Consumer theory

[Meghanjalli Routh]

An individual consumes three goods – X, Y, and Z – and the individual’s utility function satisfies the usual assumptions.  Let W(pX, py, F, z) denote the maximum utility that the individual can obtain when the prices of goods X and Y are pX and pY, F is the amount that the individual spends on goods X and Y, and z is the amount of good Z consumed.The individual’s utility is U(x, y, z) = z + ln(x) + (αz+1) ln(y) where α ≥ 0.  Determine the conditions for the optimal amounts of goods X and Y (given pX, pY, and F)

I am using lagrange, but I am still not getting the answer. The answer is  x* = F/(pX(αz + 2)) and y* = F(αz + 1)/ (αz + 2)

[Sumit]

I'm getting x* = F/(pX(αz + 2)) and Y*=F(αz + 1)/(Py(αz + 2))...I think in Y you forget to type Py...

[Meghanjalli Routh]

What is the lagrange set up you have used? And the answer does not have Py in the y*.

[Devika]

Whats the answer? I got the optimal bundle same as sumit's.

[Sumit]

@Meghanjalli Routh,
look for derivation of optimal outputs of cobb-douglus demand function through lagrange in choice chapter appendix of Hal.varian...(particularly the trick part)

[juhi]

if you not getting solution even with lagrangian... Put MRSxy=price ratio and solve by putting the value in budget constraint