Intermediate Macroeconomic Theory II, Winter 2009 Solutions to Problem Set 1

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1 Intermediate Macroeconomic Theor II, Winter 2009 Solution to Problem Set 1 1. (18 point) Indicate, when appropriate, for each of the tatement below, whether it i true or fale. Briefl explain, upporting our anwer with relevant formula and/or graph. (a) (2 point) Aume that production function i AK α 1 α, where A i the total factor productivit. At ome point in time ou oberve that the growth of population, n, i 3%, the growth of total capital, K, i 3%, α 1/3, and the growth of total output, i 3%. Uing the growth accounting, what i the growth of the total factor productivit, A? (b) (2 point) In the data, the ratio of total capital to total output, K, i fairl table over time. It i true that both the Solow model with a poitive technological and population growth, and the Solow model without technological growth but with a poitive population growth predict the contanc of thi ratio in the long run. (c) (2 point) In the data, the real interet rate i fairl table over time. The Solow model with technological growth doe a good job at predicting thi empirical obervation to hold in the long run, while the Solow model without technological growth fail to do o. (Hint: the real interet rate i defined a r MP K δ.) (d) (2 point) ou oberve that the real wage in the econom grow at the rate equal to the growth rate of output per worer. Auming that the econom i in it tead tate (on a balanced growth path), ou can infer that an adequate model for our econom in the long run i the Solow model with a poitive population growth and no technological growth. (e) (2 point) If the aggregate production function in the econom i K α 1 α, and the aving rate,, in the econom i equal to α, then the econom i at it golden rule tead tate. (f) (2 point) Suppoe that the production function in per effective labor term i α, the current aving rate i above α, and the econom i in the tead tate. If houehold reduce their aving rate to α, then conumption per worer will increae in the long run, while output per worer and capital per worer will fall in the long-run. (g) (2 point) If the 0.20, n 0.02, δ 0.01, and g 0.02, then, in the tead tate, the ratio of total capital to total output i equal to 2.0. (h) (2 point) The Permanent Income Hpothei (PIH) predict that if one ha a temporar, unpredicted reduction in income, e.g., due to a temporar illne, conumption will be reduced b the magnitude of the income reduction. (i) (2 point) The PIH predict that if one expect that hi dipoable income will increae in the next period (e.g., when the lat mortgage pament i done) the adjutment of conumption between the current (t) and next period (t + 1) will be enitive to thi predictable increae in the dipoable income: i.e., C t+1 will be enitive to a portion of t+1 due to the predicted increae in dipoable income. (a) It can be eail hown that (1 α) A A + α K K ) 0.03 (1/ /3 0.03) 0. + (1 α). Thu, A A (α K K + (b) True. Note that K K/ / K/(E) /(E). Solow model without technological but with a poitive population growth predict the contanc of K/, and /, and therefore the contanc of their ratio, K/. Solow model with a poitive growth in technolog and population predict the contanc of K/(E), and /(E), and therefore the contanc of K/. 1

2 (c) Fale. For the CRS production function we tudied, MP K α K α / K/ α /(E) K/(E). Thu, the MP K i contant both in the model with and without technological progre: in the firt cae, K/ and / are contant in the tead tate; in the econd cae, K/(E) and /(E) are contant along a balanced growth path, and o will be their ratio that determine the MP K. (d) Ambiguou. For the CRS production function we tudied, the real wage are equal to (1 α). The real wage are growing at the rate of the growth in ( ). If there i no technological growth, the real wage are contant over time ince (/) will be contant; if there i a poitive technological growth the real wage will grow at the rate pew pew + g, where pew E (ince E E pew E), or at the rate equal to g, the rate of growth in the efficienc of labor. There are two correct anwer. Firt, if ou aume that the growth rate in output per worer and therefore wage i zero, then the Solow model without technological progre i fine. Second, if the aumed growth rate in output per worer i poitive, then the right model i the Solow model with technological progre. If ou argue correctl the firt or the econd cae, either one i fine with me. (e) True. Thi i the model without technological growth. The golden rule aving rate hould reult in the tead tate capital where MP K(gold ) (n+δ), or MP K( gold ) (n+δ). Alo we now that in the SS α (n + δ). Thu, the golden rule level of aving hould atif: α MP K(gold ) α α 1 α α. Therefore, for thi equalit to hold, we need to have α. (f) True. If the current aving rate i above α, then it i above the golden rule aving rate the one that maximize conumption per worer. Thu, when the aving rate i permanentl reduced toward it golden rate, conumption per worer in the ver long, when the econom reache it new tead tate, will be higher, while capital per worer and therefore output per worer will be lower. (g) Fale. For Solow model with technological and population growth, the SS capital per effective ( worer i found from α (n + δ + g), or ( ) α/(α 1). Thu,. Since E, K E, 0.2/( ) 4 2. K ) 1/(α 1). ( ) α K. Thu, in the SS, (h) Fale. Firt, the PIH tate that conumption repond to new unpredictable change in the lifetime reource that occur due to unpredictable event. Second, the PIH tate that the magnitude of the change in conumption will be dependent on the effect of thi hoc in income on our permanent income. Since the hoc i reaonabl aumed to be temporar, we ma expect that it doe not affect much the permanent income. Therefore conumption will barel, if at all, repond to thi hoc. A PIH conumer who i not borrowing contrained will borrow or rel on hi toc of aving to cope with thi temporar hoc. (i) Fale. Conumption will not repond to thi predictable increae in the future dipoable income. A PIH conumer who i not borrowing contrained would revie hi etimate of the permanent income in period t and et hi conumption equal to the annuit value of thi etimate. In the next period, conumption will change onl if there are new about permanent income, and C t+1 hould not repond to the part of the variation in dipoable income, t+1, which originate from the (completel predictable) variation due to the lat mortgage pament at time t (20 point) Conider an econom with production function K 1/5 (E) 4/5. Aume that population growth in the econom, n 0.02; aving rate, 0.1; depreciation rate, δ 0.01; and technological growth i g (a) (2 point) Derive the per-effective-worer production function. (b) (2 point) Find the tead-tate capital per effective worer, output per effective worer, invetment per effective worer, and conumption per effective worer. 2

