Consumer equilibrium -cardinal theory, Macroeconomics

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 px.x units of money income if px 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 390_Consumer Equilibrium3.png and px 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, 1564_Consumer Equilibrium.png.Or, we get ()1821_Consumer Equilibrium2.png 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 1870_Consumer Equilibrium1.png which is ensured by the assumption of falling MUx.  

758_Consumer Equilibrium4.png

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