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Consider the following optimization problem (called cake-eating):

$$\sum\limits_{t = 0}^{\infty} u(a_t) \to \max$$

subject to

$$\sum\limits_{t = 0}^{\infty} a_t \leq s, \quad s >0, \quad a_t \geq 0$$

Show that if $u(a)$ is increasing, whose derivative at the zero point tends to infinity, and $u(a)$ being a strictly concave then this problem has no solution.

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Abbyss
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  • Cake eating? Why? It would be great if you added context to this. Questions in the imperative without context, like this one, usually get put on hold on this site. – Giuseppe Negro Oct 18 '19 at 12:30
  • @GiuseppeNegro, Because it is a well-known task in the field of optimization. I can find an approximate description of the origin of the problem, but the point is not even in the essence of the problem, but in how to approach the problem in order to solve it. – Abbyss Oct 18 '19 at 12:36
  • @GiuseppeNegro, I wrote the name of the task in case someone knows it well and can immediately give an answer. – Abbyss Oct 18 '19 at 12:39
  • Ok, I was asking out of curiosity, frankly. That's a funny name. :-) – Giuseppe Negro Oct 18 '19 at 12:41
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    @GiuseppeNegro, i found this interpretation: Suppose that you have a cake of size W. At each point of time, t=1,2,3, ....T you can consume some of the cake and thus save the remainder. Let a_t be your consumption in period t and let u(a_t) represent the flow of utility from this consumption. Here you can find a detailed description of the problem and the search for solutions: http://www2.econ.iastate.edu/tesfatsi/dpintro.cooper.pdf, but it is not very clear why there is no solution in this case. – Abbyss Oct 18 '19 at 12:48
  • Do we have $u(0)=0$? –  Oct 18 '19 at 14:39

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Hint: Essentially, if the utility function is as described in your setting, you can increase the utility by spreading out you cake eating further into smaller chunks. If the derivative of $u$ tends to infinity at zero then for sufficiently small $a$ you have $2u(a/2) > u(a)$. So for any given sequence $a_t$ you can improve utility by cutting your daily slice in half and eating it over two days.

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