Technological innovations, economic renovation, and anticipation effects

Journal of Mathematical Economics 46 (2010) 1064–1078

Contents lists available at ScienceDirect

Journal of Mathematical Economics

journa l homepage: www.e lsev ier .com/ locate / jmateco

Technological innovations, economic renovation, and anticipationeffects

Natali Hritonenkoa,∗, Yuri Yatsenkob,c

a Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USAb College of Business and Economics, Houston Baptist University, Houston, TX 77074, USAc Center for Operations Research and Econometrics, Louvain-la-Neuve, B-1348, Belgium

a r t i c l e i n f o

Article history:Received 20 May 2009Received in revised form 2 August 2010Accepted 2 August 2010Available online 6 August 2010

JEL classification:C61E22O33

Keywords:Vintage capital modelsTechnological changeTechnological innovationsAnticipation echoesOptimal capital lifetime

a b s t r a c t

Optimal replacement of a firm’s capital is described in the framework of Solow-type vintagecapital models. The firm controls the investment into new capital and scrapping of obsoletecapital. The embodied technological change involves a continuous component and techno-logical innovations (breakthroughs, technology shocks). The provided analytic and numericinvestigation reveals the qualitative structure of optimal regimes. It demonstrates thatthe optimal investment is zero immediately before and after a technological breakthrough(direct anticipation effect) and contains a set of zero-investment boundary intervals (antici-pation echoes) before the breakthrough time. The optimal capital lifetime oscillates aroundan interior balanced growth trajectory before and switches to a new balanced trajectoryafter the breakthrough.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

In mathematical economics, technological change (TC) is modeled as increasing labour productivity (the output-augmentingTC) or increasing efficiency of converting resources into useful work (the resource-saving TC). Recent research demonstratesthat “the gradual, continuous and homogeneous technical progress normally assumed in economic models cannot explainkey aspects of economic growth” (Aures, 2005) and two fundamentally different modes of technical progress coexist:

• a gradual improvement (a “normal” mode) when technological improvements occur incrementally as a result of accumu-lated experience and learning,

• the radical improvement mode (technological breakthrough) when a radically new innovation is capable of displacing anolder general-purpose technology among competing technologies.

The modern economic theory considers technological breakthroughs as radical innovations caused by the substitutionof one general-purpose technology by another. Starting with de Solla Price (1984) and Bresnahan and Trajtenberg (1995),

∗ Corresponding author. Tel.: +1 936 261 1978; fax: +1 936 261 2088.E-mail addresses: [email protected] (N. Hritonenko), [email protected] (Y. Yatsenko).

0304-4068/$ – see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2010.08.002

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1065

such breakthroughs explain economy-wide structural changes. The radical innovations are necessary for continued long-term economic growth (Aures, 2005). The steam engine, gasoline engine, electric power, semiconductors are examples ofsuch general-purpose (enabling) technologies. Many economists interpret and analyze the recent IT revolution as a majorbreakthrough (Greenwood and Yorukoglu, 1997; Jovanovic and Lach, 1997; Martínez et al., 2010).

This paper analyzes the simultaneous impact of two above-described TC modes on rational management of technologicalreplacement at a firm’s level. We employ a simple vintage capital model (VCM) to explore two powerful managerial decisionsof a firm: investing into new, more efficient capital and scrapping the oldest, least efficient capital. Such VCMs with continuousTC have been systematically investigated by Boucekkine et al. (1997, 1998) and Hritonenko and Yatsenko (1996, 2005). Per-ceived forthcoming innovations will obviously affect the rational firm’s decision on replacing or upgrading the related capital.

The importance of TC shocks in the VCMs with endogenous scrapping was first raised by Greenwood and Yorukoglu (1997)to explain the US productivity slowdown in the mid-1970s. They considered unanticipated permanent increase in the TC ratein 1974 and concluded that it caused the productivity slowdown mainly because of the costs of learning and adoption of newtechnologies. There is considerable economic evidence of substantial short-run negative effects until new IT equipment iscompletely adopted (Martínez et al., 2010). Boucekkine et al. (1998) address this issue analyzing an unanticipated permanentincrease (shock) in the TC rate in a macroeconomic VCM with nonlinear utility. They numerically simulate the productivityslowdown and explain it by increasing the scrapping age of vintages and smoothing consumption after the shock time.

Pakko (2002) analyzes an optimal anticipated response to technology shocks in a general-equilibrium stochastic-growthmodel with neutral and investment-specific TC à la Greenwood calibrated on the US data over 1949–2000. He shows thatpermanent shocks in embodied TC rate lead to a large sharp decline in investment just before the shock as agents anticipatehigher returns in the future (a negative anticipation effect). The model predicts that perceived changes in the future TCgrowth trends provide incentives to firms to alter the mix of capital and labor that leads to lower investment, output andemployment immediately before and after the shock time (with some lag).

Feichtinger et al. (2006) analyze an optimal anticipated response to a technology shock in a firm-level VCM with a givenfixed lifetime of vintages. The model involves nonlinear adjustment costs of capital, output-dependent product prices, andthe possibility of investments into older vintages. They show the existence of negative anticipation effects in the investmentbefore the technology shock occurs (in the case when the firm has market power).

Our paper contributes to the literature by analyzing how the technology shocks affect the optimal dynamics of endoge-nous capital lifetime and investment (before and after the shock time). The economic and mathematical novelty lies in theinvestigation of the complete optimal dynamics of a VCM with endogenous scrapping under both continuous and discon-tinuous TC. To obtain meaningful results, we consider a VCM with partial equilibrium setup, Leontief technology, constantreturns to scale, a price-taking firm, no learning, no adoption cost or adjustment cost. The obtained nonlinear optimal controlproblem allows us to find its exact solutions in special cases. The interpretation of the constructed solutions reveals that theadjustment of optimal model dynamics to a TC shock essentially happens before the shock time in the form of direct andechoed anticipation effects.

Follow the mainstream of VCMs (Solow et al., 1966; Benhabib and Rustichini, 1993; Boucekkine et al., 1997; Boucekkineet al., 1998; Cooley et al., 1997; Hritonenko and Yatsenko, 1996; Yorukoglu, 1998; Jovanovic and Tse, 2010), our model isdeterministic and assumes a perfect foresight, which means that the whole evolution of future technology is already knownat the present time. This purely theoretical assumption allows obtaining qualitative conclusions about the rational responseof economy to technological advances. The novelty is that this future evolution may include some technology shocks whosearrival times are known. A forthcoming technological breakthrough on macroeconomic level is obviously a big event andcan be predicted with certain accuracy. The last macro technology shock was related to IT. Currently, applied researchersargue that the next macroeconomic productivity shock based on renewable energy is forthcoming (Becerra-Lopez andGolding, 2007; Schmidt and Marschinski, 2009). On a specific firm’s level, it is also practically feasible to concentrate ononly the earliest forthcoming productivity shock (like switching to a new enterprise planning software or robotic line).That is why we restrict ourselves in this paper with the case of one future technological breakthrough, despite the fact thatour mathematical technique can handle several shocks (as shown in Section 4.3). A possible extension of our deterministicmodel would be to suggest a stochastic occurrence of innovations, similar to the one used in models of technology adoption(Doraszelski, 2004).

