Solutions Manual forIntroduction to Modern Economic GrowthINSTRUCTORS MANUAL Michael Peters Alp Simsek Princeton University Press Princeton and Oxford Copyright 2009 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock,Oxfordshire OX20 1TW

All Rights Reserved The publisher would like to acknowledge the author of this volume for providing thecamera-ready copy from which this book was produced. press.princeton.edu For you, Edna. -M.P.To my parents and my brothers, -A.S.ContentsIntroduction xiChapter 2: The Solow Growth Model 1Exercise 2.7 1Exercise 2.11 3Exercise 2.12 6Exercise 2.14* 7Exercise 2.16* 9Exercise 2.17 10Exercise 2.18* 13Exercise 2.19* 15Exercise 2.20 15Exercise 2.21 17Exercise 2.22 20Exercise 2.23 20Exercise 2.27 23Chapter 3: The Solow Model and the Data 27Exercise 3.1 27Exercise 3.2 29Exercise 3.9 29Exercise 3.10 30Chapter 4: Fundamental Determinants of Dierences in Economic Performance 31Exercise 4.3 31Chapter 5: Foundations of Neoclassical Growth 33Exercise 5.1 33Exercise 5.2 35Exercise 5.9 36Exercise 5.10 38Exercise 5.11 40Exercise 5.12 42Exercise 5.13 47Exercise 5.14* 48Chapter 6: Innite-Horizon Optimization and Dynamic Programming 51Exercise 6.2* 51Exercise 6.3* 51Exercise 6.7 52Exercise 6.8 54vvi Solutions Manual for Introduction to Modern Economic GrowthExercise 6.9 58Exercise 6.12 59Exercise 6.18* 61Chapter 7: An Introduction to the Theory of Optimal Control 63Exercise 7.1 63Exercise 7.2* 65Exercise 7.5 65Exercise 7.19 66Exercise 7.10 69Exercise 7.17* 70Exercise 7.18 71Exercise 7.23 73Exercise 7.21 73Exercise 7.26 75Exercise 7.24* 76Exercise 7.25 79Exercise 7.28 80Chapter 8: The Neoclassical Growth Model 87Exercise 8.2 87Exercise 8.7 88Exercise 8.11 90Exercise 8.13 92Exercise 8.15 93Exercise 8.19 96Exercise 8.23 97Exercise 8.25 100Exercise 8.27 101Exercise 8.30 105Exercise 8.31 108Exercise 8.33 113Exercise 8.34 117Exercise 8.37 119Exercise 8.38* 121Chapter 9: Growth with Overlapping Generations 129Exercise 9.1 129Exercise 9.3 130Exercise 9.6 132Exercise 9.7 134Exercise 9.8 137Exercise 9.15 138Exercise 9.16 141Exercise 9.17 146Exercise 9.20 147Exercise 9.21 149Exercise 9.24* 150Exercise 9.32* 151Solutions Manual for Introduction to Modern Economic Growth viiChapter 10: Human Capital and Economic Growth 155Exercise 10.2 155Exercise 10.6 157Exercise 10.7 161Exercise 10.14* 165Exercise 10.18 167Exercise 10.20 169Chapter 11: First-Generation Models of Endogenous Growth 171Exercise 11.4 171Exercise 11.8 177Exercise 11.14 178Exercise 11.15 182Exercise 11.16 182Exercise 11.17 183Exercise 11.18 184Exercise 11.21* 185Chapter 12: Modeling Technological Change 191Exercise 12.2 191Exercise 12.5 192Exercise 12.9 193Exercise 12.11 196Exercise 12.13 197Exercise 12.14 201Chapter 13: Expanding Variety Models 205Exercise 13.1 205Exercise 13.5 207Exercise 13.6 210Exercise 13.7 211Exercise 13.13* 213Exercise 13.15 216Exercise 13.19 219Exercise 13.22 227Exercise 13.24 231Chapter 14: Models of Schumpeterian Growth 237Exercise 14.2 237Exercise 14.6 238Exercise 14.7* 239Exercise 14.12* 242Exercise 14.13 246Exercise 14.14 250Exercise 14.15 253Exercise 14.18 262Exercise 14.19* 265Exercise 14.20* 270Exercise 14.21* 271Exercise 14.22* 274viii Solutions Manual for Introduction to Modern Economic GrowthExercise 14.26 275Exercise 14.27* 284Exercise 14.35 287Chapter 15: Directed Technological Change 293Exercise 15.6 293Exercise 15.11 297Exercise 15.18 301Exercise 15.19 312Exercise 15.20 316Exercise 15.24* 317Exercise 15.27 319Exercise 15.28* 323Exercise 15.29 327Exercise 15.31* 329Chapter 16: Stochastic Dynamic Programming 331Exercise 16.3* 331Exercise 16.4* 331Exercise 16.8 332Exercise 16.9 333Exercise 16.10 333Exercise 16.11* 334Exercise 16.12 341Exercise 16.13 342Exercise 16.14 344Exercise 16.15 346Exercise 16.16 347Chapter 17: Stochastic Growth Models 351Exercise 17.5 351Exercise 17.7 352Exercise 17.13 356Exercise 17.15 360Exercise 17.18 361Exercise 17.22 364Exercise 17.30* 368Chapter 18: Diusion of Technology 373Exercise 18.8 373Exercise 18.9 378Exercise 18.12 381Exercise 18.13* 382Exercise 18.16* 385Exercise 18.18 387Exercise 18.19 390Exercise 18.21 391Exercise 18.26* 394Chapter 19: Trade and Growth 401Solutions Manual for Introduction to Modern Economic Growth ixExercise 19.2* 401Exercise 19.3 404Exercise 19.4* 406Exercise 19.7 408Exercise 19.11* 408Exercise 19.13* 411Exercise 19.24 415Exercise 19.25* 416Exercise 19.26* 417Exercise 19.27* 422Exercise 19.28 427Exercise 19.29 429Exercise 19.33 432Exercise 19.34 435Exercise 19.37 437Chapter 20: Structural Change and Economic Growth 441Exercise 20.3 441Exercise 20.5 442Exercise 20.6 444Exercise 20.7* 444Exercise 20.8 446Exercise 20.9* 449Exercise 20.16* 450Exercise 20.17 455Exercise 20.18 461Exercise 20.19* 463Chapter 21: Structural Transformations and Market Failures in Development 467Exercise 21.1 467Exercise 21.2 469Exercise 21.4 472Exercise 21.6 477Exercise 21.9 481Exercise 21.10 482Exercise 21.11 484Exercise 21.12 488Chapter 22: Institutions, Political Economy and Growth 495Exercise 22.2 495Exercise 22.3 498Exercise 22.8 500Exercise 22.9 502Exercise 22.16 503Exercise 22.17 508Exercise 22.18* 511Exercise 22.19* 512Exercise 22.20* 512Exercise 22.21* 515x Solutions Manual for Introduction to Modern Economic GrowthExercise 22.22* 516Exercise 22.25* 521Exercise 22.26* 522Exercise 22.27 523Exercise 22.30 531Chapter 23: Institutions, Political Economy and Growth 533Exercise 23.4 533Exercise 23.5 534Exercise 23.12 536References 547IntroductionThis manual contains solutions to selected exercises from Introduction to Modern Eco-nomic Growth by Daron Acemoglu.This volume is the Instructor Edition of the solutionsmanual, which contains a wider range of exercises than the Student Edition. The exerciseselection for both editions is guided by a similar set of principles. First, we have tried toinclude the exercises that facilitate the understanding of the material covered in the book, forexample, the ones that contain proofs to propositions or important extensions of the baselinemodels.Second, we have included exercises which we have found relatively more useful forimproving economic problem-solving skills or building economic intuition. Third, we madean eort to include exercises which seemed particularly challenging. Fourth, we also tried tostrike a balance across the chapters. Even with these criteria, making the nal selection hasnot been easy and we had to leave out many exercises which are no doubt important andinteresting. We hope the readers will nd our selection useful and we apologize up front fornot providing the solution of an exercise which may be of interest.A word on the organization and the equation numbering of this manual may be helpful.The exercises are presented in the same chapters they belong to in the book. Our solutionsregularly refer to equations in the book and also to equations dened within the manual. Toavoid confusion between the two types of references, we use the prex I for the labels of theequations dened in the Instructor Edition of the solutions manual. For example Eq. (5.1)would refer to the rst labeled equation in Chapter 5 of the book, whereas Eq. (I5.1) wouldrefer to the rst labeled equation in Chapter 5 of this edition.Although this version of the manual went through various stages of proofreading, there areno doubt remaining errors. To partly make up for the errors, we will post an errata documenton our personal websites which we will commit to updating regularly. In particular we wouldappreciate it if readers could e-mail us concerning errors, corrections or alternative solutions,which we will include in the next update of the errata document. Our present e-mail andwebsite addresses are as follows:Michael Peters, mipeters@mit.edu, http://econ-www.mit.edu/grad/mipetersAlp Simsek, alpstein@mit.edu, http://econ-www.mit.edu/grad/alpsteinAn errata document and additional information will also be posted on the companion sitefor Introduction to Modern Economic Growth at: http://press.princeton.edu/titles/8764.htmlAcknowledgments.We would like to thank Daron Acemoglu for his help with the exercise selection and foruseful suggestions on multiple solutions. We would also like to thank Camilo Garcia Jimeno,Suman Basu and Gabriel Carroll for various contributions and suggestions, and to thankSamuel Pienknagura for providing his own solutions to some of the exercises in Chapter 22.A number of exercises have also been assigned as homework problems for various economicsclasses at MIT and we have beneted from the solutions of numerous graduate students inthese classes.xiChapter 2: The Solow Growth ModelExercise 2.7Exercise 2.7, Part (a). Assuming C (t) = :1 (t) is not very reasonable since it impliesthat consumption for a given level of aggregate income would be independent of govern-ment spending. Since government spending is nanced by taxes, it is more reasonable toassume that higher government spending would reduce consumption to some extent. As analternative, we may assume that consumers follow the rule of consuming a constant shareof their after tax income, captured by the functional form C (t) = : (1 (t) G(t)). UsingG(t) = o1 (t), this functional form is also equivalent to C (t) = (: :o) 1 (t). In Part(b), we assume a more general consumption rule C (t) = (: `o) 1 (t) with the parameter` [0, 1[ controlling the response of consumption to increased taxes. The case ` = 0 corre-sponds to the extreme case of no response, ` = : corresponds to a constant after-tax savingsrule, and ` [0, 1[ correspond to other alternatives.Exercise 2.7, Part (b). The aggregate capital stock in the economy accumulates ac-cording to1 (t 1) = 1 (t) (1 c) 1 (t)= 1 (t) C (t) G(t) (1 c) 1 (t)= (1 : o (1 `)) 1 (t) (1 c) 1 (t) , (I2.1)where the last line uses C (t) = (: `o) 1 (t) and G(t) = o1 (t). Let ) (/) = 1 (t) ,1 =1 (1, 1, ) and assume, for simplicity, that there is no population growth. Then dividing Eq.(I2.1) by 1, we have/ (t 1) = (1 : o (1 `)) ) (/ (t)) (1 c) / (t) .Given / (0), the preceding equation characterizes the whole equilibrium sequence for thecapital-labor ratio /o (t)ot=0 in this model, where we use the subscript o to refer to theeconomy with parameter o for government spending.We claim that with higher government spending and the same initial / (0), the eectivecapital-labor ratio would be lower at all t0, that is/o (t)/o0 (t) for all t, where o < ot. (I2.2)To prove this claim by induction, note that it is true for t = 1, and suppose it is true forsome t _ 1. Then, we have/o (t 1) = (1 : o (1 `)) ) (/o (t)) (1 c) /o (t) (1 : o (1 `)) ) (/o0 (t)) (1 c) /o0 (t)

