A2: Gresham's Law — When Bad Money Wins

L02 Monetary Economics — Selgin (1996) Salvaging Gresham's Law

Assignment Brief

Introduction

"Bad money drives out good money" — this famous economic principle, known as Gresham's Law, explains why people spend depreciating currency and hoard stable currency. You'll explore an agent-based model that simulates this tipping-point behavior.

The Model

Imagine 1,000 agents (simulated people) who each period must decide: spend Currency A (the "bad money" that loses 5% of its value per period) or spend Currency B (the "good money" that only loses 1% per period)?

The model uses a logit feedback rule to determine the probability that an agent spends Currency A:

P(spend A) = 1 / (1 + exp(-z_t))

where:

z_t = k_feedback × (share_A_{t-1} - 0.5) + (r_A - r_B) × 10 + base_bias × (t / T)
Logit feedback: A mathematical formula that converts any number z into a probability between 0 and 1. When z is large positive, P ≈ 1 (almost certain). When z is large negative, P ≈ 0 (almost never).
k_feedback (default: 15.0): How strongly past behavior amplifies current behavior. If many people spent A yesterday, that makes it more rational for you to spend A today (because holding onto A means you'll suffer the depreciation loss).
(r_A - r_B) × 10: The depreciation differential bias. When Currency A depreciates faster than B, agents have an incentive to spend A and hoard B. With default values (r_A=0.05, r_B=0.01), this contributes +0.4 to the logit at every step. With equal depreciation (r_A=r_B), this term is zero.
base_bias (default: 0.10): A small exogenous drift toward spending A, scaled by time progress.
share_A_{t-1}: The fraction of agents who spent A in the previous period.
t / T: Time progress (0 at start, 1 at end), which gradually increases the pressure to spend A.

This creates a tipping point: initially, agents are split 50-50, but once enough people start spending A, the feedback loop kicks in and everyone switches to spending A (and hoarding B).

Your Task

Run the simulation with three parameter variations and analyze what changes:

Variation 1: Weak Feedback (k_feedback = 5.0)

Change: Reduce k_feedback from 15.0 to 5.0.

Question: Does tipping still happen within 100 periods? If not, what happens instead? How does the rate of change in circulation shares compare to the baseline?

Variation 2: Strong Feedback (k_feedback = 25.0)

Change: Increase k_feedback from 15.0 to 25.0.

Question: What happens to the tipping point timing? How fast is the transition from 50-50 to A-dominance?

Variation 3: Equal Depreciation (r_A = r_B = 0.03)

Change: Set both depreciation rates equal: r_A = 0.03 and r_B = 0.03.

Question: Does tipping still occur? Why or why not? What does this tell you about the conditions required for Gresham's Law?

Open Extension (Optional)

Add a third currency C with r_C = 0.00 (perfectly stable, like gold-backed currency).

Question: Does C get hoarded even more than B? Does A still drive out both B and C, or does C survive in circulation?

How to Run

Download the source code: L02_Monetary_Economics/05_greshams_law_simulation/chart.py
Open Google Colab: colab.research.google.com
Upload the file, or copy-paste the code into a new notebook.

Modify parameters for each variation:

# For Variation 1 (weak feedback):
k_feedback = 5.0

# For Variation 2 (strong feedback):
k_feedback = 25.0

# For Variation 3 (equal depreciation):
r_A = 0.03
r_B = 0.03

Run the code (Shift+Enter or click Run). The chart will display inline.
Save each chart (right-click → Save Image).

Deliverables

Prepare a 7-slide presentation (PowerPoint, Google Slides, or PDF):

Title slide: Your name, course, "Gresham's Law Simulation"
The Model: Explain the logit feedback rule and what each parameter means.
Baseline Results: Show the original chart (k=15). When does tipping occur?
Variation 1: Show weak feedback chart (k=5). What changed?
Variation 2: Show strong feedback chart (k=25). What changed?
Variation 3: Show equal depreciation chart (r_A=r_B=0.03). What changed?
Key Insight: In 1-2 sentences, explain what controls WHETHER tipping happens vs HOW FAST it happens.

Time Allocation

Reading & Setup: 10 minutes
Running Variations: 20 minutes
Analysis: 15 minutes
Slides: 10 minutes
Total: 45 minutes + 10 minutes presentation

Assessment Criteria

Criterion	Weight
Correct implementation of variations	30%
Accurate description of tipping behavior	30%
Clear charts with appropriate labels	20%
Insightful explanation of k vs depreciation difference	20%

Reference

Selgin, G. (1996). Salvaging Gresham's Law: The Good, the Bad, and the Illegal. Journal of Money, Credit and Banking, 28(4), 637-649.

Your Variation Chart

Download chart_varied.py Download chart_varied.pdf

Model Answer Presentation (7 Slides)

Slide 1

Gresham's Law: Why People Spend Bad Money and Hoard Good Money

Economic Principle: "Bad money drives out good money"

Model: 1,000 agents decide each period whether to spend Currency A (depreciating 5%/period) or Currency B (depreciating 1%/period).

