weaer version of the property suffices 4

5 More recenty, discrete versions of tatonnement have been studied Codenotti et a [11] consider a tatonnement-ie process that required some coordination among different goods and showed poynomia time convergence for WGS marets Coe and Feischer [12] were the first to estabish fast convergence for a truy distributed discrete version of tatonnement, once again for a cass of WGS marets Cheung, Coe and Rastogi [10] extended this resut sighty beyond WGS marets, to CES utiities for a imited range of parameters 8 In comparison, our resuts cover the entire range of parameters for CES utiities Feischer et a [17] aso consider price dynamics that are simiar to tatonnement but they aso need coordination and further, their resuts concern the average price throughout the process rather than convergence of the sequence In a simiar spirit to this paper, Birnbaum, Devanur and Xiao [4] considered another distributed process caed the Proportiona Response (PR) dynamics for thnear utiities case, showed its equivaence to gradient descent with KL-divergence for a different convex function and obtained convergence rates for the process The PR dynamics wors in the space of offers rather than the space of prices, which is why the corresponding convex function is different For inear utiities, the PR dynamics are more appropriate than tatonnement, especiay since the demand function is not continuous [4] prove a certain convergence resut (Theorem 1) which we usn this paper to show convergence for the case of Leontief utiities EG marets were defined by Jain and Vazirani [20], after observing that many marets in the Fisher mode had simiar convex programs that captured the equiibrium The foowing is a brief ist of such marets: Eisenberg and Gae [15] gave a convex program for thnear utiities case, generaized by Eisenberg [14] to the case of homothetic utiities, Jain et a [21] for homothetic utiities with production, and Key and Vazirani [23] for certain networ-fow marets Jain and Vazirani [20] showed many agorithmic and structura properties of such marets 2 Preiminaries A Warasian maret mode has m divisibe goods and n agents Each agent i has a utiity function u i : R m + R that specifies the agent s utiity for a given bunde of goods Each agent i has an initia endowment of amount of good The suppy of good, w := i is the tota endowment of good among a the agents Wog we choose the units of measurement such that the suppies are a 1 Suppose we assign a price p to each good, then the demand of agent i is a bunde of goods (x i1, x i2,, x im ) that maximizes her utiity subect to the budget constraint, namey that she does not spend more than the vaue of her endowment It is the soution to the foowing optimization probem: maximize u i (x i1, x i2,, x im ) st p x i p,, x i 0 If the utiity function is stricty concave, then thers a unique utiity maximizing bunde when the prices are a positive, so we can ta of the demand of an agent The maret demand for 8 CES utiities are parameterized by an exponent, ρ When 0 < ρ 1 the maret is WGS, and when ρ < 0 the goods are compementary [10] anayzed the range 1 < ρ 0 5

6 a good is x = i x i, the tota demand for that good This is viewed as a function of the price vector p = (p 1, p 2,, p m ) A price p is an equiibrium pricf the maret cears, that is for a, x = w = 1 For notationa convenience, we define the excess demand for good as z = x 1 The equiibrium condition is that every excess demand be zero It is nown that equiibrium prices exist if the utiity functions are a stricty concave An aternate mods the Fisher maret mode, where thers a fixed endogenous suppy of each good (which is again chosen to be 1 unit) The agents have a fixed endowment of money, which defines their budget constraint Let the endowment of agent i be units of money The budget constraint for agent i is p x i The Fisher mods actuay a specia case of the exchange mode We now define somnteresting subcasses of marets A maret satisfies the Wea Gross Substitutes (WGS) property or equivaenty a maret is a WGS maret if increasing the price of any one good does not reduce the demand for any other good If the demand function is continuous and differentiabe, then this property can be written as x p 0, In terms of the Jacobian of the demand function, in a WGS maret a the off-diagona entries are non-negative The Leontief utiities are of the form u i = min {x i /b i } One needs b i units of good, for each good, in order to get one unit of utiity Thus Leontief utiities capture the case of perfect compements It is easy to see that the demand for good is x i = β i b i, where β i = / b i p (1) Thus the maximum utiity buyer i can obtain is u i = / b i p (2) Utiities with a Constant Easticity of Substitution (CES) utiities, are of the form u i = (a i1 x ρ i 1 + a i2 x ρ i a im x ρ i m) 1/ρ i (3) with ρ i 1 and a i 0 If 0 < ρ i 1 then the goods are substitutes; the goods are compementary when ρ i < 0 Leontief utiities are obtained in thmit, as ρ The utiity function obtained in thmit, as ρ 0, is caed the Cobb-Dougas utiity An Eisenberg-Gae-type convex program is a convex program of the form maximize og u i (x i1, x i2,, x im ) i st, x i 1, (suppy constraints) i i,, x i 0 6

