text: two small communities will merge into one community (Appendix A). In contrast, it is easy to see that ≥ D > D with m 2, which indicates the partition density D based optimization does not separate merge have the resolution problem mentioned above (Appendix A). The above analysis is conducted for the special partition wherein two smaller bicliques are merged into a community with each other biclique as a separate community. With the fact that bicliques have maximum partition density, it is easy to see that any other partition has a lower partition density than the one with each biclique as a community. Therefore, the optimal value of partition density D corresponds to the correct partition. In contrast to bipartite modularity Q, optimizing partition density D can correctly detect communities of any size. In summary, optimization of partition density D can often achieve correct partitions while Barber’s bipartite modularity not. Thus, we deduce that bipartite partition density D can act as a proper quantitative function for community detection in bipartite networks. 4 Integer Programming Model for Bipartite Community Parti- tion { · · · } { · · · } (cid:80) (cid:80) Let U = u , u , , u , V = v , v , , v be two disjoint node sets of bipartite network G = 1 2 p 1 2 q (U, V, E), and A = (a × ) × be the adjacency matrix of node sets U and V in G, where a = 1 indicates i j p q | | ij there is an edge e between nodes u and v , a = 0 otherwise. Note that E = p q a is the ij i j ij i=1 j=1 ij total number of edges in G. Let’s deﬁne bina(cid:26)ry variables xic, yjc to represent whether node ui and node v belong to community G or not, respectively: j c ∈ 1 if u G x = (cid:26) i c ic 0 otherwise, ∈ 1 if v G y = j c jc 0 otherwise. We can formulate the problem of partitioning bipartite network G into K communities into an integer (cid:80) (cid:80) programming model (Model(cid:88)-1): (cid:80) (cid:80) max D =(cid:88)K|E1| cK=1 (( pi=pi=11 xiqjc=)1·(·a·ijqjx=i1cyyjjcc))2 (4) (cid:88) x = 1 i = 1, 2, , p (5) ic s.t.  cK=1 yjc = 1 j = 1, 2, · · · , q (6) c=1∈ { } · · · · · · x 0, 1 ; i = 1, 2, , p, c = 1, 2, , K (7) ic ∈ { } · · · · · · y 0, 1 ; j = 1, 2, , q, c = 1, 2, , K (8) jc where the objective function (4) is to maximize partition density D, constraints (5) and (6) indicate that each node must belong to one and only one community, and constraints (7) and (8) ensure that the variables are binary ones. We can easily solve the integer programming model by using the Lingo (cid:80) software [30] for small-scale bipartite networks to see if the model can ﬁnd communities properly. To (cid:80) partition a bipartite network into overlapping communities, we can simply relax the constraints (5) and ≥ (6) to allow a node to belong to more than one community by changing the constraints to K x 1 · · · ≥ · · · c=1 ic for i = 1, 2, , p and K y 1 for j = 1, 2, , q. We represent such a model as Model-2. c=1 jc For example, the bipartite network in Figure 2A has graph density 0.4321. We can easily partition it into two and three communities by solving Model-1 and obtain the partition density 0.6391 and 0.9429. If we partition it into more than 3 communities, the partition density will decrease and thus the maximum partition density is 0.9429 which seems reasonable. For the bipartite network in Figure 2B, we can partition it into two overlapping communities by solving Model-2 with the maximum partition density 1. 6 ##LTLine## ##LTLine##

text: two

text: small

text: communities

text: will

text: merge

text: into

text: one

text: community

text: (Appendix

text: A).

text: In

text: contrast,

text: it

text: is

text: easy

text: to

text: see

text: that

text: D

text: D

text: with

text: m

text: which

text: indicates

text: the

text: partition

text: density

text: D

text: based

text: optimization

text: does

text: not

text: separate

text: merge

text: have

text: the

text: resolution

text: problem

text: mentioned

text: above

text: (Appendix

text: A).

text: The

text: above

text: analysis

text: is

text: conducted

text: for

text: the

text: special

text: partition

text: wherein

text: two

text: smaller

text: bicliques

text: are

text: merged

text: into

text: a

text: community

text: with

text: each

text: other

text: biclique

text: as

text: a

text: separate

text: community.

text: With

text: the

text: fact

text: that

text: bicliques

text: have

text: maximum

text: partition

text: density,

text: it

text: is

text: easy

text: to

text: see

text: that

text: any

text: other

text: partition

text: has

text: a

text: lower

text: partition

text: density

text: than

text: the

text: one

text: with

text: each

text: biclique

text: as

text: a

text: community.

text: Therefore,

text: the

text: optimal

text: value

text: of

text: partition

text: density

text: D

text: corresponds

text: to

text: the

text: correct

text: partition.

text: In

text: contrast

text: to

text: bipartite

text: modularity

text: Q,

text: optimizing

text: partition

text: density

text: D

text: can

text: correctly

text: detect

text: communities

text: of

text: any

text: size.

text: In

text: summary,

text: optimization

text: of

text: partition

text: density

text: D

text: can

text: often

text: achieve

text: correct

text: partitions

text: while

text: Barber’s

text: bipartite

text: modularity

text: not.

text: Thus,

text: we

text: deduce

text: that

text: bipartite

text: partition

text: density

text: D

text: can

text: act

text: as

text: a

text: proper

text: quantitative

text: function

text: for

text: community

text: detection

text: in

text: bipartite

text: networks.

text: Integer

text: Programming

text: Model

text: for

text: Bipartite

text: Community

text: Parti-

text: tion

text: Let

text: U

text: u

text: u

text: u

text: V

text: v

text: v

text: v

text: be

text: two

text: disjoint

text: node

text: sets

text: of

text: bipartite

text: network

text: G

text: p

text: q

text: (U,

text: V,

text: E),

text: and

text: A

text: (a

text: be

text: the

text: adjacency

text: matrix

text: of

text: node

text: sets

text: U

text: and

text: V

text: in

text: G,

text: where

text: a

text: indicates

text: i

text: j

text: p

text: q

text: ij

text: there

text: is

text: an

text: edge

text: e

text: between

text: nodes

text: u

text: and

text: v

text: a

text: otherwise.

text: Note

text: that

text: E

text: p

text: q

text: a

text: is

text: the

text: ij

text: i

text: j

text: ij

text: ij

text: total

text: number

text: of

text: edges

text: in

text: G.

text: Let’s

text: deﬁne

text: variables

text: xic,

text: yjc

text: to

text: represent

text: whether

text: node

text: ui

text: and

text: node

text: v

text: belong

text: to

text: community

text: G

text: or

text: not,

text: respectively:

text: j

text: c

text: if

text: u

text: G

text: x

text: i

text: c

text: ic

text: otherwise,

text: if

text: v

text: G

text: y

text: j

text: c

text: jc

text: otherwise.

text: We

text: can

text: formulate

text: the

text: problem

text: of

text: partitioning

text: bipartite

text: network

text: G

text: into

text: K

text: communities

text: into

text: an

text: integer

text: programming

text: model

text: max

text: D

text: x

text: i

text: p

text: ic

text: s.t.

text: yjc

text: j

text: q

text: x

text: i

text: p,

text: c

text: K

text: ic

text: y

text: j

text: q,

text: c

text: K

text: jc

text: where

text: the

text: objective

text: function

text: is

text: to

text: maximize

text: partition

text: density

text: D,

text: constraints

text: and

text: indicate

text: that

text: each

text: node

text: must

text: belong

text: to

text: one

text: and

text: only

text: one

text: community,

text: and

text: constraints

text: and

text: ensure

text: that

text: the

text: variables

text: are

text: binary

text: ones.

text: We

text: can

text: easily

text: solve

text: the

text: integer

text: programming

text: model

text: by

text: using

text: the

text: Lingo

text: software

text: for

text: small-scale

text: bipartite

text: networks

text: to

text: see

text: if

text: the

text: model

text: can

text: ﬁnd

text: communities

text: properly.

text: To

text: partition

text: a

text: bipartite

text: network

text: into

text: overlapping

text: communities,

text: we

text: can

text: simply

text: relax

text: the

text: constraints

text: and

text: to

text: allow

text: a

text: node

text: to

text: belong

text: to

text: more

text: than

text: one

text: community

text: by

text: changing

text: the

text: constraints

text: to

text: K

text: x

text: ic

text: for

text: i

text: p

text: and

text: K

text: y

text: for

text: j

text: q.

text: We

text: represent

text: such

text: a

text: model

text: as

text: jc

text: For

text: example,

text: the

text: bipartite

text: network

text: in

text: Figure

text: has

text: graph

text: density

text: We

text: can

text: easily

text: partition

text: it

text: into

text: two

text: and

text: three

text: communities

text: by

text: solving

text: and

text: obtain

text: the

text: partition

text: density

text: and

text: If

text: we

text: partition

text: it

text: into

text: more

text: than

text: communities,

text: the

text: partition

text: density

text: will

text: decrease

text: and

text: thus

text: the

text: maximum

text: partition

text: density

text: is

text: which

text: seems

text: reasonable.

