Technological innovations, economic renovation, and anticipation effects

Journal of Mathematical Economics 46 (2010) 1064–1078

Contents lists available at ScienceDirect

Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Technological innovations, economic renovation, and anticipation effects Natali Hritonenko a,∗ , Yuri Yatsenko b,c a b c

Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA College of Business and Economics, Houston Baptist University, Houston, TX 77074, USA 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 2009 Received in revised form 2 August 2010 Accepted 2 August 2010 Available online 6 August 2010 JEL classification: C61 E22 O33 Keywords: Vintage capital models Technological change Technological innovations Anticipation echoes Optimal 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 vintage capital models. The firm controls the investment into new capital and scrapping of obsolete capital. The embodied technological change involves a continuous component and technological innovations (breakthroughs, technology shocks). The provided analytic and numeric investigation reveals the qualitative structure of optimal regimes. It demonstrates that the optimal investment is zero immediately before and after a technological breakthrough (direct anticipation effect) and contains a set of zero-investment boundary intervals (anticipation echoes) before the breakthrough time. The optimal capital lifetime oscillates around an interior balanced growth trajectory before and switches to a new balanced trajectory after 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-augmenting TC) or increasing efficiency of converting resources into useful work (the resource-saving TC). Recent research demonstrates that “the gradual, continuous and homogeneous technical progress normally assumed in economic models cannot explain key 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 accumulated experience and learning, • the radical improvement mode (technological breakthrough) when a radically new innovation is capable of displacing an older general-purpose technology among competing technologies. The modern economic theory considers technological breakthroughs as radical innovations caused by the substitution of 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 longterm economic growth (Aures, 2005). The steam engine, gasoline engine, electric power, semiconductors are examples of such general-purpose (enabling) technologies. Many economists interpret and analyze the recent IT revolution as a major breakthrough (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 technological replacement at a firm’s level. We employ a simple vintage capital model (VCM) to explore two powerful managerial decisions of a firm: investing into new, more efficient capital and scrapping the oldest, least efficient capital. Such VCMs with continuous TC have been systematically investigated by Boucekkine et al. (1997, 1998) and Hritonenko and Yatsenko (1996, 2005). Perceived 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 rate in 1974 and concluded that it caused the productivity slowdown mainly because of the costs of learning and adoption of new technologies. There is considerable economic evidence of substantial short-run negative effects until new IT equipment is completely adopted (Martínez et al., 2010). Boucekkine et al. (1998) address this issue analyzing an unanticipated permanent increase (shock) in the TC rate in a macroeconomic VCM with nonlinear utility. They numerically simulate the productivity slowdown 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-growth model with neutral and investment-specific TC à la Greenwood calibrated on the US data over 1949–2000. He shows that permanent shocks in embodied TC rate lead to a large sharp decline in investment just before the shock as agents anticipate higher returns in the future (a negative anticipation effect). The model predicts that perceived changes in the future TC growth trends provide incentives to firms to alter the mix of capital and labor that leads to lower investment, output and employment 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 given fixed lifetime of vintages. The model involves nonlinear adjustment costs of capital, output-dependent product prices, and the possibility of investments into older vintages. They show the existence of negative anticipation effects in the investment before 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 endogenous capital lifetime and investment (before and after the shock time). The economic and mathematical novelty lies in the investigation of the complete optimal dynamics of a VCM with endogenous scrapping under both continuous and discontinuous TC. To obtain meaningful results, we consider a VCM with partial equilibrium setup, Leontief technology, constant returns to scale, a price-taking firm, no learning, no adoption cost or adjustment cost. The obtained nonlinear optimal control problem allows us to find its exact solutions in special cases. The interpretation of the constructed solutions reveals that the adjustment of optimal model dynamics to a TC shock essentially happens before the shock time in the form of direct and echoed anticipation effects. Follow the mainstream of VCMs (Solow et al., 1966; Benhabib and Rustichini, 1993; Boucekkine et al., 1997; Boucekkine et al., 1998; Cooley et al., 1997; Hritonenko and Yatsenko, 1996; Yorukoglu, 1998; Jovanovic and Tse, 2010), our model is deterministic and assumes a perfect foresight, which means that the whole evolution of future technology is already known at the present time. This purely theoretical assumption allows obtaining qualitative conclusions about the rational response of economy to technological advances. The novelty is that this future evolution may include some technology shocks whose arrival times are known. A forthcoming technological breakthrough on macroeconomic level is obviously a big event and can be predicted with certain accuracy. The last macro technology shock was related to IT. Currently, applied researchers argue that the next macroeconomic productivity shock based on renewable energy is forthcoming (Becerra-Lopez and Golding, 2007; Schmidt and Marschinski, 2009). On a specific firm’s level, it is also practically feasible to concentrate on only 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 that our mathematical technique can handle several shocks (as shown in Section 4.3). A possible extension of our deterministic model 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 their levels and/or growth rates (Boucekkine et al., 1998; Pakko, 2002). Boucekkine et al. (1998) analyze both the technology level shocks and shocks in TC rate and point out a crucial difference in generated optimal dynamics. Similar explorations in Operations Research (Rogers and Hartman, 2005; Feichtinger et al., 2006) consider technological breakthroughs as discontinuities in 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 the TC rates is that the optimal capital lifetime depends on the TC rates rather than TC levels (the optimal lifetime is infinite when 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 data and commonly used in the macroeconomic VCM literature. Greenwood and Yorukoglu (1997) clearly identify such macroeconomic 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 important differences between 1995–2000 and 2000–2005. They argue that “aggregate data conceal striking variations among industries and prevent analysts from tracing the evolution of productivity to its industry sources” and that “. . . only at the industry level . . . production analysts can seek to understand the specific changes in technology and choices that the firms make in response. . .”. Most recently, Martínez et al. (2010) identify the TC embedded in computer hardware as the main leading force in US productivity growth and estimate the IT-specific embodied TC in the US as 0.62% for 1980–1994, 3.2% for 1995–2004. 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 analytically and 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 investigates a more realistic case of the exponential TC and demonstrates analytically and numerically that the optimal dynamics under technological breakthroughs remains similar to the model case. Section 4 briefly discusses several possible extensions of the model. Particularly, breakthroughs in industry-specific productivity can be accompanied by jumps of the new capital price, which is addressed in Sections 4.1 and 4.2. Section 4.3 explores the case of several TC shocks. Section 5 discusses the obtained 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 and scrapping the oldest one. The dynamics of the firm can be efficiently described by vintage capital models with controlled scrapping lifetime. The economic objective is to maximize the present value of the firm profit over the infinite horizon:



