Solutions Manual for Introduction to Modern Economic Growth INSTRUCTOR’S 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 the camera-ready copy from which this book was produced.
press.princeton.edu
For you, Edna. -M.P. -M.P.
To my parents and my brothers, -A.S.
Contents Intro duction
xi
Chapter 2: Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
The Solow Growth Mo del 2.7 2.11 2.12 2.14* 2.16* 2.17 2.18* 2.19* 2.20 2.21 2.22 2.23 2.27
1 1 3 6 7 9 10 13 15 15 17 20 20 23
Chapter 3: Exercise Exercise Exercise Exercise
The Solow Model and the Data 3.1 3.2 3.9 3.10
27 27 29 29 30
Chap Chapte terr 4: Funda undame men ntal Dete Determ rmin inan ants ts of Di¤e Di¤ere renc nces es in Econ Econom omic ic Perf Perfor orma manc ncee Exercise 4.3
31 31
Chapter 5: Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
Foundations of Neoclassical Growth 5.1 5.2 5.9 5.10 5.11 5.12 5.13 5.14*
33 33 35 36 38 40 42 47 48
Chapter 6: Exercise Exercise Exercise Exercise
In…nite ite-Horizo izon Optimi imization and Dynamic Programming 6.2* 6.3* 6.7 6.8
51 51 51 52 54 v
vi
Solutions Solutions Manual Manual for Introduc Introduction tion to Modern Modern Economic Economic Growth Growth
Exercise 6.9 Exercise 6.12 Exercise 6.18*
58 59 61
Chapter 7: Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
An Intro duction to the Theory of Optimal Control 7.1 7.2* 7.5 7.19 7.10 7.17* 7.18 7.23 7.21 7.26 7.24* 7.25 7.28
63 63 65 65 66 69 70 71 73 73 75 76 79 80
Chapter 8: Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
The Neo classical Growth Model 8.2 8.7 8.11 8.13 8.15 8.19 8.23 8.25 8.27 8.30 8.31 8.33 8.34 8.37 8.38*
87 87 88 90 92 93 96 97 10 0 10 1 10 5 10 8 11 3 11 7 11 9 121
Chapter 9: Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
Growth with Overlapping Generations 9.1 9.3 9.6 9.7 9.8 9.15 9.16 9.17 9.20 9.21 9.24* 9.32*
129 129 130 132 134 137 13 8 14 1 14 6 14 7 14 9 150 151
Solutions Solutions Manual Manual for Introduc Introduction tion to Modern Modern Economic Economic Growth Growth
vii
Chapter 10: Human Capital and Economic Growth Exercise 10.2 Exercise 10.6 Exercise 10.7 Exercise 10.14* Exercise 10.18 Exercise 10.20
15 5 15 5 15 7 16 1 16 5 167 169
Chapter 11: First-Generation Mo dels of Endogenous Growth Exercise 11.4 Exercise 11.8 Exercise 11.14 Exercise 11.15 Exercise 11.16 Exercise 11.17 Exercise 11.18 Exercise 11.21*
171 17 1 17 7 178 182 182 183 184 18 5
Chapter 12: Mo deling Technological Change Exercise 12.2 Exercise 12.5 Exercise 12.9 Exercise 12.11 Exercise 12.13 Exercise 12.14
191 19 1 19 2 19 3 196 197 201
Chapter 13: Expanding Variety Models Exercise 13.1 Exercise 13.5 Exercise 13.6 Exercise 13.7 Exercise 13.13* Exercise 13.15 Exercise 13.19 Exercise 13.22 Exercise 13.24
205 20 5 20 7 21 0 21 1 21 3 216 219 227 231
Chapter 14: Mo dels of Schumpeterian Growth Exercise 14.2 Exercise 14.6 Exercise 14.7* Exercise 14.12* Exercise 14.13 Exercise 14.14 Exercise 14.15 Exercise 14.18 Exercise 14.19* Exercise 14.20* Exercise 14.21* Exercise 14.22*
23 7 23 7 23 8 239 24 2 246 250 253 262 26 5 27 0 27 1 27 4
viii
Solutions Solutions Manual Manual for Introduc Introduction tion to Modern Modern Economic Economic Growth Growth
Exercise 14.26 Exercise 14.27* Exercise 14.35
275 28 4 287
Chapter 15: Directed Technological Change Exercise 15.6 Exercise 15.11 Exercise 15.18 Exercise 15.19 Exercise 15.20 Exercise 15.24* Exercise 15.27 Exercise 15.28* Exercise 15.29 Exercise 15.31*
29 3 29 3 297 301 312 316 31 7 319 32 3 327 32 9
Chapter 16: Sto chastic Dynamic Programming Exercise 16.3* Exercise 16.4* Exercise 16.8 Exercise 16.9 Exercise 16.10 Exercise 16.11* Exercise 16.12 Exercise 16.13 Exercise 16.14 Exercise 16.15 Exercise 16.16
331 331 331 33 2 33 3 333 33 4 341 342 344 346 347
Chapter 17: Sto chastic Growth Models Exercise 17.5 Exercise 17.7 Exercise 17.13 Exercise 17.15 Exercise 17.18 Exercise 17.22 Exercise 17.30*
35 1 35 1 35 2 356 360 361 364 36 8
Chapter 18: Di¤usion of Technology Exercise 18.8 Exercise 18.9 Exercise 18.12 Exercise 18.13* Exercise 18.16* Exercise 18.18 Exercise 18.19 Exercise 18.21 Exercise 18.26*
37 3 37 3 37 8 381 38 2 38 5 387 390 391 39 4
Chapter 19: Trade and Growth
401
Solutions Solutions Manual Manual for Introduc Introduction tion to Modern Modern Economic Economic Growth Growth
Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise Exercise
19.2* 19.3 19.4* 19.7 19.11* 19.13* 19.24 19.25* 19.26* 19.27* 19.28 19.29 19.33 19.34 19.37
ix 401 40 4 406 40 8 40 8 41 1 415 41 6 41 7 42 2 427 429 432 435 437
Chapter 20: Structural Change and Economic Growth Exercise 20.3 Exercise 20.5 Exercise 20.6 Exercise 20.7* Exercise 20.8 Exercise 20.9* Exercise 20.16* Exercise 20.17 Exercise 20.18 Exercise 20.19*
441 44 1 44 2 44 4 444 44 6 449 45 0 455 461 46 3
Chap Chapte terr 21: 21: Stru Struct ctur ural al Trans ransfo form rmati ation onss and and Ma Mark rket et Failu ailure ress in Dev Develop elopme men nt Exercise 21.1 Exercise 21.2 Exercise 21.4 Exercise 21.6 Exercise 21.9 Exercise 21.10 Exercise 21.11 Exercise 21.12