text: For

text: the

text: bipartite

text: network

text: in

text: Figure

text: we

text: can

text: partition

text: it

text: into

text: two

text: overlapping

text: communities

text: by

text: solving

text: with

text: the

text: maximum

text: partition

text: density

text: ##LTLine##

text: ##LTLine##


text: 11 0.1 output price wages 0.05 consumption 0 -0.05 policy shock -0.1 time FIG. 3: Aggregate response of the economy to a policy shock represented by a drop in the interest rate ρ set by the central 0 bank (from 2% to 1.8%). The vertical dashed line gives the timing of the policy shock. All aggregate variables are represented as a relative variation from their average values prior to the shock. For a better visualization, we ampliﬁed by a factor 100 the ∗ wage and price responses. Note that the damped oscillation relaxation towards equilibrium. Parameters are Θ = 3, ρ = 2%, α = 50, α = 4, and other parameters as in Fig. 2. Γ c A. A monetary shock In order to check that our framework produces reasonable results, it is interesting to start with the simplest policy experiment, i.e. an expansionary monetary policy shock where the baseline interest rate is instantaneously decreased from 2% to 1.8%, all other parameters of the model being ﬁxed, and with no further intervention of the Central Bank (i.e. φ = φ = 0). The resulting dynamics of output, wages and prices is shown in Fig. 3 and can be compared π ε with e.g. Fig 1 of Ref. [36]. It is gratifying to see that our simple modelling strategy, in particular the way the monetary policy is channelled to the economy, does lead to a quite realistic shape of impulse response functions, at least qualitatively. In fact, the empirical hump-shaped response of the output reported in [36] is quite well reproduced by their DSGE model with nominal rigidities. Although the underlying microfundations are very diﬀerent, our ABM indeed includes frictions: all adjustments (of production, wages and prices) in our model are only made progressively. So our framework is indeed able to capture eﬀects predicted by enhanced DSGE models. No attempt was made here to adjust parameters to quantitatively match empirical data, in particular the time scales: this is beyond the scope of the present study which is primarily intended to be a description of the qualitative behavior of our toy-economy. B. Mild vs aggressive monetary policy In Fig. 4 for example, we show the result of the policy of the Central Bank that attempts to bring down the natural ≈ ∗ unemployment rate of u 0.33 (a rather large value corresponding to Θ = 2, R = 2 and ρ = 2%) to a low target − ∗ ∗ nat ∗ of 1 ε = u = 0.05. The target inﬂation is π = 0.2% per time step (corresponding to 2.4% annual if one interprets ≈ the time step to be a month), compared to a natural inﬂation that ﬂuctuates around zero: π 0. The left graph nat corresponds to a mild monetary policy, with Taylor-rule parameters set to φ = φ = 0.5. The policy is seen to be π ≈ε rather successful: the inﬂation is on target, while unemployment goes down to u 0.07, not far from the target of 0.05. But now look at the graph on the right, where the only diﬀerence is the aggressiveness of the CB that attempts → to reach target too quickly, merely doubling the value of φ = φ 1. In this case, the monetary policy has induced π ε strong instabilities, with “business cycles” of large amplitude and inﬂation all over the place. We show in Fig. 5 the result of an extensive exploration of the role of φ and φ when Θ = 2, R = 2 and ∗ π ε ρ = 2%. One sees that there is a wedge-like region around φ = φ = 0 where the policy does not induce instabilities π ε (signaled by a yellow/orange hue in Fig. 5-right). However, the region of parameters where the unemployment rate is ∼ signiﬁcantly reduced is only a subset of this wedge, corresponding to the black region in Fig. 5-left, where φ 0.5, ≤ ε φ 0.5. In other words, the Central Bank must navigate in a narrow window: too little is not enough, too aggressive π is counterproductive and leads to instabilities and wildly oscillating economies. ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

font: default

text: output

text: price

text: wages

text: consumption

text: policy

text: shock

text: time

text: FIG.

text: Aggregate

label: caption

text: response

text: of

text: the

text: economy

text: to

text: a

text: policy

text: shock

text: represented

text: by

text: a

text: drop

text: in

text: the

text: interest

text: rate

text: ρ

text: set

text: by

text: the

text: central

text: bank

text: (from

text: to

text: The

text: vertical

text: dashed

text: line

text: gives

text: the

text: timing

text: of

text: the

text: policy

text: shock.

text: All

text: aggregate

text: variables

text: are

text: represented

text: as

text: a

text: relative

text: variation

text: from

text: their

text: average

text: values

text: prior

text: to

text: the

text: shock.

text: For

text: a

text: better

text: visualization,

text: we

text: ampliﬁed

text: by

text: a

text: factor

text: the

text: wage

text: and

text: price

text: responses.

text: Note

text: that

text: the

text: damped

text: oscillation

text: relaxation

text: towards

text: equilibrium.

text: Parameters

text: are

text: Θ

text: ρ

text: α

text: α

text: and

text: other

text: parameters

text: as

text: in

text: Fig.

text: Γ

text: c

text: A.

text: A

text: monetary

text: shock

text: In

text: order

text: to

text: check

text: that

text: our

text: framework

text: produces

text: reasonable

text: results,

text: it

text: is

text: interesting

text: to

text: start

text: with

text: the

text: simplest

text: policy

text: experiment,

text: i.e.

text: an

text: expansionary

text: monetary

text: policy

text: shock

text: where

text: the

text: baseline

text: interest

text: rate

text: is

text: instantaneously

text: decreased

text: from

text: to

text: all

text: other

text: parameters

text: of

text: the

text: model

text: being

text: ﬁxed,

text: and

text: with

text: no

text: further

text: intervention

text: of

text: the

text: Central

text: Bank

text: (i.e.

text: φ

text: φ

text: The

text: resulting

text: dynamics

text: of

text: output,

text: wages

text: and

text: prices

text: is

text: shown

text: in

text: Fig.

text: and

text: can

text: be

text: compared

text: π

text: ε

text: with

text: e.g.

text: Fig

text: of

text: Ref.

text: It

text: is

text: gratifying

text: to

text: see

text: that

text: our

text: simple

text: modelling

text: strategy,

text: in

text: particular

text: the

text: way

text: the

text: monetary

text: policy

text: is

text: channelled

text: to

text: the

text: economy,

text: does

text: lead

text: to

text: a

text: quite

text: realistic

text: shape

text: of

text: impulse

text: response

text: functions,

text: at

text: least

text: qualitatively.

text: In

text: fact,

text: the

text: empirical

text: hump-shaped

text: response

text: of

text: the

text: output

text: reported

text: in

text: is

text: quite

text: well

text: reproduced

text: by

text: their

text: DSGE

text: model

text: with

text: nominal

text: rigidities.

text: Although

text: the

text: underlying

text: microfundations

text: are

text: very

text: diﬀerent,

text: our

text: ABM

text: indeed

text: includes

text: frictions:

text: all

text: adjustments

text: (of

text: production,

text: wages

text: and

text: prices)

text: in

text: our

text: model

text: are

text: only

text: made

text: progressively.

text: So

text: our

text: framework

text: is

text: indeed

text: able

text: to

text: capture

text: eﬀects

text: predicted

text: by

text: enhanced

text: DSGE

text: models.

text: No

text: attempt

text: was

text: made

text: here

text: to

text: adjust

text: parameters

text: to

text: quantitatively

text: match

text: empirical

text: data,

text: in

text: particular

text: the

text: time

text: scales:

text: this

text: is

text: beyond

text: the

text: scope

text: of

text: the

text: present

text: study

text: which

text: is

text: primarily

text: intended

text: to

text: be

text: a

text: description

text: of

text: the

text: qualitative

text: behavior

text: of

text: our

text: toy-economy.

text: B.

text: Mild

text: vs

text: aggressive

text: monetary

text: policy

text: In

text: Fig.

