Abstract
We develop a new multistep monotone map approach to characterize minimal state-space recursive equilibrium for a broad class of infinite horizon dynamic general equilibrium models with positive externalities, dynamic complementarities, public policy, equilibrium indeterminacy, and sunspots. This new approach is global, defined in the equilibrium version of the household’s Euler equation, applies to economies for which there are no known existence results, and existing methods are inapplicable. Our methods are able to distinguish different structural properties of recursive equilibria. In stark contrast to the extensive body of existing work on these models, our methods make no appeal to the theory of smooth dynamical systems that are commonly applied in the literature. Actually, sufficient smoothness to apply such methods cannot be established relative to the set of recursive equilibria. Our partial ordering methods also provide a qualitative theory of equilibrium comparative statics in the presence of multiple equilibrium. These robust equilibrium comparison results are shown to be computable via successive approximations from subsolutions and supersolutions in sets of candidate equilibrium function spaces. We provide applications to an extensive literature on local indeterminacy of dynamic equilibrium.
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Notes
See Stokey et al. (1989).
The “technology” we specify can be interpreted as a “reduced-form” for production in various nonoptimal economies, including models with monopolistic competition, taxes, learning, production externalities, and even some cash-in-advance models. See Benhabib and Farmer (1994), Greenwood and Huffman (1995), and Datta et al. (2002) for examples of other economies that fit this structure.
For recent papers, see Shimokawa (2000), Beaudry and Portier (2007), Jaimovich (2007, 2008), Wang and Wen (2008), Guo and Harrison (2010), Antoci et al. (2011), d’Albis et al. (2012), and Huang and Meng (2012), Lloyd-Braga et al. (2014), Benhabib et al. (2013), Liu and Wang (2014), and Feng and Hoelle (2017).
See also MasColell (1986) for a comprehensive discussion.
These authors use new methods in topological dynamics of Devaney, among others, which require one to prove the existence of globally continuous sequential or recursive equilibrium. As with the applications of the methods of smooth dynamical systems (a la Grobman–Hartmann), sufficient conditions for the existence of (globally) continuous equilibrium dynamics are also not known. Although we are unable to prove existence of such topological equilibrium dynamics, our methods are in the spirit of the pioneering work of Stockman (2010) and Raines and Stockman (2010).
In this sequel, to minimize notation, after defining a function space we delete the domain from the reference if the context is clear. For example, \( B^{f}(\mathbf {K)}\) is subsequently referred as \(B^{f}.\)
It is shown in Lemma 2, N is decreasing in C, and increasing in K.
We could also allow for \(u(c)=\ln (c)\). Define \(\mathbf {R}_{-}^{*}= \mathbf {R\cup -\infty }\), the Bellman equation will be an upper semicontinuous function \(V^{*}(\cdot ;C,N):\) \(\mathbf {K}\times \mathbf {K} _{*}\mathbf {\rightarrow R}_{-}^{*}\) that is continuous, when \(K>0.\) Aside from that case, the range of the Bellman operator is actually \(\mathbf { R}_{+}.\)
The point is, in any sequential equilibrium where \(\{K_{t}^{*}\},\{N_{t}^{*}\}\) where \(k_{t}^{*}=K_{t}^{*}\) and \(n_{t}^{*}=N_{t}^{*}\) (where small letters denote individual household decisions on capital and labor), the labor, consumption, and capital decisions from the economy’s initial state \((k_{0},K_{0})\) when \(k_{0}=K_{0}\) must be consistent with a value function \(V^{*}(k_{0},K_{0};\{K_{t}\},\{N_{t}\}\) has an envelope theorem in its first argument (hence, the existence of Euler equation). It is not clear how APS-type generalized Markov equilibrium methods can guarantee this for our economies (as the expanded set of state variables are constructed from a correspondence of solutions of systems that are at best only upper semicontinuous in (k, K) when \(k=K\)).
Under Assumption A2, when the equilibrium wage rate w is increasing in \(\hat{N}\) (e.g., as in the case of Benhabib and Perli (1994) and Liu and Wang (2014), among many others), the set of contingent RE labor supply decision will be a correspondence. We should note, in any sequential equilibria for Benhabib and Farmer models, for technologies evaluated at the so-called indeterminacy parameters, this similar equilibrium labor supply decisions each period is also a correspondence. This means smoothness conditions near steady states that required to apply the Grobman–Hartman Theorem and/or stable manifold theorem are going to be problematic to check. That is, it is difficult to prove the existence of smooth equilibria, which is required to check the hypotheses needed to apply smooth dynamical systems methods to characterize the local determinacy of sequential equilibrium. This is true for both discrete and continuous time models.
