Ec101: Behavioral Economics

Transcription

1 Ec: Behavioral Economics Answer Key to Homework # 4 th May 7 Question One (i Bayesian updating Let ( L p We can assume (following Kahneman & Tversky 97, sychological Review that the only two choices are lawyers or engineers, thus ( E p a sing Bayes rule, write down the posterior probability ( L Tom ( L Tom p ( Tom L ( L ( Tom L ( L + ( Tom E ( E p ( Tom L p ( Tom L p + ( Tom E( p b Graph the relationship between ( Tom and ( Tom 7 L likelihood ratios do not depend on the priors L (5 points, assuming the Substituting for p in our equation from (a gives us 7 ( L Tom ( Tom L x ( Tom L + ( Tom E( / 7 x + 7 ( L Tom ( Tom L x 7 ( Tom L + ( Tom E( 7 / x + Where ( Tom L ( Tom E x is not a function of p, by assumption We can plot the two posteriors parametrically by varying x from to ( is sufficient for graphing purposes As x is a ratio of probabilities, it cannot fall below Alternatively you can solve one equation for x and substitute back into the other to get a Cartesian equation, and plot that directly Here is my graph, using Mathematica, where ( Tom is on the y-axis and ( Tom is on the x-axis 7 L L

2 osterior Hp 7 L osterior Hp L c Now, graph the relationship between ( Tom and ( Tom assumptions as for question (b 9 L L (5 points, with the same Here is the same graph with the probabilities changed to 9 and respectively The curve bows out more osterior Hp 9 L osterior Hp L (5 points d If there is complete base rate neglect, what is the relationship between the two posteriors? Complete base rate neglect means that the prior probabilities add no information Mathematically this is accomplished by assuming ( L ( E 5 assumption, the ratio of the two posteriors is equal to the likelihood ratio i e ( Tom L ( Tom E ( L Tom ( E Tom nder this (5 points

3 Question One (ii More Bayesian updating e A disease has an overall base rate ( D sensitivity S, (ie ( positive D S C ( test negative D ( D test postive? A test for the disease has test and specificity Q (ie Q If a patient tests positive, what is the posterior ( D test postive ( test positive D ( D C C ( test positive D ( D + ( test positive D ( D S S + 999( Q f (Equivalent to (b in the question sheet Numerical examples (5 points If S Q 99 then the posterior is or 969 If S 99 and Q 9 then the posterior is Question Two Intertemporal Choice (5 points a When will a naïve hyperbolic discounter incur cost c? Begin by finding out what a naïve hyperbolic discounter (call him Naïf will choose at time t Naïf must pay the cost in one of the four periods, and will receive the benefit in the following period Naïf takes the cost Notation Discounted utility (according to Naïf s perception at in period time c + B( c + B ( c + B ( ( c + δ ( B e ( c + B( ( c + δ ( B e 4 ( c + B( 4 ( c + δ ( B e Naïf wants to maximize his utility He will choose which period to take the cost with this goal in mind Notice immediately that Naïf will not choose to take the cost in periods

4 or, because < δ and because ( t B is decreasing in t, < < Thus we need only compare utility between period and period Which is optimal will depend on the exact values of the parameters if and only if c + B c + ( δe + B( δ ( B e [] If equation [] holds true, then Naïf will take the cost immediately at time and the problem is solved To solve the rest of the question, now suppose equation [] is not true Now Naïf decides at period to take the cost in time Since Naïf is naïve, his preferences change at time, so he could possibly change his decision Naïf takes the cost in period Notation Discounted utility (according to Naïf s perception at time c + B c + B e ( ( ( c + B ( ( c + δ ( B e ( c + B( ( c + δ ( B e As before, Naïf will not choose to take the cost in period, because period is always a better choice Comparing the first two periods, we find ( B e c + ( B e c + (( δ e + B( δ [] The RHS of equation [] is strictly smaller than the RHS of equation [] Given equation [] does not hold, then the following is true c > ( δe + B( δ (( δ e + B( δ > 4

5 In other words, equation [] does not hold either so at time, Naïf will choose to take the cost in period Now consider what Naïf will do in period Naïf takes the cost Notation in period Discounted utility (according to Naïf s perception at time c + B c + B e ( ( ( c + B( 4 ( c + δ ( B e Note that if and only if ( B e c + ( B e c + (( δ e + B( δ [] The RHS of equation [] is also less than the RHS of equation [] Thus, if Naïf has to make a decision in period, he will choose to wait until period (given that equation [] must hold Thus if equation [] holds true, naïve hyperbolic discounters takes the cost at period Otherwise, they take the cost in period Note: if you numbered periods from -4, the solution will be the same (Naïf takes the cost in periods or 4, but equation [] will be slightly different [5 points well deserved] b When will a sophisticated hyperbolic discounter incur cost c? Again, we calculate the utility for a sophisticated hyperbolic discounter (call her Soph The e asiest way to find Soph s decision is to solve using backward induction The game takes on the following extensive form: 5

6 ( c, B,,, (, c, B e,, (,, c, B e, (,,, c, B e In each period, Soph can either take the cost, or defer until the next period (take the cost in the future The outcomes for each period of taking the cost in that period are depicted Working backwards, I find conditions under which Soph would take the cost at each period, assuming Soph understands her future behaviour At time t, Soph has no choice She must take the cost and receive the outcome shown At time t, Soph can choose between taking the cost now, with utility c + B e, or taking the cost at time, with utility ( c + ( B e She will choose whichever is greater ( B( δ + e( δ [4] So if equation [4] is true, Soph will take the cost in period If no t, she will defer to the next period Now we have two cases to consider 6