Technological breakthroughs affect the capital efficiency and/or price and can be described by discontinuities in theirlevels and/or growth rates (Boucekkine et al., 1998; Pakko, 2002). Boucekkine et al. (1998) analyze both the technology levelshocks and shocks in TC rate and point out a crucial difference in generated optimal dynamics. Similar explorations in Opera-tions Research (Rogers and Hartman, 2005; Feichtinger et al., 2006) consider technological breakthroughs as discontinuitiesin the level variables. While the shocks in technology levels are obviously possible for specific firms and types of machines,they do not represent all cases observed in reality. Another theoretical reason for modeling breakthroughs as jumps in theTC rates is that the optimal capital lifetime depends on the TC rates rather than TC levels (the optimal lifetime is infinitewhen the TC rate is zero). This is a fundamental feature of all vintage models with endogenous scrapping of oldest vintages,including (Solow et al., 1966; Benhabib and Rustichini, 1993; Boucekkine et al., 1997, 1998; Boucekkine and Pommeret,2004; Cooley et al., 1997; Hritonenko and Yatsenko, 1996, 2005; Jovanovic and Tse, 2010).

In this paper, we consider discontinuities in the TC growth rates, which are confirmed empirically by economic dataand commonly used in the macroeconomic VCM literature. Greenwood and Yorukoglu (1997) clearly identify such macroe-conomic shock that happened in 1974 and relate it to IT. Variations in aggregated and industry-specific TC rates have

1066 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

been exposed by subsequent research. Pakko (2002) provides the TC rates of 2.2% for 1950–1973 and 3.2% for 1974–2000.Jorgenson et al. (2007) analyze the productivity growth of 85 industries in the US over 1960–2005 and point out impor-tant differences between 1995–2000 and 2000–2005. They argue that “aggregate data conceal striking variations amongindustries and prevent analysts from tracing the evolution of productivity to its industry sources” and that “. . . only at theindustry level . . . production analysts can seek to understand the specific changes in technology and choices that the firmsmake in response. . .”. Most recently, Martínez et al. (2010) identify the TC embedded in computer hardware as the mainleading force in US productivity growth and estimate the IT-specific embodied TC in the US as 0.62% for 1980–1994, 3.2% for1995–2004. So, the numbers vary but all of them indicate technology shocks in the TC rate.

This paper is constructed as follows. Next section introduces a vintage capital model and an optimization problem (OP),provides necessary preliminary results, and extends them for the problem under study. Section 3 analyses the OP analyticallyand numerically under technological breakthroughs. We start with a model case with a linear continuous TC and single shock,where the analytical OP solution is constructed and reveals the exact structure of optimal trajectories. Section 3.2 investigatesa more realistic case of the exponential TC and demonstrates analytically and numerically that the optimal dynamics undertechnological breakthroughs remains similar to the model case. Section 4 briefly discusses several possible extensions ofthe model. Particularly, breakthroughs in industry-specific productivity can be accompanied by jumps of the new capitalprice, which is addressed in Sections 4.1 and 4.2. Section 4.3 explores the case of several TC shocks. Section 5 discusses theobtained outcomes and compares them to known literature results.

2. Model and preliminary results

Let us consider optimal policies of vintage capital replacement of a firm that controls both investing into new capital andscrapping the oldest one. The dynamics of the firm can be efficiently described by vintage capital models with controlledscrapping lifetime. The economic objective is to maximize the present value of the firm profit over the infinite horizon:

maxm,a,y

I =∫ ∞

t0

e−rt[y(t) − p(t)m(t)] dt, (1)

under the constraints-equalities:

y(t) =∫ t

a(t)

ˇ(�, t)m(�) d�, (2)

∫ t

a(t)

m(�) sd� = R(t), (3)

the constraints-inequalities:

m(t) ≥ 0, (4)

a′(t) ≥ 0, a(t) ≤ t, (5)

m(t) ≤ y(t)p(t)

, t ∈ [t0, ∞) (6)

and the initial conditions:

a(t0) = a0 < t0, m(�) = m0(�), � ∈ [a0, t0]. (7)

The endogenous variables are the investment m(t) into new capital (measured in the resource R units), the scrapping timea(t) of the oldest capital vintage used at time t, and the product output y(t). The given model functions are the efficiency ˇ(�,t) of capital vintages introduced at instant �, the consumption R(t) of a limited resource (labour, energy, etc.), the price p(t)of new capital, and the discounting factor e−rt. Following Boucekkine et al. (1997, 1998), the model uses a Leontief vintageproduction function (2) where the capital and resource are complements. Inequalities-constraints (4) and (5) reflect theirreversibility of investment and scrapping decision (scrapped vintages cannot be used again). The liquidity constraint (6)keeps the net cash y(t) − p(t)m(t) non-negative and prevents the firm from incurring debt. To hold (6) at t = t0, the givenmodel functions should satisfy

p(t0)m0(t0) ≤∫ t0

a0

ˇ(�, t0)m0(�) d�.

The given function R is assumed to be continuously differentiable, ˇ and p are piecewise-differentiable, m0 is piecewisecontinuous, and all these functions are strictly positive on [t0, ∞). The conditions∫ ∞

t0

e−rtˇ(t, t)R(t) dt < ∞,

∫ ∞

t0

e−rtp(t) dt < ∞

are imposed to guarantee the convergence of the improper integral in (1) (Hritonenko and Yatsenko, 2008a).

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1067

We consider the technological dynamics to be known in advance on the firm level and assume the existence of futuretechnological breakthroughs at the global level. Because of the embodied TC, the capital efficiency ˇ(�, t) increases in �.Specifically, we assume that both ˇ(�, t) and p(t) consist of two dynamic components:

• a smooth (exponential or linear) TC,• technological shocks in the form of irregularities in ˇ(�, t) and p(t) at certain times.

The equality-constraint (3) is not common for a firm and deserves a few comments. It suggests that the firm fully uses thegiven amount R(t) of a certain resource (labour, land, financial budget, environmental pollution, etc.). Such assumptions aremuch more common in macroeconomic VCMs (Solow et al., 1966; Benhabib and Rustichini, 1993; Boucekkine et al., 1977;Boucekkine et al., 1998; Cooley et al., 1997; Yorukoglu, 1998). We use the resource constraint (3) for three reasons. The firstone is practical; such constraints are becoming more regular in a firm management. A relevant example is a quota on energyconsumption or, equivalently, on carbon emissions.

The second reason is methodological. The VCM (1)–(7) has the simplest setup in the partial equilibrium vintage modelingframework (Leontief technology, constant returns to scale, a price-taking firm, no learning, no adoption cost or adjustmentcost), which ensures a balanced growth. If we assume R(t) in (3) to be endogenous, then the given resource price pR(t) shouldappear in the objective function (1) as

maxm,a,y

I =∫ ∞

t0

e−rt[y(t) − p(t)m(t) − pR(t)R(t)] dt. (8)

Then, as shown by Boucekkine and Pommeret (2004) for energy-saving TC, a balanced growth will occur only at a speciallychosen price pR(t) that grows with the same rate as TC. Next, if the resource price is involved into the model, it can beendogenized by general-equilibrium reasoning. As demonstrated in Section 2.2, a natural general-equilibrium set up forVCMs leads to the same optimal scrapping rule (11) as in our model. Thus, model (1)–(7) gives a stylized model setup thatis simple and suitably broad at the same time.

The third reason is mathematical. The OP (1)–(7) in the case of smooth ˇ and p has been thoroughly analyzed in Boucekkineet al. (1997, 1998) and Hritonenko and Yatsenko (1996, 2005, 2008a) and possesses remarkable properties that simplify itstheoretical analysis (see the next section).

2.1. Extremum conditions

The OP (1)–(7) includes three unknown functions m, a, and y related by the equalities (2) and (3). Following (Hritonenkoand Yatsenko, 1996, 2005, 2008a), we choose m as the decision variable (independent control) of the OP, then y and a are thedependent (state) variables expressed via m. The inequality (4) is a standard constraint on the control m, whereas (5) and(6) are constraints-inequalities on the state variables a and y. Let m ∈ L∞

loc[t0, ∞), then the unknowns a and y in (1)–(7) are

a.e. continuous on [t0, ∞) (Hritonenko and Yatsenko, 2005).

