Strategic complementarities, coordination failures, and macroeconomic fluctuations: from multiplicity of equilibria to disequilibrium dynamics

Abstract

This paper retraces the main research avenues explored by some Keynesian macroeconomists from the mid-1980s to model aggregate fluctuations with strategic complementarities. Starting from the static coordination failure framework we identify the different devices implemented to deal with dynamics in deterministic or stochastic settings. The text examines how the analysis shifted from a multiplicity of equilibria to a multiplicity of steady states or equilibrium paths. From this standpoint, we confront the properties obtained under perfect foresight and alternative expectation formation schemes focussing on disequilibrium adjustments. The article also includes a tentative exploration of the reasons for the decline of this view.

Keywords:

• Economic fluctuations
• coordination failures
• strategic complementarities
• cyclical dynamics

Jel Code:

• B22
• E12
• E30
• E32

Acknowledgements

The first version of this paper was presented at the workshop Economics and Coordination, Nice 9–10 September 2022. I wish to thank the participants and especially Peter Howitt for his comments. I am also grateful to two anonymous referees for their critique and valuable suggestions. The usual caveats apply.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In a footnote he points out that these approaches, more generally involves what Schelling (1978) in Micromotives and Macrobehavior” especially 89–110 calls the phenomenon of critical mass” (Howitt 1986, 636).

2 In international economics, they explain why location matters for trade patterns (Krugman 1991). The emergence of a new geographic economy was also part of this approach. Recall, also, the initial work on endogenous growth which show that external economies can erase the traditional distinction between factor accumulation and technical change. Chamley (1999) mentions that, beyond economics, strategic complementarities appear also in some models of social changes. See notably Adserà and Ray (1998) and Kuran (1987).

3 However, as Benassi, Chirco, and Colombo (1994) point out, the findings related to hysteresis were often obtained by imposing very specific restrictions on the models and were less robust than the persistence effects related to the speed of convergence towards a unique equilibrium.

4 For instance, industrial organisation scholars espoused the idea of the adoption of a new technology with network externalities in which the individual desirability of adoption depends on what others do.

5 The contribution was published in 1988 but it circulated from 1985 in a Cowles Foundation Discussion Paper.

6 The function V is continuously differentiable with $\frac{{\partial }^{2}V}{{\partial }^2{e}_i}<0$ and $\frac{{\partial }^{2}V}{\partial e_i\partial \theta }>0.$

7 Let us consider the first order condition for a Nash equilibrium. By total differentiation with $d\theta =0$ we obtain the slope $\rho =\frac{\mathit{\text{de}}}{d\overline{e}}=-\frac{V_{12}}{V_{11}}.$ Thus, Strategic complementarity is equivalent to $\rho >0$ and hence is necessary for multiple equilibria.

8 Notice that if there is only a strong spill-over effect for all admissible values of e, the inflection points and the S shape form of the curve disappears and there are only 2 equilibria, with null or full effort.

9 See notably Colander (1996) and for a synthesis including both theoretical and empirical findings see Cooper and Haltiwanger (1996).

10 In this perspective, see notably d’Aspremont, Dos Santos Ferreira, and Gerard-Varet (1995), Heller (1986), and Silvestre (1995), Chatterjee and Cooper (1989).

11 This issue of observability in a dynamic setting is addressed specifically by Howitt and McAfee (1988).

12 $\Phi (p_2)={\alpha }^{\alpha }{(1-\alpha )}^{(1-\alpha )}{(\frac{1}{p_2})}^{\alpha }.$ The firms have also Cobb-Douglas preferences with parameter α, where α represents the budget share devoted to sector 1 and $(1-\alpha )$ to the non-produced good.

13 Recall that α inters into the functions ${\varphi }_1$ and ${\varphi }_2$ and represents the budget share devoted to sector 1 and $(1-\alpha )$ to sector 2. It is shown that if α is large enough and close to 1, there will be a range of values of $\overline{M}$ such that multiple equilibria always exist.

14 Recall that the basic canonical RBC framework could hardly account for persistence with shocks following an auto-regressive process.

15 As, for instance, in the study by Cooper and Durlauf (1992) of the US 1939–1991 with respectively 0.58 and 0.5. In Cooper’s approach, the major driving element is the stochastic variations in the aggregate endowments of the non-produced good M. In these simulations, the distribution of the shocks is constructed so that for one-third of the time the expected value of M is above M_L, is below M_H for one-third of the time and is in the multiple equilibrium range for one-third of the time (see Cooper 1994, 1118). As a referee noted, in the absence of large shocks, the existing branch does not disappear which meant that the economy could exhibit the hysteresis effects described in the introduction.

16 The overlapping-generations framework was mobilised, also, by d’Aspremont, Dos Santos Ferreira, and Gerard-Varet (1994, 1995), to models the effects induced by imperfect competition and final demand in the emergence of deterministic endogenous fluctuations with coordination failures.

