# Economics calculus problem

#### questioner1

There is a derivation in my economics textbook: Romer - advanced macroeconomics, that he doesn't explain and I can't see why it works:

He says that:

$$\displaystyle f(t) = \int_{y=0}^t r(y)dy$$

And concludes that:

$$\displaystyle \delta f(t)/\delta t = r(t)$$

Could someone point me to the rule that allows you to do this, so that I can study it?

Edit:
I have a similar problem with the following equation:
$$\displaystyle g(t) = \int_{t=0}^\infty a*c(t)^{1-\theta}dt$$

How do you calculate the following:
$$\displaystyle \delta g(t)/\delta t$$

Last edited by a moderator:

#### CaptainBlack

He says that:

$$\displaystyle f(t) = \int_{y=0}^t r(y)dy$$

And concludes that:

$$\displaystyle d f(t)/d t = r(t)$$

Could someone point me to the rule that allows you to do this, so that I can study it?
This is one of the fundamental theorems of calculus.

Edit:
I have a similar problem with the following equation:
$$\displaystyle g(t) = \int_{t=0}^{\infty} a*c(t)^{1-\theta}dt$$

How do you calculate the following:
$$\displaystyle \partial g(t)/\partial t$$
You can't because the right hand side is not a function of $t$ or rather $g(t)$ is independent of $t$ and so its derivative wrt $t$ is zero.

CB

Last edited:

#### questioner1

Thanks, I will look it up!

You can't because the right hand side is not a function of $t$ or rather $g(t)$ is independent of $t$ and so its derivative wrt $t$ is zero.

CB
Yes of course, you are right, because I made a typo. I need instead the derivative w.r.t to c. So what is :
$$\displaystyle \delta g(t)/\delta c$$

if

$$\displaystyle g(t) = \int_{t=0}^\infty a*c(t)^{1-\theta}dt$$

Last edited by a moderator:

#### greg1313

Forum Staff
Use \infty for $$\displaystyle \infty$$.

#### CaptainBlack

Thanks, I will look it up!

Yes of course, you are right, because I made a typo. I need instead the derivative w.r.t to c. So what is :
$$\displaystyle \delta g(t)/\delta c$$

if

$$\displaystyle g(t) = \int_{t=0}^\infty a*c(t)^{1-\theta}dt$$
Is $c$ a parameter or a function? Or do you really want the derivative wrt $\theta$.

Maybe you should just post the actual problem where this arises.

CB

#### questioner1

Is $c$ a parameter or a function? Or do you really want the derivative wrt $\theta$.

Maybe you should just post the actual problem where this arises.

CB
c is a function of t.

Here is the original problem, but the thing is that they use a trick, and they're not actually calculating the lagrangian in the traditional way:

#### CaptainBlack

c is a function of t.

Here is the original problem, but the thing is that they use a trick, and they're not actually calculating the lagrangian in the traditional way: ...
I still do not see the requirement to differentiate wrt c in that, there is some advanced hand waving to get away from the fact that you need to use the calculus of variations here (where in a sense differentiating wrt c makes some sense, but it is probably way above your pay grade).

.

#### questioner1

I still do not see the requirement to differentiate wrt c in that, there is some advanced hand waving to get away from the fact that you need to use the calculus of variations here (where in a sense differentiating wrt c makes some sense, but it is probably way above your pay grade).

.
Ok. Thank you!
I am going to have to learn calculus of variations then.
Before I do this, though, I would like to understand this:
The lagrange function that I posted represents the utility maximization problem of a consumer constrained by a budget. He has to maximize his utility by choosing a certain function for c(t). Can you solve this by differentiating w.r.t. c(t) using calculus of variations?

#### CaptainBlack

Ok. Thank you!
I am going to have to learn calculus of variations then.
Before I do this, though, I would like to understand this:
The lagrange function that I posted represents the utility maximization problem of a consumer constrained by a budget. He has to maximize his utility by choosing a certain function for c(t). Can you solve this by differentiating w.r.t. c(t) using calculus of variations?
The calculus of variations will allow you to find the minimising/maximising function for an integral. So yes that what it is designed to do.

.