Chapter 18 Solutions

Contributed by Joseph Murrey and Yingjian Cao 2017

18.1 Lawyer’s Optimal effort maximizes his surplus.

\[\max_{l} \frac{1}{3}l-\frac{l^2}{2}\]

FOC:

\(\frac{1}{3} - l = 0\)

\(\Rightarrow l = \frac{1}{3}\)

Lawyer’s surplus: \((\frac{1}{3})^2 - \frac{1}{2}(\frac{1}{3})^2 = \frac{1}{18}\)

Plaintiff’s surplus: \(l - \frac{1}{3}l = \frac{1}{3} -\frac{1}{3}*\frac{1}{3} = \frac{2}{9}\)

  1. \[\max_{c} cl - \frac{l^2}{2}\]

FOC: \(c - l = 0\)

\(l^* = c\)

Lawyer’s surplus: \(c^2 - \frac{1}{2}c^2 = \frac{1}{2}c^2\)

Plaintiff’s surplus: \(l-cl=l(1-c)\)

  1. max plaintiff’s surplus, \[\max_{c} (1-c)c\]

FOC: \(1 - 2c = 0 \Rightarrow c^* = \frac{1}{2}\)

Lawyer’s surplus: \(\frac{1}{2}(\frac{1}{2})^2 = \frac{1}{8}\)

Plaintiff’s surplus: \(\frac{1}{2} - (\frac{1}{2})^2 = \frac{1}{4}\)

  1. Lawyer’s surplus:

\[\max_{l} cl - \frac{l^2}{2} - p\]

FOC: \(c - l = 0 \Rightarrow l^* = c\)

Plaintiff’s surplus: \(l - cl + p = c - c^2 + P\)

\[\max_{c} c - c^2 + P\]

s.t. \(cl - \frac{l^2}{2} - P \leq 0\) Lawyer’s participation constraint

\[\max_{c} c - c^2 + P\]
s.t. \(c^2 - \frac{c^2}{2} - P \leq 0\)

Setting the inquality equal to 0, \(P = \frac{1}{2}c^2\). Now sub the condition from the participation constraint into the Plaintiff’s surplus. Then you can find the c that maximizes the plaintiff’s surplus given that he will take \(p = \frac{1}{2}c^2\) upfront.

\[\max_{c} c - c^2 + \frac{1}{2}c^2\]

FOC: \(1 - 2c + c = 0 \Rightarrow 1 -c = 0 \Rightarrow c^* = 1\)

Now calculating the surplus of laywer and plaintiff.

\(P = \frac{1}{2}\)
Lawyer surplus \(= 0\)

18.4

  1. \[prem=\$5,000\]

\[U=\ln(20,000-5,000)=9.616\]

  1. \[prem=\$2500\]

\[EU=0.5\ln(20k-5k-2.5k)+0.5\ln(20k-2.5k)=9.602\] (a) is better.

  1. \[prem=3500\] \[EU=0.5\ln(20k-3500-1750)+0.5\ln(20k-1750)=9.706\] They would rather take more care and buy partial insurance.

(Solution to this moral hazard problem has to get resort to share in the risk)

18.5

Lefties no ins

\[EU=0.8\ln(9,000)+0.2\ln(10,000)=9.1261\]

Righties no ins

\[EU=0.2\ln(9,000)+0.8\ln(10,000)=9.1893\]

L Premium:

\[\ln(10,000-P_L)=9.1261\]

\[P_L=\$807.897\]

(monopolist gets max WTP)

R Premium:

\[\ln(10,000-P_R)=9.1893\] \[P_R=\$208\]

\(\hspace{0.3cm}max\hspace{0.3cm}0.5(P_L-0.8\cdot1000)+0.5(P_R-0.2\cdot X_R)\) (Insurers profit)

\(^{P_L,P_R,X_R}\)

Subject to participation constraint for righties \[0.2\ln(\$10k-P_R+X_R)+0.8\ln(\$10k-P_R)\geq0.91893\]

(low risk participates)

&Incentive compatability constraint for lefties \[\ln(10k-P_L)\geq0.2\ln(10k-P_R+X_R)+0.8\ln(10k-P_R)\]

\(^{full insurance designed for lefties}\hspace{1cm} ^{partial insurance designed for righties}\)

Prices and coverage are such that high risk type does not even want to pretend to be low risk type.

Notice that for moral hazard partial coverage corrected the bad behavior.

Here full coverage for the high risk corrects the bad outcome (if prices & coverage level set correctly).