In some cases individuals may care about the date at which the uncertainty
they face is resolved. Suppose, for example, that an individual knows that his
or her consumption will be 10 units today \((Q)\) but that tomorrow's
consumption \(\left(\mathrm{C}_{2}\right)\) will be either 10 or \(2.5,\)
depending on whether a coin comes up heads or tails. Suppose also that the
individual's utility function has the simple Cobb-Douglas form
\\[U\left(Q, C_{2}\right)=V^{\wedge} Q.\\]
a. If an individual cares only about the expected value of utility, will it
matter whether the coin is flipped just before day 1 or just before day \(2 ?\)
Explain.
More generally, suppose that the individual's expected utility depends on the
timing of the coin flip. Specifically, assume that
\\[\text { expected utility
}=\mathbf{f}^{\wedge}\left[\left(\mathrm{fi}\left\\{17\left(\mathrm{C},
\mathrm{C}_{2}\right)\right\\}\right) \text { ) } \text { ? } |\right.\\]
where \(E_{x}\) represents expectations taken at the start of day \(1, E_{2}\)
represents expectations at the start of day \(2,\) and \(a\) represents a
parameter that indicates timing preferences. Show that if \(a=1,\) the
individual is indifferent about when the coin is flipped. Show that if \(a=2,\)
the individual will prefer early resolution of the uncertainty - that is,
flipping the coin at the start of day 1 Show that if \(a=.5,\) the individual
will prefer later resolution of the uncertainty (flipping at the start of day
2 ). Explain your results intuitively and indicate their relevance for
information theory. (Note: This problem is an illustration of "resolution
seeking" and "resolution-averse" behavior. See D. M. Kreps and E. L. Porteus,
"Temporal Resolution of Uncertainty and Dynamic Choice Theory," Econometrica
[January \(1978]: 185-200\).
