Abstract
This paper proposes to model market mechanisms as a collective learning process for firms in a complex adaptive system, namely Jamel, an agentbased, stockflow consistent macroeconomic model. Inspired by Alchian’s (J Polit Econ: 5(3):211–221, 1950) “blanketing shotgun process” idea, our learning model is an everadapting process that puts a significant weight on exploration visàvis exploitation. We show that decentralized market selection allows firms collectively to adapt their overall debt strategies to the changes in the macroeconomic environment so that the system sustains itself, but at the cost of recurrent deep downturns. We conclude that, in complex evolving economies, market processes do not lead to the selection of optimal behaviors, as the characterization of successful behaviors itself constantly evolves as a result of the market conditions that these behaviors contribute to shaping. Heterogeneity in behavior remains essential to adaptation. We come to an evolutionary characterization of a crisis, as the point where the evolution of the macroeconomic system becomes faster than the adaptation capabilities of the agents that populate it.
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Notes
 1.
 2.
Following Cincotti et al. (2010), Kinsella et al. (2011) and Seppecher (2012a), a growing literature has emphasized the interest of combining SFC and ABM principles; see Caverzasi and Godin (2015). A nonexhaustive list of SFCAB models, besides Jamel, includes Raberto et al. (2012), Caiani et al. (2016), Riccetti et al. (2014), Russo et al. (2016).
 3.
For instance, Lainé (2016) shows the challenge posed by the heterogeneity of the observed investment behavior of firms if one seeks to derive a model of investment decisions.
 4.
 5.
Admittedly, several modifications have been proposed to make GA more suited to everchanging environments (see, e.g., Cobb and Grefenstette 1993). Classifier systems which combine GA with features taken from other types of expert systems, such as Artificial Neural Networks, are also somehow effective in changing environments. However, those algorithms are often complicated and computationally quite costly. By contrast, the BSP used in this paper is simple, parsimonious, and while being flexible in its implementation, finds an intuitive interpretation and involves a low computational burden.
 6.
 7.
This can be because the firm’s operating characteristics are not perfectly observable by its competitors, or because the firm’s routines cannot be exactly transferred to another firm, or because of control error in the implementation of the new routine. Alchian (1950, pp. 218219) uses the concept of “roughandready imitative rules”.
 8.
See e.g. Kalecki (2010), who stresses that the amount of the entrepreneurial equity is the main limitation to the expansion of a firm.
 9.
This simplifying assumption avoids complicating the model by introducing a second industrial sector. An upcoming version of the model does encompass a capital good sector.
 10.
This is a quite standard procedure in corporate finance. However, other types of investment functions could be easily envisioned, and will be considered in further developments of the model.
 11.
The price and wage could be computed in a more complicated way, such as a trend projection of past values over the next window periods. However, this would complicated the decision making of firms, without adding much to the qualitative simulation results.
 12.
We document the frequency of this event in the simulations in Section 4; see Footnote 17. This is due to the very simplistic design of the banking sector in Jamel, a feature that is intended to be abandoned in future versions of the model.
 13.
This matching order ensures that the biggest purchasers first enter the market, which appears reasonable. However, this order does not matter as all simulations show that households’ rationing in the goods market remains a rare and negligible event, which would not be realistic otherwise.
 14.
However, we have checked that our model is able to reproduce the empirical macro regularities of Seppecher and Salle (2015). This is indeed the case, along with few more stylized facts that we can seek to reproduce now that our model incorporates investment, e.g. more volatile investment than GDP, and strong positive correlation between firms’ debt.
 15.
This, first, comes from our very stylized banking system and the absence of government intervention besides the Taylor rule that is ineffective in deflationary downturns.
 16.
The contribution of each figure to our argumentation will be presented throughout the whole section.
 17.
Because the model is randomly initialized and the single bank bears alone all the costs of firms’ losses (see Appendix A), the required adjustments may be too drastic for the single bank to absorb firms’ losses, and the simulation may break off at the beginning. We observe that this is the case in roughly 15% of the simulations. We do not report those runs in Table 1. However, once the economy survives this takeoff period, we always observed the same cyclical aggregate pattern.
 18.
Recall that the model does not encompass any technological progress nor population changes. An average growth rate close to zero is therefore an expected outcome.
 19.
We are grateful to an anonymous referee for suggesting this exercise.
 20.
Brock and Hommes (1998) make a similar point by showing that “nonrational”, trendchasing traders are not driven out by fundamental ones in a financial market model but their relative shares coevolve in a nonlinear way with the dynamics of the market that can display, as a result, very complicated, and even chaotic dynamics. See also Hommes (2006) for a related discussion.
 21.
In our model, this raise stems from the Taylor rule that increases nominal rates along the boom. Another explanation is the increase in the bank’s risk premium in an attempt to control for the increasing borrowers’ financial fragility (Stockhammer and Michell 2014). For simplicity, we abstract here from modeling endogenous risk premiums.
 22.
 23.
 24.
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Acknowledgements
The authors would like to thank the two anonymous referees for their most useful comments and suggestions. They also wish to thank the guest editors of this issue and the participants of the symposia and seminars in which previous versions of this paper have been presented, namely: the 2nd Workshop on Modeling and Analysis of Complex Monetary Economies; the first Grenoble PostKeynesian Conference; the 26th “Journée évolution artificielle Thématique”; the 22nd International Conference on Computing in Economics and Finance; the 6th annual congress of the French association for political economy; the 20th Conference of the Research Network Macroeconomics and Macroeconomic Policies; the 3rd Bordeaux Workshop on AgentBased Macroeconomics; the seminar of the LEMNA; the seminar of the GREDEG.
Funding
This research was partly founded by the EU FP7 project MACFINROBODS, grant agreement No. 612796, as well as by the ‘Lavoie Chair’, Sorbonne Paris Cité.
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The authors declare that they have no conflict of interest.
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This research has been partly financed by the EU FP7 project MACFINROBODS, grant agreement No. 612796, as well as the ‘Lavoie Chair’, Sorbonne Paris Cité.
Appendices
Appendix A: Parameter values (baseline scenario)
Random draws are performed at each period and for each agent.
Appendix B: Pseudocode of Jamel
B.1 Initialization
Equities (E_{j,0}) of each firm and of the bank are divided in ten equal shares, and given to randomly drawn households.
B.2 Execution
In each period t, t = 1,...,d^{S}:

