Optimal Inflation Target: Insights from an Agent-Based Model
Jean-Philippe Bouchaud, Stanislao Gualdi, Marco Tarzia, Francesco Zamponi
OOptimal Inflation Target:Insights from an Agent-Based Model
Jean-Philippe Bouchaud, Stanislao Gualdi, Marco Tarzia, and Francesco Zamponi CFM, 23 rue de l’Universit´e, 75007 Paris, France Universit´e Pierre et Marie Curie - Paris 6, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France Laboratoire de physique th´eorique, D´epartement de physique de l’ENS,´Ecole normale sup´erieure, PSL Research University, Sorbonne Universit´es,UPMC Univ. Paris 06, CNRS, 75005 Paris, France
Which level of inflation should Central Banks be targeting? We investigate this issue in the contextof a simplified Agent Based Model of the economy. Depending on the value of the parameters thatdescribe the behaviour of agents (in particular inflation anticipations), we find a rich variety ofbehaviour at the macro-level. Without any active monetary policy, our ABM economy can be in ahigh inflation/high output state, or in a low inflation/low output state. Hyper-inflation, deflationand “business cycles” between coexisting states are also found. We then introduce a Central Bankwith a Taylor rule-based inflation target, and study the resulting aggregate variables. Our mainresult is that too-low inflation targets are in general detrimental to a CB-monitored economy. Onesymptom is a persistent under-realisation of inflation, perhaps similar to the current macroeconomicsituation. Higher inflation targets are found to improve both unemployment and negative interestrate episodes. Our results are compared with the predictions of the standard DSGE model.
Contents
I. Introduction II. Methodological remarks and scope of the paper III. A short recap on Mark-0
IV. The “native” state of the economy V. Inflation targeting
VI. Comparison with DSGE & Conclusion Acknowledgments A. Pseudo-code of Mark 0 with inflation expectations References a r X i v : . [ q -f i n . E C ] F e b I. INTRODUCTION
Most Central Banks around the world nowadays adjust their monetary policy to reach a 2%/year inflation target.The rationale for choosing 2% rather than 1% or 3% is however not clear, as with many other “magic numbers”religiously used in economic policy. The recent crisis has put to the fore the problem of negative nominal interestrates, which can be seen as a consequence of low inflation targets and thus low baseline rates. As emphasized byO. Blanchard in 2010 [1], “As a matter of logic, higher average inflation and thus higher average nominal interestrates before the crisis would have given more room for monetary policy to be eased during the crisis.”
This view ishowever disputed by many economists, who strongly argue against a raise of the inflation target (see e.g. [2, 3] for arecent overview, and for a discussion of the historical origin of the 2% target). A major argument to that effect is thecredibility of Central Banks, who have succeeded in anchoring low-inflation expectations in the minds of economicagents. If the inflation target is changed in the face of new circumstances, these expectations may un-moor, and thevery efficiency of monetary policy may suffer as a consequence. Clearly, the fear of a lurking run-away inflation isweighing heavily on the debate.Yet, the question of an “optimal” inflation level is well worth considering, and policy makers are eager to receiveinputs from academic research. As Federal Reserve Chairwoman J. Yellen recently declared [4]: “We very much lookforward to seeing research by economists that will help inform our future decisions on this”.
Of course, optimalityneeds to be defined and different criteria (i.e. welfare functions) may lead to different results. More important stillis the modelling framework used to describe the economy. A clear puzzle is that standard monetary theories implyzero or negative optimal inflation rates, at variance with Central Banks’ inflation targets [5]. The standard DSGEmachinery – the “workhorse” of monetary economists [6] – has recently been extended to cope with non-zero inflationrates, and generally concludes that the optimal inflation rate should be smaller than 2% [7]. However, DSGE modelsare based on a series of highly debatable assumptions, and have been under intense fire after the 2007 crisis: see theinsightful set of contributions in the Oxford Review of Economic Policy [8]; see also [9–12].Another route is provided by Agent Based Models (ABM) in which reasonable behavioural rules replace the represen-tative DSGE agent with a fully rational long-term plan. ABMs can include a number of economically relevant featureswhich would be very difficult to accommodate within the DSGE straight-jacket [13–17]. Many simplifying, sometimesad-hoc assumptions are of course necessary, but a considerable advantage of ABMs is that interaction-induced, collec-tive effects are present, whereas DSGE models reduce the whole economy to a small number of representative agents.As a consequence, the global “equilibrium” state of the economy is an emergent property in the former case, while itis a deus ex machina in the latter case.In particular, crises (i.e. large swings in the output) can occur endogenously within ABMs [18, 19]. DSGE models,on the other hand, only describe small, mean-reverting fluctuations around the postulated equilibrium and crises canonly result from exogenous, unpredictable shocks. As a case in point, we found in [21] that the aggregate behavior ofthe economy is not a smooth function of the baseline interest rate: the fact that firms are risk averse and fear goinginto debt leads to more unemployment that can spiral into a destabilizing feedback loop. This is one of the “darkcorners” [22] that ABMs can help uncovering.While there is a growing literature on monetary policies in ABMs [14, 23–27], the optimal inflation target questionhas not been investigated using ABMs (although see [23] where the success of targeting policies is discussed withinthe framework of an ABM). In this paper, we take on this issue using an arguably over-simplified, bare-bone ABMdubbed “Mark-0”, studied in great details in [19, 21] following previous work by the group of Delli Gatti et al. [18] (seealso [23, 28]). As discussed in [19], the Mark-0 economy can be in different “macro-states” (HIHO: high inflation/highinput, or LILO: low inflation/low output), depending on various parameters of the model. These parameters describein a phenomenological way the behaviour of agents (firms, households and banks), and their response to differenteconomic stimuli. Interestingly, small changes in the value of these parameters can indeed induce sharp variations inaggregate output, unemployment or inflation [19, 21]. This allows us to consider different baseline economies andstudy the influence of the chosen inflation target on the total output, on the real interest rate and on the probabilityof negative nominal interest rates.Our main conclusion is that in general, increasing the inflation target reduces unemployment and reduces theprobability of negative rates. Unsurprisingly, it also reduces real interest rates on savings. In fact, trying to impose R. Lucas famously argued that the 2008 crisis was not predicted because economic theory predicts that such events cannot be pre-dicted [20]. However, as discussed in [8], a benchmark macroeconomic model should be able to account for crises because it is crisesthat have the largest effects on individual well-being. In fact, as will be shown below, HIHO and LILO states can even coexist for some parameter range, see Figure 1 below. This coexistenceregion was overlooked in an earlier version of this paper, and accounts for some of the puzzling effects that we initially reported. low inflation on an economy that would naturally run at full steam with high inflation can lead to an output collapse.On the other hand, high inflation policies can be dangerous and may generate hyper-inflation if agents lose faith inthe ability of Central Banks to fulfill their mandate.
