Forecasting in a complex environment: machine

learning sales expectations in a SFC Agent based

simulation model

Ermanno Catullo, Mauro Gallegati and Alberto Russo ∗

Abstract

We analyze the micro and macro effects of introducing into an agentbased model firms that are able to formulate effective sales forecasts. Weendow firms with machine learning techniques in order to formulate expec-tations on sales variations,these techniques are able to provide predictionsthat are not biased and present a certain degree of accuracy. Simulationsshow that firms that exploit these expectations to orientate their pro-duction and sales decisions increase their profits without affecting theirriskiness. However, higher profits are associated with a reduced wageshare, that affects negatively the aggregate demand, resulting in higherunemployment and lower productivity in the long run.

1 Introduction

The aim of the paper is studying both the micro and macro effects of introducinginto an agent based simulation model firms that are able to formulate effectivesales forecasts. Thus, each firm makes expectations on the variations of itssales in order to orientate its production and price decisions. We focus just onsales expectations because the credit market and the labor market are quitesimplified, while in this model the good market is at the basis of productionand innovation decisions.

Indeed, in each period, each firm formulates a forecast on the growth rateof its sales and uses this prevision to determine its production choices. Wetested different methods to make sales forecasts: a genetic algorithm (GA), anautoregressive model (AR) and a naive approach. The GA and the AR methodsare able to provide expectations that are unbiased and that present a relativedegree of accuracy. Moreover, firms using this two predictive methods are ableto increase their profits without augmenting their riskiness. However, on theaggregate level higher profit rates lead to a lower wage share that depressesaggregate demand, thus leading to higher unemployment and hampering growthin the long run.

The model is based on Caiani et al. (2018a). However, in order to focus onfirm forecasting we slightly modified it. In fact, we simulated a closed economy.We simplified the wage formation rule and the computation of firm innovationexpenditure. Finally, we modified the rule that determines selling price and

∗Corresponding author Ermanno Catullo: ermanno.catullo@gmail.com

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production quantity to link firm decisions more directly to sales growth expec-tations.

In the first section we describe the model. The second section illustratethe predictive methods. In the third, we allow firms to exploit the predictivemethods to orientate their sales, thus we show the consequent micro and macroeffects. In the fourth section, we let interact firms that adopt different predictivemethods. The fifth section concludes.

2 The Model

3 Model Description

We employ the Agent Based-Stock Flow Consistent model first presented inCaiani et al. (2018a) to model several predictive methods applied to firm salesforecasting. The main difference with respect to Caiani et al. (2018a) is that themodel simulates a closed economy. The paper focus on the effect of expectationson sales, therefore the wage rule is simplified to avoid the necessity of formulatingexpectations on unemployment levels (Equations 5 and 14).

Moreover, in order to limit the impact of expectations on price and quantityproduced, innovation expenditure is computed as a fraction of past sales andnot from the desired production (Equation 6). Finally the behavioral rule thatgoverns the selling price and quantity produced is modified in order to giveimportance to forecasting (Equations 4 and 40).

The simulated economy is populated by a given number of households (H)and by an endogenously varying number of firms (It) and banks (Zt), depend-ing on defaults arising endogenously during the simulation and on households’equity investment in the creation of new firms and banks.

The model considers a ‘pure labor’ economy a la Adam Smith1 where pro-duction by firms is carried out using labor only.

Government collects taxes on income and profits and provide public spendingin the form of a lump-sum monetary transfer to households. Government has amaximum deficit-to-GDP ratio that it commits to comply by tuning spendingand tax rates.

The Central Bank sets the policy rate (i.e. the interest paid on cash advancesasked by commercial banks to fulfill mandatory liquidity constraints) and buythe possible residual bonds issued by the government which were not purchasedby private banks.

Firms invest in R&D in order to achieve innovations that increase laborproductivity, reducing unit costs of production. Furthermore they can imitatethe technology of their competitors so to catch up with the industry standards.This gives rise to sectoral spillovers.

Following the logic of Riccetti et al. (2015, 2016); Caiani et al. (2016, 2018b,a)model dynamics is driven by agents’ adaptive reactions and decentralized inter-actions through specific matching protocols on the various markets modeled.Five types of markets are considered: good markets, labor markets, depositmarkets, credit and bond markets.

1See also the book by Pasinetti (1993) presenting a simple model of a ‘pure labor’ economy,in which many goods are produced by labor alone.

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Next subsections sketch out the behaviors of agents and the structure oftheir interactions on the different markets.

3.1 Agents

3.1.1 Firms

Firms’ desired output level qPi,t depends on desired sales qSi,t and the level ofinventories inherited from the past invi,t. Furthermore, firms aim to keep acertain amount of inventories, expressed as a share θ of expected sales, as abuffer (Steindl, 1952; Lavoie, 1992).

qPi,t = qSi,t(1 + θ) − invi,t (1)

In the baseline scenario, prices pi,t and desired sales qSi,t are revised adap-tively from period to period according to a simple scheme depending on qi,t−1

(the output produced by firm i in t− 1), qi,t−1 (the quantities sold in t− 1).

if qi,t−1 ≥ qSi,t−1 :

{qSi,t = qSi,t−1(1 + U [0, δ])

pi,t = pi,t−1(1 + U [0, δ])(2)

if qi,t−1 ≤ qSi,t−1 :

{qSi,t = qSi,t−1(1 − U [0, δ])

pi,t = pi,t−1(1 − U [0, δ])(3)

(4)

Equation 2 states that if past sales exceeded desired sales, firms adaptivelyincrease both the sales desired and their selling price. The opposite happenswhen past sales are lower than desired sales (equation 3). Moreover, prices havea lower bound represented by unit costs of production: pi,t ≥ wi,t

φi,t, where φi,t is

firm’s i labor productivity in period t.In the fourth section we modify the production and price decision to incor-

porate selling expectations.Firm’s labor demand is then computed as: lDi,t = qDi,t/φi,t. Output can

be lower than desired if labor employed is less than needed due to financialconstraints or if the firm is not able to find workers willing to fill vacant positions.

