I do not understand how Mas-Collel, Whinston, and Green derive the Hicksian demand functions in Example 3.E.1 in their textbook.
Allow me to give further background regarding the problem: The authors give us a Cobb-Douglas utility function for L = 2, where L refers to the number of commodities. The function is u(x1, x2) = (k)x1^(α) * x2^(1-α), where k is some positive constant and alpha is some number between 0 and 1. They note that the function is strictly increasing on xi > 0, where i=1,2. Thereafter, they apply an increasing transformation of u(x1,x2) - namely, the natural logarithm. They note that since natural logarithm is strictly concave, it is strictly quasi-concave, which tells us that the Walrasian Demand correspondence will in fact be a function. Moreover, since the utility function is increasing, we know that the consumer has monotone preferences, thus her preferences satisfy local non-satiation, thus the budget constraint holds with equality; that is: p1x1 + p2x2 = w (wealth).
However, when the authors go on to derive the Hicksian demand functions, I am lost. I tried setting up the first-order conditions for minimization, by setting up the Lagrangian function; that is, take the objective function (the utility function) plus Lambda times (u minus (the utility function)). I set the partial derivative of the Lagrangian with respect to the choice variables and set them greater than or equal to zero. Next, there is the partial derivative of the Lagrangian function with respect to lambda, which I set less than or equal to zero. Lastly, there are the complementary slackness conditions.
References: Mas-Colell, Andreu, et al. Microeconomic Theory. Oxford University Press, 1995.
This is not a homework problem, I just wish to have elucidation on this example.