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there are two types of consumer- sophisticated and naive. types are private information of consumer, not know to the firm. consumers have changing preferences. let X be the set of action that consumer can take in period 2 , if he has has accepted firms price scheme. first period preferences are given by u and second period preferences are given by v.sophisticated consumer knows that his preference in period 2 will be given by v but naive consumer believes that his preferences will not change. Price scheme t : X -> R. c(x) denotes the cost that firm incurs. let X= {L,H}. monopolist offers a price scheme of the form {ts,tn}, the former aimed at sophisticated consumer and latter is aimed at naive.
L H
u 1 1
v 0 2
c 0 c(H) belongs to (1,2)
Ques: show that in an optimal monopoly menu of the form {ts, tn} no price scheme strictly dominates another. show, however, that it is possible that tn weakly dominates ts; i.e. ts(x)>= tn(x) for all x belonging to X, with at least one strict in equality. (ref : bounded rationality and industrial organization- spiegler)
Solution: optimal price scheme for sophisticated consumer
max t(x)-c(x) s.t.u(x)- t(x)>= 0
xs= arg max(u-c)
ts=u(xs)
for above case it will be
Xs=L ts(L)= 1 Ts(H)>3 since v(L)-ts(L)>= V(H)-ts(H)
optimal price scheme for naive consumer {xu,tu, xv,tv}
firm knows that in period 2 naive consumer's preference will be given by v.
so firm will maximise his profit tv - c(xv) subject to icv icu and u(x)- t(x)>= 0
for above case solution will be
xv=L xu=H tn(L)=1 and tn(H)=3
please check if my solution is correct.
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