Question

Sally lives in town with an ongoing pandemic. If she goes to work, theres a probability, p, of getting infected by the virus that is causing the epidemic, where 0 < p < 1. If she stays at home, she can't be infected and will not earn income. Suppose when she goes to work and does not get infected, she will earn $50. If she gets infected he will lose $80 in health care cost, but will still earn an income of $50. Sally has past savings of $100 and has utility function U = ln(x) where x is money. Determine the range of values for p where Sally will stay home.

Answer 1

Sally will stay at home if the utility from staying at home is greater than going out and work.

And the utilities will depend on her income.

If sally goes to work she will earn $50 and with probability p she will get infected on going out and will have to spend $80 on health care and with probability 1-p she will not get infected and will earn $50. Ans she already have $1000 savings from past.

So the expected income from going out

= p×log(1000 + 50 - 80) + (1-p)×log(1000 + 50)

= p×log(970) + (1-p) × log(1050)

= p × log(970) + log(1050) - p × log(1050)

= p×2.98 + 3.02 - p×3.02

= 3.02 + 2.98p - 3.02p

= 3.02 - 0.04p

So the expected utility from going out is equal to 3.02 - 0.04p.

Utility from staying at home will be equal to,

= log(1000)

Since will earn Nothing by staying at home and will only have 1000 to spend.

= log(1000)

= 3.

Sally will only prefer to stay at home if the utility from staying at home is greater than the expected utility from going outside.

Utility from staying at home > utility from going outside

3 > 3.02 - 0.04p

0.04p > 3.02 - 3.0

0.04p > 0.02

p > 0.02/0.04

p > 2/4

p > 0.4

So for any p > 0.5 sally will prefer to stay at home than to go outside.

I hope I was able to help you, thank you.