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.

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

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 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

Estimating the Dynamics of R&D-based Growth Models

Several R&D-based models of endogenous economic growth are investigated under the Solow-like assumption of fixed allocation of resources across activities. We identify model parameters that lead to explosive dynamics and analyse various economic techniques to avoid it. The techniques include adding stricter constraints on model trajectories and limiting factors in technology equation. In particular, we demonstrate that our vintage version of the well-known R&D-based model of economic growth (Jones, 1955) exhibits the same balanced dynamics as the original modelVintage capital models, Endogenous technological change, R&D investment, Explosive dynamics, Nonlinear Volterra integral equations

Adaptation and mitigation in long-term climate policies

The paper analytically explores the optimal policy mix between mitigation and environmental adaptation against climate change at a macroeconomic level. The constructed economic- environmental model is formulated as a social planner problem with the adaptation and abatement investments as separate decision variables. The authors prove the existence of a unique steady state and provide a comparative static analysis of the optimal investment. It leads to essential implications for associated long-term environmental policies. In particular, the dependence of the optimal ratio between abatement and adaptation investments on economic efficiency appears to have an inverted U-shape. Data calibration and numerical simulation are provided to illustrate theoretical outcomes.environmental adaptation, mitigation, optimal investment, long-term climate policies

Investment, replacement and scrapping in a vintage capital model with embodied technological change

This paper analyzes and compares two alternative policies of determining the service life and replacement demand for vintage equipment under embodied technological change. The policies are the infinite-horizon replacement and the transitory replacement ending with scrapping. The corresponding vintage capital models are formulated in the dynamic optimization framework. These two approaches lead to different estimates of the duration of replacements and the impact of technological change on the equipment service life.vintage capital equipment; embodied technological change; service life; replacement; scrapping