Key Reference: Selgin (1996) – Salvaging Gresham's Law: The Good, the Bad, and the Illegal

Slide 2

The Model: Logit Feedback Rule

Probability that an agent spends Currency A at time t:

P(spend A)_t = 1 / (1 + exp(-z_t))

where the logit argument evolves as:

z_t = k × (s_{t-1} - 0.5) + (r_A - r_B) × 10 + b × (t/T)

Variables:

s_{t-1} = share of agents who spent A in previous period
k = feedback strength (how much past behavior amplifies current behavior)
(r_A - r_B) × 10 = depreciation differential bias (higher depreciation of A relative to B incentivizes spending A)
b = small exogenous drift toward spending A (default 0.10)
t/T = time progress (0 at start, 1 at end)

Mechanism: When Currency A depreciates faster than B, agents prefer to spend A and hoard B. This depreciation differential enters the logit directly. When more people spend A, it becomes even more rational for you to also spend A (positive feedback). Together, these forces create a tipping point.

Slide 3

Baseline: k = 15, Tipping at ~Period 48

Baseline chart

Observations:

Initially 50-50 split between spending A and B
Tipping point at period ~48: Currency A reaches 80% circulation
By period 100: A dominates spending (~98.5%), B is hoarded

Note: The varied chart (next slides) uses an explicit depreciation differential term (r_A - r_B) x 10 in the logit, which accelerates baseline tipping to t=27.

Slide 4

Variation 1: Weak Feedback (k = 5)

Variation chart

Panel 2 (Top Right): Weak feedback delays tipping significantly -- tipping occurs at t=57 (vs t=27 baseline). Currency A reaches ~90% by period 100.

Why? Lower k means past behavior has less influence. The depreciation differential (4 pp) and small exogenous drift still push toward spending A, but the feedback loop is too weak to create a rapid cascade. The transition from 50-50 to A-dominance takes more than twice as long as baseline.

Key Insight: Weak feedback = delayed tipping. Even with the same depreciation difference, insufficient feedback strength stretches the transition over many more periods.

Slide 5

Variation 2: Strong Feedback (k = 25)

Variation chart

Panel 3 (Bottom Left): Strong feedback accelerates tipping to ~period 23.

Why? Higher k means past behavior strongly amplifies current behavior. Combined with the depreciation differential of 0.4 pushing the logit positive, even a small deviation from 50-50 triggers a rapid cascade. The transition from 50-50 to A-dominance happens in roughly 10 periods.

Key Insight: Strong feedback = faster tipping, earlier tipping point. The S-curve transition is sharper. Tipping at t=23 vs baseline t=27 shows diminishing returns from increasing k when the depreciation differential already provides a strong push.

Slide 6

Variation 3: Equal Depreciation (r_A = r_B = 3%)

Variation chart

Panel 4 (Bottom Right): Tipping is significantly delayed to ~period 48 (vs baseline t=27). The depreciation differential is zero, so only the small exogenous drift (base_bias) gradually pushes spending toward A.

Why? With equal depreciation (r_A = r_B = 3%), the depreciation_bias term is zero. There is no quality difference to drive Gresham's Law. The only remaining force is the small base_bias drift (0.10 * t/T), which slowly nudges the logit positive over time. This eventually triggers the feedback loop, but the transition takes nearly twice as long as baseline.

Key Insight: Removing the depreciation differential delays tipping by ~21 periods (from t=27 to t=48). The depreciation difference is the primary accelerant of Gresham's Law — without it, the feedback loop lacks an initial push and must rely on weak exogenous drift to eventually trigger.

Slide 7

Key Insight: k Controls HOW FAST, Depreciation Difference Controls WHETHER

Parameter	Effect	Evidence
k (feedback strength)	Controls tipping SPEED	k=5: tipping delayed to t=57 (~90% by end) k=15: tipping at t=27 (baseline) k=25: tipping at t=23 (rapid)
Depreciation difference (r_A - r_B)	Controls tipping ACCELERATION	r_A=5%, r_B=1%: tipping at t=27 (strong depreciation push) r_A=3%, r_B=3%: tipping delayed to t=48 (no depreciation push, only slow drift)

Economic Implication: Gresham's Law has two drivers that interact:

Depreciation difference provides the initial incentive to spend bad money. Removing it delays tipping by ~21 periods (t=27 to t=48).
Feedback strength determines how quickly a small initial bias cascades into full substitution. Weak feedback (k=5) delays tipping by ~30 periods (t=27 to t=57).

The depreciation differential is the more powerful driver: it shifts the logit argument by 0.4 at every step (vs the exogenous drift which contributes at most 0.10 over the full horizon).

Historical Example: During hyperinflation (very large depreciation difference + strong social feedback), everyone rushes to spend depreciating currency, creating rapid collapse. During mild inflation (small depreciation difference), weak feedback allows dual-currency systems to persist for decades (e.g., dollarization in Latin America).

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