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Math Help - partial derivatives in finance

  1. #1
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    partial derivatives in finance

    Hello. I am studying for my exam and just came across a problem that states I must take a partial derivative of a utility function. I am kind of confused since it's been a while since I've done this. The answer in the book appears to be missing a decimal point. But I also need a refresher on partial derivatives it seems . Please note that w1 and w2 are just terms.
    The utility function is w1^{0.6}w2^{0.4} and we must figure out it's derivative as in \frac {dW2}{dW1}

    i got dw2/dw1 =  \frac {0.6w1^{-0.4}w2^{0.4}}{0.4w2^{-0.6}w1^{0.6}}

    from here i am unsure as to why this is equal to \frac {1.5w2}{w1}

    any help is appreciated

    thank you
    chris
    Last edited by ffezz; December 7th 2011 at 05:20 PM.
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  2. #2
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    Re: partial derivatives in finance

    Quote Originally Posted by ffezz View Post
    Hello. I am studying for my exam and just came across a problem that states I must take a partial derivative of a utility function. I am kind of confused since it's been a while since I've done this. The answer in the book appears to be missing a decimal point. But I also need a refresher on partial derivatives it seems . Please note that w1 and w2 are just terms.
    The utility function is w1^{0.6}w2^{0.4} and we must figure out it's derivative as in \frac {dW2}{dW1}
    A few questions/comments:

    1. Are the 1 and 2 subscripts? If so, you can type them like this: [TEX]w_{1}, w_{2},[/TEX] which produces w_{1}, w_{2}.

    2. Also, math is case-sensitive. W_{1}\not=w_{1}, in general. Are they the same variables for you?

    3. Without a starting equation, you cannot compute a derivative. That is, unless you know that y=x^{2}, you cannot compute that y'=2x. Let's say that your utility function is denoted by u. Are you told that u=w_{1}^{0.6}w_{2}^{0.4}? If so, then precisely what derivative are you asked to compute?

    i got dw2/dw1 =  \frac {0.6w1^{-0.4}w2^{0.4}}{0.4w2^{-0.6}w1^{0.6}}

    from here i am unsure as to why this is equal to \frac {1.5w2}{w1}
    Well,

    \frac{0.6w_{1}^{-0.4}w_{2}^{0.4}}{0.4w_{2}^{-0.6}w_{1}^{0.6}}=\frac{1.5w_{2}^{0.6}w_{2}^{0.4}}{  w_{1}^{0.4}w_{1}^{0.6}}=\frac{1.5w_{2}}{w_{1}}.

    That's arithmetic and exponentiation laws.
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  3. #3
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    Re: partial derivatives in finance

    Thank you very much Ackbeet All your assumptions are correct. I will try to be more careful with case and equation setting in the future.
    I simply didn't see the re-arranging. It makes perfect sense now that you included the intermediate step.
    I appreciate your answer.
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    Re: partial derivatives in finance

    You're welcome. Have a good one!
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