Treść książki
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JustifythatifPiD(piI)CXiR+
2ithenforlinearutilityfunction:
−ifVA>oIi(alia2)iA(plip2)iAp,theproblem(P2)hasaninfinitenumber
ofoptimalsolutionsbelongingtothesegmentx-iIx1+:x2i
wherex1i(
pl
l
io)ix2i(oi
p2
l
),
VIi:?oiI+:i1,
−ifIi(alia2)±A(plip2)iAp,theproblem(P2)hasexactlyoneoptimalsolution:
x-i(
pl
l
io)orx-i(oi
p2
l
),
−fortheremainingutilityfunctionstheproblem(P2)hasexactlyoneoptimalsolution:
3lIi:>oiI+:i1x-iIx1+:x2,wherex1i(
p1
l
io),x2i(oi
p2
l
),
or:
31Ii:>oiI+:i1x-i(I
pl
l
;:
p2
l
)ł
Ad1.d
Fromageometricillustrationofasolutiontotheconsumptionutilitymaximizationproblem:
(4)
(5)
(6)
u(xlix2)im.n{alxl;a2x2}ąmax
plxl+p2x2śIi
xlix2?oi
itresultsthattheoptimalsolutionsatisfiesasystemofequations:
(7)
(8)
plx-l+p2x-2iIi
alx-lia2x-2ł
Ithastheform:
x-i(
a2pl+alp2
a2l
;
a2pl+alp2
all
)i(
(pla2+p2al)
pla2
pl
l
;
(pla2+p2al)
p2al
p2
l
)i(I
pl
l
i:
p2
l
),
where:Ii
(pla2+p2al)
pla2
>oi:i
(pla2+p2al)
p2al
>oisuchthatI+:i1ł
(9)
Conclusions
C.1SetoffeasiblesolutionsD(piI)CXiR+
2isacompactandconvexset.
C.2Theutilityfunctionisincreasing(orweaklyincreasing),concaveorstrictlyconcave.
C.3Ifthesetoffeasiblesolutionsiscompactandconvex,andtheobjectivefunctionis
strictlyconcaveandincreasing,thentheproblem(P2)hasexactlyoneoptimalsolution.
C.4Ifthesetoffeasiblesolutionsiscompactandconvex,andtheobjectivefunctionis
concaveandincreasing(orweaklyincreasing),thentheproblem(P2)hasatleastone
optimalsolution.Iftherearemorethanone,theyformacompactandconvexset.
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