A game between central banks and households involving central bank digital currencies, other digital currencies and negative interest rates

Figures & data

Figure 1. The household’s consumption $\mathit{c}$, holding of CBDC $\mathit{m}$, holding of non-CBDC $\mathit{q}$, and utility $\mathit{U}$, and the central bank’s interest rate ${\mathit{I}}_{\mathit{m}}$ and profit $\mathit{u}$, as functions of the nine parameter values $\mathit{\alpha },$${\mathit{\lambda }}_{\mathit{n}},{\lambda }_g,$${\mathit{\beta }}_{\mathit{n}},{\gamma }_n,$${\beta }_g,$${\mathit{a}}_{\mathit{n}},a_g$ relative to the benchmark parameter values $\mathit{\alpha }=\eta =1/5,\beta =\mu =2/5,\lambda =1/10,$$\mathit{\theta }=r=1,\sigma =5,I_q=0$. Division of ${\mathit{I}}_{\mathit{m}}$ with 5 and $\mathit{u}$ with 3 is for scaling purposes.

Table 1. Upward arrow $\uparrow$, sideways arrow $\to$, or downward arrow $\downarrow$, respectively, depending on whether the first order derivative (listed first) and second order derivative (listed second) in the proposition are positive, zero or negative. The derivatives are for the variable in the row with respect to the parameter in the column.

Variable	$\alpha$	$\beta$	$\mu$	$\eta$	$\lambda$	$\theta$	$\sigma$	$r$	$I_q$
$c$	$\uparrow \to$	$\to$	$\downarrow \uparrow$	$\downarrow \uparrow$	$\downarrow \downarrow$	$\to$	$\to$	$\uparrow \to$	$\to$
$m$	$\to$	$\uparrow \to$	$\uparrow \downarrow$	$\downarrow \uparrow$	$\uparrow \uparrow$	$\to$	$\to$	$\uparrow \to$	$\to$
$q$	$\downarrow \to$	$\downarrow \to$	$\downarrow \uparrow$	$\uparrow \downarrow$	$\uparrow \uparrow$	$\to$	$\to$	$\uparrow \to$	$\to$
$I_m$						$\downarrow \uparrow$	$\uparrow \uparrow$		$\uparrow \downarrow$
$U$						$\downarrow \uparrow$	$\uparrow$		$\uparrow \downarrow$

Table

Parameters
$r$	Household’s monetary energy, or resources, $r\ge 0$
$\alpha$	Household’s output elasticity for consumption $c$, $\lambda \le \alpha \le 1$
$\beta$	Household’s output elasticity for CBDC $m$, $0\le \beta \le 1$
$1-\alpha -\beta$	Household’s output elasticity for non-CBDC $q$, $0\le 1-\alpha -\beta \le 1$
$I_q$	Interest rate, $I_q\in \mathbb{R}$
$\mu$	Household’s transaction efficiency for CBDC $m$, $\mu \ge 0$
$\eta$	Household’s transaction efficiency for non-CBDC $q$, $\eta \ge \lambda$
$\lambda$	Household’s transaction efficiency for consumption $c$, $0\le \lambda \le \alpha$
$\theta$	Scaling or degree or level of the household’s transaction cost, θ ≥ 0
$\sigma$	Scaling parameter for the central bank’s profit, σ > 0
Household’s free choice variables
$c$	Household’s consumption, $0\le c\le r$
$m$	Household’s holding of CBDC, $0\le m\le r$
Central bank’s free choice variable
$I_m$	CBDC interest rate for the household’s holding of CBDC $m$, $I_m\in \mathbb{R}$
Dependent variables
$U$	Household’s utility
$u$	Central bank’s profit per household
$q=r-m-c$	Household’s holding of non-CBDC, $0\le q=r-m-c\le r$
$E$	Household’s transaction efficiency