II. METHODOLOGICAL REMARKS AND SCOPE OF THE PAPER
Before going into the technical discussion about the model and the results, we want to stress here some method-ological aspects, and state a few disclaimers. • As recalled above, we are fully aware that our stylized ABM relies on somewhat arbitrary assumptions and isunrealistic on several counts. We have discussed in detail the logic of our approach in [19]. While micro-rulesshould in fine be justified by direct empirical data on the behaviour of households and firms, many results arein fact generic and robust against changes of these micro-rules – allowing one to draw important qualitativeconclusions from such stylized models. It thus seems to us more relevant, at this stage of the ABM researchagenda, to stick with the now well-studied Mark-0 model and explore the issue of inflation target in a “proof ofconcept” manner. A number of possible improvements are listed in the conclusion of the present paper. • Although far from perfect, Mark-0 contains plausible ingredients that are most probably present in reality. Forexample, our model encodes in a schematic manner the consumption behavior of households facing inflation,that is in fact similar to the standard Euler equation for consumption in general equilibrium models [6]. We alsoaccount for the effect of inflation on the policy of indebted firms, which appears to be absent in DSGE models.The fact that our results strongly contrast with those of standard DSGE models is in our opinion enough tomotivate in-depth investigations of more realistic ABMs, and more empirical work on the micro/behaviouralassumptions that underpin these models. • Our approach is not normative, in the sense that we do not consider any specific welfare function that shouldbe maximized. Our aim is rather to provide a synthetic “dashboard” of the simulated economies, with inflation,unemployment, probability of negative rates and real rates on deposits, as a function of the target inflation level.Although not formalized, it will be clear from these dashboards that some inflation targets are qualitativelybetter than others. Remaining at the level of qualitative statements seems to us a way to avoid the “pretenseof knowledge” syndrom [10, 29]. As Keynes said, it is better to be roughly right than exactly wrong. • The Mark-0 model can exhibit very different behaviours depending on the parameters [19], with regions wherethe economy collapses. It is important to stress that the behavioral rules that define our model are supposedto be reasonable when the economy behaves normally, and we believe that they correctly describe how such anormal state can become unstable. However, once the instability happens and the economy truly collapses, itis of course unreasonable to expect that agents will keep acting according to the same rules, in particular theCentral Bank and other institutions. However, this is besides the point we want to make in this paper. • It is somehow unavoidable that complex ABMs can lead to non-intuitive results, in particular concerning thedetailed shape of the phase diagrams, for example the appearance of a region of parameters where good andbad states of the economy can coexist. Although it might seem unsatisfactory not to have a fully analyticalunderstanding of these transitions, we claim that it is in fact one of the strength of ABMs: to be able to elicitscenarios that are hard to imagine for the unaided human mind, because they are the result of non-linearitiesand sometimes antagonist feedback loops. • Finally, an important disclaimer: the parameters chosen in the following are not the result of a precise calibration.We only made reasonable guesses in order to have reasonable numbers as outputs of the model (e.g. reasonablevalues of yearly inflation). All the numbers quoted below are not intended to be taken literally (although webelieve they should be taken seriously!).
III. A SHORT RECAP ON MARK-0
The Mark-0 model with a Central Bank (CB) and interest rates has been described in full details in [19, 21], wherepseudo-codes are also provided. We will not repeat here the full logic of the model, but only focus on the elementsthat are relevant for determining inflation in the three sectors: households, firms and the CB. The pseudo-code ofMark-0, where all the modelling choices are made explicit, can be found in Appendix A.First, we need some basic notions. The model is defined in discrete time, where the unit time between t and t + 1 isplausibly of the order of months. For definiteness, we will choose in the following the unit time scale to be 6 months.Each firm i at time t produces a quantity Y i ( t ) of perishable goods that it attempts to sell at price p i ( t ), and pays awage W i ( t ) to its employees. The demand D i ( t ) for good i depends on the global consumption budget of households C B ( t ), itself determined as an inflation rate-dependent fraction of the household savings. D i is a decreasing functionof the firm price p i , with a price sensitivity parameter that can be tuned. To update their production, price and wagepolicy, firms use reasonable “rules of thumb” [19] that also depend on the inflation rate through their level of debt (seebelow). For example, production is decreased and employees are made redundant whenever Y i > D i , and vice-versa. The model is fully “stock-flow consistent” (i.e. all the stocks and flows within the toy economy are properly accountedfor).The instantaneous inflation rate π ( t ) is defined as: π ( t ) = p ( t ) − p ( t − p ( t −
1) ; p ( t ) = (cid:80) i p i ( t ) Y i ( t ) (cid:80) i Y i ( t ) , (1)where p ( t ) is the production-weighted average price. We will assume that firms, households and the CB do not reactto the instantaneous value of π ( t ), but rather to a smoothed, exponential moving average π ema ( t ) of the realisedinflation, computed as π ema ( t ) = ωπ ( t ) + (1 − ω ) π ema ( t − , (2)where we fix ω = 0 .
