Description of Mark-0

The Mark-0 model with a Central Bank (CB) and interest rates has been described in full details in [16–18], where pseudo-codes are also provided. It was originally devised as a simplification of the Mark family of ABMs, developed in [25, 26]. Since the details of the model are spread across multiple papers, for purposes of completeness we summarize the fundamental aspects of the model below.

First, we establish some basic notation. The model is defined in discrete time, where the unit time between t and t + 1 will be chosen to be ∼ 1 month. 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 a wage 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. To update their production, price and wage policy, firms use reasonable “rules of thumb” [16] that also depend on the inflation rate through their level of debt (see below). For example, production is decreased and employees are made redundant whenever Y_i > D_i, and vice-versa. As a consequence of these adaptive adjustments, the economy is on average always “close” to the global market clearing condition one would posit in a fully representative agent framework. However, small fluctuations persists in the limit of large system sizes giving rise to a rich phenomenology [16], including business cycles. The model is fully “stock-flow consistent” (i.e. all the stocks and flows within the toy economy are properly accounted for). In particular, there is no uncontrolled money creation or destruction in the dynamics, except when explicitly stated. In our baseline simulation, the total amount of money in circulation is set to 0 at t = 0. This choice is actually irrelevant in the long run, but may have important short term effects. We will actually allow some money creation below, when “helicopter money” policies will be investigated.

In Mark-0 we assume a linear production function with a constant productivity, which means that output Y_i and labor N_i coincide up to a multiplicative factor ζ, i.e. Y_i = ζN_i. The unemployment rate u is defined as

where N is the number of agents. Note that firms cannot hire more workers than available, so that u(t) ≥ 0 at all times—see Eq (11) below. The instantaneous inflation rate, denoted π(t), is defined as

where p¯(t) is the production-weighted average price p¯=∑ipiYi∑iYi. We define w¯ as the production-weighted wage as well: w¯=∑iwiYi∑iYi. We further assume that households and firms adapt their behavior not to the instantaneous value of the inflation π(t), but rather to a smoothed averaged value. We name this the “realized inflation” and it is defined as:

Households and firms form expectations for future inflation by observing the realised inflation and the target inflation set by the Central Bank. This is modeled as follows:

The parameters τ^R and τ^T model the importance of past inflation for forming future expectations and the confidence that economic actors have in the central bank’s ability to achieve its target inflation.

The Mark-0 economy is made of firms producing goods, households consuming these goods, a banking sector and a central bank. Households and the banking sector are described at the aggregate level by a single “representative household” and “representative bank” respectively. In what follows, we describe how they interact within our toy economy.

We assume that the total consumption budget of households C_B(t) is given by:

where S(t) is the savings, W(t) = ∑_i W_i(t)N_i(t) the total wages, ρ^d(t) is the interest rate on deposits, and c(t) is the “consumption propensity” of households, which is chosen such that it increases with increasing inflation:

Here α_c is a parameter that modulates the sensitivity of households to the real interest (inflation adjusted) rate on their savings ρ^d(t). With this choice of the consumption propensity, Eq (5) describes an inflation-mediated feedback on consumption similar to the standard Euler equation in DSGE models [27].

The total household savings then evolve according to:

where Δ(t) are the dividends received from firms’ profits (see below). C(t) ≤ C_B(t) is the actual consumption of households, determined by the matching of production and demand and computed as

The demand for goods D_i itself is modeled via an intensity of choice model with a parameter β which defines the dependence of the demand on the price:

The model contains N_F firms (we chose N_F = N for simplicity [16]), each firm being characterized by its workforce N_i and production Y_i = ζN_i, demand for its goods D_i, price p_i, wage W_i and its cash balance Ei which, when negative, is the debt of the firm. We characterize the financial fragility of the firm through the debt-to-payroll ratio

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 as the total payroll), the firm i is allowed to continue its activity. If on the other hand Φ_i ≥ Θ, the firm i defaults and the corresponding default cost is absorbed by the banking sector, which adjusts the loan and deposit rates ρ^l and ρ^d accordingly (see below). The defaulted firm is replaced by a new one at rate φ, initialized at random. The parameter Θ controls the maximum leverage in the economy, and models the risk-control policy of the banking sector.

