Monetary policy and dark corners in a stylized agent-based model

Abstract

We extend in a minimal way the stylized macroeconomic Agent-Based model introduced in our previous paper (Gualdi et al. in J Econ Dyn Control 50:29–61, 2015a), with the aim of investigating the role and efficacy of monetary policy of a ‘Central Bank’ that sets the interest rate such as to steer the economy towards a prescribed inflation and employment rate. Our major finding is that provided its policy is not too aggressive (in a sense detailed in the paper) the Central Bank is successful in achieving its goals. However, the existence of different equilibrium states of the economy, separated by phase boundaries (or “dark corners”), can cause the monetary policy itself to trigger instabilities and be counter-productive. In other words, the Central Bank must navigate in a narrow window: too little is not enough, too much leads to instabilities and wildly oscillating economies. This conclusion strongly contrasts with the prediction of DSGE models.

Appendices

Appendix 1: Price, production and wage updates

1.1 The households timeline

• At the beginning of the time step households are characterized by a certain amount of savings $S(t)\ge 0$.

• After each firm chooses its production $Y_i(t)$, price $P_i(t)$ and wage $W_i(t)$ for the current time step (see later) wages are paid. Since firms use a one-to-one linear technology taking only labor as input and productivity is set to 1, the production equal the workforce of the firm. Hence, the total amount of wages paid is given by

  $$\begin{aligned} W_T(t) = \sum _i W_i(t)Y_i(t) \end{aligned}$$

  (14)

• Once the total payroll of the economy is determined, interests on deposits are paid and households set a consumption budget as

  $$\begin{aligned} C_B(t) = c(t) [ S(t) + W_T(t) + \rho ^{\text {d}}(t)S(t)] \end{aligned}$$

  (15)

  where $c(t)\in [0,1]$ is the propensity to consume and may depend on inflation/interests on deposits, see Eq. (9).

• The consumption budget is distributed among firms using an intensity of choice model (Anderson et al. 1992). The demand of goods for firm i is therefore:

  $$\begin{aligned} D_i(t) = \frac{C_B(t)}{p_i(t)} \frac{e^{- \beta p_i(t) / \overline{p}(t)}}{\sum \nolimits _i e^{-\beta p_i(t) / \overline{p}(t)}} \ , \end{aligned}$$

  (16)

  where $\beta$ is the price sensitivity parameter determining an exponential dependence of households demand to the price offered by the firm. Indeed, $\beta = 0$ corresponds to complete price insensitivity and $\beta \rightarrow \infty$ means that households select only the firm with the lowest price. In this sense, as long as $\beta >0$ firms compete on prices.

• The actual consumption C(t) (limited by production) is given by

  $$\begin{aligned} C(t){:=} \sum _{i=1}^{N_\mathrm{F}}p_i(t)\min {\{Y_i(t),D_i(t)\}}\le C_B(t) = \sum _{i=1}^{N_\mathrm{F}}p_i(t)D_i(t) \end{aligned}$$

  (17)

  and households accounting therefore reads

  $$\begin{aligned} S(t+1) = [1+\rho ^{\text {d}}(t)]S(t) + W_T(t) - \sum _i p_i(t)\min {\{D_i,Y_i\}} + \Delta (t) \end{aligned}$$

  (18)

  where $\Delta (t)$ are dividends paid (see below for a definition of this last term).

Fig. 8

Scatter plot in one time step of firm profits versus the price offered. The red line correspond to a moving flat average of 100 consecutive points. Parameters are: $R=2$ (with $\eta ^0_-=0.1$), $\Theta =2$, $\gamma _p=\gamma _w=0.05$, $\alpha _\Gamma =50$, $\Gamma _0=0$, $\beta =2$, $\alpha _c=4$, $N=5000$ and $\rho ^{*}=0.5\,\%$

1.2 The firms timeline

• At the beginning of the time step t firms with $\Phi _i(t) \ge \Theta$ become inactive and are removed from the simulation. Default costs are computed as

  $$\begin{aligned} {\mathcal {D}}(t) = -\sum _{i\ \mathrm{bankrupt}}{\mathcal {E}}_i(t) \ . \end{aligned}$$

