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Parametric persistence of multiple equilibria in an economy directly calibrated to 5 equilibria
Economic Modelling 41 (2014) 356–364
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Parametric persistence of multiple equilibria in an economy directly calibrated to 5 equilibria☆ John Whalley a,1, Shunming Zhang b,c,⁎ a b c
Department of Economics, Faculty of Social Science, University of Western Ontario, London, Ontario N6A 5C2, Canada China Financial Policy Research Center, Renmin University of China, Beijing 100872, PR China School of Finance, Xingjiang University of Finance and Economics, Urumqi, 830012, PR China.
a r t i c l e
i n f o
Article history: Accepted 12 May 2014 Available online 19 June 2014 Keywords: Multiple equilibria Equilibrium manifolds Index theorem Parametric variation Pure exchange economies
a b s t r a c t In existing literature, there are few concrete examples of multiple equilibria and the only ones known to us have 3 equilibria, but multiplicity remains a major concern for applied models used in policy work. Here, we report numerical examples for a 3 individual 2 good CES/LES pure exchange economy directly calibrated to 5 equilibria. We are able to use analytical methods of the model to show that for certain parameterizations there are no more than 5 equilibria (given the parameter values). We are also able to explore the size of the regions of the parameter space for which 5 equilibria persist, and show these ranges to be very small. Other features of the equilibrium manifolds are explored. Findings are only suggestive and indicative for the special cases we consider rather than definitive, but informative relative to existing work. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Despite discussion of multiplicity of equilibria in theoretical literature (see the Mas-Colell (1985)) less is known about how severe a problem it is in practice. At a theoretical level, Mas-Colell (1985) suggests that all that can be said on the multiplicity issue is contained in the index theorem, i.e., the two propositions that equilibria have plus and minus one indices associated with them derived (in the pure exchange case) from the sign of the determinant of the negative of the Jacobian matrix with the first row and column deleted, and that their sum is always plus one. But how often multiple equilibria occur and in which types of models, and how many equilibria can be involved remain issues on which there is little literature. Does multiplicity occur over
☆ Whalley acknowledges the financial support from Ontario Research Fund and The Center for Intentional Governance Innovation (CIGI, Waterloo, Ontario). Zhang acknowledges the financial support from the Major Basic Research Plan of Renmin University of China (Grant 14XNL001) and the National Natural Science Foundation of China (NSFC Grant 71273271). Presentations of this work have been given to seminar groups at Zhongshan University, University of Warwick and University of Western Ontario. We are grateful to comments from these groups, as well as to Zeke Wang, Srihari Govindan and Timothy J. Kehoe for discussions. ⁎ Corresponding author. School of Finance, Renmin University of China, Beijing 100872, PR China. Tel./fax: +86 10 8250 0626. E-mail addresses: [email protected] (J. Whalley), [email protected], [email protected] (S. Zhang). 1 Tel.: +1 519 661 3509; fax: +1 519 661 3666.
http://dx.doi.org/10.1016/j.econmod.2014.05.015 0264-9993/© 2014 Elsevier B.V. All rights reserved.
concentrated portions of model admissible parameter spaces, or does it occur intermittently and somewhat unpredictably? Are they widely separated? Are they largely a theoretical curiosum, or are they the norm? To our knowledge, there are only two concrete examples of multiple equilibria in the literature and both only present three equilibria. One is due to Shapley and Shubik (1977), who use an ingenious representation of preferences that yields this multiplicity. The other is a later example by Kehoe (1985) in which he constructs a model parameterization which guarantees an equilibrium with an index of sign −1. Since the indices across equilibria must sum to + 1, two other equilibria must exist which Kehoe then is able to find. In this paper, we show both how to find parameter values simultaneously consistent with a larger prespecified number of equilibrium price vectors and explore how large is the range of parametric displacement that preserves multiplicity of various orders. For a two good, three individual LES pure exchange economy we are able to both show that for particular model parameterizations no more than 5 equilibria exist and to produce an example of an economy with this number of equilibria. Having constructed our example, we then are able to move around in the model parameter space to evaluate whether multiple equilibria persist in neighboring regions of the parameter space. We claim no general results, all are for one particular model structure and functional form; results are suggestive, not definitive. Nevertheless, the results are both revealing and informative relative to literature.
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
Multiplicity poses special problems for numerical simulation equilibrium exercises now widely used to guide policy by governments around the world (see Shoven and Whalley (1992)). If multiple equilibria characterize the model, which equilibrium does the economy actually go to under a given policy change? Most numerical modelers seem to believe that with simple functional forms and, effectively, small dimensions in their equilibrium structures (goods and factor models that reduce to factor space) their models have unique equilibria on the basis of various ad hoc tests. They use displacement from computed equilibria and recomputation under different speeds or step sizes and initial starting points and argue that they seemingly always return to the same equilibrium. But theorists remain skeptical and point to nonpathological examples of multiple equilibria in simple models (see Kehoe (1991)). Sonnenschein's (1973) result clearly suggests that if continuity is the only property one can ascribe to market excess demand functions that satisfy Walras' Law in a pure exchange case, then it should be possible to always calibrate some model of an economy to any arbitrary number of equilibria. Here, the issue explored is how easily this can be done for a parsimonious model using the types of functional forms widely used in applied models. We are both able to show for particular models and parameterizations that there is an upper bound to the number of equilibria, and to construct examples of multiple equilibria directly using calibration methods. What is needed is to have sufficient degrees of freedom to fit the same model to more than one equilibrium. With Cobb–Douglas preferences in a pure exchange economy and prespecified endowments this is not possible, and CES only adds one extra degree of freedom. However, with LES (or Stone–Geary) preferences (or variable endowments, effectively the same thing), sufficient degrees of freedom are created to provide for a larger number of equilibria for any given model. In the paper we show that calibration to five prespecified equilibrium price vectors is possible for this model, but not for all combinations of vectors, i.e. certain combinations of prices are not admissible of 5 equilibria for the model. Moreover, as we move away from these equilibria in the model parameter space (shares, elasticities, and endowments minus minimum requirements), the ranges of parameter values for which five equilibria persist seem to be surprisingly small. The 5 equilibria occur, but only for small portions of the admissible parameter space. Also, ranges of 5 equilibria in parameter space may be small, but much larger ranges in price space for these equilibria seem to be involved, raising the question of whether multiplicity should be judged as more or less likely to occur in parameter or price space. The five equilibria we construct all have the appropriate Kehoe indices (three +1, two − 1), but as equilibrium paths collapse to three or one equilibria under parametric perturbation signs of indices along equilibrium paths can change. Sometimes the five equilibrium paths in parameter space collapse to one central and two extreme paths, even to three upside or downside paths or only one path. Effectively, +1 equilibria can transform into −1 equilibria and vice versa. This is a phenomenon that has been speculated on, but as far as we know has gone unrecorded in previous literature for particular models. We also show cases where for large regions of the parameter space unique equilibria are found, but these are for sharply different price vectors in alternative regions of the parameter space due to deformations along the path of equilibria. Uniqueness may thus seem to hold in some applied models, but different regions of the parameter space may exhibit sharply divergent unique equilibria with, seemingly, a large jump in behavior as hard to reconcile with underlying economic theory. In the final section, we report on calculations of frequencies of multiple equilibria in model in the calibration sense discussed above. We use a single digit grid in price space, and consider all potential combinations of five grid points as collections of equilibria for model calibration. We keep track of cases where construction of 5 equilibria is possible and where it is not, yielding an estimate of what fraction of potential
357
