- #1
lfdahl
Gold Member
MHB
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Hello MHB members and friends!(Callme)
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!
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!