Derive Hicksian demand for -
$$u(x,y) = ax+ b\ln(y)$$
Explain in words what they mean?
I solved the problem with the Lagrange Multiplier Method and found Hicksian demand for $x$ only.
Solution: Suppose, the expenditure function is -
E = P1x + P2y
Subject to utility function -
u = ax + b ln y
Using Lagrange Multiplier for constrained optimization we get -
L = P_1 x+P_2 y+µ(u-ax-blny)
using first order condition for optimization - ∂L/∂x=P_1-aλ=0 …… (1) ∂L/∂y=P_2-bλ/y=0 ……. (2) ∂L/∂λ=u-ax-blny=0 ……. (3) Now, dividing equation (1) by equation (2) – y=P_1/P_2 .b/a Putting this value in equation (3), we get -
$$x = \frac1a \cdot \left[u - b\ln\left(\frac{P_1b}{P_2a}\right)\right] $$
I do not know how to explain this situation or demand function. Also, I do not understand how to find Hicksian demand for y from this. Is there any alternative method I should have used?