combinations of 5 price equilibria in the grid have admissible parameterizations for 5 equilibria. For the 3 individual, 2 good CES/LES pure exchange model we consider, this probability (in this sense) we put at 2.38% in the case of a grid over single digit price vectors, which we suggest is a small number. 2. Constructing an economy with 5 equilibria The literature thus far on multiple equilibria has been heavily influenced by theoretical work due to Kehoe (1980, 1982, 1985) which is summarized and extended by Mas-Colell (1985). This shows that any equilibrium has associated with it an index (the sign of the determinant of the negative Jacobian matrix with the first row and column deleted in the pure exchange case) which is either +1 or −1. The sum of these indices must be plus one, and so an odd number of equilibria are always present (see also Eaves (1978)). Kehoe's index theorem is reminiscent of Samuelson's (1947) separation theorem which shows that equilibria are alternately stable and unstable. Beyond this, there are only isolated examples of three equilibria. An early and neglected example of 3 equilibria due to Shapley and Shubik (1977) uses an ingenious preference representation to obtain this outcome. Beyond this, there exist an example of 3 equilibria for multi household economies with activity analysis exist due to Kehoe (1985), a numerical demonstration of uniqueness of equilibrium in a large dimensional general equilibrium US tax model due to Kehoe and Whalley (1985), and intuition by Forster and Sonnenschein (1970) as to why tax distortions may introduce multiplicity. Mas-Colell, after extensive discussion of multiplicity in pure exchange economies, is dubious that anything more than the index theorem can be stated on the issue. Sonnenschein's (1973) paper had earlier shown that the only restriction that can be placed on market demand functions that satisfy Walras' Law is continuity. In this sense, constructing examples of multiplicity of equilibria might seem an uninteresting exercise, since it should always be possible to construct underlying structural forms for economies in terms of household preferences and endowments for a pure exchange case which yield the required number of prespecified equilibria. Rather, as Mas-Colell (1985) argues, the more interesting issue is whether or not multiplicity occurs in particular types or forms of economies. Here, we focus more broadly on models of economies employing so called ‘convenient’ functional forms (Cobb–Douglas, CES, LES) widely adopted in applied models (see Shoven and Whalley (1992)) and used to analyze the impacts of such policy changes as tax reform and trade liberalization. We also use small dimensional parsimonious models so as to ease computational requirements, focusing, as will be seen below and for reasons given later, on a 3 individual 2 good pure exchange model with CES/LES preferences. Our aim is to use numerical simulation methods to generate insights as to when multiplicity occurs in specific cases, how large the regions of multiplicity within parameter spaces are, and how regions with various numbers of equilibria are connected. Our point of departure relative to earlier literature is to directly generate parameter values for models in which equilibrium conditions simultaneously hold for a series of prespecified price vectors, and then to explore the robustness of these multiple equilibria in price space. In a pure exchange economy with L goods, equilibrium is characterized by L market clearing demand–supply equalities with L − 1 relative prices being endogenously determined. We assume that I consumers (i = 1,…,I) are considered. We use a Stone–Geary/CES demand structure of the form
C il ¼ M il þ
α il
XL σi
Pl
P 0 ðW il0 −M il0 Þ l0 ¼1 l ; X L 1−σ α 0P 0 i l0 ¼1 il l
i ¼ 1; …; I; l ¼ 1; …; L
ð1Þ
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where Cil denote consumptions of good l for individual i, Mil define minimum requirements of good l for individual i, αil are share parameters of good l for individual i (∑Ll = 1 αil = 1), Wil denote endowments of good l for individual i, Pl is price of good l, and σi is the elasticity of substitution in preferences for individual i. As the demand system for individual i has (2L + 1) parameters rather than L as in the Cobb–Douglas case, simply adding consumers to a model will generate enough degrees of freedom that in principle it should be possible to calibrate a pure exchange economy model to any arbitrary number of equilibria. We could also use a CES demand structure and allow endowments to be endogenously determined through calibration (effectively the same). For the case where there are 3 individuals and 2 goods, there are 15 parameters (αil, σi, Wil − Mil) in the model, although with normalization of the αil to sum to one for any i only 12 are free parameters. If data on K equilibria are to be fitted to this system, there are 2K equilibrium conditions to be satisfied (2 market clearing conditions per equilibrium), although using Walras' Law this reduces to K restrictions. If we consider particular values of elasticities and share parameters for this model, more concrete statements as to the number of equilibria which exist can be made. Defining Xil = Cil − Mil and Eil = Wil − Mil, we have XL
α il P l0 Eil0 0 X il ¼ σ XL l ¼1 1−σ ; i Pl α 0P 0 i l0 ¼1 il l
i ¼ 1; …; I; l ¼ 1; …; L:
σ P1 i
α ðP E þ P 2 Ei2 Þ E þ PEi2 i1 1 i1 ¼ i1 ; i ¼ 1; …; I 1−σ i 1−σ i 1 þ α i P 1−σ i α i1 P 1 þ α i2 P 2
5
þ½ðα 2 þ α 3 ÞE12 þ ðα 3 þ α 1 ÞE22 þ ðα 1 þ α 2 ÞE32 q þðE12 þ E22 þ E32 Þq−ðα 1 E11 þ α 2 E21 þ α 3 E31 Þ:
If, in addition, we consider the special (symmetric) case α2 = 1, α1α3 = 1, E11 = E32, E21 = E22, and E31 = E12, then 9 F ðqÞ ¼ ðα 1 E11 þ α 2 E21 þ α 3 E31 Þ q −1 8 −ðE11 þ E21 þ E31 Þ q −q
5 4 þ½ðα 1 þ α 2 ÞE11 þ ðα 3 þ α 1 ÞE21 þ ðα 2 þ α 3 ÞE31 q −q
and defining a0 = α1E11 + α2E21 + α3E31, a1 = E11 + E21 + E31, and a2 = [(α1 + α2)E11 + (α3 + α1)E21 + (α2 + α3)E31], yields 9 8 5 4 4 F ðqÞ ¼ a0 q −1 −a1 q −q þ a2 q −q ¼ ðq−1Þq f ðqÞ
a0 −a1 N0 a0
ð12Þ
ð4Þ x1 x2 þ x1 x3 þ x1 x4 þ x2 x3 þ x2 x4 þ x3 x4 ¼ −
x1 x2 x3 þ x1 x2 x4 þ x1 x3 x4 þ x2 x3 x4 ¼ 2 ð5Þ x1 x2 x3 x4 ¼
α 1 E11 þ PE12 α E þ PE22 þ 2 21 P σ 1 1 þ α 1 P 1−σ 1 P σ 2 1 þ α 2 P 1−σ 2 α E þ PE32 þ σ33 31 : P 1 þ α 3 P 1−σ 3
ð11Þ
system f(q) = 0. If the 4 solutions are x1, x2, x3, and x4, then, for 0 b a0 b a1 b a2,
and E12 þ E22 þ E32 ¼
ð10Þ
ð3Þ
where α i ¼ ααi2 and P ¼ PP2 . i1 1 For the case of 3 individuals, market clearing requires that, for goods 1 and 2, E þ PE12 E þ PE22 E þ PE32 ¼ 11 þ 21 þ 31 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3
ð9Þ
4
−½α 1 ðα 2 þ α 3 ÞE11 þ α 2 ðα 3 þ α 1 ÞE21 þ α 3 ðα 1 þ α 2 ÞE31 q
x1 þ x2 þ x3 þ x4 ¼ −
α ðP E þ P 2 Ei2 Þ α i2 1 i1 ¼ σi X i1 ; i ¼ 1; …; I X i2 ¼ σ 1−σ 1−σ P i P 2 i α i1 P 1 i þ α i2 P 2 i
8
4 3 2 1 1 1 þ ða0 −a1 Þ q þ −ð3a0 þ a1 Þ q þ − where f ðqÞ ¼ a0 q þ q q q 2ða0 −a1 Þ q þ 1q þ ða0 þ a1 þ a2 Þ: It follows that there are at most 4 solutions q þ 1q for the equation
and
E11 þ E21 þ E31
9
F ðqÞ ¼ ðα 2 α 3 E12 þ α 3 α 1 E22 þ α 1 α 2 E32 Þq −α 1 α 1 α 3 ðE11 þ E21 þ E31 Þq
ð2Þ
For the 2 good case (L = 2), X i1 ¼
which yields a polynomial in q, F(q), for which zeros yield equilibria in q (and hence P). Thus
ð6Þ
10
3a0 þ a1 b0 a0
ð13Þ
a0 −a1 b0 a0
ð14Þ
a0 þ a1 þ a2 N0: a0
ð15Þ
Fo (P)
From Walras' Law, these equations imply that, 1−
1 1 1 þ 1− þ 1− E E E31 11 21 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3 P P P ¼ E12 þ E22 þ E32 : 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3
5
0
ð7Þ For the case where σ 1 ¼ σ 2 ¼ σ 3 ¼ 15, we can show that at most 5 equilibria will exist. Defining P ¼ q (that is, P = q5), we can rewrite Eq. (7) as
-5
1 5
α1 α2 α3 q E11 þ E21 þ E31 ¼ E12 1 þ α 1 q4 1 þ α 1 q4 1 þ α 2 q4 1 þ α 3 q4 q q þ E22 þ E32 1 þ α 2 q4 1 þ α 3 q4
ð8Þ
P
-10 0
2
4
Price Ratio Fig. 1. 5 Equilibria generated as zeros of the F0(P) function for model parameterization set out in the text.
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
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3. Ranges of parametric variation supporting multiple equilibria
From Eqs. (13) and (14) there exists at least one negative solution for the equation f(q) = 0. From Eq. (15) the negative solutions occur in pairs. Therefore the number of negative solutions is 2 or 4. That is to say, there are at most 2 positive solutions q þ 1q for the equation
Table 1 presents the model parameterization consistent with the 5 price equilibria shown in Fig. 1 for a 3 individual 2 good CES/LES pure exchange economy. The price equilibria in this particular example are chosen arbitrarily. These equilibria each have Kehoe indices as reported, and these are consistent with the index theorem and sum to +1. A feature of the equilibria in Table 1 is that they are widely separated in price, quantity and utility space for consumers. Local analysis searching for further equilibria in the neighborhood of each of these does not yield other equilibria. The model parameterization set out in Table 1 can be used for a range of experiments to provide insight on how regions of multiple equilibria in parameterization space behave for this model. How large are regions of multiplicity? How do they change as one moves between them? Are they continuous; do they have holes, for instance? How far can one move in parameter space via perturbations of shares, elasticities and endowments while still preserving five equilibria, before reverting to either three or one? Are such ranges of equilibria large or small, and for which parameters? Generating model parameterizations which support such multiple equilibria involves numerically solving a non-linear system of equations in the shares, minimum requirements and elasticities in preferences in the model. We employ GAMS optimization software and use a procedure of repeated restart through the parametric solution space (αil, σi, Eil) to search for multiple equilibria. Choice of both initialization
f(q) = 0, implying that there are at most 4 positive solutions for q. Thus, in this case, there are at most 5 positive solutions (q) for the equation f(q) = 0, and therefore at most 5 nondegenerate equilibria exist. In Fig. 1 we show an example of zeros of the F 0 ðP Þ ¼ F P function 1 5
yielding the five equilibrium price vectors (4, 1), (2, 1), (1, 1), (1, 2), and (1, 4). This case has been generated where not only are substitution elasticity parameters σ1 = σ2 = σ3 equal, but they also all equal to 0.2. We have set share parameters α11 = α32 = 4 and α12 = α31 = α12 = α22 = 1, and then calculated effective endowments (net of minimum requirements) of E11 = E32 = 1000.000, E21 = E22 = 1648.973 and E31 = E12 = −199.596 to yield these specific 5 equilibria. If, instead, σ 1 ¼ σ 2 ¼ σ 3 ¼ 14 , it can also be shown that at most 5 equilibria exist if 3a0 − 3a1 + a2 N 0 and at most 3 equilibria exist if 3a0 − 3a1 + a2 b 0. If σ 1 ¼ σ 2 ¼ σ 3 ¼ 13, at most 5 equilibria exist. If σ 1 ¼ σ 2 ¼ σ 3 ¼ 12, at most 3 equilibria exist if 3a0 − a1 + a2 b 0, and there exists a unique equilibrium if 3a0 − a1 + a2 ≥ 0. If σ1 = σ2 = σ3 = 1, 2, ⋯ (natural number), there exists a unique equilibrium. No such results on the number of equilibria for specific parameterizations of models occur, to our knowledge, in currently available literature.
