# Help with Lagrangian expression!

• Jul 30th 2010, 04:53 PM
seanP
Help with Lagrangian expression!
If someone could try to explain the lagrangian expression to me, I'd greatly appreciate it. The textbook i have is not very good at explaining things and there doesn't even seem to be an example...

also, this is the question i need to apply it to:

A person's satisfaction (or utility) from playing tennis and golf is given by:

U = ln(G) + 4 ln(T)
where G is the number of hours spent playing golf and T is the number of
hours spent playing tennis. This person has 10 hours per week to devote to
these sports. However, each hour of playing tennis typically entails one hour
of waiting for an empty court, thus using up twice the time of actual play.
As a consequence, for example, if he spent 2 hours playing golf and 4 hours
playing tennis, he would have used up the full 10 hours of his available time.

a) What equation describes this person's time constraint?

So: 2T + G <or=10
b) What is the Lagrangian expression of this constrained maximization prob-
lem?

c) Use this Lagrangian expression to nd out the satisfaction (or utility)
maximizing choice of time to play golf and tennis.
• Jul 30th 2010, 06:22 PM
pickslides
Quote:

Originally Posted by seanP
If U = ln(G) + 4 ln(T)
where G is the number of hours spent playing golf and T is the number of
hours spent playing tennis. This person has 10 hours per week to devote to
these sports. However, each hour of playing tennis typically entails one hour
of waiting for an empty court, thus using up twice the time of actual play.
As a consequence, for example, if he spent 2 hours playing golf and 4 hours
playing tennis, he would have used up the full 10 hours of his available time.

a) What equation describes this person's time constraint?

So: 2T + G <or=10
b) What is the Lagrangian expression of this constrained maximization prob-
lem?

c) Use this Lagrangian expression to nd out the satisfaction (or utility)
maximizing choice of time to play golf and tennis.

In your problem $f(t,g)=\ln(g) + 4 \ln(t), h(t,g)=2t+g-10$

The Lagrangian is $L(t,g,\lambda) =f(t,g)+\lambda h(t,g)$

To maximise find where $L'(t,g,\lambda)=0$
• Jul 30th 2010, 10:58 PM
CaptainBlack
Quote:

Originally Posted by seanP
If someone could try to explain the lagrangian expression to me, I'd greatly appreciate it. The textbook i have is not very good at explaining things and there doesn't even seem to be an example...

also, this is the question i need to apply it to:

A person's satisfaction (or utility) from playing tennis and golf is given by:

U = ln(G) + 4 ln(T)
where G is the number of hours spent playing golf and T is the number of
hours spent playing tennis. This person has 10 hours per week to devote to
these sports. However, each hour of playing tennis typically entails one hour
of waiting for an empty court, thus using up twice the time of actual play.
As a consequence, for example, if he spent 2 hours playing golf and 4 hours
playing tennis, he would have used up the full 10 hours of his available time.

a) What equation describes this person's time constraint?

So: 2T + G <or=10
b) What is the Lagrangian expression of this constrained maximization prob-
lem?

c) Use this Lagrangian expression to nd out the satisfaction (or utility)
maximizing choice of time to play golf and tennis.

I presume that you have been told to solve this using Lagrangian multipliers?

The reason I ask is because in a proper formulation of this T is almost certainly constrained to be an integer or semi-integer, so Lagrangian multipliers is an inappropriate method of solution.

(also you should assume an equality constraint (as anything less than 10hours playing can be improved by playing either game for the remaining time) which is what you need for Lagrange's method anyway)

CB