7 The base of the og does not matter for the maximization in the convex program However, ater in the paper some cacuations are simpified if we assume that the natura ogarithm is intended, and so we assume this henceforth We note that the above program satisfies Sater s conditions for strong duaity (see [5], p 226, for exampe) and consequenty an optima soution to the dua probem yieds the same optimizing vaue as the prima program An Eisenberg-Gae (EG) maret is a Fisher maret for which the optima soution and the (corresponding) Lagrange mutipiers of the suppy constraints in the above convex program are respectivey equiibrium demands and prices for the maret Conversey, equiibrium demands and prices are respectivey an optima soution and Lagrange mutipiers of the suppy constraints to the above convex program Note that any stricty monotone transformation of the utiity function eaves the maret unchanged, since the demand function is invariant under such transformations Thus one may need to appy suitabe monotone transformations to the utiity functions in order to obtain an EG maret It is nown that buyers with Leontief and CES utiities in the Fisher mode form EG marets We next present a generaized version of gradient descent and a convergence resut for this version For any stricty convex differentiabe function h, the Bregman divergence with erne h is defined as d h (p, q) = h(p) h(q) h(q) (p q) (4) For exampe, the square of the Eucidean distancs obtained as a Bregman divergence, p q 2 = d h (p, q), if h(p) = 1 2 p 2 Another we-nown examps the KL-divergence, p og p, which is obtained when h(p) = p og p p (5) For a convex function φ, define the tangent hyperpane at a given point q, thought of as a inear approximation to the function, as φ (p; q) = φ(q) + φ(q) (p q), where φ(q) denotes an arbitrary subgradient of φ at q The generaized gradient descent wrt a Bregman divergence d h on the convex function φ is a sequence p 0, p 1,, p t, defined inductivey (for any given starting point p 0 ) by p t+1 = arg min p { φ (p; p t ) + d h (p, p t )} (6) Note that if the subgradient is not unique, then this sequence need not be unique either For the quadratic erne, h(p) = 1 2 p 2, the above update rue reduces to the usua gradient descent rue: p t+1 = p t φ(p t ) If the erns the weighted entropy, h(p) = γ (p og p p ) for some weights γ, the update rus ( ) p t+1 = p t φ(p exp t ) γ, for a (7) Birnbaum, Devanur and Xiao [4] showed the foowing convergence resut for gradient descent (6) 7

8 Theorem 1 ([4]) Suppose that the convex function φ and the erne h satisfy: for a p, q, Let p be a minimizer of φ Then for a t, φ(p) φ (p; q) + d h (p, q) (8) φ(p t ) φ(p ) d h(p, p 0 ) t We need a sighty more genera version of this theorem where we require (8) to hod ony for consecutive pairs p t, p t+1 for a t, instead of requiring it for a pairs p, q It is easy to see that their proof needs ony this weaer condition, yieding the foowing theorem Theorem 2 Suppose that the sequence of prices p t obey the foowing condition: Let p be a minimizer of φ Then for a t, φ(p t+1 ) φ (p t ; p t+1 ) + d h (p t, p t+1 ) (9) φ(p t ) φ(p ) d h(p, p 0 ) t The discrete version of the tatonnement process we consider wi be equivaent to the gradient descent (6) where h is the weighted entropy function, ie, the update (7) for a suitabe choice of weights γ The potentia function φ wi satisfy φ = z The continuous versions we consider are obtained by introducing a mutipier 1/ɛ to the divergence term d h and taing the imit as ɛ 0 This wi be presented in more detai in Section 4 21 New Definitions We now define the new casses of marets being introduced in this paper A maret is said to be a Convex Potentia Function (CPF) maret if thers a convex potentia function φ of the prices such that for a prices p, φ(p) = z(p) By abuse of notation, we et φ denote the set of sub-gradients when φ is not differentiabe 9 and we et z(p) denote the set of excess demand vectors when the demand is not unique The subcass of CPF marets for which the demand function is differentiabs caed the Convex Conservative Vector Fied (CCVF) marets The foowing characterization of CCVF marets foows essentiay immediatey from Green s Theorem [18, 25] Lemma 3 A maret with a differentiabe demand function is CCVF if and ony if the Jacobian of its demand function is aways a negative semi-definite symmetric matrix Proof For a CCVF maret, the potentia function satisfies φ(p) = z(p) As x(p) and hence z(p) are differentiabe, it is now easy to chec that the Jacobian is symmetric Negative semi-definiteness foows because the potentia function φ associated with the CCVF maret is convex, and hence the Jacobian of z(p) is positive semi-definite If the Jacobian of x(p) is symmetric, by Green s Theorem [18, 25], thers a function f : R n R such that f = x Let φ = p f(p) Then φ(p) = 1 x(p) = z(p) φ(p) is convex as its Jacobian is positive semi-definite, and as φ(p) = z(p), it foows that the maret is a CPF maret with a differentiabe demand, it is a CCVF maret Marets with Leontief utiities and those with CES utiities are both CCVF marets contrast, marets with inear additive utiities are not CCVF 9 We assume throughout that φ is continuous By 8

