For a function of two or more variables, the surface at a saddle-point resembles a saddle that curves up in one or more directions, and curves down in one or more other directions. ....For example, two hills separated by a high pass will show up a saddle point, at the top of the pass,....
A saddle point is a very important idea for economics, though it is somewhat mathematical. For econ learners, it is necessary to keep it in mind.
An econ example is about the consumer's utility maximization problem(UMP):
Suppose he or she had two kinds of goods: apples and oranges. The price of an apple is p, and the price of an orange is q. The number of apples is A, and the number of oranges is O. He or she has a utility function like U(A,O). It expresses his preference and assumes that the more oranges or apples he gets the happier he becomes. His income is M. And then his or her utility maximization problem is:
max U(A,O)
such that pA+qO≦M
When an economist usually solves the above problem, she uses the Lagrangian function:
L(A,O,l)=U(A,O) - l(pA+qO-M)
where l is the Lagrange multiplier, or the marginal utility of income.(If you see it as the marginal utility of income, differenciate L with respect to M and you get l. As M increases L also increases. The multiplier l is the rate of change of L to a small change in M. )
And she differentiates L with respect to A,O and l to get the first-order necessary conditions for the maximization problem.
f.o.c.
dL/dA=dU/dA - lp=0....(1)
dL/dO=dU/dO - lq=0....(2)
dL/dl=- (pA+qO-M)≧0,
l(pA+qO-M)=0 and l≧0....(3)
The above conditions are called "the Kuhn-Tucker necessary conditions". (Note that dL/dx means the partial derivative.) The Kuhn-Tucker conditions (or the Kuhn-Tucker Theorem) say that there exists l such that dL/dA=dU/dA - lp=0, dL/dO=dU/dO - lq=0 and dL/dl=- (pA+qO-M)=0.
And the condition (3) says that l>0 and -(pA+qO-M)>0 cannot hold at the same time. (Of course, l=0 and (pA+qO-M)=0 can hold simultaneously.)
If l=0, then -(pA+qO-M)≧0. In this case, this problem is unconstrained. And so you can solve the problem just by equating the derivatives of U(A,O) with respect to A and O to zero.
If l>0, then -(pA+qO-M)=0. In this case, this problem is constrained to income. And so you can solve the problem by solving the simultaneous equations of A,O and l, (1), (2) and (3).
Well, I'll reach the conclusion soon. The optimal solutions (A*,O*,l*) are at a saddle point: These give the maxima of the utility in A and O, and the minimum in l. That is, for a function of two variables A and O, the surface at a saddle-point curves up in two directions of A and O, and curves down in one other direction of l.
The meaning is simple. If you have a little more income left, you would be better to spend it on goods in order to make your utility as large as possible. If you maximize your utility, no income is left. That's the optimum!
Appendix:
If you have a maximization problem with nonnegative constraints, you'll have the following first-order conditions:
max U(A,O)
such that pA+qO≦M, A≧0 and O≧0
f.o.c.
dL/dA=dU/dA - lp≦0, A(dL/dA)=0 and A≧0....(1)'
dL/dO=dU/dO - lq≦0, O(dL/dO)=0 and O≧0....(2)'
dL/dl=- (pA+qO-M)≧0, l(pA+qO-M)=0 and l≧0....(3)'
7 comments:
Hi, Taro!
Could you mention the book or other issue, where this interpetation of the saddle point is described, please. It seems to be a little incorrect to use the auxiliary Lagrangian function as a neu utility function. The given utility function of a consumer - U(A,O) - doesn´t depend on money M. The form of this funktion U is uncertain, but you say: "the more A and/or O, the better". That means, that a consumer spends all his money on apples and oranges,
p*A + q*O = M.
There are no Kuhn-Tacker-conditions here.
If we had a utility function with a saturation point (more goods is not always better), it would be more interesting. There could be some money left if the "best" quantity of goods would cost less than M.
p*A + q*O < M
Anyway, I think, it doesn´t deal with the utility of money. What do you mean?
By the way, that was my comment about automatic translator. I am Russian, female, I live in Germany and study economics.
Thank you very much for your critical, but very nice comment, Ms,tatiana! It is very important and it made me study more on the static optimization.