Lemma 1. (the necessary and sufficient condition for an extremum). Let ˇ(�, t) strictly increase in � at � ∈ [a0, ∞), t ∈ [t0, ∞).Then, a function m*(t), t ∈ [t0, ∞), is a solution of the OP (1)–(7) if and only if

I′(t) ≤ 0 at m∗(t) = mmin(t), I′(t) ≥ 0 at m∗(t) = y∗(t)p(t)

,

I′(t) = 0 at mmin(t) < m∗(t) <y∗(t)p(t)

, t ∈ [t0, ∞),(9)

where

I′(t) =∫ a−1(t)

t

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� − e−rtp(t) (10)

is the Freshet derivative of I in m, the state variables a* and y* are determined from (2) and (3), a−1(t) is the inverse function ofa(t), and

mmin(t) = max{0, R′(t)}.

Proof. This result was proven for the OP (1)–(7) with the constraint m(t) ≤ mmax(t) instead of (6) in Hritonenko and Yatsenko(2008a). The inconvenient constraint a′(t) ≥ 0 in (5) was replaced with the stricter constraint m(t) ≥ mmin (t) for the controlm. The constraint a(t) ≤ t is never active because of (3), R > 0, and m ≥ 0.

To complete the proof for the OP (1)–(7), we need to consider the case when the optimal m*(t) = y*(t)/p(t) at somet ∈ � ⊂ [t0,∞) and prove that then I′(t) ≥ 0. Giving small variations ım(t), ıy(t), t ∈ �1 ⊂ �, mes(�1) 1, to m* and y*, weobtain from Eqs. (2) and (3) that ıy(t) = o(ım(t)). Let us choose ım(t) < 0, then a new perturbed m(t) = m*(t) +ım(t) < m*(t) andthe perturbed y(t) ≈ y*(t), therefore the restriction m(t) ≤ y(t)/p(t) holds and m(t) is admissible. The resulting increment of

1068 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

the functional (1) ıI =∫ ∞

t0I′m(t)ım(t) dt and (10) demonstrate that if I′(t) < 0, then the admissible variation ım(t) < 0 gives a

larger value I + ıI of the functional (1), so, m*(t) = y*(t)/p(t) is not optimal. Therefore, I′(t) ≥ 0 is necessary for the optimality ofm*(t) = y*(t)/p(t). As shown in Hritonenko and Yatsenko (2008b), if ˇ(�, t) strictly increases in �, the functional (1) is concavein m, hence, the above necessary condition for an extremum is also sufficient.

The lemma is proven.�The condition of strictly increasing ˇ in � means the presence of the background continuous embodied TC (newer vintages

are always more efficient). The concavity of the OP has important theoretical implications. First, it produces the necessaryand sufficient condition and solves the issue of solution existence. Indeed, if a function m* satisfies Lemma 1, then it is an OPsolution. Also, the OP concavity also means that the OP solution m* is unique and, therefore, delivers the global optimum tothe OP.

The functional derivative I′(t) of the OP depends on a only and is denoted as I′(a, t) here and thereafter. By (9) and (10),possible interior optimal trajectories a should satisfy the integral-functional equation I′(a, t) = 0 or

∫ a−1(t)

t

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� = e−rtp(t), t ∈ [t0, ∞) (11)

with respect to a. We will refer to the solution a of (11), if it exists, as the turnpike.

2.2. On generality of the extremum conditions

2.2.1. Let us consider maximization of (8) under restrictions (2)–(7) assuming endogenous R. Following Malcomson (1975)or Boucekkine and Pommeret (2004), the optimal interior trajectory a* should satisfy two equations

∫ a−1(t)

t

e−r�[ˇ(t, �) − pR(�)] d� − e−rtp(t) = 0, (12)

ˇ(a(t), t) = pR(t), (13)

which can happen only at a specially chosen price pR(t). Moreover, then (12) and (13) coincide with (11), therefore, thetrajectory a* is the same as in OP (1)–(7). In a general case, there is no interior solution a*.

2.2.2. The general-equilibrium VCM with energy-saving TC of Azomahou et al. (2009) assumes a standard rationaleconomic behavior of a representative household, final product sector, energy sector, and government. The embodied tech-nological change is concentrated in intermediate good sectors. In each intermediate sector, a monopolistic firm solves the OP(2)–(8) where R is an endogenous energy demand determined by the supply in the energy sector. The optimality condition(12) is still valid for the firm, but now the energy price pR is endogenous and determined from a zero-profit condition (13).Combining (13) with (12), we obtain exactly condition (11) for the interior scrapping time, that is a direct sequence of theextremum condition (9) and (10).

2.2.3. Another general-equilibrium VCM that leads to the same optimal scrapping rule (11) is more recent model ofJovanovic and Tse (2010) with output-augmenting TC. The general-equilibrium setup is quite different from the one in 2.2.2and involves an elastic consumer’s demand output curve. The model of Jovanovic and Tse (2010) uses a maintenance costas an expense instead of the energy price pR in Azomahou et al. (2009) but produces the same extremum condition (11) forthe interior scrapping time.

2.2.4. Finally, a central planner problem in the macroeconomic Ramsey VCM with linear utility (Boucekkine et al., 1997)leads to the same FOC condition (11). Here, the limited resource is labor. Thus, the optimality condition (11) for the interiorscrapping time is sufficiently broad. At the same time, it is simple enough for obtaining exact solutions.

2.3. Optimal dynamics under smooth technological change

As usually in similar economic problems, the structure of the optimal trajectories involves a short-term transition part(with a corner solution) and a long–term interior regime such that I′ ≡ 0.

Lemma 2. (Hritonenko and Yatsenko, 2009). In the case of the exponential TC and capital deterioration:

b(�, t) = b0ecb�−cd(t−�), p(t) = p0ecpt, (14)

b0 > 0, p0 > 0, cb + cd > 0, cb < r, cp < r, b0(r − cb) > p0(r + cd)(r − cp)e(cp−cb)t0 .

Eq. (11) has a unique solution a such that:

(i) If cb = cp = c, then a(t) = t − A, t ∈ [t0, ∞), where the constant A > 0 is found from the nonlinear equation

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1069

(r + cd)e−(c+cd)A − (c + cd)e−(r+cd)A = (r − c)[

1 − (r + cd)p0

b0

](15)

and A =√

2p0/(b0(c + cd)) + o(r) at r 1.(ii) If cb > cp, then a(t) < t, a′(t) > 0, and A(t) = t − a monotonically decreases on [t0, ∞) and a(t) → t at t → ∞.

(iii) If cb < cp, then the solution a(t), a′(t) > 0, exists only on a finite interval [t0, tcr), t0 < tcr < ∞.

So, a strictly increasing turnpike a(t) exists on the infinite horizon [t0, ∞) only if cb ≥ cp.In the case of smooth given functions ˇ and p, the structure of the OP solution is the following.

Theorem 1. If a strictly increasing turnpike a exists, then the solution (m*, y*, a*) of the OP (1)–(7) has the following structure:

• A transition period [t0, �) with three possible cases:◦ Case 1: a0 > a(t0). Then m*(t) = mmin(t), a*(t) = a0 on [t0, �), and a ∗ (�) = a(�) at certain � > t0.◦ Case 2: a0 < a(t0). Then m*(t) = y*(t)/p(t), a*(t) increases fast and a∗(�) = a(�) at some � > t0.◦ Case 3: a0 = a(t0). Then � = t0 (no transition dynamics).