9 3 EG marets In this section we prove the foowing theorem Theorem 4 A EG marets are CPF marets The proof is by an expicit construction of a convex potentia function φ for which φ(p) = z(p) φ is actuay the dua of the corresponding EG-type convex program Reca that the EG-type convex program has variabes x i for a i and We et X denote the set of a these variabes Aso reca that the optimum soution gives the equiibrium aocation and the optima Lagrangian mutipiers of the suppy constraints in the program are the equiibrium prices The KKT conditions characterize the optima soution to a convex program and the corresponding Lagrange mutipiers We now write the KKT conditions in terms of the Lagrangian function, which is obtained by mutipying the suppy constraints by the prices and adding them to the obective function L(X, p) := i og(u i ) i, p x i + p 1, on the domain {X, p: i,, x i 0;, p 0} X and p are said to satisfy the KKT conditions if 1 X arg max X 0 L(X, p ) and 2 p arg min p 0 L(X, p), which is equivaent to for a, p (1 i x i) = 0 (10) We define the potentia function to be the dua obective of the EG-type convex program φ(p) := max L(X, p) X 0 φ is convex by construction Theorem 4 foows by showing that the gradient of φ is equa to the negative of the excess demand (Lemma 6) However, the ey property of EG marets is captured by the foowing emma Lemma 5 For an EG maret, for a p, the demand set x(p) is exacty equa to arg max X 0 L(X, p), whenever they are both finite Proof Part 1, x(p) arg max X 0 L(X, p): We first argue that if x(p) is the demand at price p then it must aso maximize L(X, p) In fact, we first argut for the specia case when the price and the demand form an equiibrium, denoted by p and x(p ) Since this is an EG maret, by definition of an EG maret, the pair (p, x(p )) must correspond to an optima soution of the corresponding convex program They must therefore satisfy the corresponding KKT conditions (10), which impy that x(p ) arg max X 0 L(X, p ) as desired This immediatey shows the same for any price p and every demand x(p), since the pair forms an equiibrium when the suppy is equa to x(p) Thus the above hods for a prices and for a demand vectors Part 2, arg max X 0 L(X, p) x(p): The argument is simiar to Part 1 Consider any p and an X that maximize L(X, p) Consider the maret instance with suppy equa to i x i for 9

10 good Note that the KKT conditions (10) are then satisfied with p and X for this instance and therefore they form an optima soution to the corresponding EG-type convex program Since any optima soution to the convex program must aso be an equiibrium, it foows that X must be a demand at price p as desired In fact it is easy to see that the converse of Lemma 5 is aso true, that if for a p the demand set is equa to arg max X 0 L(X, p) then the maret is an EG maret The KKT conditions (10) are then exacty the same as the equiibrium conditions Lemma 6 φ(p) = 1 x(p) = z(p) Proof It is we nown that if a convex function is defined as the maximum of many inear functions then the gradient is given by the gradient of thnear function providing this maximum φ is indeed defined in this way and by Lemma 5 the arg max es are given by the demands, or in other words the maximizing inear function L(X, p) is the one defined using the demands Thus φ(p) = 1 x(p) = z(p) The foowing convenient form for φ(p) was shown in [13], and wi be used in the anayses of the marets with Leontief and CES utiities Lemma 7 For EG marets with inear, CES or Leontief utiities (and others) the dua obective can be written as φ(p) = p i og(ν i ) + a constant independent of p where ν i is the ratio of to the optima utiity of i at price p, ie the minimum cost for obtaining one unit of utiity Proof Reca that { φ(p) := max L(X, p) = max p + X 0 X 0 i og u i (x i ) i, p x i }, where x i denotes the demands of buyer i From Lemma 5, for each i, an x i in the arg max above is buyer i s demand for good and therefore p x i must be equa to Hence i, p x i = i is a constant We aso rewrite i og u i (x i ) = i og[ /u i (x i )] + i og ; then setting ν i = /u i (x i ) gives φ in the desired form 4 Convergence of Continuous Time Tatonnement A continuous version of tatonnement is a traectory in the price space which, to be notationay consistent with the discrete version, is denoted by p t for a t R + Cassicay, the traectory is defined by specifying a differentia equation dp = F (t, p(t)) for a t, which we aso ca the dt update rue We define a famiy of update rues derived from gradient descent As before, 10

