1. The basic idea

What does it mean for an economy to be “complex”? Plenty of intuitive proxies — GDP per capita, value-added per worker, R&D intensity — work fine for the obvious cases (Switzerland complex, Chad not), but they don’t generalise well: a high-income oil exporter has lots of value-added per worker but is doing exactly one thing.

Hidalgo & Hausmann’s move is to look at the structure of what a country exports, not the headline numbers. The intuition has two pieces:

• Diversity. A complex economy can make many different things; a simple one can make few.
• Ubiquity. Some products can be made by almost anyone (clothing, basic agricultural goods). Others require knowledge that very few countries have (aircraft, lithography equipment). A complex economy is one that does the rare things.

These two ideas pull in opposite directions when used naively. Many countries have high diversity of common products; very few countries have low diversity of rare ones. The Method of Reflections is the iteration that resolves this tension into a single ranking.

2. The matrix and the iteration

Start with bilateral trade data: country c exports value of product in some reference year. Define Revealed Comparative Advantage (Balassa 1965) as:

In words: is country c’s share of product p’s world exports, divided by country c’s share of all world trade. If country c specialises in product p more than its overall trade footprint would predict, .

Binarise to get the specialisation matrix:

This is the only place magnitude is collapsed into a yes/no. From we get diversity and ubiquity directly as row and column sums:

The Method of Reflections refines these by iteratively averaging each side against the other:

The semantics shift at every step. At , is “the average ubiquity of my products” — high if I export common things, low if I export rare ones. At , is “the average diversity of the countries that make my products” — high if my products are made by economies that themselves make many things. Continue this and the iteration converges; the Economic Complexity Index is the converged (z-scored), the Product Complexity Index is the converged (z-scored).

3. From iteration to eigenvector

The iteration has a closed form. Stack the country values into the country-side similarity matrix:

where and are diagonal matrices of diversity and ubiquity. Then , so iterating the Method of Reflections is the same as repeatedly multiplying by .

is row-stochastic (its rows sum to 1, by construction), so its largest eigenvalue is exactly 1 and its largest eigenvector is the constant vector. The second eigenvector carries the only useful structure — the relative positions of countries within the second-largest eigendirection. That eigenvector, standardised to mean zero and unit variance, is the Economic Complexity Index.

Symmetrically, gives PCI as its second eigenvector.

4. Implementing it

Forty lines of pandas, give or take. Load the Atlas HS92 country × HS2 × year trade data, compute RCA, binarise, build and , take their second eigenvectors:

# Illustrative implementation — RCA, Mcp, and the eigendecomposition.
import numpy as np, pandas as pd

df = pd.read_csv("hs92_country_product_year_2.csv",
                 dtype={"product_hs92_code": str})

def compute_mcp_for_year(df_year, min_country_total=0.0):
    # Country filter (drop tiny exporters)
    country_totals = df_year.groupby("country_iso3_code")["export_value"].sum()
    qualifying = country_totals[country_totals >= min_country_total].index
    df_year = df_year[df_year["country_iso3_code"].isin(qualifying)]

    # Balassa RCA
    c_tot = df_year.groupby("country_iso3_code")["export_value"].sum()
    p_tot = df_year.groupby("product_hs92_code")["export_value"].sum()
    w_tot = df_year["export_value"].sum()

    d = df_year.copy()
    d["c_tot"] = d["country_iso3_code"].map(c_tot)
    d["p_tot"] = d["product_hs92_code"].map(p_tot)
    d["rca"]   = (d["export_value"] / d["c_tot"]) / (d["p_tot"] / w_tot)
    d["mcp"]   = (d["rca"] >= 1).astype(int)

    M = d.pivot(index="country_iso3_code", columns="product_hs92_code",
                values="mcp").fillna(0).astype(int)
    d_c = M.sum(axis=1); u_p = M.sum(axis=0)
    # Drop countries with no qualifying products
    M = M.loc[d_c[d_c > 0].index, u_p[u_p > 0].index]
    return M, M.sum(axis=1), M.sum(axis=0)

def eci_pci(M, d_c, u_p):
    M_np = M.values.astype(float)
    W_c = (1.0/d_c.values[:, None] * M_np) @ (1.0/u_p.values[:, None] * M_np.T)
    W_p = (1.0/u_p.values[:, None] * M_np.T) @ (1.0/d_c.values[:, None] * M_np)

    # Second-largest eigenvector of each
    ev_c, vec_c = np.linalg.eig(W_c)
    ev_p, vec_p = np.linalg.eig(W_p)
    eci_raw = vec_c[:, np.argsort(-ev_c.real)[1]].real
    pci_raw = vec_p[:, np.argsort(-ev_p.real)[1]].real

    # Standardise
    eci = (eci_raw - eci_raw.mean()) / eci_raw.std()
    pci = (pci_raw - pci_raw.mean()) / pci_raw.std()
    return pd.Series(eci, index=M.index), pd.Series(pci, index=M.columns)

