6 Equation (2) is the fundamental identity in the “poverty‐ growth‐inequality triangle”. For a given poverty line _z_ , it relates the incidence of poverty to the mean of the distribution (inversely), and to the parameters of the Lorenz curve. The Lorenz curve is deeply associated with the concept of inequality.<sup>7</sup> Given a constant real poverty line _z_ , the poverty headcount is therefore fully determined by the “shape” of the distribution (given by the Lorenz curve) and its central location (given by the mean). Analogous, if somewhat more complicated, identities relate other poverty measures to the distribution’s mean and Lorenz curve.<sup>8</sup> It is straightforward to move from the fundamental identity of the triangle in levels, to its dynamic counterpart. One simply differentiates equation (2) with respect to time. Denoting the time derivative of variable _x_ as _dx_ , one obtains: Equation (3) relates changes in poverty to the rate of economic growth ( ���⁄ ), and to changes in “inequality” or, more precisely, in the Lorenz curve ( �� ).<sup>9</sup> It tells us that any change in poverty can be mechanically decomposed into a _growth component_ (the first term on the RHS), and a _distribution component_ (the second term). Since the Lorenz curve is always increasing and convex by construction (so that �� �0, ��� �0 ), equation (3) also tells us that the partial elasticity of poverty with respect to � ������ growth ( ), is always negative: if there is no change in the Lorenz curve, an increase in mean �<sup>��</sup> � � incomes must be accompanied by a decline in poverty. In other words: growth must lead to poverty reduction, unless it is accompanied by substantial distributional change (which may be fairly, albeit somewhat imprecisely, described as “increasing inequality”).

The partial inequality elasticity ( ����� ~~)~~ , on the other hand, cannot be signed a priori: changes in the �<sup>��</sup> � distribution may increase or reduce poverty, depending on how the Lorenz curve shifts, and on the relative position of the poverty line.

> 7 The Atkinson (1970) theorem establishes that, given two distributions A and B such that �����������, �� , with the inequality holding strictly at least at one point, then any inequality index that satisfies the Pigou‐Dalton transfer principle must take a higher value in distribution A than in B. The (strong) Pigou‐Dalton transfer principle states that an inequality measure must rise in response to a transfer of income from a poorer to a richer person. It is regarded as a key axiom for inequality indices.

> 8 The most widely‐used family of poverty measures is the Foster‐Greer‐Thorbecke class, �� ����� ����� ~~�~~ � ������ . The headcount is the FGT measure when ��0 . Other commonly‐used members of the class take values of ��1,2 .

> 9 We are deliberately glossing over the empirically important question of whether one should measure economic growth as the growth in mean household incomes (from a household survey), or by growth in some concept obtained from national accounts data, such as GDP per capita. While the two should not differ systematically over long periods of time, they have often been found to diverge considerably in the short and medium runs.