Question: Consider a consumer who lives for two periods. The first period income is \(m_{1}\), the second period income is \(m_{2}\). Denote the consumer's consumption in the first period and second period by \(c_{1}, c_{2}\), respectively. The consumer's preferences are given by \(\sqrt{c_{1}}+\delta \sqrt{c_{2}}\).

1. Write down the consumer's choice problem.
2. Derive the first order condition and characterize the solution.
3. What is the solution to the problem if \(\delta=0\) ? What is the solution if \(\delta=1\) ? explain.
4. For general \(0<\delta<1\), is it possible that the consumer's consumption in each period is the same (that it, the solution is \(\left.c_{1}^{*}=c_{2}^{*}\right) ?\) If it is, find the conditions under which this is the solution.

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