∞

maxI =

m,a,y

e−rt [y(t) − p(t)m(t)] dt,

(1)

t0

under the constraints-equalities:



t

y(t) =

ˇ(, t)m() d,

(2)

a(t)



t

m() sd = R(t),

(3)

a(t)

the constraints-inequalities: m(t) ≥ 0, 

a (t) ≥ 0, m(t) ≤

(4) a(t) ≤ t,

y(t) , p(t)

(5)

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 time a(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 vintage production function (2) where the capital and resource are complements. Inequalities-constraints (4) and (5) reflect the irreversibility 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 given model functions should satisfy



t0

p(t0 )m0 (t0 ) ≤

ˇ(, t0 )m0 () d. a0

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



∞

t0



e−rt ˇ(t, t)R(t) dt < ∞,

∞

e−rt p(t) dt < ∞

t0

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 future technological 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 the given amount R(t) of a certain resource (labour, land, financial budget, environmental pollution, etc.). Such assumptions are much 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 first one is practical; such constraints are becoming more regular in a firm management. A relevant example is a quota on energy consumption or, equivalently, on carbon emissions. The second reason is methodological. The VCM (1)–(7) has the simplest setup in the partial equilibrium vintage modeling framework (Leontief technology, constant returns to scale, a price-taking firm, no learning, no adoption cost or adjustment cost), which ensures a balanced growth. If we assume R(t) in (3) to be endogenous, then the given resource price pR (t) should appear in the objective function (1) as



