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EnergyMarketsPrimer.md

E3 Modeling Concepts Primer

Foundations for Understanding GEM-E3, PRIMES, and PROMETHEUS

Purpose: This document provides an accessible introduction to all foundational concepts needed to understand the E3Modelling suite (GEM-E3, PRIMES, PROMETHEUS). It is designed for readers with a basic quantitative background but limited exposure to economics, energy systems, or optimization theory.

How to use this document: Read sequentially for a comprehensive foundation, or jump to specific chapters as needed. Each chapter is self-contained but builds on earlier material.

Part I: Economic Foundations

Chapter 1: Markets and Demand

1.1 What Is a Market?

A market is any arrangement that brings together buyers and sellers to exchange goods, services, or resources. Markets can be physical (like a farmer's market) or abstract (like the global oil market or the European electricity market).

In energy-economy models, we represent many interconnected markets:

Goods markets: Steel, chemicals, transportation services
Factor markets: Labor, capital
Energy markets: Electricity, natural gas, oil products
Emissions markets: Carbon permits (in cap-and-trade systems)

1.2 Demand

Demand describes how much of a good buyers are willing and able to purchase at various prices.

The Law of Demand: All else equal (ceteris paribus—Latin for "other things being equal"), as price increases, quantity demanded decreases. This inverse relationship is fundamental to economics.

Demand curve: Shows the maximum price each buyer is willing to pay. Different buyers value the good differently:

Some buyers value it highly (would pay a lot) → top of curve
Others value it less (would only buy if cheap) → bottom of curve

Price (P)
    │
    │\  ← Buyers who value it most (would pay high price)
    │ \
    │  \
    │   \  Demand curve (D)
    │    \
    │     \ ← Buyers who value it least
    └──────\─────── Quantity (Q)

Demand shifters (factors that move the entire curve):

Income: Higher income typically increases demand
Prices of related goods: Substitutes and complements
Preferences: Consumer tastes and habits
Expectations: Anticipated future price changes
Population: Number of potential buyers

1.3 Price Elasticity

Elasticity measures responsiveness—how much one variable changes when another changes.

Price elasticity of demand (ε_d): $$\varepsilon_d = \frac{\text{% change in quantity demanded}}{\text{% change in price}} = \frac{\Delta Q / Q}{\Delta P / P}$$

Point elasticity (calculus form): $$\varepsilon_d = \frac{dQ}{dP} \cdot \frac{P}{Q}$$

The calculus form gives the instantaneous rate of change—more precise for smooth demand curves. The percentage formula above is the discrete approximation (arc elasticity). Both forms are equivalent in the limit.

Elasticity Value	Classification	Interpretation
\|ε\| > 1	Elastic	Demand responds strongly to price changes
\|ε\| = 1	Unit elastic	Proportional response
\|ε\| < 1	Inelastic	Demand responds weakly to price changes
ε = 0	Perfectly inelastic	Demand unchanged by price (e.g., insulin)

Why elasticity matters in E3 models:

Energy demand is typically inelastic in the short run (you can't immediately change your car or heating system) but more elastic in the long run (you can buy a more efficient car, insulate your home).

Energy Type	Short-run Elasticity	Long-run Elasticity
Gasoline	-0.1 to -0.3	-0.5 to -0.8
Electricity	-0.1 to -0.2	-0.3 to -0.7
Natural gas	-0.1 to -0.3	-0.5 to -1.0

Literature ranges—values vary substantially by study, time period, region, and estimation method. These are illustrative central tendencies, not consensus estimates. See Espey (1998) for gasoline, Labandeira et al. (2017) for a meta-analysis.

Cross-price elasticity measures how demand for one good responds to prices of another: $$\varepsilon_{xy} = \frac{\text{% change in demand for x}}{\text{% change in price of y}}$$

If ε_xy > 0: goods are substitutes (gas and coal for electricity generation)
If ε_xy < 0: goods are complements (cars and gasoline)

Income elasticity measures how demand responds to income changes: $$\varepsilon_I = \frac{\text{% change in demand}}{\text{% change in income}}$$

Normal goods: ε_I > 0 (demand increases with income)
Inferior goods: ε_I < 0 (demand decreases with income)
Necessities: 0 < ε_I < 1
Luxuries: ε_I > 1

1.4 Where This Appears in E3 Models

Model	Demand Modeling
GEM-E3	Utility-based demand; responds to prices and income
PRIMES	Energy demand by sector; price-responsive
PROMETHEUS	Econometric demand with income and price elasticities

Key insight: E3 models don't assume fixed demand—they model how demand responds to price and income changes through elasticity parameters. Getting these right is crucial for realistic policy analysis.

How do producers decide what to supply? That depends on production costs—see Chapter 2.

Chapter 2: Production and Supply

Chapter 1 covered what buyers want. Now: how do firms produce, and what determines what they're willing to sell?

2.1 What Is a Production Function?

A production function describes how inputs (labor, capital, energy, materials) are transformed into outputs:

$$Y = F(K, L, E, M)$$

where:

$Y$ = output (quantity produced)
$K$ = capital (machines, buildings)
$L$ = labor (workers, hours)
$E$ = energy
$M$ = materials (intermediate inputs)

2.2 Key Concepts

Marginal product: Additional output from one more unit of input $$MP_L = \frac{\partial F}{\partial L}$$

This partial derivative measures how much extra output you get from adding one more unit of labor, holding capital fixed.

Returns to scale:

Constant returns (CRS): Double inputs → double output
Increasing returns (IRS): Double inputs → more than double output
Decreasing returns (DRS): Double inputs → less than double output

CGE models typically assume constant returns to scale (modeling choice: required for competitive equilibrium with zero profits, but many real industries have increasing returns).

Substitution: Can one input replace another?

Perfect substitutes: can trade 1-for-1 (e.g., different grades of coal)
Perfect complements: must use in fixed proportions (e.g., one driver per truck)
Imperfect substitutes: can trade off, but at varying rates

2.3 The CES Production Function

Constant Elasticity of Substitution (CES) is the workhorse of CGE models. (Modeling choice: CES is chosen for tractability, not because it's the "true" form of production.)

The core question: You combine inputs to produce output. If one input gets expensive, how much can you substitute the other?

Example (capital K vs labor L):

Wages rise → do you automate (buy machines instead of hiring)?
How responsive is that substitution to the price change?

This responsiveness is measured by the elasticity of substitution σ:

$$\sigma = \frac{\Delta (K/L) / (K/L)}{\Delta (P_L/P_K) / (P_L/P_K)}$$

In words: σ = (% change in K/L ratio) / (% change in relative price).

Real-world intuition for σ:

σ ≈ 0 (complements): Truck + driver. 10 trucks + 1 driver = 1 delivery. Can't substitute.
σ ≈ 1 (Cobb-Douglas): Factory automation. Can replace some workers with machines, but diminishing returns.
σ → ∞ (perfect substitutes): Hiring contractors vs employees for identical work. Only cost matters.

The CES formula: $$Y = A \left[ \alpha \cdot K^{\rho} + (1-\alpha) \cdot L^{\rho} \right]^{1/\rho}$$

where σ = 1/(1-ρ). The formula is chosen so that σ stays constant regardless of your current K/L mix—that's the "constant" in CES. This makes the math tractable: you don't need to track where you are on some complex curve.

Note: CES can combine ANY two inputs—not just K and L. In nested CES structures (see next section), different nests combine different things: capital vs labor, coal vs gas, energy vs materials. Each nest has its own σ.

Parameters:

$A$ = productivity multiplier (better technology → more output from same inputs)
$\alpha$ = capital's importance (how much of output is "due to" capital vs labor)
$\rho$ = substitution parameter → determines σ

Why CES is used:

Flexible: nests multiple functional forms
Empirically estimable: σ can be estimated from data
Analytically tractable: has nice mathematical properties

Limitations to keep in mind:

CES assumes substitution elasticity is constant across all input ratios—real production may have varying substitutability
Aggregating heterogeneous firms into a single CES function can bias estimated elasticities
The nesting structure (which inputs are grouped together) is a modeling choice that affects results but is rarely tested empirically
CES cannot represent situations where inputs become complements at some ratios and substitutes at others

2.4 Nested CES Production

Real production uses many inputs with different substitution possibilities. Nested CES handles this by grouping inputs:

Output (σ ≈ 0, near-Leontief)
├── Value Added (σ_KL ≈ 0.5)
│   ├── Capital
│   └── Labor
└── Intermediate Bundle (σ_M ≈ 0.3)
    ├── Energy Bundle (σ_E ≈ 0.5-1.0)
    │   ├── Electricity
    │   └── Fossil Fuels (σ_F ≈ 1.0-2.0)
    │       ├── Coal
    │       ├── Oil
    │       └── Gas
    └── Materials (σ_NE ≈ 0.2)

Each nest has its own elasticity:

Low σ at top (output needs both labor/capital AND materials)
Higher σ within energy (can substitute between fuels)
Highest σ within fossil fuels (gas for coal is easier than electricity for heat)

2.5 Elasticity Values Matter!

The substitution elasticities are crucial parameters—they determine how the economy responds to price changes.