Table 1 Characteristics of a 3 individual 2 good CES/LES pure exchange economy with the 5 equilibria represented in Fig. 1. A. Model characteristics • Pure exchange economy with 3 individuals and 2 goods • CES/LES utility function B. Model parameterization Substitution elasticity parameters σ1 = σ2 = σ3 = 0.2 Share parameters Consumer 1 Consumer 2 Consumer 3 Initial endowments Consumer 1 Consumer 2 Consumer 3
Good 1
Good 2
4 1 1
1 1 4
Good 1
Good 2
1000.000 1648.973 0.000
0.000 1648.973 1000.000
Minimum requirements Consumer 1 Consumer 2 Consumer 3
Good 1
Good 2
0.000 0.000 199.596
199.596 0.000 0.000
C. Equilibria and Kehoe indices Demands by individuals Equilibria
Price vectors
Individual 1
Individual 2
Individual 3
Equilibrium 1 Equilibrium 2 Equilibrium 3 Equilibrium 4 Equilibrium 5
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
(114.775, 221.342) (418.492, 290.670) (640.323, 359.677) (787.222, 425.678) (877.700, 489.120)
(2045.079, 1549.930) (1804.802, 1571.081) (1648.973, 1648.973) (1571.081, 1804.802) (1549.930, 2045.079)
(489.120, 877.700) (425.679, 787.222) (359.677, 640.323) (290.670, 418.492) (221.342, 114.775)
Utility values by individuals Equilibria
Price vectors
Individual 1
Individual 2
Individual 3
Kehoe indices
Equilibrium 1 Equilibrium 2 Equilibrium 3 Equilibrium 4 Equilibrium 5
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
20.289 73.902 112.918 138.768 154.767
1443.297 1402.595 1386.615 1402.595 1443.297
154.767 138.768 112.918 73.902 20.289
+1 −1 +1 −1 +1
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J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
and step size are critical in the practical application of these techniques. We finely divide the potential parameter space into small slivers using upper and lower bounds so as to exhaustively search for parameterization supporting multiple equilibria. This yields solutions similar to those that would be obtained using path following global Newton methods (see Smale, 1976). When we find supporting parameterizations for particular prespecified multiple equilibrium price vectors, we then make small parametric variations in model parameterizations once these are found and follow paths of equilibria (equilibrium manifolds in Mas-Colell's terminology) through the neighborhood. Results from such experiments are reported in Table 2. They suggest that the ranges of parameter changes that allow 5 equilibria to be preserved in this case are extremely small. For elasticity parameters, only very small changes in the neighborhood of 0.2 are possible, between 0.1976 and 0.20108 for σ1 and σ3. Dropping elasticity values further moves the model economy into a range of three equilibria, which persists until these parameters become small. Increasing elasticities produces a slightly larger (but still small) range of elasticity values with 3 equilibria, followed by an open ended range with one equilibrium. Results from perturbing share parameters again yields small ranges for 5 equilibria, but there are neighboring ranges of 3 and 1 equilibria on both sides of the parameterization reported in Table 1. Only with parametric changes in minimum requirements do regions of 5 equilibria reappear, once the initial region around the pre-specified price vectors is departed from. The results for each row of Table 2 can be represented diagrammatically showing how paths of equilibria are connected as one moves through parameter space, and how regions of multiple equilibria change and are related to each other. Fig. 2 shows results for the first row of Table 2 (sigma 1), and also the case where individual 1's endowments of good 1 are perturbed. These cases have been drawn so as to display the portion of parametric variation over which manifolds deform and serve to illustrate how equilibrium manifolds behave in these cases.
4. The behavior of equilibrium manifolds in neighboring regions of the parameter space Having generated a parameterization for a 3 individual, 2 good CES/ LES economy which supports the 5 equilibria represented in Fig. 1, we can use numerical simulation methods to explore the behavior of equilibrium manifolds in neighboring regions in the parameter space. Can we easily change the prespecified price vectors, for instance, and still find parameterizations which support 5 equilibria? How much can we change share, endowment, and other model parameters and still find 5 equilibria supporting parameterizations? Are there examples of index changes as we move between regions of multiple equilibria? How often do they occur; do some cases differ from others? And what of the behavior of equilibrium price paths across cases?
4.1. Changing prespecified price vectors The example of 5 multiple equilibria presented in Fig. 1 involves 5 prespecified equilibria which are both symmetric and dispersed in price space. Is this more likely to be a case for which 5 equilibria occur? Suppose, for instance, that three equilibria are close together and the other 2 are dispersed, some distance away in price space; would this latter case be one for which 5 equilibria also occur? In Table 3 we report maximum changes in the coordinates of the individual price vectors used in Fig. 1 which still allow five equilibria to occur. We report maximum displacements of the second and fourth price vectors such that five equilibria can be preserved. If both of these vectors are displaced together, the 3.333 and 6.667 coordinates of vectors of these vectors can be displaced in one direction to 2.112 and 7.888, relatively close to the neighboring vectors of 2.000 and 8.000, and can be displaced in the other to 4.907 and 5.093, relatively close to the neighboring vector of 5.000. This suggests a relatively wide range of displacement in price vector prespecification, but if only the
Table 2 Maximum parameter perturbations around model specification in Table 1 preserving multiple equilibria (asymmetric setting).1 Model parameters subject to perturbation Substitution elasticities Sigma 1 Sigma 2 Sigma 3 Share parameters Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 Initial endowments Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 Minimum requirements Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 1 2
Parameter values used in generating example in Table 12
Parameter range yielding 1 equilibrium
0.200 0.200 0.200
Parameter range yielding 3 equilibria
Parameter range yielding 5 equilibria
Parameter range yielding 3 equilibria
Parameter range yielding 1 equilibrium
[0.1975] [0.199615] [0.1975]
[0.1976, 0.20108] [0.199616, 0.200189] [0.1976, 0.20108]
[0.20109, 0.2030] [0.20109, 0.2030]
[0.2031] [0.20019] [0.2031]
4.000 1.000 1.000 1.000 1.000 4.000
[3.9952] [0.9965] [0.99979] [0.99979] [0.9965] [3.9952]
[3.9953, 3.9972] [0.99651, 0.99944] [0.999791, 0.999906] [0.999791, 0.999906] [0.99651, 0.99944] [3.9953, 3.9972]
[3.9973, 4.0022] [0.99945, 1.00069] [0.999907, 1.000092] [0.999907, 1.000092] [0.99945, 1.00069] [3.9973, 4.0022]
[4.0023, 4.0139] [1.0007, 1.0012] [1.000093, 1.000209] [1.000093, 1.000209] [1.0007, 1.0012] [4.0023, 4.0139]
[4.014] [1.0013] [1.00021] [1.00021] [1.0013] [4.014]
1000.000 0.000 1648.973 1648.973 0.000 1000.000
[999.199]
[999.200, 999.593]
[1000.366, 1001.018]
[1001.109]
[1648.566] [1648.566]
[1648.567, 1648.824] [1648.567, 1648.824]
[1649.134, 1649.296] [1649.134, 1649.296]
[1649.297] [1649.297]
[999.199]
[999.200, 999.593]
[999.594, 1000.365] 0.000 [1648.825, 1649.133] [1648.825, 1649.133] 0.000 [999.594, 1000.365]
[1000.366, 1001.018]
[1001.109]
[199.444]
[199.445, 199.486]
[199.681, 199.920]
[199.921]
[199.444]
[199.445, 199.486]
[199.681, 199.920]
[199.921]
0.000 199.596 0.000 0.000 199.596 0.000
0.000 [199.487, 199.680] 0.000 0.000 [199.487, 199.680] 0.000
Each row of the table represents parametric variation in the named parameter alone. The symbols [and] refer to bounds of regions in parameter space for 5, 3, and 1 equilibria. This model specification has been constructed so as to be consistent with the 5 equilibrium prices (4, 1), (2, 1), (1, 1), (1, 2), and (1, 4).
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Parametric Variation in Individual 1’s Substitution Elasticity (Row 1 of Table 2) 6
361
second price vector alone is displaced, the region of feasible displacement is restricted to 2.828 and 7.172 (Fig. 3). 4.2. Displacing parameter values so as to preserve the ability to produce 5 price equilibria
Price Ratio
A related experiment is to see how large the displacements are that can be made either singly or jointly in all parameters so as to preserve 5 equilibria under a modified parametric specification. Results from such variations are presented in Table 4. These show still small admissible parametric variations as elasticities and share parameters change, but show that wider ranges occur with joint variation. Some of these results are implicit in Table 2, where ranges for single parameter displacements are displayed as part of the overall pattern of multiple equilibrium regime change; but a wider variation is possible using combinations of parameters.
4
2
0 .196
.198
.2
.202
.204
Sigma (1)
4.3. Examples of index changes along price paths
B) Parametric Variation in Individual 1’s Endowment of Good 1 (Row 10 of Table 2)
4
2
We are also able to monitor index changes for equilibria (in the Kehoe sense) as we move along price paths corresponding to displacements of parametric settings around those set out in Table 1. Such index changes have been conjectured in previous literature, but to our knowledge no actual cases and examples are documented and hence these are of interest. Fig. 4 displays examples of index changes which occur for particular parametric displacements. As indicated, cases can arise for which multiple index changes are present. As an index changes along a price path, a zero index applies as the determinant of the negative Jacobian matrix with the first row and column deleted disappears. 4.4. The behavior of equilibrium manifolds under parameter displacement
0 999
999.5
1000
1000.5
1001
Initial Endowment (1,1) Fig. 2. Equilibrium manifolds in parameter–price space corresponding to results reported in Table 2. A. Parametric variation in individual 1's substitution elasticity (row 1 of Table 2). B. Parametric variation in individual 1's endowment of good 1 (row 10 of Table 2).
We are also able to show how equilibrium price manifolds behave under parametric displacement. These are shown in Fig. 5 for two particular cases of perturbation of share parameters and endowments. Case A show how multiple equilibria can occur even where only a single price path is present. This deforms over a narrow price range with large displacement in price space occurring over the non-deformed portions of the equilibrium manifold. A practical difficulty suggested by such cases is that for applied models ad hoc tests for uniqueness might be confirmatory for different parameters and locally such equilibria
Table 3 Bounds for asymmetric displacement of price vectors for which parameterizations preserving 5 equilibria are possible. A. Symmetric case in Table 3 giving 5 equilibria Price ratios to which equilibrium are calibrated Price vectors (P1, P2)
Price ratios PP
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
4.000 2.000 1.000 0.500 0.250
Cases
2 1
A1 A2 A3 A4 A5
B. Maximum displacements of price vectors A2 and A4 preserving 5 equilibria (B2 and B4 are the implied displacements in the price of the other good) Moving toward (2.000, 8.000) Moving toward (8.000, 2.000) Moving toward (5.000, 5.000) Moving toward (5.000, 5.000)
A2 (B2) A4 (B4) A2 (B2) A4 (B4)
= = = =
(2.112, 7.888) (7.888, 2.112) (4.907, 5.093) (5.093, 4.907)
C. Maximum displacements of price vector A2 alone preserving 5 equilibria (C2 is the implied displacements in the price of the other good) Moving toward (2.000, 8.000) Moving toward (5.000, 5.000)
A2 (C2) = (2.828, 7.172) A2 (C2) = (4.044, 5.956)
362
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Maximum Displacements of Price Vectors A2 and A4 Preserving 5 Equilibria
4
3
2
1
0 180
200
220
240
260
Minimum Requirement
B) Maximum Displacements of Price Vector A2 Alone 4
3
2
5. The frequency of multiple equilibria The procedures used to generate the parameterizations reported on above can also be used to evaluate the frequency of multiple equilibria for the structure we use (a 3 consumer, 2 good, CES/LES pure exchange economy). No discussion, to our knowledge, of this issue exists in previous literature and so we have constructed our own metric of frequency. While no general statements on multiple equilibria across models can be made, in the absence of other insights in the literature some evidence on frequency of multiple equilibria for particular cases seems helpful as a first step. Our procedure is to consider a listing of prespecified price vectors and to take combinations of some order of these (say, 5 to represent parametric fitting to 5 equilibria). We attempt to calculate a 5 equilibrium model parameterization for each combination, and if this fails we calculate a 3 equilibrium parameterization. If this fails a single model parameterization is determined. We explore the frequencies of 1, 3, and 5 equilibria in this way in price space. The dimensionalities involved can rapidly become overwhelming if large numbers of price vectors are considered as candidates to provide combinations of parameterizations supporting 5 equilibria, and so we restrict ourselves to single digit and two digit grid searches. We note that the equilibrium manifolds displayed in Figs. 2, 4 and 5 also raise the issue for multiplicity discussion that the frequency of observing multiple equilibria in parameter and price space can differ, and potentially sharply so. Thus in asking how likely multiple equilibria are in particular cases, one has to be clear as to whether one is referring to frequency relative to endogenous or exogenous variables in the model. 5.1. The frequency of 5 equilibria
1
0 6
6.5
7
7.5
P(1) Fig. 3. Bounds of asymmetry in 5 equilibria preserving parameterizations specified in Tables 3 and 4. B. Maximum displacements of price vectors A2 and A4 preserving 5 equilibria. C. Maximum displacements of price vector A2 alone.
would change little, but over progressively wider ranges of parametric variation large jumps in behavior would occur which could not be accounted for by economic theory. In Case B multiple equilibria also occur but multiple price paths merge at various points of the map. These correspond to zones of 5, 3, and 1 equilibria.
