Question EH: .lohn can consume tw'o goods, or. and 2,, at price p. and p,, respectively, with

301,305 2:- 11. His income is y 2:- D. In addition to consuming :1 and in, John can also

work out at the gym. Let t E [1}, 1] denote the intensity with which John exercises.

Exercise is free as long as it is not too intense, or t 5 %. However, any t ‘2:- % requires a gym membership, the cost of 1which is m, m E [my]. John's prelierenoes can he

described by the utility function n[o:1,org,t} = alzgtﬂ — tII. Finally, assume that it: there are two utility-maximising vectors, (zi,:;,t’} and

[mini t"), then John picks the one that involves the larger amount of exercise. [1']- Are John’s preferences non-satiated‘? Are they convex? Prove and explain

your answers. Solve John's utility maximisation problem, deriving the optimal

values of on, org, and t as functions of the parameters. Assume in ﬁle remoirsder' of the gisstierr that John potentially tinesfor two periods.

Speciﬁoally, John survives to period 2 with probability git}, where ﬁt]- E [I], l] and

edit} ‘2:- ll 2 flit}. Should he survive to period 2, John's income remains unchanged,

as do p1 and p:. Howenmr, the cost ofa gym membership changes from m to ml E [thy]

from period 1 to period 2. Barr's-1m and saving is ruled out". There is no discounting.

.lohn seeks to maximise his expected utility over the two periods. Assume that

m,m' 5kg, where k =[ —;'J§j e [I], 1}. [ii]- Solve John‘s new utility maximisation problem, deriving consumption and ex—

ercise levels in as much detail as possible in both periods [exercise in period

1 can only he implicitly characterized]. Explain how you used the restrictions

on m and no“. How does the period 1 exercise level depends on m and m“?

Explain the intuition. [iii]- Assume the ﬁrm must determine at at the beginning of period 1 and is not

allowed to change its price in period 2. Derive the optimal value of m. [in]- Jlssume now that the ﬁrm at the beginning out-period 1 can commit to prices

at and m" in period 1 and 3, respectively. Can you determine if m is diEerent

from the one you derived in part {ii-i}? Explain. Hover about 111'? lf the answer

is ambiguous, then explain the intuition. [v]- :E'acplaiu how you think the answer to part [iii] might change if the restriction that m, rrs|l if. by is dropped. lnparticular, could it be optimal to charge in. = rrs|l

marginally higher than icy? Explain the intuition.