  (19)

  Firms with $\Phi _i(t) < \Theta$ are instead allowed to continue their activity and contribute to total loans ${\mathcal {E}}^-(t)$ and total firms savings ${\mathcal {E}}^+(t)$ as

  $$\begin{aligned} {\mathcal {E}}^-= & {} -\sum _{i\ \mathrm{not \ bankrupt}}\min {\{{\mathcal {E}}_i(t),0\}}\nonumber \\ {\mathcal {E}}^+= & {} \sum _{i\ \mathrm{not \ bankrupt}}\max {\{{\mathcal {E}}_i(t),0\}}. \end{aligned}$$

  (20)

• Active firms set production, price and wage for the current time step following simple adaptive rules which are meant to represent an heuristic adjustment. In particular:

  • Price: Prices are updated through a random multiplicative process which takes into account the production-demand gap experienced in the previous time step and if the price offered is competitive (with respect to the average price). The update rule for prices reads:

    $$\begin{aligned} \text {If } Y_i(t)< D_i(t)\Rightarrow & {} {\left\{ \begin{array}{ll} &{} \text {If } p_i(t) < \overline{p}(t) \Rightarrow p_i(t+1) = p_i(t) (1 + \gamma _p \xi _i(t) ) \\ &{} \text {If } p_i(t) \ge \overline{p}(t) \Rightarrow p_i(t+1) = p_i(t) \\ \end{array}\right. }\nonumber \\ \text {If } Y_i(t)> D_i(t)\Rightarrow & {} {\left\{ \begin{array}{ll} &{} \text {If } p_i(t) > \overline{p}(t) \Rightarrow p_i(t+1) = p_i(t) (1 - \gamma _p \xi _i(t) ) \\ &{} \text {If } p_i(t) \le \overline{p}(t) \Rightarrow p_i(t+1) = p_i(t)\\ \end{array}\right. }\nonumber \\ \end{aligned}$$

    (21)

    where $\xi _i(t)$ are independent uniform U[0, 1] random variables and $\gamma _p$ is a parameter setting the relative magnitude of the price adjustment (we set it to $5\,\%$ unless stated otherwise). Fig. 8, which plots the average profit of firms as a function of the offered price, shows that these rules lead to a reasonable emergent “optimizing” behavior of firms. As expected, the profit reaches a maximum for prices slightly above the average price $\overline{p}(t)$. Higher prices are not competitive and firms lose clients, while lower prices do not cover production costs.

  • Production: Independently of their price level, firms try to adjust their production to the observed demand. When firms want to hire, they open positions on the job market; we assume that the total number of unemployed workers, which is $N_\mathrm{F} u(t)$, is distributed among firms according to an intensity of choice model which depends on both the wage offered by the firm⁹ and on the same parameter $\beta$ as it is for Eq. (16); therefore the maximum number of available workers to each firm is:

    $$\begin{aligned} u^*_i(t) = \frac{e^{\beta W_i(t) / \overline{w}(t)}}{\sum \nolimits _i e^{\beta W_i(t) / \overline{w}(t)}} N_\mathrm{F} u(t) \ . \end{aligned}$$

    (22)

    The production update is then defined as:

    $$\begin{aligned} \begin{aligned} \text {If } Y_i(t) < D_i(t)&\Rightarrow Y_i(t+1)=Y_i(t)+ \min \{ \eta ^+_i ( D_i(t)-Y_i(t)), u^*_i(t) \} \\ \text {If } Y_i(t) > D_i(t)&\Rightarrow Y_i(t+1)= Y_i(t) - \eta ^-_i [Y_i(t)-D_i(t)] \\ \end{aligned} \end{aligned}$$

    (23)

    where $\eta ^\pm \in [0,1]$ 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 $\eta _\pm$ are the sensitivity of the output change to excess demand/supply.