11 et h be a stricty convex differentiabe function The natura way to specify the differentia equation is p(ɛ) := arg min p { φ(p t ) (p p t ) + 1 ɛ d h(p; p t ) } dp dt p (ɛ) p t := im ɛ 0 ɛ However, there are thressues we need to address with this specification The first issus that in the marets we consider, the demand function of an agent can be muti-vaued at a price vector 10, and hence φ(p t ) can aso be a set of mutipe eements, namey the set of subgradients of φ at p t Since φ(p t ) can be muti-vaued, p(ɛ) and hence dp can be too To dt resove this, we need the notion of differentia incusion, which is a generaization of differentia equations In brief, a differentia incusion is a system which aows dp to tae any vaue from dt a set We specify our cass of differentia incusions in the domain R n +, as foows: { p t ( v, ɛ) := arg min v (p p t ) + 1d p ɛ h(p; p t ) } (11) { } F (p t p t ( v, ɛ) p t ) := im ɛ 0 ɛ v φ(pt ) (12) dp dt : F ( p t) (13) The existence of a soution to (13) requires F to be non-empty, convex, compact and upper semi-continuous (We wi give precise definitions and state the reevant resuts in Section 41; we refer the readers to Smirnov s text [28] for more detai on this topic) In fact, going to set-vaued maps aso heps us hande some discontinuities, since upper semi-continuity for set-vaued maps is in a sense a weaer requirement than continuity for functions The second issus reated to the requirement that F be non-empty and compact Lemma 11 p( v,ɛ) p shows that if φ is finite and bounded, then im ɛ 0 exists and is bounded The main ɛ difficuty in showing φ is bounded occurs when one of the prices tends to zero This is aso reated to the next issue Prices tending to create a simiar difficuty The third issus that we do not aow prices to be negative This imposes a boundary on the price domain Cassicay, existence theorems for differentia incusions/equations guarantee the existence of a soution up to the boundary, ie a soution may ony be guaranteed for a finite time span Yet we want goba existence, ie a soution for t [0, + ) To hep resove this, we wi extend F (p) to the negative price domain so as to remove the boundary whie ensuring that any soution remains in R n + As we mentioned, the second and the third issues are connected to the main impediment to proving the convergence of tatonnement, which is the possibe presence of zero-vaued prices on the tatonnement path If we are using a rue such as the mutipicative update rue, dpt dt = z t p t, the price p wi not changf equa to zero, precuding convergence to an equiibrium with p 0 One way to avoid this difficuty is to imit the update rue so as to ensure that if a price starts out positive, it wi remain positive Note that this does not precude a price converging to zero as t, if that is its vaue at equiibrium In this case, the price domain boundary is never reached 10 An exampe: if a buyer has utiity function u(x 1, x 2 ) = x 1 + 3x 2 and budget 40, then at prices (p 1, p 2 ) = (2, 6), the buyer optimizes her utiity by purchasing (x 1, x 2 ) = (20 3y, y), for any y [0, 20/3] 11

12 By contrast, if the price update rus additive, eg dpt = z t dt, a price might tae on a zero vaue despite being positivnitiay However, this wi sti be viabe so ong as the tatonnement avoids price vectors p with p = 0 and z = { }, which we ca unbounded demand price vectors As we sha see, for inear, CES and Leontief Fisher marets, if they start from a nonunbounded (ie from a bounded) demand price vector, they wi reach ony bounded demand price vectors, regardess of which tatonnement rus used In this case, we do need to ensure that the price domain boundary is not crossed A soution to the differentia incusion (13) coud see to eave the domain R n + if F (p t ) contains a negative vaue when p t = 0, as it may 11 But, when p t = 0, F (p t ) can aways be made to contain items with z 0, providing the hope of a soution that remains in R n + To this end, we observe that at a price vector p with p = 0, as good is free, an agent may purchase an infinite amount of good even if it does not increase her utiity We wi use this freedom of being abe to purchase additiona quantities of zero-priced goods to modify the definition of F so that it is non-empty, compact and incudes non-negative excess demands for the goods with zero prices In addition, we wi extend the domain of F to a of R n in such a way that the ony soutions are those with prices that stay in the domain R n + 41 Differentia Incusion and Semi-Continuity of Sets Definition 1 A differentia incusion is an equation of the form dp F (t, p(t)), where F (t, p) dt is a non-empty set for a t and p This generaizes standard differentia equations of the form = f(t, p(t)), where f(t, p) is singe-vaued dp dt In our setting, F is a function of p aone Let P(A) denote the power set of the set A Let Ω(a) denote an open neighborhood of a point a Definition 2 A set-vaued map F : Z P(Y ) is upper semi-continuous at z 0 Z if for any open set M P(Y ) which contains F (z 0 ), there exists Ω(z 0 ) such that for a z Ω(z 0 ), F (z) M A set-vaued map F is upper semi-continuous if it is so at every z 0 Z A set-vaued map F : Z P(Y ) is ower semi-continuous at z 0 Z if for any y 0 F (z 0 ) and any neighborhood Ω(y 0 ), there exists a neighborhood Ω(z 0 ) such that for a z Ω(z 0 ), F (z) Ω(y 0 ) A set-vaued map F is ower semi-continuous if it is so at every point z 0 Z A set-vaued map F : Z P(Y ) is continuous at z 0 Z if it is both upper and ower semicontinuous at z 0 A set-vaued map F is continuous if it is so at every z 0 Z For any sets A 1, A 2,, A, et their sumset be basic facts, which wi be usefu ater { i=1 a i a i A i } We state the foowing Lemma 8 (a) If A 1, A 2,, A are convex and compact, then their sumset is convex and compact (b) If A 1, A 2,, A : Z P(Y ) are upper semi-continuous at z Z, then their sumset is upper semi-continuous at z 11 An exampe: in a Leontief Fisher maret, it is possibe that at an equiibrium some p = 0 with a negative excess demand for good 12