Subsequent to this, we shall again omit the domain of \(\mathbf {C}^{*}\) from the notation. So, for example, \(\mathbf {C}^{*}(\mathbf {D},B^{f})\) will be denoted by \(\mathbf {C}^{*}(B^{f}).\)
Analogously, we can restrict our second-step domain by \(\bar{h}_{2}^{m}\).
Note that Feng et al. (2014) do not apply to this economy.
The Grobman and Hartmann theorem is the key result that is usually applied in this literature in justifying local methods to study determinacy of equilibrium via topological conjugacy arguments. In discrete time, this theorem requires sequential and/or recursive equilibrium dynamics near the steady state be smooth (e.g., to verify the steady state is hyperbolic and topological conjugacy). No such results on existence of smooth equilibrium dynamics in models with labor externalities (or large capital externalities) are known.
Further, given the results in Santos (1991) for Pareto optimal economies, one would assume such conditions would be very strong, requiring global strong concavity conditions for household dynamic programs along equilibrium paths. Hence, a global argument is needed to study indeterminacy.
We should mention, the case of indeterminacy occurring in economies with inelastic labor such that the equilibrium return on capital could be increasing in K is mentioned in Boldrin and Rustichini (1994).
In this case, we need to take a bound \(f^{M}\) in the definition of the second-step space \(B^{f}\) to be slightly higher: i.e., let \(\hat{N}>1,\) so \(f(K,\hat{N})\tilde{e}(K)>f(K,1)\tilde{e}(K)\). As the first-step operator uses the upper bound for output to be F(K, 1, K, 1), the first step greatest fixed point \(h^{*}(h_{2}(K))(K)<f(K,\hat{N})\tilde{e}(K)\), hence if we take \(\bar{h}_{2}=f(K,1)\tilde{e}(K)<f(K,\hat{N})\tilde{e}(K)\) in the second step, \(A^{*}(\bar{h}_{2})<\bar{h}_{2}.\)
See Hirsch et al. (2013, Chap. 7, especially Chap. 7.4).
The question of policy-induced indeterminacy have been discussed extensively in the literature. Early literature includes papers by Schmitt-Grohe and Uribe (1997), Guo and Lansing (1998), and Guo and Harrison (2004), while more recently Nourry et al. (2013), Nishimura et al. (2015), and Menuet et al. (2017) among others has addressed this issue.
Keep in mind, \(C=0\) is a trivial fixed point.
Here, “decreasing in \(\tau \)” is in the pointwise partial order: \(\tau _{1}\ge \tau _{2}\) if \(\tau _{1}(K)\ge \tau _{2}(K)\) for all K.
If the shocks are additionally countable, then all the spaces in this section are complete lattices (noting, measurability is only preserved for countable collections under pointwise partial orders).
Again, if the shocks are countable, the set of extremal recursive equilibria forms a complete lattice under pointwise partial orders.
To avoid using references to “isotone mapping,” we will often use the more traditional terminology in economics “increasing.” In the literature on partially ordered sets, an “increasing map” often denotes something slightly different (e.g., \(f(x^{\prime })\ge _{Y}f(x)\) when \(x^{\prime }>_{X}x\) for x, \(x^{\prime }\in X\)).
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Acknowledgements
We are grateful to Francesco Agostinelli, Rabah Amir, Łukasz Balbus, Robert Becker, Stefano Bosi, Dilsat Dalkiran, Jean-Pierre Drugeon, Ed Green, Martin Kaae Jensen, Takashi Kamihigashi, Robert Lucas, Olivier Morand, Ed Prescott, Manuel Santos, Yiannis Vailakis, Alain Venditti, as well as the seminar participants at Johns Hopkins, Paris I-Sorbonne, Warsaw School of Economics, 2013 SAET and DIET conferences in Paris, 2014 ANR Nuvo Tempo Workshop on Recursive Methods in Glasgow, 2015 ANR Nuvo Tempo Tempe Arizona, University of Miami Economic Theory conference, 2016 European General Equilibrium Conference in Glasgow, and two anonymous referees for their helpful comments and suggestions. We especially would like to thank M. Ali Khan for extended conversations on this project that reoriented its focus. All remaining shortcomings are our own.
