Spatial Inequality and the Kuznets Curve

Is there an inverted-U? A synthetic replication in R

inverted-Udevelopment → spatial inequality

$2.1k / $31kturning points

cubic n.s.within countries (panel FE)

Carlos Mendez

Nagoya University (GSID)

June 15, 2026

The Tension

Act I

Why do some countries have huge regional gaps and others almost none?

Kuznets (1955) and Williamson (1965) had an answer: as countries develop, spatial inequality first rises, then falls — an inverted-U.

But the data to test it — regional accounts for poor and rich countries — barely exist. Lessmann (2014) assembled them. We rebuild the exercise on synthetic data so the whole pipeline is open.

The hook: an old, intuitive hypothesis that was very hard to test because of data. We make it teachable by simulating a world calibrated to the paper.

Two pictures of the same question — and they disagree

Cross-section: a line declines, a quadratic bends, a cubic adds a high-income upturn.

Plant the puzzle: the shape you see depends on the functional form. We earn each curve in Act II, then resolve the between-vs-within tension.

Where we’re going

• Measure spatial inequality: the weighted coefficient of variation (WCV)
• Cross-section OLS — the inverted-U and a high-income upturn
• Panel two-way fixed effects with fixest — a clean inverted-U
• Semiparametric checks — Robinson and Baltagi–Li
• The twist: the upturn is between countries, not within them

The Investigation

Act II

We compute inequality, not assume it

\[\mathrm{WCV} = \frac{1}{\bar{y}}\left[\sum_{j} p_j\,(\bar{y}-y_j)^2\right]^{1/2}\]

Population-weighted spread of regional GDP per capita. A populous poor region counts; a tiny rich enclave barely moves it.

We simulate regions for 56 countries, 1980–2009, and compute the WCV from them — 890 annual observations.

Spatial inequality is related to personal inequality — but not the same

GINI = 0.31 + 0.21·WCV (t = 2.5, r = 0.32).

Motivates why we study the spatial dimension on its own: it explains a third of personal inequality, not all of it.

Cross-section: the inverted-U emerges with controls, and a cubic adds the upturn

Five specifications: bivariate → quadratic → +controls (inverted-U) → +cubic (N-shape).

Column 4: +0.34* / −0.02** — the inverted-U. Column 5: cubic significant — the N-shape upturn.

Where does the curve turn? Set the derivative to zero

∂WCV/∂ln(GDP) = β₁ + 2β₂Y + 3β₃Y² = 0 → peak ≈ $2,100, trough ≈ $31,000.

Three phases: rise during take-off, fall through convergence, rise again at the post-industrial top.

Significant ≠ a genuine bend — check the discriminant

All three cubic terms can be significant and the curve still not bend in range. The test is the discriminant \(D=\beta_2^2-3\beta_1\beta_3\):

• \(D>0\) → two turning points · \(D=0\) → inflection only · \(D<0\) → monotonic
• Cross-section: \(D=+0.006>0\), both turning points in range → genuine N-shape
• Panel cubic: insignificant, and a turning point falls far outside the data → no within-country bend
• Always also check the turning points lie inside the observed income range

The discriminant decides the shape

Same significant terms, three shapes — only the discriminant tells them apart.

Same sign pattern, vary only the squared term: D<0 monotonic, D=0 inflection, D>0 genuine N. In BMA the same trap reads “high PIP ≠ a genuine bend.” See the companion note (Mendez 2026).

Fixed effects change the story, not just the standard errors

feols(wcv ~ lnGDP + I(lnGDP^2) | country + year, vcov = "hetero")

• Within-country quadratic: +0.39** / −0.021** — a clean inverted-U
• Cubic term: insignificant — no within-country upturn
• The upturn was a between-country artefact all along

The within-country inverted-U

Fitted WCV from the two-way FE quadratic, peaking near $18,000.

Semiparametric, no polynomial assumed — same shape

Robinson (1988) partial fit with a 90% band: inverted-U with a high-income upturn.

Triangulation: a kernel-based estimator that never sees the word “cubic” recovers the same curve. Baltagi–Li (B-spline FE) agrees within countries.

The Resolution

Act III

Structural change is the mechanism

Replace income with the non-agricultural share of output — the inverted-U returns.

Development raises inequality because it reshuffles where output is made; as the modern sector spreads, gaps close.

The high-income upturn is real but fragile

It appears in income levels, vanishes in logs (and within countries).

What we learned

• Inverted-U confirmed across OLS, fixest TWFE, and two semiparametric estimators
• Turning points at ~$2,100 and ~$31,000 of GDP per capita
• The upturn is between-country, not within-country — fixed effects reveal it
• Wide regional gaps are largely a transitional feature of development

Thank you

Full tutorial, code, data and web app: carlos-mendez.org/post/r_kuznets

Reference: Lessmann, C. (2014). Spatial inequality and development — Is there an inverted-U relationship? Journal of Development Economics 106, 35–51. Synthetic data; pedagogical replication.