Stocks & Flows

POLS 3220: How to Predict the Future

P0 = transpose(pop)
tfr0 = d3.sum(P0, d => d.fert)            // today's total fertility rate (births per woman)

// births from today's women, then deaths, then everyone who survives gets a year older
step = (st, par) => {
  const n = st.m.length, m = new Array(n), f = new Array(n);
  const births = d3.sum(st.f, (x, i) => x * par.fert[i]);
  m[0] = births / 2; f[0] = births / 2;
  for (let i = 1; i < n; i++) {
    m[i] = st.m[i - 1] * (1 - par.qm[i - 1]);
    f[i] = st.f[i - 1] * (1 - par.qf[i - 1]);
  }
  m[n - 1] += st.m[n - 1] * (1 - par.qm[n - 1]);   // the 100+ group is open-ended
  f[n - 1] += st.f[n - 1] * (1 - par.qf[n - 1]);
  return {m, f};
}

makePar = (tfr = tfr0, dmult = 1) => ({
  fert: P0.map(d => d.fert * tfr / tfr0),
  qm: P0.map(d => Math.min(1, d.qm * dmult)),
  qf: P0.map(d => Math.min(1, d.qf * dmult))
})

today = ({m: P0.map(d => d.m), f: P0.map(d => d.f)})

// summary statistics for a population
describe = (st) => {
  const tot = st.m.map((x, i) => x + st.f[i]);
  const N = d3.sum(tot), share = tot.map(x => x / N);
  let cum = 0, median = 0;
  for (let i = 0; i < share.length; i++) {
    if (cum + share[i] >= 0.5) { median = i + (0.5 - cum) / share[i]; break; }
    cum += share[i];
  }
  const sumAges = (lo, hi) => d3.sum(tot.slice(lo, hi + 1));
  return {N, share, median, old: sumAges(65, 100) / N, workers: sumAges(18, 64) / sumAges(65, 100)};
}

// life expectancy at birth implied by the death rates (average of men & women)
lifeExp = (par) => {
  const e0 = (q) => {
    let l = 1, e = 0;
    for (let i = 0; i < q.length - 1; i++) { const next = l * (1 - q[i]); e += (l + next) / 2; l = next; }
    return e + l / q[q.length - 1];
  };
  return (e0(par.qm) + e0(par.qf)) / 2;
}

ageData = (share) => share.map((s, i) => ({age: i, share: s}))
todayStats = describe(today)
todayE0 = lifeExp(makePar())

Recap

• So far this semester, we’ve covered:

  • Probability (Part I). How to handle uncertainty.

  • Wisdom of Crowds (Part II). The value of thinking from multiple perspectives.

  • Election Forecasting (Part III). Using historical patterns in data to forecast future outcomes.

• To date, we’ve mainly focused on short- and medium-term predictions.

  • This was a practical choice. We needed forecasts we could score!

• But many of the most important prediction problems have longer time horizons.

• Here are a few trends where I deeply want to know what’s going to happen over the next few decades:

Artificial Intelligence

Democracy

The National Debt

Partisan Polarization

Global Surface Temperature

Agenda

• In the final few days of class, I want to give you some tools for reasoning about these sorts of long-term trends.

• Today: thinking in terms of stocks & flows.

Stocks & Flows

• A stock measures some quantity at a point in time.

• A flow measures the rate of change in a quantity.

• A useful metaphor is a bathtub (Sterman 2000).

Exercise

Predicting stocks and flows can be tricky! Sketch what you think will happen to the stock of water in these bathtubs over time. Both start at 100 liters.

Stocks & Flows

Lots of problems can be described in terms of stocks and flows:

Stock	Inflow	Outflow
$ in my bank account	Monthly income	Monthly spending
National Debt	Taxes	Government Spending
Number of Democracies	Democratization	“Democratic Backsliding”
CO2 in Atmosphere	Carbon Emissions	Carbon Sinks
Population	Births	Deaths

• Confusing stocks and flows is a common mistake.

• For example: debt is a stock, while deficit is a flow.

  • “Cutting the deficit” does not necessarily mean that the national debt will decrease!

Markov Models

• When flows are defined in terms of probabilities, we call it a Markov Model.
• For example, suppose that every autocracy has a 4% chance of democratizing each year.
• And every democracy has a 1% chance of becoming an autocracy…
• If this process continued for 1,000 years, what do you think would happen to the share of countries that are democracies?

