ارزش فعلی خالص مورد انتظار، ارزش آینده خالص مورد انتظار، و قاعده رمزی

Weitzman [1] showed that when future interest rates are uncertain, using the expected net present value implies a term structure of discount rates that is decreasing to the smallest possible interest rate. On the contrary, using the expected net future value criteria implies an increasing term structure of discount rates up to the largest possible interest rate. We reconcile the two approaches by introducing risk aversion and utility maximization. We show that if the aggregate consumption path is optimized and made flexible to news about future interest rates, the two criteria are equivalent. Moreover, they are also equivalent to the Ramsey rule extended to uncertainty.

The long term sustainability of economic growth is a crucial topic for public opinions. Still, there is much disagreement in academia about how future generations should be treated in the evaluation of public policies. Indeed, the classical cost–benefit methodology is often criticized for sacrificing distant generations. This raises questions about the level and term structure of the discount rate. There is no consensus in our profession about the rate that should be used to discount long term costs and benefits. This implies that economists disagree fundamentally about the intensity of our effort to improve the distant future. This is best illustrated in the field of climate change, but other applications exist in relation to nuclear wastes, biodiversity, genetically modified organisms, or to the preservation of natural resources. Stern [2], who implicitly uses a discount rate of 1.4% per year, estimated the current social cost of carbon around 85 dollars per ton of CO2. Nordhaus [3] criticizes the low discount rate used by Stern, and recommend alternatively a discount rate of 5%, which leads to an estimation of 8 dollars per ton of CO2. This huge difference in the estimation of the social cost of carbon yields radical discrepancies in the policy recommendations related to global changes. This is due to the exponential nature of the impact of a change of the discount rate on net present values (NPV). For example, using a rate of 5% rather than 1.4% to discount a cash flow occurring in 200 years reduces its NPV by a factor 1340. A standard arbitrage argument justifies using the rate of return of capital, hereafter denoted ρρ, in the economy as the socially efficient discount rate. Indeed, diverting productive capital to invest in environmental projects with an internal rate of social return below ρρ would reduce the welfare of future generations. Another argument is based on the NPV. At equilibrium, the interest rate in the economy equals ρρ. Suppose that one borrows today at rate ρρ an amount equal to the present value of future cash flows, which implies that these cash flows will be just enough to reimburse the initial loan. This converts all future costs and benefits into present costs and benefits. Thus, the project should be implemented only if its NPV discounted at rate ρρ is positive. However, this argument requires knowing the returns of capital for the different maturities of the cash flows of the environmental projects under scrutiny. Obviously, these future returns are highly uncertain for the time horizons that we have in mind when we think about climate change or nuclear waste. Weitzman [1] and [4] has developed a simple argument based on this fact to recommend a smaller rate to discount distant impacts. The NPV is a decreasing convex function of the interest rate, and the degree of convexity of this function is increasing in the time horizon. Therefore, by Jensen's inequality, introducing an uncertain permanent shock on future interest rates raises the expected NPV, and this effect is increasing in the maturity. It implies that the uncertainty on ρρ has an effect on the expected NPV that is equivalent to a sure reduction in ρρ. It also implies that the term structure of this reduction is increasing. Newel and Pizer [5] and Groom et al. [6] estimated the impact of the uncertainty about future interest rates on the socially efficient discount rate for different time horizons. More recently, Gollier et al. [7] estimated the term structure based on the uncertainty of the GDP-weighted world interest rates. They obtained discount rates of 4.2%, 2.3% and 1.8%, respectively, for the short term, 100 years and 200 years. We compare this approach with two alternative approaches. The first alternative approach is exactly symmetric to the one proposed by Weitzman [1]. It measures the impact of the uncertainty on the expected net future value (NFV), as described for example by Gollier [8], Hepburn and Groom [9], Freeman [10] and Buchholz and Schumacher [11]. Rather than assuming that all costs and benefits of the environmental project are transferred to the present, it assumes that they are all transferred to the terminal date of the project. Under this alternative approach the uncertainty on future interest rates raises the discount rate in an increasing way with the time horizon. The second alternative approach consists in making no intertemporal transfer of costs and benefits, which implies that consumption is modified to compensate them in real time. Under this approach, one needs to compare the impact of uncertain changes in consumption at different dates on the intertemporal welfare. Ramsey [12] derived a simple formula that equalizes the efficient discount rate net of the rate of pure preference for the present to the product of the growth rate of GDP by the index of aversion to (intertemporal) consumption inequality. Gollier [13] and [14] and Weitzman [15] extended this formula to take account of the uncertainty on the growth rate of GDP, which is itself linked to the uncertainty of future interest rates. They show that the term structure of the efficient discount rate is flat when shocks on the growth rate are temporary, and is decreasing when shocks have a permanent component. These results appear to conflict with one another. The aim of this paper is to unify them in a single framework under uncertainty. We show that these approaches are equivalent once risk aversion is properly integrated into the model. We make explicit the investor's objective function, which is assumed to be the standard expected utility criterion. We show how to generalize the discount rates based on the NPV and NFV rules to the case of risk aversion. In fact, we show that the expectations considered above must be based on risk-neutral probabilities, which are proportional to the marginal utility of consumption at the evaluation date. This is in line with the standard consumption-based methodology in finance to estimate risk premia. Introducing risk aversion alone does not solve the puzzle. One must also require the consumption path is optimized. Under the condition that the investor optimizes her consumption plan contingent to the observed interest rate, we show that the NPV and NFV approaches lead to exactly the same term structure of discount rates, which is decreasing and tends to the lowest possible interest rate. Moreover, we show that the two approaches are also perfectly compatible with the well-known Ramsey rule. This paper is mostly non-technical, in contrast to Gollier [16]. This companion paper also shows that the decreasing nature of the term structure obtained in this framework depends heavily upon the assumption that shocks are permanent. If they are purely transitory, the term structure of discount rates should be flat.

We have shown that the choice of the discount rate for marginal projects depends in general upon how the benefits and costs of the project are allocated through time. This explains the differences between the approaches examined, respectively, by Ramsey [12], Weitzman [1] and Gollier [8]. The Ramsey rule is obtained by assuming that the cash flows of the investment are consumed at the time at which they occur. In Weitzman's NPV approach, it is assumed that all costs and benefits are consumed at date 0, whereas they are all consumed at the terminal date in Gollier's NFV approach. Of course, the risk associated to the various financing strategies must be taken into account given the risk aversion of the representative agent. This must be done by using risk-neutral probability distributions that vary with the evaluation date. Our main contribution is to show that these different approaches to discounting are completely equivalent once we recognize that consumers react optimally to changes in the interest rate. Ramsey, Weitzman and Gollier's approaches lead to the same term structure of discount rates when consumption paths are optimal. When shocks on future interest rates have a permanent component, the discount rate has a decreasing term structure. It tends to the smallest possible rate for very distant time horizons. When applied to climate change, this conclusion tends to go in favor of the low implicit discount rate used in the Stern Review [2], and therefore in favor of a relatively large price for carbon. When consumption plans cannot be fully optimized, the allocation of a project's net benefits will not be neutral when evaluating the project. This will affect the discount rate that should be used to evaluate environmental projects. It is hard to believe that our collective consumption trajectory that span many different generations over the next few centuries is socially efficient given the uncertainty we face today. Obvious conflicts of interest and rigidities inhibit the economy's ability to adapt rapidly to news about the future productivity of capital and about the future growth of the economy. Thus, we conclude that one cannot separate the problem of the choice of the rate to discount the cash flows of an environmental project from the question of how it will be financed and of who will cash the benefits or pay the costs.