1 Hotelling example (with small change)

Note on “shadow prices” or “dual variables” (π) These are extremely important in economic modeling (and more generally in economics). Basic idea is that these represent opportunity costs. I will use the example of cost minimization. In our Hotelling problem, we are minimizing costs (C) subject to the various constraints. Let’s take the resource constraints (sum production < resources = R 1 ). If we do this as a Lagrangean, we get the following interesting result: ∂C/∂R 1 = π 1 = shadow price on reserve grade 1 = $ This says that if we increase the quantity of reserves by 1 unit, this lowers discounted cost by $ (I changed the sign to positive.) An important theorem from econ is that this equals the first-period royalty in competitive markets (!). This grows with the interest rate until exhausted. I calculated this numerically using a linear programming algorithm in the GAMS program and got the π i shown in the table. We can actually do this for 50,000 variables and constraints using a LP algorithm. This is done, for example, in oil refineries to optimize yield from crude oil. 2

3 Economics 331b Spring 2010 Integrated Assessment Models of Economics of Climate Change

Integrated Assessment (IA) Models of Climate Change What are IA model? –These are models that include the full range of cause and effect in climate change (“end to end” modeling). –They are necessarily interdisciplinary and involve natural and social sciences Major goals: –Project the impact of current trends and of policies on important variables –Assess the costs and benefits of alternative policies –Assess uncertainties and priorities for scientific and project/engineering research

Major Components of Models Identities Behavioral and Scientific Equations Value Judgments (markets, policies, ethics, etc.)

6 Person or nation 1 Person or nation 2 Inefficient initial (no- policy) position Bargaining region (Pareto improving) Pareto Improvement from Climate Policy

Elements of IA Models. To be complete, the model needs to incorporate the following elements: - human activities generating emissions - carbon cycle - climate system - biological and physical impacts - socioeconomic impacts - policy levers to affect emissions or other parts of cycle.

8 Fossil fuel use generates CO2 emissions Carbon cycle: redistributes around atmosphere, oceans, etc. Climate system: change in radiation warming, precip, ocean currents, etc.. Impacts on ecosystems, agriculture, diseases, skiing, golfing, … Measures to control emissions (limits, taxes, subsidies, …) The emissions -climate- impacts- policy nexus

Representative Scenarios for Models “Baseline” or uncontrolled path: - Set emissions at zero control or zero “tax” level. - Business as usual Alternative strategies: - “Optimal” where maximize objective function - Stabilize emissions, concentrations, or climate - Kyoto Protocol/ Copenhagen Accord limits

There are many kinds of IA models, useful for different purposes Policy evaluation models - Models that emphasize projecting the impacts of different assumptions and policies on the major systems; - often extend to non-economic variables Policy optimization models - Models that emphasize optimizing a few key control variables (such as taxes or control rates) with an eye to balancing costs and benefits or maximizing efficiency; - often limited to monetized variables

Two Examples 1.Reprise on Hotelling example for pset 1 2. The Samuelson-Negishi equivalence for modeling Maximize (producer + consumer surplus) = Maximize Discounted [U(c) – Cost(c)] = Competitive equilibrium 11

12

13 Outcome of efficient competitive market (however complex but finite time) Maximization of weighted utility function: Economic Theory Behind Modeling = 1. Basic theorem of “markets as maximization” (Samuelson, Negishi) 2. This allows us (in principle) to calculate the outcome of a market system by a constrained non-linear maximization.

How do we solve IA models? The structure of the models is the following: We solve using various mathematical optimization techniques. 1.GAMS solver (proprietary). This takes the problem and solves it using linear programming (LP) through successive steps. It is extremely reliable. 2.Use EXCEL solver. This is available with standard EXCEL and uses various numerical techniques. It is not 100% reliable for difficult or complex problems. 3.MATHLAB. Useful if you know it. 4.Genetic algorithms. Some like these. 14

Can also calculate the “shadow prices,” here the efficient carbon taxes Remember that in a constrained optimization (Lagrangean), the multipliers have the interpretation of d[Objective Function]/dX. So, in this problem, interpretation is MC of emissions reduction. Optimization programs (particularly LP) will generate the shadow prices of carbon emissions in the optimal path. For example, in the problem we just did, we have the following shadow prices: 15

Basic economic strategy 1.Begin with a Solow-style economic growth model 2.Add the geophysical equations: note these impose an externality 3.Then add an objective function to be optimized subject to constraints: = optimal growth model [Friday] = integrated assessment model 4. Then estimate or calibrate the various components. 5. Then do various simulations and policy runs. 16

17 Emissions trajectories: Start with data base of 70 major countries representing 97 % of output and emissions Major issue of whether to use PPP or MER (next slide) Estimate productivity growth Estimate CO2 emissions-output ratios Project these by decade for next two centuries Then aggregate up by twelve major regions (US, EU, …) Constrain by global fossil fuel resources This is probably the largest uncertainty over the long run: σ(Q) ≈.01 T, or + factor 2.5 in 100 yrs, +7 in 200 yrs Modeling Strategies (I): Emissions

