Adapting to Climate Change

This paper examines the optimal time to adapt to climate change. We take the perspective of a farmer growing a crop in a stochastic environment. The farmer faces stochastic seasonal growth, which makes harvest at the end of any season a random variable. Within a season, crop biomass is assumed to grow according to a continuoustime Itô process. The standard deviation rate of the Itô process is itself stochastically evolving, season to season, as a result of climate change. We assume the seasonal standard deviation rate follows a discrete-time random walk, with positive drift. As the seasonal standard deviation rate grows, expected biomass at harvest, and thus revenue, declines. The farmer has the option to make an investment, say in an irrigation system, which will reduce the seasonal standard deviation rate. The investment in irrigation has a fixed cost and also results in higher cultivation costs during a season. The question becomes “How large must seasonal variation become before it is optimal to make the investment and adapt to climate change?” *Jon M. Conrad is a professor and Koji Kotani is a Ph.D. candidate in the Department of Applied Economics and Management at Cornell University. Correspondence should be addressed to Professor Conrad at 455 Warren Hall, Cornell University, Ithaca, New York, 14853, or by email to [email protected]. 1 Adapting to Climate Change Potential changes in the frequency, intensity, and persistence of climate extremes (for example, heat waves, heavy precipitation, and drought) and in climate variability, ... are emerging as key determinants of future impacts and vulnerability. Intergovernmental Panel on Climate Change (2001) Introduction and Overview By the year 2100, many of the scientists studying climate change believe that the increased concentration of greenhouse gases will lead to an increase in annual global mean surface temperature in the range of 1.4 to 5.8o C. Theory and mathematical models also project an increase in climate variability, specifically the frequency of “climate extremes.” It is the increase in climate variability that raises the greatest potential for adverse impacts within human (socioeconomic) and ecological systems. The vulnerability of different human populations and plant and animal species will depend on the speed of climate change and on the ability to adapt. Adaptation to climate change can take many forms. In agriculture, it may involve the adoption of later maturing cultivars, changing the mix of crops, or altering the timing of field operations [Kaiser et al. (1993)] . In the extreme, it may involve abandonment of land and human migration, as in the dust bowl of the U.S. Midwest during the 1930s or in Africa today. This paper is concerned with the timing of capital investments undertaken as a means of adapting to climate change. We assume that such investments will involve an initial capital cost and perhaps higher variable cost during cultivation after investment. It 2 is further assumed that the investment may be so specialized that, if not technically irreversible, it would be costly to reverse, and any scrap value would be a small fraction of the initial capital cost. Two investments come to mind. Irrigation equipment, adopted because of the increased frequency or duration of drought, might be removed and sold at a later date. Investment in better drainage, for a field now subject to higher levels of precipitation, might be costly to undo, and the drainage tile itself is likely to be worthless as scrap. We assume these investments are risky, in the sense that their use and value will still vary from season to season, and thus the future net return from making the investment is not known with certainty. A farmer, contemplating an investment to cope with climate change, is faced with the classic economic problem of risky, irreversible investment [Dixit and Pindyck (1994)]. The proper evaluation of such investments requires their analysis as real options [Trigeorgis (1996)]. This paper is organized as follows. In the next section we develop two models. The first is an infinite-horizon model, which might be appropriate for a corporate or family farm with the expectation of long-term operation. The second model is a finitehorizon model, which might be appropriate for a farmer with no heirs interested in continued farming, and with a plan for selling land and equipment at some future date as a source of retirement income. This section is followed by numerical analysis for a hypothetical farm growing a single crop. In the infinite-horizon model there is a single standard deviation rate that triggers the corporate or family farm to make the investment to adapt to a more variable climate. In the finite-horizon problem, with the farmer planning divestment prior to retirement, there is a schedule of critical standard deviation 3 rates which increase as the farmer nears retirement. Quite logically, the farmer is less interested in a costly investment to cope with a more variable climate the nearer he or she is to retirement. The final section gathers conclusions and suggests future lines of research. The Infiniteand Finite-Horizon Models Let X=X(t) denote the biomass of a crop during a growing season. We assume a continuous, intra-season, stochastic growth process given by € dX = rX(1−X K)dt + σsXdz (1) where r>0 is an intrinsic growth rate, K>0 is the maximum crop biomass at the end of a season if there were no stochasticity to growth, σs is the standard deviation rate during season s, and dz is the increment of a Wiener process. Solving the Kolmogorov forward equation for the steady state density of crop biomass, Merton (1975) and Dixit and Pindyck (1994) show that X(t) will have a gamma distribution as € t→∞ . The expected crop biomass, at the end of season s, may be approximated by € E{Xs} = K(1 −σs 2 (2r)) (2) By proper selection of the time step and appropriately scaling of the intrinsic growth rate, r, Equation (2) will provide a good approximation for expected harvest during season s, when the standard deviation rate is σs. Note, if σs increases from season to season, 4 expected harvest declines. In Figure 1 we show three realizations, from a sample of 10,000 realizations, where € X0 = 0.01, Δt=0.001, r=0.08, K=1, σs=0.05 and the growing season was T=120 days. The mean harvest, for all 10,000 realizations, was 0.8935, with a standard deviation of 0.1369; not significantly different from € K(1−σs 2 2r)=0.9844. Figure 1. Three Realizations of Crop Biomass Suppose, because of climate change, that the standard deviation rate is evolving according to a discrete random walk with drift as given by € σs+1 = μ + σs + εs+1 (3) 5 where μ>0 is the drift rate in the seasonal standard deviation of crop biomass and € εs+1 ~ N(0,σε 2 ). It can be shown that the distribution of σs+1, conditional on σs, is given by € f (σs+1 |σs ) = 1 2πσε es+1s )) 2 2σε 2 (4) At the start of season s, with r, K, and σs known, the farmer has an expected net revenue of € Ns = pK(1−σs 2 (2r)) − c (5) where p>0 is the per unit commodity price at harvest and c>0 is the cost of cultivation and harvest. Suppose the farm is owned by a corporation or a family with an expectation of multi-generational succession. Suppose further that the corporation or family can make an investment which will stabilize (fix) the seasonal standard deviation in crop growth at € σI < 2r . If the investment is adopted after season s and is in place for the start of season s+1, the expected discounted net revenue with the investment would be given by € N = Ns − I + ρ [pK(1−σI 2 (2r))− k] s=1 ∞ ∑ = Ns − I + [pK(1−σI 2 (2r)) − k] δ (6)