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Appendix: fixed point theory in ordered spaces
Appendix: fixed point theory in ordered spaces
1.1 Mathematical terminology
Posets and lattices: A partially ordered set (or poset) is a set P ordered with a reflexive, transitive, and antisymmetric relation. If any two elements of \(C\subset P\) are comparable, C is referred to as a linearly ordered set, or chain. A lattice is a set L ordered with a reflexive, transitive, and antisymmetric relation \(\ge \) such that any two elements x and \(x^{\prime }\) in L have a least upper bound in L, denoted \(x\wedge x^{\prime },\) and a greatest lower bound in L, denoted \(x\vee x^{\prime }.\) \(L_{1}\subset L\) is a sublattice of L if it contains the sup and the inf (with respect to L) of any pair of points in \(L_{1}.\) A lattice is complete if any subset \(L_{1}\) of L has a least upper bound and a greatest lower bound in L. \(L_{1}\) is subcomplete if it is complete and a sublattice. In a poset P, if every subchain in \( C\subset P\) is complete, then C is referred to as a chain complete poset (or CPO). If every countably subchain in C is complete, then C is referred to as a countably chain complete poset (or CCPO). Let \([a)=\{x|x\in P,x\ge a\}\) be the upperset of a, and \((b]=\{x|x\in P,x\le b\}\) the lowerset of b. We say P is an ordered topological space if [a) and (b] are closed in the topology on P. An order interval is defined to be \([a,b]=[a)\cap (b]\), \( a\le b\).
Isotone (or order preserving) mappings on a poset: Let \((X,\ge _{X})\) and \((Y,\ge _{Y})\) be Posets. A mapping \(f{:}\,X\rightarrow Y\) is increasing (or isotone) on X if \(f(x^{\prime })\ge _{Y}f(x),\) when \(x^{\prime }\ge _{X}x,\) for \(x,x^{\prime }\in X.\) If \(f(x^{\prime })>_{Y}f(x)\) when \(x^{\prime }>_{X}x\), we say f is strictly increasing.Footnote 27 The mapping \(f{:}\,X\rightarrow Y\) is join preserving (resp, meet preserving) if we have for any countable chain C, \(f(\vee C)=\vee f(C)\) (resp, \(f(\wedge C)=\wedge f(C)\) ). A mapping that is both join and meet preserving is order continuous.
A correspondence (or multifunction) \(F{:}\,X\rightarrow 2^{Y}\) is ascending in a binary set relation \(\rhd \) on \(2^{Y}\) if \(F(x^{\prime })\rhd F(x),\) when \(x^{\prime }\ge _{X}x.\) Let \(\mathbf {X}\) be a poset, \( \mathbf {Y}\) a lattice, and define the relation \(\rhd =\ge _{v}\) on the range \(\mathbf {L(Y)}\) of all nonempty sublattices of \(\mathbf {Y}\), where for\( \ L_{1},L_{2}\in \mathbf {L(X)}\) we say \(L_{1}\ge _{v}L_{2}\) in Veinott’s Strong Set order if for all \(x_{2}\in L_{2},\) \(x_{1}\in L_{1},\) \( x_{1}\vee x_{2}\in L_{1},\) \(x_{1}\wedge x_{2}\in L_{2}.\)
Fixed points. Let \(F{:}\,X\rightarrow 2^{X}\) be a nonempty valued correspondence. \(x\in X\) is a fixed point of F if \(x\in F(x).\) If F is a function, a fixed point is \(x\in X\) such that \(x=F(x)\). For \(F{:}\,X\times T\rightarrow 2^{X}\) denote by \(\varPsi _{F}{:}\,T\rightarrow 2^{X}\) the fixed point correspondence of F.
1.2 Order and uniform topologies
Consider a mapping \(f{:}\,X\rightarrow Y,\) where X and Y are each countable chain complete partially ordered sets. We say f is order continuous if \(f(\vee X^{\prime })=\vee f(X^{\prime })\) and \(f(\wedge X^{\prime })\)=\(\wedge f(X^{\prime })\) for all countable chains \(X^{\prime }\subset X.\) If X and Y are additionally Banach spaces, say f is a compact operator if it is (a) continuous (relative to the norm topologies on X and Y), and (b) for any bounded \(X^{\prime }\) \(\subset X, \) \(f(X^{\prime })\subset X\) is relatively compact. We have the following result:
Proposition 3
Say X(S) a collection of functions on \(S=[0,1],\) \(X(S)\ \) compact in the topology of uniform convergence and endowed with the pointwise partial order, \(f{:}\,X(S)\rightarrow X(S)\) is isotone and compact. Then, f is order continuous on X(S).