The salary wi,t offered by firm i is adaptively revised following a schemesimilar to that characterizing workers’ reservation wage. Equation 5 shows thatfirms first check if they were able to fill all vacant positions in the previous pe-riod, comparing their past labor demanded lDi,t−1 and labor actually employedli,t−1. If labor employed was below the demanded level, they consider to in-crease the salary so to attract workers with a probability (υF ). If instead allvacant positions had been filled, they consider to reduce the wage offered so toincrease their profit margin with probability (1−υF ). However, the probabilityof both types of revision depends on unemployment levels: indeed, reducingwages when unemployment is low exposes the firm to the risk of not filling itsvacant positions, ending up being labor constrained; conversely, the risk is lowif many workers are unemployed.

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wi,t =

{wi,t−1(1 + U [0, δ]), if lDi,t−1 − li,t−1 > 0 with Pr(w+

i,t) = υF

wi,t−1(1 − U [0, δ]), if lDi,t−1 − li,t−1 = 0 with Pr(w−i,t) = 1 − υF e

−υut−1

(5)Firms can also increase their profit margin by improving their productivity

φi,t. Firms have a constant and equal parameter γ expressing desired R&D asa share of previous sales, thus desired innovation expenditure (R&DD

i,t) is equalto:

R&DDi,t = γqi,t−1pi,t−1 (6)

Actual R&Di,t is equal to R&DDi,t only if no financial or labor constraints

are binding.The amount of resources invested in R&D, in turn, determines the proba-

bilities of enhancing firm’s productivity by either carrying out an incrementalinnovation or by exploiting sectoral spillovers through imitation (Dosi et al.,2010). Therefore, firms are, in every period of the simulation, at the same timeinnovators and imitators, the total investment in R&D (imitative and innova-tive) being defined by equation 6. As innovators, they try to come up withtechnical improvement on their own. As imitators they try to collect informa-tion on the productive techniques in use among their competitors and, in casethey realize they are suffering a productivity gap with respect to the average oftheir competitors, they try to catch up through imitation.

Formally, the probability of success of these two types of R&D activities (i.e.innovation and imitation) is the same given the total investment in R&D, andit is given by:

Prsuccessi,t = 1 − e−νR&Di,t

ΦtPt (7)

where Pt and Φt are the average level of prices and the average productivity.Equation 7 shows that the probability of success is a non-linear increasing

function of the real investment on R&D activities (R&Di,t/Pt), divided by theaverage level of productivity (Φt).

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When successful in innovating, firms update their labor productivity asshown in 8:

φi,t+1 = φi,t(1 + U [0, δ]) (8)

If firms discover they have a productivity below the average they can also ex-ploit sectoral spillovers through imitation to narrow the gap with the standardsof production. When successful in imitating, they sample a new productivitylevel in a range between their current one and the average.

φi,t+1 = φi,t + U [0, (Φt − φi,t)] if φi,t < Φt (9)

The new level of productivity achieved thanks to an innovation and/or animitation is embeded in the firm’s production process starting from the nextperiod.

2This correction is required in order to prevent Prsuccessi,t from continuously increasing asa consequence of the continuous rise of potential output due to the higher levels of productivityΦt achieved during the simulation.

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Firms’ costs of production and R&D investment can be financed using inter-nal funds or external funding in the form of loans asked to banks (Lit). Firmsresort to bank credit only after internal funding has been exhausted, since thecost of external finance is usually higher due to market imperfections and infor-mation asymmetries (Meyers, 1984).

Firms’ demand for loans can be expressed as:

LDi,t =

{wi,tl

Di,t +R&DD

i,t −Di,t, if wi,tlDi,t +R&DD

i,t > Di,t

0, if wi,tlDi,t +R&DD

i,t ≤ Di,t

(10)

Firms can try to fulfill their funding needs asking credit to different banks.Nonetheless, they may end up being credit-constrained (Li,t ≤ LDi,t) (see section3.1.3). When this occurs, firms prioritize production over R&D. For simplicityreasons, loans are assumed to last only one period, being granted at the begin-ning and repaid at the end of each round, together with the interest accrued,similarly to the Monetary Circuit Theory (Graziani, 2003) and to Delli Gattiet al. (2010); Dosi et al. (2010); Riccetti et al. (2015).

Firms hold their funds at a randomly selected deposit bank, receiving aninterest rd,t. Firms’ profits are the sum of revenues from sales, interests ondeposits, and the nominal variation of inventories, minus wages paid to workers,R&D costs, and interests on credit:

πi,t = pi,tqi,t + rd,tDit + ∆INVi,t − wi,tli,t −R&Di,t − ri,tLi,t (11)

Firms’ net operating cash flows, indicated by π∗i,t can be obtained by sub-

tracting the variation of inventories from the definition of profits. When π∗i,t > 0

firms pay taxes (Tπi,t) and distribute dividends (Divπi,t) to equity holders, ex-

pressed as a share ρ of their residual net cash inflow.3

Tπi,t =

{τk,tπ

∗i,t, if π∗

i,t > 0

0, if π∗i,t ≤ 0

(12)

Divπi,t =

{ρ(π∗

i,t − Tπi,t), if π∗i,t > 0

0, if π∗i,t ≤ 0

(13)

Dividends are distributed to equity holders proportionally to their partici-pation in the firm’s equity.

