Global Climate Change - Economics, Science, and Pol Homework Set 2

Climate Change: Economics, Science and Policy Problem Set #2 MODELING ECONOMIC GROWTH AND EMISSIONS 1. THE ASSIGNMENT In this assignment you will apply a spreadsheet-based model of global CO2 emissions to prepare 100-year projections of economic growth and carbon emissions, and to carry out studies of emissions restriction. The purpose of the exercise is to gain insight into the ways such models are constructed and applied. The model used here is simplified by ignoring the influence of government taxation and expenditure, and international trade, and by implementing a simple structure of production and input pricing. The model is laid out in Section 2 along with parameter values. A template with most of the equations already written in is provided for download on the course website. Columns A through Q are used in this homework set. You will work with both recursive-dynamic (myopic) and forward looking versions. Analyses to be conducted are described in Section 3. Section 4 describes how the template worksheet was set up, and Section 5 provides some worksheet hints. There are nine parts: A1, A2, B1, B2, B3, C1, C2, C3 and D. You may find it convenient to include print-outs of the solved worksheets in your answers. 2. THE EMISSIONS PREDICTION MODEL 1. The Model of Economic Growth and Emissions National output is modeled as a function of three factors: labor, capital and energy. The equations for labor and capital are similar to those discussed in class on 5 March. Carbon dioxide emissions are modeled as proportional to energy consumption, and the emissionsmitigation policy instrument is direct control of the level of energy use. Analysis of other measures that might be used in emissions control, such as taxes and subsidies, is not possible with this formulation. Economic variables are stated on an annual basis ($1012 /yr). The model is solved on a 10-year time step. Homework #2 p. 2 1.1 GDP Growth World GDP is represented by a single good, y, which is modeled as a function of inputs of capital, k, labor, l, and energy, e. The units of y are trillion (1012) $US. The output of the economy is represented by a constant-elasticity-of-substitution production function as follows, [ y t = at α k t ρ KLE + β lt ρ KLE + (1− α − β ) et ] ρ KLE 1/ ρ KLE (1) where the elasticity of substitution, σKLE, is σ KLE = 1 . 1− ρ KLE The labor force grows at an annual rate λ, lt = l 0 (1 + λ ) t . (2) The capital stock changes over time as a result of depreciation and new investment. The magnitude of the capital stock is defined as of the first year of the time step, and the level in each subsequent time step is a function of the depreciation rate, δ, and the investment during the period, it, k t +1 = n × it + (1 − δ ) n k t (3) where n is the number of years in each period of the model’s solution. All-factor productivity change augments output at an annual rate γ. a t = a 0 (1 + γ ) t , (4) where this shift parameter at has an initial level a0. 1.2 Income, Savings, and Investment The output of the economy, net of the costs of supplying energy, becomes the income received by a representative global consumer. The relationship among these factors is defined by the following accounting identity: y t = c t + s t + (pe t × e t ) , (5) where pet is the price of energy. Also, all savings are invested, s t = it . (6) Homework #2 p. 3 Savings behavior is modeled differently depending on the version of the model you are working with. In the recursive-dynamic version, saving is determined as a fixed fraction, η, of income, st = η yt (7) In the forward-looking version, saving is a variable determined in the multi-period optimization, explained below. 1.3 The Cost of Energy Energy use et is measured in exajoules (EJ = 1018J). The marginal cost of energy (and therefore the price) is determined by the depletion of natural resources. As cumulative energy use rises, the cost of energy per EJ increases as follows ⎛ r pet = μ ⎜⎜ ⎝ r − cumet ν ⎞ ⎟⎟ ⎠ (8) where • r is a constant representing the fixed amount of in-ground resources, • cumet, cumulative energy consumption, evolves according to cume t +1 = n × e t + cume t , cume0 = 0 (9) • μ and ν are estimated parameters. 