Typical values used in GEM-E3:

Elasticity	Symbol	Range	Impact
Capital-Labor	σ_KL	0.4-1.0	How automation responds to wages
Energy-Value Added	σ_E,VA	0.1-0.5	Very important for climate policy
Interfuel	σ_F	0.5-2.0	Fuel switching response
Armington (trade)	σ_A	2.0-8.0	Trade response to prices (see Chapter 11)

These ranges are illustrative, drawn from econometric literature. Actual values are contested and vary by sector, region, and estimation approach. The energy-value added elasticity (σ_E,VA) is particularly uncertain and consequential—different credible estimates can change policy cost projections by factors of 2-3x.

Higher σ_E,VA → easier to reduce energy use → lower cost of carbon policy Lower σ_E,VA → harder to reduce energy use → higher cost of carbon policy

This is why sensitivity analysis on elasticities is essential—results should always be tested against alternative plausible values.

2.6 Technical Change

Autonomous Energy Efficiency Improvement (AEEI): Over time, economies become more energy-efficient even without price changes. This is captured by a trend parameter:

$$E_t = E_0 \cdot (1 - AEEI)^t \cdot f(Y_t, P_t)$$

This says energy demand at time $t$ equals base-year demand, reduced by autonomous efficiency gains $(1-AEEI)^t$, then adjusted for economic activity and prices via $f(Y_t, P_t)$. Typical AEEI values: 0.5-1.5% per year.

Endogenous technical change: Some models make efficiency improvements depend on R&D spending or learning-by-doing.

2.7 Cost Functions and Input Demands

The firm's problem: Produce Y units at minimum cost, given input prices (wage W, capital rental R).

Cost function: $C(R, W, Y)$ = minimum cost to produce Y at prices R, W.

With "constant returns to scale" (2× inputs → 2× output), cost-per-unit is constant: $C = Y \cdot c(R,W)$.

Shephard's Lemma: Optimal input demand equals the derivative of cost w.r.t. that input's price: $$K^* = \frac{\partial C}{\partial R}, \quad L^* = \frac{\partial C}{\partial W}$$

General intuition: When an input price rises by €1, total cost rises by the amount of that input used (€1 × quantity). Hence ∂C/∂(price) = quantity.

Example: 10 machines × €1 price increase = €10 cost increase, so ∂C/∂R = 10.

(Re-optimization effects are second-order—"envelope theorem.")

Why this is useful: The cost function C embeds the solution to "minimize cost subject to producing Y." Shephard's Lemma is a shortcut: once you have C, extract input demands by differentiation—no need to re-solve the optimization. CGE models specify convenient functional forms for C (e.g., CES), and input demands follow automatically.

2.8 From Costs to Supply

Now we can complete the producer's story: production function → cost function → supply curve.

The supply curve is the marginal cost curve. A profit-maximizing firm supplies another unit only if the price covers the cost of producing it:

At low quantities, marginal cost is low → willing to supply at low price
At high quantities, marginal cost rises → need higher price to justify production

Price (P)
    │           /
    │         / ← Expensive units (need high price)
    │       /  Supply = Marginal Cost
    │     /
    │   /
    │  / ← Cheap units (low marginal cost)
    │ :  ← Minimum price (no supply below this)
    └─────────────── Quantity (Q)

Caveat: This upward slope assumes diminishing returns, not economies of scale. With economies of scale (larger production → lower costs), you get natural monopolies instead of competitive markets. CGE models typically assume constant returns to scale for tractability.

Supply shifters:

Input prices: Cost of labor, raw materials, energy (affects cost function → shifts supply)
Technology: Productivity improvements (higher A in production function)
Number of sellers: Market entry/exit
Government policies: Taxes, subsidies, regulations

Carbon tax example: A carbon tax raises energy input prices → increases production costs → shifts supply curve up → at any given price, less is supplied.

2.9 Where This Appears in E3 Models

Model	Production & Supply
GEM-E3	Nested CES for all 31 sectors; supply from cost minimization
PRIMES	Technology-specific engineering detail; explicit cost curves
PROMETHEUS	Reduced-form; aggregate production relationships

Now that we understand both demand (Chapter 1) and supply (Chapter 2), we can see how they come together—see Chapter 3.

Chapter 3: Equilibrium Concepts

We've seen what buyers want (Chapter 1) and what sellers offer (Chapter 2). Now: how do they meet?

3.1 Market Equilibrium

Equilibrium occurs when quantity demanded equals quantity supplied. At the equilibrium price, the market "clears"—no excess supply or demand.

Price (P)
    │           S
    │         /
    │       / 
    │     /
P*  │----X--------  ← Equilibrium price
    │   / \
    │ /    \
    │/      \ D
    └────────────── Quantity (Q)
          Q*
          ↑
    Equilibrium quantity

Market clearing condition: $$Q^D(P^) = Q^S(P^)$$

where $Q^D$ = quantity demanded, $Q^S$ = quantity supplied, and $P^*$ = equilibrium price.

This equation is central to all economic models. In GEM-E3, market clearing must hold for all goods, factors, and permits simultaneously.

3.2 What Is Economic Equilibrium? (General Concept)

More broadly, equilibrium is a state where no agent has an incentive to change their behavior given current prices and the choices of others. It's a "rest point" of the economic system.

Think of it like a ball in a bowl: the ball settles at the bottom where forces balance. In economics, equilibrium is where supply and demand forces balance across all markets.

3.3 Partial Equilibrium vs. General Equilibrium

This distinction is crucial for understanding the difference between PRIMES and GEM-E3.

Key terminology:

Exogenous variable: Determined outside the model; taken as given (e.g., world oil prices in a national model)
Endogenous variable: Determined within the model by the equilibrium conditions (e.g., domestic prices, quantities)

Partial Equilibrium:

Analyzes one market in isolation
Holds prices in other markets constant ("ceteris paribus")
Ignores feedback effects from the rest of the economy
Simpler, more detailed for the market in question

Example: Analyzing the electricity market, taking GDP, labor costs, and other prices as given.

General Equilibrium:

Analyzes all markets simultaneously
All prices adjust together to clear all markets
Captures feedback effects and interdependencies
More complex, but more comprehensive

Example: Analyzing how a carbon tax affects electricity prices, which affects production costs, which affects wages, which affects consumption, which affects electricity demand again...

Partial Equilibrium (PRIMES approach):
+-------------------------------------+
|         Energy Sector               |
|  +---------+    +---------+         |
|  | Supply  | <> | Demand  |         |
|  +---------+    +---------+         |
|         v equilibrium v             |
|        Energy prices                |
+-------------------------------------+
        ^                   v
   GDP (fixed)         Energy prices
        ^                   v
  ---------- REST OF ECONOMY (exogenous) ----------


General Equilibrium (GEM-E3 approach):
+-------------------------------------------------+
|                 ENTIRE ECONOMY                  |
|  +----------+  +----------+  +----------+       |
|  |  Goods   |<>| Factors  |<>|  Energy  |       |
|  | markets  |  |  (L,K)   |  | markets  |       |
|  +----------+  +----------+  +----------+       |
|       ^             ^             ^             |
|       +--------------------------+              |
|                     v                           |
|           ALL PRICES ADJUST                     |
|           SIMULTANEOUSLY                        |
+-------------------------------------------------+

3.4 Walrasian General Equilibrium

The theoretical foundation for Computable General Equilibrium (CGE) models like GEM-E3 comes from Léon Walras (1874) and was formalized by Arrow and Debreu (1954).

Perfect competition — the key assumption:

Many buyers and sellers — no single agent can influence the market price
Price-takers — everyone takes the market price as given and decides only how much to buy/sell
Homogeneous goods — products are identical (no brand loyalty)
Perfect information — everyone knows all prices and qualities
Free entry/exit — firms can enter or leave the market without barriers

This is an idealization. Real markets have market power, asymmetric information, and barriers to entry. But it's a useful benchmark.

Walrasian equilibrium (theoretical framework) is a set of prices such that:

Each consumer maximizes utility given their budget constraint
Each firm maximizes profit given its technology
All markets clear (supply = demand)

Mathematically: Find price vector $\mathbf{p}^* = (p_1^, p_2^, ..., p_n^*)$ such that:

$$z_i(\mathbf{p}^*) = 0 \quad \forall i$$

where:

$\mathbf{p}$ = price vector (prices of all $n$ goods)
$z_i(\mathbf{p})$ = excess demand in market $i$: $z_i = D_i(\mathbf{p}) - S_i(\mathbf{p})$
$D_i, S_i$ = demand and supply for good $i$ (functions of all prices, not just $p_i$)

Key properties of Walrasian equilibrium:

Walras' Law: The value of total excess demand is always zero: $\sum_i p_i \cdot z_i(\mathbf{p}) = 0$
If all but one market clears, the last one clears automatically
Prices are only determined up to a numeraire (we can normalize one price to 1)

Limitations to keep in mind:

Walrasian equilibrium assumes all agents are price-takers with perfect information—real markets have market power, asymmetric information, and transaction costs
The theory says nothing about how equilibrium is reached or how long adjustment takes
Multiple equilibria may exist; the model finds one but can't tell you which one the economy would actually reach
The framework is comparative statics: it compares equilibria, not the transition path between them

Simple Example: Two-Good Exchange Economy

Consider two consumers (A and B) and two goods (apples and bread):

	Apples	Bread
A's endowment	10	0
B's endowment	0	10

Both prefer variety. If $P_{apples} = P_{bread} = 1$:

A sells 5 apples, buys 5 bread → ends with (5, 5)
B sells 5 bread, buys 5 apples → ends with (5, 5)

Market clearing check:

Apples: A supplies 5, B demands 5 ✓
Bread: B supplies 5, A demands 5 ✓

This is a Walrasian equilibrium: both maximize utility given their budget, and both markets clear.

