ECON 3029

Problem SET 1 2017-2018

Solution

Problemi1 : $400 ( laao $ ) is the level of income tat got buy subsistence

. f- 1500kcal )2. Main cuhque :

most of namxson data are not

coming from observation

Th shoo assvptim is critical.

3- $400 = 1500kcal,

which is too low

u.

the evolution of height is mainly driven by The calories Intake ofindividuals ⇒ if,

one has obswatin on height thoughthistory,

then are can get a lower bound on income =D it is according to

CLARK often larger than nappbon 's $ ha estimate

5- Urbanization can be a good proxy for gdp Per capita ,because It is a proxy

of He share of population not employed In agriculture .

↳ this is five after 1570 in England . But before ,

secondary and tertiary activities were located In the countryside .

⇒ urbanization a not always a useful proxy .

Problem #-

by = dbtpbyt i )

Mt= am -

Pm ye (2)

Yt= hytpy Pt G)

yet YHP

Ast

1- Discuss equations i see lectured slides

(3) ⇒ ¥f=dyf+pyPp÷.

⇒ yt=

hpsuftpy(5) = aaugepoduekuty

2- C) WE ) ⇒ Pte , = Pt ← be pt . meptd¥a=Py= mmsmdpoehchuf

= ( Kbt - me ) PE

stead ) State : bt=mt ⇒ dbtpby = am - pmy ⇒ yA=£1

Pbtpm

plugging y* in (5) Yat - p .=P →

Ptyfnpty

STITT : → pre =his (Pbtpm )

nb n

| #bby)

k¥1 psfn )

¥ my )if9 y

a. =D 5- ¥ thlyrey

3-Assume now db=o,

2m=2, dy - 11 py=o , pb=pn=E

then y* =L

I=%and 411

4- Dynamics : R¥(ltb+- mi . ) Pt

⇒ BE =It ¢b-

am)t€pbtpn) bi . (8)

using numerical values far parameters:(

6) : Pttrpt = yt - 2

mb n¥

#bly

)=ay2M¥74instating

main . "IttnerPtp¥t n i Aep¥t=y - a

o '¥ • =stead

,slate

. . # →

E¥h=oag

The dynamic qualm (6) can be written

Ptei - Pt = ytpt - aptin

1

⇒ |Pt←,=€a)PtT_ ⇒ Pte ,=f (Pe )

5- Convergence Po close to 0

Ptasfn0#|Pta=H(task .

1.#,#↳¥'

:-p;Po

MONOTONE CONVERGENCE

÷as - *-8

FIL Pte ,=&lPt)=1R.us Px=Pt←

,

1 -# Pte,

-ftp.t.t#w.ggg...p...

*

="

¥=:

iffPee,

s Pr=Pte ,

t¥E÷⇒¥.¥von. monotone convergence

÷a##wm↳

Fonda,

that's"⇒ '

ppg

E×P↳5wE DYNMKS ⇒ The SS is unstable

:#

Problem # L=hp length of a personborn In 0

-

density fll )= meal

Time unit : 1 year ,m= 0*01

1.

zcumlalne? PCL > f) = µmealdl

= mxf . tmeml ]|= e-

ml

The cumulative fmohnisFCL ) = P(L< e) = 1- PC↳ f)=L

.Ehl

z - P ( L > 10 ) = Em "0= e-

at

= 0,9048

4.

P K > 80) = e- M×8°= e-.8

= 0,4493

5. Probability of lining at least more Than wyeans conditional a hams hued atleast ( o years

?

Plmoliao ) = Punkish = MpY÷g}

= g. 01×20

= e- ' ° ' +10

= 0.9848

to-

The pobashht of hung at least 10

more years is th same at

birth or at 10 years old ( • rat too yo !)=

"

perpetual youth model '

G-more generally : P ( to> b) = e-

mb

PIL ) at b / ↳ a) =P lL>aebawL=

PCL > a)

= Mphhajabtg =←eYIYI = embark> b)

7- life expectancy at birth E ( see

Peotnshdes)

0

E.= µ Lfll ) DL =/} × m Le -

mL dl = mts =160 years

life expectancy at 50 yearns old : Eso = 100 because thprocess

's memoryless

.

⇒ expected age at death istherefore 150 years .

Pwbkmftksolution is almost identical to tte model of section 5

,except forth

whit denied from food by adults

• Adult optimal behavior

u=#tgn. + slogft

Budget Constant mteprcnttft ) Ewt

Lt = mttrlognetslogftxdtcwt - mtpilnttf ))

Foc 1Mt : At =I

lnt : Fr=itpt ⇒ nt=¥1ft ¥ =ttPt ⇒ ft=§

note that ft= Ent

Demand for food b) an adult : nteft.tt#! Total demand for food : Ltxcnrfr ) = rtf . Lt

. supply of food : Ytr = AECLADK

. Equilibrium on th food market : 4. x 2¥ ni . = AYE LAY

⇒ at = HE = Eye) m . Lt"

He ah difference a. a #e ,

# )"

slides is that wehave ¥ instead of 1 ftk slides correspond to

E- o ]

By assumptive , wagesare equal to

average productivity

WF = Ypf = Mfnf÷ =ME

and wF= prY¥ =pt.tt#AT=prAFLaft.Labour market equilibrium : wf=wf ⇒ P+=C¥D OF 4th f)

( using LF = 94 . )

msn.yeeimpfffnf.ge#feffIIF

⇒ Pr = ,¥¥n !?¥)¥ Ma,

# 4¥

using nt= ¥ ⇐ p+=Ft ,we have

F. =

an¥÷ to# 5¥ nr¥ 4¥

⇐ nt=rnAf÷ Exam[ again ,

this is the of difference capered to He

lecture 1 slides

•

BEMAth

.OF the , 4- =L 't + of

wereussiidy haveg. = got = go

( if not,

either O becomes I I ⇒ possibleor 9 → 0 ⇒ No food production

⇒ no children⇒ no population . )

thnta÷ =

CKSADE

Ttgna× n¥n and n 't = kglr

⇒ It ghn, =(HyDh =D of BGP with poxhe

population : gz = 0

Couldn .All He results are qualitatively similar

.

the only difference is that fntuhh here is always smalleras

save food Is used for adult consumption .