• The long-term dynamics interval [�, ∞):

a∗(t) = a(t), m∗(t) = m∗(a(t))a′(t) + R′(t), t ∈ [�, ∞), (16)

provided that the optimal m*(t) and y*(t) satisfy (2)–(6).

Proof. We assume that a solution (m*, a*) exists. Both transition and long-term dynamics have been investigated fora0 > a(t0) (Cases 1 and 3) by Hritonenko and Yatsenko (2008a).

To complete the proof, let us consider Case 2: a0 < a(t0). Since a(t) is continuous, then a∗(t) < a(t) on some interval [t0,�) such that a∗(�) = a(�) (otherwise, a∗(t) < a(t) on [t0, ∞)). Then, a−1(t) > a−1(t) for t ∈ [t0, �), I′(a, t) ≡ 0 at t ∈ [t0, ∞), bythe definition of a, and the derivative (10) can be written as

I′(a∗, t) =∫ �

t

e−r�[ˇ(t, �) − ˇ(a∗(�), �)] d� +∫ a∗−1(t)

�

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� − e−rtp(t)

=∫ �

t

e−r�[ˇ(a(�), �) − ˇ(a∗(�), �)] d� +∫ a∗−1(t)

t

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� − e−rtp(t)

>

∫ a∗−1(t)

t

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� − e−rtp(t)

>

∫ a−1(t)

t

e−r�[ˇ(t, �) − ˇ(a(�), �)] d� − e−rtp(t) = I′(a, t) = 0, t ∈ [t0, �). (17)

Hence, I′(a*, t) > 0 at t ∈ [t0, �) for any m*. Therefore, m*(t) should be maximum possible at t ∈ [t0, �). By (9), the maximalm*(t) at a fixed t is m*(t) = y*(t)/p(t). The optimal m*(t) = y*(t)/p(t) and a*(t) on [t0, �) are determined from the system of twononlinear integral Eqs. (2) and (3). Similar to (Hritonenko and Yatsenko, 1996), we can prove that this system possesses aunique positive solution m*(t), a*(t), t ∈ [t0, �). Since the kernel of Eq. (2) ˇ(�, t) > 0, both solutions m*(t) and a*(t) increase.Moreover, a*(t) increases faster than a(t), so they will intersect at some point t = �.

At t ∈ [�, ∞), an interior in the domain (4)–(7) solution (m*, y*, a*) exists such that a∗(t) = a(t) and I′(a∗, t) = I′(a, t) ≡ 0for t ∈ [�, ∞). The theorem is proven.�

The optimal investment trajectory m*(t) is boundary during the transition dynamics [t0,�]: m*(t) = mmin(t) orm*(t) = y*(t)/p(t) is maximal. By (16), the initial (boundary) dynamics of m* on [a0, �] is replicated throughout the long-termhorizon [�, ∞) following the formula m∗(t) = R′(t) + m(a(t))a′(t). This effect is known as the replacement echoes (Boucekkineet al., 1997, 1998; Hritonenko and Yatsenko, 1996, 2005). In particular, if a(t) = t − A and R′(t) = 0, then m*(t) = m*(t − A) isstrictly periodic.

Theorem 1 does not prevent the constraint-inequalities (4)–(6) from being saturated (binding) on certain parts of thelong-term dynamics interval [�, ∞). For instance, if a0 > a(t0) in addition to a(t) = t − A and R′ = 0, then, by Case 1, theoptimal investment is boundary m*(t) = 0 on the transition interval [t0, �] and is boundary later on the infinite set of intervals[� + iA, � + (i + 1)A], i = 0, . . ., ∞. However, the investment trajectory in general is not required to be boundary during thereplacement echoes. If a0 < a(t0), then, by Case 2, m*(t) = y*(t)/p(t) is boundary during transition [t0, �] but m*(t) = m*(t − A)is not necessarily boundary on [�, ∞). An unusual situation would be m*(t) > y*(t)/p(t) at some part of [�, ∞), i.e., when theendogenous bound (6) is too strict for such m*. We will mention similar exceptions only when they are relevant for our topic.

In particular, the given ˇ(�, t), R(t), and p(t) should expose a smooth monotonic behavior in order for m*(t) and y*(t) tosatisfy (4) and (6). Shocks (sharp changes) in the dynamics of ˇ(�, t), R (t), and p(t) can violate constraints (4)–(6) and the

1070 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

existence of a smooth turnpike a on [�, ∞). This is the case that we explore in the next section. We shall see that then theoptimal trajectory a* returns to a before and after the shock.

3. Optimal dynamics under technological shocks

The optimal investment and scrapping of vintage capital under technological shocks leads to the OP (1)–(7) with non-smooth functions ˇ and p. First, we focus on the long-term optimal dynamics, which appears to be much more complex thandescribed by Theorem 1. To analyze the long-term dynamics, we have to investigate the possibility of solving the nonlinearintegral Eq. (11) at non-smooth functions ˇ(�, t) and/or p(t). One can show that the transition dynamics remains the sameas in Theorem 1.

In general, both capital efficiency ˇ(�, t) and capital price p(t) can experience TC shocks at certain times. Possible tech-nological breakthroughs impact the efficiency function ˇ(�, t) first of all and can be described by discontinuities in the ˇ(�,t) level or its growth rate. Explorations in Operation Research almost exclusively consider technological breakthroughs asdiscontinuities in the level variables (see Rogers and Hartman, 2005 and the references therein). However, as discussed inIntroduction, the breakthrough shocks in levels do not appear in macroeconomics, where shocks in the growth rates aremore significant.

In this section, we consider a technological breakthrough described as the discontinuity in the embodied TC growth rate∂ˇ(�, t)/∂� at certain instant � = t. We assume t > t0, which means that the breakthrough is anticipated in the future.1 InSection 3.1, we identify a model case (the linear background TC), when an analytic solution of the OP can be constructed. InSection 3.2, we consider the more realistic exponential TC and demonstrate analytically and numerically that the structureof optimal trajectories remains essentially similar to the model case. The breakthroughs in productivity can be accompaniedby shocks of the new capital price p(t), which will be addressed in Sections 4.1 and 4.2.

3.1. Case of piecewise-linear technological progress: analytic solution

Let us start with a model case of the piecewise-linear ˇ(�, t) = ˇ(�) and constant p(t):

ˇ(�) ={

C1(� − t) + b, � ≤ tC2(� − t) + b, � > t

, t > t0, p(t) = p = const, r = 0. (18)

In the case C1 = C2, there is no jump and the OP has a unique solution (a*, m*) described by Theorem 1, whose long-termdynamic component a* coincides with the turnpike

a(t) = t − A, A =√

2p

C2> 0, t ∈ [t0, ∞). (19)

Formula (19) was first established in Hritonenko and Yatsenko (1996) and can be verified by the direct substitution of(19) into Eq. (11). The turnpike trajectory (19) with the optimal constant lifetime A represents a useful benchmark trajectoryfor analyzing the impact of TC shocks.

Here we investigate the case C1 < C2. Then Eq. (11) has the following form:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

C2

∫ a−1(t)

t

[t − a(�)] d� = p, a−1(t) ≤ t < ∞,

C2

∫ a−1(t)

t

(t − t) d� − C1

∫ a−1(t)

t

[a(�) − t] d� − C2

∫ a−1(t)

a−1(t)

[a(�) − t] d� = p, t ≤ t < a−1(t),

C1

∫ a−1(t)

t

[t − a(�)] d� = p, t < t.