text: for

text: example,

text: we

text: show

text: the

text: result

text: of

text: the

text: policy

text: of

text: the

text: Central

text: Bank

text: that

text: attempts

text: to

text: bring

text: down

text: the

text: natural

text: unemployment

text: rate

text: of

text: u

text: (a

text: rather

text: large

text: value

text: corresponding

text: to

text: Θ

text: R

text: and

text: ρ

text: to

text: a

text: low

text: target

text: nat

text: of

text: ε

text: u

text: The

text: target

text: inﬂation

text: is

text: π

text: per

text: time

text: step

text: (corresponding

text: to

text: annual

text: if

text: one

text: interprets

text: the

text: time

text: step

text: to

text: be

text: a

text: month),

text: compared

text: to

text: a

text: natural

text: inﬂation

text: that

text: ﬂuctuates

text: around

text: zero:

text: π

text: The

text: left

text: graph

text: nat

text: corresponds

text: to

text: a

text: mild

text: monetary

text: policy,

text: with

text: Taylor-rule

text: parameters

text: set

text: to

text: φ

text: φ

text: The

text: policy

text: is

text: seen

text: to

text: be

text: π

text: ≈ε

text: rather

text: successful:

text: the

text: inﬂation

text: is

text: on

text: target,

text: while

text: unemployment

text: goes

text: down

text: to

text: u

text: not

text: far

text: from

text: the

text: target

text: of

text: But

text: now

text: look

text: at

text: the

text: graph

text: on

text: the

text: right,

text: where

text: the

text: only

text: diﬀerence

text: is

text: the

text: aggressiveness

text: of

text: the

text: CB

text: that

text: attempts

text: to

text: reach

text: target

text: too

text: quickly,

text: merely

text: doubling

text: the

text: value

text: of

text: φ

text: φ

text: In

text: this

text: case,

text: the

text: monetary

text: policy

text: has

text: induced

text: π

text: ε

text: strong

text: instabilities,

text: with

text: “business

text: cycles”

text: of

text: large

text: amplitude

text: and

text: inﬂation

text: all

text: over

text: the

text: place.

text: We

text: show

text: in

text: Fig.

text: the

text: result

text: of

text: an

text: extensive

text: exploration

text: of

text: the

text: role

text: of

text: φ

text: and

text: φ

text: when

text: Θ

text: R

text: and

text: π

text: ε

text: ρ

text: One

text: sees

text: that

text: there

text: is

text: a

text: wedge-like

text: region

text: around

text: φ

text: φ

text: where

text: the

text: policy

text: does

text: not

text: induce

text: instabilities

text: π

text: ε

text: (signaled

text: by

text: a

text: yellow/orange

text: hue

text: in

text: Fig.

text: However,

text: the

text: region

text: of

text: parameters

text: where

text: the

text: unemployment

text: rate

text: is

text: signiﬁcantly

text: reduced

text: is

text: only

text: a

text: subset

text: of

text: this

text: wedge,

text: corresponding

text: to

text: the

text: black

text: region

text: in

text: Fig.

text: where

text: φ

text: ε

text: φ

text: In

text: other

text: words,

text: the

text: Central

text: Bank

text: must

text: navigate

text: in

text: a

text: narrow

text: window:

text: too

text: little

text: is

text: not

text: enough,

text: too

text: aggressive

text: π

text: is

text: counterproductive

text: and

text: leads

text: to

text: instabilities

text: and

text: wildly

text: oscillating

text: economies.

text: ##LTFigure##

text: ##LTLine##

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text: 18 ± ∈ where η [0, 1] are what we denote as the hiring/ﬁring propensity of the ﬁrms. According to this mechanism, the change in output responds to excess demand (there is an increase in output if excess demand is positive, a decrease in output if excess demand is negative, i.e. if there is excess supply). The propensities to hire/ﬁre η± are the sensitivity of the output change to excess demand/supply. - Wage: The wage update rule we chose follows (in spirit) the choices made f(cid:40)or price and production. We propose that at each time step ﬁrm i updates its wage as: − (cid:48) (cid:40)Y (t) < D (t) WT (t + 1) = W (t)[1 + γ (1 ΓΦ )εξ (t)] if Pi i i i w i i (t) > 0 i (A11) − (cid:48) Y (t) > D (t) W (t + 1) = W (t)[1 γ (1 + ΓΦ )uξ (t)] if Pi i i i w i i (t) < 0 i P (cid:48) where γ is a certain parameter, (t) is the proﬁt of the ﬁrm at time t and ξ (t) an independent U[0, 1] w i i random variable. If WT (t + 1) is such that the proﬁt of ﬁrm i at time t with this amount of wages would i P have been negative, W (t + 1) is chosen to be exactly at the equilibrium point where (t) = 0 otherwise i i W (t + 1) = WT (t + 1). i i The above rules are intuitive: if a ﬁrm makes a proﬁt and it has a large demand for its good, it will increase the pay of its workers. The pay rise is expected to be large if the ﬁrm is ﬁnancially healthy and/or if unemployment is low (i.e. if ε is large) because pressure on salaries is high. Conversely, if the ﬁrm makes a loss and has a low demand for its good, it will reduce the wages. This reduction is drastic if the company is close to bankruptcy, and/or if unemployment is high, because the pressure on salaries is then low. In all other cases, wages are not updated. The parameters γ allow us to simulate diﬀerent price/wage update timescales, i.e. the aggressivity with p,w which ﬁrms react a change of their economic conditions. In the following we set γ = 0.05 and γ = γ . p w p The case γ = 0 corresponds to removing completely the wage update rule, such that the version of the w model with constant wage is recovered. • After prices, productions and wages are set and interests paid, consumption and accounting take place. Since { } each ﬁrm has total sales p min Y , D ﬁrms proﬁts are i i i P { } − {E } {E } (t) = p (t) min Y (t), D (t) W (t)Y (t) + ρd max (t), 0 + ρ(cid:96) min (t), 0 . (A12) i i i i i i i i E P E When ﬁrms have both positive and dividends are paid as a fraction δ of the ﬁrm cash balance . The i i i update rule for ﬁrms cash balance is therefore E E P − E P E (t + 1) = (t) + (t) δ (t)θ( (t))θ( (t)) (A13) i i i i i i (cid:88) (cid:88) (cid:88) where θ(x) = 1 if x > 0 and 0 otherwise. Correspondingly, households savings are updated as − { } E P E S(t + 1) = S(t) + W (t)Y (t) p (t) min Y (t), D (t) + δ (t)θ( (t))θ( (t)). (A14) i i i i i i i i i i i (cid:88) The dividends share δ is set to 2% unless stated otherwise and the ∆(t) term in Eq. (A5) is given by E P E ∆(t) = δ (t)θ( (t))θ( (t)) (A15) i i i i • Finally, an inactive ﬁrm has a ﬁnite probability ϕ (which we set to 0.1) per unit time to get revived; when it does so its price is ﬁxed to p (t) = p(t), its wage to w (t) = w(t), its workforce is the available workforce i i E Y (t) = µu(t) and its cash-balance is the amount needed to pay the wage bill (t) = W (t)Y (t). This small i E i E i i ’liquidity’ is provided by ﬁrms with positive in shares proportional to their wealth . i i Appendix B: Firms’ adaptive behavior leads to a second order phase transition We start by analysing the eﬀect of adaptation of ﬁrms. In order to get a ﬁrst insight it is useful to consider a simpliﬁed setting where Γ = Γ (i.e. α = 0), ρ(cid:96)(t) = ρ (t) = 0, f = 1, c(t) = c = 0.5 (hence α = 0) and wages are 0 Γ 0 0 c constant and equal to 1 (γ = 0). In this case the basic model described in [1] (with constant wages) is recovered. w ##LTLine## ##LTLine##

label: list

text: where

text: η

text: are

text: what

text: we

text: denote

text: as

text: the

text: hiring/ﬁring

text: propensity

text: of

text: the

text: ﬁrms.

text: According

text: to

text: this

text: mechanism,

text: the

text: change

text: in

text: output

text: responds

text: to

text: excess

text: demand

text: (there

text: is

text: an

text: increase

text: in

text: output

text: if

text: excess

text: demand

text: is

text: positive,

text: a

text: decrease

text: in

text: output

text: if

text: excess

text: demand

text: is

text: negative,

text: i.e.

text: if

text: there

text: is

text: excess

text: supply).

text: The

text: propensities

text: to

text: hire/ﬁre

text: η±

text: are

text: the

text: sensitivity

text: of

text: the

text: output

text: change

text: to

text: excess

text: demand/supply.

text: Wage:

text: The

text: wage

text: update

text: rule

text: we

text: chose

text: follows

text: (in

text: spirit)

text: the

text: choices

text: made

text: price

text: and

text: production.

text: We

text: propose

text: that

text: at

text: each

text: time

text: step

text: ﬁrm

text: i

text: updates

text: its

text: wage

text: as:

text: (t)

text: D

text: (t)

text: WT

text: (t

text: W

text: γ

text: ΓΦ

text: )εξ

text: (t)]

text: if

text: Pi

text: i

text: i

text: i

text: w

text: i

text: i

text: (t)

text: i

text: Y

text: (t)

text: D

text: (t)

text: W

text: (t

text: W

text: γ

text: ΓΦ

text: )uξ

text: (t)]

text: if

text: Pi

text: i

text: i

text: i

text: w

text: i

text: i

text: (t)

text: i

text: P

text: where

text: γ

text: is

text: a

text: certain

text: parameter,

text: (t)

text: is

text: the

text: proﬁt

text: of

text: the

text: ﬁrm

text: at

text: time

text: t

text: and

text: ξ

text: (t)

text: an

text: independent

text: w

text: i

text: i

text: random

text: variable.