3.1.2 Households

Households play three main roles in the model: they are workers, equity holders,and consumers.

On the labor market workers interact with ψ randomly sampled potentialemployers trying to sell them their labor force lS , which is normalized to 1.The quantity of labor sold to firms lh,t is then equal to 1 if the worker is fully

3Taxes on profits generated in period t are paid in period t+1. Accordingly, also dividendsgenerated in period t are paid to equity holders in period t+ 1.

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employed, 0 < lh,t < 1 if the worker is part-time employed4, and finally lh,t = 0if the worker is unemployed. Workers can sell their unitary labor supply todifferent employers until they have exhausted it. As a consequence, they canbe employed by different employers at the same time. We define lhi,t as thequantity of labor sold by worker h to firm i, and whi,t the wage he receives inexchange for it. The total labor sold by the household h in period t is then:

lh,t =∑Ik,ti,lhi,t>0 lhi,t.

Workers choose the employer offering the highest wage but they do notaccept vacant positions below a reservation level wh,t which is adaptively revisedfrom period to period depending on the worker’s past employment condition(Equation 14). Workers who are not fully employed tend to decrease theirreservation wage with a probability (1 − υH), while full-time workers tend toincrease it with a probability (υH).

wh,t =

{wh,t−1(1 + U [0, δ]), if lS − lh,t−1 = 0 with Pr(w+

h,t) = υH

wh,t−1(1 − U [0, δ]), if lS − lh,t−1 > 0 with Pr(w−h,t) = 1 − υH

(14)

The parameter υH represents a scaling factor, whose value is calibratedin relation to the corresponding scaling parameter υF for firms’ offered wagerevision rule (equation 5 in section 3.1.1) in order to avoid having an exces-sive mismatch between the wages offered by firms and the reservation wages ofworkers which would give rise to unreasonable levels of ‘frictional’ or ‘voluntary’unemployment: the agent-specific condition which induces firms to consider thepossibility of rising wages (i.e. firm’s inability to fill all vacant positions) isin fact inevitably less frequent than the corresponding agent-specific conditioninducing workers’ to consider rising their reservation wage (i.e. having beenfully employed in the last period). Imposing υH < υF is then required to avoidworkers’ reservation wages to rise too fast compared to firms’ offers.

Workers can be employed by firms for production and R&D activities in-differently. Investment in Research and Development activities (R&Di,t, seesection 3.1.1) is thus assumed to add on to workers’ labor income, being dis-tributed according to the quantity of labor they individually supply.

In addition, households also receive interests on deposits Dh,t from banks,dividends from participated firms and banks (Divh,t), and a tax-exempt mone-tary transfer (Gt/H) from the government.

Defining by τt the tax rate charged by the government, households’ grossand net income (respectively yh,t and yDh,t, where D stands for ‘disposable’) canbe expressed as:

yh,t =

It∑i,lhi,t>0

whi,tlhi,t + rd,tDh,t +Divh,t +

Ik,t∑i,lhi,t>0

R&Di,tlhitlit

(15)

yDh,t = (1 − τk,t)yh,t +Gk,tH

(16)

4Firms labor demand in fact is formulated as a positive real number so that for each firmthere will be a ‘marginal’ worker employed only for the decimal part. Another reason why lh,tcan be between 0 and 1 is the presence of financial constraints which impede the firm fromemploying the worker at full time.

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Households’ desired nominal consumption (CDi,t) is a linear function of cur-rent disposable income and current wealth held in the form of deposits, withfixed marginal propensities cy and cd:

CDh,t = cyyDh,t + cdDh,t (17)

Consumption is given by:

Ch,t = cCh,t (18)

(19)

Consumers samples ψ potential suppliers, and rank them from the most tothe least preferred. The model employs a locational specification of consumers’preferences and firms’ offered varieties inspired by Salop’s (1979) circular refine-ment of Hotelling (1929): we assume that good varieties produced by firms’ andconsumers’ preferences are randomly located on a circle with unitary diameter.We define dhi as the distance between consumer h and a firm i.5

Consumers rank suppliers based on their distance and offered price: the lowerthe price and the distance of a given supplier, the greater the satisfaction thatthe consumer gets from consuming its products (i.e. the greater the preferencefor the supplier). Formally, household h prefers firm i to firm j if:

1

dβhi

Ptpi,t

>1

dβhj

Ptpj,t

(20)

where pi,t and pj,t are the prices charged by the two suppliers, Pt is thesector average price, and β ≥ 0 is a parameter weighting households’ preferencesfor variety: the lower β, the more consumers perceive consumption goods ashomogeneous, their consumption allocation becoming more sensitive to pricedifferentials.

In presence of supply constraints consumers can browse through their rank-ing of suppliers trying to satisfy the residual demand.

