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<math>M=e(p,u)\,</math>, so that <math>v(e(p,u),p)=u\,</math> when the expenditure function is evaluated at <math>p\,</math> and <math>u\,</math>.
<math>d/dp(v(M,p)) = \frac{dv(M,p)/}{dm } \cdot dM/dp + dv/dp = 0,\,</math> where <math>dM/dp = de(p,u)/dp\,</math>.
<math>\therefore q(m,p) = de(p,u)/dp = -\frac{dv/dp}{dv(M,p)/dm}\,</math>
<math>\therefore q(m,p) = -d/dp(v(p))\,</math>\\
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