Variations on the Theme of Scarf s Counter-Example

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1 Computational Economics 24: 1 19, Kluwer Academic Publishers. Printed in the Netherlands. 1 Variations on the Theme of Scarf s Counter-Example ALOK KUMAR 1 and MARTIN SHUBIK 2 1 Mendoza College of Business, University of Notre Dame, Notre Dame, IN 46556, U.S.A.; akumar@nd.edu 2 Cowles Foundation, Box , Yale University, New Haven, CT 06520, U.S.A.; martin.shubik@yale.edu (Accepted 11 September 2003) Abstract. We study the relation between the stability of a competitive equilibrium (CE) and the price adjustment mechanism used to attain that equilibrium point. Using two specific examples, a three-commodity exchange economy with a unique competitive equilibrium (Scarf s global instability example) and a two-commodity, two-trader type exchange economy with multiple competitive equilibria, we show that the stability of a CE depends critically upon the dynamics of the price adjustment mechanism. A particular CE may be unstable under one price adjustment mechanism but stable under another. The joint dynamics of the chosen price adjustment mechanism and the given economy determines the overall stability of its competitive equilibrium. Our results suggest that context-rich studies of economic systems which focus on a specific price adjustment mechanism may provide insights into the dynamics and stability of economic systems that are often not revealed through a context-independent analysis. Key words: Scarf s counter-example, price adjustment mechanism, feedback controller, multiple competitive equilibria, stability 1. Introduction The joint dynamics of a chosen price adjustment mechanism and the given economy determines the overall stability of its competitive equilibrium (CE). A particular CE may be unstable under one adjustment mechanism but stable under another. In this paper we consider two economies, a three-commodity exchange economy with a unqiue competitive equilibrium (Scarf s global instability example (Scarf, 1960)) and a two-commodity, two-trader type exchange economy with multiple competitive equilibria (Shapley and Shubik, 1977b). In both cases we show that a CE is unstable under one particular mechanism but achieves stability under another kind of price adjustment mechansim. Furthermore, for a given mechanism, the dynamics may be stable for some parameter values but unstable for other values of the parameters. These results suggest that context-rich studies 1 of economic systems which focus on a specific price adjustment mechanism may

2 2 ALOK KUMAR AND MARTIN SHUBIK provide insights into the dynamics and stability of economic systems that is often not revealed through a context-independent analysis. The motivation for our study comes from the domains of economic dynamics and feedback control theory (Ogata, 1970). One of the primary concerns of control theory is to design fast but stable feedback controllers. Given a system and a reference input signal that the system is expected to follow, a feedback controller attempts to reduce the difference (the error) between the output of the system and the reference input by applying a corrective control action using the difference between the desired input and the actual output. The controller design problem is to select a controller from a certain class of controllers that achieves stability and satisfies a desired set of performance criteria. A price adjustment mechanism is essentially a feedback controller. Given an economy (a system), a price adjustment mechanism (a controller) tries to minimize the difference between the aggregate demand and the aggregate supply (excess demand error) by varying the price (a control input). Just as an appropriate feedback controller can be chosen that achieves stability and satisfies a set of desired characteristics, for a given economy, it may be possible to choose an appropriate price adjustment mechanism from a class of mechanisms that provides stability to the economic system and satisfies a set of performance criteria (for example, fast speed of convergence without excess oscillations). In both the economies analyzed in our study, a stable price adjustment mechanism is identified from a class of simple (but effective) feedback controllers. Furthermore, by adjusting the parameters of the mechanism, we are able to control the speed of adjustment and the oscillatory dynamics. This suggests that control theory may provide an effective framework for the design and identification of fast and stable price adjustment mechanisms. Whether the control-theoretic framework yields mechanisms that are economically feasible must be evaluated on a case-by-case basis. 2. Scarf s Global Instability Example Revisited Scarf (1960) provides an example of a three-commodity exchange economy where prices move in closed orbits and do not converge to the unique competitive equilibrium. In Scarf s example, the economy consists of three agents, 1, 2, and 3, and three divisible commodities, x 1, x 2,andx 3. The utility functions of the agents are assumed to be: u 1 (x 1,x 2,x 3 ) = min(x 1,x 2 ), u 2 (x 1,x 2,x 3 ) = min(x 2,x 3 ), u 3 (x 1,x 2,x 3 ) = min(x 1,x 3 ) and a cyclic permutation of initial endowments are assigned to the three agents: e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1).