Flows Over Time

Let’s start with a world of 250 autocracies and 0 democracies. Year 0:

Net change: +10 democracies per year

Flows Over Time

As democracies accumulate, the inflow shrinks and the outflow grows. Year 10:

Net change: +6 democracies per year

Flows Over Time

As democracies accumulate, the inflow shrinks and the outflow grows. Year 25:

Net change: +2.75 democracies per year

Flows Over Time

As democracies accumulate, the inflow shrinks and the outflow grows. Year 50:

Net change: +0.75 democracies per year

Flows Over Time

As democracies accumulate, the inflow shrinks and the outflow grows. Year 100:

Net change: +0.05 democracies per year

Equilibrium

• Eventually, the system settles into a stable point, called an equilibrium.

• In equilibrium, inflows = outflows. Nothing changes from year to year.

• Intuitively, if the rate of democratization (4% per year) is 4x the rate of backsliding (1% per year), then in equilibrium there were be 4x more democracies than autocracies.

• 4:1 = 80%

Does the Starting Point Matter?

• What if we started with 100% democracies? Or a 50/50 split?

• Doesn’t matter! Regardless of starting point, the system converges to the same equilibrium: 80% democracies.

Practice

Every month, 2% of employed workers lose their jobs, and 30% of unemployed workers find a new job.

1. In equilibrium, what percent of workers are unemployed?

2. A recession hits, and the job-finding rate falls to 18% per month. What is the new equilibrium unemployment rate?

Ergodic Theorem

• When should we expect a system of stocks & flows to reach an equilibrium?

• The ergodic theorem states that there will be a unique equilibrium for any system that meets the following conditions:

  • There exists a path from every stock to every other stock.

  • Flow probabilities are stationary (i.e. unchanging).

• No matter how complicated the model gets, if these two conditions hold, there will be a unique equilibrium where all the inflows equal outflows.

Demographic Forecasts

• Here’s a much more complicated stock and flow model, which we can use to think about long-term demographic change.

Demographic Forecasts

Below are the age-specific mortality and fertility rates for Americans, according to the SSA and CDC.

Demographic Forecasts

• If the number of people in each age group is the stock, then those rates represent the flows between age groups.

Demographic Forecasts

If the current fertility and mortality rates remain stationary, the age distribution will eventually settle into an equilibrium (Arthur 1982).

// a play button + year slider
viewof year = {
  const max = 200;
  const form = html`<form style="display:flex; align-items:center; gap:14px; font-size:20px; margin:0 auto; width:860px">
    <button name="b" type="button" style="width:110px; font-size:18px; padding:4px">▶ Play</button>
    <input name="i" type="range" min="0" max="${max}" step="1" value="0" style="flex:1">
    <output name="o" style="width:170px"></output></form>`;
  let timer = null;
  const show = () => {
    form.value = form.i.valueAsNumber;
    form.o.value = form.value === 0 ? "Today" : `In ${form.value} years`;
  };
  const stop = () => { clearInterval(timer); timer = null; form.b.textContent = "▶ Play"; };
  form.i.oninput = () => { stop(); show(); };
  form.b.onclick = () => {
    if (timer) return stop();
    if (form.i.valueAsNumber >= max) form.i.valueAsNumber = 0;
    form.b.textContent = "❚❚ Pause";
    timer = setInterval(() => {
      form.i.valueAsNumber += 1; show();
      form.dispatchEvent(new CustomEvent("input"));
      if (form.i.valueAsNumber >= max) stop();
    }, 70);
  };
  invalidation.then(stop);
  show();
  return form;
}

// project today's population forward under today's rates, one year at a time
projection = {
  const par = makePar();
  const out = [describe(today)];
  let st = today;
  for (let t = 1; t <= 200; t++) { st = step(st, par); out.push(describe(st)); }
  return out;
}

{
  const s = projection[year];
  return html`<div>${Plot.plot({
    width: 900, height: 400, style: {fontSize: "15px"}, marginBottom: 45,
    x: {label: "Age", labelAnchor: "center", labelOffset: 38, domain: [0, 101],
        ticks: d3.range(0, 101, 10), tickFormat: d => d === 100 ? "100+" : d},
    y: {label: null, tickFormat: d3.format(".1%"), domain: [0, 0.015], grid: true},
    marks: [
      Plot.rectY(ageData(s.share), {x1: "age", x2: d => d.age + 0.9, y: "share", fill: "#4a7ab0"}),
      Plot.line(ageData(todayStats.share), {x: d => d.age + 0.45, y: "share", stroke: "#222",
        strokeWidth: 1.5, strokeDasharray: year > 0 ? "4,3" : "none", strokeOpacity: year > 0 ? 0.6 : 0}),
      Plot.ruleX([s.median], {stroke: "#d1495b", strokeWidth: 2}),
      Plot.text([s.median], {x: d => d, y: 0.0145, text: d => `median age ${d.toFixed(1)}`,
        dx: 6, textAnchor: "start", fill: "#d1495b", fontSize: 16, fontWeight: "bold"}),
      Plot.ruleY([0])
    ]
  })}
  <div style="text-align:center; font-size:18px; color:#555">
    Dashed line = today's age distribution · Population: <b>${(s.N / 1e6).toFixed(0)} million</b>
    · Share aged 65+: <b>${(s.old * 100).toFixed(1)}%</b></div></div>`;
}

