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Math Help - Tricky differentiation w.r.t. to the index

  1. #1
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    Tricky differentiation w.r.t. to the index

    Could you please help me with understanding how to take the partial derivative of

    -\lambda\left(\int_0^1 P_t(i)C_t(i) di-Z_t\right)

    with respect to C_t(i) using the standard rules of derivatives and integrals and get

    -\lambda P_t(i)

    i is an index representing firm i. There is a continuum of firms from 0 to 1. t is just an time index. The problem is from page 61 of Jordi Gali's book Monetary Policy, Inflation and the Business Cycles.
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  2. #2
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    To put it in another more simple way, could you provide a formal proof of

    \frac{\partial }{\partial C(m)} \int_{0}^{1}P(i)C(i)di=P(m)

    I understand the intuition behind the result (I guess). One can think of the integral as an infinite sum, but I would like to see the math.

    Anyone?
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  3. #3
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    To get your answer

    <br />
\displaystyle<br />
\frac{\partial}{\partial C(m)} \int_0^1 \ P(i) \ C(i) \ di \ = \int_0^1 \frac{\partial}{\partial C(m)} \ P(i) \ C(i) \ di \ = \int_0^1  \ P(i)  \ \frac{\partial}{\partial C(m)}\ C(i) \ di \ =<br />

    <br />
\displaystyle<br />
\int_0^1  \ P(i)  \ \delta(i-m)\ di \ = \ P(m)<br />

    but I don't know what P(i) and C(i) is.
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  4. #4
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    C(i) represents quantity of good i and P(i) represents price of good i. There is a continuum of goods between i=0 and i=1.

    zzzoak, could you please explain the last two steps in more detail, what is the meaning of the letter delta in the second last equation?
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