3.5 The Auctioneer Metaphor

Walras imagined a fictional "auctioneer" who:

Announces prices
Collects supply and demand from all agents
Adjusts prices (raise if excess demand, lower if excess supply)
Repeats until all markets clear

This is called tâtonnement (French for "trial and error" or "groping toward equilibrium"). CGE models effectively implement this process computationally.

3.6 Existence and Uniqueness

Arrow-Debreu Theorem: Under certain conditions (continuous preferences, no increasing returns to scale, etc.), a Walrasian equilibrium exists.

Uniqueness is not guaranteed—there may be multiple equilibria. CGE models typically find one equilibrium (the one closest to the starting point).

3.7 Comparative Statics

CGE models use comparative statics: comparing two equilibria (before and after a policy change) without modeling the transition path.

Equilibrium A          →          Equilibrium B
(no carbon tax)       Policy      (with carbon tax)
                      change

We compare A and B, but don't model the path between them

This is a limitation: CGE models tell you the new equilibrium, not how long it takes to get there or what happens during the transition.

3.8 Dynamic Extensions

Recursive dynamics (used in GEM-E3):

Solve a sequence of static equilibria
Each period, capital stocks update based on previous period's investment
Agents have adaptive expectations (don't perfectly foresee the future)

Period 1 → Period 2 → Period 3 → ...
    ↓          ↓          ↓
  K₁ → I₁ → K₂ → I₂ → K₃ → ...

Intertemporal optimization (used partly in PRIMES):

Agents optimize over entire time horizon
Perfect foresight (or rational expectations)
More computationally demanding

3.9 Labor Markets: Beyond Perfect Competition

Standard Walrasian equilibrium assumes labor markets clear → no involuntary unemployment. Reality disagrees.

GEM-E3's solution: Efficiency Wages (Shapiro-Stiglitz 1984)

Core idea: Firms can't perfectly monitor effort. If wages = market-clearing, workers have nothing to lose by shirking (jobs are easy to find). So firms pay a premium → creates unemployment → workers fear job loss → don't shirk.

The efficiency wage: $$w = w^* \cdot \left(1 + \frac{e}{b + \rho/q}\right)$$

$w^*$ = market-clearing wage; $e$ = required effort
$b$ = unemployment benefit rate; $\rho$ = discount rate; $q$ = detection probability

Intuition: Higher required effort or weaker penalties for shirking (generous benefits, low detection, high discounting) → bigger premium needed. Higher unemployment → smaller premium (job loss is scarier).

For GEM-E3: Unemployment is endogenous. Labor cost policies have employment effects. Revenue recycling (carbon tax → cut labor taxes) can deliver a "double dividend."

3.10 Where This Appears in E3 Models

Model	Equilibrium Concept
GEM-E3	Full general equilibrium (all markets clear simultaneously)
PRIMES	Partial equilibrium (energy sector only); takes GDP as exogenous
PROMETHEUS	Market clearing for world energy; partial equilibrium

Key insight from the technical summaries:

GEM-E3: "Find price vector p* such that all markets clear simultaneously"
PRIMES: "EPEC (Equilibrium Problem with Equilibrium Constraints)"—finds equilibrium within the energy sector
The models are coupled to get benefits of both: PRIMES provides energy detail, GEM-E3 provides economy-wide feedback

Chapter 4: Welfare Economics

4.1 What Is Welfare?

Welfare in economics refers to the well-being or satisfaction of individuals and society. Welfare economics asks: Is one economic outcome "better" than another? How do we measure the impact of policies on society?

This matters for E3 models because policymakers need to know not just what happens (GDP change, emissions reduction) but whether society is better or worse off.

4.2 Utility

Utility = numerical measure of satisfaction. Can't observe it directly, but infer from choices.

Utility function: $U = U(x_1, x_2, ..., x_n)$ where $x_i$ = quantity of good $i$.

Key properties: More is better (non-satiation). Diminishing marginal utility (10th slice < 1st slice). Ordinal only (rankings matter, absolute numbers don't).

Example (Cobb-Douglas): $$U(C, L) = C^{\alpha} \cdot L^{1-\alpha}$$

C = consumption (goods/services); L = leisure (time not working); α ∈ (0,1) = weight on consumption

This models the labor-leisure tradeoff: work more → more income (C) but less free time (L). Both essential (U=0 if either is zero). Constant shares: fraction α of time-budget goes to work, (1-α) to leisure.

4.3 Consumer and Producer Surplus

Markets use one price for heterogeneous buyers and sellers:

Consumer surplus: Buyers who would've paid more than P* keep the difference
Producer surplus: Sellers who would've accepted less than P* keep the difference

Price (P)
    │ D
    │  \        S
    │   \      /    Consumer Surplus (above P*, below D)
    │    \    /
P*  │------X------ ← uniform market price
    │     /  \      Producer Surplus (below P*, above S)
    │    /    \
    │ S          D
    └──────────────── Quantity (Q)
          Q*

Total surplus = gains from trade. Deadweight loss = surplus destroyed by distortions (taxes, monopoly)—trades that would've been mutually beneficial but didn't happen.

4.4 Pareto Efficiency

Allocation: Who gets what—a distribution of goods across people, each yielding some utility.

Pareto efficient: No way to make someone better off without making someone else worse off. Think: pie split with no crumbs on the table. But 50-50 and 99-1 are both Pareto efficient—it says nothing about fairness.

Pareto improvement: A change that helps at least one person without hurting anyone.

The Welfare Theorems:

First: Competitive markets reach Pareto efficient outcomes. (No gains from trade left on the table.)

Second: Any Pareto efficient allocation can be achieved via markets—if you first redistribute endowments using lump-sum (non-distortionary) transfers. This conceptually separates efficiency (let markets work) from equity (who starts with what).

The catch: Real redistribution (income tax, means-tested welfare) depends on behavior → creates distortions → efficiency costs. Lump-sum transfers are a theoretical device, not a policy tool. Hence the equity-efficiency tradeoff: in practice, more redistribution typically means some efficiency loss.

For policy analysis: Markets achieve efficiency but not fairness. Policies create winners and losers. We need welfare measures to compare outcomes (next sections).

4.5 Measuring Welfare Changes

When prices change, how do we convert utility changes into money?

Equivalent Variation (EV): The € amount (at OLD prices) that would cause the same utility change as the policy.

Compensating Variation (CV): The € amount (at NEW prices) needed to restore original utility after the policy.

Measure	Evaluated at	Question
EV	OLD prices	"How much € (at old prices) = same utility change?"
CV	NEW prices	"How much € (at new prices) = same utility change?"

Why they differ (for a price increase):

EV (old prices): €1 buys more → need fewer € taken away to reach new lower utility → EV smaller
CV (new prices): €1 buys less → need more € added to restore old higher utility → CV larger

Example: Price rises €0.10 → €0.15/kWh. You cut consumption 1000 → 800 kWh.

Bill rises €20, but welfare loss > €20 (you also gave up 200 kWh you valued)
EV (at old prices where €1 = 10 kWh): smaller € amount
CV (at new prices where €1 = 6.7 kWh): larger € amount

For small price changes, EV ≈ CV ≈ consumer surplus change. GEM-E3 uses EV for cross-scenario comparability.

4.6 Social Welfare Functions

To compare outcomes where some gain and others lose, we need a social welfare function (SWF) that aggregates individual utilities:

Utilitarian (Benthamite): $$W = \sum_i U_i$$ Sum of all utilities—treats everyone equally.

Rawlsian: $$W = \min_i {U_i}$$ Only care about the worst-off individual.

Weighted sum: $$W = \sum_i \omega_i \cdot U_i$$ Different groups have different weights (distributional preferences).

CGE models typically use utilitarian welfare (sum or average of equivalent variations), but can report distributional impacts separately.

4.7 Efficiency vs. Equity Trade-offs

Most real policies involve trade-offs:

Policy	Efficiency	Equity
Carbon tax	✅ Corrects externality, efficient	❌ Regressive (hurts low-income more)
Carbon tax + rebates	✅ Still efficient	✅ Can be made progressive
Subsidies for renewables	❓ May distort markets	✅ Benefits vary

GEM-E3 can analyze both:

Total welfare change (efficiency)
Welfare change by region, sector, or income group (distributional)

4.8 The Double Dividend Hypothesis

A key policy question: Can environmental taxes provide a "double dividend"?

First dividend: Environmental improvement (less pollution)
Second dividend: Economic improvement (if tax revenue is used to reduce distortionary taxes like labor taxes)

GEM-E3 is specifically designed to analyze this because it models:

Environmental taxes and their revenue
Labor market with unemployment
Revenue recycling options (lump-sum vs. tax cuts)

4.9 Where This Appears in E3 Models

Model	Welfare Concepts
GEM-E3	Reports Equivalent Variation by region; analyzes double dividend
PRIMES	Reports total system costs; implicit welfare through consumer choice
PROMETHEUS	No explicit welfare; focuses on price projections

End of Part I: Economic Foundations

Continue to Part II: Mathematical Methods for the mathematical tools used in these models.

