A1: How Fast Do Payment Technologies Spread?

L01 Introduction — Rogers (1962) Diffusion of Innovations

Assignment Brief

Introduction

New payment technologies don't take over the world overnight. They follow a predictable pattern called an S-curve (a curve shaped like the letter S that shows how technology adoption starts slow, accelerates, then levels off). The Rogers (1962) Diffusion of Innovations model describes this using three key parameters: the carrying capacity K (the maximum percentage of people who will ever adopt the technology), the growth rate r (how fast adoption accelerates during the steep middle phase), and the inflection point t0 (the year when adoption reaches half of its maximum level K). Understanding these parameters helps predict whether a payment technology will become mainstream or remain niche.

Your Task

Your job is to replicate the chart showing adoption S-curves for six payment technologies (credit cards, ATMs/debit, online banking, mobile payments, cryptocurrencies, and CBDCs). Then explore how changes to the model parameters affect adoption trajectories.

Required Variations:

Add a new technology: Add "Buy Now Pay Later" (BNPL) services with K=40%, r=0.20, t0=2022. How does its adoption curve compare to existing technologies? Why might BNPL have a higher growth rate r but lower ceiling K than credit cards?
Double the growth rate for crypto: Change the cryptocurrency growth rate r from 0.10 to 0.20. What year does adoption reach 50% of its ceiling (which is 15% of the population)? Compare this to the original model.
Increase CBDC ceiling: Set the CBDC (Central Bank Digital Currency) carrying capacity K to 90% instead of 50%. How does the curve change by 2050? Would a government-backed digital currency with K=90% eventually overtake mobile payments?

Open Extension (optional, for extra credit):

The S-curve (logistic function) is not the only growth model. Compare it with the Gompertz curve: G(t) = K * exp(-exp(-r*(t-t0))). The Gompertz curve has a slower initial growth phase but faster late-stage growth. Which model fits better for technologies with rapid early adoption (like mobile payments) versus slow-burn technologies (like credit cards)?

How to Run

Open Google Colab (colab.research.google.com)
Create a new notebook
Paste the Python code from chart_varied.py
Click the Run button (play icon) on each cell
Download the generated charts
Use the charts in your presentation

Time Budget

45 minutes: Code exploration + chart generation + analysis
10 minutes: Presentation (5-7 slides)

Deliverable

A 5-7 slide presentation that includes:

The S-curve model equation and parameter definitions
Your replicated baseline chart (6 technologies)
At least one variation chart you generated
Analysis: Which parameters matter most for predicting technology success?
One insight: What surprised you or challenged your assumptions?

Presentation Tips:

Use plain-text math (S = K / (1 + exp(-r*(t-t0)))) not LaTeX
Define every term when first used
Show charts full-screen (not embedded in slides with tiny text)
Compare curves visually (don't just list numbers)
End with a testable prediction: "By 2030, I predict [technology X] will reach [Y]% adoption because..."

Assessment Criteria:

Correct replication of baseline chart (20%)
All three required variations completed (30%)
Clear explanation of how parameter changes affect curves (30%)
Presentation clarity and visual design (20%)

Your Variation Chart

Download chart_varied.py Download chart_varied.pdf

Model Answer Presentation (7 Slides)

Slide 1

How Fast Do Payment Technologies Spread?

Rogers (1962) Diffusion of Innovations

In-Class Presentation Assignment — L01 Introduction

Student Name: Model Answer
Date: February 2026

Slide 2

The Model: S-Curve (Logistic Growth)

Equation: S(t) = K / (1 + exp(-r * (t - t0)))

Parameters:

K (carrying capacity) = maximum adoption percentage the technology will ever reach
r (growth rate) = how fast adoption accelerates during the steep middle phase
t0 (inflection point) = the year when adoption reaches exactly half of K

Why S-shaped? Early adopters are easy to convince (slow start), then word-of-mouth creates exponential growth (steep middle), then the market saturates as remaining holdouts resist (flat top).

Source: Rogers, E.M. (1962). Diffusion of Innovations. Free Press.

Slide 3

Baseline: Six Payment Technologies

Baseline

Key Observations:

Credit cards: slowest growth (r=0.08) but highest ceiling (K=75%)
Mobile payments: fastest growth (r=0.15) but medium ceiling (K=65%)
Cryptocurrencies: slow growth (r=0.10) and low ceiling (K=30%)
Growth rate r determines steepness through inflection, carrying capacity K determines finish line

Slide 4

Variation 1: Adding Buy Now Pay Later (BNPL)

Variations

Parameters: K=40%, r=0.20, t0=2022

Analysis:

BNPL has the fastest growth rate (r=0.20) of all technologies shown
But lowest ceiling among traditional payments (K=40%)
Interpretation: BNPL spreads quickly among young consumers (high r) but faces regulatory limits and consumer skepticism that cap long-term adoption (low K)
By 2035, BNPL reaches ~38% (near its ceiling), while credit cards continue growing toward 75%

Slide 5

Variation 2: Faster Crypto Adoption (r doubled)

Both curves: reach 15% (half of K=30%) at the inflection point t0=2022

Analysis:

Doubling r changes HOW STEEPLY the curve rises through 15%, not WHEN it crosses 15%
With r=0.10: goes from 5% to 25% (17% to 83% of K) over ~28 years
With r=0.20: goes from 5% to 25% in ~14 years (twice as steep)
But ceiling K=30% remains unchanged — crypto still plateaus at 30% regardless of r
Key Insight: Growth rate r controls the steepness of adoption during the transition phase, while carrying capacity K controls your ultimate market share

Real-world question: What would it take to increase crypto's K from 30% to 60%? (Hint: regulatory clarity, stability mechanisms, user-friendly interfaces)

Slide 6

Variation 3: CBDC with Higher Ceiling (K=90%)

Original: K=50%, reaches ~45% by 2050
Increased: K=90%, reaches ~80% by 2050

Analysis:

By 2040, K=90% CBDC (at ~70%) overtakes mobile payments (plateauing at 65%)
By 2050, K=90% CBDC dominates all technologies
Interpretation: If governments mandate CBDC for tax payments or social benefits, carrying capacity K could approach 90% (nearly universal adoption)

Counterargument: Privacy concerns and distrust of government surveillance might keep actual K closer to 50-60%, not 90%

Slide 7

Key Insight: Two Parameters, Two Questions

Every technology adoption story answers two questions:

How fast? (controlled by growth rate r)
- Mobile payments: r=0.15 (fastest)
- Credit cards: r=0.08 (slowest)
How far? (controlled by carrying capacity K)
- ATMs/debit: K=85% (highest)
- Cryptocurrencies: K=30% (lowest)

Winning strategy: High r gets you market share quickly (first-mover advantage), but high K determines whether you're still relevant in 20 years.

Testable prediction: By 2030, CBDCs will reach 25-30% adoption in countries that launch pilots before 2025 (China, Sweden, Nigeria). Countries without pilots will stay below 5%.

Slide 8

References & Data Sources

Rogers, E.M. (1962). Diffusion of Innovations. Free Press.
Chart code: Python implementation using logistic growth function
Technology parameters: Estimated from industry reports (World Payments Report 2023, McKinsey FinTech surveys)
Model limitations: Assumes single homogeneous population (reality: different adoption curves for age groups, countries, income levels)

Code availability: Full Python code in chart_varied.py

Next: Gresham's Law →