Households hold deposit accounts at commercial banks Dh,t, returning apositive interest at the rate rd,t, and participations in the equity of firms andbanks Ah,t, yielding dividends when profits are positive. In each period house-holds have to decide how to allocate their savings between these two types offinancial assets. Given the simplicity of the model financial side and the lim-ited number of financial assets considered, we implemented a simple portfoliofunction inspired by the Tobinesque approach to households portfolio allocation(Brainard and Tobin, 1968). The fundamental insight we take from this ap-proach is that households, in their role of financial investors, assess the desiredallocation of their financial wealth by comparing the expected rates of returnof the assets they can purchase: for simplicity reasons, we take the past ratesof return yielded by deposits and past equity investments as a measure of theseexpectations. While deposits are a risk-free asset, the rate of return on equityinvestment is weighted by its perceived riskiness, proxied by the past extinction

5Being ωi the radian value identifying the position of the firm i and ωh the radian valueassociated to consumer h’s location: dhi = sin(min[|ωh − ωi|, 2π − (|ωh − ωi|)]/2)

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rate of firms and banks indicated by Prdefaultt .Indicating by lph,t the share orwealth that households desire to hold in the form of deposits, we have:

lph,t =

λe−(Divh,t−1Ah,t−1

(1−Prdefaultt )−rd,t)ifDivh,t−1

Ah,t−1≥ rd,t and Ah,t−1 ≥ 0

λ ifDivh,t−1

Ah,t−1< rd,t or Ah,t−1 = 0

(21)with 0 < λ < 1 representing an exogenous upper threshold to the share of

wealth that households want to hold in the form of deposits.Households choose their deposit bank randomly, since every bank offers the

same interest rate rd,t for simplicity reasons.If we indicate by NWD

h,t = NWh,t−1 + yDh,t−CDh,t households’ expected levelof net-worth based on their planned consumption and income levels, the desiredlevel of equity and deposits can be then expressed as:

ADh,t = max{Ah,t−1, (1 − lph,t)NW

Dh,t

}(22)

DDh,t = NWD

h,t − (ADh,t −Ah,t−1) (23)

where ADh,t − Ah,t−1 is the desired investment in equity, which is bound tobe non-negative.

Households having a positive desired investment act together as an invest-ment fund to create a new firm or a new bank. If funds collected are sufficient(i.e. above the threshold represented by the initial equity value which is ran-domly sampled, see section 3.1.6), the new enterprise is created. Otherwise,households postpone investment to following periods and deposit resources orig-inally allocated to equity investment in their bank account. Conversely, if thequantity of funds collected is very high, more than one firm (bank) might enterthe market in the same period.

3.1.3 Banks

Banks offer demand deposit accounts to households and firms, paying an interestrd,t equal to a constant fraction ζ of the discount rate rt fixed by the CentralBank. In addition, banks create money endogenously providing credit to firms.In order to avoid taking excessive risks, the maximum amount of credit thatbanks are willing to supply in any given period is a multiple µ1 of their equityAz,t: L

DSz,t = µ1Az,t

Banks receive credit applications from firms. For each loan application, theycompute a probability Pr(Loani,t) to grant it and an interest rates (ri,t) tocharge. These are defined as, respectively, a decreasing and increasing functionof the borrowers’ riskiness, proxied by their target leverage (LDi,t/Ai,t), whereAi,t indicates the equity of firm i at period t:

Pr(Loani,t) = e−ιl

LDi,tAi,t (24)

ri,t = χLDi,tAi,t

+ rt (25)

Banks are subject to minimal reserve requirements, expressed as a share µ2

of their deposits: RMz,t = µ2Dz,t

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If reserves RMz,t are below the minimum level, banks apply to the CentralBank lending facility, receiving cash advances (LzCB,t) at the discount rate rtto restore the mandated liquidity ratio. If instead banks have more reservesthan needed, the excess can be invested in the purchase of bonds (Bz,t) issued,which bring an interest rate rb,t (equation 35). Banks’ probability of purchasingeach tranche of public debt depends on the public debt-to-GDP ratio (see Caianiet al. (2018a) for the details).

Banks’ profits are then equal to:

πz,t =

It∑i,Liz,t>0

ri,tLiz,t +

It∑k,Bz,t>0

rb,tBz,t + rreRz,t −BDiz,t − rd,tDz,t − rtLzCB,t

(26)

where (BDiz,t) indicates ‘the ‘bad debt”, that is non-performing loans dueto borrowers’ default.

Banks pay taxes on (positive) profits and distribute to equity holders a shareρ of net profits. These dividends are distributed between investors proportion-ally to the share of the bank’s equity they own.

Tπz,t =

{τk,tπz,t, if πz,t > 0

0, if πz,t ≤ 0(27)

Divπz,t =

{ρ(πz,t − Tπz,t), if πz,t > 0

0, if πz,t ≤ 0(28)

3.1.4 Central Bank

The Central Bank sets the discount interest rate following a Taylor rule basedon the average level of inflation (Taylor, 1993; Smets and Wouters, 2007; Geraliet al., 2010):

rt = r(1 − ξ) + ξrt−1 + (1 − ξ)ξP (Pt−1 − P ∗) (29)

where r is the exogenous long run interest rate, ξ is the parameter defining

the speed of the adjustment, ξP is the sensitivity to inflation, Pt−1 is the pastaverage level of inflation, and P ∗ is the inflation target.

The Central Banks hold reserves of commercial banks (RCB,t), accommo-dates their requests of cash advances (LCB,t), and possibly buys bonds issued bythe government (BCB,t) which remained unsold after private banks’ purchases.

Central Banks’ profits (πCB,t = rb,tBCB,t + rtLCB,t − rreRCB,t) are auto-matically redistributed to the government.

3.1.5 Government

Government collects income taxes from households (h) and taxes on past periodprofits from firms (i) and banks (z). Total taxes Tt are then equal to:

Tt =

H∑h,yh,t>0

τtyh,t +

I∑i,π∗>0

τtπi,t−1 +

Z∑z,π>0

τtπz,t−1 (30)

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Public spending Gt takes the form of a lump-sum, equally-distributed mon-etary transfer to households (Gt/H).

The government balance is the difference between revenues from taxes andgovernment spending, including interests paid on bonds. When negative, thegovernment runs a deficit DEFt. In the opposite case the government attainsa budget surplus SUt−1. Possible budget surpluses are set aside to fund publicexpenditure in the next periods, thereby reducing the quantity of bonds to beissued.

