Lecture 2: Perfect Substitutes

Perfect Substitutes

  • Given → U=x12U= x^{\frac{1}{2}} m=6m=6 Px0=2Px_{0}=2 Px1=1Px_{1}=1 Py=yPy=y
  • Bundle A → Px=2Px=2 BLA=2x+Uy=6BL_{A}=2x+Uy=6

MRSA=MUxMUy=12x12=PxPy=2U=12MRS_{A}=\frac{MUx}{MUy}=\frac{1}{2}x^-{\frac{1}{2}}=\frac{Px}{Py}=\frac{2}{U}=\frac{1}{2} x=1,y=1,U=2x=1, y=1, U=2

A(1,1)A(1,1)

  • Bundle B → U8=UA=2=X812=Y8U_{8}=UA=2=X_{8}^{\frac{1}{2}}=Y_{8}

MRSc=12x12=1UMRS_{c}=\frac{1}{2}x\frac{-1}{2}=\frac{1}{U} x8=U,y8=0,U=2x_{8}=U, y_{8}=0, U=2

B(U,0)B(U,0) BL0=x+Uy=UBL_{0}=x+Uy=U CV=2CV=-2

  • Bundle C → Px=1Px=1  BLc=x+Uy=6BL_{c}=x+Uy=6

MRSc=12x12=1UMRS_{c}=\frac{1}{2}x^-{\frac{1}{2}}=\frac{1}{U} Xc=y,Yc=12,U=52Xc=y, Yc=\frac{1}{2}, U=\frac{5}{2}

C(U,12)C(U, \frac{1}{2})

  • Bundle D → UD=Uc=52=XD12=YDU_{D}=U_{c}=\frac{5}{2}=X_{D}^{\frac{1}{2}}=Y_{D}

MRSD=MRSA=12x12=12MRS_{D}=MRS_{A}=\frac{1}{2}x^{\frac{-1}{2}}=\frac{1}{2} XD=Y,YD=32,D(1,32)X_{D}=Y, Y_{D}=\frac{3}{2}, D (1, \frac{3}{2})

BLD=2x+Uy=8BL_{D}=2x+Uy=8 CV=86=2CV=8-6=2

  • Demand → 12x12=PxPy\frac{1}{2}x^{}\frac{-1}{2}=\frac{Px}{Py} Py=UPy=U Px=X12=2Px=X^{\frac{-1}{2}}=2

x=UPx2=UPx2x=UPx^{-2}=\frac{U}{Px^{2}}

CS=12UPx2CS=\int_{1}^{2}\frac{U}{Px^{2}}  dPx=UPx112=2(4)=2dPx=-UPx^{-1}\int_{1}^{2}=-2-(-4)=2

  • When you shift the budget line parallel, there is no income effect- Quas linear utility.


Perfect Substitutes

a. Px=y,Py=2,M=12Px=y, Py=2, M=12 Ocostx=2>MRS O-cost_{x}=2> MRS Px=y,Py=2,M=12,Ocostx=2>MRSPx=y, Py=2, M=12, O-cost_{x}=2> MRSso buy yy

BLA=12,Ux+2yBL_{A}=12, Ux+2y y=122=6y=\frac{12}{2}=6 . x=0x=0 A(0,6)A(0,6) U=6U=6

b. Px=1Px=1 Py=2Py=2 M=?M=? U8=UA=6=x+yU_{8=U_{A}}=6=x+y Ocost=2O-cost=2 buy all xx .

B(6,0)B(6,0) CostB1(6)+2(0)=6Cost B1(6)+2(0)=6

CV=126=$6CV=12-6=\$ 6

BL0=6=x+2yBL_{0}=6=x+2y

c. Px=1Px=1 Py=2Py=2 M=12M=12 O=costx=12<MRSO=cost_{x}=\frac{1}{2}< MRS so all xx .

BLc=12=x+2yBL_{c}=12=x+2y y=0y=0 c(12,0)c(12,0)

d. Px=yPx=y Py=2Py=2 M=?M=? U0=UcU_{0}=Uc OCostx=2O-Cost_{x}=2

U=12U=12 y=12y=12 x=0x=0 D(0,12)D(0,12)

CostD=$2UCost D=\$ 2U

Ev=$12Ev=\$12 BLD=2U=Ux+2yBL_{D}=2U=Ux+2y

  • Approx ΔCS\Delta CS ABA\rightarrow B

MRS=OCost1=PxPyMRS=O-Cost\rightarrow 1=\frac{Px}{Py} Px=PyPx=Py OxMPx=12Px=6O\leq x\leq \frac{M}{Px}=\frac{12}{Px}=6

Px>PyPx> Py x=0x=0

Px<PyPx< Py x=12Pxx=\frac{12}{Px}

ΔCS=1212Pxd\Delta CS=\int_{1}^{2\frac{12}{Px}}d Px+2UPx+\int_{2}^{U} 0dPx0dPx

=12InPx12=8.32=12 In Px\int_{1}^{2}=8.32

Exact ΔCS\Delta CS

UA=U8=6=x+yU_{A}=U_{8}=6=x+y

Px=PyPx=Py 0x60\leq x\leq 6

HxHx \mid  Px>PyPx>Py x=0x=0

Px<PyPx<Py x=6x=6

ΔCS=126\Delta CS=\int_{1}^{2}6 dPx+2U0dPx=6Px12=$6=CVdPx+\int_{2}^{U}0dPx=6Px\int_{1}^{2}= \$ 6= CV




  • Exact ΔCS\Delta CS CDC\rightarrow D

Uc=UD=12=x+yUc=U_{D}=12=x+y

Px=PyPx=Py 0x120\leq x\leq 12

HxHx \mid Px>PyPx>Py x=0x=0

Px<PyPx<Py x=12x=12

ΔCS=1212dPx+2U0dPx\Delta CS=\int_{1}^{2}12dPx+\int_{2}^{U}0 dPx

=12Px12=12=12Px\int_{1}^{2}=12






Note Created by
Is this note helpful?
Give kudos to your peers!
00
Wanna make this note your own?
Fork this Note