1. ## Lagrange multipliers

I am having difficulty with the following problem :

A lecturer eats only two types of food and has a salary I > 0 . The quality of his lectures is given by x1^a1 * x2^a2 , where xi is the daily consumption of food i, respectively, and a i > 0 are constants, i = 1, 2 , a with a1 + a2 = 1 . The price of food i is pi > 0 , i = 1, 2 . The lecturer wishes to give lectures that are
of quality as high as possible, without spending more than his income. What is the optimal food consumption? If the salary is increased, after recent presidential elections, with ∆I , how much of an increase of the quality of lectures can be expected?

2. Originally Posted by terulic
I am having difficulty with the following problem :

A lecturer eats only two types of food and has a salary I > 0 . The quality of his lectures is given by x1^a1 * x2^a2 , where xi is the daily consumption of food i, respectively, and a i > 0 are constants, i = 1, 2 , a with a1 + a2 = 1 . The price of food i is pi > 0 , i = 1, 2 . The lecturer wishes to give lectures that are
of quality as high as possible, without spending more than his income. What is the optimal food consumption? If the salary is increased, after recent presidential elections, with ∆I , how much of an increase of the quality of lectures can be expected?

So your object function is $\displaystyle F= x_1^{a_1}x_2^{1-a_1}$ and you want to maximize that subject to $\displaystyle G= p_1x_1+ p_1x_2= I$.
Now, $\displaystyle \nabla F= a_1x_1^{a_1-1}\vec{i}+ (1-a_1)x_2^{-a_1}\vec{j}$ and $\displaystyle \nabla G= p_1\vec{i}+ p_2\vec{j}$.

Since you titled this "Lagrange multipliers", I assume you know that the optimum value will be where $\displaystyle \nabla F= \lambda\nabla G$ for some number $\displaystyle \lambda$. In this case, that means $\displaystyle a_1x_1^{a_1-1}\vec{i}+ (1-a_1)x_2^{-a_1}\vec{j}= \lambda p_1\vec{i}+ \lambda p_2\vec{j}$ so you have the two equations $\displaystyle a_1x_1= \lambda p_1$ and $\displaystyle a_2x_2= \lambda p_2$. I recommend dividing one equation by another to eliminate $\displaystyle \lambda$, that you don't really care about, and then using the constraint $\displaystyle p_1x_1+ p_2x_2= I$ to solve for $\displaystyle x_1$ and $\displaystyle x_2$. Of course, you answers will be in terms of $\displaystyle a_1$, $\displaystyle a_2$, $\displaystyle p_1$, $\displaystyle p_2$, and I.

To answer the second question, subtract the result with "I" from the result with "$\displaystyle I+ \Delta I$".

3. ## True genius

Thq, I will come back later today with the rest of the solution. I just needed a jump start.

4. Originally Posted by HallsofIvy
So your object function is $\displaystyle F= x_1^{a_1}x_2^{1-a_1}$ and you want to maximize that subject to $\displaystyle G= p_1x_1+ p_1x_2= I$.
Now, $\displaystyle \nabla F= a_1x_1^{a_1-1}\vec{i}+ (1-a_1)x_2^{-a_1}\vec{j}$ and $\displaystyle \nabla G= p_1\vec{i}+ p_2\vec{j}$.

Since you titled this "Lagrange multipliers", I assume you know that the optimum value will be where $\displaystyle \nabla F= \lambda\nabla G$ for some number $\displaystyle \lambda$. In this case, that means $\displaystyle a_1x_1^{a_1-1}\vec{i}+ (1-a_1)x_2^{-a_1}\vec{j}= \lambda p_1\vec{i}+ \lambda p_2\vec{j}$ so you have the two equations $\displaystyle a_1x_1= \lambda p_1$ and $\displaystyle a_2x_2= \lambda p_2$. I recommend dividing one equation by another to eliminate $\displaystyle \lambda$, that you don't really care about, and then using the constraint $\displaystyle p_1x_1+ p_2x_2= I$ to solve for $\displaystyle x_1$ and $\displaystyle x_2$. Of course, you answers will be in terms of $\displaystyle a_1$, $\displaystyle a_2$, $\displaystyle p_1$, $\displaystyle p_2$, and I.

To answer the second question, subtract the result with "I" from the result with "$\displaystyle I+ \Delta I$".
For the first question i got $\displaystyle x_1 = a_2p_1I/(p_1^{2}a_2+a_1p_2^{2})$ and similar for $\displaystyle x_2$ Do I need to substitute the values in $\displaystyle F= x_1^{a_1}x_2^{1-a_1}$ and then substract from the same with "$\displaystyle I+ \Delta I$" ? What am i missing?