∞

max I =

m,a,y

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

(8)

t0

Then, as shown by Boucekkine and Pommeret (2004) for energy-saving TC, a balanced growth will occur only at a specially chosen price pR (t) that grows with the same rate as TC. Next, if the resource price is involved into the model, it can be endogenized by general-equilibrium reasoning. As demonstrated in Section 2.2, a natural general-equilibrium set up for VCMs leads to the same optimal scrapping rule (11) as in our model. Thus, model (1)–(7) gives a stylized model setup that is 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 Boucekkine et al. (1997, 1998) and Hritonenko and Yatsenko (1996, 2005, 2008a) and possesses remarkable properties that simplify its theoretical 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 (Hritonenko and Yatsenko, 1996, 2005, 2008a), we choose m as the decision variable (independent control) of the OP, then y and a are the dependent (state) variables expressed via m. The inequality (4) is a standard constraint on the control m, whereas (5) and ∞ [t , ∞), then the unknowns a and y in (1)–(7) are (6) are constraints-inequalities on the state variables a and y. Let m ∈ Lloc 0 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) =

I  (t) = 0 at mmin (t) < m∗ (t) <

y∗ (t) , p(t)

where

 

I (t) =

a−1 (t)

y∗ (t) , p(t)

(9)

t ∈ [t0 , ∞),

e−r [ˇ(t, ) − ˇ(a(), )] d − e−rt p(t)

(10)

t

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 of a(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 control m. 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 some t ∈  ⊂ [t0 ,∞) and prove that then I (t) ≥ 0. Giving small variations ım(t), ıy(t), t ∈ 1 ⊂ , mes(1 ) 1, to m* and y* , we obtain 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) and the 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 =

∞ t0

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

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 of m* (t) = y* (t)/p(t). As shown in Hritonenko and Yatsenko (2008b), if ˇ(, t) strictly increases in , the functional (1) is concave in 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 necessary and sufficient condition and solves the issue of solution existence. Indeed, if a function m* satisfies Lemma 1, then it is an OP solution. Also, the OP concavity also means that the OP solution m* is unique and, therefore, delivers the global optimum to the 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)

e−r [ˇ(t, ) − ˇ(˜a(), )] d = e−rt p(t),

t ∈ [t0 , ∞)

(11)

t

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)

e−r [ˇ(t, ) − pR ()] d − e−rt p(t) = 0,

(12)

t

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, the trajectory 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 rational economic behavior of a representative household, final product sector, energy sector, and government. The embodied technological 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 the extremum condition (9) and (10). 2.2.3. Another general-equilibrium VCM that leads to the same optimal scrapping rule (11) is more recent model of Jovanovic and Tse (2010) with output-augmenting TC. The general-equilibrium setup is quite different from the one in 2.2.2 and involves an elastic consumer’s demand output curve. The model of Jovanovic and Tse (2010) uses a maintenance cost as an expense instead of the energy price pR in Azomahou et al. (2009) but produces the same extremum condition (11) for the 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 interior scrapping 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) = b0 ecb −cd (t−) ,

p(t) = p0 ecp t ,

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

(14)

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



(r + cd )e−(c+cd )A − (c + cd )e−(r+cd )A = (r − c) 1 −



(r + cd )p0 b0

1069

 (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 for a0 > 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 , ∞), by the definition of a˜ , and the derivative (10) can be written as

 I  (a∗ , t) =



 e−r [ˇ(t, ) − ˇ(a∗ (), )] d +

t

 =

e−r [ˇ(t, ) − ˇ(˜a(), )] d − e−rt p(t)

 



a∗−1 (t)

e−r [ˇ(˜a(), ) − ˇ(a∗ (), )] d +

e−r [ˇ(t, ) − ˇ(˜a(), )] d − e−rt p(t)

t

t



a∗−1 (t)

a∗−1 (t)

>

e−r [ˇ(t, ) − ˇ(˜a(), )] d − e−rt p(t)

t

 >

a˜ −1 (t)

e−r [ˇ(t, ) − ˇ(˜a(), )] d − e−rt p(t) = I  (˜a, t) = 0,

t ∈ [t0 , ).