_1 : ot (1 `)_) (/o0 (t)) (1 c) /o0 (t) = /o0 (t 1) ,where the second line uses the induction hypothesis and the fact that ) (/) is increasing in/, and the third line uses oto.This proves our claim in (I2.2) by induction.Intuitively,12 Solutions Manual for Introduction to Modern Economic Growthhigher government spending reduces net income and savings in the economy and depressesthe equilibrium capital-labor ratio in the Solow growth model.As in the baseline Solow model, the capital-labor ratio in this economy converges to aunique positive steady state level /+ characterized by) (/+)/+=c1 : o (1 `). (I2.3)The unique solution /+ is decreasing in o and increasing in ` since ) (/) ,/ is a decreasingfunction of /. In the economy with higher government spending (higher o), the capital-laborratio is lower at all times, and in particular, is also lower at the steady state. Also, the moreindividuals reduce their consumption in response to government spending and taxes (higher`), the more they save, the higher the capital-labor ratio at all times and, in particular, thehigher the steady state capital-labor ratio.Exercise 2.7, Part (c). In this case, Eq. (I2.8) changes to) (/+)/+=c1 : o (1 ` c).Since ) (/) ,/ is decreasing in /, the steady state capital-labor ratio /+ is increasing in c.With respect to o, it can be seen that /+ is increasing in o if c1 ` and decreasingin o if c < 1 `. In words, when the share of public investment in government spending(i.e. c) is suciently high, in particular higher than the reduction of individuals savingsin response to higher taxes, the steady state capital-labor ratio will increase as a result ofincreased government spending. This prediction is not too reasonable, since it obtains whenthe government has a relatively high propensity to save from the tax receipts (high c) andwhen the public consumption falls relatively more in response to taxes (high `), both of whichare not too realistic assumptions.An alternative is to assume that public investment (such as infrastructure investment) willincrease the productivity of the economy. Let us posit a production function 1 (1, 1, cG, ),which is increasing in public investment cG, and assume, as an extreme case, that 1 hasconstant returns to scale in 1, 1 and public investment cG. With this assumption doublingall the capital (e.g. factories) and the labor force in the economy results in two times theoutput only if the government also doubles the amount of roads and other necessary publicinfrastructure. Dene ) (/, cq) = 1 (/, 1, cq, ) where q = G,1. Then, the steady statecapital-labor ratio /+ and government spending per capita q+ are solved by the system ofequations) (/+, cq+)/+=c1 : o (1 `)q+= o) (/+, cq+) .The second equation denes an implicit function q+ (/+) for government spending in termsof the capital-labor ratio, which can be plugged into the rst equation from which /+ can besolved for. In this model, /+ is increasing in o for some choice of parameters. Since someinfrastructure is necessary for production, output per capita is 0 when public investment percapita is 0, which implies that /+ is increasing in o in a neighborhood of o = 0. Intuitively,when public infrastructure increases the productivity of the economy, increased governmentspending might increase the steady state capital-labor ratio.Solutions Manual for Introduction to Modern Economic Growth 3Exercise 2.11Exercise 2.11,Part (a). Recall that the capital accumulation in the Solow (1956)model is characterized by the dierential equation`1 (t) = :1 (t) c1 (t) . (I2.4)Let / (t) = 1 (t) ,1(t) denote the capital-labor ratio. Using the production function 1 (t) =1(t)o1 (t)c71coand the assumption that the population is constant, the evolution ofthe capital-labor ratio is given by`/ (t)/ (t) =`1 (t)1 (t)= :1o1 (t)c171coc= :/ (t)c1.1coc,where the rst line uses Eq. (I2.4) and the second line denes . = 7,1 as the land to laborratio. Setting `/ (t) = 0 in this equation, the unique positive steady state capital-labor ratiocan be solved as/+ = _:.1coc_1(1c). (I2.5)The steady state output per capita is in turn given byj+= :_/

