I'm trying to complete the following problem:
$\max_{x_0, x_1, x_2} 2ln(x_0) + \ln(x_1) + \ln(x_2) $ s.t $\ x_0 + p_\alpha\alpha + p_\gamma\gamma = 4, x_1 = 4 + \alpha + \gamma, x_2 = 4 + \gamma $
$p_\alpha, p_\gamma$ are constants.
I'm getting myself into a mess, however. I've substituted in the last two constraints directly, and then done a Lagrange optimization with $x_0, \alpha, \gamma$ but the algebra is horrific.
Am I missing an easier method?
You wish to maximise $2\ln(x_0)+\ln(x_1)+\ln(x_2)$, which equals $\ln(x_0^2x_1x_2)$.
As the logarithm is a monotonic function, you could try a Lagrange optimisation on maximising $x_0^2x_1x_2$ instead, and see if the algebra is more manageable.