text: If

text: WT

text: (t

text: is

text: such

text: that

text: the

text: proﬁt

text: of

text: ﬁrm

text: i

text: at

text: time

text: t

text: with

text: this

text: amount

text: of

text: wages

text: would

text: i

text: P

text: have

text: been

text: negative,

text: W

text: (t

text: is

text: chosen

text: to

text: be

text: exactly

text: at

text: the

text: equilibrium

text: point

text: where

text: (t)

text: otherwise

text: i

text: i

text: W

text: (t

text: WT

text: (t

text: i

text: i

text: The

text: above

text: rules

text: are

text: intuitive:

text: if

text: a

text: ﬁrm

text: makes

text: a

text: proﬁt

text: and

text: it

text: has

text: a

text: large

text: demand

text: for

text: its

text: good,

text: it

text: will

text: increase

text: the

text: pay

text: of

text: its

text: workers.

text: The

text: pay

text: rise

text: is

text: expected

text: to

text: be

text: large

text: if

text: the

text: ﬁrm

text: is

text: ﬁnancially

text: healthy

text: and/or

text: if

text: unemployment

text: is

text: low

text: (i.e.

text: if

text: ε

text: is

text: large)

text: because

text: pressure

text: on

text: salaries

text: is

text: high.

text: Conversely,

text: if

text: the

text: ﬁrm

text: makes

text: a

text: loss

text: and

text: has

text: a

text: low

text: demand

text: for

text: its

text: good,

text: it

text: will

text: reduce

text: the

text: wages.

text: This

text: reduction

text: is

text: drastic

text: if

text: the

text: company

text: is

text: close

text: to

text: bankruptcy,

text: and/or

text: if

text: unemployment

text: is

text: high,

text: because

text: the

text: pressure

text: on

text: salaries

text: is

text: then

text: low.

text: In

text: all

text: other

text: cases,

text: wages

text: are

text: not

text: updated.

text: The

text: parameters

text: γ

text: allow

text: us

text: to

text: simulate

text: diﬀerent

text: price/wage

text: update

text: timescales,

text: i.e.

text: the

text: aggressivity

text: with

text: p,w

text: which

text: ﬁrms

text: react

text: a

text: change

text: of

text: their

text: economic

text: conditions.

text: In

text: the

text: following

text: we

text: set

text: γ

text: and

text: γ

text: γ

text: p

text: w

text: p

text: The

text: case

text: γ

text: corresponds

text: to

text: removing

text: completely

text: the

text: wage

text: update

text: rule,

text: such

text: that

text: the

text: version

text: of

text: the

text: w

text: model

text: with

text: constant

text: wage

text: is

text: recovered.

text: After

text: prices,

text: productions

text: and

text: wages

text: are

text: set

text: and

text: interests

text: paid,

text: consumption

text: and

text: accounting

text: take

text: place.

text: Since

text: each

text: ﬁrm

text: has

text: total

text: sales

text: p

text: min

text: Y

text: D

text: ﬁrms

text: proﬁts

text: are

text: i

text: i

text: i

text: P

text: {E

text: {E

text: (t)

text: p

text: (t)

text: min

text: Y

text: (t),

text: D

text: (t)

text: W

text: (t)Y

text: (t)

text: ρd

text: max

text: (t),

text: min

text: (t),

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: E

text: P

text: E

text: When

text: ﬁrms

text: have

text: both

text: positive

text: and

text: dividends

text: are

text: paid

text: as

text: a

text: fraction

text: δ

text: of

text: the

text: ﬁrm

text: cash

text: balance

text: The

text: i

text: i

text: i

text: update

text: rule

text: for

text: ﬁrms

text: cash

text: balance

text: is

text: therefore

text: E

text: E

text: P

text: E

text: P

text: E

text: (t

text: (t)

text: (t)

text: δ

text: (t)θ(

text: (t))θ(

text: (t))

text: i

text: i

text: i

text: i

text: i

text: i

text: where

text: θ(x)

text: if

text: x

text: and

text: otherwise.

text: Correspondingly,

text: households

text: savings

text: are

text: updated

text: as

text: E

text: P

text: E

text: S(t

text: S(t)

text: W

text: (t)Y

text: (t)

text: p

text: (t)

text: min

text: Y

text: (t),

text: D

text: (t)

text: δ

text: (t)θ(

text: (t))θ(

text: (t)).

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: i

text: The

text: dividends

text: share

text: δ

text: is

text: set

text: to

text: unless

text: stated

text: otherwise

text: and

text: the

text: ∆(t)

text: term

text: in

text: Eq.

text: is

text: given

text: by

text: E

text: P

text: E

text: ∆(t)

text: δ

text: (t)θ(

text: (t))θ(

text: (t))

text: i

text: i

text: i

text: i

text: Finally,

text: an

text: inactive

text: ﬁrm

text: has

text: a

text: ﬁnite

text: probability

text: ϕ

text: (which

text: we

text: set

text: to

text: per

text: unit

text: time

text: to

text: get

text: revived;

text: when

text: it

text: does

text: so

text: its

text: price

text: is

text: ﬁxed

text: to

text: p

text: (t)

text: p(t),

text: its

text: wage

text: to

text: w

text: (t)

text: w(t),

text: its

text: workforce

text: is

text: the

text: available

text: workforce

text: i

text: i

text: E

text: Y

text: (t)

text: µu(t)

text: and

text: its

text: cash-balance

text: is

text: the

text: amount

text: needed

text: to

text: pay

text: the

text: wage

text: bill

text: (t)

text: W

text: (t)Y

text: (t).

text: This

text: small

text: i

text: E

text: i

text: E

text: i

text: i

text: ’liquidity’

text: is

text: provided

text: by

text: ﬁrms

text: with

text: positive

text: in

text: shares

text: proportional

text: to

text: their

text: wealth

text: i

text: i

text: Appendix

text: B:

text: Firms’

text: adaptive

text: behavior

text: leads

text: to

text: a

text: second

text: order

text: phase

text: transition

text: We

text: start

text: by

text: analysing

text: the

text: eﬀect

text: of

text: adaptation

text: of

text: ﬁrms.

text: In

text: order

text: to

text: get

text: a

text: ﬁrst

text: insight

text: it

text: is

text: useful

text: to

text: consider

text: a

text: simpliﬁed

text: setting

text: where

text: Γ

text: Γ

text: (i.e.

text: α

text: ρ

text: (t)

text: f

text: c(t)

text: c

text: (hence

text: α

text: and

text: wages

text: are

text: Γ

text: c

text: constant

text: and

text: equal

text: to

text: (γ

text: In

text: this

text: case

text: the

text: basic

text: model

text: described

text: in

text: (with

text: constant

text: wages)

text: is

text: recovered.

text: w

text: ##LTLine##

text: ##LTLine##


text: ))ω(I( goL 01 501 ∗ 0ω 2Γ / )ε(raV ))ω(I( goL 01 501 ∗ 0ω 20 1 2 10 0 10 0.8 -2 10 -4 10 -6 0.6 10 -8 10 Γ/ε 0 0.2 0.4 0.6 0.8 1 0.4 R N = 100, Γ = 10−3 F 0.2 N = 100, Γ = 10−4 F N = 104, Γ = 10−3 F N = 104, Γ = 10−4 F 0 0 0.2 0.4 0.6 0.8 1 R FIG. 9: Inverse of the average employment rate Γ /ε as a function of the ratio R = η0 /η−0 with η−0 = 0.1 and γ = 0.01 0 −+ − when Γ > 0. When the employment rate is rescaled with the parameter Γ (here Γ = 10 3, 10 4) the diﬀerent lines collapse 0 0 0 and Eq. (B2) agrees with numerical simulations. In the inset we also plot the rescaled variance, still as a function of η0 . + Approaching the critical point the variance of the unemployment ﬂuctuations diverges, together with their relaxation time going to inﬁnity. The other parameters are: δ = 0.02, Θ = 5, γ = 0, c = 0.5, β = 0 and ϕ = 0.1 w 4 0 10,0 7,5 2 7,5 -2 5,0 5,0 0 2,5 -4 2,5 -2 00 0.5 1 -6 00 0.5 1 R R -4 -8 -6 -10 -8 -12 -10 -14 -12 -16 -7 -6 -5 -4 -3 -2 -1 0 -7 -6 -5 -4 -3 -2 -1 0 Log (ω) Log (ω) 10 10 − FIG. 10: Logarithm of the normalized power spectrum for Mark 0 with adaptive ﬁrms (Γ = 10 3), γ = 0.05 and N = 1 0 p F (left) and N = 1000 (right). The other parameters are set as in Fig. 9. The main plot shows two examples of the spectrum for F η−0 = 0.1 and η0 = 0.05 (black line) and η0 = 0.09 (red line) in the left plot, η0 = 0.05 (black line) and η0 = 0.08 (red line) in + + + + the right plot. The time series is made of 228 time steps after T = 500 000 and the logarithm of the spectrum is averaged over eq a moving window of 100 points for a better visualization. With both system sizes the ﬁt with Eq. (B3) (blue dashed lines) is good with the only diﬀerence that when N > 1 a clear oscillatory pattern appears at high frequencies, that becomes sharper F and sharper as N increases. In the inset of each ﬁgure we plot the value of ω in Eq. (B3) obtained from the ﬁt as a function F 0 of the ratio R = η0 /η−0 . In both cases ω goes linearly to 0 as the critical value is approached. + 0 ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ))ω(I(