3 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 3 Note that agents 1, 2, and 3 have no desire for commodities 3, 2, and 1 respectively. The excess demand functions for the three commodities are given by: E 1t = D 1t S 1t = E 2t = D 2t S 2t = E 3t = D 3t S 3t = P 3t P 2t (1) P 1t + P 3t P 1t + P 2t P 1t P 3t (2) P 1t + P 2t P 2t + P 3t P 2t P 1t. (3) P 2t + P 3t P 1t + P 3t 2.1. SCARF S ECONOMY: PRICE ADJUSTMENT USING WALRASIAN MECHANISM Scarf s example uses a Walrasian price adjustment mechanism (the classical tatonnement process) where the price adjustments are proportional to the excess demand. The dynamics of the Walrasian mechanism can be described by a first order differential equation: dp t = λ 0 [D t (P t 1 ) S t (P t 1 )]=λ 0 E t, (4) dt where S t (P t 1 ) is the total quantity offered by the sellers at time t and price P t 1, D t (P t 1 ) is the total demand in the market at price P t 1 and time t, E t = D t (P t 1 ) S t (P t 1 ) is the excess demand function at time t and λ 0 is a parameter that determines the sensitivity of the market to changes in supply and demand. The behavioral model used to motivate the use of Walrasian mechanism is as follows: all agents in the market are price takers and utility maximizers. They use the previous period price (P t 1 ) to compute the optimal desired quantities of commodities by solving their individual optimization problem. All agents submit their decisions (quantity demanded or quantity supplied) to a central auctioneer who aggregates demand and supply and announces a price so as to reduce the demand-supply differential (excess demand). The auctioneer increases the price if excess demand is positive (demand > supply) and reduces the price if the excess demand is negative (demand < supply). The agents learn the new price and they recompute their desired optimal quantities of commodities. The whole process is repeated until the excess demand is zero (demand = supply) and the resulting price is the competitive equilibrium price (P ). The Walrasian price adjustment dynamics shown in Equation (4) can also be written as: t P t = P 0 + λ 0 E t dt. (5) 0

4 4 ALOK KUMAR AND MARTIN SHUBIK This representation of the Walrasian mechanism allows for another interpretation of the price adjustment process. If the excess demand is interpreted as the error in the system which must be reduced to zero, the price at any time t is the area under the E t -curve (or the error curve) up to that instant, scaled by the factor λ 0. When prices are rescaled at the end of each period so that they lie between 0 and 1, the economy in Scarf s example contains a unique equilibrium given by the price vector P = ( 1 3, 1 3, 1 ). However, this equilibrium price cannot be attained 3 using the Walrasian mechanism. For all initial prices P 0 = P, the prices show an oscillatory behavior (a limit cycle) and does not converge to the equilibrium price P. This suggests that the price dynamics under the Walrasian mechanism are globally unstable. Figure 1 shows an example of the price dynamics with limit cycles for one set of the initial price vector, P 0 = (0.08, 0.75, 0.17). Under the Walrasian price adjustment mechanism, the limit cycle observed in Scarf s economy is not at all surprising. In control system design, Walrasian type mechanisms are known to have oscillatory dynamics. In the next section we show that by a slight variation of the Walrasian mechanism, we can stabilize the price dynamics in Scarf s economy (the limit cycle disappears) so that the competitive equilibrium point can be attained by the price adjustment process SCARF S ECONOMY: PRICE ADJUSTMENT USING A VARIATION OF WALRASIAN MECHANISM The dynamics of the Walrasian mechanism (or the tatonnement process) is identical to a particular kind of feedback control mechanism known as the integral controller (Ogata, 1970). Due to its oscillatory nature, the integral controller is often used in combination with other types of controllers such as the proportional controller or the derivative controller. All these controllers belong to a general class of controllers known as the proportional-integral-derivative (PID) controller. Motivated by the form of the PID controller, we consider a slightly modified form of the Walrasian mechanism where the auctioneer adjusts prices not just using the current excess demand but also by making use of the first and second derivatives of the excess demand function. The dynamics of the PID price adjustment mechanism is given by: dp t dt = λ 0 E t + λ 1 de t dt + λ 2 d 2 E t dt 2 (6) and by integrating the above equation, we can rewrite the price adjustment dynamics as: t de t P t = P 0 + λ 0 E t dt + λ 1 E t + λ 0 }{{} 2 dt }{{}}{{} Integral term Proportional term Derivative term. (7)

