Title Linear expenditure system (1).docx ✅

Linear expenditure system: Linear expenditure system deals with group of the commodities rather than individual commodity. If we add this entire group, we get total consumer expenditure. Linear expenditure system is usually formulated on the basis of utility function. However it is different from indifference curve analysis. Because it is applied to groups of commodities between which there no substitution. Therefore the indifference curve for linear expenditure system would be right angle. It means according to this approach consumer will substitute the goods within the same group of commodities but not between the groups. It shows non substitutability of groups of commodities.
Under linear expenditure system utility is additive i.e. the total utility is the sum of the utilities derived from the various group of the commodities. :. U=∑ U i where, U i = U (A) +U (B) +U(C) +U (D) and A, B, C, D are the groups of commodities between which substitutions is not possible such as A= food, B= clothing=consumer durables, D= house operation expenditure. These categories are broadly divided such that the possibility of substitution between categories is eliminated. But substitution can occur within each group. Each group must include all substitutes and complements. It is assumed that consumer first spends to fulfill his basic minimum requirements required for subsistence and only after this the remaining income is divided among categories according to price. Consumers buy some minimum quantity from each group irrespective of prices. The minimum quantities are called subsistence quantities. Thus income is divided into two parts subsistence income and supernumerary income .subsistence income is the minimum requirement for keeping the consumer alive. Supernumerary income is the income left after the minimum expenditure is covered. Mathematically, let the utility function be U= I log (q i -y i ) Where y i = minimum quantity from group i q i = quantity purchased from group i b i = marginal budget share for group
y i >0 (no negative minimum quantity) (qi –y i )>0(some quantity above minimum is purchased) The consumer wants to maximize utility subject to budget constraint Max . U = I log (q i -y i ) Sub. to y=∑ p i q i Using Lagrangian function Ø= I log (q i -y i ) +£(y-∑ p i q i ) The first order condition for maximizing utility ∂Ø/∂ q i = b i /(qi –y i ) -£p i = 0 Or, bi/ (qi –y i ) = £p i …............ (1) ∂Ø/∂£= y-∑ p i q i = 0 Or, y-∑ p i q i Solving the above equation, Maximization of constraint utility function yield the following demand function Q i = y i + b i /p i (y-∑p i y i ) Pi q i = y i p i + b i (y-∑p i y i ) Where, q i = quantity demanded of group i y i = minimum quantity of group i b i = marginal budget share
y= consumer’s total income p i = price index of group i ∑p i y I = subsistence income (y-∑p i y i )= supernumerary income The demand function may be written as p i q i = y i p I + bi(y-∑p i y i ) (Exp. on group) = (subsis. Exp.) + (supernum. Exp.) B i is the partial derivative of expenditure on i with respect to the supernumerary income. B i = ∂ (pi q i )/ ∂(y-∑p i y i ) this is the marginal budget share.