For our calculation of frequency of 5 equilibria, we use initial endowments minus minimum requirements as 5 free parameters. If we restrict ourselves to equally spaced single digit relative price vectors (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), this yields 98765 54321 ¼ 126 combinations of 5 price vectors (0.1, 0.3, 0.5, 0.7, 0.9, for instance). For the 3 individual 2 good CES/LES pure exchange economy we consider, we arbitrarily select 5 of the 11 parameters (we assume that the initial endowment for consumer 1 and good 1 is 1000.000) given the 5 equilibria (see Table 5 for details). Only 3 trials succeed in attaining 5 equilibria through direct calibration methods. These are (0.2, 0.3, 0.5, 0.7, 0.8), (0.2, 0.4, 0.5, 0.6, 0.8), and (0.3, 0.4, 0.5, 0.6, 0.7). Thus the frequency of 5 equilibria in the sense of a combination of single digit price 3 1 vectors is 123 ¼ 42 ¼ 2:38%. We summarize both these results and the procedures we employ in Table 5. If we then consider two digit price settings, the relative price vectors are equally spaced (0.01, 0.02, …, 0.99). There are 9998979695 ¼ 54321
Table 4 Maximum single or joint parametric variation around the model specification in Table 2 which preserves 5 equilibria. Perturbation parameter setting used in example in Table 1
Single or joint perturbation
Parameter value in example in Table 1
Maximum displacement while preserving the ability to support 5 equilibria
Elasticity of substitution σ1 Elasticity of substitution σ1 = σ3 Elasticity of substitution σ2 Share parameter α11 Share parameter α11 = α32 Share parameter α12 Share parameter Α12 = Α31 Share parameter α21 Initial endowment W11 Initial endowment W12 = W32 Initial endowment W21 Initial endowment W21 = W22 Minimum requirement M12 Minimum requirement M12 = M31
Single Joint Single Single Joint Single Joint Single Single Joint Single Joint Single Joint
0.2 0.2 0.2 4 4 1 1 1 1000.000 1000.000 1648.973 1648.973 199.596 199.596
[0.19769, 0.20108] [0.198763, 0.200744] [0.199616, 0.200189] [3.9973, 4.0022] [3.99213, 4.01840] [0.99945, 1.00069] [0.99540, 1.00197] [0.999907, 1.000092] [999.594, 1000.365] [999.978, 1005.662] [1648.825, 1649.133] [1645.8010, 1650.6047] [199.487, 199.680] [199.303, 200.179]
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Index Change from + 1 to - 1 (Displacement of
363
A) Displacement of Share Parameter
Share Parameter) 6 6 4 4 2 2 0 3.995
4
0
4.005
4.01
4.015
Alpha (1,1) 3.99
4
4.01
4.02
Alpha (1,1) = Alpha (3,2)
B) Displacement of Initial Endowment
B) Index Change from - 1 to + 1 (Displacement of Elasticity of Substitution)
6
6 4
4 2
2 0 995
1000
1005
Endowment (1,1) = Endowment (3,2)
0 .198
.199
.2
.201
Sigma (1) = Sigma (3) Fig. 4. Examples of index changes under parametric displacement around the specification used for the example in Table 2. A. Index change from −1 to +1 (displacement of elasticity of substitution). B. Index change from + 1 → −1 (joint displacement of share parameter).
71; 523; 144 combinations of such price vectors. If we again restrict ourselves to the symmetric settings (e.g., 0.12, 0.34, 0.50, 0.66, 0.88), which yields 4948 21 ¼ 1176 combinations of 5 price vectors, 434 trials succeed in generating 5 equilibria. The frequency of 5 equilibria in this case 434 (restricted to symmetric cases) is now 1176 ¼ 217 586 ¼ 37:03%, a considerably larger number. 5.2. The frequency of 3 equilibria To assess the frequency of 3 equilibria we assume that the initial endowment for consumer 1 of good 1 is 1000.000, and the initial endowments for consumer 1 of good 2 and for consumer 3 of good 1 are both 0. The other 3 initial endowments are used as 3 free parameters to generate supporting parameterizations for 3 equilibria. We assume CES utility functions in this case. If we again restrict ourselves to equally spaced single digit relative price vectors (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), this yields 987 321 ¼ 126 combinations of 3 price vectors (0.1, 0.5, 0.7, for instance). 42 trials succeed in generating parameterizations supporting
Fig. 5. The behavior of equilibrium manifolds under sample parameter displacements. A. Displacement of share parameter α11. B. Displacement of initial endowment W11 = W32. 1 3 equilibria. The frequency of 3 equilibria in this case is 42 84 ¼ 2 ¼ 50:00%. 5 If σ 1 ¼ σ 2 ¼ σ 3 ¼ 14, however, the frequency of 3 equilibria is 35 84 ¼ 12 ¼ 41:67%.
Table 5 The frequency of multiple equilibria in a 3 consumer 2 good CES/LES pure exchange economy using a single digit grid in price space. A. Experiments • We consider all possible 5, 3 or single combinations of single digit relative price on the unit interval (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9); 126 in the 5 price combination case, 84 in the 3 price combination case, 9 in the single digit price case. • For each combination we assess the feasibility of parametric support for the prespecified equilibria for the model selected, taking the determination of a supporting parameterization as evidence of the presence of the required number of multiple equilibria. • We assume substitution elasticity parameters σ1 = σ2 = σ3 equal 0.2, share parameters are α11 = α32 = 4 and α12 = α31 = α12 = α22 = 1. The initial endowment (less of minimum requirement) for consumer 1 of good 1 is E11 = 1000.000 for 5 equilibrium case. We also assume that initial endowments (less of minimum requirements) for consumer 1 of good 2 and for consumer 3 of good 1 are E31 = E12 = 0 for 3 equilibrium case. B. Results (in the single digit price grid case) 3 • Frequency of 5 equilibria 126 ¼ 2:38%: • Frequency of 3 equilibria 42 84 ¼ 50:00%: • Frequency of 1 equilibrium 99 ¼ 100:00%:
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6. Concluding remarks In this paper we present an example of 5 equilibria in a 3 individual 2 good CES/LES pure exchange economy, which we use to give example specific insights on the behavior of equilibrium manifolds. While a special case, this goes beyond existing literature which to our knowledge only considers cases of 3 equilibria, and simply presents examples. We go further and also explore the characteristics of neighboring zones of multiple equilibria. We prespecify an arbitrary number of price equilibria for a chosen model, and recover parameterizations for the model consistent with these multiple equilibria. Once found, we can perturb and displace around these equilibria to see how large the zones of multiple equilibria are in parameter space. As is well known (Mas-Colell (1985)) no general analytical results are available on the multiplicity issue. Our aim here is to use numerical simulation techniques in a new way to search out insights on the issue for a specific model. Any results are suggestive (in the absence of other literature) rather than definitive. Several important insights emerge. We find cases where 5 equilibria are present for this model and show that 5 is also an upper bound. 5 equilibria, however, seem to persist in parameter space (shares, elasticities, endowments) only for very small ranges of parametric displacement. These parameter ranges, however, correspond to much larger price ranges. These 5 equilibria also deform to smaller numbers of equilibria in all manner of ways as model parameters change and equilibrium price manifolds are generated. Examples exist where indices change sign, and we show cases where unique equilibria exist at sharply different prices for different parts of a price map connected by a continuous deformation over a small parameter interval. Ad hoc tests for uniqueness in applied models may thus encounter the problem that while
confirmatory, there is no explanation in economic theory for small changes in economic behavior over some parameter ranges, but larger changes over others. We conclude with some calculations for this model structure which, while suggestive, yield estimates of the frequency of multiple equilibria of different numbers for this model. References Eaves, B. Curtis, 1978. Computing stationary points. Math. Program. Study 7, 1–14. Forster, Edward, Sonnenschein, Hugo F., 1970. Price distortion and economic welfare. Econometrica 38, 197–281. Kehoe, Timothy J., 1980. An index theorem for general equilibrium models with production. Econometrica 48, 1211–1232. Kehoe, Timothy J., 1982. Regular production economies. J. Math. Econ. 10, 147–176. Kehoe, Timothy J., 1985. A numerical investigation of multiplicity of equilibria. In: Manne, A.S. (Ed.), Economic Equilibrium: Model Formulation and Solution. Mathematical Programming Study, 23. North-Holland, Amsterdam. Kehoe, Timothy J., 1991. Computation and multiplicity of equilibria. In: Hildenbrand, W., Sonnenschein, H. (Eds.), Handbook of Mathematical Economics, vol. 4. Elsevier Science Publishers B.V. (Chapter 38). Kehoe, Timothy J., Whalley, John, 1985. Uniqueness of equilibrium in large scale numerical general equilibrium models. J. Public Econ. 28, 247–254. Mas-Colell, Andreu, 1985. The Theory of General Economic Equilibrium — A Differentiable Approach. Cambridge University Press. Samuelson, Paul Anthony, 1947. Foundations of Economic Analysis. Harvard University Press. Shapley, Lloyd S., Shubik, Martin, 1977. An example of a trading economy with three competitive equilibria. J. Polit. Econ. 85, 873–875. Shoven, John B., Whalley, John, 1992. Applying General Equilibrium. Cambridge University Press. Smale, Steve, 1976. A convergent process of price adjustment and global Newton methods. J. Math. Econ. 3, 1–14. Sonnenschein, Hugo F., 1973. Do Walras' identity and continuity characterize the class of community excess demand functions. J. Econ. Theory 6, 345–354.