3 (c) (2 point) What i the growth rate of output per worer, capital per worer, and conumption per worer in the tead tate? (d) (2 point) What i the growth rate of total output, total conumption, and total capital in the tead tate? (e) (2 point) Compute the hare of labor and capital income in total income. (f) (2 point) Compute the golden rule level of capital per effective worer for thi econom. (g) (2 point) Compute the golden rule aving rate for thi econom. I it different from the current aving rate in the econom? (h) (2 point) Baed on our anwer to (2g), i the econom over-aving or under-aving? (i) (2 point) Find the real interet rate in the tead tate, when 10%. (Hint: the real interet rate i defined a r MP K δ.) (j) (2 point) Calculate the real wage per worer in the tead tate. (Hint: ou will not be able to calculate the exact number for the wage per worer; it will be equal to the multiple of the output per effective worer in the tead tate, where the multiplication term will be related to E and the hare of labor income in total income). (a) E K1/5 (E) 4/5 E (b) In the SS, ( K 1/5 (E) 1/5 ( K E) 1/5 1/5. ) 1/(α 1) ( ) 5/ ( ) α/(α 1) ( ) α i c (1 ) i (c) K K E E E. Thu, the growth rate in output per worer i equal to the um of the growth rate in capital per effective worer (0 on a balanced growth path) and the growth rate in efficienc (g), i.e., equal to g. Similarl, the growth rate in conumption per worer and invetment per worer i equal to g %. (d) K K E (E) (E). Thu, K K. At the tead tate, the growth rate in capital per effective worer i zero, and o K K n + g %. Uing imilar logic, ou can infer that the growth rate in total output and total conumption i equal to 4%. (e) The hare of capital income i found from R K MP K K α K K α 1/5. The hare of labor income in total income i found from W MP (1 α) 4/5. (f) The golden rule capital per effective worer i found from α(gold )α (n + δ + g)gold ( ), or 1/(α 1) gold α Note that the golden rule level of capital per effective worer + E E + i larger than the tead tate level of capital per effective worer under the current aving rate. Thu, the econom i under-aving. (g) The golden rule aving rate i equal to α 0.2. e, it i 10% higher than the current aving rate. (h) The econom i under-aving, and it i poible to achieve a higher level of conumption per worer in the long run b increaing the aving rate now (b acrificing current conumption to achieve higher tandard of living in the long run, poibl enjoed b future generation). (i) The real interet rate i r MP K δ α K δ α δ α n+g+δ δ In the golden rule tead tate, r MP K δ (n + g + δ) δ n + g (An of thee two interet rate will do for the correct anwer.) 3

