Chakraborty [Journal of Economic Theory, 2004] introduces endogenous mortality in a two period overlapping generations model by postulating that the probability of surviving from the first period to the second depends on tax−funded public health. His central result on the existence of multiple steady states (including development traps) summarized in Proposition 1 is incorrect. This paper presents the correct proposition and its proof, and in the process, uncovers several new, interesting results. Contrary to Chakraborty\u27s analysis, high mortality yet high capital nations may not be able to escape the poverty trap. Interestingly, TFP growth can help economies escape the vicious cycle of poverty

Bunzel, Helle

Qiao, Xue

English

Chakraborty [Journal of Economic Theory, 2004] introduces endogenous mortality in a two period overlapping generations model by postulating that the probability of surviving from the first period to the second depends on tax−funded public health. His central result on the existence of multiple steady states (including development traps) summarized in Proposition 1 is incorrect. This paper presents the correct proposition and its proof, and in the process, uncovers several new, interesting results. Contrary to Chakraborty&apos;s analysis, high mortality yet high capital nations may not be able to escape the poverty trap. Interestingly, TFP growth can help economies escape the vicious cycle of poverty

Helle Bunzel

Xue Qiao

CiteSeerX

Endogenous Lifetime and Economic Growth Revisited

Chakraborty [Journal of Economic Theory, 2004] introduces endogenous mortality in a two period overlapping generations model by postulating that the probability of surviving from the first period to the second depends on tax−funded public health. His central result on the existence of multiple steady states (including development traps) summarized in Proposition 1 is incorrect. This paper presents the correct proposition and its proof, and in the process, uncovers several new, interesting results. Contrary to Chakraborty's analysis, high mortality yet high capital nations may not be able to escape the poverty trap. Interestingly, TFP growth can help economies escape the vicious cycle of poverty.This is an article from Economics Bulletin 15 (2004): 1. Posted with permission.</p

Digital Repository @ Iowa State University (ISU)