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Parametric persistence of multiple equilibria in an economy directly calibrated to 5 equilibria☆ John Whalley a,1, Shunming Zhang b,c,⁎ a b c
Department of Economics, Faculty of Social Science, University of Western Ontario, London, Ontario N6A 5C2, Canada China Financial Policy Research Center, Renmin University of China, Beijing 100872, PR China School of Finance, Xingjiang University of Finance and Economics, Urumqi, 830012, PR China.
a r t i c l e
i n f o
Article history: Accepted 12 May 2014 Available online 19 June 2014 Keywords: Multiple equilibria Equilibrium manifolds Index theorem Parametric variation Pure exchange economies
a b s t r a c t In existing literature, there are few concrete examples of multiple equilibria and the only ones known to us have 3 equilibria, but multiplicity remains a major concern for applied models used in policy work. Here, we report numerical examples for a 3 individual 2 good CES/LES pure exchange economy directly calibrated to 5 equilibria. We are able to use analytical methods of the model to show that for certain parameterizations there are no more than 5 equilibria (given the parameter values). We are also able to explore the size of the regions of the parameter space for which 5 equilibria persist, and show these ranges to be very small. Other features of the equilibrium manifolds are explored. Findings are only suggestive and indicative for the special cases we consider rather than definitive, but informative relative to existing work. © 2014 Elsevier B.V. All rights reserved.
1. Introduction Despite discussion of multiplicity of equilibria in theoretical literature (see the Mas-Colell (1985)) less is known about how severe a problem it is in practice. At a theoretical level, Mas-Colell (1985) suggests that all that can be said on the multiplicity issue is contained in the index theorem, i.e., the two propositions that equilibria have plus and minus one indices associated with them derived (in the pure exchange case) from the sign of the determinant of the negative of the Jacobian matrix with the first row and column deleted, and that their sum is always plus one. But how often multiple equilibria occur and in which types of models, and how many equilibria can be involved remain issues on which there is little literature. Does multiplicity occur over
☆ Whalley acknowledges the financial support from Ontario Research Fund and The Center for Intentional Governance Innovation (CIGI, Waterloo, Ontario). Zhang acknowledges the financial support from the Major Basic Research Plan of Renmin University of China (Grant 14XNL001) and the National Natural Science Foundation of China (NSFC Grant 71273271). Presentations of this work have been given to seminar groups at Zhongshan University, University of Warwick and University of Western Ontario. We are grateful to comments from these groups, as well as to Zeke Wang, Srihari Govindan and Timothy J. Kehoe for discussions. ⁎ Corresponding author. School of Finance, Renmin University of China, Beijing 100872, PR China. Tel./fax: +86 10 8250 0626. E-mail addresses: [email protected] (J. Whalley), [email protected], [email protected] (S. Zhang). 1 Tel.: +1 519 661 3509; fax: +1 519 661 3666.
http://dx.doi.org/10.1016/j.econmod.2014.05.015 0264-9993/© 2014 Elsevier B.V. All rights reserved.
concentrated portions of model admissible parameter spaces, or does it occur intermittently and somewhat unpredictably? Are they widely separated? Are they largely a theoretical curiosum, or are they the norm? To our knowledge, there are only two concrete examples of multiple equilibria in the literature and both only present three equilibria. One is due to Shapley and Shubik (1977), who use an ingenious representation of preferences that yields this multiplicity. The other is a later example by Kehoe (1985) in which he constructs a model parameterization which guarantees an equilibrium with an index of sign −1. Since the indices across equilibria must sum to + 1, two other equilibria must exist which Kehoe then is able to find. In this paper, we show both how to find parameter values simultaneously consistent with a larger prespecified number of equilibrium price vectors and explore how large is the range of parametric displacement that preserves multiplicity of various orders. For a two good, three individual LES pure exchange economy we are able to both show that for particular model parameterizations no more than 5 equilibria exist and to produce an example of an economy with this number of equilibria. Having constructed our example, we then are able to move around in the model parameter space to evaluate whether multiple equilibria persist in neighboring regions of the parameter space. We claim no general results, all are for one particular model structure and functional form; results are suggestive, not definitive. Nevertheless, the results are both revealing and informative relative to literature.
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
Multiplicity poses special problems for numerical simulation equilibrium exercises now widely used to guide policy by governments around the world (see Shoven and Whalley (1992)). If multiple equilibria characterize the model, which equilibrium does the economy actually go to under a given policy change? Most numerical modelers seem to believe that with simple functional forms and, effectively, small dimensions in their equilibrium structures (goods and factor models that reduce to factor space) their models have unique equilibria on the basis of various ad hoc tests. They use displacement from computed equilibria and recomputation under different speeds or step sizes and initial starting points and argue that they seemingly always return to the same equilibrium. But theorists remain skeptical and point to nonpathological examples of multiple equilibria in simple models (see Kehoe (1991)). Sonnenschein's (1973) result clearly suggests that if continuity is the only property one can ascribe to market excess demand functions that satisfy Walras' Law in a pure exchange case, then it should be possible to always calibrate some model of an economy to any arbitrary number of equilibria. Here, the issue explored is how easily this can be done for a parsimonious model using the types of functional forms widely used in applied models. We are both able to show for particular models and parameterizations that there is an upper bound to the number of equilibria, and to construct examples of multiple equilibria directly using calibration methods. What is needed is to have sufficient degrees of freedom to fit the same model to more than one equilibrium. With Cobb–Douglas preferences in a pure exchange economy and prespecified endowments this is not possible, and CES only adds one extra degree of freedom. However, with LES (or Stone–Geary) preferences (or variable endowments, effectively the same thing), sufficient degrees of freedom are created to provide for a larger number of equilibria for any given model. In the paper we show that calibration to five prespecified equilibrium price vectors is possible for this model, but not for all combinations of vectors, i.e. certain combinations of prices are not admissible of 5 equilibria for the model. Moreover, as we move away from these equilibria in the model parameter space (shares, elasticities, and endowments minus minimum requirements), the ranges of parameter values for which five equilibria persist seem to be surprisingly small. The 5 equilibria occur, but only for small portions of the admissible parameter space. Also, ranges of 5 equilibria in parameter space may be small, but much larger ranges in price space for these equilibria seem to be involved, raising the question of whether multiplicity should be judged as more or less likely to occur in parameter or price space. The five equilibria we construct all have the appropriate Kehoe indices (three +1, two − 1), but as equilibrium paths collapse to three or one equilibria under parametric perturbation signs of indices along equilibrium paths can change. Sometimes the five equilibrium paths in parameter space collapse to one central and two extreme paths, even to three upside or downside paths or only one path. Effectively, +1 equilibria can transform into −1 equilibria and vice versa. This is a phenomenon that has been speculated on, but as far as we know has gone unrecorded in previous literature for particular models. We also show cases where for large regions of the parameter space unique equilibria are found, but these are for sharply different price vectors in alternative regions of the parameter space due to deformations along the path of equilibria. Uniqueness may thus seem to hold in some applied models, but different regions of the parameter space may exhibit sharply divergent unique equilibria with, seemingly, a large jump in behavior as hard to reconcile with underlying economic theory. In the final section, we report on calculations of frequencies of multiple equilibria in model in the calibration sense discussed above. We use a single digit grid in price space, and consider all potential combinations of five grid points as collections of equilibria for model calibration. We keep track of cases where construction of 5 equilibria is possible and where it is not, yielding an estimate of what fraction of potential
357