2, which corresponds to an averaging time of ≈ . ω .In Mark-0 we assume a linear production function with a constant unit productivity, which means that output andemployment coincide. The unemployment rate u is defined as: u ( t ) = 1 − (cid:80) i Y i ( t ) N , (3)where N is the number of firms, which also coincides with the total workforce [19]. Note that firms cannot hire moreworkers than available, so that u ( t ) ≥ A. The Central Bank policy
In this work, for simplicity we consider a single-mandate CB that attempts to steer the economy towards a targetinflation level π (cid:63) (in [21], we in fact considered a double-mandate CB also targeting a certain employment level ε (cid:63) ).The monetary policy followed by the CB for fixing the base interest rate is described by a Taylor-like rule of theform [6, 30]: ρ ( t ) = ρ (cid:63) + φ π [ π ema ( t ) − π (cid:63) ] (4)where ρ (cid:63) is the baseline interest rate and φ π > In the following, we do notimpose a Zero Lower Bound to the base-line interest rate, in view of the recent monetary policy; imposing it does notaffect substantially our conclusions. Note that here ρ (cid:63) cannot be interpreted as the “natural” interest rate, which isitself an emergent property of the model, that depends on all the parameters. In this sense, ρ (cid:63) is another parameterof the model, that contributes to the determination of the macroeconomic state (see Fig. 2 below).We assume that the banking sector – described at the aggregate level by a single “representative bank” – sets theinterest rates on deposits and loans ( ρ d ( t ) and ρ (cid:96) ( t ) respectively) uniformly for all lenders and borrowers . Therefore, As a consequence of these adaptive adjustments, the economy is on average always ‘close’ to the global market clearing condition onewould posit in a fully representative agent framework. However, small fluctuations persists in the limit of large system sizes giving riseto a rich phenomenology [19], including business cycles. The original Taylor rule reads [31]: ρ ( t ) = ρ (cid:63) + π ema ( t ) + φ (cid:48) π [ π ema ( t ) − π (cid:63) ] which amounts to the substitution ρ (cid:63) → ρ (cid:63) + π (cid:63) and φ π → φ (cid:48) π in Eq. (4). Note that this is the only action taken by the CB to achieve the target; in particular, no actions on the quantity of circulating money,such as quantitative easing or printing money can be taken by the CB. This is, in our model, the only role played by the banking sector: a transmission belt of the CB policy. In reality, the banking sector hasmuch more freedom, and can sometimes make the CB policy ineffective, e.g. by restricting credit even in presence of a strong incentivefrom the CB. the rate ρ (cid:96) increases and ρ d decreases when the firm default rate increases, in such a way that the banking sector(i.e., the representative bank) – which fully absorbs these defaults – makes zero profit at each time step (see [21] andthe pseudo-code provided in Appendix A for more details). B. Households
The effect of inflation on households is the standard trade-off between saving (at rate ρ d ) and consumption. Wetherefore assume that the total consumption budget of households C B ( t ) is given by: C B ( t ) = c ( t ) (cid:2) S ( t ) + W ( t ) + ρ d ( t ) S ( t ) (cid:3) with c ( t ) = [[ c (cid:2) α c ( (cid:98) π ( t ) − ρ d, ema ( t )) (cid:3) ]] , (5)where S ( t ) is the savings, W ( t ) the total wages, (cid:98) π ( t ) is the expected inflation in the next period – see Eq. (7) below– and c ( t ) is the consumption propensity, which is clipped to the interval [0 , x ]] which means that the quantity x is boxed between 0 and 1, i.e. [[ x ]] = 1 if x >
1, [[ x ]] = 0 if x <
0, and[[ x ]] = x otherwise. This propensity is equal to a baseline value c when the difference between expected inflation andthe interest paid on their savings is zero, and increases (decreases) when this difference is positive (negative). Theparameter α c > S ( t + 1) = S ( t ) + W ( t ) + ρ d ( t ) S ( t ) − C ( t ) , (6)where C ( t ) ≤ C B ( t ) is the actual consumption of households, see [19].We furthermore posit that the expected inflation (cid:98) π ( t ) is given by a linear combination of the realised inflation π ema ( t ) and the CB target inflation π (cid:63) (see also [23]): (cid:98) π ( t ) = τ R π ema ( t ) + τ T π (cid:63) . (7)The parameters τ R and τ T (“R” for realised and “T” for target) can be interpreted as capturing the importance ofpast inflation and the trust of economic agents in the ability of the CB to enforce its inflation target. When τ R = 0and τ T = 1, agents fully trust that the target inflation will be realised. When τ R >
0, they are also influenced bythe past realised inflation when they form their expectations. When τ R >
1, they expect more inflation to be realisedin the next period. As we will see below, this can give rise to hyper-inflation episodes, which is the scenario thatprevents (in the mind of many policy makers and of the public opinion) higher inflation targets.In principle, τ R and τ T should depend on the commitment of the Central Bank, captured by the parameter φ π .In particular, the function τ T ( φ π ) should be, for consistency, such that τ T = 0 when φ π = 0. In fact, in the absenceof an active CB ( φ π = 0), one should assume that the inflation expectation parameter τ T is zero, since there is noanchoring force to a definite inflation target. Although we do not introduce a precise model for τ T ( φ π ), below wealways assume that τ T = 0 when φ π = 0. Also, τ R and τ T might be time dependent, as economic agents comparethe realised inflation to the target inflation and “learn” about the credibility of the CB – see below and [23] for adiscussion of this particular point. In the present paper, we will treat τ R and τ T as time independent, leaving thisinteresting development for future work. C. Firms
1. Financial fragility
The model contains N F firms, each firm being characterized by its production Y i (equal to its workforce in our zerogrowth economy), demand for its goods D i , price p i , wage W i and its cash balance E i which, when negative, is thedebt of the firm. We characterize the financial fragility of the firm through the debt-to-payroll ratioΦ i = − E i W i Y i . (8)Negative Φ’s describe healthy firms with positive cash balance, while indebted firms have a positive Φ. If Φ i < Θ,i.e. when the flux of credit needed from the bank is not too high compared to the size of the company (measured asthe total payroll), the firm i is allowed to continue its activity. If on the other hand Φ i ≥ Θ, the firm i defaults andthe corresponding default cost is absorbed by the banking sector, which adjusts the loan and deposit rates ρ (cid:96) and ρ d accordingly. The defaulted firm is replaced by a new one, initialised at random (using the average parameters ofother firms). The parameter Θ controls the maximum leverage in the economy, and models the risk-control policy ofthe banking sector.