text: goL

text: 0ω

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text: The

text: other

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text: cases

text: ω

text: goes

text: linearly

text: to

text: as

text: the

text: critical

text: value

text: is

text: approached.

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text: 21 Algorithm 1 Mark 0 ∗ ∗ ∗ Require: N Number of ﬁrms; c , β, γ , γ , η0 , η−0 , δ, Θ, ϕ, f, α , φ , φ , α , Γ , ε , π , ρ , ω; T total evolution time; F 0 p w + c π ε Γ 0 (cid:46) Initialization ← ← for (i 0; i < N ; i i + 1) do ← F W[i] 1 ← − p[i] 1 + 0.2(random 0.5) ← − Y [i] [1 + 0.2(random 0.5)]/2 ← D[i] 0.5 (cid:46) Initial employment is 0.5 E ← [i] W[i]Y [i] random P ← − [i] p[i] min(D[i], Y [i]) W[i]Y [i] ← (cid:80) a[i] 1 (cid:46) binary variable: active (1) / inactive (0) ﬁrm end for ← − (cid:80) E S N [i] (cid:46) Main loop ←F ≤i ← for (t 1; t T; t t + 1) do ← ε (cid:80)1 Y [i] u ← 1N(cid:80)F− ε i ← (cid:80) p i(cid:80)p[i]Y [i] ← i Y [i] w i(cid:80)W[i]Y [i] (cid:101)∗ ← i Y [i] (cid:101) u [i] exp(βW[i]/w) N u ← i a[i−]exp(βW[i]/w) F x ωx + (1 ω)x where x are π, ρd, ρ(cid:96), u (cid:46) Central Bank policy ∗ ← ∗ εˆ min (ε , 1.025ε˜) ← ∗ − ∗ ∗ ρ ρ + 10φ(cid:101)(π˜ (cid:101)π ) + φ log (ε˜/εˆ ) 0 ← π ε ρ max (ρ , 0) 0← { 0 − } Γ max α (ρ(cid:96) π), Γ D ← E− ← EΓ ← 0 + 0 (cid:46) Firms update prices, productions and wages ← ← for (i 0; i < N ; i i + 1) do F if a[i] == 1 then E − if [i] > ΘW[i]Y [i] then E ← E {E } + + + max [i], 0 E− ← E− − {E } min [i], 0 ← − E Φ[i] [i] ← W[{i]Y [i] } Φ[i] min Φ[i], 1/Γ ← { − } Φ[i] max Φ[i], 1/Γ ← − η η0 (1 ΓΦ[i]) + ← + η− η−0 (1 + ΓΦ[i]) if Y [i] < D[i] then P if [i] > 0 then ← − W[i] ← W[i]{[1 + γw(1 ΓΦ[i])ε random] {E } {E } } W[i] min W[i], (P[i] min [D[i], Y [i]] + ρd max [i], 0 + ρ(cid:96) min [i], 0 )/Y [i] end if ← { − ∗ } Y [i] Y [i] + min η (D[i] Y [i]), u [i] +← if p[i] < p then p[i] p[i](1 + γp random) end if else if Y [i] > D[i] then P if [i] < 0 then ← − W[i] W[i][1 γw(1 + ΓΦ[i])u random] end if ← { − − } Y [i] max 0, Y [i] η−(D[i] Y [i]) ← − if p[i] < p then p[i] p[i](1 γp random) end if end if E ≤ − else if [i] ΘW[i]Y [i] then ← a[i] 0 D ← D − E [i] end if end if end for ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Algorithm

text: Mark

text: Require:

text: N

text: Number

text: of

text: ﬁrms;

text: c

text: β,

text: γ

text: γ

text: η0

text: η−0

text: δ,

text: Θ,

text: ϕ,

text: f,

text: α

text: φ

text: φ

text: α

text: Γ

text: ε

text: π

text: ρ

text: ω;

text: T

text: total

text: evolution

text: time;

text: F

text: p

text: w

text: c

text: π

text: ε

text: Γ

text: Initialization

text: for

text: (i

text: i

text: N

text: i

text: i

text: do

text: F

text: W[i]

text: p[i]

text: Y

text: [i]

text: D[i]

text: Initial

text: employment

text: is

text: E

text: [i]

text: W[i]Y

text: [i]

text: random

text: P

text: [i]

text: p[i]

text: min(D[i],

text: Y

text: [i])

text: W[i]Y

text: [i]

text: a[i]

text: binary

text: variable:

text: active

text: inactive

text: ﬁrm

text: end

text: for

text: E

text: S

text: N

text: [i]

text: Main

text: loop

text: ←F

text: ≤i

text: for

text: (t

text: t

text: T;

text: t

text: t

text: do

text: ε

text: Y

text: [i]

text: u

text: ε

text: i

text: p

text: [i]

text: i

text: Y

text: [i]

text: w

text: [i]

text: i

text: Y

text: [i]

text: u

text: [i]

text: exp(βW[i]/w)

text: N

text: u

text: i

text: a[i−]exp(βW[i]/w)

text: F

text: x

text: ωx

text: ω)x

text: where

text: x

text: are

text: π,

text: ρd,

text: u

text: Central

text: Bank

text: policy

text: εˆ

text: min

text: (ε

text: ρ

text: ρ

text: φ

text: log

text: (ε˜/εˆ

text: π

text: ε

text: ρ

text: max

text: (ρ

text: 0←

text: Γ

text: max

text: α

text: π),

text: Γ

text: D

text: E−

text: EΓ

text: Firms

text: update

text: prices,

text: productions

text: and

text: wages

text: for

text: (i

text: i

text: N

text: i

text: i

text: do

text: F

text: if

text: a[i]

text: then

text: E

text: if

text: [i]

text: ΘW[i]Y

text: [i]

text: then

text: E

text: E

text: {E

text: max

text: [i],

text: E−

text: E−

text: {E

text: min

text: [i],

text: E

text: Φ[i]

text: [i]

text: W[{i]Y

text: [i]

text: Φ[i]

text: min

text: Φ[i],

text: Φ[i]

text: max

text: Φ[i],

text: η

text: η0

text: ΓΦ[i])

text: η−

text: η−0

text: ΓΦ[i])

text: if

text: Y

text: [i]

text: D[i]

text: then

text: P

text: if

text: [i]

text: then

text: W[i]

text: ΓΦ[i])ε

text: random]

text: {E

text: {E

text: W[i]

text: min

text: W[i],

text: (P[i]

text: min

text: [D[i],

text: Y

text: [i]]

text: ρd

text: max

text: [i],

text: min

text: [i],

text: )/Y

text: [i]

text: end

text: if

text: Y

text: [i]

text: Y

text: [i]

text: min

text: η

text: (D[i]

text: Y

text: [i]),

text: u

text: [i]

text: if

text: p[i]

text: p

text: then

text: p[i]

text: γp

text: random)

text: end

text: if

text: else

text: if

text: Y

text: [i]

text: D[i]

text: then

text: P

text: if

text: [i]

text: then

text: W[i]

text: ΓΦ[i])u

text: random]

text: end

text: if

text: Y

text: [i]

text: max

text: Y

text: [i]

text: η−(D[i]

text: Y

text: [i])

text: if

text: p[i]

text: p

text: then

text: p[i]

text: γp

text: random)

text: end

text: if

text: end

text: if

text: E

text: else

text: if

text: [i]

text: ΘW[i]Y

text: [i]

text: then

text: a[i]

text: D

text: D

text: E

text: [i]