  • Wage: The wage update rule we chose follows (in spirit) the choices made for price and production. We propose that at each time step firm i updates its wage as:

    $$\begin{aligned} \begin{aligned} W^T_i(t+1)=W_i(t)[1+\gamma _w (1 - \Gamma \Phi _i) \varepsilon \xi ^\prime _i(t)] \quad \hbox {if}\quad {\left\{ \begin{array}{ll} Y_i(t) &{}< D_i(t)\\ {\mathcal {P}}_i(t) &{}> 0 \end{array}\right. }\\ W_i(t+1)=W_i(t)[1-\gamma _w (1 + \Gamma \Phi _i) u \xi ^\prime _i(t)] \quad \hbox {if}\quad {\left\{ \begin{array}{ll} Y_i(t) &{}> D_i(t)\\ {\mathcal {P}}_i(t) &{}< 0 \end{array}\right. } \end{aligned} \end{aligned}$$

    (24)

    where $\gamma _w$ is a certain parameter, ${\mathcal {P}}_i(t)$ is the profit of the firm at time t and $\xi ^\prime _i(t)$ an independent U[0, 1] random variable. If $W^T_i(t+1)$ is such that the profit of firm i at time t with this amount of wages would have been negative, $W_i(t+1)$ is chosen to be exactly at the equilibrium point where ${\mathcal {P}}_i(t)=0$ otherwise $W_i(t+1) = W^T_i(t+1)$.

    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 $\varepsilon$ 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 $\gamma _{p,w}$ allow us to simulate different price/wage update timescales, i.e. the aggressivity with which firms react to a change of their economic conditions. In the following we set $\gamma _p=0.05$ and $\gamma _w= \gamma _p$. The case $\gamma _w=0$ corresponds to removing completely the wage update rule, such that the version of the 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 $p_i\min {\{Y_i,D_i\}}$ firms profits are

  $$\begin{aligned} {\mathcal {P}}_i(t)= & {} p_i(t)\min {\{Y_i(t),D_i(t)\}} - W_i(t)Y_i(t)\nonumber \\&+\, \rho ^{\text {d}}\max {\{{\mathcal {E}}_i(t),0\}}+ \rho ^{\ell }\min {\{{\mathcal {E}}_i(t),0\}} \ . \end{aligned}$$

  (25)

  When firms have both positive ${\mathcal {E}}_i$ and ${\mathcal {P}}_i$ dividends are paid as a fraction $\delta$ of the firm cash balance ${\mathcal {E}}_i$. The update rule for firms cash balance is therefore

  $$\begin{aligned} {\mathcal {E}}_i(t+1) = {\mathcal {E}}_i(t) + {\mathcal {P}}_i(t) - \delta {\mathcal {E}}_i(t)\theta ({\mathcal {P}}_i(t))\theta ({\mathcal {E}}_i(t)) \end{aligned}$$

  (26)

  where $\theta (x)=1$ if $x>0$ and 0 otherwise. Correspondingly, households savings are updated as

  $$\begin{aligned} S(t+1)= & {} S(t) + \sum _iW_i(t)Y_i(t) - \sum _i p_i(t)\min {\{Y_i(t),D_i(t)\}}\nonumber \\&+\, \delta \sum _i{\mathcal {E}}_i(t)\theta ({\mathcal {P}}_i(t))\theta ({\mathcal {E}}_i(t)). \end{aligned}$$

  (27)

  The dividends share $\delta$ is set to $2\,\%$ unless stated otherwise and the $\Delta (t)$ term in Eq. (18) is given by

  $$\begin{aligned} \Delta (t) = \delta \sum _i{\mathcal {E}}_i(t)\theta ({\mathcal {P}}_i(t))\theta ({\mathcal {E}}_i(t)) \end{aligned}$$

  (28)

• Finally, an inactive firm has a finite probability $\varphi$ (which we set to 0.1) per unit time to get revived; when it does so its price is fixed to $p_i(t) = \overline{p}(t)$, its wage to $W_i(t) = \overline{w}(t)$, its workforce is the available workforce $Y_i(t) = u(t)$ and its cash-balance is the amount needed to pay the wage bill ${\mathcal {E}}_i(t) = W_i(t) Y_i(t)$. This small “liquidity” is provided by firms with positive ${\mathcal {E}}_i$ in shares proportional to their wealth ${\mathcal {E}}_i$.

