Suppose$$an$$individual$$has$$two$$states$$of$$health:$$Healthy$$and$$unhealthy$.$When$$they$$are$$healthy$,$they$$earn$$$$10,000.$When$$they$$are$$unhealthy$,$they$$must$$take$$time$$off$$from$$work$$and$$now$$earn$$$$2,000.$Suppose$$the$$probability$$of$$the$$individual$$being$$healthy$$is$0\mathrm{.}90$and$$the$$probability$$of$$being$$unhealthy$$is$0\mathrm{.}10.$

Let's$$say$$the$$individual$$is$$offered$$to$$purchase$$an$$insurance$$contract$$for$$$$800$that$$pays$$the$$individual$$$$8,000$if$$they$$are$$sick$.$

a$)$How$$much$$will$$this$$individual$$earn$$when$$they$$are$$sick$$if$$they$$purchase$$insurance$?$How$$much$$will$$this$$individual$$earn$$when$$they$$are$$healthy$$if$$they$$purchase$$insurance$?$

b$)$What$$is$$the$$individual$\text{'}$s$$expected$$payoff$$if$$they$$do$$not$$purchase$$insurance$?$

c$)$If$$the$$individual$$does$$not$$purchase$$insurance$,$is$$it$$possible$$for$$them$$to$$earn$$$$9,200?$What$$if$$they$$do$$purchase$$insurance$?$

d$)$What$$does$$having$$this$$insurance$$contract$$do$$to$$the$$individual$\text{'}$s$$uncertainty$$of$$income