The right solution to Example 6.5
The treatment of Example 6.5 in Baltovic's guide is confusing. The exposition indicates a problem but does not provide the explanation. Before reading this post try to solve the exercise on your own following the economical way.
Example 6.5. Consider the cost-minimisation problem of a consumer:
minimise subject to , ,
Don't forget that in case of minimization the lambdas in the Lagrangian should be taken with negative signs. It is assumed that
Case 1. Internal solutions are impossible because the first order conditions for give
Denote the boundaries The Kuhn-Tucker conditions are:
(2) , ,
The constraint qualification condition is obviously satisfied for considered separately.
Case 2. belongs to only. Then does not belong to and and from complementary slackness Then from (5) which is nonsense.
Case 3. Similarly, if belongs to only, then which is impossible.
Case 4. belongs to only. Then and from complementary slackness (3)-(5) simplify to , , The solution to this system is
and the value of the objective function at this point is
Case 5. The only possibilities left are Don't bother checking the constraint qualification for these points because a) it may fail, in which case the Kuhn-Tucker theorem does not apply, even though any of these points can be a minimum point, and b) even if it holds, none of these points may be a minimum point (the Kuhn-Tucker theorem provides just a necessary condition). Just check directly the values of at these points:
Since are strictly positive, we see that Thus, is the minimum if is the minimum if and we have two minimum points in case of a tie
Conclusion: the Kuhn-Tucker does work in this case!