Discrete-continuous analysis of optimal equipment replacement

In Operations Research, the equipment replacement process is usually modeled in discrete time. The optimal replacement strategies are found from discrete (or integer) programming problems, well known for their analytic and computational complexity. An alternative approach is represented by continuous-time vintage capital models that explicitly involve the equipment lifetime and are described by nonlinear integral equations. Then the optimal replacement is determined via the optimal control of such equations. These two alternative techniques describe essentially the same controlled dynamic process. We introduce and analyze a model that unites both approaches. The obtained results allow us to explore such important effects in optimal asset replacement as the transition and long-term dynamics, clustering and splitting of replaced assets, and the impact of improving technology and discounting. In particular, we demonstrate that the cluster splitting is possible in our replacement model with given demand in the case of an increasinTheoretical findings are illustrated with numeric examples.vintage capital models, optimization, equipment lifetime, discrete-continuous models.

Technological breakthroughs and asset replacement

The authors analyze the optimal replacement of assets under continuous and discontinuous technological change. They investigate the variable lifetime of assets in an infinite-horizon replacement problem. Due to deterioration, the maintenance cost increases when the asset becomes older. Because of technological change, both maintenance and new capital costs decrease for a fixed asset age. The dynamics of the optimal lifetime is investigated analytically and numerically under technological change in the cases of one and several technological breakthroughs. It is shown that the breakthroughs cause irregularities (anticipation echoes) in the asset lifetime before the breakthrough time.asset replacement, technological change, optimal lifetime, anticipation echoes.

Supercontinuum Spectrum Engineering in Ge-doped Microstructured Multicore Fiber

Supercontinuum generation in a multicore Ge-doped microstructured optical fiber is experimentally studied. The output spectral extension and the presence of four-wave mixing peaks are strongly dependent on the level of doping.3 page(s

Scarcity, regulation and endogenous technical progress

This paper studies to which extent a firm using a scarce resource input and facing environmental regulation, can still manage to have a sustainable growth of output and profits. The firm has a vintage capital technology with two complementary factors, capital and a resource input subject to quota, the latter being increasingly scarce through an exogenously rising price. The firm can scrap obsolete capital and invest in adoptive and/or innovative R&D resource-saving activities. We show that there exists a threshold level for the growth rate of the resource price above which the firm will collapse. Below this threshold, two important properties are found. In the long-run, a sustainable growth is possible at a growth rate which is independent of the resource price. In the short-run, not only will the firms respond to increasing resource price by increasing R&D on average, but they will also reduce capital expenditures and speed up the scrapping of older capital goods. Finally, we identify optimal intensive Vs extensive transitional growth regimes depending on the history of the firms.Vintage capital, technological progress, dynamic optimization, Sustainability, scarcity, environmental regulation

Sustainable growth under pollution quotas: optimal R&D, investment and replacement policies

We consider an optimal growth model of an economy facing an exogenous pollution quota. In the absence of an international market of pollution permits, the economy has three instruments to reach sustainable growth: R&D to develop cleaner technologies, investment in new clean capital goods, and scrapping of the old dirty capital. The R&D technology depends negatively on a complexity component and positively on investment in this sector at constant elasticity. First, we characterize possible balanced growth paths for different parameterizations of the R&D technology. It is shown that countries with an under-performing R&D sector would need an increasing pollution quota over time to ensure balanced growth while countries with a highly efficient R&D sector would supply part of their assigned pollution permits in an international market without harming their long-term growth. Second, we study transitional dynamics to balanced growth. We prove that regardless of how large the regulation quota is, the transition dynamics leads to the balanced growth with binding quota in a finite time. In particular, we discover two optimal transition regimes: an intensive growth (sustained investment in new capital and R&D with scrapping the oldest capital goods), and an extensive growth (sustained investment in new capital and R&D without scrapping the oldest capital).Sustainable growth; vintage capital; endogenous growth; R&D; pollution quotas

Optimal firm behavior under environmental constraints

The paper examines the Porter and induced-innovation hypotheses in a firm model where : (i) the firm has a vintage capital technology with two complementary factors, energy and capital; (ii) scrapping is endogenous; (iii) technological progress is energy-saving and endogenous trough purposive R&D investment; (iv) the innovation rate increases with R&D investment and decreases with complexity; (v) the firm is subject to emission quotas which put an upper bound on its energy consumption at any date; (vi) energy and capital prices are exogenous. Balanced growth paths are first characterized, and a comparative static analysis is performed to study a kind of long-term Porter and induced-innovation hypotheses. In particular, it is shown that tighter emission quotas do not prevent firms to grow in the long-run, thanks to endogenous innovation, but they have an inverse effect on the growth rate of profits. Some short-term dynamics are also produced, particularly, to analyze the role of initial conditions and energy prices in optimal firm behavior subject to environmental regulation. Among numerous results, we show that (i) firms which are historically “small” polluters find it optimal to massively pollute in the short run : during the transition, new and clean machines will co-exist with old and dirty machines in the productive sectors, implying an unambiguously dirty transition; (ii) higher energy prices induce a shorter lifetime for capital goods but they depress investment in both new capital and R&D, featuring a kind of reverse Hicksian mechanism.matching problem, von Neumann-Morgenstern stable sets, farsighted stability

Optimal Firm Behavior under Environmental Constraints

The paper examines the Porter and induced-innovation hypotheses in a firm model where: (i) the firm has a vintage capital technology with two complementary factors, energy and capital ; (ii) scrapping is endogenous; (iii) technological progress is energy-saving and endogenous through purposive R&D investment; (iv) the innovation rate increases with R&D investment and decreases with complexity; (v) the firm is subject to emission quotas which put an upper bound on its energy consumption at any date; (vi) energy and capital prices are exogenous. Balanced growth paths are first characterized, and a comparative static analysis is performed to study a kind of long-term Porter and induced-innovation hypotheses. In particular, it is shown that tighter emission quotas do not prevent firms to grow in the long-run, thanks to endogenous innovation, but they have an inverse effect on the growth rate of profits. Some short-term dynamics are also produced, particularly, to analyze the role of initial conditions and energy prices in optimal firm behaviour subject to environmental regulation. Among numerous results, we show that (i) firms which are historically “small” polluters find it optimal to massively pollute in the short run: during the transition, new and clean machines will co-exist with old and dirty machines in the productive sectors, implying an unambiguously dirty transition; (ii) higher energy prices induce a shorter lifetime for capital goods but they depress investment in both new capital and R&D, featuring a kind of reverse Hicksian mechanism.Vintage capital, R&D, Emission quotas, Porter hypothesis, Induced innovation hypothesis, Optimization