(17)

t

Hence, I (a* , t) > 0 at t ∈ [t0 , ) for any m* . Therefore, m* (t) should be maximum possible at t ∈ [t0 , ). By (9), the maximal 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 two nonlinear integral Eqs. (2) and (3). Similar to (Hritonenko and Yatsenko, 1996), we can prove that this system possesses a unique 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) ≡ 0 for t ∈ [, ∞). The theorem is proven.  The optimal investment trajectory m* (t) is boundary during the transition dynamics [t0 ,]: m* (t) = mmin (t) or m* (t) = y* (t)/p(t) is maximal. By (16), the initial (boundary) dynamics of m* on [a0 , ] is replicated throughout the long-term horizon [, ∞) following the formula m∗ (t) = R (t) + m(˜a(t))˜a (t). This effect is known as the replacement echoes (Boucekkine et 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) is strictly periodic. Theorem 1 does not prevent the constraint-inequalities (4)–(6) from being saturated (binding) on certain parts of the long-term dynamics interval [, ∞). For instance, if a0 > a˜ (t0 ) in addition to a˜ (t) = t − A and R = 0, then, by Case 1, the optimal 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 the replacement 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 the endogenous 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) to satisfy (4) and (6). Shocks (sharp changes) in the dynamics of ˇ(, t), R (t), and p(t) can violate constraints (4)–(6) and the m* (t)

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 the optimal 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 nonsmooth functions ˇ and p. First, we focus on the long-term optimal dynamics, which appears to be much more complex than described by Theorem 1. To analyze the long-term dynamics, we have to investigate the possibility of solving the nonlinear integral Eq. (11) at non-smooth functions ˇ(, t) and/or p(t). One can show that the transition dynamics remains the same as in Theorem 1. In general, both capital efficiency ˇ(, t) and capital price p(t) can experience TC shocks at certain times. Possible technological 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 as discontinuities in the level variables (see Rogers and Hartman, 2005 and the references therein). However, as discussed in Introduction, the breakthrough shocks in levels do not appear in macroeconomics, where shocks in the growth rates are more 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 In Section 3.1, we identify a model case (the linear background TC), when an analytic solution of the OP can be constructed. In Section 3.2, we consider the more realistic exponential TC and demonstrate analytically and numerically that the structure of optimal trajectories remains essentially similar to the model case. The breakthroughs in productivity can be accompanied by 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, C2 ( − t¯ ) + b,

 ≤ t¯ ,  > 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-term dynamic component a* coincides with the turnpike



a˜ (t) = t − A,

A=

2p > 0, C2

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 trajectory for analyzing the impact of TC shocks. Here we investigate the case C1 < C2 . Then Eq. (11) has the following form:

⎧  a−1 (t) ⎪ ⎪ [t − a()] d = p, a−1 (t¯ ) ≤ t < ∞, C2 ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪  a−1 (t¯ ) ⎨  a−1 (t) (t − t¯ ) d − C1

C



a−1 (t)

[a() − t¯ ] d − C2

2 ⎪ ⎪ t¯ t ⎪ ⎪ ⎪  a−1 (t) ⎪ ⎪ ⎪ ⎩ C1 [t − a()] d = p, t < t¯ .

