In this session, we're going to look at integrated assessment models, IAMs. What are models? How do they work? How do they differ from the net present value, the NPV approach, and how can they be used? These are some of the questions that we're going to try to answer in this session. To give concreteness to our discussion, we're going to look at one particular IAM, integrated assessment model, which is with DICE model. The DICE model is probably the best known of IAMs. It has been created by Professor Nordhaus, Nobel Prize winner exactly for his work with an integrated assessment model on the policy of climate change. He works at Yale University and his work has been highly influential. We will discuss, in the following sessions, the output of the DICE model and compare it with the output of another very well-known model, which is the model which was at the basis of the so-called Stern Report. But let's go one step at a time. What is a IAM, an integrated assessment model? Is an integrated, as the name suggests, combination of various different models which are supposed to cover the economics, the physics, the technology of climate change. It is extremely ambitious, and not surprisingly, given the extremely high bar of ambition which we have set for the module, all these different elements, the climate, the physics, the technology, have to be somewhat simplified, but not so simplified as to make them unusable in order to get real practical suggestions as to the optimal policy. All these things are integrated together to answer a number of policy related questions. For instance, there is one thing that I could do with a IAM. Give me an arbitrary schedule. For the moment it is arbitrary. An arbitrary schedule of future CO_2 emission. Give me a simplified model of the Earth's climate. So given this pattern, the schedule of emissions, I can gauge, given my knowledge of physics as I've put in the models that go into the IAM, what the future temperature will be and future climate in general will be. Then I can do a transformation from this future climate to economic damage. I can work out what losses to GDP that would entail and when in time these losses would occur. Notice that I started from an arbitrary future schedule of CO_2 emission. But actually, a IAM can do a lot more. With a IAM, I can optimize. What do I mean by that? Given a utility function, given what I call for the moment, a rate of impatience, and I'm going to clarify this term in a second, but bear with me. Given a utility function, a rate of impatience, and a simplified description of the climate and the economy, a IAM can provide the optimal policy to follow. What did I say a rate of impatience, what do they qualify? We discussed this at length when we looked at the social discount factor. Since we are looking at such long horizons, it is not really the rate of impatience of one particular investor regarding his or her own consumption, but it has more to do with intergenerational equity, so social time preference. In the future, in this session, for brevity, I will just use the word impatience, which is much shorter, but always keep this in mind. Also, very, very important to keep in mind, it is absolutely key that what we are discounting here, this impatience, always plays a role of a discount factor, is discounting utility, is not discounting consumption, we will see what that means. So as I said, when I put in these ingredients, when I put in the utility function, when I put in the impatience, and when I put in the simplified description of a universe basically, I get an optimal policy. What does optimal policy mean? Well, an optimal policy tells me such thing as when should I optimally invest? Should I spend a lot of money, and therefore, divert a lot of consumption today into climate change abating initiatives, or should I wait until later? What is the optimal carbon tax? How much should I invest today? When should I invest? Optimal means that given all the assumptions that I have made, there is no other solution that will give me a higher discounted possibly expected utility. I will explain later why I say possibly expected. IAM, an integrated assessment model used in full glory, maximizes a target function. The target function is the discounted possibly expected utility. If we want a simplified schematic view of how an integrated assessor model works, this is the picture. I have on the left-hand side, my inputs, my GDP, my population, my policies, other assumptions. They flow into the descriptive structural equations that describe the economy, the climate, the energy system, etc. The outputs are the economic outcomes, the emissions, the energy pathways, the land use, etc. It is staggering in a way in its ambition, but it is something very, very interesting, and we should really look at what these things deliver and what perhaps they fail to the deliver. In order to understand what a IAM does, it is important to distinguish between state variables, parameters, and control variables. What does this mean? Well, state variable, as the name suggests, describe the state of the system. They can be either deterministic, so I assume that today I know perfectly what the future part of the state variable, or they can be stochastic. I know some moments of the distribution over time, I know their expected value and variance, etc. But I don't know the exit value over time. The state variables can be either assigned from the outside, in which case their colleagues exogenous. They are derived and influence other parts of the model derived from an influencing other part of a model, in which case they're called endogenous. Examples, in DICE model, the population growth is exogenous, it's taken for granted. It might not be perfect, and we shall see later on potential criticism on this aspect, but it is how with DICE mobile is being constructed. The population growth is assumed to be known over time up to very, very far horizons. But temperature is another state variable, but it is endogenous because it is created by the actions that we take through in the development of the model, and at the same time, it influences other variables. The CO_2 concentration similarly is endogenous, in place there's double role of being influenced by and influencing other variables in the model. Then we have some parameters which can be either fixed or time-dependent. These are quantities that we assume to know from our expert knowledge. For instance, we assume to know the rate of growth of a population. The DICE model has been calibrated. Calibration is the process by means of which I set all these parameters. The DICE model has been calibrated in a very, very careful way by looking at the consensus opinion for all these parameters, doing extensive literature searches. These literature searches present results which are not perfect, but it is difficult to do better than what the DICE model has done. Rate of growth of population, rate of growth of productivity, the rate of growth of carbon intensity, the DICE model assumed we will become more and more efficient over time, and therefore, the carbon intensity, the amount of carbon needed to produce a unit of energy will decrease over time. Most important, that DICE model assumes to know for sure the exponent that links future damages to temperature increases. In the DICE model that exponent happens to be two, but it doesn't matter if it