San Francisco State University ECON 560 Summer 2018 Problem set 3 Due Monday, July 23

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1 San Francisco Sae Universiy Michael Bar ECON 56 Summer 28 Problem se 3 Due Monday, July 23 Name Assignmen Rules. Homework assignmens mus be yped. For insrucions on how o ype equaions and mah objecs please see he noes Typing Mah in MS Word. 2. Homework assignmens mus be prepared wihin his emplae. Save his file on your compuer and ype your answers following each secion. 3. Do no delee he quesions. 4. Lae homework assignmens will no be acceped under any circumsances 5. All he graphs should be fully labeled, i.e. wih a ile, labeled axis and labeled curves. 6. For insrucions on how o plo daa wih Excel, see he noes Insrucions for Excel. 7. In all he quesions ha involve calculaions, you are required o show all your work. Tha is, you need o wrie he seps ha you made in order o ge o he soluion. 8. This page mus be par of he submied homework.

2 Malhusian Model. (5 poins). The following quesions are based on he Malhusian discussed in class (and in he noes). a. Derive he equaion of oupu per capia (yy ) and he law of moion of oupu per capia (yy + as a funcion of yy ) for his model. y = Y L Oupu per capia: A Λ = L L Λ = A L y Law of moion of oupu per capia: + = A + + L = = + g( y ) L / A + ( A / A ) A y A g( y ) b. Suppose ha in 6 h cenury France, he echnology level is fixed a AA = 5, he populaion growh funcion is gg(yy ) =.2 yy, he land share parameer is =.4, and he land is Λ =. Find he seady sae level of oupu per worker (yy ) and populaion (LL ). The seady sae oupu per worker (y*) is found by solving g ( y*) =. Thus.2 y * =, y * = 5 y * = 25 Now, o find he seady sae populaion we use he oupu per capia equaion: y* = A 25 = 5 L* = 7.89 c. Suppose ha during he same ime China was much more echnologically advanced, wih level of echnology fixed a AA =. Find he seady sae level of oupu per worker (yy ) and populaion (LL ). The seady sae oupu per worker (y*) is found by solving g ( y*) =. Thus.4.2 y * =, y * = 5 * = 25 y Now, o find he seady sae populaion we use he oupu per capia equaion: 2

3 y* = A 25 = L* =.9 d. Based on your answer o par c, wha does he Malhusian model predic abou he cross counry differences in sandard of living in he long run? Are more echnologically advanced counries enjoying higher sandard of living han counries ha are echnologically backward? The Malhusian model predics ha in he long run, echnologically advanced counries will enjoy he same sandard of living as echnologically backward. The only difference according o he Malhusian model, he more advanced counries will have higher populaion. e. Now suppose ha France caches up echnologically wih China. Tha is, from period ττ +, he echnology jumps o he same level as in China: AA = 5, AA = 5,, AA ττ = 5, AA ττ+ =, AA ττ+2 =, Find he new seady sae level of oupu per worker (yy ) and populaion (LL ) and show he ime pahs of oupu per capia, populaion growh rae and oal populaion in France. The seady sae oupu per worker (y*) is found by solving g ( y*) =. Thus.4.2 y * =, y * = 5 y * = 25 Now, o find he seady sae populaion we use he oupu per capia equaion: y* = A 25 = L* =.9.4 3

4 y_ Time g(y_) Time L_ Time 4

5 f. Based on your answer in par e, wha is he shor run effec of he echnological improvemen on he sandard of living in he Malhusian model? In he shor run, here is improvemen in he sandard of living as a resul of a echnological improvemen. Bu evenually, populaion increases and reduces he oupu per capia o he subsisence level. I is imporan o undersand why his is happening in his model. The oupu per capia, when A is fixed, is given by y = A L In he shor run, when A goes up, here is an increase in oupu per capia. Bu noice ha oupu per capia depends also on he land per capia. We assumed ha as long as y exceeds he subsisence level, he populaion will grow, and he erm Λ / L will decline. This decline in resources per worker is he force ha brings he level of oupu per worker back o subsisence afer any emporary increase. g. Based on your answer in par e, wha is he long run effec of he echnological improvemen on he sandard of living? In he long run, he oupu per capia will reurn o he subsisence level. Thus, echnological improvemen leads o only emporary, bu no permanen increase in he sandard of living in he Malhusian model. h. Now, suppose ha he feriliy behavior in he economy changed, so ha he relaionship beween populaion growh and oupu per capia becomes gg(yy ) =. yy Find he seady sae oupu per worker in China. The seady sae oupu per worker (y*) is found by solving g ( y*) =. Thus. y * =, y * = y * = i. According o he Malhusian model, wha should be he effec of he one-child-law on he sandard of living in China? The Malhusian model predics ha a permanen increase in GDP per capia as a resul of he law. When we changed he funcion g in he previous secion such ha for every income per capia he populaion growh declines, he resul was an increase in he seady sae income per capia. The nex graph shows he ln of GDP/capia in China since 95. 5

