## 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:

(1) ,

(2) , ,

(3) ,

(4)

(5)

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!