1.
(Interest rate adjustment:)
$$ {i_{t}} = \max \left( {\phi_{\pi}} ({\pi_{t}}  {\pi^{T}}), 0 \right) $$(1)where π_{t} is the price inflation computed over past win periods, ϕ_{π} and π^{T} are parameters.

2.
(Fixed capital stock depreciation:) Each machine m of each firm j is depreciated by \(\frac {I_{j,m}}{{d^k}}\) where I_{j,m,} is the initial value of the machine paid by j and d^{k} the expected life time of the machine (in months, straightline depreciation method).

3.
(Payment of dividends:)
Each firmj i) computes \({\tilde {F}_{j,t}}\), its average past net profits F_{j} over win periods, ii) computes the share of net profits to be distributed as \(\frac {{E_{j,t}}}{{E^T_{j,t}}}\), and iii) distributes to its owners the amount\({{FD}_{j,t}} = \min \left (\frac {{E_{j,t}}}{{E^T_{j,t}}} {\tilde {F}_{j,t}} , {\kappa _d} {E_{j,t}}\right )\), in proportion to their relative share holding.The bank distributes \({{FD}_{B,t}} = \max ({E_{B,t}}{E^T_{B,t}},0)\)

4.
(Price:)
$$ \begin{array}{ll} \text{if} \ ({s_{j,t1}} > {s^T_{j,t1}} \text{ and\ } {{in}_{j,t}} < {{in}^T_{j,t}}) & \left\{ \begin{array}{l} {\overline{P}_{j,t}} = {\overline{P}_{j,t1}} (1 + {\delta^P}) \\ {\underline{P}_{j,t}} = {{P}_{j,t1}} \\ {{P}_{j,t}} \hookrightarrow \mathcal{U}({\underline{P}_{j,t}}, {\overline{P}_{j,t}}) \end{array} \right.\\ \\ \text{else if} \ ({s_{j,t1}} < {s^T_{j,t1}} \text{ and} {{\ in}_{j,t}} > {{in}^T_{j,t}}) & \left\{ \begin{array}{l} {\overline{P}_{j,t}} = {{P}_{j,t1}} \\ {\underline{P}_{j,t}} = {\underline{P}_{j,t1}} (1  {\delta^P}) \\ {{P}_{j,t}} \hookrightarrow \mathcal{U}({\underline{P}_{j,t}}, {\overline{P}_{j,t}}) \end{array} \right.\\ \\ \text{else} & \left\{ \begin{array}{l} {\overline{P}_{j,t}} = {\overline{P}_{j,t1}} (1 + {\delta^P}) \\ {\underline{P}_{j,t}} = {\underline{P}_{j,t1}} (1  {\delta^P})\\ {{P}_{t,j} }= {{P}_{j,t1}} \end{array} \right. \end{array} $$(2)with\({\overline {P}_{j,t}}\), the ceiling price,\({\underline {P}_{j,t}}\), the floor price.

5.
(Wage offer:) Each firm j observes a random sample of g’ other firms. If the observed sample contains a firm k such that k_{k,t} > k_{j,t}, then:
