Stocks Flows

264. Stocks Flows

The foundation of system dynamics modeling. Two element types:

Stocks (or levels) — accumulations: inventory, population, money
Flows (or rates) — rates of change: orders, births, spending

Stocks change only through flows. Flows change instantaneously based on system state.

264.1. Mathematical foundation

For a stock $𝑆 (𝑡)$ with inflow rate $𝑓_{in} (𝑡)$ and outflow rate $𝑓_{out} (𝑡)$ :

\frac{𝑑 𝑆}{𝑑 𝑡} = 𝑓_{in} (𝑡) - 𝑓_{out} (𝑡)

In integral form:

𝑆 (𝑡) = 𝑆 (𝑡_{0}) + \int_{𝑡_{0}}^{𝑡} [𝑓_{in} (𝜏) - 𝑓_{out} (𝜏)] 𝑑 𝜏

Each stock obeys this conservation law: change in stock = net flow.

264.2. Units

Dimensional consistency is mandatory:

[stock] = [flow] \cdot [time]

Inventory in units; production rate in units / day; integration over time recovers units.
Money in dollars; spending in dollars / month.
Population in people; birth rate in people / year.

A model with mismatched units is wrong; checking units catches half of all SD modeling errors.

264.3. Three classes of stock-flow systems

First-order linear:

\cdot 𝑆 = - 𝑘 𝑆 + 𝑓_{in}

Equivalently: stock decays toward $\frac{𝑓_{in}}{𝑘}$ with time constant $\frac{1}{𝑘}$ . Examples: exponential decay (radioactive substance), goal-seeking (workforce hiring toward target).

Second-order linear: two stocks, coupled

\begin{matrix} \cdot 𝑆_{1} = 𝑎 𝑆_{1} + 𝑏 𝑆_{2} \\ \cdot 𝑆_{2} = 𝑐 𝑆_{1} + 𝑑 𝑆_{2} \end{matrix}

Eigenvalues determine behavior: stable / unstable spirals, exponential growth/decay, oscillation. Predator-prey, mass-spring, etc.

Nonlinear: $\cdot 𝑆 = 𝑓 (𝑆)$ with $𝑓$ nonlinear — limit cycles, chaos, bifurcations. Most real-world systems.

264.4. Bathtub metaphor

A common teaching tool: a bathtub with a tap (inflow) and drain (outflow). Water level = stock. Tap and drain rates = flows.

Even simple bathtubs are non-intuitive: most people overestimate how fast filling a bathtub increases water level, because they fail to integrate the flow over time.

264.5. Stocks have memory; flows don’t

If you turn off the tap (set inflow to zero), the bathtub doesn’t suddenly empty — water remains. Same with inventory after halting production, or accumulated CO2 after stopping emissions. Stocks persist.

Conversely, a flow can change instantaneously: stop the inflow → inflow rate is zero immediately.

This distinction is fundamental: policy interventions on flows have delayed effects on stocks.

264.6. Connection to differential equations

Stocks and flows = the language of ordinary differential equations (ODEs). SD diagrams are just an alternative notation for systems of ODEs. The visual language helps:

Identify state variables (= stocks)
Track causal pathways (= flow equations)
Communicate model structure to non-modelers

264.7. See also

Causal Loop Diagrams — qualitative version
Feedback Loops
Numerical Integration — for simulation
Differential Equations