Appendix 2: 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 $\Gamma =\Gamma _0$ (i.e. $\alpha _\Gamma =0$), $\rho ^{\ell }(t)=\rho _0(t)=0$, $f=1$, $c(t)=c_0=0.5$ (hence $\alpha _c=0$) and wages are constant and equal to 1 ($\gamma _w=0$). In this case the basic model described in Gualdi et al. (2015a) (with constant wages) is recovered.

Intuitively, the coupling between financial fragility and hiring/firing propensity should have a stabilizing effect on the economy. Moreover, the full unemployment phase at $R< R_c$ is deeply affected by the presence of $\alpha _\Gamma$: for $\alpha _\Gamma \ne 0$ the unemployment rate in this phase is no longer one, but becomes smaller than one and continuously changing with R. In order to derive an estimate of these continuous values we use an intuitive argument (at $\Theta =\infty$) which is justified a posteriori by the good match with numerical results. Given the critical ratio $R = \eta ^0_+/\eta ^0_- = R_c$ separating the high/low unemployment phases when there is no adaptation (i.e. $\Gamma _0=0$) one can expect that equilibrium values of the unemployment rate different from 0 and 1 can only be stable if $\eta _+^{i}/\eta _{-}^{i}$ remains around the critical value $R_c$ at $\Gamma _0=0$. Near criticality therefore we enforce that:

$$\begin{aligned} \frac{\eta _+^i}{\eta _-^i} = \frac{\eta ^0_+ (1 - \Gamma _0 \Phi _i)}{\eta ^0_- (1 + \Gamma _0 \Phi _i) } = R_c\Rightarrow -\Gamma _0 \Phi \approx \frac{R_c\eta _-^0-\eta _+^0}{R_c\eta _-^0+\eta _+^0}. \end{aligned}$$

(29)

An explicit form of $\Phi$ in terms of the employment rate $\varepsilon ={\overline{Y}}$ can be obtained with the additional assumption that the system is always close to equilibrium (i.e. $p\approx 1$ and $D \approx Y$, at least when $\eta ^i_+/\eta ^i_-\sim R_c$), which allows one to express households savings in terms of the firms’ production. Indeed (see the discussion in Gualdi et al. 2015a) at equilibrium $W = C_B = N_\mathrm{F} Y = c(W + S)$, from which it follows that $N_\mathrm{F} Y = W = S c/(1-c)$. For $c=0.5$ as in our simulations one thus has $S=N_\mathrm{F} Y$. Since the total amount of money is conserved (in our simulations $N_\mathrm{F} \overline{{\mathcal {E}}} + S = N_\mathrm{F}\overline{{\mathcal {E}}} + N_\mathrm{F}Y= N_\mathrm{F}$, see Appendix 3) one finally obtains that $\overline{{\mathcal {L}}} = 1 - Y$ and $\Phi = (Y-1)/Y = (\varepsilon -1)/\varepsilon$, hence

$$\begin{aligned} \frac{\Gamma _0}{\varepsilon } = \frac{R_c\eta _-^0-\eta _+^0}{R_c\eta _-^0+\eta _+^0} + \Gamma _0 =\frac{R_c- R}{R_c+ R} + \Gamma _0\ . \end{aligned}$$

(30)

Note that according to this formula the employment goes to $\varepsilon =1$ at the critical point $R = R_c$. Above this value, the economy is in the “good” state and employment sticks to $\varepsilon =1$ (this is because in the argument the effect of $\Theta$ has been neglected). Moreover, when $R < R_c$, $\varepsilon$ is proportional to $\Gamma _0$ and therefore in the limit $\Gamma _0\rightarrow 0$ one has $\varepsilon =0$ for all $R < R_c$. This is the “bad” phase of full unemployment at $\Gamma _0=0$, which becomes in this case a phase where employment grows steadily but remains of order $\Gamma _0$ except very close to the critical point.