text: end

text: if

text: end

text: if

text: end

text: for

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text: 4 which focuses on a simple ABM of ﬁnancial markets coupled with a New Keynesian DSGE model, which allows one to reproduce endogenously stock price bubbles and business cycles and to study the introduction of ﬁnancial taxes. Calibration and validation procedures have been explored in [31–33]. Although similar in spirit, the present study builds upon a slightly diﬀerent perspective, following the framework and philosophy developed in [1]. In particular, we model an idealized closed economy with linear production capabilities, no capital (labor is the only input for production), no innovation and growth, no ﬁnancial sector, and only a minimal set of additional behavioural rules which couple the Central Bank monetary policy to ﬁrms’ and agents’ choices. In this sense, our model is much more parsimonious than what is found in the recent ABM literature and voluntarily overlooks important actors and processes at play in the real world. Our central concern is instead to characterize the qualitative behaviour of the emerging economy and investigate its dynamical stability with respect to the Central Bank’s policy. The outline of the paper is as follows. In Section II we give a brief description of the Mark-0 model proposed in [1] and detail the minimal additional rules we need to couple the monetary policy with ﬁrms’ and agents’ decisions. In Section III, we investigate the “natural” state of our toy-economy, in the presence of interest rates but without any Central Bank intervention. In Section IV, we perform several policy experiments: an expansionary monetary shock, and the implementation of a Taylor-rule monitoring of the interest rate by the Central Bank to achieve given employement and inﬂation targets. We establish the phase diagram of our model in the presence of this Taylor-rule based intervention. We brieﬂy compare our ﬁndings to the prediction of DSGE models. We conclude in Section V. Appendix A gives further details about Mark-0; Appendix B investigates the continuous phase transition induced by the wary behaviour of indebted ﬁrms; ﬁnally, Appendix C gives a full pseudo-code of our model that should allow easy duplication of our results. II. DESCRIPTION OF THE MODEL A. Brief summary of the minimal “Mark-0” model The Mark-0 closed economy is made of ﬁrms and households. While the latter sector is represented at an aggregate level, ﬁrms are heterogeneous and treated individually. Each ﬁrm i = 1, . . . , N at time t produces a quantity Y (t) of F i perishable goods that it attempts to sell at price p (t). It needs a number of N (t) = Y (t)/ζ of employees1 to produce i i i i Y (t), and pays a wage W (t). The demand D for good i depends on the global consumption budget of households i i i C (t), itself determined as a fraction of the household savings (that include the last wages), and it is a decreasing B function of the asked price p (t), with a price sensitivity parameter that can be tuned – see Appendix A. i To update their production, price and wage policy, ﬁrms use reasonable “rules of thumb” [1] that we detail in Appendix A and that were already (partly) justiﬁed in the original work of Delli Gatti et al. [2]. For example, production is decreased and employees are made redundant whenever Y > D , and vice-versa.2 The adjustment i i speed can however be asymmetric, i.e. the ratio R of hiring adjustement speed to ﬁring adjustement speed is not necessarily equal to one, for example because of labour laws. This turns out to be one of the most important control parameter that determines the fate of the overall economy. In the initial state of the economy, ﬁrms are heterogenous in size and prices (with a uniform distribution around the average size and price, see Appendix C for details), but all oﬀer the same wage and see the same demand. This is arbitrary and of little importance, because the stationary state of the model is statistically independent of the initialization. When the Mark-0 economy is set in motion, it soon becomes clear that some ﬁrms have to take up loans in order to stay in business. One therefore immediately has to add further rules for this to take place. In the zero-interest rate world of Mark-0, we let ﬁrms freely accumulate a total debt up to a threshold that is a multiple Θ of total payroll, beyond which the ﬁrm is declared bankrupt (its debt is then repaid partly by households and partly by surviving ﬁrms, such that there is no net creation of money). From this point of view the parameter Θ determines the maximum credit supply available to ﬁrms. Fixing the value of Θ plays the role of a primitive monetary policy, since the total amount of money circulating in the economy (‘broad money’) directly depends on Θ [1]. When Θ = 0, no debt is allowed → ∞ (zero leverage), while when Θ , ﬁrms have not limit on the loans they need to continue business (unbounded leverage). While there are several other parameters needed to deﬁne completely Mark-0 (9 parameters in total, see Appendix 1 ζi is the productivity of ﬁrm i. We chose ζi = 1 in [1] and we will stick to this choice throughout the present paper as well. 2 As a consequence of these adaptive adjustments, the economy can reach (in some regions of the parameter space) equilibrium, corre- sponding to the market clearing condition one would obtain in a fully representative agent framework. However, ﬂuctuations around equilibrium persists in the limit of large system sizes giving rise to a rich phenomenology, see [1, 2]. ##LTLine##

text: which

text: focuses

text: on

text: a

text: simple

text: ABM

text: of

text: ﬁnancial

text: markets

text: coupled

text: with

text: a

text: New

text: Keynesian

text: DSGE

text: model,

text: which

text: allows

text: one

text: to

text: reproduce

text: endogenously

text: stock

text: price

text: bubbles

text: and

text: business

text: cycles

text: and

text: to

text: study

text: the

text: introduction

text: of

text: ﬁnancial

text: taxes.

text: Calibration

text: and

text: validation

text: procedures

text: have

text: been

text: explored

text: in

text: Although

text: similar

text: in

text: spirit,

text: the

text: present

text: study

text: builds

text: upon

text: a

text: slightly

text: diﬀerent

text: perspective,

text: following

text: the

text: framework

text: and

text: philosophy

text: developed

text: in

text: In

text: particular,

text: we

text: model

text: an

text: idealized

text: closed

text: economy

text: with

text: linear

text: production

text: capabilities,

text: no

text: capital

text: (labor

text: is

text: the

text: only

text: input

text: for

text: production),

text: no

text: innovation

text: and

text: growth,

text: no

text: ﬁnancial

text: sector,

text: and

text: only

text: a

text: minimal

text: set

text: of

text: additional

text: behavioural

text: rules

text: which

text: couple

text: the

text: Central

text: Bank

text: monetary

text: policy

text: to

text: ﬁrms’

text: and

text: agents’

text: choices.

text: In

text: this

text: sense,

text: our

text: model

text: is

text: much

text: more

text: parsimonious

text: than

text: what

text: is

text: found

text: in

text: the

text: recent

text: ABM

text: literature

text: and

text: voluntarily

text: overlooks

text: important

text: actors

text: and

text: processes

text: at

text: play

text: in

text: the

text: real

text: world.

text: Our

text: central

text: concern

text: is

text: instead

text: to

text: characterize

text: the

text: qualitative

text: behaviour

text: of

text: the

text: emerging

text: economy

text: and

text: investigate

text: its

text: dynamical

text: stability

text: with

text: respect

text: to

text: the

text: Central

text: Bank’s

text: policy.

text: The

text: outline

text: of

text: the

text: paper

text: is

text: as

text: follows.

text: In

text: Section

text: II

text: we

text: give

text: a

text: brief

text: description

text: of

text: the

text: model

text: proposed

text: in

text: and

text: detail

text: the

text: minimal

text: additional

text: rules

text: we

text: need

text: to

text: couple

text: the

text: monetary

text: policy

text: with

text: ﬁrms’

text: and

text: agents’

text: decisions.

text: In

text: Section

text: III,

text: we

text: investigate

text: the

text: “natural”

text: state

text: of

text: our

text: toy-economy,

text: in

text: the

text: presence

text: of

text: interest

text: rates

text: but

text: without

text: any

text: Central

text: Bank

text: intervention.

text: In

text: Section

text: IV,

text: we

text: perform

text: several

text: policy

text: experiments:

text: an

text: expansionary

text: monetary

text: shock,

text: and

text: the

text: implementation

text: of

text: a

text: Taylor-rule

text: monitoring

text: of

text: the

text: interest

text: rate

text: by

text: the

text: Central

text: Bank

text: to

text: achieve

text: given

text: employement

text: and

text: inﬂation

text: targets.

text: We

text: establish

text: the

text: phase

text: diagram

text: of

text: our

text: model

text: in

text: the

text: presence

text: of

text: this

text: Taylor-rule

text: based

text: intervention.

text: We

text: brieﬂy

text: compare

text: our

text: ﬁndings

text: to

text: the

text: prediction

text: of

text: DSGE

text: models.

text: We

text: conclude

text: in

text: Section

text: V.

text: Appendix

text: A

text: gives

text: further

text: details

text: about

text: Appendix

text: B

text: investigates

text: the

text: continuous

text: phase

text: transition

text: induced

text: by

text: the

text: wary

text: behaviour

text: of

text: indebted

text: ﬁrms;

text: ﬁnally,

text: Appendix

text: C

text: gives

text: a

text: full

text: pseudo-code

text: of

text: our

text: model

text: that

text: should

text: allow

text: easy

text: duplication

text: of

text: our

text: results.