(20)

Starting with the solution (15) a(t) = t − A to the first equation of (20) at t > a−1(t), we can solve the other two equationsfrom right to left. To analyze the solution structure, let us differentiate (20):

t − a(t) = a−1(t) − t, t ∈ [ a−1(t), ∞), (21)

a(t) − t = C2

C1[2t − a−1(t) − t], t ∈ [t, a−1(t)], (22)

t − a(t) = a−1(t) − t, t < t. (23)

1 Case t = t0 corresponds to the unanticipated breakthroughs studied in Boucekkine et al. (1998).

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1071

Substituting the inverse a−1(t) = t + A of a(t) = t − A on [a−1(t), ∞) into (22), we obtain a solution a(t) as a straight lineon [t, a−1(t)]. Substituting its inverse on [a(t), t] into (23), we obtain the next straight line a(t), and so on. Continuing thisprocess, we obtain that the solution a(t) of Eqs. (22) and (23) on [t0, a−1(t)) coincides with one of the straight lines

ak(t) = [kC2 − (k − 1)C1)]t − (C2 − C1)t − C2A

(k − 1)C2 − (k − 2)C1, k = 1, 2, . . . (24)

that intersect at the point (t + AC2/(C2 − C1), t + AC2/(C2 − C1)). Namely, ak+1(t) is a solution of (22) and (23) on [t1, t2],t1 < t2, if ak(t) is its solution on [ak

−1(t1), ak−1(t2)]. The trajectories (24) are shown in Fig. 1 with dotted lines.

Therefore, the integral Eq. (11) at C1 < C2 has no continuous solution on the whole interval [t0, ∞), but it has continuoussolutions on certain parts of [t0, ∞). A similar situation was analyzed by Hritonenko and Yatsenko (2008a) for the OP (1)–(7)with continuous ˇ over a finite horizon [t0, T], where we connected separate pieces of ai by horizontal lines (along whicha′ = 0) to construct the continuous optimal trajectory a*. Analogously, we prove the following theorem.

Theorem 2. In the case (18) and C1 < C2, the long-term dynamics a*(t), t ∈ [�, ∞), of the OP solution (a*, m*) has the followingform

a∗(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

a(t), I′(a, t) = 0, t ∈ [a−1(t), ∞)

a1(t), I′(a, t) = 0, t ∈ [ˇ1, a−1(t))

ai(˛i), I′(a, t) < 0, t ∈ [˛i, ˇi)

ai+1(t), I′(a, t) = 0, t ∈ [ˇi+1, ˛i)

, i = 1, 2, . . . , t ∈ [m0, ∞), (25)

where the trajectories ak, k = 1, 2, 3, . . . are defined by (24), the parameters ˛k, ˇk, k = 1, 2, . . ., � < ˇk+1 < ˛k < ˇk, ˛1 < t < ˇ1,are uniquely determined as

˛k = t + AC2

C2 − C1− A

(k + C1

C2 − C1

)√C2

C1

k(C2 − C1) + C1

k(C2 − C1) + C2, (26)

ˇk = t + AC2

C2 − C1− A

(k − 1 + C1

C2 − C1

)√C2

C1

k(C2 − C1) + C2

k(C2 − C1) + C1, k = 1, 2, 3, . . . (27)

At the given a*(t), the optimal m*(t) on [�, ∞) is determined by (16).

Proof. Since the long-term solution a∗(t) = a(t) = t − A at t ∈ [a−1(t), ∞), then by (22) a∗(t) = a1(t) = C2(t − A − t)/C1 + tat t ∈ [t, a−1(t)]. Because the inverse function a−1(t) switches from t + A to a1

−1(t) at t = t (see the gray line in Fig. 1), thetrajectory a* should leave the line a*(t) = a1(t) at some point ˇ1, ˇ1 ≥ t (Fig. 1, the solid line). We set a*(t) = a1(ˇ1) at ˛1 ≤ t ≤ ˇ1.

Fig. 1. The optimal trajectory a and its inverse a−1 in the case (18) of piecewise-linear efficiency ˇ. The horizontal parts of a are the anticipation echoescaused by the TC shock at t = t. The optimal a(t) oscillates around the first turnpike ˜a(t) (the dotted line) before t and converges to the second turnpike a(t)(the dashed line) after t.

1072 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

By (24), a*(t) switches to a2(t) at some point ˛1 < ˇ1 such that a1(ˇ1) = a2(˛1) or

C2

C1(ˇ1 − A) − t

C2 − C1

C1= 2C2 − C1

C2˛1 − t

C2 − C1

C2− A, (28)

and I′m(a∗, ˛1) = 0. By (29), equation I′(a*, ˛1) = 0 can be rewritten as

C1

∫ ˇ1

˛1

[˛1 − C2

C1(ˇ1 − A) − t

C2 − C1

C1

]d� + C1

∫ a1−1(˛1)

ˇ1

[˛1 − C2

C1(� − A) + t

C2 − C1

C1

]d� = p. (29)

Solving (28) and (29), we determine points ˛1 and ˇ1 in the form of (26) and (27) and verify that ˛1 < t < ˇ1. It can beshown that I′(a*, t) < 0 at ˛1 < t < ˇ1. Now a(t) = a2(t) and I′(a*, t) = 0 to the left of ˛1 on some interval [ˇ2, ˛1], ˇ2 < ˛1. Then,because of the symmetry of the inverses about y = x, the trajectory a should leave a2(t) at some point ˇ2, before a−1(t) jumpsdown at point t = a(˛2), and so on. Let us find the “switch” points ˛k, ˇk, k = 2, 3, . . ., where a(t) leaves the old curve ak(t) forthe new one. Since ak+1(˛k) = ak (ˇk), we can express ˇk as a function of ˛k:

ˇk = C2 − C1

(kC2 − (k − 1)C1)2

[((k + 1)C2 − kC1)((k − 1)C2 − (k − 2)C1)

C2 − C1˛k + (C2 − C1)t + AC2

]. (30)

Substituting (24) and (30) to the last equation of (20) we obtain an equation with respect to ˛k, from which (26) isuniquely determined. Relation (27) follows from (26), (30), and I′(a*, t) < 0 on (˛i, ˇi). By Lemma 1, the condition I′(a*, t) ≤ 0is necessary and sufficient for the optimality of the OP solution a*.

The theorem is proven.�Thus, the optimal capital lifetime a* possesses irregular intervals where a* is boundary (a*(t) ≡ const if R′(t) ≤ 0). By Lemma

1, the optimal investment m*(t) = mmin(t) determined by (16) is also boundary on these intervals. We refer to these intervalsas anticipation echoes (see also anticipation waves in Feichtinger et al., 2006) because they are caused by the anticipation ofa future technological breakthrough, that is, the future jump in the efficiency function ˇ(�) at some instant t. These echoespropagate backward throughout the interval [t0, t] and become smaller as t − t increases. The adjustment of the long-termoptimal trajectory a* to the initial conditions (7) follows the transition dynamics of Theorem 1.

Corollary. The long-term optimal trajectory a* of the OP oscillates around the turnpike ˜a(t) = t − A1, A1 =√

2p/C1, at t < t. At

large t − t0 � 1, a(t) strives to ˜a(t) as t − t increases.

The proof follows from the analysis of (25)–(27).

3.2. Case of piecewise-exponential efficiency: analytic structure and numeric simulation

Let us consider the case of a piecewise-exponential ˇ(�, t) = ˇ(�) and exponential p(t):

ˇ(�) ={

Bec1(�−t), � ≤ tBec2(�−t), � > t

, p(t) = Pecp,t, (31)

B, P > 0, 0 < c1 ≤ c2 < r, cp < r.

