Consumer Equilibrium:

According to our assumption for 'x' units consumption of the commodity, gross utility obtained by the consumer is U(x).But for this, the consumer must spend p_x.x units of money income if p_x be the price of the commodity 'x', which is given to the consumer. Since from assumption 6, λ represents fall in utility due to one unit fall in money income, the net utility of the consumer is given by  and p_x are given to the consumer. So consumer's objective is to maximise N(x) by choosing 'x'. For that we take the first derivative of N(x) and set that equal to zero, .Or, we get () From this first order condition, we can derive the optimum value of 'x' which is (say) x* = x*(px,λ). The second order condition for utility Maximisation requires  which is ensured by the assumption of falling MU_x.