U L 1 L 1 1 92400 p 4 9960 p 4 9 480 p 1 9 1 9 n 2400 p 2400 p 4 9 n 2400 p 960

U l 1 l 1 1 92400 p 4 9960 p 4 9 480 p 1 9 1 9 n 2400

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U ( L 1 | L 1 ) = (1 / 9)(2400 - p ) + (4 / 9)(960 - p ) + (4 / 9)( - 480 - p )+ + (1 / 9) [(1 / 9) n (2400 - p | 2400 - p ) + (4 / 9) n (2400 - p | 960 - p ) + (4 / 9) n (2400 - p | - 480 - p )] + (4 / 9) [(1 / 9) n (960 - p | 2400 - p ) + (4 / 9) n (960 - p | 960 - p ) + (4 / 9) n (960 - p | - 480 - p )] + (4 / 9) [(1 / 9) n ( - 480 - p | 2400 - p ) + (4 / 9) n ( - 480 - p | 960 - p ) + (4 / 9) n ( - 480 - p | - 480 - p )] = 480 - p + (4 / 81)( μ (1440) + μ (2880) + μ ( - 1440) + μ ( - 2880)) + (16 / 81)( μ (1440) + μ ( - 1440)) = 480 - p + (4 / 81)(1440 + 2880 - 3 · 1440 - 3 · 2880 + 4 · 1440 - 4 · 3 · 1440) = 480 - p - (4 / 81)(20 , 160) = 480 - p - (4 / 9)(2240) U ( L 0 | L 1 ) = 0 + (1 / 9) n (0 | 2400 - p ) + (4 / 9) n (0 | 960 - p ) + (4 / 9) n (0 | - 480 - p ) = (1 / 9)( μ ( - 2400 + p ) + 4 μ ( - 960 + p ) + 4 μ (480 + p ) = (1 / 9)( - 16 , 800) + (1 / 9)(19) p = - (1 / 3)(5600) + (19 / 9) p U ( L 1 | L 1 ) > U ( L 1 | L 0 ) 480 - p - (4 / 9)(2240) > (19 / 9) p - (16 , 800 / 9) 12160 / 28 434 . 3 > p (b) Explain intuitively how your answers to 2(a) would differ for a risk-averse (but not loss-averse) agent. Problem 2. Juanjuan works for an hourly wage w , and has full flexibility over the number of hours she works each day H . Her daily income is wH , and she has a daily income target of £ 120. Her total daily utility is then U ( H | w ) = v ( wH ) - c ( H ) where: v ( wH ) = wH if wH 120 120 + 1 3 ( wH - 120) if wH > 120 and c ( H ) = 2 H 2 2
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EC 310 - Behavioural Economics 1. Suppose JJ arrives at work on Monday to find her hourly wage that day is £ 48. Solve for JJ’s optimal number of hours worked H * . (Note: given the piecewise-differentiable utility function, you must solve the FOC separately in each section and determine whether it yields a valid answer. If not, it is sufficient to show that the marginal utility is negative when moving in either direction away from the ‘kink’). How much does JJ earn? Suppose JJ will earn less than £ 120. Then the FOC from her maximisation is w - 4 H = 0, so H * = 12 and wH * = 576. Since 576 > 120, this is not a valid solution. What if we thought JJ would earn more than £ 120? Then the FOC from this part of the max- imisation is 1 3 w - 4 H = 0, so H * = 48 / 12 = 4 , wH * = 192 > 120 and we have a valid interior solution on our hands. Noting that the problem is globally concave, we don’t have to worry about second-order conditions. 2. On Tuesday, JJ learns her hourly wage has decreased to £ 20. How much does she work that day? How much does she earn? H * (20) = 5, wH * (20) = 100 . 3. On Wednesday, JJ learns her hourly wage has increased again to £ 24. How much does she work and earn that day? Suppose JJ will earn less than £ 120. Then the FOC from her maximisation is w - 4 H = 0, so H * = 6 and wH * = 144. Since 144 > 120, this is not a valid solution. What if we thought JJ would earn more than £ 120? Then the FOC from this part of the maximi- sation is 1 3 w - 4 H = 0, so H * = 24 / 12 = 2 , wH * = 48 < 120 and we again have an invalid solution. What if we thought that JJ would earn exactly £ 120? This is achieved by working 120 / 24 = 5 hours, so the marginal disutility of effort is 20. She will not deviate from 5 hours, then, because ( dv/dH ) - = 24 > 20 > 8 = ( dv/dH ) + .
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