_c(.+)1co(I2.6)= _:c_c(1c).(1co)(1c)To prove that the steady state is globally stable, let us dene q (/) = :.1co/c1c. Sinceq (/) is a decreasing function of / and since q (/+) = 0, we haveq (/ (t))0 for / (t) (0, /+) andq (/ (t)) < 0 for / (t) (/+, ) .Since `/ (t) = / (t) q (/ (t)), the previous displayed equation implies that / (t) increases when-ever 0 < / (t) < /+ and decreases whenever / (t)/+. It follows that starting from any/ (0)0, the capital-labor ratio converges to the unique positive steady state level /+ givenin Eq. (I2.). Intuitively, the land to labor ratio remains constant since there is no populationgrowth. This in turn implies that there is a unique steady state with a positive capital-laborratio despite the fact that the production function exhibits diminishing returns to jointlyincreasing capital and labor.Exercise 2.11, Part (b). As Eq. (I2.6) continues to apply, the capital-labor ratioevolves according to`/ (t) = :. (t)1co/ (t)c(c :) / (t) . (I2.7)In this case the land to labor ratio . (t) = 7,1(t) is decreasing due to population growth,that is` . (t). (t) = :. (I2.8)The equilibrium is characterized by the system of dierential equations (I2.8) and (I2.7) alongwith the initial conditions / (0) = 1 (0) ,1(0) and . (0) = 7,1(0).First, we claim that the only steady state of this system is given by /+ = .+ = 0. ByEq.(I2.8), limto. (t) = 0 hence .+ = 0 is the only steady state.Plugging .+ = 0 in Eq.(I2.7) and solving for `/ (t) = 0, the only steady state capital-labor ratio is /+ = 0, proving4 Solutions Manual for Introduction to Modern Economic Growthour claim. Next, we claim that starting from any initial condition, the system will convergeto this steady state. Note that Eq. (I2.8) has the solution . (t) = . (0) oxp(:t). Pluggingthis expression in Eq. (I2.7), we have the rst-order nonlinear dierential equation`/ (t) = :. (0)1cooxp(:(1 c ,) t) / (t)c(c :) / (t) .To convert this to a linear dierential equation, dene r(t) = / (t)1cand note that theevolution of r(t) is given by` a(t)a(t) = (1 c) `I(t)I(t), or equivalently` r(t) = : (1 c) . (0)1cooxp(:(1 c ,) t) (1 c) (c :) r(t) .The solution to this linear rst-order dierential equation is given by (see Section B.4)r(t) = oxp ((1 c) (c :) t)_r(0) _t0 : (1 c) . (0)1cooxp_(:, (1 c) c) tt_dtt_=_r(0) : (1 c) . (0)1co:, (1 c) c_oxp((1 c) (c :) t): (1 c) . (0)1co oxp(:(1 c ,) t):, (1 c) cUsing r(t) = / (t)(1c), the previous equation implies/ (t) =___/ (0)1c c(1c):(0)1ocao(1c)c_oxp((1 c) (c :) t)c(1c):(0)1ocao(1c)coxp(:(1 c ,) t)__1(1c), (I2.9)which provides an explicit form solution for / (t). Since c , < 1, this expression alsoimplies that limto/ (t) = 0, proving that the economy will converge to the steady statecapital-labor ratio /+ = 0 starting from any initial condition.Eq.(I2.0) demonstrates a number of points worth emphasizing.First, since 1 c0the rst component always limits to zero, hence the initial condition has no impact on thelimiting value of capital-labor ratio in the Solow model. Second, the second component limitsto zero if c, < 1, but limits to a positive value if c, = 1 or if : = 0 (which correspondsto the case studied in Part (a) of this problem). Hence, the assumptions that drive the resultsof this exercise are the joint facts that the production function has diminishing returns incapital and labor and that the population is increasing. Intuitively, as the population grows,each unit of labor commands less land for production and the output of each worker declines(and limits to zero) since land is an essential factor of production.We next claim that the aggregate capital and output limit to innity. To see this, notethat limto/ (t) 1(t) =limto___/ (0)1c c(1c):(0)1ocao(1c)c_oxp((1 c) (c :) t) c(1c):(0)1ocao(1c)coxp(:(1 c ,) t)__1(1c)1(0) oxp(:t)= limto___/ (0)1c c(1c):(0)1ocao(1c)c_oxp((1 c) ct) c(1c):(0)1oc[ao(1c)c[oxp (:,t)__1(1c)1(0) = .Consequently, 1 (t) = 1 (1 (t) , 1(t) , 7) also limits to innity, since both 1 (t) and 1(t)limit to innity. The previous displayed equation also shows that the aggregate capital growsat rate :,, (1 c) < :, that is, the aggregate variables still grow at an exponential rate butSolutions Manual for Introduction to Modern Economic Growth 5just not fast enough to compensate for the population growth and sustain a positive level ofcapital-labor ratio and output per capita.We claim that the returns to land also limit to innity. Land is priced in the competitivemarket, hence returns to land are given byj:(t) = (1 c ,) 1(t)o1 (t)c7co,which limits to innity since 1 (t) and 1(t) are increasing. Alternatively, one can also see thisby noting that the share of land in aggregate output is constant due to the Cobb-Douglasform of the production function, that is, j:(t) 7 = (1 c ,) 1 (t). Since output grows,returns to land also grow and limit to innity. Intuitively, land is the scarce factor in thiseconomy and as other factors of production (and output) grow, the marginal product of landincreases. We nally claim that the wage rate limits to zero. The wage rate is given byn = ,1o11c71co= ,/c.1co,which limits to zero since both / and . limit to zero. Labor complements land and capital inproduction, therefore, as capital-labor ratio and land-labor ratio shrink to zero, wages alsoshrink to zero. Intuitively, every worker has less machines and less land to work with, hencehas lower productivity and receives lower wages in the competitive equilibrium.An alternative (simpler and more elegant) analysis.Dene the normalized vari-able

1(t) = _1(t)o71co_1(1c),which grows at the constant rate ,:, (1 c) < :. The production function can be rewrittenin terms of this normalized variable as1 _1 (t) , 1(t)_ = 1 (t)c 1(t)1c.Then, if we interpret 1(t) as the labor force in a hypothetical economy, the textbook analysisof the Solow model shows that this hypothetical economy has a unique steady state capital-labor ratio /+ = _1 (t) ,

1(t)_+, and starting at any 1 (t)0 and 1(t)0, the economyconverges to this level of capital-labor ratio. By construction, the aggregate capital in theoriginal economy is equal to the aggregate capital in the hypothetical economy. Thus, capitalin the original economy satiseslimto1 (t)

