Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Year 12s: what will be the first way you research your future options? >>

Economic Maths

Hi everyone,

Could someone point me in the direction of the rule which allows for the below transformation.

C^* = r \bullet A^\frac{-1}{a+B} \bullet \frac{a}{B}^\frac{B}{a+B} \bullet \frac{w}{r}^\frac{B}{a+B} \bullet Q^\frac{1}{a+B} + w \bullet A^\frac{-1}{a+B} \bullet \frac{B}{a}^\frac{a}{B+a} \bullet \frac{r}{w}^\frac{a}{B+a} \bullet Q^\frac{1}{a+B}[br][br]\Rightarrow[br][br]C^* = \frac{1}{A^\frac{1}{a+B}} \bullet \left[ \frac{a}{B}^\frac{B}{a+B} + \frac{B}{A}^\frac{a}{a+B} \right] \bullet r^\frac{a}{a+B} \bullet w^\frac{B}{a+B} \bullet Q^\frac{1}{a+B}[br][br]

Thanks for any replies.

Original post by FishFullofRum

Hi everyone,

Could someone point me in the direction of the rule which allows for the below transformation.

C^* = r \bullet A^\frac{-1}{a+B} \bullet \frac{a}{B}^\frac{B}{a+B} \bullet \frac{w}{r}^\frac{B}{a+B} \bullet Q^\frac{1}{a+B} + w \bullet A^\frac{-1}{a+B} \bullet \frac{B}{a}^\frac{a}{B+a} \bullet \frac{r}{w}^\frac{a}{B+a} \bullet Q^\frac{1}{a+B}[br][br]\Rightarrow[br][br]C^* = \frac{1}{A^\frac{1}{a+B}} \bullet \left[ \frac{a}{B}^\frac{B}{a+B} + \frac{B}{A}^\frac{a}{a+B} \right] \bullet r^\frac{a}{a+B} \bullet w^\frac{B}{a+B} \bullet Q^\frac{1}{a+B}[br][br]

Thanks for any replies.

I presume the bullet is just multiplication. There are a few rules involved:

Index manipulation:

\dfrac{1}{x^y}=x^{-y}

x^r\times x^s = x^{r+s}

\left(\dfrac{x}{y}\right)^s = \dfrac{x^s}{y^s}

(Note:

w=w^1

)

and factorisation:

xy+xz = x(y+z)

generalised.

(edited 10 years ago)

OP

Excuse me for sounding like an eejit, but it isn't obvious to me why the r and w terms should be outside of the brackets if they're not found in common on both sides of the addition sign... any suggestions?

OP

Excuse me for sounding like an eejit, but it isn't obvious to me why the r and w terms should be outside of the brackets if they're not found in common on both sides of the addition sign; any suggestions?

Original post by FishFullofRum

Excuse me for sounding like an eejit, but it isn't obvious to me why the r and w terms should be outside of the brackets if they're not found in common on both sides of the addition sign; any suggestions?

I'll look at one of them, the "r" terms. Your confusion may be that the exponent applies to both the numerator and the denominator in each fraction, rather than just the numerator.

In your original formula you have before the "+" sign:

r

and

\frac{w}{r}^{\frac{B}{a+B}}

Which gives us

r^1

and

w^{\frac{B}{a+b}}r^{-\frac{B}{a+B}}

Just considering the r terms we then have:

r^{1-\frac{B}{a+B}}

which equals:

r^{\frac{(a+B)-B}{a+B}}

Hence:

r^{\frac{a}{a+B}}

Which is also the power of r in the second term, and hence we can take it out as a factor.

OP

You're an absolute star. Perfectly explained, thanks for taking the time.

Quick Reply

Related discussions

Latest

Trending

Trending

Articles for you

What can you do with an arts degree?

What can you do with an arts degree?

Writing an economics personal statement: expert advice from universities

Writing an economics personal statement: expert advice from universities

The Student Room Group Privacy Notice

What is an LLM and how could it benefit your career?

What is an LLM and how could it benefit your career?