is exactly two. The point is, it's a parameter, and it is known. These are all the things about the world that we assume to know. They are a fixed background. Clearly, all these numbers did not come down with the 10 laws with Moses. They have been estimated with error, with uncertainty, with lot of difficulty, etc. One thing that one should always do is sensitivity analysis. You want to change these values over reasonable ranges and see how much your answers changes. By answer, I mean the optimal policy. Now we come to the crux, now we come to the most important bit, the control variables. These are time-dependent quantities that we vary in order to optimize our total expected discounted utility. For instance, what could be a control variable? A very typical control variable is, how much should we invest over time, not just today, but over schedule of investment in climate abating initiatives. Or how much should we invest in productive initiatives, again, at every point in time. Given the state variables, given the parameters, given the control variables, I can work out my value of a target function and I can vary my control variables, that's why they're called control variables, in order to maximize my target function. This is how the things play together. I have control variables that affect the state variables. I have a parameters that affect the state variables, the parameters are totally fixed. The control variables are totally dynamic. The state variables, population, GDP growth, CO_2 emission, temperature, etc., influence with target variables, and what I change is the control variables that determine the target function, and I keep on changing that until I obtain a maximum. IAMs, integrated assessment models differ from NPV, net present value and approaches in several important ways. With net present value approaches, one discounts future payoffs which are expressed effectively in money. The discounting that we use in NPV approaches takes into account of impatience, of risk aversion, of our desire for consumption smoothing, of uncertainty in outcomes, precautionary savings, and in theory, should also take into account the covariance between the payoffs and consumption, which is the risk premium. Normally, we don't like investments that pay well in good states of the world and pay badly when we need the money. We pay less for these investments, and therefore the risk premium goes up. The discount factor that has to be used in an NPV approach must contain all these things. Very different what is discounted in an integrated assessment model is future utility. Another fundamental difference is that with an NPV approach, I assume already that I am at equilibrium. Then I make a little perturbation around equilibrium. That is what the first-order condition does. Remember that we have done that. We take the derivative and we know that I am a stationary point. I'm just exploring the properties in the neighborhood of this optimum. However, it is a very strong assumption to hope that we will be following optimal decisions when it comes to climate change. Therefore, the validity of an approach tells me how various quantities are related based on perturbation of these theoretical optimum can be of dubious value when we might be such a long way away from optimum. The logic of the net present value is to assume that we are at the optimum and to find relationships between quantities that describe preferences in the world. For instance, we can find a relationship between the marginal productivity of capital, something about the world, and the inter-temporal marginal substitution of utility, something about our preferences. As I said, these relationships come from a first-order condition and the logic is with an NPV approach, since we are at the optimum, we take a first derivative and set it to zero. Also, very important with an NPV approach, it is a perturbation around an optimal position. It is not correct to call it a marginalist approach, but you understand what I mean by this. I'm just perturbing at the margin. The NPV approach can only deal with small investments. But arguably, climate change investment is not small and by itself this should raise a big question regarding the validity of an NPV approach. There is a great simplicity behind the NPV. For small investment, the procedure behind the NPV approach is super simple; take the cashflows and discount them at the social rate of discount. The price to pay in order for the simplicity, is the building the social rate of discount is difficult as we have seen. IAMs are different. With IAM, I consider a consumption stream at different points in time. I associate a utility to this consumption stream. This can be easy if we use separable utility functions, or it can be more difficult if you use recursive utility functions. However, the principle remains the same, and then we discount utility undertaken impatience what I call with the shorthand impatience into account. All the additional factors, all the additional components that go into the discount factor that is used in the NPV calculations are automatically taken care of by the chosen utility function and by the consumption pattern. Deterministic or stochastic, correlated or otherwise with the payoffs, etc. With IAMs, if I have a consumption stream or rather two consumption streams, one that I have a superscript a and one superscript b, c^a and c^b, we choose say c^a, that is to say the policy option, that produce the best stream c^a if the utility discounted, etc, or c^a is greater than the total utility of the stream c^ b. In the case of a separable utility function that we've seen already, is a great simplification is become the sum of the utilities associated with the consumption stream a discounted by the delta in patients rate if that is greater than the similar sum of the utility associated with consumptions b discounted with the same rate then I choose stream a. There is a price to pay for everything. The price to pay for not having to specify explicitly all the components that go into the social discount factor is that I have to specify now a full utility function. Furthermore, in order to obtain the optimal consumption stream, or rather the control variables that affect the consumption stream, one has to do it numerically, almost invariably there is no analytical solution. With IAMs, the optimum is a product of a procedure not with starting point. With net present value, we start from the optimum, with IAMs, we obtain the optimum as the outcome of a procedure. This optimum, as I said, does not rely on investment being small and no first order condition can be used. The optimization can be burdensome. In principle, we should vary consumption at every single point in time in a way which is compatible with the exogenous constraints, for instance, budget constraints. One last observation that I've already mentioned, but I want to stress again, with the NPV approach, one discount cash flows. Therefore, one must account both for impatience and for how these cash flows are translated into utility. With IAM such as DICE, one discounts utility, therefore, one needs only account for in patients. This is why in dice, the discount rate is only 1.5 percent and we will discuss this discount rates at great length later on but that is why it looks so different from the discount rate that you would obtain from an NPV approach.