Let me answer your questions briefly;
tatiana's question(1): Could you mention the book or other issue, where this interpetation of the saddle point is described, please.
Taro's question(1): Avinash K. Dixit(1990),Optimization in Economic Theory, Second Edition, pp.104,Exercise7.3., More on linear programming.
tatiana's question(2):The given utility function of a consumer - U(A,O) - doesn´t depend on money M. The form of this function U is uncertain, but you say: "the more A and/or O, the better".
Taro's answer(2): I should have assumed that the form of this function U be concave, that is, "the more A and/or O, the better"(marginal utility is positive) and "the much more A and/or O, the less better"(decreasing marginal utility).
The assumption of a concave utility function is the most general case in an elementary microeconomics. But I said that it implied that "the more A and/or O, the better". As you said, it is incorrect, and more likely to lead you to misunderstand and I should have put the assumption.
tatiana's question(3): ...but you say: "the more A and/or O, the better". That means, that a consumer spends all his money on apples and oranges,
p*A+ q*O = M.
There are no Kuhn-Tucker-conditions here.
Taro's answer(2): The Kuhn-Tucker theorem says that if (A*,O*) solves the above problem and the constraint qualification holds(the set of gradients of the binding constraint at (A*,O*) is linearly independent), then there is the Lagrange multiplier,l>=0 such that dU/dA=lp and dU/dO=lq.(The term dU/dA is a partial derivative.)
-See Varian(1992),Microeconomic Analysis,Third Edition,pp.503.
And the Kuhn-Tucker theorem says that the Kuhn-Tucker necessary conditions holds when there's a solution for the maximum problem. That is, if we have the maximum solution, then there exists a marginal utility of income,l such that
A>=0, dU/dA<=lp, A(dU/dA)=0
O>=0, dU/dO<=lq, O(dU/dO)=0
l>=0, dU/dl=pA + qO>=0, l(dU/dO)=0
As you know, they are called complementary slackness conditions. They tell us whether the constraint will affect the optimum solution.
If you have a maximum solution for this problem, it satisfies l>0 and it is said the Kuhn-Tucker multiplier.
When some texts(for example, David Romer's Advanced Macroeconomics, Second Edition,pp.52-53.)refer to the utility maximization problem(UMP), they use the reason that "the more A and/or O, the better" and thus a consumer spends all his money on apples and oranges, p*A + q*O = M, to solve the problem with inequality constraints.
In this process they seem to have no Kuhn-Tucker necessary conditions, but, to say correctly, they find the maximum solution that satisfies l>=0, which is the Kuhn-Tucker necessary condition. (If in this case the Kuhn-Tucker necessary condition doesn't hold, it won't be maximum solution.)
When we solve the optimization problem with inequality conditions, we have to use the Kuhn-Tucker theorem to satisfy the optimum solution.
tatiana's question(3): If we had a utility function with a satiation point (more goods is not always better), it would be more interesting. There could be some money left if the "best" quantity of goods would cost less than M.
p*A + q*O < M
Taro's answer(3): Good pointer, thank you. It satisfies l=0. That is, the constraint is not binding. This can be thought as an unconstrained optimization problem. And it doesn´t deal with the utility of money.
You say that the given utility function of a consumer - U(A,O) - doesn´t depend on money M, but more generally U(A,O) - does depend on money and sometimes doesn´t. To know wether U depends on money or not,or wether the constraint is in effect or not, we usually use the Kuhn-Tucker theorem, I think.
If you have more questions on that, please feel free to comment again because I may have made some mistakes. Thank you for your good insight.
Additional Explanation:
I said that the assumption of a concave utility function is the most general case in an elementary microeconomics, but I forgot to say a much more important thing:
As you know, the assumption of concave objective function(utility function) and convex constraints(income constraint)is sufficient to have maximum solutions for the optimization problems. That is, it is a sufficient conditon.
(See the proof in Varian(1992),Microeconomic Analysis,Third Edition,pp.503-504.)
And so when we solve the maximization problem, we only have the first-order-conditions by assuming of concave objective function and convex constraints. We can skip tedious tasks of showing sufficient conditions.
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