combinations of 5 price equilibria in the grid have admissible parameterizations for 5 equilibria. For the 3 individual, 2 good CES/LES pure exchange model we consider, this probability (in this sense) we put at 2.38% in the case of a grid over single digit price vectors, which we suggest is a small number. 2. Constructing an economy with 5 equilibria The literature thus far on multiple equilibria has been heavily influenced by theoretical work due to Kehoe (1980, 1982, 1985) which is summarized and extended by Mas-Colell (1985). This shows that any equilibrium has associated with it an index (the sign of the determinant of the negative Jacobian matrix with the first row and column deleted in the pure exchange case) which is either +1 or −1. The sum of these indices must be plus one, and so an odd number of equilibria are always present (see also Eaves (1978)). Kehoe's index theorem is reminiscent of Samuelson's (1947) separation theorem which shows that equilibria are alternately stable and unstable. Beyond this, there are only isolated examples of three equilibria. An early and neglected example of 3 equilibria due to Shapley and Shubik (1977) uses an ingenious preference representation to obtain this outcome. Beyond this, there exist an example of 3 equilibria for multi household economies with activity analysis exist due to Kehoe (1985), a numerical demonstration of uniqueness of equilibrium in a large dimensional general equilibrium US tax model due to Kehoe and Whalley (1985), and intuition by Forster and Sonnenschein (1970) as to why tax distortions may introduce multiplicity. Mas-Colell, after extensive discussion of multiplicity in pure exchange economies, is dubious that anything more than the index theorem can be stated on the issue. Sonnenschein's (1973) paper had earlier shown that the only restriction that can be placed on market demand functions that satisfy Walras' Law is continuity. In this sense, constructing examples of multiplicity of equilibria might seem an uninteresting exercise, since it should always be possible to construct underlying structural forms for economies in terms of household preferences and endowments for a pure exchange case which yield the required number of prespecified equilibria. Rather, as Mas-Colell (1985) argues, the more interesting issue is whether or not multiplicity occurs in particular types or forms of economies. Here, we focus more broadly on models of economies employing so called ‘convenient’ functional forms (Cobb–Douglas, CES, LES) widely adopted in applied models (see Shoven and Whalley (1992)) and used to analyze the impacts of such policy changes as tax reform and trade liberalization. We also use small dimensional parsimonious models so as to ease computational requirements, focusing, as will be seen below and for reasons given later, on a 3 individual 2 good pure exchange model with CES/LES preferences. Our aim is to use numerical simulation methods to generate insights as to when multiplicity occurs in specific cases, how large the regions of multiplicity within parameter spaces are, and how regions with various numbers of equilibria are connected. Our point of departure relative to earlier literature is to directly generate parameter values for models in which equilibrium conditions simultaneously hold for a series of prespecified price vectors, and then to explore the robustness of these multiple equilibria in price space. In a pure exchange economy with L goods, equilibrium is characterized by L market clearing demand–supply equalities with L − 1 relative prices being endogenously determined. We assume that I consumers (i = 1,…,I) are considered. We use a Stone–Geary/CES demand structure of the form
C il ¼ M il þ
α il
XL σi
Pl
P 0 ðW il0 −M il0 Þ l0 ¼1 l ; X L 1−σ α 0P 0 i l0 ¼1 il l
i ¼ 1; …; I; l ¼ 1; …; L
ð1Þ
358
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where Cil denote consumptions of good l for individual i, Mil define minimum requirements of good l for individual i, αil are share parameters of good l for individual i (∑Ll = 1 αil = 1), Wil denote endowments of good l for individual i, Pl is price of good l, and σi is the elasticity of substitution in preferences for individual i. As the demand system for individual i has (2L + 1) parameters rather than L as in the Cobb–Douglas case, simply adding consumers to a model will generate enough degrees of freedom that in principle it should be possible to calibrate a pure exchange economy model to any arbitrary number of equilibria. We could also use a CES demand structure and allow endowments to be endogenously determined through calibration (effectively the same). For the case where there are 3 individuals and 2 goods, there are 15 parameters (αil, σi, Wil − Mil) in the model, although with normalization of the αil to sum to one for any i only 12 are free parameters. If data on K equilibria are to be fitted to this system, there are 2K equilibrium conditions to be satisfied (2 market clearing conditions per equilibrium), although using Walras' Law this reduces to K restrictions. If we consider particular values of elasticities and share parameters for this model, more concrete statements as to the number of equilibria which exist can be made. Defining Xil = Cil − Mil and Eil = Wil − Mil, we have XL
α il P l0 Eil0 0 X il ¼ σ XL l ¼1 1−σ ; i Pl α 0P 0 i l0 ¼1 il l
i ¼ 1; …; I; l ¼ 1; …; L:
σ P1 i
α ðP E þ P 2 Ei2 Þ E þ PEi2 i1 1 i1 ¼ i1 ; i ¼ 1; …; I 1−σ i 1−σ i 1 þ α i P 1−σ i α i1 P 1 þ α i2 P 2
5
þ½ðα 2 þ α 3 ÞE12 þ ðα 3 þ α 1 ÞE22 þ ðα 1 þ α 2 ÞE32 q þðE12 þ E22 þ E32 Þq−ðα 1 E11 þ α 2 E21 þ α 3 E31 Þ:
If, in addition, we consider the special (symmetric) case α2 = 1, α1α3 = 1, E11 = E32, E21 = E22, and E31 = E12, then 9 F ðqÞ ¼ ðα 1 E11 þ α 2 E21 þ α 3 E31 Þ q −1 8 −ðE11 þ E21 þ E31 Þ q −q
5 4 þ½ðα 1 þ α 2 ÞE11 þ ðα 3 þ α 1 ÞE21 þ ðα 2 þ α 3 ÞE31 q −q
and defining a0 = α1E11 + α2E21 + α3E31, a1 = E11 + E21 + E31, and a2 = [(α1 + α2)E11 + (α3 + α1)E21 + (α2 + α3)E31], yields 9 8 5 4 4 F ðqÞ ¼ a0 q −1 −a1 q −q þ a2 q −q ¼ ðq−1Þq f ðqÞ
a0 −a1 N0 a0
ð12Þ
ð4Þ x1 x2 þ x1 x3 þ x1 x4 þ x2 x3 þ x2 x4 þ x3 x4 ¼ −
x1 x2 x3 þ x1 x2 x4 þ x1 x3 x4 þ x2 x3 x4 ¼ 2 ð5Þ x1 x2 x3 x4 ¼
α 1 E11 þ PE12 α E þ PE22 þ 2 21 P σ 1 1 þ α 1 P 1−σ 1 P σ 2 1 þ α 2 P 1−σ 2 α E þ PE32 þ σ33 31 : P 1 þ α 3 P 1−σ 3
ð11Þ
system f(q) = 0. If the 4 solutions are x1, x2, x3, and x4, then, for 0 b a0 b a1 b a2,
and E12 þ E22 þ E32 ¼
ð10Þ
ð3Þ
where α i ¼ ααi2 and P ¼ PP2 . i1 1 For the case of 3 individuals, market clearing requires that, for goods 1 and 2, E þ PE12 E þ PE22 E þ PE32 ¼ 11 þ 21 þ 31 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3
ð9Þ
4
−½α 1 ðα 2 þ α 3 ÞE11 þ α 2 ðα 3 þ α 1 ÞE21 þ α 3 ðα 1 þ α 2 ÞE31 q
x1 þ x2 þ x3 þ x4 ¼ −
α ðP E þ P 2 Ei2 Þ α i2 1 i1 ¼ σi X i1 ; i ¼ 1; …; I X i2 ¼ σ 1−σ 1−σ P i P 2 i α i1 P 1 i þ α i2 P 2 i
8
4 3 2 1 1 1 þ ða0 −a1 Þ q þ −ð3a0 þ a1 Þ q þ − where f ðqÞ ¼ a0 q þ q q q 2ða0 −a1 Þ q þ 1q þ ða0 þ a1 þ a2 Þ: It follows that there are at most 4 solutions q þ 1q for the equation
and
E11 þ E21 þ E31
9
F ðqÞ ¼ ðα 2 α 3 E12 þ α 3 α 1 E22 þ α 1 α 2 E32 Þq −α 1 α 1 α 3 ðE11 þ E21 þ E31 Þq
ð2Þ
For the 2 good case (L = 2), X i1 ¼
which yields a polynomial in q, F(q), for which zeros yield equilibria in q (and hence P). Thus
ð6Þ
10
3a0 þ a1 b0 a0
ð13Þ
a0 −a1 b0 a0
ð14Þ
a0 þ a1 þ a2 N0: a0
ð15Þ
Fo (P)
From Walras' Law, these equations imply that, 1−
1 1 1 þ 1− þ 1− E E E31 11 21 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3 P P P ¼ E12 þ E22 þ E32 : 1 þ α 1 P 1−σ 1 1 þ α 2 P 1−σ 2 1 þ α 3 P 1−σ 3
5
0
ð7Þ For the case where σ 1 ¼ σ 2 ¼ σ 3 ¼ 15, we can show that at most 5 equilibria will exist. Defining P ¼ q (that is, P = q5), we can rewrite Eq. (7) as
-5
1 5
α1 α2 α3 q E11 þ E21 þ E31 ¼ E12 1 þ α 1 q4 1 þ α 1 q4 1 þ α 2 q4 1 þ α 3 q4 q q þ E22 þ E32 1 þ α 2 q4 1 þ α 3 q4
ð8Þ
P
-10 0
2
4
Price Ratio Fig. 1. 5 Equilibria generated as zeros of the F0(P) function for model parameterization set out in the text.
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
359
3. Ranges of parametric variation supporting multiple equilibria
From Eqs. (13) and (14) there exists at least one negative solution for the equation f(q) = 0. From Eq. (15) the negative solutions occur in pairs. Therefore the number of negative solutions is 2 or 4. That is to say, there are at most 2 positive solutions q þ 1q for the equation
Table 1 presents the model parameterization consistent with the 5 price equilibria shown in Fig. 1 for a 3 individual 2 good CES/LES pure exchange economy. The price equilibria in this particular example are chosen arbitrarily. These equilibria each have Kehoe indices as reported, and these are consistent with the index theorem and sum to +1. A feature of the equilibria in Table 1 is that they are widely separated in price, quantity and utility space for consumers. Local analysis searching for further equilibria in the neighborhood of each of these does not yield other equilibria. The model parameterization set out in Table 1 can be used for a range of experiments to provide insight on how regions of multiple equilibria in parameterization space behave for this model. How large are regions of multiplicity? How do they change as one moves between them? Are they continuous; do they have holes, for instance? How far can one move in parameter space via perturbations of shares, elasticities and endowments while still preserving five equilibria, before reverting to either three or one? Are such ranges of equilibria large or small, and for which parameters? Generating model parameterizations which support such multiple equilibria involves numerically solving a non-linear system of equations in the shares, minimum requirements and elasticities in preferences in the model. We employ GAMS optimization software and use a procedure of repeated restart through the parametric solution space (αil, σi, Eil) to search for multiple equilibria. Choice of both initialization
f(q) = 0, implying that there are at most 4 positive solutions for q. Thus, in this case, there are at most 5 positive solutions (q) for the equation f(q) = 0, and therefore at most 5 nondegenerate equilibria exist. In Fig. 1 we show an example of zeros of the F 0 ðP Þ ¼ F P function 1 5
yielding the five equilibrium price vectors (4, 1), (2, 1), (1, 1), (1, 2), and (1, 4). This case has been generated where not only are substitution elasticity parameters σ1 = σ2 = σ3 equal, but they also all equal to 0.2. We have set share parameters α11 = α32 = 4 and α12 = α31 = α12 = α22 = 1, and then calculated effective endowments (net of minimum requirements) of E11 = E32 = 1000.000, E21 = E22 = 1648.973 and E31 = E12 = −199.596 to yield these specific 5 equilibria. If, instead, σ 1 ¼ σ 2 ¼ σ 3 ¼ 14 , it can also be shown that at most 5 equilibria exist if 3a0 − 3a1 + a2 N 0 and at most 3 equilibria exist if 3a0 − 3a1 + a2 b 0. If σ 1 ¼ σ 2 ¼ σ 3 ¼ 13, at most 5 equilibria exist. If σ 1 ¼ σ 2 ¼ σ 3 ¼ 12, at most 3 equilibria exist if 3a0 − a1 + a2 b 0, and there exists a unique equilibrium if 3a0 − a1 + a2 ≥ 0. If σ1 = σ2 = σ3 = 1, 2, ⋯ (natural number), there exists a unique equilibrium. No such results on the number of equilibria for specific parameterizations of models occur, to our knowledge, in currently available literature.