Proof
As X(S) is compact in the topology of uniform convergence, X(S) is compact in the topology of pointwise convergence (as pointwise and uniform convergence coincide in X(S)); hence, \(X(S)\ \)is chain complete in the pointwise partial order on X(S) (Amann 1976, Corollary 3.2). As \( f{:}\,X(S)\rightarrow X(S)\) is isotone and continuous on X(S) in the topology of uniform convergence, f is continuous in the pointwise topology. This implies f is continuous in the interval topology of X(S) associated with pointwise partial orders (as interval topology in this case coincides with the uniform topology/pointwise topology in X(S)). Hence, f is order continuous in pointwise partial orders. \(\square \)
1.3 Some useful fixed point theorems
One critical result we use through the paper is Veinott’s version of Tarski’s theorem. His result is stated in the next proposition.
Proposition 4
(Veinott 1992, Chap. 4, Theorem 14). Let X be a nonempty complete lattice, T a poset, \(F{:}\,X\times T \rightarrow 2^{X}\) a nonempty, subcomplete-valued correspondence that is Veinott’s strong set order ascending. Then, (i) \(\varPsi _{F}(t)\) is a nonempty complete lattice, and (ii) \(\vee \varPsi _{F}(t)\) and \(\wedge \varPsi _{F}(t)\) are isotone selections.
Tarski’s original theorem (Tarski 1955, Theorem 1) occurs as a special case of Proposition 4, where \(F(x,t)=f(x),\) and \(f{:}\,X\rightarrow X \) is a function. An important extension of Tarski’s theorem is given by Markowsky (1976, Theorem 9) and is stated in the next proposition. The fixed point comparative statics result in the proposition per least (resp, greatest) fixed points is a corollary of a theorem proven in Heikkila and Reffett (2006, Theorem 2.1) which in turn implies \(t\rightarrow \varPsi _{F}(t)\) is weak-induced ascending upward and downward.
Proposition 5
Let X be a CPO, T a poset, \(f{:}\,X\times T\rightarrow X\) increasing in the product order on \(X\times T\). Then, (i) for each \(t\in T\) \(\varPsi _{F}(t)\) is nonempty CPO, (ii) \(t\rightarrow \wedge \varPsi _{F}(t)\) (resp, \(\vee \) \(\varPsi _{F}(t))\) are increasing selections.
For our results, we will need constructive versions of Propositions 4 and 5. For this, we will assume for each \(t\in T,\) the partial map \(f_{t}{:}\,X\rightarrow X\) is order continuous. For this case, we have the following version of Tarski–Kantorovich–Markowsky theorem. The characterization of the fixed point set in (i) is from Balbus et al. (2015). The computability result is the classic Tarski–Kantorovich theorem (e.g., Dugundji and Granas 1982, Theorem 4.2). There is a dual version for the greatest selections.
Proposition 6
Let X be a CCPO, T a poset, \(f{:}\,X\times T\rightarrow X\) order continuous in x, each t, and \(\exists \) a \( x_{L}\in X\) such that \(x_{L}\le f(x_{L},t).\) Denote by \(\varPsi _{f}(t){:}\,T\rightarrow 2^{X}\) the fixed point correspondence of f at \(t\in T. \) Then, (i) \(\varPsi _{F}(t)\) is nonempty CCPO. Further, \(\sup _{n}f^{n}(x_{L},t)=\wedge \varPsi _{f}(t).\) Finally, if in addition, X and T are each continuous domains, and f is additionally order continuous on T, then the mapping \(t\rightarrow \wedge \varPsi _{f}(t)\) is order continuous.
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Datta, M., Reffett, K. & Woźny, Ł. Comparing recursive equilibrium in economies with dynamic complementarities and indeterminacy. Econ Theory 66, 593–626 (2018). https://doi.org/10.1007/s00199-017-1046-z
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DOI: https://doi.org/10.1007/s00199-017-1046-z