iii Well \u03c6 w pu w a H r H r 1 p u w a L r L r is ambiguous since H r 0 but L r

Iii well φ w pu w a h r h r 1 p u w a l r l r is

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(iii) Well, φ w = pu ′′ ( w + a ( H - r ))( H - r ) + (1 - p ) u ′′ ( w + a ( L - r ))( L - r ) is ambiguous, since H - r > 0 but L - r < 0. (If u ′′′ () < 0, however, it turns out that if w then a ) (iv) We use the envelope theorem. In particular, let V ( H, L, p ) = [ pu ( w + a ( H - r )) + (1 - p ) u ( w + a ( L - r ))] a = a * Then ∂V ( H, L, p ) ∂H = pu ( w + a ( H - r )) a which has the same sign as a . ∂V ( H, L, p ) ∂L = pu ( w + a ( L - r )) a which has the same sign as a , ∂V ( H, L, p ) ∂p = u ( w + a ( H - r )) - u ( w + a ( L - r )) Since u () is increasing, this is positive if w + a ( H - r ) > w + a ( L - r ), or a H > a L . So if a > 0, ∂V/∂p > 0, but if a < 0, ∂V/∂p < 0. 4
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6. Suppose an agent maximizes max x f ( x, c ) yielding a solution x ( c ). Suppose the parameter c is perturbed to c . (i) Use a second-order Taylor series expansion to characterize the loss that arises from using x ( c ) instead of the new maximizer, x ( c ). Show that for small changes in c , the loss to the objective function is proportional to ( x ( c ) - x ( c )) 2 . (ii) Explain the strengths and weaknesses of using this as a model of “sub- rational” decision-making, where agents fail to re-optimize after an unexpected shock. How would you construct such a model? Linearize f ( x, c ) in x ( c ) around x ( c ) to get f ( x, c ) = f ( x ( c ) , c ) + f x ( x ( c ) , c )( x - x ( c )) + f xx ( ξ, c ) ( x - x ( c )) 2 2 where ξ is between x and x ( c ). Since f x ( x ( c ) , c ) = 0 from the FONCs, we can rearrange to get f ( x, c ) - f ( x ( c ) , c ) = f xx ( ξ, c ) ( x - x ( c )) 2 2 Evaluate at x = x ( c ), and f ( x ( c ) , c ) - f ( x ( c ) , c ) = f xx ( ξ, c ) ( x ( c ) - x ( c )) 2 2 So the loss that the agent incurs by not re-optimizing is proportional to ( x ( c ) - x ( c )) 2 , so the loss is “not too large” as long as x ( c ) and x ( c ) are close. One strength is that it is simple. We can easily see how the second-derivative is related to the approximate loss in welfare, and the model can be applied to many situations. One weakness is that even though c and c might be close, the optimal policies may be very far apart, or f xx ( ξ, c ) might be very large. This makes it hard to judge whether or not the loss is large without looking at a particular problem. I would introduce a fixed cost to changing policies, so that whenever the agent wants to move from x ( c ) to the new optimum x ( c
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