Table 1 Characteristics of a 3 individual 2 good CES/LES pure exchange economy with the 5 equilibria represented in Fig. 1. A. Model characteristics • Pure exchange economy with 3 individuals and 2 goods • CES/LES utility function B. Model parameterization Substitution elasticity parameters σ1 = σ2 = σ3 = 0.2 Share parameters Consumer 1 Consumer 2 Consumer 3 Initial endowments Consumer 1 Consumer 2 Consumer 3
Good 1
Good 2
4 1 1
1 1 4
Good 1
Good 2
1000.000 1648.973 0.000
0.000 1648.973 1000.000
Minimum requirements Consumer 1 Consumer 2 Consumer 3
Good 1
Good 2
0.000 0.000 199.596
199.596 0.000 0.000
C. Equilibria and Kehoe indices Demands by individuals Equilibria
Price vectors
Individual 1
Individual 2
Individual 3
Equilibrium 1 Equilibrium 2 Equilibrium 3 Equilibrium 4 Equilibrium 5
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
(114.775, 221.342) (418.492, 290.670) (640.323, 359.677) (787.222, 425.678) (877.700, 489.120)
(2045.079, 1549.930) (1804.802, 1571.081) (1648.973, 1648.973) (1571.081, 1804.802) (1549.930, 2045.079)
(489.120, 877.700) (425.679, 787.222) (359.677, 640.323) (290.670, 418.492) (221.342, 114.775)
Utility values by individuals Equilibria
Price vectors
Individual 1
Individual 2
Individual 3
Kehoe indices
Equilibrium 1 Equilibrium 2 Equilibrium 3 Equilibrium 4 Equilibrium 5
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
20.289 73.902 112.918 138.768 154.767
1443.297 1402.595 1386.615 1402.595 1443.297
154.767 138.768 112.918 73.902 20.289
+1 −1 +1 −1 +1
360
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
and step size are critical in the practical application of these techniques. We finely divide the potential parameter space into small slivers using upper and lower bounds so as to exhaustively search for parameterization supporting multiple equilibria. This yields solutions similar to those that would be obtained using path following global Newton methods (see Smale, 1976). When we find supporting parameterizations for particular prespecified multiple equilibrium price vectors, we then make small parametric variations in model parameterizations once these are found and follow paths of equilibria (equilibrium manifolds in Mas-Colell's terminology) through the neighborhood. Results from such experiments are reported in Table 2. They suggest that the ranges of parameter changes that allow 5 equilibria to be preserved in this case are extremely small. For elasticity parameters, only very small changes in the neighborhood of 0.2 are possible, between 0.1976 and 0.20108 for σ1 and σ3. Dropping elasticity values further moves the model economy into a range of three equilibria, which persists until these parameters become small. Increasing elasticities produces a slightly larger (but still small) range of elasticity values with 3 equilibria, followed by an open ended range with one equilibrium. Results from perturbing share parameters again yields small ranges for 5 equilibria, but there are neighboring ranges of 3 and 1 equilibria on both sides of the parameterization reported in Table 1. Only with parametric changes in minimum requirements do regions of 5 equilibria reappear, once the initial region around the pre-specified price vectors is departed from. The results for each row of Table 2 can be represented diagrammatically showing how paths of equilibria are connected as one moves through parameter space, and how regions of multiple equilibria change and are related to each other. Fig. 2 shows results for the first row of Table 2 (sigma 1), and also the case where individual 1's endowments of good 1 are perturbed. These cases have been drawn so as to display the portion of parametric variation over which manifolds deform and serve to illustrate how equilibrium manifolds behave in these cases.
4. The behavior of equilibrium manifolds in neighboring regions of the parameter space Having generated a parameterization for a 3 individual, 2 good CES/ LES economy which supports the 5 equilibria represented in Fig. 1, we can use numerical simulation methods to explore the behavior of equilibrium manifolds in neighboring regions in the parameter space. Can we easily change the prespecified price vectors, for instance, and still find parameterizations which support 5 equilibria? How much can we change share, endowment, and other model parameters and still find 5 equilibria supporting parameterizations? Are there examples of index changes as we move between regions of multiple equilibria? How often do they occur; do some cases differ from others? And what of the behavior of equilibrium price paths across cases?
4.1. Changing prespecified price vectors The example of 5 multiple equilibria presented in Fig. 1 involves 5 prespecified equilibria which are both symmetric and dispersed in price space. Is this more likely to be a case for which 5 equilibria occur? Suppose, for instance, that three equilibria are close together and the other 2 are dispersed, some distance away in price space; would this latter case be one for which 5 equilibria also occur? In Table 3 we report maximum changes in the coordinates of the individual price vectors used in Fig. 1 which still allow five equilibria to occur. We report maximum displacements of the second and fourth price vectors such that five equilibria can be preserved. If both of these vectors are displaced together, the 3.333 and 6.667 coordinates of vectors of these vectors can be displaced in one direction to 2.112 and 7.888, relatively close to the neighboring vectors of 2.000 and 8.000, and can be displaced in the other to 4.907 and 5.093, relatively close to the neighboring vector of 5.000. This suggests a relatively wide range of displacement in price vector prespecification, but if only the
Table 2 Maximum parameter perturbations around model specification in Table 1 preserving multiple equilibria (asymmetric setting).1 Model parameters subject to perturbation Substitution elasticities Sigma 1 Sigma 2 Sigma 3 Share parameters Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 Initial endowments Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 Minimum requirements Individual 1, good 1 Individual 1, good 2 Individual 2, good 1 Individual 2, good 2 Individual 3, good 1 Individual 3, good 2 1 2
Parameter values used in generating example in Table 12
Parameter range yielding 1 equilibrium
0.200 0.200 0.200
Parameter range yielding 3 equilibria
Parameter range yielding 5 equilibria
Parameter range yielding 3 equilibria
Parameter range yielding 1 equilibrium
[0.1975] [0.199615] [0.1975]
[0.1976, 0.20108] [0.199616, 0.200189] [0.1976, 0.20108]
[0.20109, 0.2030] [0.20109, 0.2030]
[0.2031] [0.20019] [0.2031]
4.000 1.000 1.000 1.000 1.000 4.000
[3.9952] [0.9965] [0.99979] [0.99979] [0.9965] [3.9952]
[3.9953, 3.9972] [0.99651, 0.99944] [0.999791, 0.999906] [0.999791, 0.999906] [0.99651, 0.99944] [3.9953, 3.9972]
[3.9973, 4.0022] [0.99945, 1.00069] [0.999907, 1.000092] [0.999907, 1.000092] [0.99945, 1.00069] [3.9973, 4.0022]
[4.0023, 4.0139] [1.0007, 1.0012] [1.000093, 1.000209] [1.000093, 1.000209] [1.0007, 1.0012] [4.0023, 4.0139]
[4.014] [1.0013] [1.00021] [1.00021] [1.0013] [4.014]
1000.000 0.000 1648.973 1648.973 0.000 1000.000
[999.199]
[999.200, 999.593]
[1000.366, 1001.018]
[1001.109]
[1648.566] [1648.566]
[1648.567, 1648.824] [1648.567, 1648.824]
[1649.134, 1649.296] [1649.134, 1649.296]
[1649.297] [1649.297]
[999.199]
[999.200, 999.593]
[999.594, 1000.365] 0.000 [1648.825, 1649.133] [1648.825, 1649.133] 0.000 [999.594, 1000.365]
[1000.366, 1001.018]
[1001.109]
[199.444]
[199.445, 199.486]
[199.681, 199.920]
[199.921]
[199.444]
[199.445, 199.486]
[199.681, 199.920]
[199.921]
0.000 199.596 0.000 0.000 199.596 0.000
0.000 [199.487, 199.680] 0.000 0.000 [199.487, 199.680] 0.000
Each row of the table represents parametric variation in the named parameter alone. The symbols [and] refer to bounds of regions in parameter space for 5, 3, and 1 equilibria. This model specification has been constructed so as to be consistent with the 5 equilibrium prices (4, 1), (2, 1), (1, 1), (1, 2), and (1, 4).
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Parametric Variation in Individual 1’s Substitution Elasticity (Row 1 of Table 2) 6
361
second price vector alone is displaced, the region of feasible displacement is restricted to 2.828 and 7.172 (Fig. 3). 4.2. Displacing parameter values so as to preserve the ability to produce 5 price equilibria
Price Ratio
A related experiment is to see how large the displacements are that can be made either singly or jointly in all parameters so as to preserve 5 equilibria under a modified parametric specification. Results from such variations are presented in Table 4. These show still small admissible parametric variations as elasticities and share parameters change, but show that wider ranges occur with joint variation. Some of these results are implicit in Table 2, where ranges for single parameter displacements are displayed as part of the overall pattern of multiple equilibrium regime change; but a wider variation is possible using combinations of parameters.
4
2
0 .196
.198
.2
.202
.204
Sigma (1)
4.3. Examples of index changes along price paths
B) Parametric Variation in Individual 1’s Endowment of Good 1 (Row 10 of Table 2)
4
2
We are also able to monitor index changes for equilibria (in the Kehoe sense) as we move along price paths corresponding to displacements of parametric settings around those set out in Table 1. Such index changes have been conjectured in previous literature, but to our knowledge no actual cases and examples are documented and hence these are of interest. Fig. 4 displays examples of index changes which occur for particular parametric displacements. As indicated, cases can arise for which multiple index changes are present. As an index changes along a price path, a zero index applies as the determinant of the negative Jacobian matrix with the first row and column deleted disappears. 4.4. The behavior of equilibrium manifolds under parameter displacement
0 999
999.5
1000
1000.5
1001
Initial Endowment (1,1) Fig. 2. Equilibrium manifolds in parameter–price space corresponding to results reported in Table 2. A. Parametric variation in individual 1's substitution elasticity (row 1 of Table 2). B. Parametric variation in individual 1's endowment of good 1 (row 10 of Table 2).
We are also able to show how equilibrium price manifolds behave under parametric displacement. These are shown in Fig. 5 for two particular cases of perturbation of share parameters and endowments. Case A show how multiple equilibria can occur even where only a single price path is present. This deforms over a narrow price range with large displacement in price space occurring over the non-deformed portions of the equilibrium manifold. A practical difficulty suggested by such cases is that for applied models ad hoc tests for uniqueness might be confirmatory for different parameters and locally such equilibria
Table 3 Bounds for asymmetric displacement of price vectors for which parameterizations preserving 5 equilibria are possible. A. Symmetric case in Table 3 giving 5 equilibria Price ratios to which equilibrium are calibrated Price vectors (P1, P2)
Price ratios PP
(2.000, 8.000) (3.333, 6.667) (5.000, 5.000) (6.667, 3.333) (8.000, 2.000)
4.000 2.000 1.000 0.500 0.250
Cases
2 1
A1 A2 A3 A4 A5
B. Maximum displacements of price vectors A2 and A4 preserving 5 equilibria (B2 and B4 are the implied displacements in the price of the other good) Moving toward (2.000, 8.000) Moving toward (8.000, 2.000) Moving toward (5.000, 5.000) Moving toward (5.000, 5.000)
A2 (B2) A4 (B4) A2 (B2) A4 (B4)
= = = =
(2.112, 7.888) (7.888, 2.112) (4.907, 5.093) (5.093, 4.907)
C. Maximum displacements of price vector A2 alone preserving 5 equilibria (C2 is the implied displacements in the price of the other good) Moving toward (2.000, 8.000) Moving toward (5.000, 5.000)
A2 (C2) = (2.828, 7.172) A2 (C2) = (4.044, 5.956)
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J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Maximum Displacements of Price Vectors A2 and A4 Preserving 5 Equilibria
4
3
2
1
0 180
200
220
240
260
Minimum Requirement
B) Maximum Displacements of Price Vector A2 Alone 4
3
2
5. The frequency of multiple equilibria The procedures used to generate the parameterizations reported on above can also be used to evaluate the frequency of multiple equilibria for the structure we use (a 3 consumer, 2 good, CES/LES pure exchange economy). No discussion, to our knowledge, of this issue exists in previous literature and so we have constructed our own metric of frequency. While no general statements on multiple equilibria across models can be made, in the absence of other insights in the literature some evidence on frequency of multiple equilibria for particular cases seems helpful as a first step. Our procedure is to consider a listing of prespecified price vectors and to take combinations of some order of these (say, 5 to represent parametric fitting to 5 equilibria). We attempt to calculate a 5 equilibrium model parameterization for each combination, and if this fails we calculate a 3 equilibrium parameterization. If this fails a single model parameterization is determined. We explore the frequencies of 1, 3, and 5 equilibria in this way in price space. The dimensionalities involved can rapidly become overwhelming if large numbers of price vectors are considered as candidates to provide combinations of parameterizations supporting 5 equilibria, and so we restrict ourselves to single digit and two digit grid searches. We note that the equilibrium manifolds displayed in Figs. 2, 4 and 5 also raise the issue for multiplicity discussion that the frequency of observing multiple equilibria in parameter and price space can differ, and potentially sharply so. Thus in asking how likely multiple equilibria are in particular cases, one has to be clear as to whether one is referring to frequency relative to endogenous or exogenous variables in the model. 5.1. The frequency of 5 equilibria
1
0 6
6.5
7
7.5
P(1) Fig. 3. Bounds of asymmetry in 5 equilibria preserving parameterizations specified in Tables 3 and 4. B. Maximum displacements of price vectors A2 and A4 preserving 5 equilibria. C. Maximum displacements of price vector A2 alone.
would change little, but over progressively wider ranges of parametric variation large jumps in behavior would occur which could not be accounted for by economic theory. In Case B multiple equilibria also occur but multiple price paths merge at various points of the map. These correspond to zones of 5, 3, and 1 equilibria.
