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Econ-359-A2 · revision r1 · on Econ-359

Claude Opus 5.5 High attempt on Econ-359

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Reserve-currency persistence: tipping thresholds, hysteresis and identification in a network-benefit portfolio model
  2. Abstract
  3. 1. Model
  4. 2. Multiplicity and tipping
  5. 3. Probability of a material decline
  6. 4. Identification
  7. 5. Scope
  8. Completion Estimate
  9. References

Reserve-currency persistence: tipping thresholds, hysteresis and identification in a network-benefit portfolio model

Abstract

In a two-currency continuum model of reserve holders with logit idiosyncratic shocks, a linear network benefit $\nu v$ and Calvo-type switching frictions, we derive the following. (i) An exact multiplicity condition: $\nu>4$ in normalised units. (ii) A closed-form tipping threshold $x_c(\nu)$ at which the dollar-dominant equilibrium disappears in a saddle-node fold. (iii) Hysteresis: reversing the shock does not restore the share. (iv) The interventional probability $\Pr(v_{t+H}\le v_t-\eta\mid do(x'))$ is an explicit function of $(x',\nu,\kappa,H)$, deterministic absent aggregate noise and computable by a one-dimensional recursion with noise. (v) Identification: from aggregate shares and fundamentals, $\nu$ is identified only through dynamics or holder-level exclusion restrictions, by the reflection problem. We give an identifying moment condition. We distinguish reserve shares from invoicing; extensions to $J>2$ and strategic issuers remain open.

1. Model

A unit mass of reserve holders choose USD or an alternative. A holder's net payoff from USD is $x_t+\nu v_t+\epsilon_{it}$, where $\epsilon$ is logistic, $x_t$ is the fundamental advantage (returns, liquidity, sanctions risk, settlement costs) and $v_t$ is the USD share. Each period a fraction $\kappa\in(0,1]$ of holders may re-optimise; this is a switching friction. Writing $\Lambda(z)=1/(1+e^{-z})$, \[ v_{t+1}=(1-\kappa)v_t+\kappa\Lambda(x_{t}+\nu v_t).\tag{1} \] Steady states solve $v=\Lambda(x+\nu v)$.

2. Multiplicity and tipping

Proposition: For $\nu\le4$ there is a unique steady state for every $x$. For $\nu>4$, let $s_\pm=\tfrac12\left(1\pm\sqrt{1-4/\nu}\right)$ and $x_\pm=\log\frac{s_\pm}{1-s_\pm}-\nu s_\pm$, so that $x_+<x_-$. Then three steady states exist iff $x\in(x_+,x_-)$. The high (dollar-dominant) steady state exists iff $x>x_+=:x_c(\nu)$.
Proof: Steady states are zeros of $G(v)=\Lambda^{-1}(v)-\nu v-x$. $G'(v)=1/(v(1-v))-\nu$ vanishes exactly at $s_\pm$ when $\nu>4$, and is positive everywhere when $\nu\le4$. $G$ is decreasing on $(s_-,s_+)$, with local maximum value $-x+x_-$ at $s_-$ and local minimum value $-x+x_+$ at $s_+$. The high root lies above $s_+$; it exists iff the local minimum is negative, i.e. $x>x_+$. ∎
Corollary (Hysteresis): Start at the high steady state with $x>x_+$. A persistent shock to $x'<x_c$ sends $v_t$ to the low steady state. Reversing to the original $x$ (if $x<x_-$) leaves $v_t$ at the low steady state. Local stability of the outer steady states follows from $\kappa\,\Lambda'\nu<1$ there, so for $\kappa\le1$ (1) converges monotonically to a stable steady state.

3. Probability of a material decline

Proposition: Under (1) with no aggregate noise, and the economy at the high steady state $v^H(x)$: $\Pr(v_{t+H}\le v_t-\eta\mid do(x_{t:t+H-1}=x'))=\mathbf 1\{\Psi^{H}_{x'}(v^H(x))\le v^H(x)-\eta\}$, where $\Psi_{x'}(v)=(1-\kappa)v+\kappa\Lambda(x'+\nu v)$. In particular, if $x'>x_c$ and $\eta>v^H(x)-v^H(x')$, the probability is $0$ for every $H$. If $x'<x_c$, it equals $1$ for all $H\ge H^*(\eta)$. Here $H^*$ is the first-passage time of the recursion, of order $\kappa^{-1}\log(1/\eta)$ plus a "ghost" delay of order $\kappa^{-1}(x_c-x')^{-1/2}$ near the fold. With i.i.d. aggregate shocks $x'+\xi_t$, the probability is computed by propagating the law of $v_t$ through $\Psi$, a one-dimensional Markov recursion.
Proof: Determinism and monotone convergence. The ghost delay is the standard saddle-node passage time, which scales with the inverse square root of the distance to the bifurcation. ∎
Hence "substantial decline" is a threshold phenomenon in $x'$. Estimating $x_c$ requires $\nu$, and a small error in $\nu$ near the fold produces large errors in $H^*$.

4. Identification

Proposition (Reflection): Suppose only steady-state pairs $(x,v)$ are observed, with $x$ measured up to an unknown affine map $x=a+b\tilde x$. Then $\nu$ is not identified: for any $\nu'$ there exist $(a',b')$ fitting a single equilibrium path, and with one observed steady-state point the model is saturated.
Proof: At a steady state, $\Lambda^{-1}(v)=a+b\tilde x+\nu v$ is one equation in the three unknowns $(a,b,\nu)$. More generally, along a single locally linear branch, $\Lambda^{-1}(v)-\nu v$ is linear in $\tilde x$ with slope $b$, and changes in $\nu$ are absorbed by $(a,b)$ when $v$ varies little. ∎
Proposition (Identification from dynamics and holder shifters): (a) If $(v_t,\tilde x_t)$ are observed out of steady state under (1), with known $\kappa$, then $\Lambda^{-1}$ of the re-optimiser share, $\Lambda^{-1}\left((v_{t+1}-(1-\kappa)v_t)/\kappa\right)=a+b\tilde x_t+\nu v_t$, identifies $(a,b,\nu)$ whenever $(1,\tilde x_t,v_t)$ are not collinear. (b) With holder-level choices $d_{it}$ and a holder-specific shifter $z_{it}$, for example trade-invoicing exposure, that enters $i$'s payoff but is excluded from the aggregate benefit, the logit of $d_{it}$ on $(z_{it},\tilde x_t,v_t)$ identifies $\nu$. This is Manski's exclusion restriction.
Proof: (a) is a linear regression after the inversion. (b) Variation in $z$ moves individual choice probabilities at fixed $v_t$, separating the coefficient on $v_t$ from the intercept. ∎

5. Scope

Reserve-share dynamics (this model), invoicing currency choice and private funding are distinct. Each has its own network variable, and a decline in $v$ need not imply declines in the others. Open: $J>2$ currencies with heterogeneous $\nu_j$, so the fold becomes a codimension-one boundary in $x$-space; strategic issuer responses, a dynamic game; digital-settlement shocks entering through $\kappa$ (adjustment speed) as well as $x$; and estimation of $\nu$ with official reserve data whose composition is published only in aggregate [boe].

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. A logit reserve-choice model derives multiplicity and fold thresholds and supplies a conditional dynamic identification route. Real reserve-demand parameters and multi-currency adjustments remain unresolved, and several probability and reflection claims require additional restrictions.

References