In fact, (31) reflects the main case of the TC shocks discussed in the macroeconomic VCM literature. Greenwood andYorukoglu (1997, p. 76) explicitly identified such shock in the US economy at t = 1974 as c1 ≈ 3% and c2 ≈ 5%. Successiveresearchers avoid being so specific but also indicate the presence of technology shocks in the TC rate. Thus, Martínez et al.(2010) estimate the IT-specific embodied TC in the US as 0.62% for 1980–1994, 3.2% for 1995–2004.

The qualitative dynamics of the optimal long-term solution in the case (31) of piecewise-exponential ˇ(�) appears to beremarkably similar to the one described in Section 3.1 for the piecewise-linear ˇ. The major differences are that the “interior”trajectories ai, i = 1, 2, . . ., are not straight lines and the switching points ˛k, ˇk can be only found implicitly using numericmethods.

In the case c1 = c2 with no jump, the solution (a*, m*) of the OP is described by Theorem 1. Its long-term dynamic componenta* coincides with the unique turnpike a(t) that exists on [t0, ∞) at c1 ≥ cp. By Lemma 2, the optimal capital lifetime t − a(t)is constant if c1 = cp, and decreases if c1 > cp.

In the case c1 < c2, Eq. (11) leads to the following recurrent formulas:⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

Be−c2 t

∫ a−1(t)

t

e−r�[ec2t − ec2a(�)] d� = e−rtPecpt, a−1(t) ≤ t < ∞, (32)

Bec1[a(t)−t] = Bec2[t−t] − Bc2ec2[t−t] [1 − e−r[a−1(t)−t]]r

+ (cp − r)Pecpt, t ≤ t < a−1(t), (33)

Bec1[a(t)−t] = Bec1[t−t] − Bc1ec1[t−t] [1 − e−r[a−1(t)−t]]r

+ (cp − r)Pecpt, t < t. (34)

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1073

Fig. 2. The optimal scrapping time a(t) (solid curve) in the case (31) of continuous exponential TC at c1 = c2 = 0.08, cp = 0.07, and r = 0.1. The dashed curveis the inverse a−1 of function a and the straight 45◦ line indicates the symmetry between a and a−1. The optimal capital lifetime t − a(t) decreases becausec1 > cp .

As in the previous section, since a(t) < t, then a−1(t) > t and Eqs. (33) and (34) can be solved in a from right to left. Namely,starting with the solution a∗(t) = a(t) of (32) on [a−1(t), ∞) and substituting the inverse a−1(t) on [t, ∞) of a*(t) into (33),we obtain a unique solution a1(t) of (33) on [t, a−1(t)]. Substituting its inverse into (34), we obtain the next curve a2(t) on[a−1(t), t], and so on. This process results in a set of trajectories ai, i = 1, 2, . . ., such that a*(t) coincides with one of ai(t) on� ⊂ [t0, a−1(t)] in order to be interior to the domain (6) at t ∈ �.

As in Section 3.1, the OP solution a* is obtained by connecting separate pieces of ai with the horizontal boundary-valuedtrajectories a′(t) ≡ 0 in such a manner that I′(a*, t) = 0 while a*(t) = ai(t) for each i. The obtained continuous a* represents thelong-term trajectory in the sense of Theorem 1. In general, a result similar to Theorem 2 is valid in the piecewise-exponentialcase (31), but its proof is considerably more challenging.2 In this paper, we will employ numeric simulation to explore thestructure of OP solutions.

As the basic input data set, we use the numeric example identified in Hritonenko and Yatsenko (2009) for capital replace-ment at a “typical” US manufacturing plant. It includes the exponential TC (31) with the rates c1 = 0.08 and cp = 0–0.09, theinterest rate r = 0.1, and the initial productivity/cost ratio B/P adjusted to produce the capital lifetime t − a(t) of approximately24 years at the initial modeling year t0 = 0.

Basic simulation run (with no TC shocks). Let us start our simulation with the exponential TC (31) c1 = c2 with no jumps.By Lemma 2, if the TC rates c1 and cp in the capital efficiency and price are different, then the long-term optimal capitallifetime t − a(t) decreases at c1 > cp and increases at c1 < cp. We have simulated the optimal dynamics for several scenariosc1 > cp, c1 = cp, c1 < cp. An example is shown in Fig. 2 for c1 = c2 = 0.08 and cp = 0.07.

Simulation with technological breakthroughs. Let the piecewise-exponential ˇ(�) in (31) have a jump c1 < c2 at t = 60(years). There are two turnpikes for the scrapping time in this case: ˜a before the jump and a after the jump. The trajectory˜a(t) is the unique solution of (11) at ˇ(�) = Bec1(�−t), and trajectory a(t) is the solution of (11) at ˇ(�) = Bec2(�−t).

A simulation example shown in Fig. 3 corresponds to c1 = 0.08 before t, c2 = 0.09 after t, and cp = 0.08. The optimal trajectorya* possesses the following properties:

• The turnpike ˜a(t) produces a constant lifetime t − ˜a(t) (because c1 = cp) and the turnpike a corresponds to a decreasinglifetime (because c2 > cp). The optimal a*(t) oscillates around ˜a(t) at t < t and coincides with the turnpike a(t) at t > a∗−1(t).

• The jump in ˇ(�) leads to the appearance of intervals, where the optimal a* is boundary (a*(t) ≡ const). The largest such aninterval appears around the jump time and reflects the direct anticipation effect: the optimal solution is no scrapping (so,the optimal lifetime t − a*(t) increases) immediately before and after the jump.

• The optimal trajectory a* also exposes the intervals with boundary solutions during the regeneration periods precedingthe TC jump (anticipation echoes). The echoes become smaller when the distance to the jump increases.

2 Such proof for a more detailed vintage model would be the subject of a separate paper.

1074 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

Fig. 3. The optimal a(t) (the solid curve) in the case of the TC jump at t = 60 with the efficiency rate c1 = 0.08 before t and c2 = 0.09 after t, the capital costrate cp = 0.08, and r = 0.1. The dotted straight line corresponds to the turnpike with the constant lifetime before the jump. The optimal lifetime t − a(t) aftert decreases because c2 > cp .

Summarizing the above results of Sections 3.1 and 3.2, a single shock in the TC rate causes the following effects in theoptimal dynamics of capital lifetime and new capital investment:

1. Direct anticipation effect: The optimal regime is boundary (no scrapping and increasing optimal lifetime) on some antic-ipation interval before and after the shock. After this interval, the optimal regime switches to a new balanced trajectory(turnpike) in a finite time. Before this interval, the optimal regime involves a series of non-interior boundary parts(anticipation echoes) and converges to the old turnpike as the time to the shock increases.

2. Anticipation echoes: The optimal lifetime is boundary on a set of intervals of increasing length during the regenerationperiods preceding the TC shock. These echoes are caused by the anticipation of a future jump in capital productivity or price(a future technological breakthrough). Both optimal capital lifetime a* and investment m* are boundary (no scrapping,a*(t) ≡ const) during the anticipation echoes. The optimal lifetime a* before the shock converges to a balanced growthpath as the distance to the shock increases and coincides with the new balanced growth path soon after the shock time.In contrast, the disturbances in the optimal investment m* disseminate after the shock time up to ∞ because of thereplacement echoes phenomenon. The optimal investment is of non-monotonic bumpy character and does not convergeto any balanced path.

4. Some extensions

Here, we briefly address other possible situations with shocks in the capital efficiency and the capital cost.