18 CO2-GDP: Three countries (PPP v. MER)

19 Climate model Idea here to use “reduced form” or simplified models. For example, large models have very fine resolution and require supercomputers for solution.* We take two-layers (atmosphere, deep oceans) and decadal time steps. Calibrated to ensemble of models in IPCC TAR and FAR science reports. Modeling Strategies (II): Climate Models *

20 Actual and predicted global temperature history

21 Projected DICE and IPCC: two scenarios

22 Modeling Strategies (III): Impacts Central difficulty is evaluation of the impact of climate change on society Two major areas: –market economy (agriculture, manufacturing, housing, …) –non-market sectors human (health, recreation, …) non-human (ecosystems, fish, trees, …)

23 Early studies contained a major surprise: Modest impacts for gradual climate change, market impacts, high- income economies, next years: - Impact about 0 (+ 2) percent of output. - Further studies confirmed this general result. BUT, outside of this narrow finding, potential for big problems: -many subtle thresholds -abrupt climate change (“inevitable surprises”) -ecological disruptions -stress to small, topical, developing countries -gradual coastal inundation of 1 – 10 meters over 1-5 centuries OVERALL: “…global mean losses could be 1-5% Gross Domestic Product (GDP) for 4 ºC of warming.” (IPCC, FAR, April 2007) Summary of Impacts Estimates

24 Estimated Damages from Yale Models and IPCC Estimate

25 Major problems of impacts analysis Most impacts analyses impose climate changes on current social-economic-political structures. Example: impact of temp/precip/CO2 on structure of Indian economy in 2005 However, need to consider what society will look like when climate change occurs. Example looking backward: –2 ˚C increase in 6-7 decades – that was Nazism, period of Great Depression, Gold Standard, pre-Keynesian macro –4 ˚C increase in 15 decades –Ming Dynasty, lighting with whale oil, invention of telegraph, no cars, many horses….

26 Modeling Strategies (IV): Abatement costs IA models use different strategies: –Some use econometric analysis of costs of reductions –Some use engineering/mathematical programming estimates –DICE model generally uses “reduced form” estimates of marginal costs of reduction as function of emissions reduction rate

Derivation of mitigation cost function Start with a reduced-form cost function: (1) C = Qλμ  where C = mitigation cost, Q = GDP, μ = emissions control rate, λ,  are parameters. Take the derivative w.r.t. emissions and substitute σ = E 0 /Q (2) dC/dE = MC emissions reductions = Qλβμ  -1 [dμ/dE] = λβμ  -1 /σ Taking logs: (3) ln (MC) = constant + time trend + ( β-1) ln(μ) We can estimate this function from microeconomic/engineering studies of the cost of abatement. 27

28 Example from McKinsey Study

29 Reduced form equation: C=.0657*miu^1.66*Q

30 Further discussion However, there has been a great deal of controversy about the McKinsey study. The idea of “negative cost” emissions reduction raises major conceptual and policy issues. For the DICE model, we have generally relied on more micro and engineering studies. The next set of slides shows estimates based on the IPCC Fourth Assessment Report survey of mitigation costs. The bottom line is that the exponent is much higher (between 2.5 and 3). This has important implications that we will see later.

Note that the MC is much more convex than McKinsey: much more diminishing returns 31 Source: IPCC, AR4, Mitigation.

Source for estimates of  (elasticity of cost function) 32 Source: IPCC, AR4, Mitigation, p. 77.

Using the IPCC as data for the cost function 33 Conclusion is that the cost function is EXTREMELY convex.

34 Alternative abatement cost functions: From IPCC Parameterized as C/Q = aμ 2.8, with backstop price(2005) = $1100/tC

35 Alternative abatement cost functions Parameterized as C/Q = aμ 2.8, with backstop price(2005) = $1100/tC

36 Alternative abatement cost functions: IPCC and MK Parameterized as C/Q = aμ 2.8, with backstop price(2005) = $1100/tC

37 Applications of IA Models How can we use IA models to evaluate alternative approaches to climate-change policy? I will illustrate analyzing the economic and climatic implications of several prominent policies. For these, I use the recently developed DICE-2007 model.

38 1. No controls ("baseline"). No emissions controls. 2. Optimal policy. Emissions and carbon prices set for economic optimum. 3. Climatic constraints with CO2 concentration constraints. Concentrations limited to 550 ppm 4. Climatic constraints with temperature constraints. Temperature limited to 2½ °C 5. Kyoto Protocol. Kyoto Protocol without the U.S. 6. Strengthened Kyoto Protocol. Roughly, the Obama/EU policy proposals. 7. Geoengineering. Implements a geoengineering option that offsets radiative forcing at low cost. Illustrative Policies for DICE-2007

Per capita GDP: history and projections 39

CO2-GDP ratios: history 40

IPCC AR4 Model Results: History and Projections 41 DICE-2007 model 2-sigma range DICE model