GDP, DICE & the Social Cost of Carbon

GDP basics

GDP definition

What: GDP (Gross Domestic Product) = the market value of final goods and services produced within a country in a period. Three measurement approaches that should agree:

• Production / value-added approach: sum value added at each stage (subtract intermediates to avoid double-counting — e.g. the coffee shop's value added excludes the paper cups, which are counted at the cup factory).
• Expenditure / demand approach: $Y = C + I + G + X - M$ (consumption + investment + government + exports − imports).
• Income approach: sum what everyone earned (wages, profits, rents).

Why: Standard yardstick; all course discussions of growth, damages, and SCC are framed in real GDP terms.

Remember: Investment counts only NEW capital (selling an old bicycle is not investment). Durable goods (a house, a car) blur consumption/investment. A firm building a new factory = investment. Informal transactions can fall into the informal economy, which is unmeasured and large in emerging markets.

Why we care about GDP

Remember: GDP is flawed but correlates with things we objectively value: life expectancy vs. GDP per capita is tight (and shifted up recently, esp. sub-Saharan Africa); life satisfaction vs. GDP per capita is strongly positive in survey data. "Money doesn't buy happiness" is usually said from a high material baseline. These correlations are NOT necessarily causal.

Real vs. nominal GDP

What: Nominal uses CURRENT prices; real uses CONSTANT base-year prices. In this course, GDP/income is always REAL.

Why: Without it we can't tell whether GDP rose because we produced more (good) or because prices rose (not necessarily good).

PPP adjustment

Purchasing Power Parity (PPP)

What: To compare income across places/currencies, adjust for price differences:

$$\text{PPP multiplier} = \frac{\text{price of consumption basket in A}}{\text{price of consumption basket in B}}$$

Why: The same money buys different amounts in different places ($2 buys more in Indonesia than in Manhattan).

Remember (calc): Two job offers — Schaffhausen CHF 120k vs. Zurich CHF 150k. Higher rent/childcare/restaurants in Zurich. If the basket costs CHF 8000/month in Zurich and CHF 6000 in Schaffhausen, PPP = 8000/6000 = 1.31. In PPP-adjusted Zurich francs, the Schaffhausen job is worth $120{,}000 \times 1.31 \approx$ CHF 157k — MORE purchasing power than Zurich's 150k. Calc rule: multiply the salary by the basket-price ratio to convert to the other location's equivalent. This is also why poverty is measured in PPP dollars — $2.15/day = "what it would be like to live in the US on $2.15/day."

Growth and the production function

Standard aggregate production function

What:

$$Y_t = A_t \cdot F(K_t, H_t), \qquad H_t = L_t \cdot h_t$$

• $A_t$ = TFP (Total Factor Productivity) — technology, knowledge, policy, institutions, infrastructure, MANAGEMENT. "Doing more with less."
• $K_t$ = physical capital (the factory).
• $L_t$ = labor (workers or hours); $h_t$ = human capital per worker. $H_t = L_t h_t$ = efficiency units of labor.

Why: Decomposes growth sources. $F$ is often Cobb-Douglas.

Remember: What grows each term? $K$ ↑ via investment; $L$ ↑ via labor-force participation (e.g. rising female participation since the 1960s) — and FALLS with aging / rising dependency ratio; $h$ ↑ via education/training; $A$ ↑ via better tech/policy/institutions/management.

The only engine of long-run growth

What: In the standard Ramsey-Cass-Koopmans / Solow growth model, with decreasing returns to both capital and labor and no ever-expanding population, the ONLY source of sustained per-capita GDP growth is productivity ($A_t$) growth.

Why: More machines alone hit diminishing returns; more people alone (without more capital/land) also do. Productivity = "doing more with less."

Remember: Productivity growth can COMPLEMENT sustainability. Examples: producing the same food now needs ~72% less land than decades ago; less water and less CO₂ per unit of crop. This is why Malthus was wrong — he predicted population would outrun food and cause misery, but underestimated the exponential rate of TFP/technological improvement.

Endogenous direction of innovation

What: The DIRECTION of technological change is endogenous to market incentives — technology isn't manna from heaven.

Why: With the right incentives (e.g. carbon prices), markets shift R&D and patents toward low-carbon tech, making clean production cheaper → decoupling.

Remember: Many countries have decoupled CO₂ from GDP growth, e.g. Ireland: +43% GDP while −50% CO₂.

Market failures (micro refresher)

Demand and supply curves

Remember: The demand curve read one way = at price $p$, how many people buy; read the other way (inverse) = at quantity $q$, the marginal consumer's willingness to pay (WTP). The supply curve = how much sellers will sell at each price; in competitive markets it equals the marginal cost (MC) curve. Prices carry critical information that lets the market allocate efficiently.