467 46 7 46 9 47 2 47 7 48 1 482 484 488
Chapter 22: Institutions, Political Economy and Growth Exercise 22.2 Exercise 22.3 Exercise 22.8 Exercise 22.9 Exercise 22.16 Exercise 22.17 Exercise 22.18* Exercise 22.19* Exercise 22.20* Exercise 22.21*
495 49 5 49 8 50 0 50 2 503 508 51 1 51 2 51 2 51 5
x
Solutions Solutions Manual Manual for Introduc Introduction tion to Modern Modern Economic Economic Growth Growth
Exercise Exercise Exercise Exercise Exercise
22.22* 22.25* 22.26* 22.27 22.30
51 6 52 1 52 2 523 531
Chapter 23: Institutions, Political Economy and Growth Exercise 23.4 Exercise 23.5 Exercise 23.12
533 53 3 53 4 536
References
547
Introduction This manual contains solutions to selected exercises from Introduction to Modern Economic Growth by Daron Daron Acemoglu. Acemoglu. This volume volume is the Instructor Edition of the solutions manual, which contains a wider range of exercises than the Student Edition . The exer exercis cisee select selection ion for both editions editions is guided guided by a simila similarr set of princi principle ples. s. First, First, we have have tried tried to include the exercises that facilitate the understanding of the material covered in the book, for example, the ones that contain proofs to propositions or important extensions of the baseline models. Second, Second, we have have included included exercises which which we have have found relativel relatively y more useful for improving improving economic problem-solv problem-solving ing skills or building building economic intuition. intuition. Third, we made an e¤ort to include include exercises exercises which seemed particularly particularly challenging. challenging. Fourth, ourth, we also tried to strike a balance across the chapters. Even with these criteria, making the …nal selection has not been easy and we had to leave out many exercises which are no doubt important and interesti interesting. ng. We hope the readers will …nd our selection useful and we apologize up front for not 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 exercises are presented presented in the same chapters chapters they belong to in the book. Our solutions solutions regularly refer to equations in the book and also to equations de…ned within the manual. To avoid confusion between the two types of references, we use the pre…x “I” for the labels of the equations de…ned in the Instructor Edition of the solutions manual. manual. For example Eq. (5.1) would refer to the …rst labeled equation in Chapter 5 of the book, whereas Eq. (I5.1) would refer to the …rst labeled equation in Chapter 5 of this edition. Although this version of the manual went through various stages of proofreading, there are no doubt remaining errors. To partly make up for the errors, we will post an errata document on our personal personal websites websites which we will commit to updating updating regularly regularly.. In particular we would appreciate it if readers could e-mail us concerning errors, corrections or alternative solutions, which which we will includ includee in the next next update update of the errata errata document document.. Our presen presentt e-m e-mail ail and website addresses are as follows: Michael Peters,
[email protected], http://econ-www.mit.edu/grad/mipeters Alp Simsek,
[email protected], http://econ-www.mit.edu/grad/alpstein An errata document document and additional additional information information will also be posted on the companion companion site for Introduction Introduction to Modern Economic Growth at: Growth at: http://press.princeton.edu/titles/8764.html Acknowledgments. We would like to thank Daron Acemoglu for his help with the exercise selection and for useful useful suggestions suggestions on multiple multiple solutions. solutions. We would also like to thank Camilo Garcia Jimeno, Jimeno, Suman Basu and Gabriel Carroll for various contributions and suggestions, and to thank Samuel 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 economics classes at MIT and we have bene…ted from the solutions of numerous graduate students in these classes. xi
Chapter 2: The Solow Growth Model Exercise 2.7 Exercise 2.7, Part (a). Assuming C (t) = sY (t) is not very reasonable since it implies that consumption for a given level of aggregate income would be independent of government spending. Since government spending is …nanced by taxes, it is more reasonable to assume that higher government spending would reduce consumption to some extent. As an alternative, we may assume that consumers follow the rule of consuming a constant share of their after tax income, captured by the functional form C (t) = s (Y (t) G (t)). Using G (t) = Y (t), this functional form is also equivalent to C (t) = (s s) Y (t). In Part (b), we assume a more general consumption rule C (t) = (s ) Y (t) with the parameter [0; 1] controlling the response of consumption to increased taxes. The case = 0 corresponds to the extreme case of no response, = s corresponds to a constant after-tax savings rule, and [0; 1] correspond to other alternatives.
