ECON 1723 Fall 2013 Problem Set Solution 2 - Economics 1723...

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Economics 1723: Problem Set 2 Solutions 1. (a) We draw an event tree which includes a node for whether regulation is present (its fine to draw separate trees): Regulation No regulation p λy 1 - p y p - ǫ (1 - k ) λy 1 - p + ǫ (1 - k ) y (b) For a general utility function, we have the following expressions for expected utility: EU ( NR ) = pU ( λy ) + (1 - p ) U ( y ) No regulation EU ( R ) = ( p - ) U ((1 - k ) λy ) + (1 - p + ) U ((1 - k ) y ) Regulation Specializing to the case where the utility function is logarithmic: without regulation we have EU ( NR ) = p log( λy ) + (1 - p ) log( y ) = p (log λ + log y ) + (1 - p ) log y = p log λ + log y and similarly with regulation we have EU ( R ) = ( p - ) log((1 - k ) λy ) + (1 - p + ) log((1 - k ) y ) = ( p - )(log(1 - k ) + log λ + log y ) + (1 - p + )(log(1 - k ) + log y ) = ( p - ) log λ + log(1 - k ) + log y (c) We conclude that regulation is preferable if EU ( R ) EU ( NR ): ( p - ) log λ + log(1 - k ) + log y p log λ + log y and hence k * is defined as the level at which these utility levels are exactly equal: ( p - ) log λ + log(1 - k * ) + log y = p log λ + log y log(1 - k * ) = log λ 1 - k * = λ = k * = 1 - λ Note that k * is a decreasing function of λ (i.e. increasing in the severity of the crisis) and an increasing function of (i.e. increasing in how much the probability of the crisis is reduced), but does not depend on p . For the specific numbers provided in the question we find k * = 1 - 0 . 8 0 . 01 0 . 0022 = 22 bps (recall that 1% = 100 basis points). So if national output is $16 trillion, the greatest acceptable dollar cost of financial regulation for these numbers is 22bps × 16 × 10 12 $35 billion 1
(d) Note that the reduction in loss of output due to regulation (not taking the cost of regulation into account) is p (1 - λ ) y - ( p - )(1 - λ ) y = (1 - λ ) y For the numerical case we have been analyzing, this is 0 . 01 × (1 - 0 . 8) = 20 bps whereas the calculation above suggests up to 1 - λ = 22 bps should be paid to avoid this cost. The politician’s criticism is therefore valid if you think risk neutrality is appropriate here. However, given that financial crises involve a reduction in national wealth, it is quite natural and intuitive to expect risk aversion. One might think of 2 bps as being the insurance premium worth paying over fair value to avoid a crisis. (e) Recall from lecture that the coefficient of relative risk aversion is the fraction of wealth an agent is willing to pay to avoid a small risk. This means that if a utility function with a higher coefficient of relative risk aversion than the logarithmic utility function was used, then the highest acceptable cost as a fraction of national wealth would be higher, and vice versa. The use of logarithmic utility simplifies the question by providing simply expressions for expected

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