$$ \left\{ \begin{array}{l} {W_{j,t}} = {W_{k,t}}\\ {\overline{W}_{j,t}} = {W_{j,t}} (1 + {\delta^W}) \\ {\underline{W}_{j,t}} = {W_{j,t}} (1  {\delta^W}) \end{array} \right. $$(3)else:
$$ \begin{array}{ll} \text{if\ } ({\rho_{j,t1}} > {\rho^T}) &\left\{ \begin{array}{l} {\overline{W}_{j,t}} = {\overline{W}_{j,t1}} (1 + {\delta^W}) \\ {\underline{W}_{j,t}} = {W_{j,t1}} \end{array} \right.\\ \\ \text{else} &\left\{ \begin{array}{l} {\overline{W}_{j,t}} = {W_{j,t1}} \\ {\underline{W}_{j,t}} = {\underline{W}_{j,t1}} (1  {\delta^W}) \end{array} \right.\\ \\ \text{and then} &{W_{j,t}} \hookrightarrow \mathcal{U}({\underline{W}_{j,t}}, {\overline{W}_{j,t}}) \end{array} $$(4)with:

\({\rho _{j,t1}} = \frac {{n^T_{j,t1}}{n_{i,t1}}}{{n^T_{j,t1}}}\), the vacancy rate previously observed by the firm,

\({\overline {W}_{j,t}}\), the ceiling wage,

\({\underline {W}_{j,t}}\), the floor wage.


6.
(Labor demand:)\({n^T_{j,t}}\) (within the lower bound 0 and the upper bound k_{j,t}):
$$ {n^T_{j,t}} = (1+\delta_{j,t}^{h}) {n^T_{j,t1}} $$(5)where\({n^T_{j,t1}}\) is the labor demand of the firm in period t − 1, and δ_{j,t} is the size of the adjustment, computed as:
$$ \delta_{j,t}^{h} = \left\{\begin{array}{ll} \alpha_{j,t} {\nu_{F}} & \text{if\ } 0 \leq \alpha_{j,t} \beta_{j,t} < \frac{{{in}^T_{j,t}}  {{in}_{j,t}}}{{{in}^T_{j,t}}} ,\\ \alpha_{j,t} {\nu_{F}} & \text{if} \ 0 \leq \alpha_{j,t} \beta_{j,t} < \frac{{{in}_{j,t}}  {{in}^T_{j,t}}}{{{in}^T_{j,t}}} ,\\ 0 & \text{else.} \end{array}\right. $$(6)with α_{j,t}, \(\beta _{j,t} \hookrightarrow \mathcal {U}(0,1)\) and ν_{F} > 0.
$$ \left\{ \begin{array}{ll} \text{if\ } {n_{j,t}}>{n^T_{j,t}} &\text{fires\ } {n_{j,t}}{n^T_{j,t}} \text{ (on a lasthiredfirstfired basis)} \\ \text{else} &\text{posts\ } {n^T_{j,t}}{n_{j,t}} \text{ job offers.} \end{array} \right. $$(7) 
7.
(Financing of current assets): according to the existing job contracts, the workforce target \({n^T_{j,t}}\), and the new wage rate offered on the labor market W_{j,t}:

computes the anticipated wage bill \({{WB}^T_{j,t}}\);

borrows \(\max ({{WB}^T_{j,t}}{M_{j,t}}, 0)\) (nonamortized shortterm loan).


