Please Derive Line(30)

consumer i 's utility maximization problem: xiR+nmaxs.t.:ui=ui(xi)pximi;miwzi+k=1nikk,iIik=1k,p0n,ziR+m, given : p,mi. - zi=(z1i,,zhi,,zmi)T:i 's input supply vector standard properties of utility function ui(xi) (assumed throughout): - real-valued: ui(xi)R. - continuous: the graph of ui(xi) does not have any jumping part - strongly increasing: ui(xi1)>ui(xi0) for xi1xi0,xi1=xi0. (uki(xi)>0k: marginal utility of goodk is positive ) - strictly quasiconcave: ui((1t)xi1+txi2)>min[ui(xi1),ui(xi2)]xi1,xi2,xi2=xi1t(0,1). Lagrangian method (see section 7.2, Math Apps): L(xi,)ui(xi)+(mipxi). ui(xi) is strictly increasing mipxi=0 (budget balancedness): Lk(xi,)=uki(xi)pk=0,k=1,,n,L(xi,)=mipxi=0. (24) : uki(xi)/uli(xi)=pk/pl,k,l=1,,n. (25),(26) Marshallian (ordinary) demand function: xi=xi(p,mi) 3.3 Equilibrium market-clearing conditions: ykiIzhi=iIxki,k=1,,n,=k=1nzhk,h=1,,m. (23): 0=k=1npk(iIxkiyk)+h=1mwh(k=1nzhkiIzhi) equilibrium: pn1. (18), (19), (20), (23), (27), (28), (29) Walrasian (competitive) equilibrium: yk(pk,w)iIzhi=iIxki(p,h=1mwhzhi+k=1nikk(pk,w)),k=1,,n1,=k=1nzhk(pk,w),h=1,,m. (21), (26): uki(xi)/uli(xi)=pk/pl=fhl(zl)/fhk(zk)iIh=1,,mk,l=1,,n. (17): fhk(zk)/fgk(zk)=wh/wgk=1,,nh,g=1,,m. 6 consumer i 's utility maximization problem: xiR+nmaxs.t.:ui=ui(xi)pximi;miwzi+k=1nikk,iIik=1k,p0n,ziR+m, given : p,mi. - zi=(z1i,,zhi,,zmi)T:i 's input supply vector standard properties of utility function ui(xi) (assumed throughout): - real-valued: ui(xi)R. - continuous: the graph of ui(xi) does not have any jumping part - strongly increasing: ui(xi1)>ui(xi0) for xi1xi0,xi1=xi0. (uki(xi)>0k: marginal utility of goodk is positive ) - strictly quasiconcave: ui((1t)xi1+txi2)>min[ui(xi1),ui(xi2)]xi1,xi2,xi2=xi1t(0,1). Lagrangian method (see section 7.2, Math Apps): L(xi,)ui(xi)+(mipxi). ui(xi) is strictly increasing mipxi=0 (budget balancedness): Lk(xi,)=uki(xi)pk=0,k=1,,n,L(xi,)=mipxi=0. (24) : uki(xi)/uli(xi)=pk/pl,k,l=1,,n. (25),(26) Marshallian (ordinary) demand function: xi=xi(p,mi) 3.3 Equilibrium market-clearing conditions: ykiIzhi=iIxki,k=1,,n,=k=1nzhk,h=1,,m. (23): 0=k=1npk(iIxkiyk)+h=1mwh(k=1nzhkiIzhi) equilibrium: pn1. (18), (19), (20), (23), (27), (28), (29) Walrasian (competitive) equilibrium: yk(pk,w)iIzhi=iIxki(p,h=1mwhzhi+k=1nikk(pk,w)),k=1,,n1,=k=1nzhk(pk,w),h=1,,m. (21), (26): uki(xi)/uli(xi)=pk/pl=fhl(zl)/fhk(zk)iIh=1,,mk,l=1,,n. (17): fhk(zk)/fgk(zk)=wh/wgk=1,,nh,g=1,,m. 6