5 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 5 Figure 1. Unstable price dynamics using Walrasian mechanism in Scarf s economy: (a) Dynamics in price space; (b) Price variation over time. P 0 = (0.08, 0.75, 0.17), λ 0 = This equation is behavioristically reasonable if we regard the adjustment as being controlled by an intermediary such as an auctioneer or a specialist. It might be considered as somewhat sophisticated for an individual trader. By choosing the set of parameters (λ 0,λ 1,λ 2 ), different types of price dynamics can be obtained, some of which may be stable and some unstable. Note that the Walrasian mechanism contains only the first term of Equation (7). Figure 2 shows the price dynamics in Scarf s economy under the PID mechanism for different

6 6 ALOK KUMAR AND MARTIN SHUBIK Figure 2. Price adjustment using the PID price adjustment mechanism in Scarf s economy: as the magnitude of the proportional term (λ 1 ) is increased, the unstable price dynamics becomes stable but for a larger value of λ 1, instability re-emerges. P 0 = (0.08, 0.75, 0.17). values of (λ 0,λ 1,λ 2 ). Figure 2(a) shows the dynamics for (0.00, 0.20, 0.00), which is identical to the Walrasian mechanism with λ 0 = 0.20 (see Figure 1). In the next plot, a small proportional term is introduced (λ 1 = 0.105). As the magnitude of the proportional term is increased, the price adjustment process converges towards the equilibrium price vector. In Figure 2(d), with λ 1 = 0.275, the convergence to equilibrium is fast and without a lot of oscillations. However, as the value of λ 1 is increased further, the dynamics becomes unstable once again (see Figures 2(e, f)). The time series of prices is shown in Figure 3. By controlling the magnitude ( gain ) of the proportional term, we are able to obtain different types of dynamics, both stable and unstable. The gain term also affects the speed of price adjustment. The third term in the mechanism, namely, the derivative term, primarily controls the oscillations in the adjustment process. In Figure 4 we show that the oscillations can be reduced significantly by adding a derivative term (λ 2 = 0.20) in the mechanism. Overall, by an appropriate choice of (λ 0,λ 1,λ 2 ), a desired price dynamics can be obtained.

7 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 7 Figure 3. Price variation in Scarf s economy for different parameters of the PID mechanism. The phenomenon of instability stability instability as λ 1 is varied is quite evident. P 0 = (0.08, 0.75, 0.17). It is easy to see why the price dynamics stabilizes under the PID type mechanism. We can write the price adjustment dynamics of one of the PID type mechanisms, namely, the proportional-integral (PI) mechanism, as: dp dt = λ 0E 1 + λ 1 de dp. (8) Compare this equation with the price dynamics under the Walrasian mechanism (Equation (4)) where dp = λ dt 0 E. The dynamics of the PI mechanism contains one extra term in the denominator which can be interpreted as providing the auctioneer with an additional degree of freedom. Instead of reacting directly and only to the excess demand, the auctioneer measures the impact of price change on excess demand (the de term) and scales the price using this information. If the market reacts dp positively to a price change (a positive de term), the auctioneer makes a smaller dp (compared with the Walrasian auctioneer) price adjustment in the next time period. The adjustment is larger (compared with the Walrasian auctioneer) if the market responds in a negative manner. In a similar way, PD and PID mechanisms use information about the higher order derivatives of excess demand to adjust prices. The extra bits of information available to the auctioneer can improve the stability of the price adjustment process SCARF S ECONOMY: PRICE ADJUSTMENT USING COURNOT-SHUBIK MECHANISM A different way in which we can approach price change is via the strategic game market mechanism (the bid-offer model) where the desired quantity is offered as