Endogenous lifetime and economic growth revisited Helle Bunzel Xue QiaoIowa State University Iowa State UniversityAbstractChakraborty [Journal of Economic Theory, 2004] introduces endogenous mortality in a twoperiod overlapping generations model by postulating that the probability of surviving fromthe first period to the second depends on tax−funded public health. His central result on theexistence of multiple steady states (including development traps) summarized in Proposition1 is incorrect. This paper presents the correct proposition and its proof, and in the process,uncovers several new, interesting results. Contrary to Chakraborty's analysis, high mortalityyet high capital nations may not be able to escape the poverty trap. Interestingly, TFP growthcan help economies escape the vicious cycle of poverty.Citation: Bunzel, Helle and Xue Qiao, (2005) "Endogenous lifetime and economic growth revisited." Economics Bulletin, Vol.15, No. 8 pp. 1−8Submitted: October 10, 2004.  Accepted: February 7, 2005.URL: http://www.economicsbulletin.com/2005/volume15/EB−04O10012A.pdf1 IntroductionA question that continues to intrigue macroeconomists and policymakers alike is this: Why isAfrica so different from the rest of the world?1 Indeed, in a recent speech, Alan Greenspan2touches on this question: “While, from a global perspective, wealth and the overall quality oflife have risen, that success has not been evenly distributed across regions or countries. Theeconomies of East Asia are often-repeated success stories. Some, including China, Malaysia,South Korea, and Thailand, stand out not only as growing very strongly, but also as having seenthe greatest declines in poverty rates. ....But, sadly, the story in Africa has been quite different.Levels of per capita income in that continent have actually fallen. The poverty rate, which in1970 matched the rate in Asia at the time, is estimated to have doubled to 40 percent by 1998.”In fact, as Haber, North, and Weingast (2003) point out, two-thirds of African countries haveeither stagnated or shrunk in real per capita terms since the onset of independence in the early1960s.Easterly and Levine [1997] were among the first to pose this question; to them, the highethnic fractionalization in the “dark continent” largely explains Africa’s woes. In a fairly influ-ential recent paper “Endogenous Lifetime and Economic Growth”, Chakraborty [2004] indirectlyrevisits this question by exploring a new connection between pervasive ill-health and economicgrowth. As he points out, the probability that a average 15-year old would die before reachingage 60, was three times as high in sub-Saharan Africa as in the richer OECD economies. Hegoes on to suggest that when life expectancy is low, agents would place little emphasis on thefuture, and hence, would invest little in productive long-term assets, thereby getting stuck in alow level of real activity. In turn, poor health is largely explained by low public health spendingwhich in turn is a direct outcome of the low level of real activity. In short, Africa is caught in adevelopment trap induced by poor health.More specifically, Chakraborty introduces endogenous mortality in an otherwise standardoverlapping generations model with production of the classic Diamond [1965] variety. In par-ticular, the probability with which young agents survive on to the second period depends onpublic health expenditures which are in turn funded by income taxes on labor income. The mainresult in Chakraborty [2004] contained in his Proposition 1 states that when the output elasticityof capital is high a development trap appears and countries differing in health and/or physicalcapital may not converge to similar living standards. It is important to note that such povertytraps do not arise in the standard Diamond model without the aid of several strong assumptionson preferences and technology that Chakraborty does not make.3In this paper, we point out a crucial error in his statement of Proposition 1. We go on tocorrect the omission and in the process we uncover several new, interesting results. Contrary tohis analysis, it turns out that high mortality nations even if they have high levels of capital maynot be able to escape the poverty trap. This implies that the vicious cycle of poverty is far morepersistent than what his analysis suggests. In addition, we show that the level of technological1In the parlance of modern growth theory, this is often summarized as the puzzle of the persistent negative“Africa dummy” in cross-country growth studies.2Remarks by Federal Reserve Chairman Alan Greenspan at Banco de Mexico’s Second International Conference”Macroeconomic Stability, Financial Markets, and Economic Development,” Mexico City, Mexico, November 12,2002.3See the discussion in Azariadis (2004).2development plays a crucial role in determining the persistence of the development trap. First,when technological development increases, a new long run equilibrium with higher real activityand reduced mortality appears. Second, this increase in technical efficiency drastically reducesthe level of capital