2. Production update
If the firm is allowed to continue its business, it adapts its price, wages and production according to reasonable(but of course debatable) “rules of thumb” – see [19, 21]. In particular, the production update is chosen as:If Y i ( t ) < D i ( t ) ⇒ Y i ( t + 1) = Y i ( t ) + min { η + i ( D i ( t ) − Y i ( t )) , u (cid:63)i ( t ) } If Y i ( t ) > D i ( t ) ⇒ Y i ( t + 1) = Y i ( t ) − η − i [ Y i ( t ) − D i ( t )] (9)where u (cid:63)i ( t ) is the maximum number of unemployed workers available to the firm i at time t (see [21, AppendixA]). The coefficients η ± ∈ [0 ,
1] express the sensitivity of the firm’s target production to excess demand/supply. Wepostulate that the production adjustment depends on the financial fragility Φ i of the firm: firms that are close tobankruptcy are arguably faster to fire and slower to hire, and vice-versa for healthy firms. In order to model thistendency, we posit that the coefficients η ± i for firm i (belonging to [0 , η − i = [[ η − (1 + ΓΦ i ( t ))]] η + i = [[ η +0 (1 − ΓΦ i ( t ))]] , (10)where η ± are fixed coefficients, identical for all firms. The factor Γ > i then leads to a faster downward adjustment ofthe workforce when the firm is over-producing, and a slower (more cautious) upward adjustment when the firm isunder-producing.In [21] we argued that Γ should in fact depend on the difference between the interest rate and the inflation: high costof credit makes firms particularly wary of going into debt and their sensitivity to their financial fragility is increased.Therefore, we postulate that interest rates influence the firm’s policy through the financial fragility sensitivity Γ, as:Γ = max { α Γ ( ρ (cid:96), ema ( t ) − (cid:98) π ( t )) , Γ } , (11)where α Γ (similarly to α c above) captures the influence of the real interest rate on loans on the hiring/firing policy ofthe firms. This feedback of inflation on firms policy is one of the important features of our model.
3. Price update
Following the initial specification of the Mark series of models [18], prices are updated through a random multi-plicative process which takes into account the production-demand gap experienced in the previous time step and ifthe price offered is competitive (with respect to the average price). The update rule for prices reads:If Y i ( t ) < D i ( t ) ⇒ (cid:40) If p i ( t ) < p ( t ) ⇒ p i ( t + 1) = p i ( t )(1 + γξ i ( t ))(1 + (cid:98) π ( t ))If p i ( t ) ≥ p ( t ) ⇒ p i ( t + 1) = p i ( t )(1 + (cid:98) π ( t ))If Y i ( t ) > D i ( t ) ⇒ (cid:40) If p i ( t ) > p ( t ) ⇒ p i ( t + 1) = p i ( t )(1 − γξ i ( t ))(1 + (cid:98) π ( t ))If p i ( t ) ≤ p ( t ) ⇒ p i ( t + 1) = p i ( t )(1 + (cid:98) π ( t )) (12)where ξ i ( t ) are independent uniform U [0 ,
1] random variables and γ is a parameter setting the relative magnitude ofthe price adjustment, chosen to be 0 . (cid:98) π ( t )) factor implies that firms also anticipateinflation when they set their prices. This is precisely the dreaded self-reflexive mechanism that may lead to hyper-inflation when expected future inflation is dominated by past realised inflation (the parameter τ R ), rather than bythe CB inflation target (the parameter τ T ). Number of firms N F c β γ η − η Rη − Fraction of dividends δ ϕ f α c α Γ ω g
4. Wage update
The wage update rule follows the choices made for price and production. Similarly to workforce adjustments, weposit that at each time step firm i updates the wage paid to its employees as: W Ti ( t + 1) = W i ( t )[1 + γ (1 − ΓΦ i )(1 − u ( t )) ξ (cid:48) i ( t )][1 + g (cid:98) π ( t )] if (cid:40) Y i ( t ) < D i ( t ) P i ( t ) > W i ( t + 1) = W i ( t )[1 − γ (1 + ΓΦ i ) u ( t ) ξ (cid:48) i ( t )][1 + g (cid:98) π ( t )] if (cid:40) Y i ( t ) > D i ( t ) P i ( t ) < P i ( t ) is the profit of the firm at time t and ξ (cid:48) i ( t ) an independent U [0 ,
1] random variable. If W Ti ( t + 1) is suchthat the profit of firm i at time t with this amount of wages would have been negative, W i ( t + 1) is chosen to beexactly at the equilibrium point where P i ( t ) = 0; otherwise W i ( t + 1) = W Ti ( t + 1). Finally, g is a certain parametermodulating the way wages are indexed to inflation. We will assume in the following full indexation ( g = 1), butchoosing g < u (cid:63)i ( t ) that appears in Eq. (9), which represents the shareof unemployed workers accessible to firm i , is an increasing function of W i . Hence, firms that want to produce more(hence hire more) do so by increasing W i , as to attract more applicants (see [21, Appendix A] for details).The above rules are meant to capture the fact that deeply indebted firms seek to reduce wages more aggressively,whereas flourishing firms tend to increase wages more rapidly: • If a firm makes a profit and it has a large demand for its good, it will increase the pay of its workers. The payrise is expected to be large if the firm is financially healthy and/or if unemployment is low because pressure onsalaries is high. • Conversely, if the firm makes a loss and has a low demand for its good, it will attempt to reduce the wages.This reduction is more drastic if the company is close to bankruptcy, and/or if unemployment is high, becausepressure on salaries is then low. • In all other cases, wages are not updated.
D. Parameters of the model
The model, as presented above, has several free parameters, that are specified at the beginning of the pseudo-codepresented in Appendix A. Some values are fixed throughout this work, using values that have been found in previouswork to yield reasonable results [19, 21]: their list is given in Table I.The parameters that remain to be specified, and whose value is varied in this work to explore the correspondingphase diagram, are the ratio of hiring/firing propensities R , the Taylor rule parameters φ π , π ∗ and ρ (cid:63) , and the inflationexpectation parameters τ R and τ T . IV. THE “NATIVE” STATE OF THE ECONOMY
In [19, 21], we have shown that the Mark-0 economy, once set in motion, can settle in a variety of stationary macro-states, where the aggregate variables behave very differently. The strength of Agent Based modelling is precisely toshow that very different macro-states can emerge from very similar micro-rules, as parameters are varied. We will notrepeat such an analysis in full here, but focus on the role of a few variables, relevant to the topic of this paper. Westart by analyzing the case where the CB does not react to inflation (i.e. the Taylor rule Eq. (4) is with φ π = 0 and,as a consequence, τ T = 0). We will see later how a Taylor-rule based policy of the CB allows it to steer the economytowards a target level of inflation, and when such a policy fails.Fig. 1 shows the phase diagram of the model in the plane ( ρ (cid:63) , R ), where ρ (cid:63) is the baseline interest rate and R = η +0 /η − is the ratio of the hiring propensity to the firing propensity, that was shown in Ref. [19] to play a crucialrole for determining the overall state of the economy. We indeed see that for R (cid:46) .
