17.8 In period 1, Mr. Consistency and Mr. Hyperbola are each trying to come up with a plan for how much they will exercise in period 2. Exercise is less enjoyable than other leisure activities, leading to a loss of 100 in terms of period 2 utils. Exercise provides health benefits, realized in period 3, leading to a gain of 250 in terms of period 3 utils. They put the following weights on utilities each period:
CURRENT PERIOD AFTER TWO PERIODS AFTER
Period CURRENT One CURRENT ONE
Mr. Consistency 1 0.5 0.25
Mr. Hyperbola 1 0.35 0.175
According to this table, Mr. Consistency’s weight on future utility falls by half each period. Mr. Hyperbola’s weights are related to Mr. Consistency’s; the difference is that Mr. Hyperbola’s are reduced a further 30% for periods after the current one.
- Would Mr. Consistency plan to exercise in period 1? Would he follow through on this plan in period 2?
In period 1, Mr. Consistency utility = 250-100 = 150
Since, Mr. Consistency utility of 150 is more than the utility of 0 (Not doing any exercise) so he would exercise in period 1.
In period 2, Mr. Consistency Expected utility = 0.5*250 -100 = 125-100 = 25
So, he would follow through on this plan in period 2
- Show that Mr. Hyperbola would not follow through on his exercise plan.
Mr Hyperbola expected utility = 0.35 * 250 – 100 = 87.5-100 = -12.5
The expected utility is less than 0 and so Mr. Hyperbola would not follow through on his exercise plan.
- Suppose Mr. Hyperbola could sign a contract in period 1 that forced him to give up an amount of money valued at x utils in period 2 if he does not stick with his exercise plan. How
high would x have to be to help him commit to his plan?
Mr. Hyperbola’s contract value x= 12.5 utils