Question

In a two-person, two commodity, pure exchange economy Person A’s utility is given by U^A (q₁^A, q₂^A) = q₁^A * q₂^A + 2q₁^A + 5q₂^A and Person B’s utility is given by U^B (q₁^B, q₂^B) = q₁^B * q₂^B + 4q₁^B + 2q₂^B. Where commodity one is w₁ and commodity two is w₂, if Person A has w^A = (w₁^A, w₂^A) = (78,0) and Person B has w^B = (w₁^B, w₂^B) = (0,164), what is the uncompensated demand function for w₁ and w₂ for Person A?

Answer 1

Uncompensated demand curve is the simple Marshallian demand curve I.e. quantity as a function of income and prices.

The uncompensated demand function of A can be derived as follows -

The income of A is what A can earn by selling his endowment at market price P1 and P2. Where P1 is the price of good 1 and P2 is price of good 2

Hence income of A=

M(A) = (78×P1) +(0×P2).

M(A) = 78×P1

The Uncompensated demand function can be derived by solving the lagrangian of the utility maximizing problem of A, as shown in following -

Here eq(6) and eq(7) represents the Uncompensated or Marshallian demand functions of A.