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Question EH:lohn can consume tw'o goods, or. and 2,, at price p. and p,, respectively, with 301,305 2: His income is y 2: In addition to consuming :1...

Doc 2.pngFurther, what is the difference between local-non satiation and strict monotonicity?

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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.

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