4.1. A shock in the cost of new capital

In this case, the piecewise-exponential p and exponential ˇ are in the following form:

p(t) ={

Pecp1(t−t), t ≤ tPecp2(t−t), t > t

, ˇ(�) = Bec�, B, P > 0, cp2 < cp1 < r, c < r. (35)

A positive breakthrough (technological progress) corresponds to a decrease in the capital cost rate, i.e., cp2 < cp1. A simula-tion example shown in Fig. 4 uses the basic data set, cp1 = 0.05 before t = 60, cp2 = 0.03 after t, and c = 0.05. The demonstratedstructure of the optimal trajectory is qualitatively similar to the ˇ jump case.

4.2. Simultaneous shock in capital efficiency rate and new capital cost

Let us also analyze the case of a simultaneous jump of the same relative magnitude in both capital efficiency rate ˇ′(t) andthe capital cost rate p′(t). This case is interesting because then both turnpikes (before and after the jump time) correspond

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1075

Fig. 4. The optimal scrapping time a(t) at the jump in the capital cost rate with cp1 = 0.05 before t = 60 and cp2 = 0.03 after t (and c = 0.05). The (dotted)turnpike lifetime t − a(t) before t is constant because cp1 = c. The optimal lifetime after t decreases because cp2 < c.

to constant capital lifetimes. It allows us to clearly distinguish between the effects of continuous and discontinuous TC andillustrate the convergence character of the optimal trajectory to the turnpikes.

Specifically, we assume the continuous TC with c1 = cp = 0.05, r = 0.1 and a TC jump at t = 60 such that:

• the rate ˇ’(t) of the piecewise-exponential efficiency (31) is c1 = 0.05 before t and c2 = c after t;• the rate p′(t) of the piecewise-exponential capital cost (35) is cp1 = 0.05 before t and c p2 = c after t, that is:

p(t) = P

{e0.05(t−t), t ≤ 60ec(t−t), t > 60

, ˇ(�) = B

{e0.05(�−t), t ≤ 60ec(�−t), t > 60

. (36)

Fig. 5 illustrates the simulated optimal capital lifetime A(t) = t − a(t) for several scenarios with different values c = 0.05,0.07, 0.08, 0.09, 0.1 (shown from top to bottom). As predicted theoretically (Lemma 2), the optimal lifetime is constant atc1 = cp1 = c2 = cp2 (if there is no jump). In the cases with jumps, we have two turnpikes with constant lifetimes, one for theinterval before the jump and the other one for the interval after the jump. Both turnpikes are visible in Fig. 5 and one can

Fig. 5. The optimal capital lifetime A(t) = t − a(t) in the case of a jump of the same magnitude in both capital efficiency and cost rates at t = 60. Both ratesare c1 = cp1 = 0.05 before the jump and c2 = cp2 = c after the jump, for different values of c = 0.05, 0.07, 0.08, 0.09, 0.1 (shown from top to bottom). The constantlifetime corresponds to c = 0.05 (no jump).

1076 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

Fig. 6. The optimal scrapping time a(t) in the case of two TC jumps at t1 = 60 and t2 = 90 with the efficiency rate c1 = 0.08 before t1, c2 = 0.09 after t1, andc3 = 0.1 after t2.

easily see the character of the convergence of the optimal lifetime to the turnpikes. If there is no jump, then both turnpikesare the same as in the case c = 0.05 in Fig. 5.

To illustrate different structural dynamics, Fig. 5 also includes a case with optimal non-constant lifetime. The upper(dotted) trajectory corresponds to a case when the cost rate p′(t) = 0.1 is higher than the efficiency rate ˇ′(t) = 0.05 after t,then the lifetime of the second turnpike increases after the jump as predicted by Lemma 2.

4.3. Case of several shocks

The presence of several TC breakthroughs obviously complicates the obtained qualitative picture. Also, as argued inIntroduction, the case of one major forthcoming breakthrough is of most applied interest. Nevertheless, our analytic andnumeric techniques can handle cases of several TC shocks in the same uniform manner as the above cases with single shocks.In particular, an extension of the structural Theorem 2 of Section 3.1 can be proved for the piecewise-linear TC with severalTC shocks but the corresponding analytic expressions will be obviously more complicated.

The analysis shows that the anticipation effects caused by earlier shocks superimpose on the top of the echoes caused bylater shocks. Let us illustrate the solution structure in the situation of a piecewise-exponential ˇ(�, t) = ˇ(�) with two shocksand an exponential p(t):

ˇ(�) =

⎧⎪⎨⎪⎩

Bec1(�−t), � ≤ t1

Bec2(�−t), t1 < � ≤ t2

Bec3(�−t), � > t2

, p(t) = Pecpt, (37)

B, P > 0, 0 < c1 < c2 < c3 < r, cp < r.

The optimal dynamics of the OP (1)–(7), (37) is numerically simulated for the shocks at t1 = 60 and t2 = 90 with c1 = 0.08,c2 = 0.09, and c3 = 0.1. The simulation resuts are shown in Fig. 6, where the anticipation echoes caused by the later shock att2 are indicated by solid arrows while the echoes caused by the earlier shock at t2 are indicated by dashed arrows.

Similarly to the cases with one TC shock (Section 3), the anticipation echoes in capital lifetime caused by every futureshock deteriorate when the distance to the shock increases, so, the optimal lifetime remains stable with respect to possibleTC shocks.

5. Summary and discussion

1. We have analyzed the optimal replacement of vintage capital under continuous embodied TC and technological shocks(breakthroughs) at certain given (predicted) times. The model specifications involve endogenous scrapping of oldest

N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078 1077

vintages, partial equilibrium setup, no disembodied TC, Leontief technology, constant returns to scale, no market power,no learning, no adoption cost or adjustment cost. The technology is described by capital efficiency (labour productivity) andthe cost of new capital. The technological breakthroughs are modeled as jumps in the TC growth rates, which are observedin vintage literature starting with Greenwood and Yorukoglu (1997). Another reason for modeling breakthroughs as jumpsin the TC rates is that the optimal capital lifetime depends on the TC rates rather than TC levels. This is a fundamental featureof all vintage models with endogenous scrapping of oldest vintages (Solow et al., 1966; Benhabib and Rustichini, 1993;Boucekkine et al., 1977; Boucekkine et al., 1998; Cooley et al., 1997; Hritonenko and Yatsenko, 1996, 2005; Yorukoglu,1998; Jovanovic and Tse, 2010).

2. An exact solution of the constructed nonlinear optimal control problem is obtained in the case of piecewise-linear TC andits structure is analytically and numerically confirmed in the case of piecewise-exponential TC. The interpretation of thesesolutions reveals how the technology shocks affect the optimal dynamics of endogenous capital lifetime and investmentbefore and after the shock time. Namely, the optimal regime is no investment and no scrapping on a set of anticipationintervals (echoes) of increasing length during the period preceding the TC shock. The rightmost and largest anticipationinterval includes the shock instant. The optimal lifetime before the shock strives to a balanced growth path as the distancefrom the jump increases and coincides with the new balanced growth path after the shock.

3. Our results develop and extend the analysis by Boucekkine et al. (1998) of unanticipated permanent TC shocks in macroe-conomic vintage models with endogenous capital scrapping. They demonstrate an increase of the optimal scrapping age ofvintages and investment decrease after the shock. It is specifically pointed that “the productivity slowdown comes fromthe fact that the economy keep on using the initial machines for a while after the occurrence of the shock” (p. 375). Weshow that the optimal reaction to a future TC shock essentially happens before the shock time t in the form of the above-described direct and echoed anticipation effects in optimal capital lifetime. More specifically, the anticipation effects inthe capital lifetime completely stop after the time a−1(t) while the effects in investments disseminate up to ∞ because ofthe replacement echoes phenomenon. The stylized VCM (1)–(7) with endogenous lifetime does not possess any cushionedeffect of nonlinear adjustment costs, market power, or decreasing returns to scale and, therefore, produces everlastinginvestment echoes. In Boucekkine et al. (1998), the echoes in the optimal investment eventually deteriorate because ofnonlinear utility.