2
2
Exercise 2.7, Part (b). The aggregate capital stock in the economy accumulates according to
K (t + 1) = I (t) + (1 ) K (t) = Y (t) C (t) G (t) + (1 ) K (t) = (1 s (1 )) Y (t) + (1 ) K (t) ,
where the last line uses C (t) = (s ) Y (t) and G (t) = Y (t).
(I2.1)
Let f (k) Y (t) =L = F (K; 1; A) and assume, for simplicity, that there is no population growth. Then dividing Eq. (I2:1) by L, we have k (t + 1) = (1
s (1 )) f (k (t)) + (1 ) k (t) :
Given k (0), the preceding equation characterizes the whole equilibrium sequence for the capital-labor ratio k (t) 1 t=0 in this model, where we use the subscript to refer to the economy with parameter for government spending. We claim that with higher government spending and the same initial k (0), the e¤ective capital-labor ratio would be lower at all t > 0, that is
f
g
k (t) > k0 (t) for all t, where < 0 .
(I2.2)
To prove this claim by induction, note that it is true for t = 1, and suppose it is true for some t 1. Then, we have
k (t + 1) = (1 > (1 > 1
s (1 )) f (k (t)) + (1 ) k (t) s (1 )) f (k (t)) + (1 ) k (t) s 0 (1 ) f (k (t)) + (1 ) k (t) = k (t + 1) , 0
0
0
0
0
where the second line uses the induction hypothesis and the fact that f (k) is increasing in k, and the third line uses 0 > . This proves our claim in (I2:2) by induction. Intuitively, 1
2
Solutions Manual for Introduction to Modern Economic Growth
higher government spending reduces net income and savings in the economy and depresses the equilibrium capital-labor ratio in the Solow growth model. As in the baseline Solow model, the capital-labor ratio in this economy converges to a unique positive steady state level k characterized by f (k ) = k 1
s
(1
) .
(I2.3)
The unique solution k is decreasing in and increasing in since f (k) =k is a decreasing function of k. In the economy with higher government spending (higher ), the capital-labor ratio is lower at all times, and in particular, is also lower at the steady state. Also, the more individuals 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, the higher the steady state capital-labor ratio. Exercise 2.7, Part (c). In this case, Eq. (I2:3) changes to f (k ) = k 1
s
(1
) .
Since f (k) =k is decreasing in k, the steady state capital-labor ratio k is increasing in . With respect to , it can be seen that k is increasing in if > 1 and decreasing in if < 1 . In words, when the share of public investment in government spending (i.e. ) is su¢ciently high, in particular higher than the reduction of individuals’ savings in response to higher taxes, the steady state capital-labor ratio will increase as a result of increased government spending. This prediction is not too reasonable, since it obtains when the government has a relatively high propensity to save from the tax receipts (high ) and when the public consumption falls relatively more in response to taxes (high ), both of which are not too realistic assumptions. An alternative is to assume that public investment (such as infrastructure investment) will increase the productivity of the economy. Let us posit a production function F (K;L;G;A), which is increasing in public investment G, and assume, as an extreme case, that F has constant returns to scale in K; L and public investment G. With this assumption doubling all the capital (e.g. factories) and the labor force in the economy results in two times the output only if the government also doubles the amount of roads and other necessary public infrastructure. De…ne f (k;g) = F (k; 1;g;A) where g = G=L. Then, the steady state capital-labor ratio k and government spending per capita g are solved by the system of equations
f (k ; g ) = k 1 s (1 ) g = f (k ; g ) .
The second equation de…nes an implicit function g (k ) for government spending in terms of the capital-labor ratio, which can be plugged into the …rst equation from which k can be solved for. In this model, k is increasing in for some choice of parameters. Since some infrastructure is necessary for production, output per capita is 0 when public investment per capita is 0, which implies that k is increasing in in a neighborhood of = 0. Intuitively, when public infrastructure increases the productivity of the economy, increased government spending might increase the steady state capital-labor ratio.
