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.