1(t) = /+,which shows that the aggregate capital 1 (t) asymptotically grows at rate ,:, (1 c) (whichis the growth rate of 1(t)).Since ,:, (1 c) < :, population grows faster than aggregatecapital, hence the capital-labor ratio limits to zero. The remaining results are obtained as inthe above analysis.Exercise 2.11, Part (c). We would expect both : and : to change. When we endogenizesavings as in Chapter 8, we see that : in general depends on a number of factors includingpreferences for intertemporal substitution and factor prices. Nevertheless, the analysis in thepreceding parts applies even when : = 1 (i.e. individuals save all their income), thus thecapital-labor ratio and the output per capita would limit to zero also in the economy withendogenously determined saving rate.Intuitively, savings cannot provide enough of a forceto overcome diminishing returns and immiseration in this economy.6 Solutions Manual for Introduction to Modern Economic GrowthThe stronger stabilizing force comes from endogenizing the demographics in the model,that is, endogenizing :. A simple way of doing this is to use the idea proposed by Malthus(1798), which we can incorporate in our model as:`1(t)1(t) = :(j (t)) , (I2.10)where :t (j)0, limjo:(j) = :0 and limjo:(j) =:< 0. The intuition behindEq.(I2.10) is that when output per capita is higher, people live longer, healthier and theyhave more children (abstracting from a lot of considerations such as birth control measures)which increases the population growth. Note that when the output per capita is very lowpopulation may shrink, and note also that there is a unique value of output per labor, j+,that satises :(j+) = 0, i.e. population remains constant when output per labor is at j+.The system that describes the equilibrium in this economy constitutes of Eqs. (I2.10),(I2.8), and (I2.7). This system has a unique steady state, (j+, .+, 1+), where j+ is the uniquesolution to :(j+) = 0, .+ is the unique solution toj+ = _:c_c(1c)(.+)(1co)(1c),and 1+ = .+7.Starting from any value of 1(0), the level of population will adjust, that islimto1(t) = 1+ = .+7 so that land per labor is .+, the output per labor is j+, and popu-lation growth is :(j+) = 0. Intuitively, as output per capita limits to 0, population growthslows down, which increases the amount of land that each person commands, and conse-quently increases output per capita.1Hence endogenizing demographics creates a stabilizingforce that sustains positive levels of output per capita. The result of Part (b), in particularthe result that output per capita and the capital-labor ratio limit to zero, are largely artifactsof taking : and : constant, which suggests that we should be careful in using the Solow modelsince the model relies on reduced form assumptions on population dynamics and consumerbehavior.Exercise 2.12Exercise 2.12, Part (a). The aggregate return to capital in this economy is given by1(t) 1 (t) = 11 (1 (t) , 1(t) , ) 1 (t), which is also the aggregate income of the capitalists.Then, capital accumulates according to`1 (t) = :111 (1 (t) , 1(t) , ) 1 (t) c1 (t) . (I2.11)1On the other hand, with Assumption (I2.10), sustained increases in output per capita are not possibleeither,even with modest amounts of technological progress. An increase in output per capita increasespopulation which in turn decreases and stabilizes output per capita. This is the so-called Malthusian trap: Ina Malthusian world, modest amounts of technological progress result in higher population but not necessarilyhigher output per capita.The Malthusian model roughly matches the evolution of output per capita beforethe Industrial Revolution. For example, despite technological progress, the real wages in England in the 17thcentury were similar to those in the 13th century (Clark (2004)). Again consistent with this model, measuresof urbanization and population density are good proxies for technological progress of ancient societies (seeAcemoglu, Johnson, Robinson (2002)).However, a suciently fast technological change might overturn thisresult, in particular, once we add labor-augmenting technological change in the model, the Malthusian trapis less likely the larger the labor-augmenting technological progress and the smaller n (the maximum rateof population growth). Hence, one can argue that the Industrial Revolution (which increased technologicalprogress) and the demographic transition (which one may interpret as reducing n) were crucial for the humansocieties to get out of the Malthusian trap.Solutions Manual for Introduction to Modern Economic Growth 7Let ) (/) = 1 (/, 1, ) and note that we have`/ (t)/ (t)=`1 (t)1 (t) := :1)t (/ (t)) c :, (I2.12)where the second line uses Eq. (I2.11) and the fact that )t (/ (t)) = 11 (1 (t) , 1(t) , ).The equilibrium path of the capital-labor ratio, [/ (t)[ot=0, is the solution to Eq.(I2.12)with the initial condition / (0). In the steady state equilibrium, the capital-labor ratio,/ (t) = /+, is constant for all t. By Eq. (I2.12), the steady state capital-labor ratio solves::1)t (/+) = c : (I2.13)This equation has a unique solution since )t (/) is decreasing in / with limI0)t (/) = and limIo)t (/) = 0 from Assumption 2. Moreover, we claim that the unique steady stateequilibrium is globally stable, that is, starting from any / (0)0, limto/ (t) = /+. To seethis, note that the fact that )t (/) is decreasing in / implies:1)t (/ (t)) c :_ 0 if / (t) < /+< 0 if / (t)/+,which shows that / (t) converges to the unique steady state /+, proving global stability.Exercise 2.12, Part (b). Recall that the golden rule capital-labor ratio /+jc|o maximizessteady state consumption per capita subject to a constant savings rule. Equivalently, /+jc|omaximizes the steady state net output, ) (/) (c :) /, and is found by)t_/+jc|o_ = c :. (I2.14)Comparing Eqs. (I2.18) and (I2.14), we see that /+ < /+jc|o since :1< 1 and )t (/) isdecreasing in /. In this economy, the steady state capital-labor ratio is always less than thegolden rule capital-labor ratio.To see the intuition, note that the golden rule capital-laborratio /+jc|o obtains in an economy when aggregate savings are equal to aggregate returns tocapital since:) _/+jc|o_ = (c :) /+jc|o = )t_/+jc|o_/+jc|o = 1+/+jc|o.When only capitalists save, it is impossible to save all the of the return to capital sincethis would require the capitalists to consume nothing. Hence, in an economy in which onlycapitalists save, the capital-labor ratio is always less than the golden rule level.Exercise 2.14*Exercise 2.14, Part (a). We will construct an example in which 1 (t) , 1 (t) and C (t)asymptotically grow at constant but dierent rates. Consider paths for 1 (t) , C (t) given by1 (t) = 1 (0) oxp(qt) , C (t) = C (0) oxp_q2t_where q0 and C (0) < 1 (0), and dene 1 (t) as the solution to`1 (t) = 1 (t) C (t) c1 (t). Note that `1 (t) ,1 (t) = q and`C (t) ,C (t) = q,2 for all t. Dene (t) = 1 (t) ,1 (t)and note that` (t)(t) =`1 (t)1 (t) q =1(t) C (t)1 (t)1(t) c q (I2.15)hence` (t) = 1 C (0)1 (0) oxp_q2t_(c q) (t) .8 Solutions Manual for Introduction to Modern Economic GrowthAs t , the middle term on the right hand side goes to zero and (t) = 1 (t) ,1 (t)converges to the constant 1, (c q), so we have limto `1 (t) ,1 (t) = q. Hence, in thisexample 1 (t) and 1 (t) asymptotically grow at rate q while C (t) asymptotically grows atrate q,2, proving that Part 1 of Theorem 2.6 is not correct without further conditions.Note that this example features C (t) growing at a constant rate slower than both 1 (t)and 1 (t) so in the limit all output is invested and both capital and output grow at the sameconstant rates. To rule out such examples, let us assume thatlimtoC (t) ,1 (t) = j+ (0, 1) (I2.16)so that qC = qY . Taking the limit of Eq. (I2.1), we have as t ` (t) - 1 j+(c q) (t) .This equation shows that limto(t) = (1 j+) , (c q) (0, ), which in turn showsthat 1 (t) and 1 (t) asymptotically grow at the same constant rates, that is q1 = qY . HenceCondition (I2.16) is sucient to ensure that the limiting growth rates of 1 (t) , 1 (t) andC (t) are equal to each other.Exercise 2.14, Part (b). We assume that Condition (I2.16) is satised so qC = qY =q1 = q. We also assume that both qY (t) and q1 (t) converge to q at a rate faster than 1,t,that is, there exists a sequence -ToT=1 with limTo-TT = 0 such that, [qY (t) qY[ < -T,2and [q1 (t) qY[ < -T,2 for all T and t _ T.Repeating the steps as in the proof of Theorem 2.6 as suggested in the exercise gives1 (t) = 1 _oxp__tT (qY (:) q1 (:)) d:_1 (t) , oxp__tT (qY (:) :) d:_1(t) , (T)_.(I2.17)For each T, we let (t) = oxp ((qY :) t) and we dene the production function1T (1 (t) , (t) 1(t)) = 1 _1 (t) , (t) 1(t)(T), (T)_,and the production function 1 (1 (t) , (t) 1(t)) as the limit1 (1 (t) , (t) 1(t)) =limTo1T (1 (t) , (t) 1(t)) .We claim that 1 provides an asymptotic representation for

1, that islimto`1(1(t),1(t), `(t))1(1(t),(t)1(t)) = 1. To see this, we rst claim thatoxp(-T (t T)) _

1 _1 (t) , 1(t) , (t)_1T (1 (t) , (t) 1(t)) _ oxp(-T (t T)) . (I2.18)To prove the right hand side, note that

11T=

1 _oxp__tT (qY (:) q1 (:)) d:_1 (t) , oxp__tT (qY (:) :) d:_1(t) , (T)_

1 _1 (t) , oxp__tT (qY :) d:_1(t) , (T)__

1 _1 (t) oxp(-T (t T)) , oxp__tT (qY :) d:_1(t) oxp(-T (t T)) , (T)_

1 _1 (t) , oxp__tT (qY :) d:_1(t) , (T)_= oxp (-T (t T)) ,Solutions Manual for Introduction to Modern Economic Growth 9where the rst line uses Eq. (I2.17), the inequality follows since [qY (:) qY[ < -T,2 and[q1 (:) qY[ < -T,2 for : _ T, and the last line follows since 1 is constant returns to scale.The left hand side of Eq. (I2.18) is proved similarly. Letting t = T for some 1 andtaking the limit of Eq. (I2.18) over T, we havelimTooxp(( 1) -TT) _limTo

1 _1 (T) , 1(T) , (T)_1T [1 (T) , (T) 1(T)[ _limTooxp(( 1) -TT) .Since limTo-TT = 0, the limits on the left and the right hand side of the inequality areequal to 1, which implies that the middle limit is also equal to 1. Using t = T, the middlelimit can be rewritten aslimto

1 _1 (t) , 1(t) , (t)_1T=t [1 (t) , (t) 1(t)[ = 1,which holds for all 1. Taking the limit of the above expression over we have1 = limo limto