13 (c) If F 1, F 2 : Z P(Y ) are two set-vaued maps which are upper semi-continuous at z Z, the map F : Z P(Y ), defined as F (z) = F 1 (z) F 2 (z), is aso upper semi-continuous at z Z The foowing Maximum Theorem is we-nown in mathematica economics It provides resuts on set-vaued map semi-continuity, which are among the required conditions for the existence of a soution to our differentia incusions Theorem 9 (Maximum Theorem, [3, p 116]) Let u : P X R be a continuous function, and C : P P(X) be a compact set-vaued map Let C (p) = arg max x C(p) u(p, x) and u (p) = max x C(p) u(p, x) If C is continuous at some p, then u is continuous at p and C is non-empty, compact and upper semi-continuous at p Let B(p 0, ρ) denote the cosed ba around p 0 with radius ρ Theorem 10 ([28, p ]) Let dp F (p(t)) be a differentia incusion, where F : P dt P(R) is upper semi-continous at every p B(p 0, ρ) for some ρ > 0 Suppose that F (p ) is convex and compact for every p B(p 0, ρ), and there exists a finite κ such that sup z F (p ) z κ for every p B(p 0, ρ) Then for 0 t ρ/κ, there exists an absoutey continuous soution p(t) to the differentia incusion with p(0) = p 0 42 Existence of a Soution for (13) We wi imit the study to the specia case where h is a separabe function, ie, it is of the form h(p ), for a 1-dimensiona function h : R R Now the minimization in (11) separates out into independent minimization probems for each good We wi use d h (p, q ) to denote h(p ) h(q ) h (q )(p q ), the one dimensiona version of Bregman divergence Note that as h is convex, d h (p, q ) 0, (14) and, by the strict convexity of h, if p q, d h (p, q ) > 0 (15) As we wi see shorty in Lemma 11, dp = dt φ(p t )/h (p t ) if φ(p t ) and h (p t ) are finite In order to appy Theorem 10 on a ba B around a price vector p, we need this term to be bounded on B And, in order to mae progress, we wi aso need that h (p t ) (Otherwise, we may get stuc at a non-equiibrium price since dp woud be 0) These ead us to mae dt assumptions on the aowabe h and on the behavior of the tatonnement, namey that it is controabe, as defined in the subsequent subsections 421 Aowabe h We wi aso need h to be twice differentiabe It may be that h (0) =, but by the convexity of h, this is the ony argument for which h might bnfinite And if h (0) = then h (0) = Lemma 11 For a, if φ(p t ) and h (p t ) are finite, then p(ɛ) p t im ɛ 0 ɛ = φ(p t ) h (p t ) 13

14 Proof The minimizer in (11) must have a zero derivative: Since d(d h(p,p t )) dp φ(p t ) + 1 ɛ d(d h (p,p t )) dp = 0 (16) = h (p ) h (p t ), substituting in (16) and soving for p gives p (ɛ) = h 1 ( h (p t ) ɛ φ(p t ) ) Note that since h is stricty convex, h is stricty increasing and hencs invertibe For notationa convenience, et g(y) = h 1 (y) Then h (g(y)) = y, h (g(y)) g (y) = 1, therefore g (p ) = 1 h (g(p )) Aso note that g(h (y)) = y Using these we obtain g (h (p )) = 1 h (p ) (17) Stricty speaing, the above argument is not vaid for p = 0 if h (0) = But in this case, we can chec directy that (17) is sti correct, for then g ( ) = 0 and h (0) = Now, p (ɛ) p t im ɛ 0 ɛ g(h (p t ) ɛ φ(p t )) g(h (p t )) = im ɛ 0 ɛ = g (h (p t )) φ(p t ) = φ(p t )/h (p t ) (by (17)) We mae the foowing additiona assumptions on h Definition 3 h(p) is aowabf h is twice differentiabe and stricty convex (hence h (p) > 0), h (p) is finitf p > 0, 1/h is continuous, and either A1 The maret is a Fisher maret, or A2 p h (q)dq = for a p > 0, and in addition either B1 h (p) is finite for a p, or B2 p 0 h (q)dq = for a p > 0; in this case, we say h is controing Henceforth, we assume that h is aowabe We note that two of the most commony used Bregman divergences satisfy the above assumptions The first one uses h(p ) = 1 2 p2 ; thus h (p ) = 1; hence dp = dt φ(p) Aso, for p > 0, h (q)dq =, so conditions A2 and B1 are satisfied The second one, which is p the KL-divergence, uses h(p ) = p og p p, h (p ) = og p and h (p ) = 1/p Hence dp = p dt φ(p) Aso, p dq = og p og 0 = and for p > 0, dq = og og p =, 0 q p q so conditions A2 and B2 are satisfied The reason for the condition B2 in Definition 3 is to ensure that if the tatonnement starts at a point with finite h it wi never reach a point with infinite h When h (p ) =, by Lemma 11, dpt = 0 no matter what the vaue of φ(p t ) is, ie p dt remains constant hereafter This is an unreasonabe tatonnement rue 14