17 Reacting to the presentation of this paper during the 2022 Nice Workshop, Peter Howitt reiterated this view. For him, the term sunspot in its modern sense” remains an insult to Jevons who took sunspots seriously”, referring, as H.L. Moore also did later, to the actual effects of climatic variations induced by these spots on crops.

18 Firm size is negligible so that they treat n_t as given.

19 According to Bayesian rule, the independence of priors implies that there is no information available for updating the probabilities associated with the state that is not observed.

20 Recall that, generally, these kinds of assumptions about the labour supply or saving function were necessary for the existence of sunspot equilibria and endogenous cycles in an OLG framework, as in Azariadis (1981), Azariadis and Guesnerie (1982, 1986), Grandmont (1985), and Reichlin (1986). In contrast, the model developed by Woodford (1986, 1988) with infinite-lived agents and financial constraint, did not require these conditions on the elasticity of the labour supply. Nevertheless, the expected rate of inflation $\frac{P_{t+1}^e}{P_t}$ is still an important element of this model. In this literature, cycles were related, also, to the indeterminacy of a stationary state with perfect foresight. On these approaches in relation to Keynes, see, e.g., Raybaut (1992).

21 It is well known that this issue was heavily debated in the analysis of economic fluctuations in the 1930s and re-emerged in the 1990s.

22 The agents f are indexed in the increasing order of their opportunity cost, i.e., $k_1\leqq k_2\leqq \dots \leqq k_F.$

23 With the exception of the exogenous fixed opportunity cost, all active agents in a sector are identical.

24 This is defined by a function W_f, decreasing with $N_t^i.$ This property characterizes” a congestion effect”, a larger number of active agents as sellers, have to share the same industry demand, which implies less profit for all agents in the sector.

25 That is, if, and only if $W_f(N_t^i+1,A_t^i)>0.$

26 Based on continuity arguments about utility and equilibria with respect to transition probabilities, Azariadis and Guesnerie had shown that in OLG models sunspot equilibria exist in the neighbourhood of a cycle of order 2.

27 Comments made by Howitt during the Economics and Coordination workshop, Nice, 9–10 September 2022. He added that this question came up repeatedly during discussions and critiques at the time.

28 In a corrective note, Diamond and Fudenberg (1991) mention that Matsuyama had stressed the need for more precision in relation to use of the Hopf bifurcation theorem for the existence of cycles and they responded that they had” left the impression that [they] claimed that cycles occur precisely at the critical value of the interest rate where the real part of the eigenvalues is zero” (Diamond and Fudenberg 1991, 218).

29 This contribution adds to the series of macrodynamics models developed at the University of Bielefied starting in the early 1990s, using advances in nonlinear and complex dynamics. From a theoretical perspective, these works build on the classical, Wicksellian and Keynesian traditions, the approaches proposed by Kalecki, Kaldor, Goodwin and Schumpeter and some New Keynesian developments. On this point, see Rosser and Rosser (2023) who refer to the Faculty of Economics at Bielefeld University as an intellectual centre. However, its members were located in several world cities, although, principally, in Bielefeld, New York, Sydney, and Tokyo. Thus, the group’s scientific output was mainly the result of visits to each other’s institutions.

30 From this perspective, Franke points to the similarity between this argument and the mechanism proposed by Kaldor (1940) in his business cycle model, the “S” shaped best response function echoing Kaldor’s investment function. However, conceptually, in Kaldor (1940), the shifts in the investment (and also the saving) functions are explained endogenously, notably by the dynamics of the stock of capital.

31 An analysis of a variant of this model with a continuously differentiable “S” shaped reaction function is developed in Raybaut (2022).

32 These extensions on multiplicity and learning are mentioned by Franke (2001), quoting some recent related works, especially Evans, Honkapohja, and Romer (1998) and Evans and Honkapohja (2001). Franke also introduced small shocks to the parameters of the best response function to contrast his results with the dynamics generated by Cooper (1994) with large shocks, and analysed the impact of their timing on the dynamics.

33 A connection could be made to the approach developed at the same time by Creedy, Lye, and Martin (1996) to identify nonlinearities and large swings in real exchange rates time series. The authors developed a distributional dynamics exchange rate approach. The stationary distribution of the Kolmogorov equation can be derived analytically and represents the stochastic analogue of a long run equilibrium or steady state. In a context of multiple equilibria, this distribution is multimodal, with the modes corresponding to the stable equilibria and the antimodes to the unstable equilibria. In this case, it can be shown that shocks can induce discrete jumps in the real exchange rate that are not necessarily reversible, which gives rise to hysteresis. Finally, it can be shown that the distributions within the generalised exponential class can be estimated using the maximum likelihood procedure. I thank a referee for suggesting this reference, which, to my knowledge, has never been included in the macroeconomic empirical literature on multiple equilibria and coordination failure-based models.

34 This group, directed by Cooper, was part of the larger NBER programme on economic fluctuations and growth which developed some major contributions on strategic complementarities. The NBER documentation refers to a programme that “featured many spirited scientific discussions but never debates between schools of thought”.