[a() − t¯ ] d = p,

t¯ ≤ t < a−1 (t¯ ),

(20)

a−1 (t¯ )

t

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 equations from right to left. To analyze the solution structure, let us differentiate (20): t − a(t) = a−1 (t) − t, a(t) − t¯ =

C2 [2t − a−1 (t) − t¯ ], C1

t − a(t) = a−1 (t) − t,

1

t ∈ [ a−1 (t¯ ), ∞), t ∈ [t¯ , a−1 (t¯ )],

t < t¯ .

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

(21) (22) (23)

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 line on [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 this process, 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¯ − C2 A , (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 continuous solutions 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 which a = 0) to construct the continuous optimal trajectory a* . Analogously, we prove the following theorem. Theorem 2. form

a∗ (t) =

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

⎧ a˜ (t), I  (a, t) = 0, t ∈ [a−1 (t¯ ), ∞) ⎪ ⎪ ⎪ ⎨  −1 a1 (t),

I (a, t) = 0,

t ∈ [ˇ1 , a

⎪ ai (˛i ), I  (a, t) < 0, t ∈ [˛i , ˇi ) ⎪ ⎪ ⎩  ai+1 (t),

I (a, t) = 0,

(t¯ ))

,

i = 1, 2, . . . , t ∈ [m0 , ∞),

(25)

t ∈ [ˇi+1 , ˛i )

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

  C k(C − C ) + C 2 2 1 1

˛k = t¯ + A

C2 C1 −A k+ C2 − C1 C2 − C1

ˇk = t¯ + A

C2 C1 −A k−1+ C2 − C1 C2 − C1

C1 k(C2 − C1 ) + C2

,

  C k(C − C ) + C 2 2 1 2 C1 k(C2 − C1 ) + C1

(26)

,

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 + t¯ at 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), the trajectory 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 echoes caused 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 2C2 − C1 C2 − C1 C2 (ˇ1 − A) − t¯ = ˛1 − t¯ − A, C1 C1 C2 C2

(28)

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



ˇ1



C1

˛1 −

˛1

C2 C2 − C1 (ˇ1 − A) − t¯ C1 C1





a1 −1 (˛1 )

d + C1 ˇ1



˛1 −

C2 C2 − C1 ( − A) + t¯ 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 be shown 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) jumps down 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) for the new one. Since ak+1 (˛k ) = ak (ˇk ), we can express ˇk as a function of ˛k : ˇk =

 ((k + 1)C − kC )((k − 1)C − (k − 2)C ) 2 1 2 1

C2 − C1 (kC2 − (k − 1)C1 )

C2 − C1

2



˛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) is uniquely determined. Relation (27) follows from (26), (30), and I (a* , t) < 0 on (˛i , ˇi ). By Lemma 1, the condition I (a* , t) ≤ 0 is 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 intervals as anticipation echoes (see also anticipation waves in Feichtinger et al., 2006) because they are caused by the anticipation of a future technological breakthrough, that is, the future jump in the efficiency function ˇ() at some instant t¯ . These echoes propagate backward throughout the interval [t0 , t¯ ] and become smaller as t¯ − t increases. The adjustment of the long-term optimal 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 = large t¯ − t0  1, a(t) strives to a˜˜ (t) as t¯ − t increases.



2p/C1 , at t < t¯ . At

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() =

B, P > 0,

Bec1 (−t¯ ) , Bec2 (−t¯ ) ,

 ≤ t¯ ,  > t¯

0 < c1 ≤ c2 < r,

p(t) = Pecp ,t ,

(31)

cp < r.