Consumer & producer surplus

Remember: Consumer surplus = WTP − price paid, summed over all buyers. Producer surplus = price − marginal cost, summed over units sold. The competitive market equilibrium (demand = supply) MAXIMIZES total social surplus — given resources, costs, and preferences. This is WHY economists normally like free markets.

Market failure & externalities

What: A market failure is anything that makes the market, left to itself, deliver an inefficient outcome. An externality exists when the private marginal cost of an action differs from the social marginal cost:

$$\text{social MC} = \text{private MC} + \text{external MC}$$

Why: When prices don't reflect the full social cost/benefit, the market over- (or under-) produces. Getting oil out of the ground costs $60 (a real resource cost) — but if burning it also causes climate change, air pollution, biodiversity/water loss NOT reflected in the driller's incentives, that's the externality.

Remember: This is "not news" — Milton Friedman and Friedrich von Hayek, foundational free-market economists, already wrote about it. The economics hasn't changed, only the politics around it. The four canonical market failures (later chapters): externalities, public goods, market power, information asymmetries.

Pigouvian tax — internalizing the externality

What: Idea from Arthur Cecil Pigou (1920 book): charge a pollution tax equal to the marginal external cost. Then private MC + tax = social MC, and the market delivers the efficient quantity. Positive externalities → symmetric subsidies.

Why: Polluters then face the true cost and decide for themselves "is it still worth it?"

Remember: Modern view: a pollution price is a NECESSARY core but NOT SUFFICIENT by itself for most environmental problems (coordination issues etc.) — yet economists overwhelmingly support some carbon price (one of the most-agreed-upon policies in expert surveys). The statutory incidence (who legally pays) doesn't matter for the outcome under competitive markets. Upstream (oil & gas extraction) is administratively easy to levy.

Building an Integrated Assessment Model

Integrated Assessment Models (IAMs)

What: IAMs couple an economic model with an environmental/natural system. Always picture a CIRCLE: economy → affects environment → environment feeds back → affects economy. Macro-based IAMs are simpler on technology/regional detail but have a proper economic EQUILIBRIUM at their core.

Why: Economists emphasize forward-looking equilibrium behavior; "simple" models trade detail for this.

Three steps to integrate climate into the macro model

Remember:

1. Add fossil energy $E_t$ as an input: more fossil energy today → more GDP (short-run benefit). $Y_t = A_t F(K_t, H_t, E_t)$.
2. Add a climate model: global mean surface temperature change over pre-industrial is a function of the history of fossil-energy/CO₂ use since the industrial revolution: $T_t = H(E_0, \ldots, E_t)$.
3. Let temperature feed back onto output: $Y_t = A_t(T_t)\,F(K_t, H_t, E_t)$. Same farmer, same land, same machine → different output depending on the weather/climate.

Where to put the climate damage

Remember: DICE puts it in the PRODUCTIVITY term (intuitive: same inputs, different output as climate changes — like a farmer's yield depending on weather). But in the DATA climate hits EVERY component: capital stock (disaster destruction), population/labor supply (mortality, fewer work hours in hot poor regions), and human capital (learning). Standardized US test-score evidence: the SAME student in the SAME school learns LESS in a year with more extreme-heat classroom days — but the effect VANISHES in air-conditioned schools.

Whose damages? Market vs. non-market

Remember: Economists typically take a benevolent-world-government view: sum damages AND benefits across every person, country, and generation. HOW to weight different people's damages is a separate (political) question. Climate also affects things NOT in GDP (species existence, non-working-population mortality/morbidity, lost snowy childhoods) — these enter via WTP, giving "GDP-EQUIVALENT" losses.

The DICE model

DICE — origin and structure

What: DICE (Dynamic Integrated Climate-Economy model) by William Nordhaus, who won the 2018 Nobel Prize for first introducing climate into macroeconomic analysis. He wrote in 1972 that "sea-level rise and the consequences of climate change are much more likely to constrain long-run growth than limited natural-resource availability." The economics has been "singing this song" since the 1970s.

Remember: Svante Arrhenius (1896) — first Swedish Nobel laureate (not for this!) — first wrote down a version of the equation linking CO₂ concentration to global heat retention, computing it by hand and already getting the CONCAVITY right. Climate change is "not new."

Production and emissions

What: Gross output is Cobb-Douglas: $Y_t^{\text{Gross}} = A_t K_t^{\gamma} H_t^{1-\gamma}$. Baseline emissions:

$$E_t^{\text{Baseline}} = \sigma_t \cdot Y_t$$

where $\sigma_t$ = emissions intensity (CO₂ per franc of GDP).

Remember: Global CO₂ per unit of GDP has fallen ~2% per year for 50+ years. DICE assumes this continues but slows. Globally CO₂ still rises with GDP (a "shark-fin" shape), but the link weakens; many European countries have fully decoupled.

