Problem
Consider a consumer who will live today and tomorrow. On each date, this consumer consumes a single good (manna); whose price on each date is $100 per unit. We write c0 for the amount of manna this consumer consumes today and c1 for the amount he consumes tomorrow. This consumer faces some uncertainty in how much manna he will be able to afford (to be explained in the next paragraph), and his preferences over uncertain consumption pairs (c0, c1) are given by the expectation of the von Neumann-Morgenstern utility function u(c0, c1) = In (c0)+ In (c1) r where In is the logarithm to the base e.
This consumer currently possesses $100 of wealth. He can spend it all today, or he can put some of it in the bank to take out and use to buy manna tomorrow. This bank pays no interest and has no deposit or withdrawal charges - $1 put in today will mean $1 taken out tomorrow. In addition to any savings that he carries forward, the consumer has a random amount of extra income, which he will receive tomorrow. If there is any uncertainty about this income, that uncertainty doesn't resolve until tomorrow - until after today's levels of consumption and savings must be fixed.
(a) Suppose the consumer is sure to receive an extra $34 tomorrow. What consumption level does he choose today, and what is his overall expected u tility?
(b) Suppose that, tomorrow, the consumer will receive an extra $100 with probability 1/2, and nothing more with probability 1/2. What consumption level does he choose today, and what is his overall expected utility? (To solve this analytically, you will have to solve a quadratic equation.)
(c) Suppose that, tomorrow, the consumer will receive nothing with probability 1/4, $34 with probability 1/2, and $100 with probability 1/4. What consumption level does he choose today, and what is his overall expected utility? (A good approximate answer is acceptable here, because to get the exact solution you have to solve a cubic equation. You will need to be quite accurate in your calculations; the third significant digit is important.)
(d) Assuming you get this far, what strikes you about the answers to parts (a), (b), and (c)? In particular, if we asked for this consumer's preferences over lotteries in the "extra income," where it is understood that all uncertainty resolves tomorrow, would this consumer satisfy the von Neumann-Morgenstern axioms?
The response should include a reference list. Double-space, using Times New Roman 12 pnt font, one-inch margins, and APA style of writing and citations.