8.
(Reservation wages:)
Each household i updates his reservation wage \({W^r_{i,t}}\).

If i is unemployed:
$$ {W^r_{i,t}} = {W^r_{i,t1}} (1  \delta^w_{i,t}) $$(8)where\(\delta ^w_{i,t} \geq 0\) is the size of the downward adjustment, and is computed as:
$$ \delta_{i,t}^{w}= \left\{\begin{array}{ll} \beta_{i,t} \cdot {\eta_{H}} & \text{if\ } \alpha_{i,t} < \frac{{d^{u}_{i,t}}}{{d^r}} \\ 0 & \text{ else.} \end{array}\right. $$(9)where α_{i,t}, β_{i,t} are \(\mathcal {U}(0,1)\), and η_{H} > 0 and d^{w} ≥ 1 are parameters.

Else:
$$ {W^r_{i,t}} = {W_{i,t1}} $$(10)where W_{i,t− 1} is the wage earned by household i at the previous period t − 1.


9.
(Labor market:) Each unemployed household i) consults a random sample of g job offers; ii) selects the job offer with the highest offered wage, denoted by W_{j,t}; iii) if \({W_{j,t}}>={W^r_{i,t}}\), accepts the job for a duration of d^{w} months, else, remains unemployed for the period t.

10.
(Production): Each firm distributes uniformly the hired workers on its machines (one per machine). Once a production process of a machine is completed (after d^{p} iterations by a worker), it adds pr^{k} goods to the firm’s inventories in_{j,t}, the value of which is then incremented by the production costs of pr^{k} goods.
This process updates i) firms’ wage bills and vacancy rates, ii) production levels, and iii) households’ cashonhand M_{i,t} = W_{i,t} + FD_{i,t} + M_{i,t− 1} (where W_{i,t} + FD_{i,t} represents its income flow, made of FD_{i,t}, the dividends that household i may receive if it owns shares in the bank or a firm, see Step 1., and W_{i,t} its labor income, and M_{i,t− 1} is its cashonhand transferred from t − 1).

11.
(Goods supply:) Each firm j puts \({s^T_{j,t}}\) goods in the goods market:
$$ {s^T_{j,t}} = \max({\mu_F} \cdot {{in}_{j,t}}, {d^{m}}\cdot{{pr}^k}\cdot{k_{j,t}}) $$(11) 
12.
(Individual experimentation:) With a probability pb_{BSP}, for each firm j, \(\ell _{j,t + 1} \hookrightarrow \mathcal {N}(0, {\sigma _{{BSP}}})\), else ℓ_{j,t,+ 1} = ℓ_{j,t}.

13.
(Investment decision):

(a)
selects a random sample of g suppliers (other firms);

(b)
if (k_{j,t} = 0) buys m = 1 new machine, for a value I_{j,t};

(c)
else if (\({E^T_{j,t}} > {E_{j,t}}\)):

i.
computes the vector of the prices of each investment project I_{m} (m the number of new machines to be bought), with m = 0, 1, 2,...;

ii.
computes \({\tilde {s}_{j,t}}\), average of the sales s_{j} over the past window periods;

iii.
computes \({s^e_{j,t}} = {\beta } \cdot {\tilde {s}_{j,t}}\), its sales expansion objective;

iv.
given its sales expansion objective \({s^e_{j,t}}\), the expected life time of a machine d^{k}, the current price P_{j,t}, the current wage W_{j,t}, the discount factor \(r=i_{t}  \tilde {\pi }_{t}\) (\(\tilde {\pi }_{t}\) is the average past inflation computed over the window last periods), and the price I_{m} of each investment project m, computes the net present value NPV_{m} of each investment project m:
$$ {NPV}_{m} \equiv \frac{{CF}_{m}}{{{r}_{t}}} \left( 1  \frac{1}{{{r}_{t}}(1+{{r}_{t}})^{{d^k}}} \right)  I_{m} $$where CF_{m} is the expected cashflow of the project:
$$ {CF}_{m} = \min({s^e_{j,t}},m \cdot {{pr}^k})\cdot{{P}_{j,t}}  m\cdot{W_{j,t}} $$where the min term ensures that the future sales cannot exceed the maximum market capacity of the firms.