4 Note that the real interet rate i maller in the golden rule tead tate impl becaue the uppl of capital i higher in the golden rule tead tate. (j) The real wage per worer i found from the MP, and i equal to (1 α) 4/5 E E 4/5 E 4/ E E. ou cannot determine the exact number here ince ou do not now the level of the labor efficienc at the tead tate. 3. (8 point) Suppoe the econom aving rate i 25%, population growth rate i 1%, technological growth i 2%, and the depreciation rate i 3%. The econom production function in per-effectiveworer term i 1/3. Aume that the econom i in it tead tate (on a balanced growth path). At time T, the econom find a wa to ue it capital in a more economizing wa o that the depreciation rate fall to 1%; at the ame time technological growth experience a jump to 3%; population growth i left unchanged. (a) (2 point) Compute the growth rate of capital per effective worer and output per effective worer before the hoc in the econom occurred. (b) (2 point) Compute the growth rate of capital per effective worer at time T, i.e., immediatel after the hoc occurred. Compare it to the one ou obtained in (4a). (c) (2 point) Compute the growth rate of output per effective worer at time T, i.e., immediatel after the hoc occurred. Compare it to the one ou obtained in (4a). (d) (2 point) Compute the growth rate of capital per worer at T, and up to the point in time when the econom reache it new tead tate. Pleae draw a graph for the growth rate of capital per worer. (Put time on our horizontal axi, the growth rate of capital per worer on our vertical axi, and be ure to draw the graph both before T (i.e., when the econom wa at the old tead tate before the hoc) and after ome point T, when the econom i at it new tead tate.) (a) Before the hoc, the growth rate in capital and output in per effective worer term i zero. (b) Denote the before-hoc tead tate capital per effective worer a. At time T, the gro aving are determined b ( ) 1/3. However, the invetment needed to eep up the capital per effective worer at it contant level i lower at T, ince (n + g + δ) ( ) 0.06 > (n T + g T + δ T ) , and, therefore, capital per effective worer will grow tarting in T up to the point when the econom reache it new tead tate. At T, f( ) (n T + g T + δ T ) (n + g + δ) (n T + g T + δ T ) The growth rate of capital per effective worer at T, i equal to / % > 0. (c) At T, the growth rate of output per effective worer i α 1/ % > 0. (d) The growth rate in capital per worer at T i equal to + E E g T (It follow from the fact that K K E E E). At point T, when the econom i at it new tead tate, the growth rate in capital per worer will be equal to g T 0.03 (ince the capital per effective worer will be contant again at T and forever after). During the period of tranition, i.e., between T and T, capital per worer will grow at the rate greater than 3% but maller than 4% teadil declining toward 3%. 4. (10 point) Conider omeone deciding how to allocate her conumption over two period. She ha utilit function U(C 1, C 2 ) 0.5(C C 1 ) 2 0.5β(C C 2 ) 2, 0 β 1. β i called the time dicount factor, and meaure how one value current conumption veru future conumption, or, in other word, one impatience. (The lower the β, the more impatient ou are, i.e., the more ou value current conumption relative to future conumption, other thing being equal.) Aume that income in the firt period i 1, income in the econd period 2 2, the real interet rate i r and C i the bli level of conumption. (Hint: for thi utilit function, MU 1 C C 1, and MU 2 C C 2.) Aume that the real interet rate i r 0, and β 1 1+r ; income in the firt period i 1 40, in the econd period i

5 (a) (5 point) Solve for optimal conumption in each period, auming the conumer can freel ave or borrow at the 0% interet rate. Determine the optimal aving in the firt period of life. (b) (3 point) Now uppoe that, intead, the agent can ave at the rate r, but i unable to borrow at all. Solve for the optimal agent conumption in each period. (2 point) Who i better off: the borrowing-contrained conumer in (4b) or the conumer in (4a)? And wh? (a) The Euler optimalit condition i C C1 β(1 + r)(c C2 ), or C1 C2 ince we aumed that β(1 + r) 1. With thee preference, conumer prefer mooth conumption path if /he i not borrowing contrained. Optimal conumption hould atif the intertemporal budget contraint: C1 + C 2 1+r 1 1+r r, or C 1 + C ince we aumed that r 0 (and β1). Utilizing the Euler equation, C1 C2 80. S1 1 C To achieve optimal conumption path, conumer will borrow 40 againt hi/her future income. (b) In thi cae, the borrowing contraint i binding conumer i not able to borrow againt hi/her future income, even though /he would lie to do o a we jut found. Thu, C 1,bc min( 1, C1,nbc ) min(40, 80) 40, where nbc and bc tand for the cae with no borrowing and borrowing contraint, repectivel. Since there i no reaon for conumer to wate reource in the econd period, C 2,bc 120. (c) The no-borrowing contrained conumer i clearl better off ince /he i allowed to mooth conumption acro the period, a /he would lie to do for the preference we potulated. ou could alo compare the utilit level achieved b a conumer whoe borrowing contraint i binding, U bc, and conumer who can freel borrow at the prevailing interet rate, U nbc. Utilizing our aumption that the time dicount factor, β, i equal to 1, U bc 0.5(C 40) 2 0.5(C 120) 2 0.5(C C 40 + C C 120); U nbc 0.5(C C 80 + C C 80). The difference between the two i U nbc U bc 0.5(C C 80 + C C 80 C C 40 C C 120) 0.5( ) > 0 lifetime utilit achieved b a conumer who i not borrowing contrained i higher relative to the lifetime utilit of the ame conumer facing the binding borrowing contraint. 5