Part II: Mathematical Methods

A note on reading this part: The sections below mix three types of statements:

Definitions — what terms mean (e.g., "elasticity is the ratio of percentage changes")

Modeling choices — how E3 models implement concepts (e.g., "GEM-E3 uses CES production functions")

Empirical findings — what data suggests (e.g., "energy demand is typically inelastic")

These are different kinds of claims with different epistemic status. Definitions are conventions; modeling choices are decisions made for tractability; empirical findings are contestable summaries of evidence.

Chapter 5: Optimization Primer

5.1 Why Optimization?

Economic models assume agents optimize: consumers maximize utility, firms minimize costs. Combined with market clearing (supply = demand), this generates equilibrium.

Different models, different mathematical structures:

Model	What's optimized	Mathematical form
PRIMES	Total energy system cost	LP/MILP — single objective, find minimum
PROMETHEUS	Market clearing	Nonlinear equations — find prices where supply = demand
GEM-E3	Many agents simultaneously	MCP — no single objective; find prices where everyone's optimal choices are mutually consistent

The key difference: PRIMES has one decision-maker (a social planner minimizing cost). GEM-E3 has many decision-makers (consumers, firms, government), each optimizing their own objective. You can't write "many agents optimizing" as a single optimization problem—but you can write down the conditions that must hold when all are at their optima simultaneously. That's what MCP does.

5.2 The Basic Optimization Problem

Unconstrained optimization: $$\max_{x} f(x) \quad \mathrm{or} \quad \min_{x} f(x)$$

First-order condition (FOC): At an optimum, the derivative is zero: $$\frac{df}{dx} = 0$$

Second-order condition (SOC): For a maximum, $f''(x) < 0$; for a minimum, $f''(x) > 0$.

5.3 Constrained Optimization and Lagrange Multipliers

Most economic problems involve constraints (budget constraints, resource limits, etc.).

General form: $$\max_{x} f(x) \quad \mathrm{s.t.} \quad g(x) = 0$$

Lagrangian method: $$\mathcal{L}(x, \lambda) = f(x) - \lambda \cdot g(x)$$

First-order conditions: $$\frac{\partial \mathcal{L}}{\partial x} = 0, \quad \frac{\partial \mathcal{L}}{\partial \lambda} = g(x) = 0$$

The Lagrange multiplier (λ) has a powerful interpretation:

λ is the shadow price of the constraint—the marginal value of relaxing the constraint by one unit.

Why? At the optimum, ∇f = λ∇g (gradients parallel). Moving in the direction that relaxes g by 1 unit increases f by λ. So λ = df*/dc where c is the constraint bound.

Connection to duality: λ is also the dual variable. Every constrained optimization has a "dual" problem where λ becomes the decision variable (interpreted as the "price" of the constraint). Under convexity, primal and dual have the same optimal value—this is why λ simultaneously measures: (1) how much you'd pay to relax the constraint, and (2) the "price" that makes the constraint worth respecting.

Example: In a carbon-constrained economy: $$\max \ GDP \quad \mathrm{s.t.} \quad Emissions \leq \bar{E}$$

The Lagrange multiplier on the emissions constraint is the marginal abatement cost—the GDP sacrifice from reducing emissions by one more ton.

5.4 Types of Optimization Problems

Linear Programming (LP): $$\min_{x} c^T x \quad \mathrm{s.t.} \quad Ax \leq b, ; x \geq 0$$

where $c$ = cost vector, $x$ = decision variables, $A$ = constraint matrix, $b$ = constraint bounds.

Objective and constraints are linear
Fast to solve (polynomial time)
Used in simple dispatch models

Mixed-Integer Linear Programming (MILP): Same as LP, but some variables must be integers.

Models yes/no decisions (build a power plant or not)
Much harder to solve (NP-hard)
Used in PRIMES for investment decisions

Nonlinear Programming (NLP): $$\min_{x} f(x) \quad \mathrm{s.t.} \quad g(x) \leq 0, ; h(x) = 0$$

Objective or constraints are nonlinear
CES production functions are nonlinear!
May have multiple local optima

Quadratic Programming (QP): Quadratic objective with linear constraints—a special case of NLP that's easier to solve.

5.5 Shadow Prices (Dual Variables)

Every constraint in an optimization problem has an associated shadow price (also called dual variable or Lagrange multiplier).

Interpretation: The shadow price tells you the marginal value of relaxing the constraint.

In energy models:

Constraint	Shadow Price
Electricity demand = supply	Wholesale electricity price (€/MWh)
Emissions ≤ cap	Carbon permit price (€/tCO₂)
Capacity ≤ installed capacity	Scarcity rent (€/MW)
Renewable share ≥ target	Cost of RES constraint (€/MWh)

GEM-E3 reports shadow prices for all constraints—these are the equilibrium prices.

5.6 Complementarity Problems

CGE equilibrium conditions are typically cast as a Mixed Complementarity Problem (MCP) and solved by specialized solvers.

The complementarity condition: $$0 \leq x \perp f(x) \geq 0$$

The symbol $\perp$ ("perp") denotes complementarity. This notation means: $x \geq 0$, $f(x) \geq 0$, and $x \cdot f(x) = 0$.

Interpretation: Either $x = 0$ or $f(x) = 0$ (or both). They "complement" each other.

In market context: $$0 \leq P \perp (S - D) \geq 0$$

If there's excess supply ($S > D$), price must be zero
If price is positive, the market must clear ($S = D$)

Why MCP instead of standard optimization?

A single agent's optimum satisfies the Karush-Kuhn-Tucker (KKT) conditions:

Gradient condition (∇f = λ∇g)
Feasibility (constraints satisfied)
Complementary slackness (if constraint slack, multiplier = 0)

In a market, each agent has their own KKT conditions. The equilibrium is where:

All agents satisfy their individual KKT conditions
Markets clear (supply = demand for each good)

You can't write this as "minimize some function"—there's no single objective. But you CAN collect all agents' KKT conditions plus market clearing into one big system of complementarity conditions. That's the MCP.

Why specialized solvers (PATH)? Standard optimization solvers (gradient descent, Newton) assume you're minimizing something. MCP solvers handle the "either-or" structure of complementarity (either x=0 or f(x)=0) directly, using pivoting methods similar to how simplex solves LPs.

Limitations to keep in mind:

MCP finds an equilibrium, but multiple equilibria may exist—the solution found depends on the starting point
Convergence is not guaranteed for all problem structures; poorly specified models may fail to solve
The equilibrium found is a mathematical fixed point; whether it represents real-world market outcomes depends on whether the underlying behavioral assumptions hold

5.7 Solving Optimization Problems

Iterative methods:

Start from an initial guess
Compute search direction (gradient, Newton step)
Update solution
Repeat until convergence

GAMS (General Algebraic Modeling System):

Industry-standard language for optimization
Declarative: you describe the problem, solver finds the solution
Multiple solvers: CPLEX (LP/MIP), CONOPT (NLP), PATH (MCP)

Example GAMS structure:

VARIABLES x, z;
EQUATIONS obj, constraint;

obj..           z =e= c * x;           * Objective
constraint..    A * x =l= b;           * Constraint

MODEL mymodel /all/;
SOLVE mymodel USING LP MINIMIZING z;

5.8 Where This Appears in E3 Models

Model	Optimization Approach
GEM-E3	MCP (market equilibrium as complementarity)
PRIMES	LP/MILP for power sector; Equilibrium Problem with Equilibrium Constraints (EPEC) overall
PROMETHEUS	Nonlinear equation system (market clearing)

Chapter 6: Discrete Choice Models

6.1 The Problem: Heterogeneous Choices

In reality, not everyone makes the same choice even when facing the same prices. Some buy electric cars, others buy gasoline cars. Some install heat pumps, others stick with gas boilers.

Why?

Different preferences
Different constraints (budget, space, access)
Different information
Behavioral factors (risk aversion, habits)

PRIMES models this heterogeneity using discrete choice theory.

6.2 Random Utility Models

Basic idea: Each option has a "utility" that includes:

Observable components (price, performance)
Unobservable components (personal taste, hidden costs)

$$U_i = V_i + \varepsilon_i$$

where:

$U_i$ = total utility of option $i$
$V_i$ = systematic (observable) utility
$\varepsilon_i$ = random (unobservable) component

Consumer chooses option with highest total utility.

6.3 The Logit Model

If the random components follow an extreme value (Gumbel) distribution, the probability of choosing option $i$ is:

$$P_i = \frac{e^{V_i / \mu}}{\sum_j e^{V_j / \mu}}$$

where $\mu$ is a scale parameter.

Properties:

Probabilities sum to 1
Higher utility → higher probability
But not deterministic: even expensive options get some market share

Limitations to keep in mind:

IIA (Independence of Irrelevant Alternatives): Adding a new option doesn't change relative shares of existing options. This is often unrealistic—adding a third car brand should affect similar brands more than dissimilar ones.
Assumes unobserved heterogeneity follows a specific distribution (Gumbel). If real heterogeneity differs, market share predictions can be biased.
Nested logit (Section 6.5) partially addresses IIA but requires specifying the nest structure, which is itself a modeling choice.

