Intertemporal Maximization

[kangkan]

Hi,this question if from ISI sample paper..i am not sure how to approach it.A community has a fixed stock X of oil that it has to consume over an infinite horizon. The utility function to be maximized is U=U=∑▒〖(δ^t)ln⁡(C_(t)) 〗 where Ct represents consumption of the resource at time t. Also δ ( between 0 to 1) is the discount factor. Find the optimal consumption of at time t.

Is there a way to apply lagarangean when the function and the constraints are in the form of integrals?

Thanks

[economist]

Its a question of consumption smoothing. As utility is concave person will choose same consumption every year. Simply use the budget constraint  to find that constant c. ASSUMING OFCOURSE PERSON IS ALLOWED TO BORROW AND LEND FREELY.
 Its a solved example in hammond , you can refer.

[kangkan]

Hi..Are you referring to Example 17.14 on page 643.. i looked it up...but in this case the the function to be maximized shoud be Integral(0 to infinity) Delta^t* ln(ct)..and the budget constraint will be integral(0 to infinity) e^-delta*t*c(t)...hence the the special case condition mentioned in the example seems inapplicable..thats how it seems to me..would appreciate your input..thanks