I need to minimize the following quantity:
$$\min x_1^{-1/n}- \left(1-x_2 \right)^{-1/n}$$
subject to:
$1-x_1-x_2=\gamma$ and $0<x_1+x_2<1$
$\gamma$ being a constant.
Had it been two equality constraints the problem could have been easily solved using two Lagrangian multipliers but how do I proceed now?
All help is greatly appreciated, thank you.
Ok as I said the inequality constraint can be removed. And moreover we can replace $x_2=1-\gamma-x_1$. So we got unconstrained optimization problem (taking in account that $x_1>0$,$x_2<1$,however,though this is automatically constrained via property of the utility function. So we have to optimize $$ \min x_1^{-1/n}-(\gamma+x_1)^{-1/n} $$ So making simple derivative we obtain $$ -1/n ( x_1^{-\frac{n+1}{n}}- (\gamma+x_1)^{-\frac{n+1}{n}}) $$ Obviously this derivative is always negative since $\gamma>0$. Thus actually the problem does not have a mininum. It asympotically tends to 0 as $x_1\to\infty$ and $x_2=1-\gamma - x_1$. But... I am not sure that this what you wanted. If say you want to bound $x_1$ by $1$. The infimum $1-(\gamma+1)^{-1/n}$ will be achieved with $x_1\to 1$.