1.4 Carbon Accounting The carbon intensity of energy, ε, is fixed over time. Emissions of CO2, denoted mt, are stated in gigatons (GT = 109 tons) of carbon per year. They are calculated by multiplying the carbon intensity of energy by the amount of energy consumed: m t = ε × et (10) 1.5 Parameter Values The parameter values are: n = 10 (number of years per period) a0 = 0.8 (initial value of technological shift factor) r = 140000 EJ (energy resource base) α = 0.25 (capital share in production function) β = 0.65 (labor share in production function) δ = 0.05 (annual depreciation rate of capital) Homework #2 p. 4 ε = 0.018 GT/EJ (carbon coefficient on energy use) μ = 0.01 (slope parameter in energy cost function) ν=1 (exponent parameter in energy cost function) σ = 1.25 (elasticity of substitution) ρ = (σ − 1) / σ = 0.2 (substitution parameter in production function) θ = 0.03 (annual discount rate) λ = 0.01 (annual growth rate of labor supply) γ = 0.01 (annual growth rate of all factor productivity change) The initial-year values of the variables in the model are the following: l0 = 22.65 (109 worker-hours) k0 = 130 (1012 $US) e0 = 600 (EJ) The time t = 0 is the year 2000, and the spreadsheet provided for Parts A, B, and C of the homework set is already set up in 10-year periods. 3. TASKS WITH THE MODEL You are to prepare calculations with two versions of this model, and use it to predict global emissions under various assumptions. For each version, please hand in 1. A copy of each worksheet where appropriate. 2. Plots of the time paths of consumption, carbon emissions, and atmospheric concentrations. Feel free to plot any other variables that you find of interest. 3. Comments on the solutions as requested below. Once you have set up the worksheet, you may want to explore the effect on the results of alternative values of key parameters, e.g., γ, η, σKLE. Note that the model may not solve if you depart far from the reference parameters specified above. Part A. Tasks with the Recursive-Dynamic Version The Excel template incorporates a myopic version of the model, and shows a projection of global CO2 emissions up to 2100 (column N). This version assumes that energy use grows from its year 2000 level of 600 EJ according to an elasticity relationship with GDP, y, with a one-period lag (column F). That is, Homework #2 p. 5 [ ] [ χ 1 e1 = 0 .5 e 0 y 0 χ K e t = 0 .5 e 0 y t ] where χ = 0.25 is the elasticity of energy use to output. Solve this model in separate cases for Problem A1. A higher rate of productivity growth (γ=0.015), and Problem A2. A lower rate of labor force growth (λ=0.005). (Return γ to its original value of 0.01 before running this case.) Comment briefly on the way these changes influence the other variables in the model. Part B. A Tasks with the Forward-Looking Version Solve the model assuming that the level of energy use (the control variable) is set over all periods to maximize the present value of welfare. In any period t, the index describing welfare is Wt = ln(Ct). This functional form, (column K) implements an assumption that the marginal utility of consumption declines as consumption rises. Thus the function to be maximized is ⎛ 1 ⎞t max PV (W t ) = ∑ lnc t ⎜ ⎟ ⎝1+ θ ⎠ t (11) where θ is the utility discount rate (rate of time preference), which is set to θ = 0.03 on an annual basis in the worksheet. The model maximizes equation (11) by choosing the path of et. The initial value of this variable is fixed, with e0 = 600 (EJ). Problem B1. Prepare a reference (“business as usual”) scenario on the assumption there is no carbon constraint. (Make sure productivity growth and labor force growth are reset (γ=0.01; λ=0.01). Problem B2. Prepare a control scenario applying an intensity target of the form announced by the Bush Administration on 14 February 2002. Assume the energy intensity of GDP (m/y) declines by 15% per decade, beginning from the intensity in the 2000 base year. (Hint: Set up the intensity constraint in Cells P24-P33. Solver is sensitive to starting conditions here. If you get repeated errors, reload the template and start again.) Problem B3. Construct a curve of marginal cost, defined in terms of lost (nondiscounted) consumption, for global emissions reductions of up to 4 GtC, for the periods 2010 and 2020. This may be done by solving the model for reductions below reference values (from Problem B1) of 1.0, 2.0, 3.0, and 4.0 GtC and then plotting the marginal decrease in consumption. Comment briefly on the shapes of the curves. Remember that consumption is in units of trillions of dollars, and so a 0.1 change in the consumption Homework #2 p. 6 level is actually 100 billion dollars. (Note