Yilmaz–Godin flexible exchange rates and the Leeds continuous-time design

Written 2026-10-03. Primary reference: Yilmaz and Godin (2020), AFD Research Papers 125, official paper. This check concerns that prototype, rather than every GEMMES country model. Downloaded PDF, extracted text and SHA-256 manifest are in output/calibration/gemmes_reference_20261003/.

What the prototype does

Its nominal quote \(e\) is domestic currency per foreign currency. It responds to desired flow demand \(D\) and supply \(S\) rather than imposing instantaneous FX clearing:

\[\dot e=e\beta_e(D-S)/S. \qquad\text{(96, printed p.21)}\]

Expectations have an independent speed and a risk-adjusted interest-rate target:

\[\dot e^a=\beta_a\left[\Upsilon\left(\frac{1+i^F}{(1+i^P)(1-r)}\right)^\sigma e-e^a\right]. \qquad\text{(99, p.22)}\]

Banks maintain no open FX position; central-bank reserves absorb the remaining reserve change (49–50, p.16). Desired intervention is distinguished from that residual (p.19). Baseline speeds are 0.7 and 0.75 (Table 13, p.30). Faster FX adjustment moderates the tested boom–bust episode, conditional on expectations and the other institutions; it does not guarantee stability (§6.1, pp.37–38). Source.

Printed income-account sign needs reconciliation

My algebraic check found an apparent inconsistency, confirmed visually against the PDF (physical pages 27–28). The printed formulas are:

\[D=IMp^W+IA/e+\dot R^{B,NOP},\] \[S=X/e+WFF_D+\dot L_B^{FX}-\dot R^{CB,I},\] \[\dot R^{FX}=X/e-IMp^W+WFF_D+\dot L_B^{FX}+IA/e.\]

Consequently their subtraction gives

\[D-S=\dot R^{B,NOP}-\dot R^{FX}+\dot R^{CB,I}+2IA/e.\]

Footnote 20 instead states the same expression without \(2IA/e\). Equation 100 defines \(IA\) as net receipts, and equation 102 includes it positively in reserve accumulation. Changing the demand term to \(-IA/e\) would reconcile this identity, but that is a candidate correction, not a verified author implementation. A zero-income-account fixture would conceal the discrepancy. Retain the published bytes and test nonzero net receipts/payments before any transcription or adaptation. No runnable author-code reproduction was completed here.

Consequences for Leeds — our design decisions

Leeds currently solves its quote from joint bill-market and wealth constraints. Replacing that quote with an ODE while retaining every original clearing identity would generally overdetermine the system. Giving the quote a speed therefore requires an explicit institution that holds unmatched transactions or supplies financing, with matching counterparty entries and feasibility limits.

Build two labeled experimental closures: an accounting-consistent continuous-time version retaining instantaneous Leeds clearing, and a separately derived flow-based FX adjustment closure. The second may use the GEMMES architecture, but must obtain demand, supply and settlement from Leeds’s two-region ledgers. GEMMES’s small-economy rest-of-world structure and cross-border bank loans cannot simply replace Leeds’s two-region bill portfolios. Document any new expectations or foreign-bank mechanism as changed economics.

Use a logarithmic quote state for positivity; define one currency orientation and derive the reciprocal. Keep trading flows, inventory adjustment and portfolio stock adjustment distinct. Define the time unit before converting rates. Changing all speeds equally rescales time in an autonomous system; varying relative speeds is the substantive experiment. Shock durations must remain fixed in economic time.

The retained source’s private profits already embed holding gains. Apply the continuous valuation identity \(d(pK)/dt=p\dot K+K\dot p\) once, then derive the income/revaluation bridge explicitly. Avoid counting those gains twice. Government and sector-level asset valuation remain unresolved choices and must appear in the conversion contract.

The existing admissible-history experiment demonstrates one-step quote amplification and rejects four-stock transition sufficiency on a five-control family. It proves neither a full unstable eigenspectrum nor unbounded fluctuations. The conversion should preserve the complete real, financial and technology state required by its equations, and measure finite-horizon attraction independently of numerical integration convergence.

Implementation: qmd/plans/2026-10-03-openflex-continuous-time.md. Begin with the 4–8 hour inventory and accounting pilot. No continuous-time model implementation was started during this planning session.