Christian Groth, University of Copenhagen 16.0 16.02. 2.20 2012 12
Econ Econom omic ic Gro Growth wth
Errata to Acemoglu’s textbook The list below refers to what looks like typos or logical errors (as far as I can see) in our textbook, Acemoglu’s Introduction to Modern Economic Growth , Princeton 2009. Symbol glossary: “l.” means “line”; “f.b.” means “from below”; “eq.” means “equation”; “n” means footnote. In the third column, in square brackets, occasionally appears a comment. page 17, l. 2 f.b. 17, l. 2 f.b. 18, 18, titl titlee of of figure 29, l. 23 23-24 -24
reads
should read (or my comment)
as economies below grow toward and and av average erage growt growth h of of inv invest estme men nts to GDP ratio dimin diminish ishing ing return returnss to capita capitall disti distinnguis guishe hess the the Solo Solow w gro growth wth mo mode dell from from its its ante antece cede den nt, the the Ha Harr rrododDomar model
as economies below or above adjust toward and averag veragee inv invest estme ment nt to GDP ratio [this is misleading. In the Harrod-Domar model odel the the produ product ctio ion n func functi tion on is Le Leon onti ti-ef (iso (iso-q -qua uan nts are are of L form form,, ther theree is no substitutability between capital and labor). It is in the opposite case − the case of perfect perfect substitutability − that diminishing returns to capital is absent.] absent. ] 33, 33, l. 3 f.b. f.b. Moreo oreov ver, er, (0 (0 )) = 0 for all all [This is implied, is implied, and and thus not an additioand nal assumptio assumption, n, as soon as the other other part part of Assumption 2 is assumed together with Assumption1.] Assumption1.] 53, 53, l. 12 f.b f.b.. In add addit itio ion, n, is increasing in in [This is true only if (( ( + + )) )) 1] 1] 54, l. 1 f.b. min { () ();(1 − ) ()()} min { () (); ()()} +1 56, figure ( − − − )() () [since ( + 1) is 1) is on the ver1+ tical axis, the figure must be about the discrete time case] case] 57, l. 2 f.b. f.b. This This estim estimate ate ignore ignoress the the share share of [The estimate 1/3 represents the income land; share of physical capital and land; calling it “income share of capital” is a kind of shortening of “income share of nonhuman wealth”] wealth”] 78, 78, l. 19 f.b. f.b. as “tot “total al fact factor or prod produc ucti tiv vity” ity” as “gro “growt wth h in tota totall fact factor or prod produc ucti tiv vity” ity” 79, l. 4 +1 +1 81, l. 12 around log () around log around log 100, l. 3 TFP diff erence diff erence erence in TFP growth 303, 303, l. 1 f.b. then then cons consum umpti ption on woul would d reac reach h zero zero in then then [delete the remainder since it is wrong] wrong ] finite time, and thus ∗
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page 313, l. 12
reads () = (1 − )( (() − )
331, l. 16 336, l. 16
Total savings in the economy with log preferences
0
362, l. 13Integrating both sides of this equa11 f.b. tion ...
should read (or comment) [To avoid confusion, note that this () is the after-tax interest rate, which in the standard notation in this course would be written (1 − )()] Total savings by the young with CRRA preferences [that is, the theorem holds not only for log preferences] [This is wrong since, generally, (10.10) is valid only at = not in an interval around ; see Lecture Note 6] is increasing if (0) neoclassical model with = 0 as well. [wrong; negative gross investment in human capital is impossible; (11.25) is conditional on an interior solution] + (1 − )( ( ) − ) + [Delete] ∗
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366, l. 8 f.b. 387, l. 10 394, l. 6-9
394, l. 23 401, l. 13-14 402, l. 9 f.b. 434, l. 14 436, l. 10 f.b. 437, l. 15 439, l. 6 439, l. 165 f.b. 440, l. 5 f.b.
is always increasing neoclassical model as well. there are no constraints on human and physical capital investments .... and physical capital. + (1 − )( ( ) − ) + (with output reaching infinity in finite time ...transversality condition). will not be possible. 0
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though the results are identical without this assumption; Let us normalize the marginal costs recall that one unit and the transversality condition is satisfied.
Maximization gives
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0
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will not be possible without technical progress. [Delete] Let us specify the marginal costs recall that on average one unit and provides scope for satisfaction of the transversality condition. [See Comment to §13.1.4 below.] In the special case = 1 − maximization gives
Continued next page.
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page
reads
should read (or comment)
441, l. 13 f.b.
Hamiltonian is strictly concave and satisfies the conditions of Theorem 7.14.
442, l. 1413 f.b.
the gap between the decentralized equilibrium and the Pareto optimal allocation,
444-445
[The truth is that the Hamiltonian is concave in ( () ()), which is enough to ensure that the FOCs together with the TVC are su ffi cient conditions for optimality; the Hamiltonian is not strictly concave; moreover, the maximized Hamiltonian is concave in () but not strictly concave; so the conditions of Theorem 7.14 are not satisfied. ] the gap between the decentralized equilibrium and the social planner’s optimal allocation, [it is not clear what the Pareto optimal allocation is − there are infinitely many Pareto optimal allocations as soon as there are more than one household] [(1326) and (13.27) presuppose = 1 − ] See Comment to §13.2 below. equilibrium (see Exercise 13.16). [Delete]
445, eq. (13.31) 445, l. 6 f.b. equilibrium (see Exercise 13.17). 446, l. 20 , and output per capita would reach infinity in finite time (“explode”) 447, eq. (13.38) = = 448, l. 12-13 It is not clear whether the data sup[See “Afterthoughts” as of May 19] ports these types of scale eff ects either. 448, l. 18-21 “semi-endogenous growth” models ... [Here “semi-endogenous growth” is not respond to taxes or other policies. defined diff erently than in the lectures and in Lecture Note 7 and 8; see “Afterthoughts” as of June 9.] ∗
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Comment to §13.1.4 (p. 439) It is true that in general equilibrium with positive R&D, there is no transitional dynamics. In my view, however, Acemoglu does not provide a valid proof. In connection with one of the exercise problems on this model we show that the relevant approach is analogue to that applied for the simple AK model (pp. 390-391). A related error in Acemoglu is the claim p. 439, l. 6, that the last inequality in (13.21) ensures that the transversality condition (TVC) is satis fied. The point is that (13.21) only opens up for the possibility that the TVC can be satisfied. What then ensures satisfaction of the TVC is that (0) is at a certain level determined by (0) This level in turn ensures balanced growth from the beginning, i.e., absence of transitional dynamics. 3
Comment to §13.2 (p. 445) One might wonder whether the formula (13.31) is valid for 0 (the general case) or only in the special case = 1 − ? The answer is that the formula, fortunately, is valid also in the general case 0 as are (13.29) and (13.30). This is because cancels out anyway in these formulas. But when stating (13.26) and (13.27), = 1 − is presupposed. This is another illustration that assuming = 1 − may easily lead to confusions. –-
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