28 the economy is in a LILOstate (low inflation and low output/high unemployment). For sufficiently large R and small ρ (cid:63) , the economy is in aHIHO state (high inflation and high output/low unemployment) and tips over to a LILO state when ρ (cid:63) > ρ † ( R ). Thistransition is driven by a drop in household consumption and an increased wariness of firms, induced by high yieldon savings and high cost of loans, see Eqs. (5) and (11). As noted in the Introduction, the HIHO/LILO transitionoccurs discontinuously while the change of interest rate is continuous. The transition line ρ † ( R ) is, as expected, anincreasing function of R (the economy is more stable where the hiring rate is larger than the firing rate); it is alsoa decreasing function of α Γ since firms refrain from taking loans to continue their business when α Γ increases [21].In the proximity of the phase boundaries, the native state of the economy may thus display endogeneous “businesscycles” corresponding to jumps between these two states.An interesting observation is the presence of a coexistence region where both the LILO state and the HIHO state ρ∗ (HIHO-LILO) FIG. 1: Phase diagram of the model in the ( ρ (cid:63) , R ) plane, with τ R = 0 . τ T = 0 and φ π = 0). TheHIHO phase in the top region of the graph is separated from the LILO phase in the bottom region by a discontinuous transitionline ρ † ( R ). This transition becomes a coexistence region for intermediate values of R , where the dynamics is non-ergodic:depending on the initial condition, the system ends up either in the LILO state or in the HIHO state. ρ∗ τ R LILOHIHO
COEX (HIHO-LILO) HY FIG. 2: Phase diagram in the ( ρ (cid:63) , τ R ) plane, with R = 1 . τ T = 0 and φ π = 0), showing a HIHOregion, a LILO region and a coexistence region. Note that hyper-inflation is avoided when agents’ expectations are sufficientlymean-reverting. can be selected by the dynamics, depending on initial conditions. For a large system, the dynamics is non-ergodic and appears to be permanently trapped in one of these two states.It is interesting to investigate the role of inflation expectations in this framework, by varying the value of τ R . Fig. 2shows the phase diagram of the model in the plane ( ρ (cid:63) , τ R ) for a fixed value of R = 1 .
3. We find again a HIHO region,a LILO region and a coexistence region for large enough ρ (cid:63) and τ R . As anticipated, a transition to a hyper-inflationstate (HY) occurs when expectations amplify inflation, more precisely when τ R > τ † ≈ .
9. The full phase diagram isquite complex, with possible coexistence between hyper-inflation and full employment (HYHO), hyper-inflation andcollapse (HYLO) or even hyper-deflation.Note that the HIHO region is characterized, on average, by negative firm profits, and negative real interest rateon deposits (see Figs. 3 and 4). This might appear counterintuitive, but it happens because our closed economy ischaracterised by imperfect market clearing (inducing some waste of goods), and a constant productivity of firms, suchthat the growth rate of the economy is zero. One could introduce exogenously a non zero growth rate by allowingproductivity to increase as (1 + G ) t , where G is the growth rate. By slightly amending the rules of the model, thisgrowth rate can be exactly rescaled away by simply shifting all interest rates by G , and inflating the money supplyby (1 + G ) t . The real interest rate on deposits, in particular, is shifted by G , and the purchasing power of householdsincreases with time. V. INFLATION TARGETING
We now pick two representative native states of the economy, both for R = 1 .
3, in order to be outside of thecoexistence region. The first state is with ρ (cid:63) = 1%/year, corresponding to the HIHO state, and the second one iswith ρ (cid:63) = 3%/year corresponding to the LILO state. The inflation level of these native states is, respectively, 4 . (cid:104) ρ d − π (cid:105) is,respectively, − .
0% and 0%. The HIHO state discourages long term savings while the LILO state is vastly inefficient In fact, very specific initial conditions can also lead to a near collapse of the economy. The possible coexistence between a good stateand a bad state of the economy was in fact discussed recently by Carlin and Soskice [32]. φ π :1 . , . .
5, i.e. an increase of inflation by 1% leads to the CB increasing the nominalbase-line rate by 2 . We assume that firms and agents form their inflation expectations by giving an equal weightto the target inflation π (cid:63) and the realised inflation π ema ; in other words we set τ R = τ T = . The results whenagents fully trust the CB policy ( τ R = 0, τ T = 1) are not radically different, although, as expected, realised inflationis closer to target. In the other extreme case ( τ R = 1, τ T = 0), the economy is fraught with instabilities.The resulting states of the monitored economy are summarized in Figs. 3 and 4. In these “dashboards” we show,as a function of the inflation target: the average unemployment (cid:104) u (cid:105) , the average realised inflation (cid:104) π (cid:105) , the probability P neg that the CB rate is negative and finally the average real interest rate paid on deposits (cid:104) ρ d − π (cid:105) , for φ π = 1 . , . Starting from the coexistence region (say R = 0 .
8) leads to a more complicated discussion, where history plays a roleand where economies may be fragile to small perturbations. We do not investigate further this intriguing possibilityhere, but we note that if bistability were to occur in real situations, it would probably urge a radical rethinking ofmacroeconomic policy. f p l l l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l ll l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l (a) une m p l o y m en t l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l ll l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l (b) i n f l a t i on l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l ll l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l (c) P neg l l l l l l l l ll l l l l l l l l l l l l l l ll l l l l ll l l l ll l l l l l l l l l ll l l l l l l l l l l l l l l (d) −3−2−101 0 1 2 3 4 5target inflation r ea l i n t e r e s t on depo s i t s FIG. 3: HIHO native state: Average unemployment (panel a), average realised inflation (panel b), probability that the CBmust set nominal rates to negative values (panel c) and average real interest rate paid on deposits (panel d) as a function of theCB target inflation π (cid:63) for the native state ( φ π = 0, τ T = 0, τ R = 0 .