For our calculation of frequency of 5 equilibria, we use initial endowments minus minimum requirements as 5 free parameters. If we restrict ourselves to equally spaced single digit relative price vectors (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), this yields 98765 54321 ¼ 126 combinations of 5 price vectors (0.1, 0.3, 0.5, 0.7, 0.9, for instance). For the 3 individual 2 good CES/LES pure exchange economy we consider, we arbitrarily select 5 of the 11 parameters (we assume that the initial endowment for consumer 1 and good 1 is 1000.000) given the 5 equilibria (see Table 5 for details). Only 3 trials succeed in attaining 5 equilibria through direct calibration methods. These are (0.2, 0.3, 0.5, 0.7, 0.8), (0.2, 0.4, 0.5, 0.6, 0.8), and (0.3, 0.4, 0.5, 0.6, 0.7). Thus the frequency of 5 equilibria in the sense of a combination of single digit price 3 1 vectors is 123 ¼ 42 ¼ 2:38%. We summarize both these results and the procedures we employ in Table 5. If we then consider two digit price settings, the relative price vectors are equally spaced (0.01, 0.02, …, 0.99). There are 9998979695 ¼ 54321
Table 4 Maximum single or joint parametric variation around the model specification in Table 2 which preserves 5 equilibria. Perturbation parameter setting used in example in Table 1
Single or joint perturbation
Parameter value in example in Table 1
Maximum displacement while preserving the ability to support 5 equilibria
Elasticity of substitution σ1 Elasticity of substitution σ1 = σ3 Elasticity of substitution σ2 Share parameter α11 Share parameter α11 = α32 Share parameter α12 Share parameter Α12 = Α31 Share parameter α21 Initial endowment W11 Initial endowment W12 = W32 Initial endowment W21 Initial endowment W21 = W22 Minimum requirement M12 Minimum requirement M12 = M31
Single Joint Single Single Joint Single Joint Single Single Joint Single Joint Single Joint
0.2 0.2 0.2 4 4 1 1 1 1000.000 1000.000 1648.973 1648.973 199.596 199.596
[0.19769, 0.20108] [0.198763, 0.200744] [0.199616, 0.200189] [3.9973, 4.0022] [3.99213, 4.01840] [0.99945, 1.00069] [0.99540, 1.00197] [0.999907, 1.000092] [999.594, 1000.365] [999.978, 1005.662] [1648.825, 1649.133] [1645.8010, 1650.6047] [199.487, 199.680] [199.303, 200.179]
J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
A) Index Change from + 1 to - 1 (Displacement of
363
A) Displacement of Share Parameter
Share Parameter) 6 6 4 4 2 2 0 3.995
4
0
4.005
4.01
4.015
Alpha (1,1) 3.99
4
4.01
4.02
Alpha (1,1) = Alpha (3,2)
B) Displacement of Initial Endowment
B) Index Change from - 1 to + 1 (Displacement of Elasticity of Substitution)
6
6 4
4 2
2 0 995
1000
1005
Endowment (1,1) = Endowment (3,2)
0 .198
.199
.2
.201
Sigma (1) = Sigma (3) Fig. 4. Examples of index changes under parametric displacement around the specification used for the example in Table 2. A. Index change from −1 to +1 (displacement of elasticity of substitution). B. Index change from + 1 → −1 (joint displacement of share parameter).
71; 523; 144 combinations of such price vectors. If we again restrict ourselves to the symmetric settings (e.g., 0.12, 0.34, 0.50, 0.66, 0.88), which yields 4948 21 ¼ 1176 combinations of 5 price vectors, 434 trials succeed in generating 5 equilibria. The frequency of 5 equilibria in this case 434 (restricted to symmetric cases) is now 1176 ¼ 217 586 ¼ 37:03%, a considerably larger number. 5.2. The frequency of 3 equilibria To assess the frequency of 3 equilibria we assume that the initial endowment for consumer 1 of good 1 is 1000.000, and the initial endowments for consumer 1 of good 2 and for consumer 3 of good 1 are both 0. The other 3 initial endowments are used as 3 free parameters to generate supporting parameterizations for 3 equilibria. We assume CES utility functions in this case. If we again restrict ourselves to equally spaced single digit relative price vectors (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9), this yields 987 321 ¼ 126 combinations of 3 price vectors (0.1, 0.5, 0.7, for instance). 42 trials succeed in generating parameterizations supporting
Fig. 5. The behavior of equilibrium manifolds under sample parameter displacements. A. Displacement of share parameter α11. B. Displacement of initial endowment W11 = W32. 1 3 equilibria. The frequency of 3 equilibria in this case is 42 84 ¼ 2 ¼ 50:00%. 5 If σ 1 ¼ σ 2 ¼ σ 3 ¼ 14, however, the frequency of 3 equilibria is 35 84 ¼ 12 ¼ 41:67%.
Table 5 The frequency of multiple equilibria in a 3 consumer 2 good CES/LES pure exchange economy using a single digit grid in price space. A. Experiments • We consider all possible 5, 3 or single combinations of single digit relative price on the unit interval (0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9); 126 in the 5 price combination case, 84 in the 3 price combination case, 9 in the single digit price case. • For each combination we assess the feasibility of parametric support for the prespecified equilibria for the model selected, taking the determination of a supporting parameterization as evidence of the presence of the required number of multiple equilibria. • We assume substitution elasticity parameters σ1 = σ2 = σ3 equal 0.2, share parameters are α11 = α32 = 4 and α12 = α31 = α12 = α22 = 1. The initial endowment (less of minimum requirement) for consumer 1 of good 1 is E11 = 1000.000 for 5 equilibrium case. We also assume that initial endowments (less of minimum requirements) for consumer 1 of good 2 and for consumer 3 of good 1 are E31 = E12 = 0 for 3 equilibrium case. B. Results (in the single digit price grid case) 3 • Frequency of 5 equilibria 126 ¼ 2:38%: • Frequency of 3 equilibria 42 84 ¼ 50:00%: • Frequency of 1 equilibrium 99 ¼ 100:00%:
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J. Whalley, S. Zhang / Economic Modelling 41 (2014) 356–364
6. Concluding remarks In this paper we present an example of 5 equilibria in a 3 individual 2 good CES/LES pure exchange economy, which we use to give example specific insights on the behavior of equilibrium manifolds. While a special case, this goes beyond existing literature which to our knowledge only considers cases of 3 equilibria, and simply presents examples. We go further and also explore the characteristics of neighboring zones of multiple equilibria. We prespecify an arbitrary number of price equilibria for a chosen model, and recover parameterizations for the model consistent with these multiple equilibria. Once found, we can perturb and displace around these equilibria to see how large the zones of multiple equilibria are in parameter space. As is well known (Mas-Colell (1985)) no general analytical results are available on the multiplicity issue. Our aim here is to use numerical simulation techniques in a new way to search out insights on the issue for a specific model. Any results are suggestive (in the absence of other literature) rather than definitive. Several important insights emerge. We find cases where 5 equilibria are present for this model and show that 5 is also an upper bound. 5 equilibria, however, seem to persist in parameter space (shares, elasticities, endowments) only for very small ranges of parametric displacement. These parameter ranges, however, correspond to much larger price ranges. These 5 equilibria also deform to smaller numbers of equilibria in all manner of ways as model parameters change and equilibrium price manifolds are generated. Examples exist where indices change sign, and we show cases where unique equilibria exist at sharply different prices for different parts of a price map connected by a continuous deformation over a small parameter interval. Ad hoc tests for uniqueness in applied models may thus encounter the problem that while
confirmatory, there is no explanation in economic theory for small changes in economic behavior over some parameter ranges, but larger changes over others. We conclude with some calculations for this model structure which, while suggestive, yield estimates of the frequency of multiple equilibria of different numbers for this model. References Eaves, B. Curtis, 1978. Computing stationary points. Math. Program. Study 7, 1–14. Forster, Edward, Sonnenschein, Hugo F., 1970. Price distortion and economic welfare. Econometrica 38, 197–281. Kehoe, Timothy J., 1980. An index theorem for general equilibrium models with production. Econometrica 48, 1211–1232. Kehoe, Timothy J., 1982. Regular production economies. J. Math. Econ. 10, 147–176. Kehoe, Timothy J., 1985. A numerical investigation of multiplicity of equilibria. In: Manne, A.S. (Ed.), Economic Equilibrium: Model Formulation and Solution. Mathematical Programming Study, 23. North-Holland, Amsterdam. Kehoe, Timothy J., 1991. Computation and multiplicity of equilibria. In: Hildenbrand, W., Sonnenschein, H. (Eds.), Handbook of Mathematical Economics, vol. 4. Elsevier Science Publishers B.V. (Chapter 38). Kehoe, Timothy J., Whalley, John, 1985. Uniqueness of equilibrium in large scale numerical general equilibrium models. J. Public Econ. 28, 247–254. Mas-Colell, Andreu, 1985. The Theory of General Economic Equilibrium — A Differentiable Approach. Cambridge University Press. Samuelson, Paul Anthony, 1947. Foundations of Economic Analysis. Harvard University Press. Shapley, Lloyd S., Shubik, Martin, 1977. An example of a trading economy with three competitive equilibria. J. Polit. Econ. 85, 873–875. Shoven, John B., Whalley, John, 1992. Applying General Equilibrium. Cambridge University Press. Smale, Steve, 1976. A convergent process of price adjustment and global Newton methods. J. Math. Econ. 3, 1–14. Sonnenschein, Hugo F., 1973. Do Walras' identity and continuity characterize the class of community excess demand functions. J. Econ. Theory 6, 345–354.