text: II.

text: DESCRIPTION

text: OF

text: THE

text: MODEL

text: A.

text: Brief

text: summary

text: of

text: the

text: minimal

text: model

text: The

text: closed

text: economy

text: is

text: made

text: of

text: ﬁrms

text: and

text: households.

text: While

text: the

text: latter

text: sector

text: is

text: represented

text: at

text: an

text: aggregate

text: level,

text: ﬁrms

text: are

text: heterogeneous

text: and

text: treated

text: individually.

text: Each

text: ﬁrm

text: i

text: N

text: at

text: time

text: t

text: produces

text: a

text: quantity

text: Y

text: (t)

text: of

text: F

text: i

text: perishable

text: goods

text: that

text: it

text: attempts

text: to

text: sell

text: at

text: price

text: p

text: (t).

text: It

text: needs

text: a

text: number

text: of

text: N

text: (t)

text: Y

text: (t)/ζ

text: of

text: to

text: produce

text: i

text: i

text: i

text: i

text: Y

text: (t),

text: and

text: pays

text: a

text: wage

text: W

text: (t).

text: The

text: demand

text: D

text: for

text: good

text: i

text: depends

text: on

text: the

text: global

text: consumption

text: budget

text: of

text: households

text: i

text: i

text: i

text: C

text: (t),

text: itself

text: determined

text: as

text: a

text: fraction

text: of

text: the

text: household

text: savings

text: (that

text: include

text: the

text: last

text: wages),

text: and

text: it

text: is

text: a

text: decreasing

text: B

text: function

text: of

text: the

text: asked

text: price

text: p

text: (t),

text: with

text: a

text: price

text: sensitivity

text: parameter

text: that

text: can

text: be

text: tuned

text: see

text: Appendix

text: A.

text: i

text: To

text: update

text: their

text: production,

text: price

text: and

text: wage

text: policy,

text: ﬁrms

text: use

text: reasonable

text: “rules

text: of

text: thumb”

text: that

text: we

text: detail

text: in

text: Appendix

text: A

text: and

text: that

text: were

text: already

text: (partly)

text: justiﬁed

text: in

text: the

text: original

text: work

text: of

text: Delli

text: Gatti

text: et

text: al.

text: For

text: example,

text: production

text: is

text: decreased

text: and

text: employees

text: are

text: made

text: redundant

text: whenever

text: Y

text: D

text: and

text: The

text: adjustment

text: i

text: i

text: speed

text: can

text: however

text: be

text: asymmetric,

text: i.e.

text: the

text: ratio

text: R

text: of

text: hiring

text: adjustement

text: speed

text: to

text: ﬁring

text: adjustement

text: speed

text: is

text: not

text: necessarily

text: equal

text: to

text: one,

text: for

text: example

text: because

text: of

text: labour

text: laws.

text: This

text: turns

text: out

text: to

text: be

text: one

text: of

text: the

text: most

text: important

text: control

text: parameter

text: that

text: determines

text: the

text: fate

text: of

text: the

text: overall

text: economy.

text: In

text: the

text: initial

text: state

text: of

text: the

text: economy,

text: ﬁrms

text: are

text: heterogenous

text: in

text: size

text: and

text: prices

text: (with

text: a

text: uniform

text: distribution

text: around

text: the

text: average

text: size

text: and

text: price,

text: see

text: Appendix

text: C

text: for

text: details),

text: but

text: all

text: oﬀer

text: the

text: same

text: wage

text: and

text: see

text: the

text: same

text: demand.

text: This

text: is

text: arbitrary

text: and

text: of

text: little

text: importance,

text: because

text: the

text: stationary

text: state

text: of

text: the

text: model

text: is

text: statistically

text: independent

text: of

text: the

text: initialization.

text: When

text: the

text: economy

text: is

text: set

text: in

text: motion,

text: it

text: soon

text: becomes

text: clear

text: that

text: some

text: ﬁrms

text: have

text: to

text: take

text: up

text: loans

text: in

text: order

text: to

text: stay

text: in

text: business.

text: One

text: therefore

text: immediately

text: has

text: to

text: add

text: further

text: rules

text: for

text: this

text: to

text: take

text: place.

text: In

text: the

text: zero-interest

text: rate

text: world

text: of

text: we

text: let

text: ﬁrms

text: freely

text: accumulate

text: a

text: total

text: debt

text: up

text: to

text: a

text: threshold

text: that

text: is

text: a

text: multiple

text: Θ

text: of

text: total

text: payroll,

text: beyond

text: which

text: the

text: ﬁrm

text: is

text: declared

text: bankrupt

text: (its

text: debt

text: is

text: then

text: repaid

text: partly

text: by

text: households

text: and

text: partly

text: by

text: surviving

text: ﬁrms,

text: such

text: that

text: there

text: is

text: no

text: net

text: creation

text: of

text: money).

text: From

text: this

text: point

text: of

text: view

text: the

text: parameter

text: Θ

text: determines

text: the

text: maximum

text: credit

text: supply

text: available

text: to

text: ﬁrms.

text: Fixing

text: the

text: value

text: of

text: Θ

text: plays

text: the

text: role

text: of

text: a

text: primitive

text: monetary

text: policy,

text: since

text: the

text: total

text: amount

text: of

text: money

text: circulating

text: in

text: the

text: economy

text: (‘broad

text: money’)

text: directly

text: depends

text: on

text: Θ

text: When

text: Θ

text: no

text: debt

text: is

text: allowed

text: (zero

text: leverage),

text: while

text: when

text: Θ

text: ﬁrms

text: have

text: not

text: limit

text: on

text: the

text: loans

text: they

text: need

text: to

text: continue

text: business

text: (unbounded

text: leverage).

text: While

text: there

text: are

text: several

text: other

text: parameters

text: needed

text: to

text: deﬁne

text: completely

text: parameters

text: in

text: total,

text: see

text: Appendix

text: ζi

text: is

text: the

text: productivity

text: of

text: ﬁrm

text: i.

text: We

text: chose

text: ζi

text: in

text: and

text: we

text: will

text: stick

text: to

text: this

text: choice

text: throughout

text: the

text: present

text: paper

text: as

text: well.

text: As

text: a

text: consequence

text: of

text: these

text: adaptive

text: adjustments,

text: the

text: economy

text: can

text: reach

text: (in

text: some

text: regions

text: of

text: the

text: parameter

text: space)

text: equilibrium,

text: corre-

text: sponding

text: to

text: the

text: market

text: clearing

text: condition

text: one

text: would

text: obtain

text: in

text: a

text: fully

text: representative

text: agent

text: framework.

text: However,

text: ﬂuctuations

text: around

text: equilibrium

text: persists

text: in

text: the

text: limit

text: of

text: large

text: system

text: sizes

text: giving

text: rise

text: to

text: a

text: rich

text: phenomenology,

text: see

text: ##LTLine##


text: pγ / noitalfni 5 10 1 1 - FU 0.1 2 - RU 8 0.8 3 - EC 0 4 - FE 6 4 - FE -0.1 1 - FU inflation 0.6 Θ deflation -0.2 u 4 0.4 decreasing γw 2 3 - EC alternating inflation / deflation 0.2 2 - RU null inflation increasing employment 0 0 1 2 3 4 0 R time − FIG. 1: (Left) Phase diagram in the R Θ plane of the basic Mark-0 model as obtained in [1] with wage update. There are four distinct phases separated by critical lines. The Full Employment (FE) phase (R > R , Θ large) is characterized by c positive average inﬂation, while there is deﬂation in the Full Unemployment (FU) phase (R < R ). Endogenous Crises (EC) c are characterized by alternating cycles of inﬂation and deﬂation, and occur for R > R , Θ intermediate. Finally, R > R , small c c Θ correspond to a region of small inﬂation and Residual Unemployment (RU). The location of phase boundaries is only weakly aﬀected by the choice of the other parameters of Mark-0, see [1]. (Right) Typical trajectories of the unemployment rate u(t) for each of the phases. In the inset, the price variations are shown, displaying either inﬂation (in the FE phase) or deﬂation (in the − − FU phase). γ 1 (resp. γ 1) sets the characteristic time scale for price adjustments (resp. wages) in the model, see Appendix p w A. The surprising occurrence of endogenous oscillations in the EC phase can be fully understood analytically, see [34]. A), the detailed investigation of [1] has established that only R and Θ are the ones determining the phase-diagram of the model, shown in Fig. 1, where we also plot typical trajectories of the economy in each phase. It is important to stress that this diagram is extremely robust against both details of the model speciﬁcation and the value of the other parameters, which only aﬀect the above phenomenology quantitatively, but leave the qualitative emergent behavior essentially unchanged. Its salient features are [1]: • ∞ When Θ = the economy is characterized by two distinct phases separated by a ﬁrst order (discontinuous) phase transition as a function of the parameter R. When R < R (i.e. fast downward production adjustments), one c ﬁnds at long times a collapse of the economy towards a deﬂationary/low demand/full unemployment state (FU). For R > R , on the other hand, the long run state of the economy is characterized by a positive inﬂation/high c demand/full employment phase (FE). • ∞ When Θ < the above description holds but must be reﬁned to allow for the appearance of three sub-phases for R > R : c ∞ 1. a full employment and inﬂationary phase for high values of Θ (the FE phase, similar to the Θ = case); 2. a phase for intermediate values of Θ characterized by high employment and inﬂation on average, which is, however, intermittently disrupted by “endogenous crises” (EC), that temporarily bring deﬂation and high unemployment spikes; 3. a phase with zero inﬂation and residual unemployment for small Θ (the RU phase), where the impossibility to obtain loans creates a positive stationary level of bankruptcies. Note that the unexpected occurrence of purely endogenous oscillations in the EC phase can in fact be understood fully analytically [34] and it is a robust feature that only relies on very mild assumptions about the destabilizing feedback mechanisms present in the economy. B. Introducing interest rates in Mark-0 We now introduce in the model a banking system made up of a Central Bank (CB) which sets the base interest rate ρ (t) (and, as part of a prudential policy, the parameter Θ that controls the maximum credit supply available to 0 ﬁrms), and a private banking system that will act as a transmission belt for the CB policy, by setting interest rates ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: pγ