Example: Technology choice in buildings $$V_i = -\alpha \cdot Cost_i - \beta \cdot Hassle_i + \gamma \cdot Efficiency_i$$

6.4 The Weibull Distribution (in PRIMES)

PRIMES uses Weibull-based market shares:

$$S_i = \frac{e^{-\nu \cdot C_i}}{\sum_j e^{-\nu \cdot C_j}}$$

where:

$S_i$ = market share of technology $i$
$C_i$ = generalized cost
$\nu$ = heterogeneity parameter

The generalized cost includes "intangible costs": $$C_i = (CAPEX_i + OPEX_i) + \mu_i$$

where:

CAPEX = capital expenditure (upfront cost: buying the equipment)
OPEX = operating expenditure (ongoing costs: fuel, maintenance)
μ_i = intangible costs (non-monetary factors—see below)

Intangible costs capture:

Risk aversion (new technology is risky)
Hidden costs (installation complexity, learning time)
Behavioral inertia (familiarity with current technology)
Market barriers (lack of information, financing constraints)

6.5 Nested Logit (for Transport)

When choices have natural groupings, nested logit is used:

Travel choice
├── Private car
│   ├── Gasoline car
│   ├── Diesel car
│   └── Electric car
├── Public transit
│   ├── Bus
│   └── Train
└── Active (walk/bike)

Two-level choice:

Choose mode (car, transit, active)
Choose specific option within mode

Formula: $$P(mode\ m) = \frac{e^{V_m / \lambda_m}}{\sum_{m'} e^{V_{m'} / \lambda_{m'}}}$$

$$P(option\ i\ |\ mode\ m) = \frac{e^{V_i / \mu}}{\sum_{j \in m} e^{V_j / \mu}}$$

where $\lambda_m$ = nest-specific scale parameter (captures correlation within mode $m$), $\mu$ = scale parameter for options within a nest, and $V$ = systematic utility.

6.6 Calibration of Intangible Costs

The intangible costs ($\mu_i$) are calibrated, not estimated:

Observe actual market shares in base year
Calculate what intangible costs would be needed to match these shares given known financial costs
Use these calibrated values for projections

This is a limitation: We're inferring behavior from outcomes, not from direct measurement of preferences.

6.7 Why This Matters for Policy

Discrete choice modeling affects policy analysis:

Without heterogeneity:

Carbon tax makes EVs cheapest
Everyone switches to EVs immediately
Unrealistic!

With heterogeneity:

Some early adopters switch quickly
Others need larger price signals
Technology diffusion is gradual
More realistic policy impact

6.8 Where This Appears in E3 Models

Model	Discrete Choice Application
GEM-E3	Not directly; uses CES (continuous substitution)
PRIMES	Technology choice, mode choice, vehicle choice
PROMETHEUS	Not directly; uses aggregate demand functions

Chapter 7: Stochastic Methods

7.1 The Challenge of Uncertainty

Energy projections face deep uncertainty in:

Future oil prices
Technology costs (will solar keep getting cheaper?)
Economic growth (especially in emerging economies)
Policy evolution
Resource availability

PROMETHEUS explicitly addresses this by treating key parameters as probability distributions rather than point estimates.

7.2 Types of Uncertainty

Aleatory uncertainty: Inherent randomness (weather, accidents)

Can be characterized probabilistically
Won't disappear with more research

Epistemic uncertainty: Lack of knowledge (future technology costs)

Could be reduced with more information
But often we must act before uncertainty resolves

Deep uncertainty: Fundamental unknowns (paradigm shifts, black swans)

Hard to assign probabilities
Scenario analysis may be more appropriate

7.3 Probability Distributions

PROMETHEUS uses distributions for uncertain parameters:

Normal distribution: $X \sim N(\mu, \sigma^2)$

Symmetric, bell-shaped
Used for: demand elasticities, growth rates

Lognormal distribution: $X \sim LN(\mu, \sigma^2)$

Always positive, right-skewed
Used for: prices, resource estimates

Triangular distribution: Defined by min, mode, max

Easy to elicit from experts
Used for: technology learning rates

Uniform distribution: Equal probability in range

Maximum ignorance within bounds
Used when we only know plausible range

7.4 Monte Carlo Simulation

Monte Carlo method:

Draw random values from input distributions
Run the model with these values
Record outputs
Repeat many times (1000-10000)
Analyze distribution of outputs

Input: θ ~ distribution
       │
       ▼
┌──────────────────┐
│     MODEL        │
│  (PROMETHEUS)    │
└──────────────────┘
       │
       ▼
Output: Oil price, demand, etc.

Repeat N times → Output distribution

Output: Not a single forecast, but a probability distribution

Mean, median
Standard deviation
Percentiles (P10, P50, P90)
Full distribution

7.5 Latin Hypercube Sampling (LHS)

Simple random sampling can miss parts of the parameter space. Latin Hypercube Sampling ensures better coverage:

Divide each parameter's range into N equal-probability intervals
Sample exactly once from each interval
Randomly pair samples across parameters

Advantage: More efficient—variance reduction is roughly O(1/N) vs O(1/√N) for random sampling. For expensive models, this can cut required runs by 5-10×.

Simple Random:          Latin Hypercube:
X₂                     X₂
│  •    •              │     •  
│    •     •           │  •     
│  •    •              │        •
│      •  •            │     •  
└──────────── X₁       └──────────── X₁
(clumpy coverage)      (stratified coverage)

7.6 Sensitivity Analysis

Question: Which uncertain inputs drive the most uncertainty in outputs?

Note: Local sensitivity is a deterministic concept (just calculus). We include it here as context before the stochastic method (Sobol indices), which requires Monte Carlo runs.

Local sensitivity: Change one input slightly, see output change $$S_i = \frac{\partial Y}{\partial X_i}$$

Global sensitivity (Sobol indices): Decompose output variance

First-order Sobol index: $$S_i = \frac{V[E[Y|X_i]]}{V[Y]}$$

where $V[\cdot]$ = variance, $E[\cdot]$ = expected value, and $X_{-i}$ = all inputs except $X_i$.

Fraction of output variance explained by input $X_i$ alone.

Total-effect index: $$S_{T,i} = 1 - \frac{V[E[Y|X_{-i}]]}{V[Y]}$$

Includes all interactions involving $X_i$.

Typical PROMETHEUS findings:

Parameter	First-order Index
Oil Ultimate Recoverable Resources (URR)	0.2-0.4
GDP growth	0.1-0.3
OPEC behavior	0.1-0.2
Demand elasticity	0.05-0.15

7.7 Interpreting Stochastic Results

Don't:

Treat the mean as "the forecast"
Ignore the distribution width

Do:

Report ranges: "Oil prices in 2040: €60-120/barrel (80% probability)"
Use for stress-testing policies
Identify which uncertainties matter most

7.8 Where This Appears in E3 Models

Model	Stochastic Treatment
GEM-E3	Deterministic; uses scenario analysis for uncertainty
PRIMES	Deterministic; can use PROMETHEUS price distributions
PROMETHEUS	Full Monte Carlo with LHS; explicit uncertainty quantification

End of Part II: Mathematical Methods

Continue to Part III: Energy Systems for energy-specific concepts.

Part III: Energy Systems

Chapter 8: Energy Fundamentals

8.1 Why Energy Matters for Economic Modeling

Energy is unique among economic goods:

Essential input for virtually all production
Limited substitutability in many uses
Environmental externalities (emissions)
Strategic importance (national security)
Infrastructure-intensive (long-lived capital)

Understanding energy fundamentals is essential for interpreting E3 models.

8.2 Energy Forms and Conversions

Primary energy: Energy as found in nature

Fossil fuels (coal, crude oil, natural gas)
Nuclear (uranium)
Renewables (solar, wind, hydro, biomass, geothermal)

Secondary energy: Transformed/refined energy

Electricity (from any primary source)
Refined petroleum products (gasoline, diesel, jet fuel)
Hydrogen (produced from various sources)

Final energy: Energy delivered to end users

What you buy (electricity at the meter, gasoline at the pump)
Before losses in end-use equipment

Useful energy: Energy service actually provided

Heat delivered to room
Motion of vehicle
Light from bulb

Primary Energy (100 units)
    │
    │ Extraction, refining, generation
    │ (losses: ~30-40%)
    ▼
Secondary/Final Energy (60-70 units)
    │
    │ End-use conversion
    │ (losses: ~30-70%)
    ▼
Useful Energy (20-40 units)

PRIMES models the entire chain from primary to useful energy, capturing losses at each stage.

8.3 Energy Units

Unit	Definition	Typical Use
Joule (J)	SI unit of energy	Scientific
kWh	3.6 MJ	Electricity billing
toe	Tonne of oil equivalent (41.868 GJ)	Energy statistics
Mtoe	Million toe	National/EU level
PJ	Petajoule (10¹⁵ J)	Energy balances
TWh	Terawatt-hour (3.6 PJ)	Electricity statistics
BTU	British Thermal Unit (~1055 J)	US/UK
MMBtu	Million BTU	Natural gas (US)

Conversion factors:

1 toe = 11.63 MWh = 41.868 GJ
1 TWh = 0.086 Mtoe
1 barrel of oil ≈ 0.136 toe

8.4 Energy Balances

An energy balance is an accounting framework showing all energy flows in an economy.

