# Lecture 4: Slutsky SE

### Slutsky SE

• We take away enough ﻿$M$﻿ to make ﻿$A$﻿ just affordable.
• Since P changed, we can get to a higher utility for ﻿$B$﻿ .

﻿$A\rightarrow B$﻿ is ﻿$SE$﻿ , ﻿$B\rightarrow C$﻿ is ﻿$IE$﻿

• Figure cost ﻿$A$﻿ with the new ﻿$BL$﻿ conditions. Adjust ﻿$M$﻿ to get comp ﻿$BL$﻿ . Than may utility for this ﻿$BL$﻿ .Net b-base year & + equal some other year. How did consumption change from year b to year a If we use price at time b, we get the Laspeyres index. If we use price at time, we get the Paasche index.

Paasche ﻿$\rightarrow$﻿ ﻿$q$﻿ index ﻿$\rightarrow$﻿ ﻿$P_{q}=\frac{Px^{+}x^{+}+Py+y^{+}}{Px^{+}x^{b}+Py+y^{b}}$﻿

• If its ﻿$>1,$﻿ Consumer is better at time ﻿$+$﻿
• If its ﻿$<1,$﻿ Just means unaffordable

Laspeyres ﻿$\rightarrow$﻿ ﻿$q$﻿ index ﻿$\rightarrow$﻿ ﻿$Lq=\frac{Px^{b}x^{+}+Py^{b}y^{+}}{Px^{b}x^{b}+Py^{b}y^{b}}$﻿

• If its ﻿$<1,$﻿ the consumer is better off at time ﻿$b$﻿ .
• If we use ﻿$q$﻿ of time ﻿$+$﻿ we get the Paasche ﻿$P$﻿ index ﻿$\rightarrow$﻿ ﻿$Pp=\frac{Px^{+}x^{+}+Py^{+}y^{+}}{Px^{b}x^{+}+Py^{b}y^{b}}$﻿
• If we use ﻿$q$﻿ of time b, we get the Laspeyres ﻿$L$﻿ index ﻿$\rightarrow$﻿ ﻿$Lp=\frac{Px^{+}x^{b}+Py^{b}y^{b}}{Px^{b}x^{+}+Py^{b}y^{b}}$﻿
• To compare the change in welfare, we have to set an index of change in total expenditure.

﻿$M=\frac{Px^{+}x^{b}+Py^{b}y^{b}}{Px^{b}x^{+}+Py^{b}y^{b}}$﻿

• If ﻿$Pp>M$﻿ , consumer is better off at time ﻿$b$﻿ . If ﻿$Lp﻿ , the consumer is better off at time ﻿$+$﻿ .

### Perfect Complements

• Income Effect﻿$=$﻿ Total Effect- ( Because you can't substitute ﻿$\sigma=0$﻿ )

### Perfect Substitutes

• Substitution Effect﻿$=$﻿ Total Effect- (Because you will always substitute ﻿$\sigma=0$﻿

### Quasi-Linear Preferences

• Sub Effect﻿$=$﻿ Total Effect- (Because good ﻿$c(x)$﻿ is income independent).