5, black circles) and for τ R = τ T = 0 . φ π = 1 . φ π = 2 . φ π = 5 (green circles). In the last case, large oscillations around these average valuesappear, as discussed in [21]. Other parameters are: ρ (cid:63) = 1% and R = 1 .
3. Both inflation and rates are expressed as %/year,unemployment is expressed in % of the workforce. It would be interesting to extend the present study to dual-mandate CBs. Note however that for φ π = 5, there are large oscillations (“business cycles”) around these average values. As discussed in [21], anaggressive CB policy can destabilise the economy. A. High inflation/high output (HIHO) native state
Starting from a HIHO native state, one sees that targeting a low inflation rate by increasing ρ ( t ) according tothe Taylor-rule has a destabilizing effect on output. Unemployment rockets to 40%, while realised inflation is indeedlow and on target (see Fig. 3, panel a). For our particular “from-the-hip” choice of parameters, realised inflationsignificantly overshoots target when π (cid:63) (cid:38) φ π = 2 . P neg. plummets from ≈ .
25 when π (cid:63) = 0 .
25% to zero when π (cid:63) > − . π (cid:63) (cid:46) .
5% in the monitored economy.The situation improves slightly when agents fully trust the ability of the CB to reach its target (i.e. τ T = 1 and τ R = 0). Hence, stronger anchoring of inflation expectations is beneficial in our ABM setting, in agreement with theintuition gained from DSGE models. At variance with DSGE models, however, large Taylor coefficients (e.g. φ π = 5)may lead to instabilities, see [21], and increases significantly the probability of negative nominal rates, see Fig. 3,panel c). B. Low inflation/low output (LILO) native state
Let us now assume that the underlying economic mechanisms (as described by the parameters of Mark 0) are suchthat the native state of the economy is LILO, for example when R is small (firms are more reluctant to hire than tofire) or when ρ (cid:63) or α Γ are large (firms are reluctant to take loans). In this case, the role of the CB is to kick start theeconomy by lowering the interest rate.The results of a Taylor-rule based policy are shown in Fig. 4 as a function of the inflation target π (cid:63) , again for φ π = 1 . , . τ T = τ R = 0 .
5. Surprisingly, the dependence of (cid:104) u (cid:105) on π (cid:63) is found to be non-monotonic .For φ π = 2 . < π (cid:63) ≤ . higher than in the native state, while realised inflation isbelow target.Unemployment only dips below 10% when π (cid:63) is large enough. For example, when π (cid:63) = 4%, unemployment isaround 7 .
5% (down from 40% in the native state), long term real savings rate is 0 .
5% and the probability of negativenominal rates is zero. Realised inflation is however above target: (cid:104) π (cid:105) ≈ (cid:104) π (cid:105) < π (cid:63) it is likely that π (cid:63) is too low,in the sense that output can be increased by increasing the inflation target. C. Economic interpretation
The transmission channels that are responsible for the positive impact of inflation in our model are the following.First of all, from the update equation for the consumption budget, Eq. (5), it is clear that stirring the economy toa state with higher inflation increases the agents’ propensity to consume, thereby boosting demand. Perhaps moreimportantly, increased inflation reduces the real interest rate on loans, and thus the sensivity of firms on their financialfragility: indebted firms are less likely to fire and more likely to hire when inflation is larger. Similarly, higher inflationis favourable to the wage policy, see Eq. (13). These effects result in a decrease of the unemployment level and ahigher demand that generates a feedback loop that stabilizes the economy and increases the total output.In a sense, this is the classical “Keynesian” positive feedback between the increase of consumption → increase ofoutput → decrease of unemployment. A clear indication that such mechanism is at play is that whenever the CB issuccessful in stabilizing the economy, reducing the unemployment, and increasing the total output, we always findthat the realised inflation is larger than π (cid:63) , due to the fact that the economy tends to amplify the effect of the CB.As we will argue below, this feedback is absent in DSGE models.Another general argument which allows to rationalize our results is based on our previous observation that the badstates of the economy of the Mark-0 model and of its generalizations are often associated with a large amount of“inactive” money, stored in the agents’ and firms’ savings [19, 21]. Increasing the inflation rate induces an erosion ofsavings and an increased demand, thereby increasing the total amount of money circulating in the economy.2 f p l l l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l ll l l l l l l ll l l l l lll ll l l l (a) une m p l o y m en t l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l ll l l l l l l ll l l l l lll ll l l l (b) i n f l a t i on l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l ll l l l l l l ll l l l l lll ll l l l (c) P neg l l l l l l l l l l l l l l l l l ll l l l l l l l l l l l ll l l l l l l l l ll l l l l l l ll l l l l lll ll l l l (d) −1.0−0.50.00.51.0 0 1 2 3 4 5target inflation r ea l i n t e r e s t on depo s i t s FIG. 4: LILO native state: Average unemployment (panel a), average realised inflation (panel b), probability that the CB mustset nominal rates to negative values (panel c) and average real interest rate paid on deposits (panel d) as a function of the CBtarget inflation π (cid:63) for the native state ( φ π = 0, τ T = 0, τ R = 0 .
5, black circles) and for τ R = τ T = 0 . φ π = 1 . φ π = 2 . φ π = 5 (green circles). In the last case, large oscillations around these average valuesappear, as discussed in [21]. Other parameters are: ρ (cid:63) = 3% and R = 1 .