15 Lemma 12 Suppose that h (p 0 ) is finite If h is aowabe then h (p t ) is finite for a t 0 Proof If condition B1 of Definition 3 hods then the resut is immediate So suppose that condition B2 hods By assumption, h (p) = ony if p = 0 As z 1 aways, φ(p) 1 aways Consequenty, by Lemma 11, dpt 1/h (p t dt ) Suppose that p 0 > 0 Then et t > 0 be the eariest time at which p coud be zero We use condition B2 to ustify the ast equaity beow: p 0 t 0 dp t p 0 dp t /dt h (p)dp = 0 Thus ony at time t = can p be 0, and hence ony at time t = can h (p ) be The reason for the condition A2 in Definition 3 is to ensure that if the tatonnement starts at a point with finite vaue, no price wi bow up to + in finite time Lemma 13 Suppose that p 0 is finite If h is aowabe then p t is finite for a t 0 Proof If the maret is a Fisher maret then prices remain bounded by the maximum of their initia vaue and the amount of money in the maret So suppose the maret is not a Fisher maret; then, by assumption, p h (q)dq = for a p > 0 Let p max = max p Define M t = pt p max n Then z max n So d dt pt max n/h (p t max) Let t be the eariest time at which p t max coud bnfinite Let tmin = arg min t< t p t max If p tmin max > 0, then by Condition A2, t 1 n p tmin max h (p)dp =, and if p tmin max = 0, then the same bound hods by Conditions A2 and B2 The foowing exampe shows that a price may bow up to + in finite timf condition A2 is vioated Exampe 1 Consider an Arrow-Debreu maret with one agent and two goods The agent has one unit of each good as initia endowment The agent wants ony good 1 So the equiibrium price vector is (p 1, p 2) = (p, 0) for any p > 0 At any (p 1, p 2 ), the excess demand for good 1 is (p 1 + p 2 )/p 1 1 = p 2 /p 1 and the excess demand for good 2 is 1 Suppose the tatonnement starts at (p 1, p 2 ) = (2, 1) and h satisfies h (p) = 1/p for p 1 and h (p) = 1/p 3 for p 1 Then dpt 2 = p t dt 2 and dpt 1 = (p t dt 1) 2 p t 2 The soution is p 1 (t) = 2 and 2e t 1 p 2 (t) = e t Note that p 1 (t) bows up to + at t = og Loca Convergence of (13) Next we show that thers a soution to (13) for some timnterva [0, t], under additiona assumptions Later, we wi show how to extend the soution to arbitrariy arge t and remove the assumptions In order to appy Theorem 10 to (13), we need its right hand side ( φ(p)/h (p ) when φ(p) is finite) to be convex, compact and upper semi-continuous in any ba B(p 0, ρ) we consider The difficuty we facs that when some prices are zero, the corresponding demands can bnfinite, and then compactness wi not hod for such price vectors 15