Demographic Forecasts

• This has implications for:

  • the cost of social insurance programs

  • military recruitment

  • higher education

  • the national debt

  • interest rates

  • the “gerontocracy”

• And the model lets us test out different scenarios. What happens to the equilibrium if fertility keeps falling? If medical advances help Americans live longer? If more immigrants arrive each year?

Try It Yourself

viewof tfr = Inputs.range([0.8, 3.0], {step: 0.01, value: Math.round(tfr0 * 100) / 100,
  label: "Fertility (births per woman)"})
viewof dmult = Inputs.range([0.25, 2], {step: 0.05, value: 1, label: "Death rates (× today's)"})

html`<table class="stat-box">
  <tr><td></td><td class="muted">equilibrium</td><td class="muted">today</td></tr>
  <tr><td>Life expectancy</td><td><b>${eqm.e0.toFixed(1)}</b></td><td class="muted">${todayE0.toFixed(1)}</td></tr>
  <tr><td>Median age</td><td><b>${eqm.stats.median.toFixed(1)}</b></td><td class="muted">${todayStats.median.toFixed(1)}</td></tr>
  <tr><td>Aged 65+</td><td><b>${(eqm.stats.old * 100).toFixed(1)}%</b></td><td class="muted">${(todayStats.old * 100).toFixed(1)}%</td></tr>
  <tr><td>Workers / retiree</td><td><b>${eqm.stats.workers.toFixed(1)}</b></td><td class="muted">${todayStats.workers.toFixed(1)}</td></tr>
</table>`

// the equilibrium: run the model forward for a long, long time
eqm = {
  const par = makePar(tfr, dmult);
  let st = today;
  for (let t = 1; t <= 1000; t++) {
    st = step(st, par);
    const N = d3.sum(st.m) + d3.sum(st.f);          // rescale each year: only the shares matter
    st = {m: st.m.map(x => x / N), f: st.f.map(x => x / N)};
  }
  return {stats: describe(st), e0: lifeExp(par)};
}

Plot.plot({
  width: 640, height: 470, style: {fontSize: "14px"}, marginBottom: 45,
  x: {label: "Age", labelAnchor: "center", labelOffset: 38, domain: [0, 101],
        ticks: d3.range(0, 101, 10), tickFormat: d => d === 100 ? "100+" : d},
  y: {label: null, tickFormat: d3.format(".1%"), grid: true,
      domain: [0, Math.max(0.015, d3.max(eqm.stats.share) * 1.08)]},
  marks: [
    Plot.rectY(ageData(eqm.stats.share), {x1: "age", x2: d => d.age + 0.9, y: "share", fill: "#4a7ab0"}),
    Plot.line(ageData(todayStats.share), {x: d => d.age + 0.45, y: "share", stroke: "#222",
      strokeWidth: 1.5, strokeDasharray: "4,3", strokeOpacity: 0.7}),
    Plot.ruleX([eqm.stats.median], {stroke: "#d1495b", strokeWidth: 2}),
    Plot.text([eqm.stats.median], {x: d => d, frameAnchor: "top", text: d => `median age ${d.toFixed(1)}`,
      dx: 6, dy: 4, textAnchor: "start", fill: "#d1495b", fontSize: 15, fontWeight: "bold"}),
    Plot.text(["dashed line = today"], {frameAnchor: "top-right", fill: "#555", fontSize: 13, dy: 4}),
    Plot.ruleY([0])
  ]
})

Takeaways

• Keep in mind the distinction between stocks (the level) and flows (the rate of change).

• A system of stocks and flows is ergodic if the flow probabilities remain fixed and there’s a path from every stock to every other stock.

  • Ergodic systems always converge to an equilibrium state.

• Next time, we’ll discuss positive and negative feedback.

  • i.e. What happens when the size of a stock affects the rate of inflow/outflow?

• We’ll conclude the semester with lectures on tipping points and path dependence.

References

Arthur, W. Brian. 1982. “The Ergodic Theorems of Demography: A Simple Proof.” Demography 19 (4): 439–45. https://doi.org/10.2307/2061011.

Sterman, John D. 2000. Business Dynamics: Systems Thinking and Modeling for a Complex World. Boston: Irwin/McGraw-Hill.