(a) Given the following monotonically transformed utility function faced by the consumer
lnU(X1X2) = ∝lnX1+βlnX2
The price of good X1 is P1 and the price of good X2 is P2.
Construct the corresponding Langregian function
Derive the optimal demand (Marshallian demand) function for X1 and for X2
The Lagrangian function is a technique that combines the function being optimized with functions describing the constraint or constraints into a single equation. Solving the Lagrangian function allows you to optimize the variable you choose, subject to the constraints you can’t change. Marshallian demand (dX1) is a function of the price of X1, the price of X2 (assuming two goods) and the level of income or wealth (m): X*=dX1(PX1, PX2, m) Optimal demand (Marshallian demand) function for X1 and for X2 will be: X = (0.5I/P1, 0.5I/P2)