The government determines in each period the level of public spending (Gt)and the tax rate (τt) following an adaptive scheme, based on the discrepancybetween desired and past levels of public expenditure on the one hand, andexpected and admissible levels of public deficit on the other hand. The desiredlevel of public expenditure GDt is simply defined as the initial (exogenously set)real value of public spending G, adjusted for the average level of prices Pt andaverage productivity Φt, so to ensure that the dimension of GDt remains roughlystable compared to aggregate GDP: GDt = PtΦtG. In addition, governments arecommitted to make efforts not to exceed a deficit-to-GDP threshold indicatedby dmax. Public expenditure and tax rates are then revised according to thefollowing scheme:6

if dt−1 ≥ dmax and GDt ≤ Gt−1 :

{Gt = Gt−1(1 − U [0, δ])

τt+1 = τt(1 + U [0, δ])(31)

if dt−1 ≥ dmax and GDt > Gt−1 :

{Gt = Gt−1

τt+1 = τt(1 + U [0, δ])(32)

if dt−1 < dmax and GDt ≤ Gt−1 :

{Gt = Gt−1(1 − U [0, δ])

τt+1 = τt(1 − U [0, δ])(33)

if dt−1 < dmax and GDt > Gt−1 :

{Gt = Gt−1(1 + U [0, δ])

τt+1 = τt(34)

In each period, government repays bonds previously issued and pay intereststo bond holders. The interest rate on bonds is set as a premium on the CentralBank discount rate depending on the debt-to-GDP ratio of the country:

rb,t = rt + χBt/Yt (35)

Newly issued bonds (for a total value of Bt) are split into 100 tranches (bk,t =Bk,t/100) and put on the bond market where they are purchased by commercialbanks, and by the Central Bank for the possible residual part.

Finally, government steps in to guarantee depositors in case of default bya bank. For this sake, government issues an additional batch of bonds, whichis directly purchased by the Central Bank, and uses the liquidity collected toreimburse households and firms who lost their deposits in the default.

6To avoid unreasonable high or low values, the tax rate is bound to vary within therange {τmin, τmax}, whereas Gt is bound between a minimum and maximum share of GDP:{gminYt, gmaxYt}.

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3.1.6 Firms’ and banks’ endogenous entry and exit

Part of households’ savings is invested in the creation of new firms and newbanks (see section 3.1.2).

The new entrant will be a bank when either the ratio between banks’ andfirms’ number, or the ratio between banks’ and firms’ total net worth are belowa given percentage η. Otherwise, the new entrant will be a firm. In this way weaim to avoid excessive imbalances in the relative dimension of the banking andfirm sectors.

The initial equity level of the new organization is sampled in a range betweenthe net worth of the smallest and the net worth of the larger incumbents: thefirst h investors required to collect this level of funds become its shareholders.If funds invested by households are lower than the randomly sampled initial networth, no firm (bank) is created and funds originally allocated by households toequity investment are deposited at banks, being available to fund households’investment in the next period.

New firms’ initial productivity (φ), price (pi,t), and offered wage (wi,t) arealso sampled within a range going from the lowest to the highest values of in-cumbent firms in the sector. Sales expectations (qei,t) are the maximum betweenthe random value sampled in the range between the lowest and highest valuesof incumbents and

Ai,twi,t

φi,t, this latter representing the amount of goods feasibly

producible, given the initial values of equity, wage, and productivity of the newfirm.

Firms whose net worth is below a threshold level, defined as the wage theyoffer to workers Ft = wi,t, default. Similarly, banks having a net-worth belowthe average wage default.7

A default by a firm implies a non performing loan for creditors. The largerthe bad debt suffered by banks, the worse the effect on their balance sheet. Inthe rare case of a default by a bank the government steps-in and issues additionalbonds to reimburse depositors, as discussed in section 3.1.5.

3.2 Simulation scheduling

The sequence of events taking place within each period of the simulations is thesame presented in Caiani et al. (2018a):

1. Firms determine their desired production, their labor demand, the priceof their output, the wage offered, and their desired R&D investment.

2. Firms interact with banks on the credit market and possibly receive loans.Banks possibly ask cash advances to the Central Bank to satisfy themandatory liquidity ratio.

3. Firms interact with workers on the labor market.

4. Workers are paid and employed to produce firms’ output and to performR&D. Dividends generated in the previous period are distributed to equityholders, summing up to their current income.

7In this way we remove from the simulation not only firms having a negative net-worthbut also microscopic firms whose contribution to the dynamics of the model is negligible, thusrepresenting just a computational burden.

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5. Government calculates revenues from taxes (on past period profits andcurrent period households’ income), determines the level of public spend-ing and the tax rate for the next period, repays bonds plus interests tobond holders, and determines the quantity of bonds to be issued.

6. Bonds are put on the bond market where commercial banks buy it. Thepossible residual part is purchased by the Central Bank.

7. After having paid taxes and received the tax-exempt monetary transferfrom the government, households compute their demand for consumptiongoods and interact with firms on the good market.

8. Firms and banks compute their profits and update their net worth andshareholders’ equity accordingly. Taxes and dividends to be paid in thenext period, respectively, to the government and to equity holders are thencomputed.

9. Defaulted firms and banks exit the market. Households equity investmenttakes place and, if enough financial resources are collected, new firms andbanks are created.

4 Sale variation expectations

Even if the rules that govern the behaviors of the agents are simple, the finalgood market is quite turbulent. Indeed each firm faces the competition of theother ones which may vary their prices and selling quantity in each periods.Moreover each incumbent firm competes with new entrants that may producegood varieties that are similar to the one it sales. Besides, sales are determinedby aggregate demand fluctuations. While production is influenced by both pro-ductivity and wages variations.