┌──────────────────────────────────────────────────────────────┐
│                         SUPPLY                               │
│  Domestic production + Imports - Exports - Stock changes     │
└──────────────────────────────────────────────────────────────┘
                              ↓
                      Primary Energy Supply
                              ↓
┌──────────────────────────────────────────────────────────────┐
│                     TRANSFORMATION                           │
│  Power plants, refineries, heat plants                       │
│  (input - output = losses)                                   │
└──────────────────────────────────────────────────────────────┘
                              ↓
                      Final Energy Consumption
                              ↓
┌──────────────────────────────────────────────────────────────┐
│                         DEMAND                               │
│  Industry + Transport + Residential + Services + Agriculture │
└──────────────────────────────────────────────────────────────┘

Key identity: $$Primary\ Supply = Final\ Consumption + Transformation\ Losses + Own\ Use$$

PRIMES and GEM-E3 are calibrated to Eurostat energy balances for the base year.

8.5 Energy Intensity

Energy intensity measures how much energy is used per unit of economic output:

$$Energy\ Intensity = \frac{Total\ Primary\ Energy\ Supply}{GDP}$$

Usually expressed in toe/€million or MJ/€.

Decomposition: $$\frac{E}{GDP} = \sum_s \frac{E_s}{Y_s} \cdot \frac{Y_s}{GDP}$$

This decomposes aggregate energy intensity into: (1) sector-level intensity ($E_s/Y_s$) and (2) sectoral shares of GDP ($Y_s/GDP$). A country can reduce energy intensity either by making each sector more efficient or by shifting toward less energy-intensive sectors.

Energy intensity depends on:

Sectoral composition (services vs. heavy industry)
Technology efficiency (better equipment)
Behavior (conservation practices)

Trends:

Developed economies: ~1-2% annual decrease in energy intensity
Driven by structural change + efficiency improvements
Autonomous Energy Efficiency Improvement (AEEI) parameter in models captures this

8.6 Emissions and Carbon Intensity

Carbon intensity of energy: $$Carbon\ Intensity = \frac{CO_2\ Emissions}{Energy\ Consumption}$$

Carbon content by fuel:

Fuel	kg CO₂/GJ	Relative
Coal	94-96	Highest
Oil	73-75	Medium
Natural gas	56-58	Lowest fossil
Biomass	0*	Net zero (if sustainable)
Nuclear	0	Zero direct
Renewables	0	Zero direct

*Biomass carbon is considered biogenic (part of natural cycle).

Emissions accounting in GEM-E3: $$EMI_{f,s,r} = \epsilon_f \cdot E_{f,s,r}$$

where $\epsilon_f$ = emission factor for fuel $f$ (kg CO₂/GJ), $E_{f,s,r}$ = energy consumption of fuel $f$ in sector $s$ and region $r$. This simple multiplication links the economic model to physical emissions.

8.7 Exhaustible Resources: Hotelling Theory

The Hotelling Rule (1931) is fundamental to understanding fossil fuel supply in PROMETHEUS.

Core insight: Exhaustible resources (oil, gas, coal) have a finite stock. Extracting today means less available tomorrow. Owners must consider the opportunity cost of extraction.

The Hotelling condition: $$\frac{dP}{dt} = r \cdot P$$

In words: the price of an exhaustible resource rises at the interest rate $r$.

Intuition:

Resource owner can either extract now (get price P, invest at interest r) or wait (get price P_{t+1})
In equilibrium, they must be indifferent: $P_{t+1} = P_t(1+r)$
Otherwise arbitrage: if prices rise faster than r, everyone waits; if slower, everyone extracts now

With extraction costs: $$\frac{d(P - MC)}{dt} = r \cdot (P - MC)$$

The resource rent (price minus marginal cost) rises at rate r.

Why prices actually fluctuate: The simple Hotelling model predicts smooth price rises, but real prices are volatile because:

Demand shocks (recessions, growth spurts)
Supply shocks (discoveries, wars, technology)
Market power (OPEC decisions)
Uncertainty about reserves

Limitations to keep in mind:

Hotelling assumes rational, forward-looking resource owners with perfect information about reserves—real actors have limited information and heterogeneous expectations
Historical oil prices have not followed Hotelling paths; empirical tests generally reject the simple model
The theory works better as a benchmark for understanding long-run tendencies than as a short-run price predictor
Political factors (sanctions, nationalization, OPEC quotas) often dominate economic logic

PROMETHEUS models this by:

Tracking cumulative extraction relative to URR (Ultimate Recoverable Resources—the total amount ultimately extractable)
Making extraction costs rise as easy reserves deplete
Treating URR as uncertain (probability distributions)

8.8 Market Power in Energy: OPEC

Global oil markets are not perfectly competitive. OPEC (Organization of Petroleum Exporting Countries) has significant market power.

PROMETHEUS models OPEC as a "dominant firm with competitive fringe":

Market demand = OPEC supply + Non-OPEC supply
                    ↓               ↓
            (strategic)      (price-taking)

The dominant firm model:

Competitive fringe supplies according to marginal cost: Fringe supply: $S^{fringe}(P)$ = supply from non-OPEC producers at price $P$
OPEC faces residual demand: $$D^{OPEC}(P) = D^{world}(P) - S^{fringe}(P)$$
OPEC maximizes profit: $$\max_P (P - MC^{OPEC}) \cdot D^{OPEC}(P)$$

Result: OPEC restricts output below competitive level, raising prices.

PROMETHEUS captures this by:

Modeling OPEC production decisions
Including capacity constraints and market share targets
Allowing for different OPEC behavior scenarios (aggressive vs. accommodating)

Why this matters: Oil price projections are highly sensitive to assumptions about OPEC behavior—a major source of uncertainty.

8.9 Where This Appears in E3 Models

Model	Energy Data Use
GEM-E3	Energy balances for calibration; emissions by fuel
PRIMES	Detailed energy flows; transformation sector
PROMETHEUS	Global energy supply/demand; Hotelling dynamics; OPEC behavior

Chapter 9: Power Sector Basics

9.1 Why Power Sector Gets Special Treatment

The electricity sector is modeled in detail because:

Central to decarbonization (electrification of transport, heat)
Highly capital-intensive (long-lived assets)
Complex operations (real-time balancing)
Subject to extensive regulation

PRIMES has a dedicated power sector module with sub-annual temporal resolution (see Section 9.6 on time slices).

9.2 Generation Technologies

Technology	Type	Dispatchable?	Capacity Factor
Coal	Thermal	Yes	40-85%
Natural gas CCGT (Combined Cycle Gas Turbine)	Thermal	Yes	30-60%
Nuclear	Thermal	Baseload	80-95%
Hydro (reservoir)	Renewable	Yes	30-50%
Wind onshore	Renewable	No	20-35%
Wind offshore	Renewable	No	35-50%
Solar PV	Renewable	No	10-25%
Solar CSP	Renewable	Partly	20-40%

Dispatchable: Can increase/decrease output on demand (gas, coal, hydro) Variable (non-dispatchable): Output depends on weather (wind, solar) Baseload: Technically dispatchable, but economically optimal to run at constant high output (nuclear)

9.3 Key Technical Parameters

Capacity factor: $$CF = \frac{Actual\ Generation}{Maximum\ Possible\ Generation} = \frac{E}{P \cdot 8760}$$

where E = annual generation (MWh), P = capacity (MW), 8760 = hours/year.

Efficiency: $$\eta = \frac{Electricity\ Output}{Fuel\ Input}$$

Typical efficiencies:

Coal: 35-45%
CCGT: 55-62%
Nuclear: 33-37%
Solar/Wind: N/A (no fuel)

Heat rate: Inverse of efficiency, often used for thermal plants $$Heat\ Rate = \frac{1}{\eta} \quad (GJ/MWh\ or\ BTU/kWh)$$

9.4 Cost Structure

Capital cost (CAPEX): €/kW installed

Nuclear: €4,000-8,000/kW
Offshore wind: €2,500-4,000/kW
Onshore wind: €1,000-1,500/kW
Solar PV: €400-800/kW
CCGT: €600-900/kW

Operating costs:

Fixed O&M: €/kW/year (maintenance regardless of operation)
Variable O&M: €/MWh (depends on generation)
Fuel: €/MWh (depends on fuel price and efficiency)

Levelized Cost of Electricity (LCOE): $$LCOE = \frac{\sum_t \frac{I_t + M_t + F_t}{(1+r)^t}}{\sum_t \frac{E_t}{(1+r)^t}} = \frac{\text{NPV of lifetime costs}}{\text{NPV of lifetime electricity}}$$

where I = investment, M = maintenance, F = fuel, E = electricity output, r = discount rate.

Where this comes from: Set NPV(revenues) = NPV(costs), assume a constant selling price P, and solve for P. The result is the LCOE—the unique break-even price.