Abatement and the policy lever

What: Choose an abatement/mitigation rate $\mu_t \in [0,1]$ — the fraction of baseline emissions you avoid. Actual emissions:

$$E_t = (1 - \mu_t) \cdot E_t^{\text{Baseline}}$$

Abatement is costly — a fraction of GDP $\Lambda_t(\mu_t)$, CONVEX:

$$\Lambda_t(\mu_t) = \theta_{1,t}\, \mu_t^{\theta_2}, \quad \theta_2 \approx 2.6 \text{ (convex)}$$

Remember (DICE-2023 internals): Abatement costs are calibrated from a model-ensemble comparison. Headline anchor: instantly abating ALL industrial CO₂ now would cost ~11% of GDP — but costs FALL over time because of (a) learning-by-doing, (b) innovation/R&D, and (c) capital-stock turnover (gradually replacing capital as it wears out is far cheaper than forcing everything at once — like being told to cut your apartment's energy 80% by next WEEK vs. over 5 YEARS).

The climate module (DICE-2023)

Remember: DICE-2023 uses the FAIR (Finite-Amplitude Impulse-Response) emulator. Carbon cycle = four boxes (biosphere, upper ocean, deep ocean, etc.); CO₂ in each decays at a different rate, and the overall absorption rate SLOWS as the ocean warms (dissolves less CO₂) and ACIDIFIES. Rising concentrations → rising radiative forcing → warming. DICE accounts for industrial CO₂, land-based CO₂, AND non-CO₂ greenhouse gases (methane, nitrous oxide, F-gases).

The damage function

What: Closes the circle — temperature feeds back to the economy. In simple DICE, one damage function $D(T)$, QUADRATIC:

$$D(T_t) = \psi\, T_t^2$$

summarized as some % of GDP-EQUIVALENT loss per degree of warming. GDP-EQUIVALENT, not GDP — it includes monetized non-market losses (premature mortality, non-working-population morbidity, lost environmental services) via WTP.

Net output

What:

$$Y_t^{\text{Net}} = [1 - \Lambda_t(\mu_t)] \cdot [1 - D(T_t)] \cdot Y_t^{\text{Gross}}$$

Why/Remember (key tradeoff): Higher abatement today ($\Lambda \uparrow$, lower current GDP) ⇔ lower future damages ($D \downarrow$, higher future GDP). This is a standard intertemporal consumption-savings problem applied to climate: abate-now-costs vs. avoided-future-damages.

Discounting and the SCC

Three reasons we value income today more than the future

Remember:

1. Impatience / pure time preference ($\delta$ or $\rho$): humans prefer things now; also mortality and extinction risk.
2. Income growth + concave utility: if you expect to be richer later, the marginal utility of a future franc is LOWER. Reflected in the CONCAVITY of $u$.
3. Opportunity cost of capital: money invested today earns a real return $r$; by no-arbitrage you won't pay more than the bank-equivalent. If you can earn 5% real, a franc next year is worth $1/(1.05) \approx 0.95$ today.

The objective: present discounted lifetime utility

What:

$$U = \sum_{t} \frac{1}{(1+\rho)^t}\, u(c_t)$$

Remember: $\rho$ (pure rate of time preference / utility discount rate) captures how much LESS we value a util just because it's in the future. Even mild utility discounting compounds: $\rho = 1$%/yr ⇒ a util in 2100 is worth ~HALF a util today. DICE takes a MARKET-BASED view (opportunity-cost-of-capital focus).

Social Cost of Carbon (SCC)

What: The present value of marginal damages from emitting +1 ton of CO₂ today, summed over all affected agents and all time:

$$\text{SCC}_t = \sum_{j=0}^{T_{\max}} L_{t+j} \cdot \frac{1}{(1+\rho)^j} \cdot \frac{\Delta u(c_{t+j})}{\Delta u(c_t)} \cdot \frac{\Delta Y_{t+j}}{\Delta T_{t+j}} \cdot \frac{\Delta T_{t+j}}{\Delta E_t}$$

Why: It's the marginal external cost of carbon — the "right" Pigouvian tax / ETS price.

Remember: Computed MANY times under different assumptions → a DISTRIBUTION. DICE-2023 headline SCC ≈ $100/tCO₂ in 2025 USD. Highly sensitive to the discount rate.

Discounting drives SCC differences

Remember: Lower $\rho$ (more patient) ⇒ HIGHER SCC, because damages accrue over centuries and you weight them more. When you set DICE's discount rate equal to other models' rates, you get essentially the SAME SCC — so the headline disagreement (e.g. Stern vs. Nordhaus) is mostly about DISCOUNTING/ETHICS, not climate science. Two takeaways: (1) discounting is critical for SCC; (2) the media-portrayed disagreement is exaggerated once you adjust for discounting.