Therefore, sales forecasting are not so easy. This is one of the main reasonwhy we choose to focus only on forecasting the sign of the sales variation in eachperiod. In effect we try to provide firms with predicting techniques that mayforecast just if the sales growth is negative or positive. Moreover, the agents inthis model adapt gradually their behavior, thus a punctual prediction of salesgrowth would not be so useful in determining their choices. Instead knowing ifsales will increase or decrease is an information that may easily be incorporatedin producing and sales rules.

We implemented four forecasting methods: a naive choice (N), a genetic al-gorithm (GA), an autoregressive model (AR) and a a random choice (Random).The naive choice (N) means that firm forecasts are equal to their past periodresult: if in the past period sales increased (decreased) firms predict a positive(negative) growth in the next period. The genetic algorithm (GA) uses as in-formative set four variables: firm past sales, the ratio between firm price andthe average price, the inflation growth rate and the unemployment growth rate.The first two variables are measures of competitiveness while the other two tryto capture some aggregate dynamics. We use the same variables to implementa vectorial autoregressive model (AR), which provides a punctual sales growthone-step ahead predictions. Both the output of the genetic algorithm (GA)and of the autoregressive model (AR) are translated in a simpler prediction ofjust the sign of the growth rate of sales. Moreover, in order to increase the

12

effectiveness of these two methods, which rely on the volume of data that theymay process, new entering firms inherit data processed by a randomly chosenincumbent firm. Finally, according to the random method (Random) firmschoose their prediction randomly with equal probability for positive or negativegrowth. The random choice provides us as a sort of zero intelligence scenariouseful to test the effectiveness of the other forecast methods.

We compute the prediction mean squared error (MSEt) and the averageerror (ERt). Firms have only to try to predict if in their next period saleswill grow or decline, thus we define a synthetic sales growth indicator xit thatis equal to 1 if sales are positive and, on the contrary, is equal to -1 if salesare negative. The mean squared error is given by the average square differencebetween the predicted and the effective value of the synthetic growth indicatorof each firm i of the N firms in the market at time t:

MSEt =

∑Ni (xit − xEit)

2

N(36)

The average error (ERt) is the sum of the prediction errors of firms:

ERt =

∑Ni (xit − xEit)

N(37)

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GA AR N Random

Figure 1: On the left side panel average error. On the right side average meansquared error. GA in red, AR in blue, N in gray, Random in green. Solid linesrepresent averages over 50 simulation, dotted line are 95% confidence interval.

We calibrated both the genetic algorithm (GA) and the autoregressive method(AR) in order to let their forecasting error converge to zero as shown in the leftpanel of figure 1. While the Naive method is negatively biased and the Ran-dom method is even more negatively biased. The negative bias of the last twomethods is due to the fact that the real output of the economy grows, thus onaverage also firm sales tend to increase through time. While random predictionuse the same probability for both positive and negative sale variations, thus itwill predict more frequently negative growth with respect to what happens inthe economy. To a lesser extent, the same happens for the naive methods, whichis only partially able to capture the intrinsic growing tendency of the economy.

13

Consequently, the mean squared error is higher for the random and the naivemethods (Figure 1, right Panel). While the genetic algorithm outperforms theautoregressive method. Indeed, the autoregressive method to be effective needsa more stable environment and a huge set of data for each firm. While thegenetic algorithm needs only to associate the sign of the variation of the inputvariables with the sign of the expected sales variation.

5 Exploiting expectations for sales decisions

In this section, we define a new sales rule that incorporating sales expectationsto improve firm performance.

In the baseline model, when sales are higher than desired production, firmsincrease prices and sale. While when sales are lower than the desired ones, firmsreduce them.

When sales are higher than the desired production firms may not have acorrect measure of the demand that they receive, i.e. when the effective demandthat a firm receives is larger than the supply, the demand is underestimated.Therefore, even according the new sales rule, in this case firms simply increasetheir desired production.

While the new rule diverges from the baseline one when sales are lower thanthe the desired ones. Indeed, in this last case firms have a correct measure of thedemand received in the last period and may effectively exploit the expectationon next sales. Therefore, even if sales were lower than the desired ones, whenpositive growth is expected in the following period, it may be more effective tonot decrease the desired sales or to reduce them by a smaller amount only. Onthe contrary, when sales are lower than the desired one and negative growth isexpected in the following period, firms may exploit expectations reacting morerapidly, thus reducing by a larger amount both the price and quantity sold.

Thus, with sales expectation (∆qEi,t+1) and an adjusting parameter associ-ated to the expectations δE , the production and prices decisions become:

if qi,t−1 ≥ qSi,t−1 :

{qSi,t = qSi,t−1(1 + U [0, δ])

pi,t = pi,t−1(1 + U [0, δ])(38)

if qEi,t−1 ≤ qSi,t−1 :

if ∆qEi,t+1 ≥ 0 :

{qSi,t = qSi,t−1(1 − U [0, δ − δE ])

pi,t = pi,t−1(1 − U [0, δ − δE ])

if ∆qEi,t+1 < 0 :

{qSi,t = qSi,t−1(1 − U [0, δ + δE ])

pi,t = pi,t−1(1 − U [0, δ + δE ])

(39)

(40)

When sales are larger than the desired production and their expected varia-tion is positive, firms increase the desired sales and prices by δ as in the baselinescenario. While, when sales are lower than the desired production and the ex-pected variation of sales is positive, firms decrease production and prices at aslower pace (δ − δE). On the contrary, when sales expectations are negative,firms faster reduce prices and production than in the baseline scenario (δ+δE)).