Validation: the Atlas of Economic Complexity ships its own ECI per country-year and PCI per product-year. Both should match a clean implementation up to a sign flip (eigenvectors are defined up to sign) and a standardisation convention. Run this across 30 years (1995–2024) and we get:

A few things to notice. First, no implementation of the linear Method of Reflections at HS2 resolution will match Atlas to numerical precision, because Atlas’s published ECI is computed at HS4 (~1240 product categories) and uses a non-linear refinement (Tacchella et al. 2012) — more on that in section 7. A ~0.72 correlation on the linear, HS2 version is what the paper-canonical algorithm produces. Second, the country filter helps a lot in recent years and modestly in older ones; we’ll see why in section 6.

For context, the top 15 countries by ECI in 2022 under the country-filtered regime are:

The qualitative agreement with Atlas is very strong — Taiwan, Japan, Korea, Germany, Switzerland are the canonical “complex economies” of the trade literature, and they’re at the top under both implementations.

5. China × HS85: tracing the iteration

The empirical headline of the original paper was that high-ECI economies grow faster than their income would predict. China is the most-cited example: ECI rising sharply 1995–2010, GDP per capita following with a lag. Let me trace that rise mechanistically using the iteration values, picking one specific product to track alongside.

China’s biggest manufactured export in 2024 is HS85 — electrical machinery and electronics (27.9% of world exports of HS85 originated in China). Tracking and across iterations and years, here’s what happens to the key inputs:

Three observations:

1. China’s raw diversity barely moves. 46 sectors in 1995, 47 in 2024. The number of HS2 chapters China specialises in is essentially flat over the entire period.
2. HS85 ubiquity rises: 15 countries in 1995 → 24 in 2024. More countries entered electronics over the period.
3. Mcp[CHN, HS85] flipped from 0 to 1 between 1995 and 2000. China didn’t have RCA in electrical machinery in 1995; it did from 2000.

If you stopped at the raw counts, the obvious story would be “China stayed at the same complexity; HS85 got less exclusive”. Both wrong. Look at iteration specifically:

The iteration alternates between “averaging the ubiquity of my products” (odd , row) and “averaging the diversity of my products’ countries” (even ). In 1995, China’s value is z-score — its products are slightly below the global mean of ubiquity. By 2024, is — China’s products are much rarer than average. That single number is the whole story.

Concretely: in 1995, China specialised in textiles, apparel, and basic manufactured goods — the kind of products that 30+ other countries also export with RCA. By 2024, China’s basket still includes those, but also HS84 (machinery), HS85 (electronics), HS86 (railway equipment), HS90 (instruments). These are products only 15–25 countries specialise in. The composition shift — not the count — is what drives the iteration deeper into the “low ubiquity” direction.

By iteration , this has compounded through the recursive structure (“rare products of countries that themselves make rare products”) and China’s complexity z-score has climbed from to , with a peak of around 2020.

The full iteration trajectory across years:

And the resulting ECI trajectory for China, against Atlas’s shipped values:

The vanilla and country-filtered curves diverge in the 2005–2015 window — that’s the period where the vanilla algorithm starts amplifying micro-state noise (next section).

6. Where it bends and where it breaks

The two failure modes the original paper does not address. Both are intrinsic to the Balassa-RCA-plus-binarisation construction.

7. What works better, and where the field went

The micro-state fix from section 6.1 (country-level threshold) is necessary and sufficient for clean rankings on the linear Method of Reflections. The hidden-giant issue is not fixable inside the Balassa + binarisation paradigm — it’s a property of the construction. The follow-up literature has gone in two directions.

Non-linear iterations. Tacchella, Cristelli, Caldarelli, Gabrielli & Pietronero (2012, Scientific Reports) propose a non-linear update that dampens the micro-state amplification algorithmically rather than via a pre-filter. Their “fitness” measure also handles some hidden-giant cases more gracefully because a country’s fitness depends on the complexity of its products in a multiplicative way that doesn’t binarise on RCA in quite the same way.

Higher resolution. The Harvard Atlas computes its canonical ECI at HS4 (~1240 products) rather than HS2 (~97 chapters). At HS4, the hidden-giant problem shrinks: HS85 contains sub-codes like HS8542 (integrated circuits), HS8517 (telephones), HS8528 (monitors) — each its own product. China’s share of HS8542 is much higher than 15.46%, so RCA clears comfortably and Mcp = 1. The micro-state problem also softens at higher resolution, because random products at HS4 are far less likely to hit RCA ≥ 1 spuriously than at HS2.

In other words: the original paper’s algorithm works best on fine product resolution, after country-level filtering, possibly with a non-linear refinement. Each of these three choices is its own research thread.