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” [16, 18]. In particular, the production update is chosen as follows:

where ui⋆(t) is the maximum number of unemployed workers available to the firm i at time t, which depends on the wage the firm pays,

where w¯ is the production-weighted average wage. Firms hire and fire workers according to their level of financial fragility Φ_i: firms that are close to bankruptcy are arguably faster to fire and slower to hire, and vice-versa for healthy firms. The coefficients η^± ∈ [0, 1] express the sensitivity of the firm’s target production to excess demand/supply. We posit that the coefficients ηi± for firm i (belonging to [0, 1]) are given by:

where η0± are fixed coefficients, identical for all firms, and 〚x〛 = 1 when x ≥ 1 and 〚x〛 = 0 when x ≤ 0. The factor Γ > 0 measures how the financial fragility of firms influences their hiring/firing policy, since a larger value of Φ_i then leads to a faster downward adjustment of the workforce when the firm is over-producing, and a slower (more cautious) upward adjustment when the firm is under-producing. Γ itself depends on the inflation-adjusted interest rate and takes the following form:

where α_Γ is a free parameter, similar to α_c that captures the influence of the real interest rate.

Price update. Following the initial specification of the Mark series of models [25], 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:

where ξ_i(t) are independent uniform U[0, 1] random variables and γ is a parameter setting the relative magnitude of the price adjustment. The factor (1+π^(t)) models firms’ anticipated inflation π^(t) when they set their prices and wages.

Wage update. The wage update rule follows the choices made for price and production. Similarly to workforce adjustments, we posit that at each time step firm i updates the wage paid to its employees as:

where Pi is the profit of the firm at time t, ξi′(t) an independent U[0, 1] random variable and g modulates how wages are indexed to the firms’ inflation expectations. If WiT(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 Pi(t)=0; otherwise Wi(t+1)=WiT(t+1). Here, Γ is the same parameter introduced in Eq (13).

Note that within the current model the productivity of workers is not related to their wages. The only channel through which wages impact production is that the quantity ui⋆(t) that appears in Eq (11), which represents the share of 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.

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 pay rise is expected to be large if the firm is financially healthy and/or if unemployment is low because pressure on salaries 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, because pressure on salaries is then low.

In all other cases, wages are not updated.

Profits and dividends. Finally, the profits of the firm Pi are computed as the sales minus the wages paid with the addition of the interest earned on their deposits and the interest paid on their loans:

If the firm’s profits are positive and they possess a positive cash balance, they pay dividends as a fraction δ of their cash balance Ei:

where θ is the Heaviside step-function. These dividends are then reduced from the firms’ cash balance and added to the households savings in Eq (7).

The banking sector in Mark-0 consists of one “representative bank” and a central bank which sets baseline interest rates. The central bank also has an inflation targeting mandate. The central bank sets the base interest rate ρ₀ via a Taylor-like rule:

Here, ϕ_π modulates the intensity of the central bank policy, ρ* is the baseline interest rate and π* is the inflation target for the central bank. The banking sector then sets interest rates for deposits ρ^d (for households) and loans ρ^l (for borrowing by firms). Defining E+=∑imax(Ei,0) (equivalent to firms’ positive cash balance) and E-=-∑imin(Ei,0) (equivalent to firms’ total debt), the interest rates are set as:

where D(t) are the total costs accrued due to defaulting firms. The parameter f then determines how the impact of these defaults fall upon lenders and depositors—f interpolates between these costs being borne completely by the borrowers (f = 1) or fully by the depositors (f = 0). The total amount of (central-bank) money M in circulation is kept constant and the balance sheet of the banking sector reads:

This is simply a restatement of the fact that the sum of households savings and the deposits and debts of firms is equal to the total amount of money in circulation.