DICE policy scenarios and results

The five named scenarios

Remember:

1. Baseline / business-as-usual: keeps current climate-policy trends — assumes today's IMF-estimated average global carbon price ~$6/tCO₂, growing ~2.5%/yr in real terms.
2. Extended Paris: meet all unconditional Nationally Determined Contributions (NDCs) by 2030, then a new agreement continuing moderate mitigation.
3. Cost-benefit optimal: maximize net present value of utility for the average global person (benefits AND costs of mitigation).
4. Cost-effectiveness: cheapest path that respects the 2°C target.
5. Alternative discounting: vary the near-term discount rate.

Emissions and warming results

Remember: Current/baseline emissions are FAR above the 2°C path. Extended Paris takes a good chunk out but stays above both 2°C and the cost-benefit-optimal path — because the modeled "Paris" uses the UNCONDITIONAL NDC pledges, which fell short of the treaty's ~2°C ambition. Warming under BAU ≈ 3.5°C by end of century.

SCC results

Remember: Headline DICE-2023 SCC ≈ $100/tCO₂ (2025 USD). A LOWER discount rate (more patient) ⇒ much HIGHER SCC. When harmonizing discount rates, DICE agrees closely with other models. "Trillions of dollars at stake."

Key formulas & one-line takeaways

Key formulas

$$Y = C + I + G + X - M \qquad \text{(expenditure approach to GDP)}$$

$$\text{PPP multiplier} = \frac{\text{basket price in A}}{\text{basket price in B}}; \quad \text{equiv. salary}_A = \text{salary}_B \times \text{PPP multiplier}$$

$$Y_t = A_t F(K_t, H_t), \quad H_t = L_t h_t \qquad \text{(production function; long-run growth} \Leftarrow A_t)$$

$$\text{social MC} = \text{private MC} + \text{external MC}; \quad \text{Pigouvian tax} = \text{marginal external cost}$$

$$E_t^{\text{Baseline}} = \sigma_t Y_t; \quad E_t = (1-\mu_t)E_t^{\text{Baseline}}; \quad \Lambda_t(\mu_t) = \theta_{1,t}\mu_t^{\theta_2}\,(\theta_2 \approx 2.6)$$

$$D(T) = \psi T^2; \quad Y_t^{\text{Net}} = [1-\Lambda_t(\mu_t)][1 - D(T_t)]\, Y_t^{\text{Gross}}$$

$$U = \sum_t \frac{1}{(1+\rho)^t} u(c_t)$$

$$\text{SCC}_t = \sum_{j=0}^{T_{\max}} L_{t+j}\cdot\frac{1}{(1+\rho)^j}\cdot\frac{\Delta u(c_{t+j})}{\Delta u(c_t)}\cdot\frac{\Delta Y_{t+j}}{\Delta T_{t+j}}\cdot\frac{\Delta T_{t+j}}{\Delta E_t}$$

One-line takeaways

• GDP = market value of final goods; three approaches (production/value-added, expenditure $C+I+G+X-M$, income).
• Rising NOMINAL GDP ≠ rising REAL GDP — the Appleville money-illusion trap.
• PPP: multiply a salary by the basket-price ratio (A/B) to get the other location's equivalent.
• Production function $Y = A F(K, H)$; in the long run only productivity ($A$) growth sustains per-capita growth. Malthus was wrong because he underestimated TFP growth.
• Externality ⇒ social MC > private MC; the Pigouvian tax = marginal external cost internalizes it. Not news — Friedman/Hayek knew.
• IAMs = circular economy↔environment models. Three integration steps: add energy input, add a climate model, let temperature affect output.
• DICE (Nordhaus, 2018 Nobel): $E^{Baseline}=\sigma Y$, abatement rate $\mu$, CONVEX abatement cost $\Lambda=\theta_1\mu^{\theta_2}$ ($\theta_2\approx 2.6$), QUADRATIC damages $D=\psi T^2$, net output $Y^{Net}=(1-\Lambda)(1-D)Y^{Gross}$.
• DICE-2023: FAIR climate emulator, four-box carbon cycle (absorption slows as ocean warms/acidifies); instantly abating all industrial CO₂ ≈ 11% of GDP; Arrhenius (1896) already got the concavity.
• Damages are % of GDP-EQUIVALENT (includes non-market via WTP) and OMIT pieces like wildfire; ~3% looks small but is COVID-sized every year.
• Key climate-policy tradeoff: abate-now costs vs. avoided-future-damages — a standard intertemporal savings problem.
• Three reasons to discount: impatience ($\rho$, mortality/extinction), income growth + concave utility, opportunity cost of capital ($r$).
• SCC = PV of marginal damages of +1 tCO₂ over all agents and time; DICE-2023 ≈ $100/tCO₂ (2025 USD); extremely sensitive to the discount rate — lower $\rho$ ⇒ higher SCC. Stern vs. Nordhaus differ on ETHICS, not climate science.