1 _1 (t) , 1(t) , (t)_1T=t [1 (t) , (t) 1(t)[= limto

1 _1 (t) , 1(t) , (t)_limo1T=t [1 (t) , (t) 1(t)[= limto

1 _1 (t) , 1(t) , (t)_1 [1 (t) , (t) 1(t)[ ,where the last line follows from denition of 1. This proves limto 1,1 = 1, that is, 1provides an asymptotic representation for 1 as desired.Note that our proof relies on the inequality in (I2.18), which does not necessarily holdwhen either q1 (t) or qY (t) converges to q at a rate slower than 1,t. In this case, 1 does notnecessarily have an asymptotic representation with labor-augmenting technological progress.Exercise 2.16*Exercise 2.16, Part (a). Let jt = dj,d/ and rewrite the equation j = c(j /jt)oasc1odjj1o c1oj = d// .The right hand side is readily integrable but the left hand side is not. After dividing thenumerator and the denominator on the left hand side with j1o, and multiplying both sidesby (o 1) ,o, we haveo1oc1oj1odj1 c1oj(o1)o= o 1od// .In this form, the left hand side is equal to d log_1 c1oj(o1)o_,dj so that integratingboth sides giveslog_1 c1oj (/)(o1)o_ = log_/1_C,where C is a constant of integration. Solving this equation, we havej (/) = _c1oc1ooxp(C) /1_1.Letting c0 = c1ooxp(C) gives the desired expression for j (/).10 Solutions Manual for Introduction to Modern Economic GrowthExercise 2.16, Part (b). Dividing Eq.(2.88) by 1(t) and dropping the time depen-dence, we havej (/) = _ (11)1/1 (1 ) (11)1_1.Hence, for the two expressions to be identical, we needc0= (11)1c1o= (1 ) (11)1,which can be simplied toc = (11)1o(1 )o, and c0 = c1o1 .If c and c0 satisfy these equations, then we obtain the exact form of the CES function in(2.88).Exercise 2.17Exercise 2.17, Part (a). Let 1 take the Cobb-Douglas form, that is, assume1 [11, 11[ = C (11)c(11)1c,for some constants C and c. Then, 1 can be rewritten as1 [11, 11[ = C1c__1c(1c)1_1_1c.Note that, when written in this form, the technological change is essentially labor-augmenting.Then the textbook analysis for the Solow model with technological progress applies in thiscase as well. In particular, dene (t) = 1 (t) 1 (t)c(1c)as the labor-augmenting tech-nological progress and / (t) = 1 (t) , ((t) 1(t)) as the eective capital-labor ratio, and notethat`/ (t)/ (t)=:1 [1 (t) 1 (t) , 1 (t) 1(t)[ c1 (t)1 (t)`(t)(t) `1(t)1(t)= :C/ (t)c1c q1c1 cq1.Solving for `/ (t) = 0, there exists a globally stable steady state with eective capital-laborratio/+ =_:Cc q1 c1cq1_ 11o.It follows that the economy admits a balanced growth path in which the eective capital-labor ratio is constant and the capital-labor ratio and output per capita grow at the constantrateq = q1 c1 cq1.Starting from any level of eective capital-labor ratio, the economy converges to this eectivecapital-labor ratio, that is, if / (0) < /+, then the economy initially grows faster than q and/ (t) /+, and similarly, if / (0)/+, then the economy initially grows slower than q and/ (t) | /+.Solutions Manual for Introduction to Modern Economic Growth 11Exercise 2.17, Part (b). We rst prove a general result that will be useful to solve thisexercise. We claim that the eective capital-labor ratio in this economy limits to innity,that islimto/ (t) = 1 (t) 1 (t) , (1 (t) 1) = . (I2.19)The intuition for this result is as follows: the capital stock would asymptotically grow atrate q1 if 1 (t) were constant. Hence, with the added technological progress in 1 (t), theeconomy does not do worse and capital stock continues to grow at least at rate q1. It followsthat the eective capital stock, 1 (t) 1 (t) grows strictly faster than q1, leading to Eq.(I2.10). The following lemma and the proof formalizes this idea.Lemma I2.1. Suppose that the production function takes the form 1 (t) =1 (1 (t) 1 (t) , 1 (t) 1(t)) and suppose 1 (t) grows at the constant rate q1 and 1 (t) _1 (0) for all t. Let / (t) = 1 (t) , (1 (t) 1(t)) denote the capital to eective labor ra-tio in this economy and / (t) denote the capital to eective labor ratio in the hypotheti-cal economy which has the same initial conditions but in which the production function isgiven by 1 (t) = 1 (1 (0) 1 (t) , 1 (t) 1(t)), that is, the hypothetical economy has labor-augmenting technological change at the same rate q1 but it has no capital-augmenting techno-logical change. Then, / (t) _ / (t) for all t. In particular, limto / (t) _ /+, and moreover,limto/ (t) = limto1 (t)/ (t) = whenever limto1 (t) = .Proof. Let ) _/_ = 1 _/, 1_ and note that / accumulates according tod/,dt = :) _1/_(c :)/_ :) _1 (0)/_(c :)/, (I2.20)where the inequality follows since 1 (t) _ 1 (0). Similarly, capital to eective labor ratioin the hypothetical economy, /, satisesd

/,dt = :) _1 (0)

/_(c :)

/, (I2.21)with the same initial condition, that is, / (0) = / (0). Suppose, to get a contradiction, that/ (t) _ / (t) for some t0. Since both / and / are continuously dierentiable in t, and since/ (0) = / (0), there exists some tt [0, t[ where / just gets ahead of /, that is / (tt) = / (tt)and d/ (tt) ,dt < d