What I built and what I’d build next

For a reader interested in actually running this — the full pipeline is ~150 lines: data download from Harvard Dataverse, RCA + Mcp + eigendecomposition, validation against Atlas, one figure of China’s iteration trajectory. Total compute is a few seconds on a laptop for the full 30-year × 232-country × 97-product matrix at HS2.

The natural next steps from here, in increasing difficulty:

1. Repeat at HS4 to see whether China’s ECI rise looks the same, and to recover the hidden giants. The Atlas ships an HS4 file; the same code reads it after changing one filename.
2. Implement the Tacchella et al. (2012) non-linear iteration and compare its rankings with both vanilla MoR and Atlas.
3. Replace Balassa with an alternative RCA that doesn’t scale with country size — Hoen & Oosterhaven (2006) propose an additive version. Compare ECI under each.
4. Apply the framework to a different question entirely — for example, ranking cities within a country by complexity of their patent applications. The method generalises to any bipartite specialisation matrix.

The deepest takeaway from running this end-to-end is that the Method of Reflections is simple, robust, and computationally trivial — but it requires two non-obvious pre-processing choices to give defensible answers (country filter, fine resolution). The original paper does not flag either of these, and the failure modes you observe without them are not subtle.

Sources

• Hidalgo, C. A., & Hausmann, R. (2009). The building blocks of economic complexity. PNAS, 106(26), 10570–10575. DOI
• Balassa, B. (1965). Trade liberalisation and “revealed” comparative advantage. The Manchester School, 33(2), 99–123.
• Tacchella, A., Cristelli, M., Caldarelli, G., Gabrielli, A., & Pietronero, L. (2012). A new metrics for countries’ fitness and products’ complexity. Scientific Reports, 2, 723.
• Hoen, A. R., & Oosterhaven, J. (2006). On the measurement of comparative advantage. The Annals of Regional Science, 40(3), 677–691.
• Harvard Growth Lab. Atlas of Economic Complexity — International Trade Data (HS, 92). Harvard Dataverse, doi:10.7910/DVN/T4CHWJ, v18.0.

1. The headline contradiction

The single chart that frames everything: for fiscal 2025, each segment’s share of revenue sits next to its share of operating profit.

2. Where the money goes

The cleanest way to show a company’s cost structure is a waterfall from net sales down to operating income — each step is a category of spend that Amazon discloses on the face of its income statement.

A couple of things jump out. Cost of sales (≈50% of revenue) is the physical reality of being a retailer — the goods themselves and shipping. But notice technology & infrastructure: at ~$109B it has nearly caught up with fulfillment (~$109B) and is the fastest-growing line, reflecting the capex and depreciation behind AWS and the current AI build-out.

3. The trend: how the mix shifted, 2021–2025

A single year is a snapshot. The interesting story is the trajectory.

Revenue growth is broad-based — every segment is bigger. But revenue is the boring part. Operating income is where the real regime change lives:

To make the margin story explicit, here are segment operating margins — operating income as a percentage of that segment’s own sales:

AWS runs at a ~35% operating margin — software economics. North America has clawed its way to ~7%, and International to ~3%. These are structurally different businesses stapled together.

4. What the market did with all this

The daily chart is noise on top of signal, though. To see what actually drives the price over years rather than days, the right lens is per-share metrics — because an investor doesn’t own “Amazon,” they own a share of it.

5. Do the profits actually turn into cash?

There’s an old auditor’s saying: “Profit is an opinion; cash is a fact.” Everything above — revenue, operating income, margins — comes from accrual accounting, which records sales when they’re earned and costs when they’re incurred, not when money actually changes hands. That’s the right way to measure economic activity, but it opens a gap between the profit a company reports and the cash it collects. A healthy skeptic always asks: do the earnings show up as real money?

There are three distinct things people worry about behind “great results on paper, but where’s the cash?” — and each has a different diagnostic:

The way to settle the question is to walk from net income down to free cash flow — the cash left after running and maintaining and expanding the business. Here is that bridge for Amazon in FY2025.

Read the bridge left to right and the answer falls out:

• The operating cash is real — emphatically so. Operating cash flow ($139.5B) is nearly double reported net income ($77.7B). That is the opposite of the fictional-revenue signature, where cash lags profit. Most of the gap is depreciation & amortization ($65.8B): a genuine expense on the income statement, but one where no cash leaves the building this year — so it’s added back. Stock-based comp ($19.5B) works the same way (it dilutes shareholders rather than spending cash).
• “Delayed payments” actually works in Amazon’s favour. Amazon runs on negative working capital: customers pay instantly (card on file), while Amazon pays its suppliers weeks later. So its own slow payments are a source of cash — suppliers and customers effectively finance the business. The dangerous version (Amazon’s own customers stalling, receivables ballooning faster than sales) simply isn’t visible in the numbers.

So the popular “they have no cash!” worry, in the fraud sense, does not apply to Amazon. But there is a real catch — it’s just a different one: