Global Climate Change: Homework Set 2 Solution

Problem Set 2: Problem A1, A2: Key Variables: PV Consumption: Myopic Reference PV Cons High Productivity Growth 15.1154 Low Labor Growth 15.7567 14.7826 Ratio of year 2010 values to the myopic reference: GDP hiprod lowlabor (y) 1.05 0.97 Tech Chg (a) 1.05 1.00 Labor Capital Energy Inv Cons (l) 1.00 0.95 (k) 1.00 1.00 (e) 1.00 1.00 (i) 1.05 0.97 (c) 1.06 0.97 Labor Capital Energy Inv Cons Energy Price Carbon Emiss (l) 1.00 0.61 (k) 1.79 0.75 (e) 1.18 0.92 (i) 2.09 0.70 (c) 2.17 0.68 (pe) 1.19 0.93 (m) 1.18 0.92 Ratio of year 2100 values to the myopic reference: GDP hiprod lowlab. (y) 2.09 0.70 TechCh g (a) 1.64 1.00 A1: Productivity growth directly influences the model through the all factor productivity (techn. change) parameter. An increase from 0.01 to 0.015 in productivity growth leads to a 64% increase in technical change by 2100 (1.005^100). GDP scales linearly with the technical change parameter, and if we look at 2010 ratios (before investment and energy effects kick in) we can see this. Investment is proportional to GDP, and is included in the capital stock of the subsequent period. So an increase in GDP leads to an increase in the capital stock, and the increased capital stock feeds back into GDP. Energy use increases with the increase of GDP, also with a lag, leading to a total increase in energy of 18% by the end of the century. This is another positive feedback that leads to yet larger GDP. The increase in energy use leads to an increase in carbon emissions, and also in carbon price as energy resources are consumed. Consumption is related to GDP minus investments and energy purchases, and because energy purchases grow less than GDP grows, consumption grows even faster than GDP. Note that because of the log nature of the net present value calculation, the net present value of consumption increases by only 4.3%. Carbon intensity also improves significantly, as the economy grows much faster than the increase in emissions. A2: Labor growth obvious affects labor: a decrease from 0.01 to 0.005 in labor growth leads to a decrease to 61% of reference in the labor parameter (.995^100). Labor is an input into GDP through the CES function: we see that the decrease in GDP in 2010 is not quite as much as the decrease in labor, because labor has only a 65% share in the production function, and because of the elasticity of substitution. In a parallel fashion to A1, a change in GDP changes investment that ends up creating a negative feedback. By the end of the century GDP has decreased by 30%, capital by 25%, energy by 8%, and consumption by 32%. NPC decreases by 2.2%. Because energy decreases less than GDP, carbon intensity does not improve as much as in the reference case. Problem B1: Ratio of values in 2100 for forward-looking model over myopic model GDP (y) 1.010 TechCh g (a) 1.000 Labor Capital Energy Inv Cons Energy Price (l) 1.000 (k) 1.021 (e) 1.018 (i) 1.010 (c) 0.967 (cume) 1.114 (pe) 1.348 NPV for forward looking model: 15.1216 Note that the NPV is greater than for the myopic model, which is to be expected because the forward looking model should be optimizing for the largest possible NPV. This increase is small, because the energy use in the myopic model was designed in an attempt to approximate the forward looking model (though cumulative energy use does increase by 10%). B2: Time Step 2000 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100 Carbon Intensit y (m/y) 0.2424 0.2060 0.1751 0.1489 0.1265 0.1075 0.0914 0.0777 0.0660 0.0561 0.0477 Reference PV Carbon Cons Emiss ln(c)*df (m) ac 3.36 10.80 2.71 13.52 2.17 16.13 1.73 18.43 1.37 20.98 1.08 22.31 0.84 22.78 0.66 22.94 0.51 23.20 0.40 23.78 0.31 24.93 15.12 Carbon Intensity Decline 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 Intensity Target PV Carbon Cons Emiss ln(c)*dfa (m) c 3.36 10.80 2.71 11.90 2.17 13.02 1.72 14.20 1.36 15.46 1.07 16.82 0.84 18.31 0.66 19.94 0.51 21.75 0.40 23.76 0.31 25.99 15.11 Note that the intensity target in this model is tightest in the first four periods when the reference version has intensity declines of less than 15%: after that, the reference actually has intensity improve by more than 15% every period. This leads to the odd result that emissions intensity in the constraint case is actually higher by the end of the century than it was in the reference case, though cumulative energy use is still significantly lower (92000 compared to 108000). B3 Marginal Reduction in Consumption (billion $) 100 80 2010 2020 60 40 20 0 0 1 2 3 4 5 -20 Carbon Abatement (GtC) B3 involved constructing a curve of marginal cost. The term “marginal” implies the additional increment of cost. So the 2010 curve was generated by taking the consumption in 2010 (constraint of 12.99 GtC) – consumption in BAU (13.99 GtC), followed by the consumption (11.99 GtC) – consumption (12.99 GtC) and so on. The most important feature to note in the curves: The marginal loss due to each additional ton of reductions is more expensive than the previous ton. This result matches the presumption that a society will choose the least expensive options first, and the most expensive options later. Marginal Reduction in NPV*1000 (units not defined) 7 6 2010 2020 5 4 3 2 1 0 0 1 2 3 4 5 Carbon Abatement in 2010 or 2020 (GtC) Several people noted that it was a little odd that a reduction of 1 GtC below BAU cost more in 2020 than in 2010. This is due to two factors: the first is a discount rate issue. Money in 2020 is worth less than money in 2010. The second is the use of the logarithm in determining present value – and since consumption is higher in 2020 than in 2010, and additional dollar is worth even less. However, the overriding reason is that the model optimizes over the entire century, so costs (and benefits) may be spread out over multiple time periods. This is clear when looking at the marginal reduction in net present value as shown in the chart above (not required for problem set). This matches intuition somewhat better: reductions later in the future are less expensive, and reductions start at positive net present cost. 18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 600.0 500.0 400.0 300.0 200.0 100.0 (y) (conc) (c) (m) 20 00 20 10 20 20 20 30 20 40 20 50 20 60 20 70 20 80 20 90 21 00 0.0 m (GtC/yr) C1 Year Concentration in the base case rises to 861 ppm. In order to stabilize concentrations at 550 ppm, it is necessary to reduce emissions to 9.63 GtC per year. The yearly carbon uptake in 2100 can be solved in one of two ways: one is to note that in 2100, the system is at equilibrium, therefore inputs (emissions) equals output (carbon sink), therefore the sink is also 9.63 GtC The other way is to solve this analytically, by recognizing that the sink in the equation stkm = nxm + (1-1/tau)^n * stkm for a 10 year period can be calculated by looking at the reduction in stock in one time period. . If the concentration is 550, the stock has to be 2.181*550 (1199.55). (1-1/120)^10 = 0.9197. Therefore the reduction in stock is (1-0.9197)*stock, divided by 10 to get one year – which also gives us 9.63 GtC. Is this sink realistic? Clearly, the answer is no. Under stabilization, we actually expect the ocean and terrestrial sink to drop to near zero as the ecosystem and ocean enter equilibrium with the atmosphere. This will of course require emissions to drop to near zero, which is a significant challenge for the future. 18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 600.0 500.0 400.0 300.0 200.0 100.0 (y) (conc) (c) (m) 20 00 20 10 20 20 20 30 20 40 20 50 20 60 20 70 20 80 20 90 21 00 0.0 m (GtC/yr) C2: Year With the discount rate at 0.01, the final emissions are still the same (9.63 GtC) for physical reasons: however, the timing of emissions reductions changes. There are fewer emissions from 2010 to 2040, and more emissions from 2060 to 2080. This is because future consumption is discounted less, and therefore more valuable, than in the case where the discount rate is 3%. Note that the increase in NPV is mostly due to accounting: by changing the discount rate you aren’t changing people’s welfare, you are just changing how you weight their welfare. And then when you optimize you might choose to allocate resources differently so that people in the future are better off at the cost of some consumption in the near term. C3: In C1, 2060 and 2070 had emissions reduction of close to 20%. If we restrict energy use reductions in any time period to only 10%, then the model has to adjust so that a drop of that size isn’t necessary. It does that by decreasing emissions in 2040 and 2050. However, this allows some increases of emissions at the beginning and end of the period… NPV also decreases a little, as is to be expected when adding additional constraints. 500.0 400.0 300.0 200.0 100.0 (y) (conc) (c) (m) 20 00 20 10 20 20 20 30 20 40 20 50 20 60 20 70 20 80 20 90 21 00 0.0 m (GtC/yr) 18.00 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 600.0 Year D: There are many possible answers for this section. My personal favorite list (given the model form, these should be additions that would be fairly easy to implement): Include a damage function Improve the carbon cycle Add a “clean energy” technology Allow “energy research investment” I would also like a model that could handle price instruments (like taxes) or report prices from quantity instruments (eg, cap and trade), but this would be more challenging to implement. A lot of people mentioned: Adding other GHGs Multiple regions The difficulty of handling political realities Most common mistakes: Not including the energy equation in problem A (we’ll write the problem more clearly next year) Reporting total cost rather than marginal cost in problem B3: marginal is the additional cost resulting from the last unit of change – similar to the slope of the total cost curve. Grading: A: B: C: D: 20 Pts 30 Pts 30 Pts 20 Pts