Why it's useful: Allows apples-to-apples comparison of technologies with different cost profiles (nuclear: high upfront, low fuel; gas: low upfront, high fuel).

9.5 Load and Dispatch

Load: Electricity demand varies over time

Demand (MW)
    │     ┌──────┐
    │    /        \        Peak
    │   /          \
    │──/            \───   Shoulder
    │ /              \
    │/                \    Baseload
    └──────────────────────── Time (hours)
      6am    12pm   6pm

Merit order dispatch: Plants dispatched from lowest to highest marginal cost

Marginal Cost
(€/MWh)
    │              ┌──┐ Peak (gas turbines)
    │         ┌────┘  │
    │    ┌────┘       │ Mid (CCGT)
    │ ───┘            │ Baseload (nuclear, coal)
    │                 │
    └─────────────────┴──── Cumulative Capacity (GW)

Wholesale price = marginal cost of most expensive unit running

9.6 Time Slices in PRIMES

PRIMES doesn't model all 8760 hours—too computationally expensive. Instead, it uses representative time slices:

Time slices = Season × Day type × Hour type

Seasons: Winter, Summer, Intermediate (3)
Day type: Peak day, Average day (2)  
Hour type: Peak, Shoulder, Off-peak, Night (4)

Total: 3 × 2 × 4 = 24 representative periods

Each time slice has:

Typical demand level
Renewable availability (wind, solar)
Weight (hours it represents)

9.7 Reserve Margin and Reliability

Reserve margin: $$RM = \frac{Available\ Capacity - Peak\ Demand}{Peak\ Demand}$$

Typically 10-20% to ensure reliability.

Capacity credit: Fraction of nameplate capacity that counts as "firm" for reliability planning.

Thermal plants: ~90-95% (high availability)
Wind: ~5-15% (may not blow at peak)
Solar: ~0-30% (depends on when peak occurs—high if summer afternoon, zero if winter evening)

Example: 100 MW wind farm with 10% capacity credit → only 10 MW counts toward meeting peak demand.

9.8 Electricity Pricing: Ramsey-Boiteux

Electricity pricing is complex because it involves both competitive wholesale markets and regulated retail distribution.

Wholesale market pricing (competitive): The wholesale price equals the marginal cost of the most expensive plant needed to meet demand: $$P_{wholesale} = MC_{marginal_unit}$$

Why? Grid operators solve an optimization: minimize total generation cost subject to meeting demand. The Lagrange multiplier on the demand constraint is the cost of serving one more MW—which equals the marginal unit's cost.

The problem with regulated infrastructure:

Distribution networks (grids) are natural monopolies:

High fixed costs, low marginal costs
Marginal cost pricing doesn't recover investment costs
But monopoly pricing exploits consumers

Ramsey-Boiteux pricing solves this by setting prices to recover costs while minimizing welfare loss.

The Ramsey-Boiteux rule: $$\frac{P_i - MC_i}{P_i} = \frac{k}{\varepsilon_i}$$

where:

$P_i$ = price charged to consumer class $i$
$MC_i$ = marginal cost of serving class $i$
$\varepsilon_i$ = price elasticity of demand for class $i$
$k$ = constant ensuring total revenue = total cost

Interpretation:

Charge higher markups to consumers with lower elasticity
Industrial users (elastic) pay close to marginal cost
Residential users (inelastic) pay higher markups
This minimizes the total deadweight loss from above-marginal-cost pricing

Why this matters for PRIMES:

PRIMES models retail electricity prices as wholesale price + network tariffs + policy costs + taxes
Network tariffs follow Ramsey-Boiteux principles
Different consumer classes face different prices
This affects technology adoption decisions

Retail price decomposition (from PRIMES): $$P_{retail} = P_{wholesale} + T_{network} + T_{policy} + Taxes$$

where:

$T_{network}$ = grid tariffs (Ramsey-Boiteux pricing)
$T_{policy}$ = RES support levies, capacity payments
$Taxes$ = VAT, energy taxes

9.9 Where This Appears in E3 Models

Model	Power Sector Treatment
GEM-E3	Aggregate electricity sector; substitution between fuels
PRIMES	Detailed technology-by-technology; hourly dispatch approximation
PROMETHEUS	Global power generation projections; technology mix

Chapter 10: Technology Representation

10.1 Why Technology Detail Matters

Energy transitions are fundamentally about technology change:

From coal to gas to renewables
From internal combustion to electric vehicles
From gas boilers to heat pumps

Modeling technology choice and evolution is crucial for policy analysis.

10.2 Technology Vintages and Stock Turnover

Energy-using capital has long lifetimes:

Power plants: 30-60 years
Buildings: 50-100 years
Vehicles: 10-20 years
Appliances: 5-20 years

Stock turnover model: $$K_t = K_{t-1} \cdot (1 - \delta) + I_t$$

where:

$K_t$ = capital stock in year $t$
$\delta$ = depreciation/retirement rate
$I_t$ = new investment

Implication: Even with zero new fossil fuel investment, old plants keep running. Decarbonization takes decades.

10.3 Learning Curves

Experience curve: Technology costs decline as cumulative production increases

$$C_t = C_0 \cdot \left(\frac{Q_t}{Q_0}\right)^{-b}$$

where:

$C_t$ = unit cost at time $t$
$Q_t$ = cumulative production (total units ever made)
$b$ = learning parameter (higher = faster cost decline)

The negative exponent means costs fall as production accumulates—"learning by doing."

Learning rate (LR): Percentage cost reduction per doubling of capacity $$LR = 1 - 2^{-b}$$

If $b = 0.32$, then $LR = 1 - 2^{-0.32} \approx 0.20$ (20%)—each time cumulative production doubles, unit cost falls by 20%.

Typical learning rates:

Technology	Learning Rate
Solar PV modules	20-24%
Wind turbines	10-15%
Batteries (Li-ion)	15-20%
Nuclear	~0% (no learning in recent decades)
CCS (Carbon Capture & Storage)	Uncertain (5-15%?)

Historical estimates that may not persist. Solar's high learning rate is well-documented but unprecedented in energy history. Whether it continues, and whether other technologies follow similar curves, is genuinely uncertain.

In PROMETHEUS: Learning rates are treated as uncertain parameters with probability distributions—reflecting that past learning rates are imperfect guides to future cost reductions.

10.4 Intangible Costs

Real technology adoption is slower than pure cost optimization would predict. Intangible costs capture this:

$$C_i^{total} = C_i^{financial} + \mu_i$$

where $\mu_i$ represents:

Factor	Description
Risk premium	Uncertainty about new technology performance
Hidden costs	Installation complexity, training needs
Hassle factor	Time and effort to research, purchase
Financing constraints	Limited access to capital
Information barriers	Lack of awareness
Behavioral inertia	Preference for familiar options

PRIMES calibrates intangible costs to match observed market shares in the base year.

10.5 Technology Choice in PRIMES

PRIMES uses a discrete choice framework for technology selection:

Calculate generalized cost for each technology (financial + intangible)
Apply Weibull distribution to get market shares
Result: Technology mix, not winner-take-all

Example: Residential heating choice

Gas boiler: Low capital, high fuel cost, familiar
Heat pump: High capital, low running cost, unfamiliar

Even if heat pumps have lower lifecycle cost, gas boilers retain market share due to intangible costs.

10.6 Autonomous vs. Price-Induced Technical Change

AEEI (Autonomous Energy Efficiency Improvement):

Efficiency improves over time independent of prices
Represents ongoing technological progress
Exogenous parameter (typically 0.5-1.5%/year)

PIEEI (Price-Induced Energy Efficiency Improvement):

Higher energy prices → more investment in efficiency
Endogenous response to policy
Captured through substitution elasticities

Debate: How much is autonomous vs. induced? Important for policy analysis—if most improvement is autonomous, carbon prices matter less.

10.7 R&D and Innovation

Some models include explicit R&D and innovation:

Government R&D spending
Private R&D induced by carbon prices
Knowledge spillovers

GEM-E3 and PRIMES mostly treat technical change as exogenous (given by assumptions), though learning curves provide some endogeneity.

10.8 Where This Appears in E3 Models

Model	Technology Representation
GEM-E3	Aggregate; AEEI parameter; no explicit technologies
PRIMES	Hundreds of explicit technologies; learning curves; intangible costs
PROMETHEUS	Learning curves with uncertainty; technology cost projections

End of Part III: Energy Systems

Continue to Part IV: Trade and Environment for international and environmental aspects.

Part IV: Trade and Environment

Chapter 11: International Trade

11.1 Why Trade Matters for E3 Models

International trade is crucial for energy-economy analysis:

Energy trade: Oil, gas, coal are globally traded commodities
Carbon leakage: Production may shift to countries with weaker climate policy
Competitiveness: Energy-intensive industries face international competition
Technology diffusion: Trade spreads efficient technologies

11.2 Comparative Advantage

Ricardo's principle: Countries benefit from specializing in goods where they have comparative (not absolute) advantage.

Absolute advantage: Produce more with same resources (Country A makes wine faster than B)
Comparative advantage: Produce at lower opportunity cost (what you give up)

Key insight: Even if Country A is better at making everything, both countries gain from trade if each specializes where it's relatively best.