3. Both inflation and rates are expressed as %/year,unemployment is expressed in % of the workforce.
D. Phillips Curves
It is interesting to draw the Phillips curves predicted by our framework. There are two ways to think about thesecurves:1. One is to draw realised inflation as a function of unemployment for a fixed set of parameters, as the economyfluctuates over time around a unique equilibrium.2. The second is to represent the average realised inflation versus the average unemployement rate as the parametersof the economy, such as the inflation target of the CB, are changed. This amounts to represent parametricallythe results of panel b) versus those of panel a) in Figs. 3 and 4.In both cases we find a downward sloping relation (see Fig. 5), compatible with the standard wisdom. Note howeverthat the dynamical scatter plot obtained with procedure 1 leads to a noisy blob of points, with a downward slopingregression line. The second procedure leads to a nice looking graph, since each point is itself an average quantity.Reality should be in between, as one expects the underlying parameters of the economy to be themselves timedependent. But in any case, substantial deviations from a perfect downward sloping relation are expected.3 i n f l a t i on l l l lllll llllllll lll llllllllllll l lllllllllllllll llllll lllllllll i n f l a t i on r * ll FIG. 5: Phillips curves, following two procedures: (Left) scatter plot of inflation vs. unemployement during the time evolutionof the economy around a single equilibrium, obtained with R = 1 . ρ (cid:63) = 3%, φ π = 2 . π (cid:63) = 4%, and τ T = τ R = 0 . φ π = 2 . E. Discussion
The results of this section suggest that independently of the nature of its native state, low inflation targets aredetrimental to a CB-controlled economy. Interestingly, our results show that a situation where the realised inflationis lower than the target inflation cannot be optimal; in fact, realised inflation should rather overshoot target inflationon average (at least up to the point where savings are wiped out by inflation). This is the case for example in theLILO state discussed above, when unemployment reaches 8% for a target inflation of 4%, and a realised inflation of5% (see Fig. 4 for φ π = 2 . τ R and τ T , which here have been assumed to be constant for simplicity, should in realitybe time dependent and related to other quantities in the model. For example, persistent inflation overshoots mayresult in a loss in the credibility of CB, which in the present model means an increase of the value of τ R and/or adecrease of τ T , as economic agents start looking for guidance in past realised inflation rather than in the CB target.Such an increase can lead to a run-away inflation state (see Fig. 2), which cannot be controlled anymore usingTaylor-rule based policies. Within our model, such an hyper-inflation scenario can be tamed if firms do not fullyindex wages on expected inflation (i.e. set the parameter g to a value less than unity in Eq. (13)). This has the effectof reducing realised inflation as households reduce consumption, pushing the hyper-inflation threshold τ † to highervalues (for example, τ † ≈ . g = 0 . ω might itself depend on inflation, as morevolatility inflation could lead agents to become more short-sighted, such that ω →
1. It would therefore be extremelyinteresting to extend the present model to include a dynamic coupling between τ R , τ T , and the target and realisedinflation, as well as a dynamic dependence of g and ω on inflation. We leave such a study for future work. VI. COMPARISON WITH DSGE & CONCLUSION
The issue of an optimal inflation target has only recently been considered within the mainstream DSGE macroe-conomic model [33, 34] (see also [27] for a direct comparison of ABMs and DSGE models on a related topic). Inthis framework, the main cost of inflation comes from price dispersion and is a consequence of the following string ofassumptions [33]: a) firms face friction costs and cannot update their prices as often as they would like; b) inflationleads to a stronger dispersion of (stale) prices across different sectors of the economy; c) stronger price deviations We should again insist that the numbers quoted should not be taken at face value since no attempt has been made to calibrate Mark-0on real data. In particular, the chosen elementary time scale of 6-months is quite arbitrary but directly scales all inflation and interestrates. l l l l l l l l l l l l l l l l l l l l l l l l l l −0.4−0.20.00.2 0.0 2.5 5.0 7.5 10.0 Target Inflation (% / year) E xcess O u t pu t ( % ) b l FIG. 6: Effect of inflation on output in the DSGE model, following [7, 36]. All parameters are as in [7], but the subjectivediscount factor β is changed from 0 .
998 (corresponding to a horizon of 125 years) to 0 .
95 (corresponding to a horizon of 5years). In the former case, inflation causes output to decrease except for a very small window π < . π < . from equilibrium lowers economic efficiency.However, while crucial in determining the optimal inflation rate within DSGE, such a dispersion induced cost haslittle empirical support [35]. Embracing the choice of parameters and welfare function made by Coibion et al. [7], theoptimal inflation rate is found to be 1 . β used by the representative household, i.e. how far in the future do economic agents assessthe consequences of their present decision. In many DSGE calibrations, the discounting horizon is extremely long, forexample 125 years (!) in Ref. [7]. Although rooted in market efficiency arguments and based on the value of historicalrates, such an enormous time scale is in our eyes totally unreasonable. In line with the behavioral arguments usedto construct ABMs, where agents are assumed to be myopic, we believe that this time scale should be rather on thescale of a few – perhaps 5 – years. This substantially changes the conclusions of DSGE models, as the total outputwould then be an increasing function of inflation up to 5 . itself an output of the ABM. This emergent state can change radically whenthe parameters characterizing the micro-behaviour of the different agents are only slightly modified. For examplethe LILO, HIHO or hyper-inflation states considered in this paper are emergent properties of the model, and notpostulated a priori. Not surprisingly, a rough knowledge of where the economy is “naturally” poised to go is neededto determine an adequate monetary policy. Trying to steer the economy too far from its native state is detrimental(as for the HIHO state) or even lead to instabilities and crises (see [21]). Quite interestingly, we have even foundnon-ergodic regions where the economy can be either in a LILO state or in a HIHO state depending on the initialconditions. This corresponds to potentially fragile situations where monetary policy could become extremely tricky.Mark-0 is a bare-bone ABM where many important effects are left out, that need to be considered in future studies.For example the network structure of firms [37, 38] and the feedback of investment on growth are clearly among themost urgent ingredients to be added in Mark-0. The difference with DSGE is that missing effects are straightforwardto include in an ABM, while quite a bit of arm-twisting is usually necessary to include them in a DSGE frameworkwithout ruining the mathematical tractability of the model. In this sense, the much touted “micro-founded” nature ofDSGE is quickly buried under a number of ad-hoc assumptions (such as Calvo’s sticky price mechanism [39]), whichare not much more convincing than the equally ad-hoc assumptions made in ABMs.In any case, the main result of our study is that the optimal inflation rate could be somewhat higher than thecurrently accepted 2% target. One clear symptom of a too-low target is a persistent under-realisation of inflation,perhaps similar to the current macroeconomic situation in the U.S. and in Europe. In our model, this predicament5is alleviated by higher inflation targets that are found to improve both unemployment/output and negative interestrate episodes, up to the point where persistent over-realisation of inflation would lead to a loss of faith in the CentralBank and potential instabilities.Although our results are based on an arguably over-simplified model, it certainly militates for more work along theselines [12, 40]. After all, DSGE models are themselves over-simplified and, as recently emphasized by O. Blanchard [8,41], they have to become less imperialistic and accept to share the scene with other approaches to modelisation. Acknowledgments
The input and comments of O. Blanchard, R. Bookstaber, H. Dawid, D. Delli Gatti, D. Farmer, R. Farmer,X. Gabaix, C. Hommes and A. Kirman have been extremely useful. We also thank the anonymous referees andcommentators of the first version of this paper, their comments have helped improving its quality.
