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#### lfdahl

##### Well-known member

- Nov 26, 2013

- 722

An economy student asked me, if I could explain the following partial differentiation:

\[\frac{\partial}{\partial C(i)}\int_{i\in[0;1]}[C(i)]^\frac{\eta - 1}{\eta}di

=\int_{j\in[0;1]}[C(j)]^\frac{\eta - 1}{\eta}dj\frac{\eta - 1}{\eta}[C(i)]^{-\frac{1}{\eta}}

\]

I am not sure, why the differentiation is performed as shown above ($\eta$ is a constant).

If it can be of any help in understanding the identity, the following should be added:

The function C(i) may or may not take a specific form. Whether or not, the C(i) is usually implicitly defined by the so called “felicity function”, which in this case takes the form:

$ u(C)=\frac{[C(i)]^{1-a}-1}{1-a}$, where a is a constant.

The function u(C) is a measure of the instantaneous utility a consumer has of the consumption amount C. The variable i is a time measure. The theory states, that the consumer prefers consumption instantaneously (“here and now”) instead of saving up for the future.

I presume, that the appearance of the partial derivative is a part of Lagranges optimization.

Thankyou in advance for any help in the matter. I´d also like to thank the MHB staff for a very exciting and interesting homepage!