8 8 ALOK KUMAR AND MARTIN SHUBIK Figure 4. Addition of a derivative term to the price adjustment mechanism can reduce oscillatory behavior and speed-up price convergence: (a) Price dynamics without a derivative term (λ 2 = 0); (b) Price dynamics with a derivative term (λ 2 = 0.200). P 0 = (0.08, 0.75, 0.17). the strategic variable (Shapley and Shubik, 1977b). The bid-offer model is selected as the simplest decentralized price formation mechanism. The one period game with this mechanism calls for a single and simultaneous move trade made by all players each of whome has a single information set. 2 We assume that after each trader has named quantities offered to the market no actual trade takes place but a new price is calculated and the individuals move again until an equilibrium price is reached which is signaled when they repeat their actions. If b i (t) ( 0) represents the bid of trader i and q j (t) ( 0) is the offer of trader j, then a new price is formed as the ratio of the aggregate bid and the aggregate offer: P t = Nb i=1 b it Ns j=1 q. (9) jt

9 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 9 If either the aggregate bid on the aggregate offer is zero, the price is set to zero. In Scarf s economy, the price formation equations under the bid-offer mechanism (Cournot-Shubik (CS) mechanism) can be obtained as: P 1t = P 3t 1(P 1t 1 + P 2t 1 ) (10) P 2t 1 (P 3t 1 + P 1t 1 ) P 2t = P 1t 1(P 2t 1 + P 3t 1 ) P 3t 1 (P 1t 1 + P 2t 1 ) (11) P 3t = P 2t 1(P 3t 1 + P 1t 1 ) P 1t 1 (P 2t 1 + P 3t 1 ). (12) Under the modified price dynamics of the CS mechanism, the competitive equilibrium in Scarf s economy is stable. Figure 5 shows the price dynamics under the CS mechanism for one initial condition, P 0 = (0.08, 0.75, 0.17). The fast convergence of prices to the competitive equilibrium is quite evident. The convergence under the CS mechanism is significantly faster than the convergence under the PID class of price mechanisms. In Scarf s economy, following the CS mechanism, the convergence to equilibrium is complete ( P t P <ɛ)within 25 iterations while under PID type mechanisms, typically more than 50 iterations are needed to achieve the same amount of convergence. 3. A Trading Economy with Three Competitive Equilibria We continue our investigation of the relationship between the stability of the competitive equilibrium (CE) and price adjustment mechanisms and analyze the dynamics of price adjustment in a two-commodity, two-trader type exchange economy which is known to have multiple equilibria. Two previous papers (Shapley and Shubik, 1977a; Kumar and Shubik, 2000) provide details of the structure of this exchange economy but here, in order to let this paper be self-contained, we reproduce the basic structure of the model for one initial endowment. Figure 6 shows the economy graphically in an Edgeworth box. In this figure, I represents the initial allocations to both trader types, O 1 and O 2 are the origins for type I and type II traders respectively, IC 1 and IC 2 are the zero gain indifference curves, CE 1, CE 2 and CE 3 are the three competitive equilibrium points and V is the value solution. We assume that all traders of one type have identical preferences and identical endowments (initial allocations) at the beginning. The preferences of traders can be represented by the following utility functions: u 1 (x, y) = x + 100(1 e y/10 ); u 2 (x, y) = y + 110(1 e x/10 ). (13) Both the utility functions are not only concave and smooth (C ) but also additively separable, with one good entering linearly in each case. The Pareto set for this

10 10 ALOK KUMAR AND MARTIN SHUBIK Figure 5. Stable price dynamics using Cournot Shubik mechanism in Scarf s economy: (a) Dynamics in price space; (b) Price variation over time. P 0 = (0.08, 0.75, 0.17). economy is the locus of the points of tangency of the indifference curves of the two trader types and it is given by: y = x log(110) = x (14) This is represented as O 1 C 1 D 1 C 2 D 2 O 2 in Figure 6. It includes the maximal points along the perimeter of the Edgeworth box (O 1 D 1 and O 2 D 2 ). The portion of the Pareto set, C 1 C 2, which lies between the zero gain indifference curves is the con-

