A little help solving an equation

[Windows2000]

Hello,

This may seem trivial, but I have been trying to work out the solution to an equation. It is a Lagrange problem. I am following the textbook to get the solution on one of the decision variables, but I cannot see how they solved for those variable.

In short, I want to know how they solve for M and D from Eqs (6.b) and (6.c) respectively, to Eqs (6.18) and (6.19).

I would really appreciate any help with this

 

[Dr.Peterson]

Hello,

This may seem trivial, but I have been trying to work out the solution to an equation. It is a Lagrange problem. I am following the textbook to get the solution on one of the decision variables, but I cannot see how they solved for those variable.

In short, I want to know how they solve for M and D from Eqs (6.b) and (6.c) respectively, to Eqs (6.18) and (6.19).

I would really appreciate any help with this

Please show your attempt, so we can see where you are going wrong. It is fairly straightforward, though complicated: Isolate the term containing the variable you want, then divide by the coefficient, then raise to the reciprocal power, and clean things up.

 

[Windows2000]

Sure. This is what I got so far... but I'm not sure where I'm going wrong.

I'd appreciate if you could walk me through the answer.

 

[Dr.Peterson]

You're working on

​

But you've changed d and q from exponents into multipliers, and lost a d:

​

Then you changed the exponent on the parenthesis:

​

There's more wrong in the last steps, but they can't be fixed until you correct these details.

 

[Windows2000]

I should have been more clear with the variables from the beginning.

The "d" in "p^d" is actually a superscript, to denote the price of D. Just like the "q" in "p^q" for the price of Q. And the deltaD is one parameter as well. So I just wrote it alone.

About the exponent, I thought you could just simplify it before solving it, but
I'm really rusty on my algebra honestly. Do you think you could please point out where I'm going wrong with it.

 

[Dr.Peterson]

I should have been more clear with the variables from the beginning.

The "d" in "p^d" is actually a superscript, to denote the price of D. Just like the "q" in "p^q" for the price of Q. And the deltaD is one parameter as well. So I just wrote it alone.

About the exponent, I thought you could just simplify it before solving it, but
I'm really rusty on my algebra honestly. Do you think you could please point out where I'm going wrong with it.

Given the explanations of the notation, the one actual mistake is in changing "n*(1-n)/n" to "1/n". It should be "1-n". Do you see that? Or is there yet another quirk of notation here?

I'd like to see you take it from here, so we can talk through the remaining errors.

One thing I would do is to rewrite [imath]\left(\frac{Q}{\gamma}\right)^{1-\eta}[/imath] as [imath]\frac{Q^{1-\eta}}{\gamma^{1-\eta}}[/imath] and do a little more simplifying before solving. Then you'll have something nicer to divide by, and the result will look closer to their form. Also, seeing that they have [imath]\frac{1}{1-\eta}[/imath] in the answer rather than [imath]\frac{1}{\eta-1}[/imath], presumably because it is known that [imath]\eta<1[/imath], we can expect to take a reciprocal, changing the sign of the exponent.

 

[Windows2000]

Well noted. The eta is an elasticity parameter that should be = or < than 1.

I could solve it using your advices. Still if you want to take a look I'm attaching the solution.

Thank you very much for taking your time to help me. I really appreciate it.

 

[Dr.Peterson]

Well noted. The eta is an elasticity parameter that should be = or < than 1.

I could solve it using your advices. Still if you want to take a look I'm attaching the solution.

Thank you very much for taking your time to help me. I really appreciate it.

Good. I was right to assume the exercise would be good for you, and well within your ability.