required to escape the development trap. Our results therefore suggest thathigh mortality and low capital nations can escape the low activity trap by raising their TFP.2 The ModelWe use the exact model outlined in Chakraborty [2004]. Here, young agents are born eachperiod and inelastically supply one unit of labor, earning a wage w. The probability of a youngagent surviving to the next period is given by the non-decreasing concave function φt ≡ φ (ht) ,where ht denotes her health capital. We assume that φ (0) = 0, limh→∞ φ (h) = β ≤ 1 andlimh→0 φ (h) = γ <∞. Public health expenditure in period t is financed through a proportionaltax τ t ∈ (0, 1) such that ht = τ twt. At the end of each period, the young agents deposit theirsavings in a mutual fund, which earns a gross return on its investments of Rt+1, thus guaranteeinga gross return of R̂t+1 = Rt+1/φt for the surviving old. The young agents born in period t + 1are not affected by the health capital of the previous generation.A person born in period t maximizes her expected lifetime utilityUt = ln ctt + φt ln ctt+1,subject to the budget constraintsctt ≤ (1− τ t)wt − zt, ctt+1 ≤ R̂t+1zt.Optimal savings takes the form zt = (1− τ t)σtwt, where σt ≡ φt1+φt .Final goods are produced using a constant returns to scale Cobb-Douglas technology F (K,L) =AKαL1−α, where A > 0 and α ∈ (0, 1) . Perfect competition ensures that wt = (1− α)Akαt andRt = 1+αAkα−1t − δ, where k is the capital-labor ratio and δ is the depreciation rate of physicalcapital.In the model set up above, using zt = kt+1, it follows that the general equilibrium law ofmotion for the capital-labor ratio is given bykt+1 = (1− τ)(1− α)σ (kt)Akαt , (1)given k0 > 0 and h = τw (k) = τA (1− α) kα. We are now ready to re-state Proposition 1 ofChakraborty [2004].3 ResultsBefore we proceed to make corrections, we restate his Proposition 1. (i) below.Chakraborty’s Proposition 1. (i). The dynamic system described by (1) [his equation(10)] possesses two steady states {0, k̄} when α < 1/2, only the positive one being asymptoticallystable. When α > 1/2, three steady states exist©0, k̄1, k̄2ªwith k̄2 > k̄1; the two extreme steadystates are asymptotically stable, the intermediate one is not.3By means of a simple counterexample, it is easy to demonstrate that the second part of thisresult is incorrectly stated. Suppose A = 5, α = 0.55, β = 0.5, τ = 0.2, and φ(h) = βh/(1 + h),a functional form for φ(h) that satisfies all of Chakraborty’s assumptions on φ (as stated by himin his footnote 4). Then, it is easy to check that there exists a unique steady state, k = 0. 1 2 3 4 5k 1 2 3 4 5  J(k) 450 Figure 1: J(k) versus kAs Figure 1 illustrates, the error appears because Chakraborty ignores the possibility that(1) may forever lie below the 45o line thereby producing only the trivial steady state, k = 0. Tosee this, define J(k) to be the right hand side of (1). Then, as proven in Lemma A.1 (AppendixA), J(0) = 0, J 0(k) ≥ 0 ∀k ≥ 0, limk→∞J(k)/k < 1 and limk→0J 0(k) = 0 if α > 1/2. In other words,the J locus starts at 0, is non-decreasing, and eventually falls below the 45o line; additionally,the 0 steady state is locally stable. From this Chakraborty erroneously concludes that the Jlocus “intersects the 45o line from below at least once before falling below it.” Indeed as wedemonstrate below, α > 1/2 is simply necessary but not sufficient for three steady states toexist.Since the first part of the proposition (the case of α < 1/2) is correct, henceforth we willfocus our discussion only on the second part (the case of α > 1/2).The correct statement should read:Proposition 1. (i). Suppose α > 1/2 and suppose k∗ = argmax(J (k) − k) satisfiesJ(k∗) > k∗. Then at least three steady states exist. Naming the three smallest©0, k̄1, k̄2ªwithk̄2 > k̄1; the two extreme steady states are asymptotically stable, the intermediate one is not.4If J(k∗) = k∗, there are exactly two steady states©0, k̄1ª, where k̄1 is neither a repellor nor anattractor.It turns out that even with the specific assumptions that Chakraborty makes, it is not possibleto write down necessary and sufficient parametric conditions under which J(k∗) > k∗. Howeverwe can provide a sufficient condition under which there exists a k̂ such that J(k̂) > k̂, implyingthe presence of at least three steady states. This sufficient condition is provided in the corollarybelow, and the proof is in the appendix.4Note that Chakraborty [2004] ignores the possibility that there might be more than 3 steady states. Withoutspecifying the φ function precisely, it is not possible to determine the maximum number of steady states.4Corollary to Proposition 1. (i). Suppose α > 1/2 andA >1(1− α)µ(1− τ) φ (τ)1 + φ (τ)¶−α. (2)Then at least three steady states exist. Naming the three smallest©0, k̄1, k̄2ªwith k̄2 > k̄1; thetwo extreme steady states are asymptotically stable, the intermediate one is not.This corollary provides a condition which is easy to check for a given φ function and set ofparameters. For example, if φ = β h1+hand for the parameter values provided above, namelyα = 0.55, β = 0.5 and τ = 0.2, condition (2) becomes A > 