We present the results of the new rule in two scenarios: the first with δE equalto half δ (WA: Low Adjustment) and the second with δE equal to δ (SA: Strong

14

Adjustment). Thus, when sales are lower than the desired production and theexpected sales variation is positive, in the first scenario (WA) the reduction ofprice and sales is reduced by a half, while in the second (SA) firms simply do notdecrease prices and sales. On the contrary, when sales are lower than the desiredproduction and the expected sales variation is negative, in WA the reduction ofprice and sales is faster by a half, while in SA the adjustment speed is doubled.

Changing the sales rule does not affect the predictive capabilities of both thegenetic and the autoregressive methods. Nevertheless, the weak adjustment rule(WA) and, in particular, the strong adjustment one (SA) are able to translatepredictive capabilities in higher profits without affecting firm riskiness. Indeed,the left panel of Figure 2 shows that profits increase when firms adopt the au-toregressive model (AR) and the genetic algorithm (GA) to predict the variationof future sales. At the same time, the right panel of Figure 2 shows that thefailure rate of firms remains unchanged with respect to the baseline scenario.

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SA0.

050.

100.

150.

20

Profit Rate

300 400 500 600 700 800 900 1000

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0.05

0.06

0.07

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0.09

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Failure Rate

300 400 500 600 700 800 900 1000

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0.05

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Failure Rate

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Figure 2: On the left side panels WA scenario. On the right side panels SAscenario. On the upper panels, profit rate. On the bottom panels firm failureprobability. GA in red, V AR in blue, N in gray, Random in green. Solid linesrepresent averages over 50 simulation, dotted line are 95% confidence interval.

However, on the aggregate level (Figure 6), higher firm profitability is as-sociated with higher mark-ups, that results in a weaker dynamic of real wagesin both the GA and AR implementations. Therefore, the wage share shrinksaffecting negatively the aggregate demand. Moreover, firms’ innovative effort isproportional to past sales, thus a weaker demand reduces their innovativeness.Therefore, in the long run, higher wages come at the price of higher unemploy-ment levels and lower output growth.

15

WA

0.72

0.74

0.76

0.78

0.80

0.82

Wage Share

300 400 500 600 700 800 900 1000

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● ●

●

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● ●

●

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● ●

●

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●

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●

●●

●●

●

●

●

●

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●

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●

●

●

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●

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●

●

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●

●

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●●

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●

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●

●

●

●

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●

●

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●

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●

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●

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●

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●

● ●

●

●

●

●

●

●

●

●

●

●

●

SA

0.65

0.70

0.75

0.80

Wage Share

300 400 500 600 700 800 900 1000

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

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● ●

●

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●

●

●

●

●

●

●

●●

●

●

● ●

●

●

● ●

●● ● ●

●●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●●

●

●

●

●

●

● ●

●

● ● ●

●

●

●●

●

●

●●

●

●

●●

●

● ●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

● ●●

●

●●

●

● ● ●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

●

●

● ●●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

● ●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●● ●

●

●

●

●

●

●

●

●

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●

●

●

●

●●

●

● ●

●

●●

●

●

●

●

●

●

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●

● ●

●

●

●

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●

●● ●

●●

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●

●

●

●●

●

●

●

●

●

●●

●

●

● ●

●

●

●●

●

●●

● ●

●

●●

● ●

●

●

●

●

●

●

●

●●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

● ●

●

●●

●

● ●

●

●

●●

●●

●

●

●

● ●

●

●

● ●●

●

●●

●

●

● ●

●●

●

● ●

●

●

● ●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●●

●

●

●●

●

●●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ● ●

●

●

●

●

● ●● ●

●

●

●●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

● ●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●●

●

●

●

●●

●

●

●

●

● ● ●

●●

●

●

●

●

● ●

●●

●

●

●

●

●

●

●

● ●

●

●

●

● ●

● ●

●●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

● ●

●

● ●●

●

●

●

●

●

●

0.04

0.05

0.06

0.07

Unemployment

300 400 500 600 700 800 900 1000

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

● ●

●

●

●

●●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●● ●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●●

●●

●

● ●●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

0.04

0.05

0.06

0.07

0.08

0.09

Unemployment

300 400 500 600 700 800 900 1000

●

●

●

●

●

●●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●●

●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●●

●●

●

●

●

● ●

●

●

●

●

●

●

●●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

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●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

● ●

●

●

●●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

● ●

●

● ●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

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●

●

●

●

●

●

●

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●

●

●

●

● ●

●

●

●

●

●

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●

●

●

●

●

●

●

●

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●

●

●

●

●

●

●

●

●

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● ●

●

●●

●

●

●

●

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●

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●

●

●

●

●

●

●

●

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●

●

●

●

●

●

●

●

●

●

●

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●

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●

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●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

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●

● ●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

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●

●

●

●

●

● ●

●

●

●

●

●

●

●●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●●

● ●

●

●

●

●

●

●

●

●

●

●

24

68

1012

14

Productivity

300 400 500 600 700 800 900 1000

●

●●

●

●●

●●

●

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Figure 3: On the left side panel WA scenario. On the right side panel SAscenario.On the upper panels wage share. On the middle upper panels unem-ployment. On the middle lower panels productivity. On the lower panels output.GA in red, AR in blue, N in gray, Random in green, the baseline model in black.Solid lines represent averages over 50 simulation, dotted line are 95% confidenceinterval.

