David is thinking about investing $10,000 in Zoom. He might earn $2,000 over the next year with probability 0.8, or lose all money with probability 0.2. David’s utility function is given by U(W) = log W. 

a. Show the endowment point for no insurance case in the commodity space.

 b. An insurance company offers a policy that would pay x dollars in case of loss at a premium 1/5 x. Graph David’s budget constraint. 

c. Find the optimal level of insurance and the level of wealth under two contingencies. Does David insure fully? 

d. Repeat the previous part with insurance premium 2/5x What is the maximum amount David would pay for insurance? What is the minimum amount that the insurance company would accept? Determine the range of values where insurance may take place.