6 ln(gdp/cap) I seems ha unil he lae 7 s he GDP per capia was growing a slower rae han since he lae 7 s. Remember ha he slope of he ln(gdp/capia) ells us approximaely how fas does i grow. The one-child law was enaced in 979, and i looks like he predicion of he Malhusian model is suppored by he daa. j. Compare he predicion of he Malhusian model wih ha of he Solow model concerning he once-and-for-all increase in echnology. In paricular, discuss wha happens o he seady sae oupu per worker in he Solow model as a resul of he once-and-for-all increase in echnology. In he Malhusian model a once-and-foe-all increase in echnology leads o only emporary increase in he sandard of living. In he Solow model, on he oher hand, a once-and-for-all increase in echnology leads o permanen increase in he sandard of living. Recall he seady sae level of oupu in he Solow model: Higher A leads o higher SS y s y SS = A n + δ, ha is permanenly higher oupu per worker. Demography 2. (5 poins). In his quesion you need o use he Daa for HW3 posed on he web. a. The spreadshee called LifeExp conains daa on age specific moraliy raes, age specific survival raes, probabiliy of being alive a any age, and ages specific feriliy raes in Ausralia for years 92 and 23. The spreadshee demonsraes 6

7 how o calculae he life expecancy for 92. Follow he example o calculae he life expecancy in Ausralia in 23. i= LE = π i = 8.33 b. Use he daa on age specific feriliy raes o compue he Toal Feriliy Rae (TFR) in Ausralia, in 92. = TFR i= F i = 4 c. Use he daa on age specific feriliy raes and he probabiliies of being alive a any age, o compue he Ne Reproducion Rae (NRR) in Ausralia in 92. Assume ha half of he newborns are girls. = NRR π 2 i= i F i = ( poins) The following able shows daa for a counry of Fanasia. Fanasians live for a maximum of five years. In addiion, all he people are women and here are no men. Age (from las Birhday) Probabiliy of surviving o nex age Populaion in 2 Age specific feriliy raes Calculae he populaion of Fanasia in 2. Age (from las Birhday) Populaion in 2 Age specific feriliy raes Probabiliy of surviving o nex age Populaion In = 6.8 = 2.8 = 3.5 = 4.5 = 5 Toal ( poins). For his quesion he Gapminder bubble char ool. 7

8 a. Selec Income per person as he x-variable and Babies per woman as he y- variable. Presen he rail of feriliy and income per person since 8 for your counry of choice, oher han he Unied Saes. The nex graph is for he Unied Saes. b. Briefly describe wha happened o feriliy and income per person since 8 in your counry of choice. Overall, here has been a decline in Toal Feriliy Rae since 8. The big hump is he baby boom during he 5s early 6s. 5. ( poins). Observe he nex able and answer he following quesions. a. Beween he years boh he oal feriliy rae and life expecancy in India changed. A he same ime he ne reproducion rae sayed roughly consan. Provide a clear and brief explanaion how his is possible. In your explanaion use he formula of NRR. The ne reproducion rae depends boh on feriliy and moraliy, as he formula indicaes 8

9 NRR = 2 π On he one hand feriliy declined, which refleced by F i, and on he oher hand moraliy declined as well (life expecancy increased), so he probabiliy of being alive a cerain age increased, which is refleced by π i. As he daa shows, i urns ou ha he wo forces canceled each oher. i= b. Afer 98, he life expecancy increased, bu neverheless he ne reproducion rae declined. Provide a clear and brief explanaion how his is possible. In your explanaion, use he formula of NRR. Alhough life expecancy increased, he decline in feriliy was significan ( F i ) and was he dominaing facor ha drove he decline in NRR. i F i 9