San Francisco State University ECON 560 Fall Midterm Exam 2. Tuesday, October hour, 15 minutes

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1 San Francisco Sae Universiy Micael Bar ECON 560 Fall 207 Miderm Exam 2 Tuesday, Ocober 3 our, 5 minues Name: Insrucions. Tis is closed book, closed noes exam. 2. No calculaors or elecronic devices of any kind are allowed. 3. Sow all e calculaions, and explain your seps. 4. If you need more space, use e back of e page. 5. Fully label all graps. 6. Use a ruler o draw nea graps. Good Luck

2 . (40 poins). Consider e Malusian model discussed in class, and described as follows. Consumers: Like o consume food (YY ). Eac consumer supplies uni of labor. Producers: Produce food using land and labor. Oupu of food a ime is given by YY AA ΛΛ LL, 0 < <, were AA is produciviy level a ime, ΛΛ is (fixed) land, and LL is e number of workers, wic is also e size of e populaion. Populaion: evolves according o LL gg(yy )LL, were gg(yy ) is e grow rae of populaion as a funcion of oupu per capia yy YY /LL. I is assumed a ere is some subsisence level of consumpion per capia yy suc a yy > yy gg(yy ) >, yy < yy gg(yy ) < and yy yy gg(yy ). a. Derive e equaion of oupu per capia (yy ) and e law of moion of oupu per capia (yy as a funcion of yy ) for is model. yy YY YY AA ΛΛ LL Oupu per capia: AA ΛΛ LL LL LL AA Λ LL Law of moion of oupu per capia: ΛΛ (AA /AA )AA gg(yy )LL AA /AA gg(yy ) yy

3 b. Suppose a in some counry e produciviy level is fixed a AA 0, e populaion grow funcion is gg(yy ) aayy, aa 2, bb 0. bb yy Te land is Λ 2000 and land sare is 0.5. Solve for e seady sae level of oupu per capia (yy ) and e seady sae populaion level (LL ). Seady sae oupu per capia gg(yy ) aayy bb yy aayy bb yy yy bb aa Seady sae populaion yy AA Λ LL yy AA Λ LL LL Λ AA yy

4 c. Suppose a a ime ττ e produciviy riples (i.e. AA ττ AA ττ 30), and says a is new level forever (once-and-for-all increase in produciviy). Assuming a prior o e cange, e economy was a a seady sae; calculae e immediae impac of e produciviy cange. Ta is, find e oupu per capia yy ττ, e ne populaion grow rae gg(yy ττ ), and e populaion size in e following period, i.e. LL ττ. Oupu per capia a ime ττ: yy ττ AA ττ Λ LL ττ Alernaively, using e law of moion of oupu per capia: yy ττ AA ττ/aa ττ gg(yy ττ ) yy ττ 30/ Populaion grow rae a ime ττ: gg(yy ττ ) aayy ττ 2 30 bb yy ττ Te ne grow rae of populaion is: gg(yy ττ ) % Populaion size a ime ττ : LL ττ gg(yy ττ )LL ττ

5 d. Given e cange in e las secion, solve for e long-run (seady sae) level of oupu per capia (yy ) and e seady sae populaion level (LL ). Seady sae oupu per capia gg(yy ) aayy bb yy aayy bb yy yy bb aa Seady sae populaion LL Λ AA 2 30 yy ,000 4

6 e. Now suppose a e populaion grow funcion canged, and is now gg(yy ) aayy, aa 2, bb 20. bb yy Solve for e new seady sae oupu per capia, and illusrae e effec of e prevenive ceck on a fully labeled grap of gg(yy ). In your grap, use e numerical values calculaed in is a previous secions. Seady sae oupu per capia aayy gg(yy ) bb yy yy bb aa Grow rae of populaion gg( ) 0 Solid line is e original populaion grow funcion, and e dased line is e new one. 20 5

7 2. (0 poins) Te following able sows daa for a plane of Ocampa, populaed by species a live for a maximum of five years. In addiion, all e people are female, wo are noneeless able o reproduce. Age (from las Birday) Populaion in 206 Age specific feriliy raes Probabiliy of surviving o nex age Populaion in ( ) Toal Calculae e populaion a eac age and e oal populaion in

8 3. (20 poins). Age specific feriliy raes (FF ii ) and probabiliy of being alive a age ii (ππ ii ) in India are given in e following able. Age F i π F i i π i a. Calculae e life expecancy in India in 950 and in 207. You mus presen e formula of LE, before plugging any numbers. LLLL ππ ii ii0 LLEE LLEE b. Calculae e oal feriliy rae (TFR) in India for e years 950 and 207. You mus presen e formula of TFR, before plugging any numbers. TTTTTT FF ii ii0 TTTTRR cildren per woman TTTTRR cildren per woman 7

9 c. Calculae e ne reproducion rae (NRR) in India for e years 950 and 207, assuming a alf of e babies are girls. You mus presen e formula of NRR, before plugging any numbers. NNNNNN 2 ππ iiff ii NNNNRR 950 [ ].25 2 ii0 NNNNRR 207 [ ].3 2 d. Explain briefly, wy e Ne Reproducion Rae increased beween e years 950 and 207, despie e fac a Toal Feriliy Rae during ese years decreased dramaically. Your answer mus be based on e formula of NRR. Te Ne Reproducion Rae is a join measure of feriliy and moraliy: NNNNNN 2 ππ iiff ii During e period under discussion, feriliy and moraliy bo declined, wic means a: FF ii, ππ. In oer words, women are aving fewer cildren, bu ere is also a iger cance a ey survive roug eir cildbearing years. Te second force (iger survival) urns ou o be a lile sronger an e firs one (lower feriliy). ii0 8

10 4. (20 poins). Assume a e aggregae oupu is produced according o YY AA KK ( LL ), 0 < <, were YY is e oal real GDP, AA is e Toal Facor Produciviy, KK is e oal pysical capial, LL is e number of workers, and is uman capial per worker. a. Te nex able presens daa on wo counries. yy ii yy jj ii jj AA ii AA jj ii jj kk ii kk jj 28? Based on e above able, if e only difference beween e wo counries was produciviy, wa would be e raio of counry ii o counry jj GDP per capia? yy ii ii AA ii ii kk ii yy jj jj AA jj jj kk jj 28 AA ii AA jj AA ii AA jj 4 b. Te nex able sows ow e average wage increases in years of educaion in a sample of counries. Years of scooling ,0, Marginal reurn Based on e above able, ow would you esimae e uman capial per worker in a counry were e average worker as 4.5 years of educaion? You only need o wrie e formula a you would use if you ad a calculaor. (4.5)

11 0 5. (0 poins). Assume a e aggregae oupu is produced according o YY AA KK ( LL ), 0 < <. Derive e approximae grow accouning formula for oupu per capia, were e fracion of workers in populaion is LL /NN. Oupu per worker is ( ) L A k L L A K y and oupu per capia is / L N A k y L Y N Y y. Tus, N N A k k A y y Using x x x x o denoe grow raes, e above becomes: ( )( )( ) ( ) k A y N Taking logs: ( ) ( ) ( ) ( ) ( ) ( ) k A y N ln ln ln ln ln Using e approximaion ( ) g g ln for small g: ( ) k A y N