16 To restore compactness we modify F as foows Let b > 0 We define F b (p t ) = F (p t ) {v b1 v b1} We then define the foowing differentia incusion on R n +: dp dt = F b(p t ) (18) This introduces the possibiity that F b (p) is empty for some p which maes the differentia incusion triviay unsatisfiabe We assume for now that F b (p) is non empty in a sma neighborhood around p, and remove this assumption ater Definition 4 We say that F is bounded near p if there exists some neighborhood Ω(p) of p and a finite positive number b such that for a q Ω(p) R n +, F b (q) is non-empty and h (q) is finite 12 Lemma 14 Suppose that h is aowabe and that F is bounded near p Then F b (p) is convexvaued, compact-vaued and upper semi-continuous at p Proof Let Ω(p) be the neighborhood of p given by the assumption that F is bounded near p (Definition 4), and et B R n + be a compact neighborhood of p such that B Ω(p) and every positive pricn p is positivn B By our choice of B h (q ) is positive and finite for a q B and for a, so there exists a positive number h such that h (q ) h for a q B and for a Then on B, b z (q)/h (q ) z (q)/ h, ie x (q) = z (q) + 1 b h + 1 Let b denote b h + 1 We appy Theorem 9 with P = Ω(p), X = [0, b] n u is the utiity function of an agent, which we assume to be continuous and concave For any q Ω(p), C(q) is the set of a affordabe bundes in X of the agent at price q It is we nown that C(q) is continuous, and sincts rangs confined to the compact set X, C(q) is compact-vaued By Theorem 9, C (p), the set of a affordabe optima bundes of the agent at price p contained in X, is compact and upper semi-continuous at p By our assumption that F b (p) is non-empty, C (p) is aso a subset of a affordabe optima bundes of the agent at price p gobay (ie without confinement to X) Aso, since u is concave, C (p) is convex By the definition of C (p) and φ, φ(p) is the sumset of C (p) over a agents and the set { 1} As C (p) is non-empty for each agent, φ(p) is aso non-empty By Lemma 8(a) and (b), φ(p) is convex and compact, and it is upper semi-continuous at any p (F b ) is φ(p) divided by h (p ), whie 1/h is continuous and positive at any p P So the division by h wi not affect convexity, compactness and upper semi-continuity The foowing emma is immediate Lemma 15 Any soution to system (18) over timnterva [0, t] starting at a price vector p 0 such that F is bounded near p 0 is aso a soution to system (13) As discussed previousy, we want to extend the domain for the differentia incusion to R n We wi wor with F b rather than F, however To hep specify the new differentia incusion system, for any price vector p, wntroduce the foowing notation: etting p = (p ), we define p + = max{0, p } and p + = (p + ) The new system is given by with G b defined as foows: dp t dt G b(p t ), (19) 12 We remar that this condition is satisfied automaticay when p > 0 16

17 1 For p R n +, G b (p) = F b (p) 2 For p / R n +, et J(p) = { p < 0}, then set G b (p) = G b (p + ) {z J(p), z 0} Lemma 16 Let p R n Suppose that h is aowabe and F is bounded near p + Then G b (p) is convex, compact and upper semi-continuous at p Proof As G b F b in R n +, by Lemma 14, the resut is immediate for p > 0 For the other p s, note that G b (p) = G b (p + ) {z J(p), z 0} is thntersection of two sets, the first being convex and compact and the second being convex and cosed So G b (p) is convex and compact What remains is to chec upper semi-continuity at these p s There are two cases: p R n + but it has some zero prices, or p R n + Case 1: p R n + but it has some zero prices For any open set M which contains G b (p) = F b (p), by Lemma 14, we can tae a sufficienty sma neighborhood B(p, δ) of p such that for a q B(p, δ) R n +, F b (q) M Then, for any q B(p, δ) \ R n +, note that q + B(p, δ) since q +, p q, p, and, of course, q + R n + Thus F b (q + ) M; and G b (q) G b (q + ) = F b (q + ) M So G b is upper semi-continuous at p Case 2: p R n + For any q R n, et V (q) denote the set {v J(q), v 0} For any q R n +, G b (q) = G b (q + ) V (q) By Case 1 and our conditions on p, G b (p + ) is upper semi-continuous at p + p + is continuous in p Hence G b (p + ) is upper semi-continuous at p Next, we observe that there exists a sma δ > 0 such that for a q B(p, δ), if p 0, then sign(q ) = sign(p ) and consequenty V (q) V (p); it immediatey foows that V (p) is upper semi-continuous at p Now, by Lemma 8(c), G b (p) is upper semi-continuous at p Lemma 17 Any soution to system (19) over timnterva [0, t] starting at price vector p 0 is aso a soution of (13) if F is bounded near p 0 Proof We wi show that any soution of (19) is a soution of (18) The resut then foows from Lemma 15 In the definition of G b, at a price vector p with p < 0, G b, (p) is aways positive or zero, so it is impossibe for any tatonnement traectory satisfying (19) to enter the region p < 0 Hence, a prices remain positive or zero, ie p t R n + for a t In R n +, (18) is identica to (19), so we are done Lemma 18 Suppose that h is aowabe, and F is bounded near p 0 Then thers a time t > 0 such that (13) has an absoutey continuous soution for timnterva [0, t] with p(0) = p 0 Proof By Lemma 16, G b (p) is convex, compact and upper semi-continuous at p in thnterior of Ω(p 0 ) Now, by Theorem 10, (19) has an absoutey continuous soution with p(0) = p 0 for some timnterva [0, t], where t > 0 And by Lemma 17, this is aso a soution to (13) Lemma 18 gives us a oca soution (ie, upto some time t > 0) under the assumption that F is bounded near p 0 We need to remove this assumption and we need a soution for arbitrariy arge t For these we need the notion of controabiity 17

