321$intermediate$macroeconomics$ …jpd48/presentation9_two_period.pdf ·...
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321$Intermediate$Macroeconomics$Two$Period$Consumer$Model$:$Consump:on$versus$Savings$
ECON$321$
Prac:ce$Problem$
• Combina:on$of$Homework$3b,$#6$and$#7$on$board.$
Two$Period$Consumer$Model:$Consump:on$versus$Savings$
• Consumer$U"="ln(c1)"+"βln(c2)"
c1$=$consump:on$in$period$one.$
c2$=$consump:on$in$period$two.$
0$<$β$<$1$is$the$subjec:ve$discount$factor.$
$ $ $empirically$≈$.95$to$.99$
Budget$Constraints$1st$$period:$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(resources$=$uses)$2nd$period:$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$(then$you$die)$
$$ $a$=$ini:al$wealth$
$$ $y1$=$period$one$income.$
$$ $y2$=$period$two$income.$
$$ $b$=$wealth$(stock$of$assets)$carried$between$$ $ $$$periods.$$$ $r$=$real$interest$rate.$
$$ $(y1$–$c1)$=$savings$or$the$flow$of$assets.$
€
a + y1 = c1+ b
€
y2 + (1+ r)b = c2
Life:me$Budget$Constraint$
1st$$period:$2nd$period:$
Life:me:$
€
a + y1 = c1+ b
€
b = a + y1 − c1
€
c2 = y 2 + (1+ r)(a + y1 − c1)
€
a + y1+y 21+ r
= c1+ c21+ r
Present$Value$of$life:me$resources.$
Present$Value$of$life:me$consump:on.$
€
y2 + (1+ r)b = c2
Two$Period$Consumer$Model$
C2$
C1$
Inefficient$$$$$$$$$$Any$point$on$the$BC$line$is$$$$$$$$$$feasible$
Super$Saver$
Party$Animal$
Not$Feasible$$
Shape$of$indifference$curves$shows$a$preference$for$variety.$$i.e.$We$will$likely$end$up$near$the$middle.$
Household$
"s.t."
"" "1st""period:"
"" "2nd"period:"
"" "Life9me:"
€
Maxc1U = ln(c1) + βln(y 2 + (1+ r)(a + y1 − c1))
€
Maxc1,c2,b
U = ln(c1) + βln(c2)
€
a + y1 = c1+ b
€
c2 = y 2 + (1+ r)(a + y1 − c1)
€
y2 + (1+ r)b = c2
Household$
FOC:$
€
Maxc1U = ln(c1) + βln(y 2 + (1+ r)(a + y1 − c1))
€
1c1−β(1+ r)c2
= 0
c2c1
= β(1+ r)
$$MU$of$C1$ $$MC$of$foregone$C2$
€
1c1−
β(1+ r)y 2 + (1+ r)(a + y1 − c1)
= 0
Consump:on$Over$Time:$$Preferences$and$Rates$of$Return$
€
c2c1
= β(1+ r)
β(1+r)>1$
β(1+r)=1$
β(1+r)<1$
c$
t$
DATA$β$≈$.98$
(1+r)$≈$1.07$(long$term)$
Household$
€
c1* = c1(a,y1,y2,β)c2 *1+ r
= c2(a,y1,y2,β)
b* = (a,y1,y 2,β)
The$Math$
€
c1β(1+ r) = y 2 + (1+ r)(a + y1 − c1)
c1 +c1β
=y 2
β(1+ r)+(a + y1)β
c11+1β
$
%
& & &
'
(
) ) )
=y 2
β(1+ r)+(a + y1)β
c1 1+ ββ
$
%
& & &
'
(
) ) )
=y 2
β(1+ r)+(a + y1)β
c1* =1
1+ βy2
(1+ r)+ (a + y1)
*
+
, , ,
-
.
/ / /
The$Math$
€
c2* = y2 + (1+ r)(a + y1 − c1*)c 2 *(1+ r)
=β
1+ βy2
(1+ r)+ (a + y1)
$
%
& & &
'
(
) ) )
b* = a + y1 − c1*
b* =β
1+ β(a + y1) − 1
1+ βy 21+ r*
+ ,
-
. /
Posi:ve$Income$Effects$
C2$
C1$
Inefficient$
Not$Feasible$$
Posi:ve$Income$Effects$
Y1$ C1$ C2$ S$
Y2$ C1$ C2$ S$
Subs:tu:on$Effect$of$an$$$$in$r$$
C2$
C1$
No$Borrowing$or$lending$point$;$doesn’t$depend$on$r.$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$c1=a+y1$;$c2=y2$
$$$$$$$$$$$$
$
Income$Effect$of$an$$$$in$r$$
• Net$Borrower:$
• Net$Saver$
r$ C1$ C2$ S$
r$ C1$ C2$ S$
Net$Effect$of$an$$$$in$r$$
• Net$Borrower:$
• Net$Saver$
r$
C1$
C2$ S$
r$ C2$ S$
$
$ $
C1$
Household:$Lump$Sum$Taxes$
"s.t.""1st""period:"
"" "2nd"period:"
"" "Government:"
"" "Life:me:"
€
Maxc1,c2,b
U = ln(c1) + βln(c2)
€
a + y1 = c1+ b +T1
€
y2 + (1+ r)b +G2 = c 2
€
T1+ T 21+ r
=G1+G2
(1+ r)
T1 =G2
(1+ r)
€
c2 = y 2 +G2 + (1+ r)(a + y1 − c1 −T1)
Lump$Sum$Taxes$
€
Maxc1U = ln(c1) + βln(y 2 +G2 + (1+ r)(a + y1 − c1 −T1))
€
FOC : 1c1−
β(1+ r)y2 +G2 + (1+ r)(a + y1 − c1 −T1)
= 0
€
c1* =1
1+ βy 2 +G2
(1+ r)+ (a + y1 −T1)
$
%
& & &
'
(
) ) )
c1* =1
1+ βy2
(1+ r)+ (a + y1)
$
%
& & &
'
(
) ) )
Where T1 =G2
(1+ r), and they cancel each out,
leaving no effect on consumption.