/ (tt) ,dt.Since / (tt) = / (tt), this yields a contradiction to Eqs.(I2.20)and (I2.21), showing that / (t) _ / (t) for all t. Note that the textbook analysis of the Solowmodel with labor-augmenting technological progress shows that limto / (t) = /+0, whichin turn implies limto / (t) _ /+0. Finally, this also implies that Eq. (I2.10) holds whenlimto1 (t) = , as desired. We next turn to the present problem. We prove the result by contradiction, that is, wesuppose there is a steady state equilibrium and we show that the production function musthave a Cobb-Douglas representation.Consider a BGP equilibrium in which both 1 and 1grow at constant rates q1and qY. We use superscripts for these growth rates so that thegrowth rates of capital and output are not confused with the productivity growth rates.We rst show that 1 and 1 must grow at the same rate, that is q1= qY. To see this,consider the capital accumulation equation`1 = :1 c1.12 Solutions Manual for Introduction to Modern Economic GrowthSince 1and 1 grow at constant rates,we have 1 (t)=1 (0) oxp_q1t_ and 1 (t)=1 (0) oxp_qYt_. Plugging these expressions in the previous displayed equation, we haveq11 (0) oxp_q1t_ = :1 (0) oxp_qYt_c1 (0) oxp_qYt_,which further impliesq11 (0) c1 (0):1 (0)= oxp__qYq1_t_.The left hand side is constant, hence this equation can only be satised if qY= q1. We referto the common growth rate of 1 and 1 as q.Second, we dene ) (/) = 1 (/, 1) and we claim that ) (/) = C/cfor some constants Cand c (0, 1). To see this, consider1 (t)1= 1 (t) 1 _1 (t)1 (t) 1 (t)1(t) , 1_= 1 (t) ) _1 (t)1 (t) 1 (t)1(t)_,Plugging 1 (t) = 1 (0) oxp(qt) and 1 (t) = 1 (0) oxp(qt), 1 (t) = 1 (0) oxp(q1t) and1 (t) = 1 (0) oxp(q1t) in this expression, we have1 (0)1 (0) 1 oxp((q q1) t) = ) (/ (0) oxp((q1 q1 q) t)) . (I2.22)By Lemma I2.1, / (t) = / (0) oxp((q1 q1 q) t) is growing. Then, considering thefollowing change of variables between t and // (0) oxp((q1 q1 q) t) = /in Eq. (I2.22), ) (/) can be calculated for all / _ / (0). In particular, we have) (/) =1 (0)1 (0) 1 oxp_q q1q1 q1 q ln// (0)_=1 (0)1 (0) 1_1/ (0)_111+/111+= C/cfor some constant C, where the last line denes c =jj1j1j1j.Finally, note that ) (/) = C/cimplies1 (11, 11) = 11) (/) (I2.23)= C (11)(jj1)(j1j1j)(11)j1(j1j1j),proving that the production function takes the Cobb-Douglas form.An alternative proof based on the fact that factor shares are constant. Suppose,as before, that we are on a BGP on which 1 and 1 grow at constant rates qYand q1. Thesame argument as above shows that we must have qY= q1= q. We rst claim that thefactor shares should also be constant on any such BGP. Letc1 = 111= 11111and c1 = n11= 12111,denote the shares of capital and labor in output. Here, 11 and 12 denote the rst and secondderivatives of the function 1 (11, 11).Solutions Manual for Introduction to Modern Economic Growth 13We rst claim that c1 (t) is a constant independent of time. Dierentiating 1 (t) =1 (1 (t) 1 (t) , 1 (t) 1(t)) with respect to t and dividing by 1, we have`11=11111_ `11 q1_ 12111_ `11 q1_= c1 (t)_ `11 q1_c1 (t)_ `11 q1_= c1 (t)_ `11 q1_ (1 c1 (t))_ `11 q1_, (I2.24)where the last line uses c1 (t) c1 (t) = 1. By assumption, we have`1 ,1 = q,`1,1 = q,and`1,1 = 0. Moreover, Lemma I2.1 shows that q _ q1, which also implies q1 qq1.Consequently, by Eq. (I2.24), c1 (t) can be solved in terms of the growth rates and is givenbyc1 (t) = c1 =q q1q q1 q1. (I2.25)This expression is independent of t, which proves our claim that c1 (t) is constant.Second, we use Eq. (I2.2) to show that 1 takes the Cobb-Douglas form. Note that wehavec1 (t) = 11111= )t (/)) (/) 1111= )t (/) /) (/),where recall that we have dened / = (11) , (11). Using the fact that c1 (t) is constant,we haved log ) (/)d/= )t (/)) (/) = c1/ .Note that by Lemma I2.1, we have that / (t) is growing. Then, the previous equation issatised for all / _ / (0), thus we can integrate it to getlog ) (/) = c1 log / log C,where log C is a constant of integration. From the previous expression, we have ) (/) =C/c1, whichagainleads totheCobb-Douglas productionfunction1 (11, 11) =C (11)c1(11)1c1. In view of the expression for c1 in Eq. (I2.2), the representa-tion obtained in the alternative proof is exactly equal to the representation obtained earlierin Eq. (I2.28).The second proof brings out the economic intuition better. From the growth accountingequation (I2.24), when eective factors grow at dierent constant rates (in particular, wheneective capital grows faster than eective labor, as implied by Lemma I2.1), output cangrow at a constant rate only if factor shares remain constant. But when eective factorsgrow at dierent rates, the only production function that keeps factor shares constant is theCobb-Douglas production function.Exercise 2.18*We rst note that, by Lemma I2.1, the eective capital-labor ratio in this economy limitsto innity, that islimto1 (t) 1 (t) , (1 (t) 1) = . (I2.26)14 Solutions Manual for Introduction to Modern Economic GrowthNext, we claim that capital, output, and consumption asymptotically grow at rate q1. Tosee this, let / (t) = 1 (t) , (1 (t) 1) denote the capital to eective labor ratio and note thatd/ (t) ,dt = :_1_1 (t)/ (t)_(o1)o1_o(o1)(c :)/ (t) .Using the limit expression in (I2.26) and the fact that o < 1, this dierential equationapproximatesd/ (t) ,dt - :o(o1)1(c :)/ (t) .Hence, we havelimtod/ (t) ,dt = 0 and limto/ (t) = :o(o1)1, (c :) .Since / (t) asymptotes to a constant, we have that 1 (t) = 1 (t) 1/ (t) asymptotically growsat rate q1. Moreover, we have1 (t) = 1 (t) 1) _1 (t)/ (t)_= 1 (t) 1_1_1 (t)/ (t)_(o1)o1_o(o1)o(o1)11 (t) 1 as t , (I2.27)hence asymptotically 1 (t) also grows at the constant rate q1.Finally, consumption in theSolow model is a constant share of output and hence also grows at rate q1, proving our claim.Finally, we claim that the share of labor in national income tends to 1. Note that thewages can be solved fromn(t) =dd1__1 (1 (t) 1 (t))(o1)o1 (1 (t) 1)_(o1)o_o(o1)= 11 (t)(o1)o11o1 (t)1o.The share of labor in national income is then given byn(t) 11 (t)=11 (t)(o1)o1(o1)o1 (t)1o1 (t)=_o(o1)11 (t) 1_(o1)o1 (t)(o1)o,which limits to 1 from Eq. (I2.27), proving our claim.Intuitively, when o < 1, capital and labor are not suciently substitutable and labor be-comes the bottleneck in production. Hence, despite deepening of eective capital to eectivelabor, capital and output can only grow at the same rate as eective labor. A comple-mentary intuition comes from considering the approximation in Eq. (I2.27). With o < 1,capital deepening causes an abundance of eective capital so that the limit production isessentially determined by how much eective labor the economy has. This exercise providesa robust counter-example to the general claim sometimes made in the literature that capital-augmenting technological progress is incompatible with balanced growth. Note, however,that the share of labor in this economy goes to one which suggests that the claims in theliterature can be remedied by adding the requirement that the shares of both capital andlabor stay bounded away from 0.Solutions Manual for Introduction to Modern Economic Growth 15Exercise 2.19*Exercise 2.19, Part (a). Similar to the construction in the proof of Theorem 2.6, notethat, in this case we have

1 _1 (t) , 1(t) , (t)_= 1 (t) `(t)1(t)1`(t)= 1 (t) `(T)1(t)1`(T).where the second line uses the fact that 1 (t) = 1(t) = oxp(:t) 1 (0). Dening (t) = 1for all t, and1T (1 (t) , (t) 1(t)) = 1 (t) `(T)((t) 1(t))1`(T), (I2.28)we have 1 _1 (t) , 1(t) , (t)_ = 1T (1 (t) , (t) 1(t)), hence the expression in (I2.28) pro-vides a class of functions (one for each T) as desired.Exercise 2.19, Part (b). The derivatives do not agree sinced1T (1 (t) , (t) 1(t))d1 (t)= (T)_1 (t)(t) 1(t)_`(T)1= (T) ,where we have used (t) = 1 and 1 (t) = 1(t), whiled

1 _1 (t) , 1(t) , (t)_d1 (t)= (t)_1 (t)1(t)_`(t)1= (t) .Hence, for any xed T, the derivatives of

1 _1 (t) , 1(t) , (t)_ and 1T (1 (t) , (t) 1(t))will be dierent as long as

(t) ,= (T).Exercise 2.19, Part (c). Note that, in this economy, capital, labor, output, and con-sumption all grow at rate :. However, the share of capital is given by

11_1 (t) , 1(t) , (t)_1 (t)

1 _1 (t) , 1(t) , (t)_ =

(t) 1 (t)1 (t) `(t)1(t)1`(t) = (t) ,where we have used 1 (t) = 1(t). Hence even though all variables grow at a constantrate, the share of capital will behave in an arbitrary fashion. When, for example,

