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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 $$U=∑_t \delta^t \ln (C_t)$$ where $C_t$ represents consumption of the resource at time $t$. Also $\delta_t \in [0,1]$ is the discount factor for time $t$. Find the optimal consumption of at time $t$.

I have tried to apply Jensen's inequality since $\delta^t \ln (C_t)$ will be a concave function, but cannot figure out how to go beyond that.

gt6989b
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kangkan
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  • That thing totally didn't render right in mathjax. – Batman Feb 11 '14 at 18:17
  • i tried my best to represent what you wanted to write in intelligible MathJax. Please check... – gt6989b Feb 11 '14 at 18:24
  • It sounds like you want to pick the function $C_t$ to maximize $U$ while satisfying $\int_0^\infty C_t dt = X$. Is that correct? – gt6989b Feb 11 '14 at 18:27
  • Yes..that will be correct..thanks – kangkan Feb 11 '14 at 20:00
  • I suppose you meant that the discount factor is $\delta^t$. If so, you can solve this optimization problem using Lagrange multipliers. I got $C_t = \delta^t (1-\delta) X$. – JHF Feb 12 '14 at 01:00
  • Hi JHF, i am not familiar with Lagranges that involve integral constraints (or functions).Would be very grateful if you can show how to set up the lagrangian..thanks – kangkan Feb 12 '14 at 06:21

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