4. On the other side, our results complement the outcomes obtained by Pakko (2002) and Feichtinger et al. (2006) aboutnegative anticipation effects in investment for vintage models without endogenous scrapping. Pakko (2002) numericallydemonstrates a decline in investment, output and employment before and after a stochastic permanent shock in embodiedTC rate because agents anticipate higher returns in the future (a direct anticipation effect). Feichtinger et al. (2006) obtaina similar direct anticipation effect in response to a technology shock in a VCM with a fixed capital lifetime and investmentsinto new and old vintages in the case when the firm has a market power. They also obtained an echoed anticipation effect(anticipation waves) when the investments into old vintages are not allowed. No anticipation effects appear for a firmwith no market power. These results do not intercept with ours because we analytically prove anticipation effects for theoptimal capital lifetime that is taken as given by Feichtinger et al. (2006). Also, they consider a shock in the technologylevel rather than in the TC rate.

5. All these results indicate the existence of a certain fundamental anticipation property in economic systems. The antici-pation echoes seem to represent a general structural pattern of the optimal economic policy in response to various futuredisturbances in the model (discontinuities of exogenous parameters, the end of planning horizon, and so on). The antic-ipation echoes in endogenous scrapping and investment were first discovered for a finite-horizon version of problem(1)–(7) by Hritonenko and Yatsenko (1996) and are caused by the anticipation of the future policy change at the end ofplanning horizon (see also Hritonenko and Yatsenko, 2005, 2008a). Specifically, a zero-investment period at the horizonend generates a chain of zero-investment anticipation echoes. Also, similar echoes appear in the problem (1)–(7) withpiecewise-linear utility function (Hritonenko and Yatsenko, 2006).

Acknowledgements

The authors express their gratitude to two anonymous referees and participants of the Workshop on Dynamics, OptimalGrowth and Population Change: Theory and Applications (Milan, September 2008) and the AIM Workshop on Stochasticand Deterministic Spatial Modeling in Population Dynamics (Palo Alto, California, May 2009). Special thanks are to RaoufBoucekkine (Belgium), Cuong Le Van (France), and Alberto Bucci (Italy) for their interest and useful remarks. The work of YuriYatsenko was supported in part by Center for Operations Research and Econometrics (Louvain-la-Neuve, Belgium). NataliHritonenko would like to acknowledge the support of National Science Foundation grant DMS-1009197.

References

Azomahou, T., Boucekkine, R., Nguyen-Van, P., 2009. Promoting clean technologies under imperfect competition. CORE Discussion Paper 2009/11, Universitécatholique de Louvain, Louvain-la-Neuve, Belgium.

Aures, R.U., 2005. Resources, scarcity, technology and growth. In: Simpson, D., Toman, M.A., Aures, R.U. (Eds.), Scarcity and Growth Revisited: NaturalResources and the Environment in the New Millennium. Resources for the Future, Washington, DC, pp. 142–154.

Becerra-Lopez, H.R., Golding, P., 2007. Dynamic exergy analysis for capacity expansion of regional power-generation systems: case study of far West Texas.Energy 32, 2167–2186.

1078 N. Hritonenko, Y. Yatsenko / Journal of Mathematical Economics 46 (2010) 1064–1078

Benhabib, J., Rustichini, A., 1993. A vintage capital model of investment and growth. In: General Equilibrium, Growth, and Trade. II. The Legacy of Lionel W.McKenzie. Academic Press, New York, pp. 248–301.

Boucekkine, R., Germain, M., Licandro, O., 1997. Replacement echoes in the vintage capital growth model. Journal of Economic Theory 74, 333–348.Boucekkine, R., Germain, M., Licandro, O., Magnus, A., 1998. Creative destruction, investment volatility and the average age of capital. Journal of Economic

Growth 3, 361–384.Boucekkine, R., Pommeret, A., 2004. Energy saving technical progress and capital stock: the role of embodiment. Economic Modelling 21, 429–444.Bresnahan, T.F., Trajtenberg, M., 1995. General purpose technologies ‘Engines of growth’? Journal of Econometrics 65, 83–108.Cooley, T., Greenwood, J., Yorukoglu, M., 1997. The replacement problem. Journal of Monetary Economics 40, 457–499.de Solla Price, D., 1984. The science/technology relationship, the craft of experimental science, and policy for the improvement of high technology innovation.

Research Policy 13, 3–20.Doraszelski, U., 2004. Innovations, improvements, and the optimal adoption of new technologies. Journal of Economic Dynamics and Control 28, 1461–1480.Feichtinger, G., Hartl, R., Kort, P., Veliov, V., 2006. Anticipation effects of technological progress on capital accumulation: a vintage capital approach. Journal

of Economic Theory 126, 143–164.Greenwood, J., Yorukoglu, M., 1997. 1974, Carnegie-Rochester Conference Series on Public Policy 46, 49–95.Hritonenko, N., Yatsenko, Yu., 1996. Modeling and Optimization of the Lifetime of Technologies. Kluwer Academic Publishers, Dordrecht.Hritonenko, N., Yatsenko, Yu., 2005. Turnpike properties of optimal delay in integral dynamic models. Journal of Optimization Theory and Applications 127,

109–127.Hritonenko, N., Yatsenko, Yu., 2006. Optimization in a vintage capital model with piecewise linear cost function. Nonlinear Analysis 65, 2302–2310.Hritonenko, N., Yatsenko, Yu., 2008a. Anticipation echoes in vintage capital models. Mathematical and Computer Modeling 48, 734–748.Hritonenko, N., Yatsenko, Yu., 2008b. Optimal control of Solow vintage capital model with nonlinear utility. Optimization 57, 581–590.Hritonenko, N., Yatsenko, Yu., 2009. Integral equation of optimal replacement: analysis and algorithms. Applied Mathematical Modeling,

doi:10.1016/j.apm.2008.08.007.Jorgenson, D.W., Ho, M.S., Samuels, J.D., Stiroh, K.J., 2007. Industry origins of the American productivity resurgence. Economic Systems Research 19, 229–252.Jovanovic, B., Lach, S., 1997. Product innovation and the business cycle. International Economic Review 38, 3–22.Jovanovic, B., Tse, C.-Y., 2010. Entry and exit echoes. Review of Economic Dynamics 13, 514–536.Malcomson, J., 1975. Replacement and the rental value of capital equipment subject to obsolescence. Journal of Economic Theory 10, 24–41.Martínez, D., Rodríguez, J., Torres, J.L., 2010. ICT-specific technological change and productivity growth in the US: 1980–2004. Information Economics and

Policy 22, 121–129.Pakko, M.R., 2002. What happens when the technology growth trend changes? Transition dynamics, capital growth, and the “New Economy”. Review of

Economic Dynamics 5, 376–407.Rogers, J., Hartman, J., 2005. Equipment replacement under continuous and discontinuous technological change. IMA Journal of Management Mathematics

16, 23–36.Schmidt, R.C., Marschinski, R., 2009. A model of technological breakthrough in the renewable energy sector. Ecological Economics 69, 435–444.Solow, R., Tobin, J., von Weizsacker, C., Yaari, M., 1966. Neoclassical growth with fixed factor proportions. Review Economic Studies 33, 79–115.Yorukoglu, M., 1998. The information technology productivity paradox. Review of Economic Dynamics 1, 551–592.