Appendix A: Pseudo-code of Mark 0 with inflation expectations
We present here the pseudo-code for the Mark 0 code described in Sec. III. The source code is available on demand.6
Algorithm 1
Mark 0
Require: N F (10000); c (0 . β (2); γ (0 . R ; η ( Rη − ); η − (0 . δ (0 . ϕ (0 . f (0 . α c (4); φ π ; α Γ (50); Γ (0); π ∗ ; ρ (cid:63) ; ω (0 . g (1); τ R ; τ T ; T (10000). Numbers between paretheses indicate the value used for the present work, the parameterswith no default number have been varied in this work. We start computing averages after T eq (5000) time steps. (cid:46) Initialization Y ← . . random for ( i ← i < N F ; i ← i + 1) do p [ i ] ← . random − Y [ i ] ← Y + 0 . random − D [ i ] ← Y W [ i ] ← (cid:46) Initial employment is random E [ i ] ← W [ i ] Y [ i ] random P [ i ] ← p [ i ] min( D [ i ] , Y [ i ]) − W [ i ] Y [ i ] a [ i ] ← (cid:46) binary variable: active (1) / inactive (0) firm end for S ← N F − (cid:80) i E [ i ] if φ π == 0 then π ∗ ← τ T ← end if (cid:46) Main loopfor ( t ← t ≤ T ; t ← t + 1) do ε ← N F (cid:80) i Y [ i ] u ← − εp ← (cid:80) i p [ i ] Y [ i ] (cid:80) i Y [ i ] w ← (cid:80) i W [ i ] Y [ i ] (cid:80) i Y [ i ] u ∗ [ i ] ← exp( W¯ [ i ] /w ) (cid:80) i a [ i ] exp( W¯ [ i ] /w ) N F ux ema ← ωx + (1 − ω ) x ema where x are π, ρ d , ρ (cid:96) , u (cid:46) Central Bank policy (cid:98) π ← τ R π ema + τ T π ∗ ρ ← ρ (cid:63) + φ π ( π ema − π ∗ )Γ ← max { α Γ ( ρ (cid:96), ema − (cid:98) π ) , Γ }D ← E − ← E + ← (cid:46) Firms update prices, productions and wagesfor ( i ← i < N F ; i ← i + 1) doif a [ i ] == 1 thenif E [ i ] > − Θ W [ i ] Y [ i ] then E + ← E + + max {E [ i ] , }E − ← E − − min {E [ i ] , } Φ[ i ] ← − E [ i ] W [ i ] Y [ i ] η + ← [[ η (1 − ΓΦ[ i ])]] η − ← [[ η − (1 + ΓΦ[ i ])]] if Y [ i ] < D [ i ] thenif P [ i ] > then W [ i ] ← W [ i ][1 + γ (1 − ΓΦ[ i ]) ε random ] W [ i ] ← min { W [ i ] , ( P [ i ] min [ D [ i ] , Y [ i ]] + ρ d max {E [ i ] , } + ρ (cid:96) min {E [ i ] , } ) /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 + γ random ) end ifelse if Y [ i ] > D [ i ] thenif P [ i ] < then W [ i ] ← W [ i ][1 − γ (1 + ΓΦ[ i ]) u random ] end if Y [ i ] ← max { , Y [ i ] − η − ( D [ i ] − Y [ i ]) } if p [ i ] < p then p [ i ] ← p [ i ](1 − γ random ) end ifend if p [ i ] ← p [ i ](1 + (cid:98) π ) W [ i ] ← W [ i ](1 + g (cid:98) π ) W [ i ] ← max ( W [ i ] , else if E [ i ] ≤ − Θ W [ i ] Y [ i ] then a [ i ] ← D ← D − E [ i ] end ifend ifend for Algorithm 2
Mark0 (continued) u ← − N F (cid:80) i Y [ i ] (cid:46) Update u and pp ← (cid:80) i p [ i ] Y [ i ] (cid:80) i Y [ i ] (cid:46) Private bank sets interest rates ρ (cid:96) = ρ + (1 − f ) D / E − ρ d = ρ (cid:96) E − −D S + E + (cid:46) Households decide the demand S ← (1 + ρ d ) S + (cid:80) i W [ i ] Y [ i ] c ← c [1 + α c ( (cid:98) π − (cid:101) ρ d,ema )] C B ← cS for ( i ← i < N F ; i ← i + 1) do D [ i ] ← C B a [ i ] exp( − p¯ [ i ] /p ) p [ i ] (cid:80) i a [ i ] exp( − p¯ [ i ] /p ) (cid:46) Inactive firms have no demand end for (cid:46)
Accounting E + ← for ( i ← i < N F ; i ← i + 1) doif a [ i ] == 1 then S ← S − p [ i ] min { Y [ i ] , D [ i ] }P [ i ] ← p [ i ] min { Y [ i ] , D [ i ] } − W [ i ] Y [ i ] + ρ d max {E [ i ] , } + ρ (cid:96) min {E [ i ] , }E [ i ] ← E [ i ] + P [ i ] if P [ i ] > E [ i ] > then (cid:46) Pay dividends S ← S + . E [ i ] E [ i ] ← E [ i ] − . E [ i ] end if E + ← E + + max {E [ i ] , } end ifend for (cid:46) Revivals
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