(t) =(2 sin (t)) ,4, the share of capital will oscillate.Exercise 2.20Exercise 2.20, Part (a). Let / (t) = 1 (t) ,1 denote the capital-labor ratio in thiseconomy. Note that n(/) = ) (/) /)t (/) is increasing in /. There are two cases toconsider. First, supposelimIon(/) < n,that is, the minimum wage level is so high that, even with abundant levels of capital-laborratio, labors productivity would be short of n (this is the case, for example, with the CESproduction function with o < 1 when n is suciently large). In this case, no rm can aordto pay wages n regardless of the capital used by each unit of labor, hence the equilibriumemployment is always zero, that is 1o(t) = 0 and equilibrium unemployment is 1. The moreinteresting case is when limIo) (/) /)t (/)n, so there exists a unique / such thatn_/_ = ) _/_/)t_/_ = n.16 Solutions Manual for Introduction to Modern Economic GrowthIn this case, suppose rst that / (t) < /. As each employed worker commands capital /,that is 1 (t) ,1o(t) = /, the employment rate |o(t) is given by|o(t) = 1o(t)1= 1 (t)1/= / (t)/.Then, output per capita is given byj (t) = |o(t) ) _/_= / (t) ) _/_/< / (t) ) (/ (t))/ (t)= ) (/ (t)) ,where the inequality follows since ) (/) ,/ is a decreasing function. The second line showsthat the production function is essentially linear when / (t) < /. The inequality shows thatoutput per capita is depressed by the minimum wage requirement since some laborers in theeconomy remain unemployed. Next, suppose that / (t)/. Then each employed workercommands capital / (t), all labor is employed, that is |o(t) = 1, and output per capita isgiven by j (t) = ) (/ (t)).Combining these two cases, capital-labor ratio in this economy evolves according to`/ =_: min_) _/_/, ) (/)/_c_/, (I2.29)given the initial condition / (0) = 1 (0) ,1. Recall that /+ < /, somin_) _/_/, ) (/)/__ ) _/_/< ) (/+)/+= c:.By Eq. (I2.20), this implies that `/ (t) < 0 for any / (t), that is / (t) is always decreasing, andin particular,limto/ (t) = 0.Hence the capital-labor ratio and output per capita in this economy converges to 0 startingfrom any initial condition. Note that the unemployment rate, given by 1 |o(t) = 1 min_/ (t) ,/, 1_, is weakly increasing and tends to 1 in the limit.Intuitively, output per capita and capital accumulation is depressed due to the minimumwage requirement since not all labor can be competitively employed at the required minimumwages.Somewhat more surprisingly the dynamic eects of the minimum wage requirementare so drastic that the capital-labor ratio and output per capita in the economy tend to 0 andunemployment rate tends to 1. The minimum wage requirement is equivalent to requiringeach employed worker to command a minimum amount of machines, /, regardless of thecapital-labor ratio in this economy. Consequently, as aggregate capital falls, fewer people areemployed which reduces aggregate savings and further reduces aggregate capital, leading toimmiseration in the long run.22In contrast with the standard Solow model, marginal productivity of capital does not increase as thecapital-labor ratio falls. By requiring that each labor commands a capital level I, the minimum wage laweectively shuts down the diminishing returns to capital channel, which would typically ensure an equilibriumwith positive capital-labor ratio.Solutions Manual for Introduction to Modern Economic Growth 17Exercise 2.20, Part (b). In this case, the dynamic equilibrium path for capital-laborratio is identical to the textbook Solow model. More specically, since all agents in thiseconomy save a constant share : of their income, the distribution of income between employeesand employers does not change the capital accumulation equation, which is still given by`/ (t) = :) (/ (t)) c/ (t) .Hence, starting with any / (0), capital-labor ratio in this economy converges to /+0 thatis the unique solution to ) (/+) ,/+ = c,:. However, the distribution of income betweencapital owners and workers will be dierent since the wages along the equilibrium path arenow given by `) (/ (t)) instead of ) (/ (t)) / (t) )t (/ (t)). Depending on ` and the form ofthe production function, the workers could be better or worse o relative to the case withcompetitive labor markets.Exercise 2.21Exercise 2.21, Part (a). Capital accumulates according to1 (t 1) = : (/ (t)) 1 (1 (t) , 1(t)) (1 c) 1 (t) ,which, after dividing by 1(t 1) = 1(t) (1 :), implies/ (t 1) =: (/ (t)) ) (/ (t)) (1 c) / (t)1 :=_:0/ (t)11_/ (t) (1 c) / (t)1 :=:01 : 1 c 1 :/ (t) =:01 : / (t) ,where the last equality uses the assumption c :=2. Then, for any / (0) (0, :0, (1 :)) we have/ (t) = _/ (0) , if t is even:0, (1 :) / (0) , if t is odd,hence the capital-labor ratio in this economy uctuates between two values.Exercise 2.21, Part (b). Deneq (/) = : (/) ) (/) (1 c) /1 :as the function that determines the next periods capital-labor ratio given the capital-laborratio /. As we have seen in Part (a), there exist production functions ) that result in discretetime cycles, that is, there exist ) (.) and values /1 < /2 such that q (/1) = /2 and q (/2) = q1.Consider the function /(/) = q (/) /. We have,/(/1) = q (/1) /1 = /2/10,and/(/2) = q (/2) /2 = /1/2 < 0.Since the function / is continuous, by the intermediate value theorem, there exists / (/1, /2)such that /_/_ = 0, that isq_/_ = /.This shows that whenever there is a cycle (/1, /2), there exists a steady state / (/1, /2).18 Solutions Manual for Introduction to Modern Economic GrowthWe next turn to the stability of the steady state. Let / be the rst intersection of /(/)with the zero line, so / crosses the zero line from above and /t_/_ < 0, which is equivalentto saying qt_/_ < 1. Even with this choice of /, the steady state is not necessarily stable.If qt_/_ is smaller than 1, then when the capital-labor ratio starts very close to the steadystate, it will overshoot the steady state value and might diverge away from the steady state.By Theorem 2.3, a sucient condition for local stability of / is qt_/_ < 1. Since we alreadyhave qt_/_ < 1, we only need to guarantee that qt_/_1. Writing this condition interms of : and ), we have qt_/_ = _:t_/_) _/_:_/_)t_/_ 1 c_, (1 :)1, orequivalently,:t_/_) _/_:_/_)t_/_c 2 :,that is, : (/) ) (/) is not decreasing too fast at the capital-labor ratio /.3If this condition issatised at /, then / is a stable steady state.Exercise 2.21, Part (c). In continuous time, capital accumulates according to`/ = :) (/) (: c) /. (I2.30)Since the right hand side is continuous, we have that / is a continuous (in fact, continuouslydierentiable) function of t. Suppose that there is a cycle, that is suppose there exists t1 < t2such that / (t1) = / (t2) = / and / (tt) ,= / for some tt (t1, t2). Without loss of generality,suppose that / (tt)/ (the other case is identical).Then there exists t [t1, tt) such that`/_t_0 and /_t_ (/, / (tt)). Dene

t = inf_t _tt, t2 [ / (t) = /_t__. (I2.31)The continuous function / (t) must decrease from / (tt) towards / (t2) = / and has to cross/_t_ (/, / (tt)) at least once in the interval [tt, t2[, hence the set over which we take theinmum in (I2.81) is non-empty and t is well dened.Moreover, by continuity of / (t), theinmum of the set is indeed attained, hence /_

t_ = /_t_. Since the system in (I2.80) isautonomous (independent of time), it must be the case that`/_

t_ = `/_t_0,that is, / (t) is increasing in a suciently small neighborhood of t. Then, there exists -0suciently small such that /_

t -_ < /_

t_ and t -tt. This implies, by continuityof / (t) and the fact that / (tt)/_t_ = /_

