Changes

'''A note on the derivation of demand'''

Recall that <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

Taking the derivitive with respect to <math>p\,</math>: <center> <math>\frac{d(v(M,p))}{dp} = \frac{dv(M,p)}{dm} \cdot \frac{dM}{dp} + \frac{dv(M,p)}{dp} = 0,\,</math> where <math>\frac{dM}{dp} = \frac{de(p,u)}{dp}\,</math>.

<math>\therefore q(m,p) = \frac{de(p,u)}{dp} = -\frac{\frac{dv(M,p)}{dp}}{\frac{dv(M,p)}{dm}}\,</math>

<math>\therefore q(m,p) = -\frac{d}{dp(v(p))}\,</math> in our case. </center>