Theorem 1 of page 12 of the Econ 510 notes gives a statement of
the Karush-Kuhn-Tucker
theorem in its most generally stated form. The theorem states:
(A) (Sufficiency) If (x(0),lam(0)) is a saddle point for the
function Z(x,lam), then x(0) solves the constrained maximization
problem.
(B) (Necessity) If "Constraint Qualification" holds, whatever
that means, and if x(0) solves the constrained maximization problem,
then there is a lam(0) such that (x(0),lam(0)) is a saddle point.
Where the function Z(x,lam) is:
Z(x,lam) = f(x) + lam (dot) g(x)
and lam (dot) g(x) is the dot product of lam and g(x), the sum
over i of the products of the i th component of lam and g(x).
This theorem is quite general. General is good in that it applies to many special cases at once. General is bad in that we do not know which particular case we are talking about. A general statement applies to all cases, so anything specific that we say must apply to all cases at once, and very few things apply to all cases at once. Even the first proposition in micro, the so-called Law of Demand, that raising the price reduces demand, does not apply to all cases. To say anything more specific, which is the goal in Macro, we have to add structure. When we add structure we lose generality, but we can say more things because we can say things that are true in our special cases even if they are false in other cases.
As we add structure, we gain the ability to say more things, but we lose generality. So we try to only add a little structure at a time. The first thing to add is the assumption of concavity to the functions f and g. This throws out many cases of the Karush-Kuhn-Tucker (KKT) theorem. In the KKT theorem, the functions f and g do not have to be continuous or differentiable or anything except being functions. A concave function defined on an open set is continuous throughout the open set and is pretty close to being differentiable throughout also. So concavity is pretty restrictive. It is also pretty convenient. So we can write down the following theorem.
Theorem 2 If f and g are (a) both concave and (b) both continuously differentiable then the following four conditions are sufficient for x(0) to be the (in there is only one) x that maximizes f(x) subject to g(x) >=0.
Since item (1) in this list is a derivative, we have to assume that the derivative exists. Hence, we assume differentiability.
This theorem is convenient because it brings us into the familiar territory of take the derivative, set to zero and solve. This result is something that macroeconomists use about 1000 times per year.
Once you get the idea of what the theorem says, you can put it to work for you. In macro, the goal is usually to say something like: "Firms do such and such" or "individuals do such and such." The wise guy in the back of the room (and there is always a wise guy) will say: "Prove it." So you (1) Set up the firm or individual's maximization problem, including constraints, (2) show that the objective and constraint functions are continuously differentiable and concave (by the way, budget lines, being linear, are always continuously differentiable and concave), and (3) crank through conditions (1) through (4) from the above, invoke the theorem and declare yourself done. The wise guy in the back of the room now has to shut up.
The KKT theorem has two sides, sufficiency and necessity. Sufficiency means, roughly, that being a saddle point is sufficient for being a maximum for the constrained optimization problem. Necessity means, roughly, that if you have a constrained optimum, there needs to be a saddle point to go with it. The qualifier "roughly" is thrown in here because constraint qualification condition needs to be added for the necessity statement and because a saddle point and a point of constrained maximization are not strictly comparable because the saddle has two objects (x(0),lam(0)) and the point of constrained maximization has on one, x(0).
In macro, sufficiency is the direction that we use 85 percent of time, and if you are at Harvard or MIT, what you would use 99.8 percent of the time. We set up the problem and solve. We show that the Kuhn-Tucker (Karush always gets slighted in Macro) conditions hold. Doing all this proves that the behavior is the optimizing behavior. That is all we usually care about.
Macro guys use the necessity side of the argument when they know that some x(0) solves the constrained maximization problem and they want to show that there are prices that support that solution. The second welfare theorem says that in some circumstances an efficient allocation can be supported by a price system. The lambda's of the Kuhn-Tucker theorem give us those prices. The theory runs as follows: Solve the social planner's problem, forgetting about prices and the competitive system. Find the real allocations. Then read off the prices that are needed to support those prices. This is backward logic in a sense. We start with what the government would do if it controlled everything, and assume that this is what would happen in a competitive equilibrium. It is very bothersome for some economists (hence its rare use at MIT and Harvard), but it is very convenient.