11 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 11 Figure 6. An Edgeworth box diagram with three competitive equilibria. tract curve. It includes a short piece D 1 D 1 along the boundary of the box. C 1 and C 2 represent the worst contracts for type I and type II traders respectively. The conditions for a competitive allocation reduce by elementary calculus to the following transcendental equation: x(1 + 11e x/10 ) = 10 log(110). (15) This equation has three roots in the region of interest which determine the three competitive equilibrium solutions, CE 1, CE 2 and CE 3, as shown in Figure 6 and Table I. Of the three competitive equilibria, two are stable (CE 1 and CE 3 )andthe third one (CE 2 ) is unstable in the sense that raising the price of either good creates a positive excess demand for that good MULTIPLE EQUILIBRIA ECONOMY: PRICE ADJUSTMENT USING WALRASIAN MECHANISM We first analyze the price dynamics in the multiple equilibria exchange economy using the Walrasian mechanism. Solving an individual utility maximization problem, each trader obtains the desired next period allocations, given the current

12 12 ALOK KUMAR AND MARTIN SHUBIK Table I. Numerical data for Figure 6. Point in Holdings of Exchange Utility payoff to Utility payoff to the figure type I traders ratio type I traders type II traders Initial allocation I (50, 0) Core solution: endpoints of the core C 1 (40, 44.89) C 2 (4.83, 7.83) Competitive solutions CE 1 (7.74, 10.74) CE 2 (26.83, 29.82) CE 3 (36.78, 39.77) Value solution V (23.00, 25.99) For holdings of type II traders, subtract the holdings of type I traders from (40, 50). Exchange ratio is the price of the first commodity in the unit of the second commodity. allocations of the two commodities, [x i(t 1),y i(t 1) ], and the previous period price, P t 1. The solution to individual optimization yields (see Appendix A for details): x 1t = x 1t [y 1t 1 10 log(10p t 1 )] P t 1 x 2t = 10 log(11) 10 log(p t 1 ). The excess demand is given by E t =[x 1t x 1t 1 ]+[x 2t x 2t 1 ] and a new price is obtained using P t = P t 1 + λe t = P t 1 + λ[x 1t x 1t 1 + x 2t x 2t 1 ]. Substituting the values of x 1t and x 2t in the equation above and simplifying, we obtain the price map for the Walrasian mechanism: [ P t = P t 1 + λ y 1t 1 P t 1 x 2t 1 10 log P t 1 { }] (1+P 10P t 1 ) t 1. (16) 11 P t 1 When the allocations are reinitialized to the initial endowments at the end of each time period, y 1t 1 = y 10 = 0, x 2t 1 = x 20 = 0, and we obtain a simpler price map: P t = P t 1 10λ P t 1 [log(10) + (1 + P t 1 ) log(p t 1 ) log(11)p t 1 ]. (17)

13 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 13 Figure 7. Price map for the two-trader, two-commodity type exchange economy under the Walrasian mechanism. (a) A section of the price map around CE 1 and CE 2, (b) the entire price map along with the three competitive equilibria. A price map is simply a mapping between two successive prices, P t 1 and P t. The points of intersection of the price map with the 45-degree line represent the competitive equilibrium points. Figure 7 is a graphical repesentation of this price map for the two-trader, two-commodity type exchange economy with the Walrasian mechanism λ and Stability of CE The market sensitivity parameter λ determines the stability of the CE and the speed of convergence towards the competitive equilibrium point. If λ is small, the prices converge gradually without any over-shooting and so the convergence is fast. On the other hand, if λ is large, the step sizes in price adjustments are bigger but the price dynamics become oscillatory and hence the convergence to the equilibrium point can get slower. A proper choice of the parameter λ is also critical to the stability of the price adjustment process. We perform a sensitivity analysis on the parameter λ to study its impact on stability. In Figure 8 the price maps for different values of λ are shown graphically. We find that the adjustment process can become unstable even for a very small value of λ (for example, 0.090). Interestingly, for this value of λ the convergence to CE 1 is stable while the convergence towards CE 3 results in a limit cycle. The market always over-adjusts in order to close the demand-supply gap and as a result the gap is never closed. When λ is increased further to 0.15, the adjustment process blows up (price becomes infinite) for P 0 < This is illustrated in Figure 8(b). Note that the shape of the price map changes as λ is modified and this