v.
chooses the project m for which NPV_{m+ 1} <NPV_{m}, for a value I_{j,t}.

vi.
adds \(\frac {I_{m}}{m}\) per new machine to its assets.

i.

(a)

14.
(Financing of fixed assets):

(a)
borrows (amortized longrun loan) the amount: \({\ell ^T_{j,t}} {I_{j,t}}\);

(b)
borrows (amortized shortrun loan) the amount: \(\max ((1{\ell ^T_{j,t}}) {I_{j,t}} {M_{j,t}}, 0)\);

(a)

15.
(Saving/consumption plan:)
Each household computes

(a)
his average monthly income flow over the last window periods, denoted by \({\tilde {Y}_{i,t}}\);

(b)
his cashonhand target \({M^T_{i,t}} ={\kappa _{S}} \cdot {\tilde {Y}_{i,t}}\);

(c)
is targeted consumption expenditures as:
$$ {C^T_{i,t}}= \left\{\begin{array}{ll} (1{\kappa_{S}}) {\tilde{Y}_{i,t}} & \text{if} {\ M_{i,t}} \leq {M^T_{i,t}} \\ {\tilde{Y}_{i,t}} +{\mu_H}({M_{i,t}}  {M^T_{i,t}} ) & \text{ else.} \end{array}\right. $$(12)where μ_{H} ≥ 0 is a parameter. The budget constraint always gives \({C_{i,t}} \leq \min ({C^T_{i,t}}, {M_{i,t}})\).

(a)

16.
(Goods market:)

(a)
matches first the firms’ demand, then the households’ demand with the firms’ supply;

(b)
goods bought by firms are transformed into new machines, while goods bought by households are consumed.

(a)

17.
(Loans:) The firms pay back part of their loans and the interests to the bank. Interest is due at each period. For an amortized loan, principal is repaid by equal fractions at each period, while for a nonamortized loan, the total principal is due the term. If the cashonhand M_{j,t} of a firm j cannot fully cover the debt repayments, it benefits of an overdraft facility, ie a new short term loan at an higher rate including the risk premium of the bank (i_{t} + rp).

18.
(Foreclosure:) If a firm has become insolvent (A_{j,t} < L_{j,t}), the bank starts the foreclosure procedure, a new \({\ell ^T_{j,t}}\) is copied from a surviving firm (\(+\mathcal {N}(0, {\sigma _{{BSP}}})\)), the firm is recapitalized up to E_{j,t} = κ_{s}A_{j,t}, and new households become owners as follows: all households that have at least 20% of E_{j,t} as cashonhand are solicited for at most 50% of their wealth, and the firm’s shares are distributed in proportion to their contribution. If the collected cashonhand is lower than E_{j,t}, the selection threshold is decreased to 10% of E_{j,t}. If the cashonhand on the solicited households is still not enough, the threshold is decreased to 4%, and then 2%. If this is still not enough to buy all the shares of the firm, the price of the shares is decreased by 10% until enough cashonhand can be collected. In case of more than 10 decreases, the simulation would stop.

19.
This process starts all over again for a given length of d^{S} periods.
Appendix C: Stockflow consistency
C.1 Stocks
C.2 Balance sheet matrix
C.3 Flows
C.4 Transactionflows matrix
C.5 Fullintegration matrix
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Seppecher, P., Salle, I. & Lang, D. Is the market really a good teacher?. J Evol Econ 29, 299–335 (2019). https://doi.org/10.1007/s0019101805717
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Keywords
 Evolutionary economics
 Learning
 Firms’ adaptation
 Business cycles
JEL Classification
 B52
 C63
 D21
 D83
 E32