In fact, (31) reflects the main case of the TC shocks discussed in the macroeconomic VCM literature. Greenwood and Yorukoglu (1997, p. 76) explicitly identified such shock in the US economy at t¯ = 1974 as c1 ≈ 3% and c2 ≈ 5%. Successive researchers 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 be remarkably 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 numeric methods. In the case c1 = c2 with no jump, the solution (a* , m* ) of the OP is described by Theorem 1. Its long-term dynamic component a* 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:

⎧  a−1 (t) ⎪ ⎪ ⎪ Be−c2 t¯ e−r [ec2 t − ec2 a() ] d = e−rt Pecp t , a−1 (t¯ ) ≤ t < ∞, ⎪ ⎪ ⎨ t [1 − e

−r[a−1 (t)−t]

]

Bec1 [a(t)−t¯ ] = Bec2 [t−t¯ ] − Bc2 ec2 [t−t¯ ] + (cp − r)Pecp t , t¯ ≤ t < a−1 (t¯ ), ⎪ ⎪ r −1 ⎪ ⎪ −r[a (t)−t] ] ⎪ ⎩ Bec1 [a(t)−t¯ ] = Bec1 [t−t¯ ] − Bc ec1 [t−t¯ ] [1 − e + (cp − r)Pecp t , t < t¯ . 1 r

(32) (33) (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 curve is 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 because c1 > 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-valued trajectories 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 the long-term trajectory in the sense of Theorem 1. In general, a result similar to Theorem 2 is valid in the piecewise-exponential case (31), but its proof is considerably more challenging.2 In this paper, we will employ numeric simulation to explore the structure of OP solutions. As the basic input data set, we use the numeric example identified in Hritonenko and Yatsenko (2009) for capital replacement at a “typical” US manufacturing plant. It includes the exponential TC (31) with the rates c1 = 0.08 and cp = 0–0.09, the interest rate r = 0.1, and the initial productivity/cost ratio B/P adjusted to produce the capital lifetime t − a(t) of approximately 24 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 capital lifetime t − a˜ (t) decreases at c1 > cp and increases at c1 < cp . We have simulated the optimal dynamics for several scenarios c1 > 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 trajectory a* 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 decreasing lifetime (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 an interval 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 preceding the 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 cost rate 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) after t¯ 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 the optimal 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 anticipation 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 regeneration periods 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 growth path 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 the replacement echoes phenomenon. The optimal investment is of non-monotonic bumpy character and does not converge to 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¯ ) , Pecp2 (t−t¯ ) ,

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 simulation 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 demonstrated structure 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) and the 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 and illustrate 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 ≤ 60 , ec(t−t¯ ) , t > 60



ˇ() = B

e0.05(−t¯ ) , t ≤ 60 . ec(−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 at c1 = cp1 = c2 = cp2 (if there is no jump). In the cases with jumps, we have two turnpikes with constant lifetimes, one for the interval 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 rates are 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 constant lifetime 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 t¯1 = 60 and t¯2 = 90 with the efficiency rate c1 = 0.08 before t¯1 , c2 = 0.09 after t¯1 , and c3 = 0.1 after t¯2 .

easily see the character of the convergence of the optimal lifetime to the turnpikes. If there is no jump, then both turnpikes are 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 in Introduction, the case of one major forthcoming breakthrough is of most applied interest. Nevertheless, our analytic and numeric 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 several TC 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 by later shocks. Let us illustrate the solution structure in the situation of a piecewise-exponential ˇ(, t) = ˇ() with two shocks and an exponential p(t):

ˇ() =

⎧ c1 (−t¯ ) ,  ≤ t¯1 ⎪ ⎨ Be ⎪ ⎩

B, P > 0,

Bec2 (−t¯ ) ,

t¯1 <  ≤ t¯2 ,

Bec3 (−t¯ ) ,

 > t¯2

0 < c1 < c2 < c3 < r,

p(t) = Pecp t ,

(37)

cp < r.

The optimal dynamics of the OP (1)–(7), (37) is numerically simulated for the shocks at t¯1 = 60 and t¯2 = 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 at t¯2 are indicated by solid arrows while the echoes caused by the earlier shock at t¯2 are indicated by dashed arrows. Similarly to the cases with one TC shock (Section 3), the anticipation echoes in capital lifetime caused by every future shock deteriorate when the distance to the shock increases, so, the optimal lifetime remains stable with respect to possible TC 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

2.

3.

4.

5.