text: noitalfni

text: FU

text: RU

text: EC

text: FE

text: FE

text: FU

text: inflation

text: Θ

text: deflation

text: u

text: decreasing

text: γw

text: EC

text: alternating

text: inflation

text: deflation

text: RU

text: inflation

text: increasing

text: employment

text: R

text: time

text: FIG.

text: (Left)

text: Phase

text: diagram

text: in

text: the

text: R

text: Θ

text: plane

text: of

text: the

text: basic

text: model

text: as

text: obtained

text: in

text: with

text: wage

text: update.

text: There

text: are

text: four

text: distinct

text: phases

text: separated

text: by

text: critical

text: lines.

text: The

text: Full

text: Employment

text: (FE)

text: phase

text: (R

text: R

text: Θ

text: large)

text: is

text: characterized

text: by

text: c

text: positive

text: average

text: inﬂation,

text: while

text: there

text: is

text: deﬂation

text: in

text: the

text: Full

text: Unemployment

text: (FU)

text: phase

text: (R

text: R

text: Endogenous

text: Crises

text: (EC)

text: c

text: are

text: characterized

text: by

text: alternating

text: cycles

text: of

text: inﬂation

text: and

text: deﬂation,

text: and

text: occur

text: for

text: R

text: R

text: Θ

text: intermediate.

text: Finally,

text: R

text: R

text: small

text: c

text: c

text: Θ

text: correspond

text: to

text: a

text: region

text: of

text: small

text: inﬂation

text: and

text: Residual

text: Unemployment

text: (RU).

text: The

text: location

text: of

text: phase

text: boundaries

text: is

text: only

text: weakly

text: aﬀected

text: by

text: the

text: choice

text: of

text: the

text: other

text: parameters

text: of

text: see

text: (Right)

text: Typical

text: trajectories

text: of

text: the

text: unemployment

text: rate

text: u(t)

text: for

text: each

text: of

text: the

text: phases.

text: In

text: the

text: inset,

text: the

text: price

text: variations

text: are

text: shown,

text: displaying

text: either

text: inﬂation

text: (in

text: the

text: FE

text: phase)

text: or

text: deﬂation

text: (in

text: the

text: FU

text: phase).

text: γ

text: (resp.

text: γ

text: sets

text: the

text: characteristic

text: time

text: scale

text: for

text: price

text: adjustments

text: (resp.

text: wages)

text: in

text: the

text: model,

text: see

text: Appendix

text: p

text: w

text: A.

text: The

text: surprising

text: occurrence

text: of

text: endogenous

text: oscillations

text: in

text: the

text: EC

text: phase

text: can

text: be

text: fully

text: understood

text: analytically,

text: see

text: A),

text: the

text: detailed

text: investigation

text: of

text: has

text: established

text: that

text: only

text: R

text: and

text: Θ

text: are

text: the

text: ones

text: determining

text: the

text: phase-diagram

text: of

text: the

text: model,

text: shown

text: in

text: Fig.

text: where

text: we

text: also

text: plot

text: typical

text: trajectories

text: of

text: the

text: economy

text: in

text: each

text: phase.

text: It

text: is

text: important

text: to

text: stress

text: that

text: this

text: diagram

text: is

text: extremely

text: robust

text: against

text: both

text: details

text: of

text: the

text: model

text: speciﬁcation

text: and

text: the

text: value

text: of

text: the

text: other

text: parameters,

text: which

text: only

text: aﬀect

text: the

text: above

text: phenomenology

text: quantitatively,

text: but

text: leave

text: the

text: qualitative

text: emergent

text: behavior

text: essentially

text: unchanged.

text: Its

text: salient

text: features

text: are

text: When

text: Θ

text: the

text: economy

text: is

text: characterized

text: by

text: two

text: distinct

text: phases

text: separated

text: by

text: a

text: ﬁrst

text: order

text: (discontinuous)

text: phase

text: transition

text: as

text: a

text: function

text: of

text: the

text: parameter

text: R.

text: When

text: R

text: R

text: (i.e.

text: fast

text: downward

text: production

text: adjustments),

text: one

text: c

text: ﬁnds

text: at

text: long

text: times

text: a

text: collapse

text: of

text: the

text: economy

text: towards

text: a

text: deﬂationary/low

text: demand/full

text: unemployment

text: state

text: (FU).

text: For

text: R

text: R

text: on

text: the

text: other

text: hand,

text: the

text: long

text: run

text: state

text: of

text: the

text: economy

text: is

text: characterized

text: by

text: a

text: positive

text: inﬂation/high

text: c

text: demand/full

text: employment

text: phase

text: (FE).

text: When

text: Θ

text: the

text: above

text: description

text: holds

text: but

text: must

text: be

text: reﬁned

text: to

text: allow

text: for

text: the

text: appearance

text: of

text: three

text: sub-phases

text: for

text: R

text: R

text: c

text: a

text: full

text: employment

text: and

text: inﬂationary

text: phase

text: for

text: high

text: values

text: of

text: Θ

text: (the

text: FE

text: phase,

text: similar

text: to

text: the

text: Θ

text: case);

text: a

text: phase

text: for

text: intermediate

text: values

text: of

text: Θ

text: characterized

text: by

text: high

text: employment

text: and

text: inﬂation

text: on

text: average,

text: which

text: is,

text: however,

text: intermittently

text: disrupted

text: by

text: “endogenous

text: crises”

text: (EC),

text: that

text: temporarily

text: bring

text: deﬂation

text: and

text: high

text: unemployment

text: spikes;

text: a

text: phase

text: with

text: zero

text: inﬂation

text: and

text: residual

text: unemployment

text: for

text: small

text: Θ

text: (the

text: RU

text: phase),

text: where

text: the

text: impossibility

text: to

text: obtain

text: loans

text: creates

text: a

text: positive

text: stationary

text: level

text: of

text: bankruptcies.

text: Note

text: that

text: the

text: unexpected

text: occurrence

text: of

text: purely

text: endogenous

text: oscillations

text: in

text: the

text: EC

text: phase

text: can

text: in

text: fact

text: be

text: understood

text: fully

text: analytically

text: and

text: it

text: is

text: a

text: robust

text: feature

text: that

text: only

text: relies

text: on

text: very

text: mild

text: assumptions

text: about

text: the

text: destabilizing

text: feedback

text: mechanisms

text: present

text: in

text: the

text: economy.

text: B.

text: Introducing

text: interest

text: rates

text: in

text: We

text: now

text: introduce

text: in

text: the

text: model

text: a

text: banking

text: system

text: made

text: up

text: of

text: a

text: Central

text: Bank

text: (CB)

text: which

text: sets

text: the

text: base

text: interest

text: rate

text: ρ

text: (t)

text: (and,

text: as

text: part

text: of

text: a

text: prudential

text: policy,

text: the

text: parameter

text: Θ

text: that

text: controls

text: the

text: maximum

text: credit

text: supply

text: available

text: to

text: ﬁrms),

text: and

text: a

text: private

text: banking

text: system

text: that

text: will

text: act

text: as

text: a

text: transmission

text: belt

text: for

text: the

text: CB

text: policy,

text: by

text: setting

text: interest

text: rates

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