Eq. (30) is plotted in Fig. 9 together with numerical results. Note that in this case the representative firm approximation ($N_\mathrm{F}=1$) is in good agreement with numerical results also for $N_\mathrm{F}=10,000$, as it was for the discontinuous transition obtained for $\Gamma =0$. In the inset of Fig. 9 one can see that the variance of the fluctuations of employment rate is diverging as long as the critical value of R is approached. This is confirmed by a spectral analysis of the unemployment time series (see Fig. 10). In order to obtain the power spectrum we apply the GSL Fast Fourier Transform algorithm to the time series $\varepsilon (t)-\langle \varepsilon \rangle$. As one can see in Fig. 10 the power spectrum is well approximated by an Ornstein-Uhlenbeck form:

$$\begin{aligned} I(\omega )=I_0\frac{\omega _0^2}{\omega _0^2+\omega ^2} \end{aligned}$$

(31)

with $\omega _0$ going linearly to 0 when $\eta ^0_+$ approaches its critical value, meaning that the relaxation time $\omega _0^{-1}$ diverges as one approaches the critical point. Note that this is not the case for the Mark 0 model with $\Gamma _0=0$ which instead has a white noise power spectrum even in proximity of the transition line. The first order (discontinuous) transition for $\Gamma _0=0, \Theta =\infty$ is thus replaced by a second order (continuous) transition when the firms adapt their behavior as a function of their financial fragility.

Fig. 9

Inverse of the average employment rate $\Gamma _0/\overline{\varepsilon }$ as a function of the ratio $R=\eta ^0_+/\eta ^0_-$ with $\eta ^0_-=0.1$ and $\gamma =0.01$ when $\Gamma _0>0$. When the employment rate is rescaled with the parameter $\Gamma _0$ (here $\Gamma _0=10^{-3},\ 10^{-4}$) the different lines collapse and Eq. (30) agrees with numerical simulations. In the inset we also plot the rescaled variance, still as a function of $\eta ^0_+$. Approaching the critical point the variance of the unemployment fluctuations diverges, together with their relaxation time going to infinity. The other parameters are: $\delta =0.02$, $\Theta =5$, $\gamma _w=0$, $c=0.5$, $\beta =0$ and $\varphi =0.1$

Fig. 10

Logarithm of the normalized power spectrum for Mark 0 with adaptive firms ($\Gamma _0=10^{-3}$), $\gamma _p=0.05$ and $N_\mathrm{F}=1$ (left) and $N_\mathrm{F}=1000$ (right). The other parameters are set as in Fig. 9. The main plot shows two examples of the spectrum for $\eta _-^0=0.1$ and $\eta _+^0=0.05$ (black line) and $\eta ^0_+=0.09$ (red line) in the left plot, $\eta _+^0=0.05$ (black line) and $\eta ^0_+=0.08$ (red line) in the right plot. The time series is made of $2^{28}$ time steps after $T_{eq}=500\, 000$ and the logarithm of the spectrum is averaged over a moving window of 100 points for a better visualization. With both system sizes the fit with Eq. (31) (blue dashed lines) is good with the only difference that when $N_\mathrm{F}>1$ a clear oscillatory pattern appears at high frequencies, that becomes sharper and sharper as $N_\mathrm{F}$ increases. In the inset of each figure we plot the value of $\omega _0$ in Eq. (31) obtained from the fit as a function of the ratio $R=\eta _+^0/\eta _-^0$. In both cases $\omega _0$ goes linearly to 0 as the critical value is approached

Finally, note that the presence of a continuum of states for the unemployment rate whenever $\Gamma _0>0$ and $R<R_c$ holds also with $\gamma _w>0$ (when wages are not constant). It was however simpler to perform analytical computations with constant wages.

Appendix 3: Pseudo-code of Mark 0

We present here the pseudo-code for the Mark 0 code described in Sect. 2.2 and Appendix 1. The source code of the baseline Mark-0 is available on demand.