Is Manufacturing India's Engine of Growth? Kaldor's First Law, 1961-2025, an Exact Identity Separating Spillover from Arithmetic, and 126 Economies

Kaldor's first law says that GDP grows faster when manufacturing grows faster, and the slope of GDP growth on manufacturing growth is often read as evidence that manufacturing is an engine of growth. Because manufacturing is part of GDP, part of that slope is arithmetic. I prove that, with non-manufacturing output defined as GDP minus manufacturing, GDP growth equals non-manufacturing growth plus the lagged manufacturing share times the growth gap, exactly, so the least-squares slope splits exactly into a spillover slope (non-manufacturing growth on manufacturing growth) and an arithmetic slope; with a constant share the slope equals the share plus one minus the share times the spillover slope. Using World Bank constant-price data for India, 1961 to 2025 (65 annual growth rates), the slope of GDP growth on manufacturing growth is 0.373 (Newey-West standard error 0.071), of which 0.100 is arithmetic and 0.274 is spillover (Newey-West standard error 0.083; the two parts sum to the slope before rounding). The spillover slope fell from 0.362 in 1961 to 1990 to 0.177 in 1991 to 2025. For services the slope is 0.835, with a spillover of 0.659 (0.118). In 126 economies with at least 30 years of data, the median manufacturing spillover slope is 0.265 and is positive in 112. Slopes depend on each sector's volatility, so I also compare correlations: the median correlation of other output with manufacturing (0.475) exceeds that with services (0.362), and manufacturing has the higher correlation in 76 of 126 economies, but not in India (0.346 against 0.485). Kaldor's alternative regression on the growth gap gives negative slopes for India (minus 0.191), which I show follows from the variance of non-manufacturing growth rather than from any causal effect. The evidence identifies comovement, not causation.