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) and the cost of new capital. The technological breakthroughs are modeled as jumps in the TC growth rates, which are observed in vintage literature starting with Greenwood and Yorukoglu (1997). Another reason for modeling breakthroughs as jumps in the TC rates is that the optimal capital lifetime depends on the TC rates rather than TC levels. This is a fundamental feature of 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). An exact solution of the constructed nonlinear optimal control problem is obtained in the case of piecewise-linear TC and its structure is analytically and numerically confirmed in the case of piecewise-exponential TC. The interpretation of these solutions reveals how the technology shocks affect the optimal dynamics of endogenous capital lifetime and investment before and after the shock time. Namely, the optimal regime is no investment and no scrapping on a set of anticipation intervals (echoes) of increasing length during the period preceding the TC shock. The rightmost and largest anticipation interval includes the shock instant. The optimal lifetime before the shock strives to a balanced growth path as the distance from the jump increases and coincides with the new balanced growth path after the shock. Our results develop and extend the analysis by Boucekkine et al. (1998) of unanticipated permanent TC shocks in macroeconomic vintage models with endogenous capital scrapping. They demonstrate an increase of the optimal scrapping age of vintages and investment decrease after the shock. It is specifically pointed that “the productivity slowdown comes from the fact that the economy keep on using the initial machines for a while after the occurrence of the shock” (p. 375). We show that the optimal reaction to a future TC shock essentially happens before the shock time t¯ in the form of the abovedescribed direct and echoed anticipation effects in optimal capital lifetime. More specifically, the anticipation effects in the capital lifetime completely stop after the time a−1 (t¯ ) while the effects in investments disseminate up to ∞ because of the replacement echoes phenomenon. The stylized VCM (1)–(7) with endogenous lifetime does not possess any cushioned effect of nonlinear adjustment costs, market power, or decreasing returns to scale and, therefore, produces everlasting investment echoes. In Boucekkine et al. (1998), the echoes in the optimal investment eventually deteriorate because of nonlinear utility. On the other side, our results complement the outcomes obtained by Pakko (2002) and Feichtinger et al. (2006) about negative anticipation effects in investment for vintage models without endogenous scrapping. Pakko (2002) numerically demonstrates a decline in investment, output and employment before and after a stochastic permanent shock in embodied TC rate because agents anticipate higher returns in the future (a direct anticipation effect). Feichtinger et al. (2006) obtain a similar direct anticipation effect in response to a technology shock in a VCM with a fixed capital lifetime and investments into 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 firm with no market power. These results do not intercept with ours because we analytically prove anticipation effects for the optimal capital lifetime that is taken as given by Feichtinger et al. (2006). Also, they consider a shock in the technology level rather than in the TC rate. All these results indicate the existence of a certain fundamental anticipation property in economic systems. The anticipation echoes seem to represent a general structural pattern of the optimal economic policy in response to various future disturbances in the model (discontinuities of exogenous parameters, the end of planning horizon, and so on). The anticipation 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 of planning horizon (see also Hritonenko and Yatsenko, 2005, 2008a). Specifically, a zero-investment period at the horizon end generates a chain of zero-investment anticipation echoes. Also, similar echoes appear in the problem (1)–(7) with piecewise-linear utility function (Hritonenko and Yatsenko, 2006).

Acknowledgements The authors express their gratitude to two anonymous referees and participants of the Workshop on Dynamics, Optimal Growth and Population Change: Theory and Applications (Milan, September 2008) and the AIM Workshop on Stochastic and Deterministic Spatial Modeling in Population Dynamics (Palo Alto, California, May 2009). Special thanks are to Raouf Boucekkine (Belgium), Cuong Le Van (France), and Alberto Bucci (Italy) for their interest and useful remarks. The work of Yuri Yatsenko was supported in part by Center for Operations Research and Econometrics (Louvain-la-Neuve, Belgium). Natali Hritonenko 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, Université catholique 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: Natural Resources 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.