14 14 ALOK KUMAR AND MARTIN SHUBIK Figure 8. Impact of the market sensitivity parameter λ on the stability of the Walrasian price adjustment process. (a) The adjustment process is stable for small value of λ. Prices converge to the competitive equilibrium. (b) The adjustment process can become unstable (left plot) even for a very small value (0.075) of λ.pricesdo not converge to the competitive equilibrium point and keeps oscillating. (c) When λ is increased to 0.15, the adjustment process blows up (left plot) and the prices becomes infinite. shows a clear relationship between the choice of λ and the stability as well as the speed of convergence of the price adjustment process λ and Selection of CE In the presence of multiple competitive equilibria, it is often difficult to determine which of the many equilibria will be chosen by the price adjustment process. In our multiple equilibria exchange economy we find that the market sensitivity parameter λ influences the domains of attraction (DOA) of the competitive equilibrium points

15 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 15 and determines which CE is finally chosen. The domains of attraction of the three CEs in our economy are clearly defined: DOA 1 =[P min, 0.75), DOA 2 = 0.75, DOA 3 = (0.75,P max ]. These domains, however, are not robust to variations in λ. WhenP 0 DOA 1,for very small values of λ(λ < 0.025), P P1 (= 0.28), the location of the first CE. For slightly higher values of λ, (λ 0.025), P P3 (= 5.07), the location of the third CE, for all values of P 0.Soforλ>0.025, CE 3 is the globally stable competitive equilibria. Figure 9 shows how the price adjustment process escapes from DOA 1 as λ is increased from to MULTIPLE EQUILIBRIA ECONOMY: PRICE ADJUSTMENT USING COURNOT SHUBIK MECHANISM We now investigate price dynamics in our multiple equilibria exchange economy under a different mechanism where price change is via the strategic game market mechanism illustrated in Equation (9). With this mechanism the prices are formed according to: P t = x 1t x 1t 1. y 2t y 2t 1 Individual utility maximization yields: x 1t = x 1t [y 1t 1 10 log(10p t 1 )] P t 1 y 2t = y 2t 1 + P t 1 [x 2t 1 10 log(11) + 10 log(p t 1 )]. Substituting these values in the price adjustment equation, we get: y 1t 1 10 log[10p t 1 ] P t = Pt 1 2 [x 2t 1 10 log(11) + 10 log(p t 1 )]. (18) As in the case of the Walrasian mechanism, when the allocations are reinitialized to the initial individual endowments at the end of each time period, y 1t 1 = y 10 = 0, x 2t 1 = x 20 = 0, and we get a simpler price map: P t = log(10p t 1) Pt log( P t 1 ). (19) Figure 10 shows this price map graphically. The functional form of the Cournot price map is quite different from the Walrasian price map (see Equation (17)) but comparing Figures 7 and 10 we can see that they both exhibit qualitatively similar dynamics. Their stability properties are also quite different. In our exchange economy we find that the Cournot mechanism has better stability properties than the Walrasian mechanism. The adjustment

16 16 ALOK KUMAR AND MARTIN SHUBIK Figure 9. The Walrasian price adjustment process escapes from the domain of attraction (DOA) of CE 1 as λ is increased. CE 3 becomes the globally stable CE for λ process under the Cournot mechanism never becomes unstable while with the Walrasian mechanism, as shown in Figure 8, an improper choice of the market sensitivity parameter (λ) can lead to instabilities. The Cournot mechanism does not explicitly define a market sensitivity parameter. Figure 11 shows the convergence of the Cournot price adjustment mechanism toward the CEs for the same set of initial conditions as in Figure 8. A comparison of Figures 8 and 11 shows clearly that the adjustment process has better stability characteristics under the Cournot Shubik mechanism than under the Walrasian mechanism.

17 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 17 Figure 10. Price map for the two-trader, two-commodity type exchange economy under the Cournot Shubik mechanism. (a) A section of the price map around CE 1 and CE 2 ;(b)the entire price map along with the three competitive equilibria. Figure 11. In the two-trader, two-commodity type exchange economy, the price adjustment process is stable under the Cournot Shubik mechanism for all initial conditions (choices of initial prices) MULTIPLE EQUILIBRIUM ECONOMY: PRICE ADJUSTMENT USING THE PID CLASS OF MECHANISMS The application of the PID class of mechanisms to our two-commodity, two-trader type exchange economy yields results that are qualitatively similar to the results for the Walrasian mechanism except that when proportional and derivative terms are present a higher value of the λ 0 (the market sensitivity parameter) can be used

