# Lecture 2: Perfect Substitutes

### Perfect Substitutes

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

﻿$MRS_{A}=\frac{MUx}{MUy}=\frac{1}{2}x^-{\frac{1}{2}}=\frac{Px}{Py}=\frac{2}{U}=\frac{1}{2}$﻿ ﻿$x=1, y=1, U=2$﻿

﻿$A(1,1)$﻿

• Bundle B → ﻿$U_{8}=UA=2=X_{8}^{\frac{1}{2}}=Y_{8}$﻿

﻿$MRS_{c}=\frac{1}{2}x\frac{-1}{2}=\frac{1}{U}$﻿ ﻿$x_{8}=U, y_{8}=0, U=2$﻿

﻿$B(U,0)$﻿ ﻿$BL_{0}=x+Uy=U$﻿ ﻿$CV=-2$﻿

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

﻿$MRS_{c}=\frac{1}{2}x^-{\frac{1}{2}}=\frac{1}{U}$﻿ ﻿$Xc=y, Yc=\frac{1}{2}, U=\frac{5}{2}$﻿

﻿$C(U, \frac{1}{2})$﻿

• Bundle D → ﻿$U_{D}=U_{c}=\frac{5}{2}=X_{D}^{\frac{1}{2}}=Y_{D}$﻿

﻿$MRS_{D}=MRS_{A}=\frac{1}{2}x^{\frac{-1}{2}}=\frac{1}{2}$﻿ ﻿$X_{D}=Y, Y_{D}=\frac{3}{2}, D (1, \frac{3}{2})$﻿

﻿$BL_{D}=2x+Uy=8$﻿ ﻿$CV=8-6=2$﻿ • Demand → ﻿$\frac{1}{2}x^{}\frac{-1}{2}=\frac{Px}{Py}$﻿ ﻿$Py=U$﻿ ﻿$Px=X^{\frac{-1}{2}}=2$﻿

﻿$x=UPx^{-2}=\frac{U}{Px^{2}}$﻿

﻿$CS=\int_{1}^{2}\frac{U}{Px^{2}}$﻿ ﻿$dPx=-UPx^{-1}\int_{1}^{2}=-2-(-4)=2$﻿ • When you shift the budget line parallel, there is no income effect- Quas linear utility.

### Perfect Substitutes a. ﻿$Px=y, Py=2, M=12$﻿ ﻿$O-cost_{x}=2> MRS$﻿ ﻿$Px=y, Py=2, M=12, O-cost_{x}=2> MRS$﻿so buy ﻿$y$﻿

﻿$BL_{A}=12, Ux+2y$﻿ ﻿$y=\frac{12}{2}=6$﻿ . ﻿$x=0$﻿ ﻿$A(0,6)$﻿ ﻿$U=6$﻿

b. ﻿$Px=1$﻿ ﻿$Py=2$﻿ ﻿$M=?$﻿ ﻿$U_{8=U_{A}}=6=x+y$﻿ ﻿$O-cost=2$﻿ buy all ﻿$x$﻿ .

﻿$B(6,0)$﻿ ﻿$Cost B1(6)+2(0)=6$﻿

﻿$CV=12-6=\ 6$﻿

﻿$BL_{0}=6=x+2y$﻿

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

﻿$BL_{c}=12=x+2y$﻿ ﻿$y=0$﻿ ﻿$c(12,0)$﻿

d. ﻿$Px=y$﻿ ﻿$Py=2$﻿ ﻿$M=?$﻿ ﻿$U_{0}=Uc$﻿ ﻿$O-Cost_{x}=2$﻿

﻿$U=12$﻿ ﻿$y=12$﻿ ﻿$x=0$﻿ ﻿$D(0,12)$﻿

﻿$Cost D=\ 2U$﻿

﻿$Ev=\12$﻿ ﻿$BL_{D}=2U=Ux+2y$﻿ • Approx ﻿$\Delta CS$﻿ ﻿$A\rightarrow B$﻿

﻿$MRS=O-Cost\rightarrow 1=\frac{Px}{Py}$﻿ ﻿$Px=Py$﻿ ﻿$O\leq x\leq \frac{M}{Px}=\frac{12}{Px}=6$﻿

﻿$Px> Py$﻿ ﻿$x=0$﻿

﻿$Px< Py$﻿ ﻿$x=\frac{12}{Px}$﻿

﻿$\Delta CS=\int_{1}^{2\frac{12}{Px}}d$﻿ ﻿$Px+\int_{2}^{U}$﻿ ﻿$0dPx$﻿

﻿$=12 In Px\int_{1}^{2}=8.32$﻿

#### Exact ﻿$\Delta CS$﻿

﻿$U_{A}=U_{8}=6=x+y$﻿

﻿$Px=Py$﻿ ﻿$0\leq x\leq 6$﻿

﻿$Hx$﻿ ﻿$\mid$﻿ ﻿$Px>Py$﻿ ﻿$x=0$﻿

﻿$Px﻿ ﻿$x=6$﻿

﻿$\Delta CS=\int_{1}^{2}6$﻿ ﻿$dPx+\int_{2}^{U}0dPx=6Px\int_{1}^{2}= \ 6= CV$﻿ • Exact ﻿$\Delta CS$﻿ ﻿$C\rightarrow D$﻿

﻿$Uc=U_{D}=12=x+y$﻿

﻿$Px=Py$﻿ ﻿$0\leq x\leq 12$﻿

﻿$Hx$﻿ ﻿$\mid$﻿ ﻿$Px>Py$﻿ ﻿$x=0$﻿

﻿$Px﻿ ﻿$x=12$﻿

﻿$\Delta CS=\int_{1}^{2}12dPx+\int_{2}^{U}0 dPx$﻿

﻿$=12Px\int_{1}^{2}=12$﻿