7 Case : Assume equation [4] holds ie Soph chooses over at time At time t Soph faces the choice of taking the cost now, and receiving utility ( B e c + or deferring to the next period where she knows (because she is sophisticated that her future self will take the cost in period, with utility (in time of ( B e c + She will choose whichever is greater ( B( δ + e( δ [5] It is fairly easy to show that if equation [4] is true, so too is equation [5] Thus at time t Soph will choose to take the cost in period (knowing that if she delayed, she would take the cost in period At time t Soph faces the choice of taking the cost now and receiving utility ( c + B or deferring to the next period, where she knows she will take the cost for sure because that is what is best for her future self, with utility c + ( B e ( B( δ + eδ [6] Again note that if equation [4] is true, so too is equation [6] Thus Soph will choose to take the cost at time So we can summarize case by saying if equation [4] holds, Soph chooses to take the cost at time Case : Assume equation [4] does not hold ie Soph chooses over at time As before, now consider Soph s decision at time t Soph faces the choice of taking th e cost now, and receiving utility c + ( B e or deferring to the next period 7

8 where she knows that her future self will take the cost in period, with utility (at time of c + ( B e ( B( δ + e( δ [7] So if equation [7] holds, Soph will choose to take the cost at time Otherwise Soph will delay, and eventually take the cost at time Now we can evaluate Soph s decision at time t, having calculated her decisions further down the game (assuming of course that equation [4] does not hold Case a: Assume equation [7] holds ie Soph chooses over at time Soph knows her choice her is between time and time Her utility, measured at time of taking the cost at time is c + ( B e cost now, with utility c + B Alternatively, she can take the ( B( δ + δe [8] She will take the cost at time if equation [8] holds Otherwise, she will take the cost at time Case b: Assume equation [7] does not hold ie Soph chooses over at time Here Soph has a choice between time and time, with utilities ( B e 4 c + respectively ( B( δ + e( 4δ c + B and [9] She will take the cost at time if equation [9] holds Otherwise, she will take the cost at time 8

9 To prove that we cannot rule out any of the options in Case, let B δ 5, B, and e nder these parameters, the following is true c Equation [4] Equation [7] Equation [8] Equation[9] Outcome 5 False True True Time 8 False True False Time False False True Time 4 False False False Time Thus a sophisticated hyperbolic discounter will take the cost in either periods,, or, with the exact tim e being specified by the following conditions Equation [4] Equation [7] Equation [8] Equation[9] Outcome True N/A N/A N/A Time False True True N/A Time False True False N/A Time False False N/A True Time False False N/A False Time [5 points also well deserved] Question Three Endowment Effects a Solve for such that the consumer is indifferent between the money and a pen c The utility from receiving a single pen is and the utility from receiving $ c is (,;, b + ϖ b u, (, c, c c c u, +ϖ Setting the two equations equal and solving gives us c b 9

10 (5 points b Find the value b which is the consumer s maximum willingness to pay The utility from receiving a single pen and giving up $ is u b (, b,, b b b +ϖb μϖb To find out the maximum price the consumer would be willing to pay to receive the pen, you must consider the alternative What is the consumer s utility if she pays nothing and receives nothing? Clearly, The consumer will pay $ only up to the point her utility from the transaction is Setting the equation above equal to and solving gives us ( ϖ b + b + μϖ b (5 points c Find the value s which is the consumer s minimum willingness to accept (the selling price The utility from giving up a pen and receiving $ s for it is u (, s,, s s μϖ b + ϖs The alternative (if you didn t engage in the transaction is that you would just have a pen (,,, b u Setting the two equations equal and solving gives ( + μϖ b s + ϖ d Show the parameter values which yield an endowment effect ( s > b (5 points s > b ( + μϖ ( + b b ϖ > + ϖ + μϖ ( + μϖ > ( + ϖ

11 + μϖ > + ϖ μ > [Note I assume ϖ > because the model does not make sense otherwise] e Where does the choosing value lie compared to buying and selling prices? c (5 points Since μ, we have + μϖ + ϖ Thus b + ϖ + μϖ b < b c b < b s + μϖ + ϖ Question Four Neuroeconomics (5 points This question has infinite possible answers ( points (9 OINTS TOTAL Some Notes on Grading (b If you graphed the posteriors against the likelihood ratio, instead of against each other, you lost points (Same for (c If you graphed against other things (there were quite a variety you generally lost points or more per graph (d No relationship got just point Most other answers got points or more (a You could achieve a maximum of 7 points if you only analysed Naïf s utilities at time (or time depending on which scale you used You lost one point for doing an analysis assuming no benefit if you take the cost in the last period (Mainly because this assumption simplified the analysis If you did not identify that Naif would never take periods & (or & then I subtracted 5 points each (ie 5 points if you missed both (b You could achieve a maximum of 8 points if you just examined payoffs at time (or without thinking about Soph s future tradeoffs You could get up to 4 points for any reasonable backward induction analysis, but 5 points was only given for positing

12 conditions under which periods, and (or, & 4 could exist (three people achieved this (a -