# Midterm Exam. Thursday, April hour, 15 minutes

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1 Economcs of Grow, ECO560 San Francsco Sae Unvers Mcael Bar Sprng 04 Mderm Exam Tursda, prl 0 our, 5 mnues ame: Insrucons. Ts s closed boo, closed noes exam.. o calculaors of an nd are allowed. 3. Sow all e calculaons. 4. If ou need more space, use e bac of e page. 5. Full label all graps. Good uc

2 . (40 pons). Consder e Malusan model dscussed n class, and descrbed as follows. Consumers: e o consume food ( Y ). Eac consumer supples un of labor. Producers: Produce food usng land and labor. Oupu of food a me s gven b Y, 0, were s producv level a me, s (fxed) land, and s e number of worers, wc s also e sze of e populaon. Populaon: evolves accordng o g( ), were g ( ) s e grow rae of populaon as a funcon of oupu per capa Y /. I s assumed a ere s some subssence level of consumpon per capa * suc a g ( ) wen *, g ( ) wen *, and g ( ) wen *. a. Derve e equaon of oupu per capa ( ) and e law of moon of oupu per capa ( as a funcon of ) for s model. Y Oupu per capa: aw of moon of oupu per capa: g( ) / / g( )

3 b. Suppose a n some counr e producv level s fxed a, e populaon grow funcon s g( ) 0. 5, e land s 000 and land sare s Solve for e sead sae level of oupu per capa ( * ) and e sead sae populaon level ( *). Sead sae oupu per capa g( *) 0.5 * * * * * * * * 4 Sead sae populaon / * /

4 c. Suppose a a me, producv ncreased o 7, and saed a s new level forever (once-and-for-all ncrease n producv). ssumng a pror o e cange, e econom was a a sead sae, fnd e oupu per capa, e ne populaon grow rae a me and populaon n e nex perod,. Oupu per capa: * lernavel, usng e law of moon of oupu per capa: / 7 / g( ) 3 Populaon grow rae: g ( ) e grow rae = 50% Populaon nex perod: g( )

5 d. Gven e cange n e las secon, solve for e sead sae level of oupu per capa ( * ) and e sead sae populaon level ( *). Snce ou don ave calculaors, plug e numbers n e formula for *, wou provdng e fnal number. Sead sae oupu per capa g( *) 0.5 * * * 4 Sead sae populaon 7 * 000 * 4 If ou use calculaor, ou ge e new sead sae populaon of * / 4

6 e. ow suppose a e populaon grow funcon canged, and s now g( ) 0. 5 (prevenve cec). Solve for e new sead sae oupu per capa, and llusrae e effec of e prevenve cec on a full labeled grap of g ( ). Use e numercal resuls from s and pervous secons. Sead sae oupu per capa g( *) 0.5 * * 4 * 6 Grow rae of populaon g ( ) 4 6 Sold lne s e orgnal populaon grow funcon, and e dased lne s e new one. 5

7 . (0 pons) Te followng able sows daa for a counr of Fanasa. Fanasans lve for a maxmum of fve ears. lso, all e people are women, wo are noneeless able o reproduce. Probabl ge Populaon ge specfc of (from las Populaon n 04 n 03 ferl raes survvng o Brda) nex age Toal Calculae e populaon a eac age n 04, as well as oal populaon n 03 and 04. 6

8 3. (0 pons). ge specfc ferl raes ( F ) and probabl of beng alve a age ( ) n Inda are gven n e followng able. ge F F a. Calculae e lfe expecanc n Inda n 950 and n 00. You mus presen e formula of E, before pluggng an numbers. E E E b. Calculae e oal ferl rae (TFR) n Inda for e ears 950 and 00. You mus presen e formula of TFR, before pluggng an numbers. TFR TFR TFR F

9 c. Calculae e ne reproducon rae (RR) n Inda for e ears 950 and 00, assumng a alf of e babes are grls. You mus presen e formula of RR, before pluggng an numbers. RR RR RR F d. Explan brefl w e e Reproducon Rae dd no cange beween e ears 950 and 00, despe e fac a Toal Ferl Rae durng ese ears decreased dramacall. Your answer mus be based on e formula of RR. Te e Reproducon Rae s a on measure of ferl and moral: RR F 0 Durng e perod under dscusson, ferl and moral bo declned, wc means a:, F. In oer words, women are avng fewer cldren, bu ere s also a ger cance a e survve roug er cld bearng ears. Tese wo opposng forces us appen o cancel eac oer ou. 8

10 4. (0 pons). ssume a e aggregae oupu s produced accordng o, 0 Y K Facor Producv, and s uman capal per worer., were Y s e oal real GDP, s e Toal K s e oal pscal capal, s e number of worers, a. Te nex able presens daa on wo counres. 5?.5 Based on e above able, f e onl dfference beween e wo counres was producv, wa would be e rao of counr o counr GDP per capa? b. Te nex able sows ow e average wage ncreases n ears of educaon n a sample of counres. Years of scoolng ,0, Margnal reurn Based on e above able, ow would ou esmae e uman capal per worer n a counr were e average worer as 3.5 ears of educaon? You onl need o wre e formula a ou would use f ou ad a calculaor. ( 3.5)

11 0 5. (0 pons). Derve e approxmae grow accounng formula for oupu per capa, based on e producon funcon gven n s queson, w e oal populaon, and e fracon of worers n populaon /. Oupu per worer s K and oupu per capa s / Y Y. Tus, Usng x x x x o denoe grow raes, e above becomes: Tang logs: ln ln ln ln ln Usng e approxmaon g g ln for small g:

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