also that, because of its simplicity, this model will yield costs higher than you will see in other analyses, and a non-zero intercept.) Part C. Stabilization of Atmospheric CO2 Concentration A simple model of CO2 accumulation (in GtC) in the atmosphere is ⎛ 1⎞ stkmt = n × mt + ⎜1− ⎟ stkmt−1 ⎝ τe ⎠ n (12) where stkm0 = 806.86 GtC (carbon in atmosphere in 2000) and τe = 120 years (e-folding time of carbon in the atmosphere). The atmospheric concentration, in ppmv, is then conct = stkmt / 2.181. The template already includes two columns for the calculation of atmospheric concentrations of CO2, and a figure to display the results is provided in the worksheet. Implement these equations and conduct the following exercises using the forward-looking model. Problem C1. Solve for the pattern of energy use (and associated carbon emissions) that stabilizes atmospheric carbon concentrations at or below 550 ppmv for every period of the model. What is the yearly carbon uptake in 2100 under this scenario? Comment on the model assumption about terrestrial and ocean carbon uptake. Problem C2. Solve the same case as C1, only with the discount rate reduced from 0.03 to 0.01. Comment briefly on the difference this makes to the solution. (Do not forget to rerun the solver, otherwise the optimization doesn’t happen and all you do is change the net present value.) Problem C3. Solve the C1 case again (i.e., <= 550ppmv for all periods), only assume that because of political constraints the level of energy use cannot be reduced more rapidly than 10% per decade. Comment briefly on how the solutions differ. Part D. How to Improve the Model List three additional features you would add to the model to make it more realistic, commenting briefly on 1. Which you believe are the most important for understanding the climate issue? 2. What problems (data, analysis methods) would arise in implementation? 3. Are there issues that are important that this type of model would never be able to address, even with significantly more complexity? Homework #2 p. 7 4. SETUP OF THE SAMPLE WORKSHEET TEMPLATE The problem is set up with 11 ten-year periods, t* = 0, ...,10. For each period, t* the value of the economic variables (y, k, l, c, i) are set at the appropriate annual levels of the first year in the 10-year calculation step. That is, y(t* = 0) ≡ y(t = 0) y(t* = 1) ≡ y(t = 10) etc. Thus equations (3), (9) and (12) assume that in the first year of the 10-year time step investment, energy and emissions (respectively) apply for all years of that step, so that these (annual) quantities must be multiplied by n = 10. Similarly, in equations (3) and (12) the annual survival factors (1 - δ) and (1 -1/τe) must be updated over an entire decade, which is accomplished by raising them to the power n. 5. WORKSHEET HINTS 1. Before beginning work save a master copy of the worksheet (to be left unchanged) and set up a working version. You may need to exit and start over and this will save you having to go back to the web. 2. This note is written as if the dynamic version of the model is to be solved in Excel using Solver. Equivalent facilities should be available in other spreadsheet programs. The policies limiting emissions and/or concentrations must be entered as constraints in Solver. 3. When solving the dynamic model, bounds sometimes must be placed on values of the variables to ensure that Solver finds an optimum that in fact makes economic sense. If you run into problems try ensuring that energy use, energy prices, consumption and investment cannot be negative. Rule out these possibilities by adding the constraints et ≥ 0, pet ≥ 0, ct ≥ 0 and it ≥ 0 for all t. 4. When solving the dynamic model under constraint, always re-run the unconstrained version before running with a different constraint. Solver does not always move accurately from one constrained solution to another. Also, sometimes Solver gets stuck (e.g., when confronted with a very restrictive constraint), and cannot return to the noconstraint solution. Therefore, you should always run an unconstrained version of any problem before moving from one constrained solution to another. 5. Sometimes Solver returns a solution that is not an optimum. If you run into this problem, often you can recover by limiting the zone of search for the optimal solution and performing successive solves, using the solution of one as the initial point for the next. This process should converge to a sensible answer.