18 423 Controabiity Given a starting price vector p 0 and any finite time t 0, we need to ensure that thers a sufficienty arge b = b(p 0, t) guaranteeing that the tatonnement remains in the domain with F b φ during the timnterva [0, t] (so that the differentia incusion is defined for a points encountered during the tatonnement) This wi be ensured by the assumption of controabiity To understand this, we first need to characterize the set of optima bundes of an agent at price vector p There are two possibiities: 1 Every bundncudes at east one good having infinite demand Then we say that p is an unbounded demand price vector Note that this good must then have price zero, and by Lemma 12 this can occur ony if h (0) is finite Via the controabiity requirement, we wi ensure that in this case the tatonnement traectory does not reach any unbounded demand price vector (If h (0) is infinite then this is aready ensured by Lemma 12) 2 A the demands in at east one bunde are finite Then we say that p is a bounded demand price vector Note that if p incudes a zero price, p = 0 say, then an optima bunde can have an infinite demand for good ; but p is a bounded demand price vector if for a such, the demand for good coud be finite For instance, in a Leontief Fisher maret, an equiibrium price vector may incude a zero price but it wi be a bounded demand price vector; ceary, we want the tatonnement traectory to be abe to converge to it Furthermore, in this case, as the tatonnement proceeds, we want the agent s sequence of optima bundes to aways have bounded demands, and further these bounds shoud appy throughout the tatonnement process We are now ready to define controabiity Definition 5 Let φ be a potentia function and T a continuous tatonnement rue The pair (φ, T ) is controed, if for any bounded demand starting price vector p 0 and any finite time t 0, there are finite bounds b(p 0, t) and c(p 0, t) such that for any tatonnement traectory induced by (18), there exists a neighborhood Ω of the traectory in which for any p Ω and for any, 1 φ(p)/h (p ) b(p 0, t) and p c(p 0, t) for a 0 t t; 2 im t t b(p 0, t) and im t t c(p 0, t) are finite 13, ie both the prices and the rate of change of the prices remain bounded throughout the tatonnement process up to and incuding time t We wi show that if h is controing (reca Definition 3) then (φ, T ) is controed We wi aso show that controabiity is obeyed by Fisher marets with CES, Leontief and inear utiities aong with any tatonnement rue (ie even if h is not controing) But it is not cear if the atter resut appies to a marets or even to a EG marets One exampe for which we have not resoved this question are Fisher marets with nested CES utiities (see [22] for a definition); these are EG marets 13 Without oss of generaity, we may assume that b(p 0, t), c(p 0, t) arncreasing functions of t, so thmits exist 18

19 Lemma 19 If h is controing then (φ, T ) is controed Proof As h is controing, in finite time t, the traectory is both upper-bounded and bounded away from zero 14, say 0 < p( t) p t p( t) < +, for a and for a 0 t t Then there exists a neighborhood Ω of the traectory up to time t such that a prices in Ω are between p( t)/2 and p( t) + 1 Set c(p 0, t) = p( t) + 1 For a p Ω, 0 < p( t)/2 p p( t) + 1 < +, so h (p ) is bounded away from 0 As φ is convex, φ is finite except possiby at the boundary, ie when one or more prices is zero When a prices are between p( t)/2 and p( t) + 1, φ is bounded Combined with the ast paragraph, φ(p)/h (p ) is bounded on Ω Set b(p 0, t) to be an upper bound of φ(p)/h (p ) on Ω The proof of the next emma is in the appendix Lemma 20 Fisher marets with CES, Leontief and inear utiities aong with any tatonnement rue are a controed Now we are ready to compete the proof of the existence of a soution to the differentia incusion system (13) Lemma 21 Suppose that h (p 0 ) is finite, h is aowabe and (φ, T ) is controed Then for any bounded demand starting price vector p 0 there exists a soution p t to (13) for time range [0, ), with p t an absoutey continuous function for any bounded time span, and p t (t = 0) = p 0 Proof We wi prove the resut for differentia incusion (19) and then the resut foows from Lemma 17 The controabiity assumption aows us to pic b = b(p 0, t) for some t > 0 and have F be bounded near p We can therefore appy Lemma 18 to get a soution for some timnterva [0, t ] with t > 0 By Lemma 17, this is aso a soution for (13) Once again, due to the assumption of controabiity, the soution path cannot end at a point with φ/h = for any By the continuity of φ/h, thers then a ba around p t in which φ/h is bounded So we can repeatedy extend the path by additiona appications of Lemma 18 Suppose that this yieds an open path ending at but possiby not reaching some time t We first argue that it can be extended to t and then can be extended yet further By the controabiity assumption, for any t [0, t), a prices in p t are bounded by im t t c(p 0, t), which is finite; then the sequence {p t } 0 t< t has a custer point p Then by the controabiity assumption again, a dpt are bounded by im dt t t b(p 0, t), which is again finite Hence, {p t } 0 t< t has at most one custer point So p is the unique custer point of the sequence {p t } 0 t< t Setting p t (t = t) = p extends the soution to t = t Again by the controabiity assumption, there exists a neighborhood of p such that a q in the neighborhood has finite φ(q)/h (q) By Lemma 18, we can extend the path P beyond time t by at east a positive time period 14 These foow easiy from the proofs of Lemma 12 and Lemma 13 19