t_, that there exists ttt _tt,

t -_ such that/ (ttt) = /_t_. But since ttt _tt,

t -_, we have a contradiction in view of the denition of tin (I2.81), proving that there cannot be a cycle.Intuitively, to have a cycle in continuous time, one has to cross a level of capital bothon the way up in the cycle and on the way down in the cycle. But this implies that theautonomous system in (I2.80) that describes the behavior of / must have a positive and anegative derivative at the same level of capital, which yields the desired contradiction.A simpler and more intuitive argument is as follows. Suppose there is a cycle as describedabove. Once can then show that there exists t [t1, t2[ such that `/_t_ = 0. This implies/ (t) = /_t_ for all t _ t, which yields a contradiction to the fact that there is a cycle. How-ever, this more intuitive argument is not entirely correct, since `/_t_ = 0 does not necessarilyimply that / (t) = /_t_ for all t _ t. Even though the path _/ (t) = /_t_ for all t _ t_ is asolution to the dierential equation starting at t, there may also be other solutions since we3Note that we need c (I) ) (I) to be decreasing- at least for some Is- to have a cycle, but we also need itto be decreasing not too fast to have a stable steady state in between.Solutions Manual for Introduction to Modern Economic Growth 19have not made strong enough assumptions to guarantee the uniqueness of solutions to thedierential equation in (I2.80).4If we assume that ) is Lipschitz continuous at each /, thenthe dierential equation in (I2.80) has a unique solution and the more intuitive argumentalso applies.5For example, if we assume that ) is continuously dierentiable with boundedrst derivative, then this implies that ) is Lipschitz continuous over the relevant range andthe more intuitive argument applies and shows that there cannot be cycles.Exercise 2.21, Part (d). This exercise shows that approximations of discrete timewith continuous time are not always without loss of generality since some qualitative resultschange after the approximation. In particular, the Solow model in discrete time may havecycles while cycles cannot exist in the Solow model in continuous time. There are two ways tointerpret this nding. If one views cycles as pathological cases, then the continuous approxi-mation is good since it removes the cycles that are artifacts of our modeling choices. On theother hand, one may also view the cycles in this model as interesting economic phenomena(even though that view requires extreme assumptions and a really good imagination!).Forexample, suppose there are overlapping generations, that each generations capital level isdetermined by the past generations savings, and that each generations savings rate respondsstrongly (and counter-cyclically) to the capital-labor ratio. Then, the discrete time model ofthis exercise suggests that the capital-labor ratio in this economy may cycle over dierentgenerations, while the continuous time model cannot capture this behavior. However, thisinterpretation is somewhat of a stretch. In reality generations are not discretely overlappingas in this interpretation. Hence the capital-labor ratio would move more smoothly, which isbetter modeled in continuous time. Moreover, the assumption that the saving rate is stronglycounter-cyclical, which is necessary to generate the cycles, is not in line with empirical evi-dence that suggests that investment is pro-cyclical over the business cycle (see, for example,Stock and Watson (1999)).Exercise 2.21, Part (e). The cycles in this problem are better viewed as pathologicalcases that are artifacts of the discrete time modeling, hence we probably should not takethese cycles too seriously. Business cycles are very important real life phenomena, but thediscrete time cycles of this problem are far from satisfactory in explaining business cycles.Exercise 2.21, Part (f ). Let q (/ (t)) = (:) (/ (t)) (1 c) / (t)) , (1 :) and recallthat the capital accumulation equation is given by / (t 1) = q (/ (t)). When : is constantand ) is nondecreasing, q (/) is also nondecreasing. Suppose, to reach a contradiction, thatthere is a cycle, i.e. suppose that there exists /1 < /2 such thatq (/1) = /2 and q (/2) = /1.Since q (/) is nondecreasing, we have/2 = q (/1) _ q (/2) = /1,which contradicts /1 < /2, proving that there are no cycles in the baseline Solow model. Toget the pathological cycles in discrete time, we need to endogenize the saving rate such that: (/) is decreasing (in some range) over /.4For example, consider the dierential equation `I = _I. This has the solution I (t) = 0 but also thesolution I (t) = t24 .5Recall that ) is Lipschitz continuous at I if there exists a neighborhood 1 of I and a constant 10such that for all I1, I2 1, [) (I1) ) (I2)[ 1[I1 I2[.20 Solutions Manual for Introduction to Modern Economic GrowthExercise 2.22We consider the continuous time version of the Solow model. Output per capita is givenby ) (/) = 1/ 1, and the capital-labor ratio accumulates according to`/ = :) (/) c/= (:1 c) / :1. (I2.32)First consider the degenerate case :1 = c. Eq.(I2.82) implies that / (t) grows and limits toinnity.Note also that limto `/ (t) ,/ (t) = limto:1,/ (t) = 0, that is, the asymptoticgrowth rate of / (t) is equal to 0. Next suppose :1 ,= c. Given any / (0), the lineardierential equation in (I2.82) is solved as (cf. Section B.4)/ (t) =:1c :1 _/ (0) :1c :1_oxp((:1 c) t) . (I2.33)There are two cases to consider. If :1 < c, then the second term in Eq. (I2.88) limits to 0and we have limto/ (t) = /+ =c1cc1. That is, starting with any / (0)0, / (t) convergesto the globally stable steady state /+. In this case, even though Assumption 2 does not hold,the capital-labor ratio still converges to a constant. In the second case, we have :1c andthe capital-labor ratio in the limit grows at rate :1 c0. More formally, Eq. (I2.88)implieslimto/ (t)oxp((:1 c) t) = / (0) :1:1 c0.Hence, with suciently large 1, the Solow/AK model generates sustained growth withouttechnological progress.Exercise 2.23Exercise 2.23, Part (a). We consider the Solow model in continuous time and notethat output per capita is given by the CES production function (rst introduced by Arrow,Chenery, Minhas, Solow (1961))) (/) = 1_ (1/)1 (1 ) (1)1_ 1. (I2.34)The capital-labor ratio accumulates according to`/ (t) = :) (/ (t)) (c :) / (t) (I2.35)= :1_ (1/ (t))1 (1 ) (1)1_ 1(c :) / (t) .Since o1, we have that ) (/) ,/ = 1_ (1)1 (1 ) (1)1/1_ 1is decreas-ing in / with limitslimI0) (/) ,/ = and limIo) (/) ,/ = 111.Then there are two cases to consider.First, if the following condition holds,111 < c ::, (I2.36)then there is a unique /+0 that solves ) (/+) ,/+ = (c :) ,:, which is the unique steadystate capital-labor ratio in the economy.Moreover, From Eq.(I2.8), when / (t)/+, weSolutions Manual for Introduction to Modern Economic Growth 21have `/ (t) < 0 and when / (t) < /+, we have `/ (t)0, which implies that the steady stateis globally stable. Hence, this case is very similar to the baseline analysis and the economyconverges to the unique steady state starting from any initial capital-labor ratio.Second, if Condition (I2.86) fails, that is, if 111 _ (c :) ,:, then Eq. (I2.8)implies that `/ (t)0 for any / (t)0, hence limto/ (t) = starting from any initialcondition. Moreover, we havelimIo) (/)11o(o1)/ = 1. (I2.37)Then, as t , the system in Eq. (I2.8) approximates`/ (t) =_:11o(o1)c :_/ (t), and the asymptotic growth rate of / (t) is qI=:1o(o1) c :. ByEq. (I2.87), the asymptotic growthrate of output andconsumption is also qI.Hence, if the productivity and the saving rate are suciently high, the production functionin the limit resembles the AK production function in Exercise 2.22, the economy behavessimilarly and features sustained growth. Intuitively, when o1, part of Assumption 2 failsand the marginal product of capital remains positive if there is an abundance of capital.Consequently, when the productivity is suciently high, sustained growth is possible justlike in the AK economy.Exercise 2.23, Part (b). Before we start the present exercise, for completeness we alsocharacterize the equilibrium with the CES production function when o _ 1. When o = 1,the production function is Cobb-Douglas and satises Assumptions 1 and 2 in the text, hencethe analysis in the text applies without change, proving that there is a unique steady stateequilibrium with positive capital-labor ratio.Next consider the same CES production function (I2.84) in Part (a) with o < 1. We have) (/) ,/ is decreasing in / with limitslimI0) (/) ,/ = 111, and limIo) (/) ,/ = 0.There are two cases to consider.First, if the opposite of Condition (I2.86) holds, that is, if 111(c :) ,:, thenthere is a unique /+0 that solves ) (/+) ,/+ = (c :) ,:, which is the unique steady statecapital-labor ratio in the economy. Moreover, from Eq. (I2.8), when / (t)/+, we have`/ (t) < 0 and when / (t) < /+, we have `/ (t)0, which implies that the steady state isglobally stable. This case is very similar to the baseline analysis and the economy convergesto the unique steady state starting from any initial capital-labor ratio.Second, if Condition (I2.86) holds as a weak inequality, that is, if 111 _ (c :) ,:,then Eq. (I2.8) implies that `/ (t) < 0 for all / (t)0 and there is a unique, globally stablesteady state at /+ = 0. In this case, the productivity in the economy and the saving rateis suciently low that, even for very low levels of capital-labor ratio, new investment is notsucient to cover the eective depreciation of the capital and the capital-labor ratio limitsto 0 in the long run.We next turn to the present exercise with the Leontief production function, ) (/) =min1/; 1, which is the limit of the CES production function (I2.84) as o 0.6In this6There is a typo in Chapter 2 and the exercise statement. As o 0, the correct limit of the CESproduction function in Eq. (I2.84) is this expression.22 Solutions Manual for Introduction to Modern Economic Growthcase, the capital-labor ratio accumulates according to`/ (t) = :1 min1/ (t) ; 1 (c :) / (t) . (I2.38)There are three cases to consider.First, since this case is the limit of the case analyzed in Part (b), we conjecture that whenthe analogue of the opposite of Condition (I2.86) as o 0 holds, i.e. when11(c :) ,:, (I2.39)there is a steady state with positive capital-labor ratio. In this case, we have1/+ _ 1(I2.40)(veried below) hence from Eq. (I2.88), the steady state capital-labor ratio can be solved as/+ = :11c :.Plugging in the expression for /+, we verify that Eq.(I2.40) holds since Eq.(I2.80) holds,proving that there is a steady state with positive capital-labor ratio. From Eq. (I2.88), it canalso be seen that, starting from any / (0), the economy converges to the capital-labor ratio/+.Note that, at this steady state, Eq.(I2.40) holds with strict inequality.Hence there isidle capital and the price of capital at the steady state is zero, that is 1+ = 0. The price oflabor at steady state is given by n+ = 11.Second, we claim that when the opposite of Condition (I2.80) hold, that is, if 11