16

6 Firm adopting different predictive methods

In the previous sections we show both the micro and macro effects of some pre-dictive methods, but what happens if in the same market firms that implementdifferent methods compete among each others? Therefore, in this section weshows two experiments. In the first one entering firms may adopt with the sameprobability one of this three predictive methods: naive (N), genetic algorithm(GA) and autoregressive (AR). Firms follow the methods that have chosen atthe beginning, thus the predictive method is fixed (FPM). In the second experi-ment ,since period 500, firms can change their predictive method in each period(CPM). In both the experiments we adopt the strong adjustment rule (SA) inorder to have a clearest glimpse of the impact of expectations on the economy.

In the changing predictive method (CPM) experiment, following Anufrievand Hommes (2012), firms may choose among the three different rules choos-ing the more effective one according to a performance measure Uhi,t−1, whichdetermines the performance of the method h by firm i at time t− 1.

Uhi,t = −(xi,t − xEhi,t)2 + ηUh,t−1 (41)

where xit is the synthetic indicator of sales variations and xEhi,t the predictedvariation according to the method h. Thus the probability of choosing a methodh is equal to:

p(Uhi,t) = δUp(Uhi,t−1) + (1 − δU )βe(Uhi,t)∑h Uhi,t)

(42)

where δU gives the memory of the performance of the different methods.The two left panels of figure 4 shows that in the fixed predictive method

experiment (FPM) the forecast results are in line with the baseline scenario,but the mean square error is higher in the FPM experiment for naive (N) andgenetic algorithm expectations (GA). In fact, the co-presence of firms applyingdifferent predictive methods inevitably increases forecast errors.

In the changing predictive method experiment (CPM), when firms are al-lowed to change their rule (period 500), more firms choose the GA method thatis the more effective in reducing the mean squared error (figure 5), indeed firmschoose the strategy that reduce the square of the prediction error (equation 41).Consequently, firms with the higher mean squared error have higher probabilityof leaving the naive (N) and autoregressive methods (AR) in favor of the geneticalgorithm (GA), thus the mean squared error of N and AR goes down while themean squared error of GA grows slightly (lower right panel of figure 4).

At the same time, all the three predictive methods become less correct: theautoregressive method (AR) becomes positivly unbiased, the genetic algorithmdoes not continue to converge to zero and the naive method augments its neg-ative bias (upper right panel of figure 4).

The possibility of changing the predictive methods increase the complexityof the system. Moreover, incorporating expectations in the sales rule impactson the effective future sales. This may lead to self-fulfilling predictions. Forinstance, when sales are lower than the desired ones and the expected salesvariation is negative, firms reduce rapidly the desired sales and so production,this may conduce to a reduction in future sales due to the reduced supplymore than as a consequence of the low demand. When firms may change their

17

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GA AR N

Figure 4: On left side panel the FPM experiment, on the right panels the CPMexperiment. In the upper panels the average error. On the bottom panel theaverage mean squared error. GA in red, AR in blue, N in gray. Solid linesrepresent averages over 50 simulation, dotted line are 95% confidence interval.

18

predictive methods, it is more difficult for all these methods to adjust to theself-fulfilling nature of their predictions. This contributes to produce biasedforecasts.

Besides, in the CPM experiments it seems that the profit rate for firmsthat adopt the GA methods increases with respect to the the other two tech-niques (figure 5 middle panels), while the failure probability for firms that adoptGA seems to slightly increase after firms are allowed to change their predictivemethods.

Macro results seems in line with the ones emerging in the previous sections:both the FPM and the CPM lead to higher unemployment and lower productiv-ity in the long run, even if these effect are stronger with respect to the scenariowhere all firms adopt the GA method (Figure 6).

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Figure 5: On the left side panels FPM experiment. On the right side panelsCPM experiment. On the middle panels, firms predictive method shares. Onthe middle panels, profit rate. On the bottom panels firm failure probability.GA in red, V AR in blue, N in gray. Solid lines represent averages over 50simulation, dotted line are 95% confidence interval.

19

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68

1012

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Figure 6: On the left upper panel wage share. On the right upper panel un-employment. On the left middle panel productivity. On the right middle paneloutput. GA in red, AR in blue, FPM in brown, CPM in orange, the baselinemodel in black. Solid lines represent averages over 50 simulation, dotted lineare 95% confidence interval.

20

7 Conclusions

Adopting computational techniques, even in a complex economic system, itis possible to endow agents with expectations that are not biased and showa certain degree of accuracy. Indeed, in the paper we show that firms thatimplement a generic algorithm (GA) or an autoregressive process (AR) are ableto formulate sales variation predictions with errors that converge to zero andwith a relatively low mean squared error.

Firms may easily exploit these expectations to orientate their productionand price decisions in order to increase their profit. In fact, using the GA andAR methods, firms increase their profit rate without augmenting their riskiness.

However, increasing profits reduce the wage share affecting negatively theaggregate demand, leading in the long run to higher unemployment and lowerproductivity.

Besides, we made different experiment allowing the competition betweenfirms adopting different prediction methods. In these experiments, as expected,the complexity of the environment increases resulting in a reduced accuracy offorecasts for all the predictive techniques tested.

In this model the financial sectors and the labor market are quite simplified,deepening them may allow also to model banks and consumers endowed withforecast capabilities. Moreover, it may be possible to test the relations betweenpolicy interventions and agents’ expectations.

21

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23

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GA AR N Random

200 400 600 800 1000

1.0

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1.8

2.0

Mean Squared Error

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GA AR N Random

Figure 7: On left side panel the WA scenario, on the right panels the SA scenario.In the upper panels the average error. On the bottom panel the average meansquared error. GA in red, AR in blue, N in gray, Random in green. Solid linesrepresent averages over 50 simulation, dotted line are 95% confidence interval.

24