Appendix A: A Gentle Mathematical Lens

1 Purpose of this appendix

Everything in this book can be understood without calculus.

This appendix exists for readers who want:

a slightly more formal lens,
a bridge to advanced modeling,
or a preview of how these ideas look in equations.

Nothing here is required for the main text.

Think of it as a translation guide between:

verbal system thinking, and
the language used inside Earth–economy models.

2 Stocks and flows in symbols

Throughout the book we used the idea:

Next period’s stock = current stock + inflows − outflows

In symbols:

S_{t+1} = S_t + I_t − O_t

Where:

S_t is the stock at time t,
I_t is the inflow,
O_t is the outflow.

Examples:

Carbon:

C_{t+1} = C_t + Emissions_t − NaturalRemoval_t

Forest:

F_{t+1} = F_t + Growth(F_t) − Harvest_t − Conversion_t

This is the backbone of every Earth–economy model.

3 Growth functions

A renewable resource often follows a curve like:

G(S) = r S (1 − S/K)

You do not need to memorize this.

It simply says:

small stocks grow slowly,
medium stocks grow quickly,
large stocks slow as they approach a limit K.

Graphically, this produces a “hill” shape.

The key insight is qualitative:

Regeneration is not constant.
It depends on the state of the system.

That is why thresholds and tipping points exist.

4 Harvest and decision rules

A simple behavioral rule might be:

H_t = h · S_t

Where:

h is the harvest rate.

Then:

S_{t+1} = S_t + G(S_t) − h S_t

You can see immediately:

if h is too large,
the stock declines,
even if growth is positive.

Policy acts on h.

Institutions decide:

who sets it,
how it is enforced,
and whose interests it reflects.

5 Prices and behavior

In economic models, behavior often takes the form:

Quantity = f(Price, Income, Technology, Rules)

For example:

EnergyUse = a − b·Price + c·Income

This is not a claim about reality.

It is a machine:

change a price,
observe how use responds,
propagate the effect through the system.

Earth–economy models connect many such machines:

production,
consumption,
land allocation,
emissions,
and ecosystem change.

6 Inclusive wealth in symbols

In simplified form:

W = p_K K + p_H H + p_N N

Where:

K = produced capital
H = human capital
N = natural capital
p_* = shadow prices

Sustainability asks:

W_{t+1} / Population_{t+1} ≥ W_t / Population_t

This is the formal version of:

Wealth per person should not decline.

Every integrated model is, in some sense, an engine for computing W_t under different futures.

7 Why math matters—and why it is not enough

Equations:

enforce consistency,
expose assumptions,
and allow simulation.

They do not:

choose values,
define justice,
or resolve conflict.

They answer:

If the world works this way, what follows?

Earth–economy modeling is not about worshiping equations.

It is about using them as:

microscopes for feedback,
stress-tests for futures,
and bridges between disciplines.

The math is a language.

The system is the story.

8 Exercises

Translate.
Take this sentence and rewrite it in stock–flow form:
“Overfishing today reduces tomorrow’s catch.”
Interpret.
If:

C_{t+1} = C_t + E_t − R_t

explain what happens if E_t > R_t for many periods.

Reflect.
Why is it dangerous to use equations without making assumptions visible?

This appendix completes the book’s arc:

from intuition,
to systems,
to models,
to equations—
and back to judgment.

Earth–economy modeling is not about replacing thinking.

It is about giving thinking a scaffold.

This appendix gives mathematically inclined students a bridge into formal modeling without breaking the book’s accessible, systems-first ethos.