10.3. Setting A = 15, this exampleprovides three steady states 0, k̄1 = 2.456× 10−6, and k̄2 = 1.90. Note that condition (2) alwaysrestricts A non-trivially, since 1(1−α)³(1− τ) φ(τ)1+φ(τ)´−αis always strictly greater than 0.While it is only possible to provide a sufficient condition under which there are at least threesteady states in the general case, it is possible to provide a complete characterization of conditionsunder which (1) would admit at least three steady states for the special functional form for φ(.)mentioned in Chakraborty [2004] footnote 4. This characterization is provided in the corollarybelow. The proof of the corollary is relegated to the appendix.Corollary to Proposition 1. (ii). Suppose φ(h) = βh/(1 + h). Also suppose α > 1/2 and(A(1− α)) 12α−1 ((1− α)α(1− τ)βτ) α2α−1µ2α− 11− α¶> τ (1 + β) (3)holds. Then at least three steady states exist. Naming the three smallest©0, k̄1, k̄2ªwith k̄2 >k̄1; the two extreme steady states are asymptotically stable, the intermediate one is not.Notice that while (3) reveals that α > 1/2 is necessary for the result to hold, since the lefthand side of (3) is negative if α < 1/2, clearly α > 1/2 is not sufficient; for example, a sufficientlyhigh value for A is needed.In light of the amended proposition, some changes to the discussion on page 124 in Chakraborty[2004] are in order. While it is true that a poverty and ill-health trap exists for α > 1/2, it isno longer given that it is possible to escape this trap if a country starts out with a high enoughcapital stock. In fact, as equation (3) reveals, unless there is a sufficiently high level of techno-logical development (high enough A), the zero capital poverty trap is the only steady state, andit is stable.Chakraborty limits his discussion of the implications of increasing A to the case where α <1/2. Our analysis reveals significant new benefits of doing so even when α > 1/2 obtains. Forgeneral φ, it is easy to verify that ∂J (k) /∂A > 0. The implication, as depicted in Figure 2, isstrong. Consider two countries identical in all respects except that countryM has a higher Athan country N . In such a setting, it is possible that country N is forever caught in the povertytrap (the only equilibrium) while countryM seizes the potential to approach a high long run levelof real activity (an equilibrium unavailable to countryN ). This situation corresponds to countriesN and M having A = A0 and A = A1 respectively in Figure 2. Perhaps more interestingly,when there are three steady states, namely©0, k̄1, k̄2ª, increasing A decreases the intermediatesteady state, k̄1, and this lowers the initial capital stock required to escape the twin traps ofpoverty and ill-health! This last benefit is in addition to the increase in k̄2, a point discussed by5Chakraborty. These last two effects are illustrated in Figure 2 as a country increases A from A1to A2. A = A0 A0 < A1 < A2 A = A2 A = A1 k J(k) Figure 2: J(k) versus k for α > 12and different values of A.6A AppendixA.1 Proof of Corollary to Proposition 1. (i).Choose k̂ =³1(1−α)A´ 1αand note that w(k̂) = 1. We wish to prove that J(k̂) > k̂. Using thedefinitions of J and k̂, we obtainJ³k̂´= (1− τ)σ³τw³k̂´´w(k̂) = (1− τ)σ (τ) = (1− τ) φ (τ)1 + φ (τ).For J³k̂´> k̂ we need(1− τ) φ (τ)1 + φ (τ)>µ1(1− α)A¶ 1α⇔A >1(1− α)µ(1− τ) φ (τ)1 + φ (τ)¶−α,which is exactly condition (2). ¥A.2 Proof of Corollary to Proposition 1. (ii).Proof: Rewrite J(k) as J(k) = (1 − τ)σ (k)w(k). Then steady states are fixed points to theequation J(k) = k. Note that 0 is a fixed point. It is easy to check that given φ(h) = βh/(1+h),σ (k) =βτw(k)1 + τw(k) (1 + β).Then J(k) = k simplifies to(1− τ)βτw(k) = k·1w(k)+ τ (1 + β)¸and finally to H(k) = c whereH(k) ≡ (1− τ)βτA(1− α)kα−1 − 1A(1− α)k−αand c ≡ τ (1 + β) . Straightforward algebra establishes that H 0(k̆) = 0 wherek̆ =·αA2β(1− α)3(1− τ)τ¸ 12α−1.Also, limk→0H(k) = −∞ and limk→∞H(k) = 0. It remains to identify conditions under which H(k̆)> c. Again straightforward but tedious algebra establishes that H(k̆) > c⇔(1− τ)βτA(1− α)·αA2β(1− α)3(1− τ)τ¸ α−12α−1− 1A(1− α)·αA2β(1− α)3(1− τ)τ¸ −α2α−1> τ (1 + β)7which simplifies toA12α−1 ((1− τ)βτ) α2α−1h(1− α) 2−α2α−1α α−12α−1 − (1− α) α+12α−1α −α2α−1i> τ (1 + β)and finally toµ1− αα¶ α2α−1(1− α) 12α−1A 12α−1 ((1− τ)βτ) α2α−1µ2α− 11− α¶> τ (1 + β)¥References[1] Chakraborty, S. (2004) “Endogenous Lifetime and Economic Growth,” Journal of EconomicTheory, 116, 119-137[2] Azariadis, C (2004) “The Theory of Poverty Traps: What Have We Learned” Forthcomingin Poverty Traps, Russell Sage Foundation.[3] Diamond, P.A. (1965), “National Debt in a Neoclassical Growth Model”, American EconomicReview, 55 (5), 1126-1150.[4] Easterly, W., and R. Levine (1997), “Africa’s Growth Tragedy: Policies and Ethnic Divi-sions”, Quarterly Journal of Economics, 112 (4), 1203-50[5] Stephen Haber, Douglass C. North, and Barry R. Weingast (2003) “If Economists Are SoSmart, Why Is Africa So Poor?”, Wall Street Journal, July 30.8

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Endogenous Lifetime and Economic Growth Revisited

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