18 18 ALOK KUMAR AND MARTIN SHUBIK without destabilizing the adjustment process. To avoid repetition, we do not report the results for the PID class of mechanisms. 4. Summary and Conclusion Using two specific examples, we have shown that the stability of a competitive equilibrium (CE) depends critically upon the dynamics of the price adjustment mechanism. A given CE may be unstable under one adjustment mechanism but stable under another. These results suggest that for a given economy, it may be possible to design a price adjustment mechanism that achieves stability and satisfies a specified set of performance criteria such as speed of convergence, stability and robustness. The control-theoretic framework provides one direction for the design and identification of fast and stable price adjustment mechanisms but there may be others. Overall, our results suggest that context-rich studies of economic systems which focus on a specific price adjustment mechanism may provide insights into the dynamics and stability of economic systems that is often not revealed through a context-independent analysis. In our current work we are trying to develop a control-theoretic framework for synthesizing an optimal price adjustment mechanism, given an economy and a specified set of performance criteria. For systems with linear dynamics, well developed techniques such as H control exists for the identification of optimal controllers (Zhou and Doyle, 1998), but no such general framework exists for systems with nonlinear dynamics. Unfortunately, linearity of dynamics cannot be guaranteed for most economic systems. In such cases it may be possible to use a Lyapunov function based controller design framework to design fast and stable price adjustment mechanisms for a given class of exchange economies. Acknowledgements We would like to thank Herbert Scarf for helpful discussions and valuable comments. This work has as its basis the program at the Santa Fe Institute involving the collaboration of Martin Shubik, Ioannis Karatzas, William Sudderth, Per Bak and Kai Nagel on the analysis of the equilibria and dynamics of money and markets utilizing dynamic programming and simulation methods. Appendix A: Individual Utility Maximization Given their current allocations, [x i(t 1),y i(t 1) ], and the previous period price, P t 1, both trader types solve the following utility maximization problem to compute [x it,x it ], their desired allocation at time t: max u i[x i (t), y i (t)] (20) [x it,y it ]

19 VARIATIONS ON THE THEME OF SCARF S COUNTER-EXAMPLE 19 such that x it = x i(t 1) + 1 [y i(t 1) y it ]. (21) P t 1 The utility functions of the traders are assumed to be: u 1 (x, y) = x + 100(1 e y/10 ) u 2 (x, y) = y + 110(1 e x/10 ). Applying the budget constraint and simple maximization of u i yields: x 1t = x 1(t 1) + 1 [y 1(t 1) 10 log(10p t 1 )] (22) P t 1 x 2t = 10 log(11) 10 log(p t 1 ) (23) y 2t = y 2(t 1) + P t 1 [x 2(t 1) 10 log(11) + 10 log(p t 1 )] (24) y 1t = 10 log[10p t 1 ]. (25) Notes 1 For an example of such a study, see Beckmann and Ryder (1969) which provides an elegant analysis of the disequilibrium dynamics and stability of a market with price and quantity adjustment. 2 A perceptive discussion of time and dynamic equilibrium is given in Diamond (1994). References Backmann, Martin J. and Ryder, Harl E. (1969). Simultaneous price and quantity adjustment in a single market. Econometrica, 37, Diamond, Peter (1994). On Time. Cambridge University Press, Cambridge, U.K. Kumar, Alok and Shubik, Martin (2000). A computational analysis of the core of a trading economy with three competitive equilibria and a finite number of traders. Cowles Foundation Discussion Paper #1290, Department of Economics, Yale University, January Ogata, Katsuhiko (1970). Modern Control Engineering. Prentice-Hall,Inc., Englewood Cliffs, NJ. Scarf, Herbert S. (1960). Some examples of global instability of competitive equilibria. International Economic Review, 1, Shapley, Lloyd S. and Shubik, Martin (1977a). An example of a trading economy with three competitive equilibria. Journal of Political Economy, 85, Shapley, Lloyd S. and Shubik, Martin (1977b). Trade using one-commodity as a means of payment. Journal of Political Economy, 85, Zhou, Kemin and Doyle, John C. (1998). Essentials of Robust Control. Prentice-Hall, Inc., Upper Saddle River, NJ.