Example: A lawyer types faster than their secretary (absolute advantage in both law and typing). Should they type their own documents? No—their comparative advantage is law (where the gap is largest). They gain by specializing in law and hiring the secretary for typing.

Why this matters:

Total gains from trade: If both countries specialize where they're relatively best, total output rises—more for everyone in aggregate.
But: winners and losers within countries. When a country specializes, the sector it doesn't specialize in shrinks. Workers in that sector lose, even if the country overall gains. (This is why trade is politically contentious.)
Climate policy connection: Comparative advantage depends on relative costs. A carbon price changes these—countries with clean energy may gain comparative advantage in energy-intensive goods.

11.3 The Armington Assumption

Problem: Simple trade theory predicts complete specialization, but we observe:

Countries both import and export similar goods
Trade flows respond gradually to price changes

Armington (1969) solution: Domestic and imported goods are imperfect substitutes.

$$X = \left[ \delta \cdot D^{\rho} + (1-\delta) \cdot M^{\rho} \right]^{1/\rho}$$

where:

$X$ = composite good consumed (a CES aggregate of domestic and imported varieties)
$D$ = domestic good
$M$ = imported good
$\sigma = 1/(1-\rho)$ = Armington elasticity

This CES form means consumers view domestic and foreign goods as differentiated—not perfectly interchangeable—so trade adjusts gradually to price changes rather than switching entirely.

Limitations to keep in mind:

Armington is a modeling convenience, not a deep theory—it's chosen because it prevents unrealistic corner solutions
The assumption that goods differ only by country of origin ignores firm-level heterogeneity within countries
Armington elasticities are difficult to estimate reliably; values vary widely across studies and affect trade flow predictions substantially
The approach cannot represent situations where trade is blocked entirely (e.g., sanctions) without ad-hoc modifications

11.4 Armington Elasticity

The Armington elasticity $\sigma_A$ measures how much trade responds to price changes:

$$\sigma_A = \frac{\Delta (M/D) / (M/D)}{\Delta (P_M/P_D) / (P_M/P_D)}$$

In words: % change in import/domestic ratio divided by % change in relative price.

Typical values in GEM-E3:

Good	Armington Elasticity
Crude oil	5-10 (highly substitutable)
Electricity	2-4 (less traded)
Manufactured goods	2-6
Services	1-2 (hard to trade)
Agriculture	2-4

Higher σ_A:

Trade responds strongly to prices
Small price advantage → large trade flow shift
More "competitive" goods

Lower σ_A:

Trade responds weakly
Prices can differ substantially between regions
More "differentiated" goods

11.5 Two-Level Armington Structure

GEM-E3 uses a nested structure:

First level: Domestic vs. imports $$X = f(D, M; \sigma_D)$$

Second level: Imports by origin $$M = g(M_1, M_2, ..., M_n; \sigma_M)$$

where $M_i$ = imports from region $i$.

This allows different substitutability between:

Domestic and any import (σ_D)
Imports from different countries (σ_M)

11.6 Trade Balance and Current Account

Trade balance: $$TB = X - M$$

where X = exports, M = imports.

Current account includes:

Trade in goods and services
Factor income (returns on foreign investment)
Transfers

GEM-E3 enforces:

Trade flows balance globally (world exports = world imports)
Current account constraints by region (over time)

11.7 Carbon Leakage

Carbon leakage occurs when climate policy in one region causes emissions to increase elsewhere:

Mechanisms:

Competitiveness channel: Energy-intensive industries relocate
Energy market channel: Lower demand reduces world fuel prices, increasing consumption elsewhere

Leakage rate: $$Leakage = \frac{\Delta E_{rest}}{\Delta E_{policy}} \times (-1)$$

where $\Delta E_{policy}$ = emission reduction in the policy region (negative), and $\Delta E_{rest}$ = emission change in the rest of the world.

If leakage = 100%, global emissions unchanged by policy.

GEM-E3 is designed to analyze leakage because it models:

Multi-regional trade flows
Energy-intensive sectors explicitly
World energy prices

11.8 Border Carbon Adjustments

Border Carbon Adjustment (BCA): Apply carbon cost to imports based on their embodied emissions

Types:

Import tariff based on carbon content
Export rebate for domestic carbon costs

GEM-E3 can model BCA by adding tariffs linked to emission coefficients.

11.9 Where This Appears in E3 Models

Model	Trade Treatment
GEM-E3	Full bilateral trade; Armington; carbon leakage analysis
PRIMES	EU focus; imports from outside EU as exogenous
PROMETHEUS	Global energy trade flows; regional energy balances

Chapter 12: Environmental Policy

12.1 The Problem: Externalities

Externality: A cost or benefit not reflected in market prices

Negative externality from emissions:

Burning fossil fuels causes climate change
Costs fall on society, not the polluter
Market outcome has too much pollution

Efficient outcome: Polluter pays the social cost of emissions

12.2 Why Markets Fail

Without intervention:

Private cost < Social cost
Production is higher than socially optimal
Result: Deadweight loss from over-pollution

Price
    │
    │      Social MC
    │        /
    │       / \
    │      /   \ Private MC
    │     /     \
    │────/───────\─── Demand
    │   /         \
    │  / Deadweight\
    │ /   loss      \
    └─────────────────── Quantity
       Q*   Q_market

12.3 Carbon Pricing: Tax vs. Cap-and-Trade

Two main approaches to internalize the externality:

Carbon Tax:

Government sets price, market determines quantity
Revenue to government
Price certainty, quantity uncertainty

Cap-and-Trade (ETS):

Government sets quantity (cap), market determines price
Permits allocated or auctioned
Quantity certainty, price uncertainty

Feature	Carbon Tax	Cap-and-Trade
Price certainty	✓	✗
Quantity certainty	✗	✓
Revenue predictability	✓	Variable
Political acceptability	Lower?	Higher?
Implementation complexity	Lower	Higher

EU Emissions Trading System (ETS): Europe's main climate policy instrument since 2005

12.4 Marginal Abatement Cost (MAC)

MAC: The cost of reducing emissions by one more ton

$$MAC = -\frac{\partial C}{\partial E} > 0$$

where C = total abatement cost, E = emissions. The negative sign ensures MAC is positive: reducing emissions (lowering E) increases cost.

MAC curve: Shows cost of emission reductions

MAC
(€/tCO₂)
    │                    /
    │                   /
    │                  /
    │                 /
    │                / Steep = expensive further cuts
    │               /
    │              /
    │             /
    │    ────────/  Flat = cheap initial cuts
    │
    └─────────────────────── Abatement (Mt CO₂)

In GEM-E3: Carbon price = shadow price on emissions constraint = MAC at equilibrium

12.5 Revenue Recycling and Double Dividend

Revenue recycling options:

Lump-sum rebates to households
Reduce other taxes (labor, capital)
Fund green investments
Reduce government debt

Double dividend hypothesis:

First dividend: Environmental improvement
Second dividend: Economic gain from reducing distortionary taxes

GEM-E3 can test this because it models:

Labor market with unemployment
Detailed tax system
Welfare effects of different recycling options

12.6 Carbon Border Adjustments

To prevent carbon leakage, countries may impose border carbon adjustments:

$$Import\ tariff = Carbon\ price \times Embodied\ emissions$$

EU Carbon Border Adjustment Mechanism (CBAM): Being phased in from 2026

GEM-E3 is used to analyze CBAM impacts on trade, competitiveness, and emissions.

12.7 Policy Interactions

Real policies come in packages:

Carbon pricing + renewable subsidies + efficiency standards + R&D support

Interactions can be:

Complementary: Carbon price + R&D = faster innovation
Conflicting: ETS + Renewable Energy Sources (RES) target = lower permit price
Redundant: Multiple policies targeting same behavior

GEM-E3 and PRIMES can model policy packages to understand interactions.

12.8 Co-Benefits

Climate policies often have co-benefits:

Air quality improvement (less SO₂, NOx, PM)
Energy security (less import dependence)
Health benefits (less pollution)
Innovation stimulus

GEM-E3 reports:

CO₂, CH₄, N₂O (greenhouse gases)
SO₂, NOx, PM (air pollutants)

This allows analysis of climate-air quality co-benefits.

12.9 Where This Appears in E3 Models

Model	Environmental Policy Features
GEM-E3	Carbon tax, ETS, revenue recycling, CBAM, multi-pollutant
PRIMES	Carbon price input; technology response; RES targets
PROMETHEUS	Global climate policy scenarios; emission projections

End of Part IV: Trade and Environment

Continue to Part V: Integration for data and model coupling.

Part V: Integration

Chapter 13: Data and Calibration

13.1 The Data Challenge

E3 models require vast amounts of data:

Economic flows between sectors and regions
Energy production, transformation, consumption
Technology costs and performance
Emissions by source
Trade flows by commodity and partner

Getting this data consistent and comprehensive is a major challenge.

13.2 Social Accounting Matrix (SAM)

A Social Accounting Matrix is the data foundation for CGE models like GEM-E3.

What is a SAM?

A square matrix showing all value flows in an economy for a base year
Rows = income (receipts)
Columns = expenditure (payments)
Row total = Column total for each account (everything balances)

SAM structure:

                              EXPENDITURES
             ┌──────────┬──────────┬─────────┬──────────┬─────┬─────┐
             │Activities│Commodit. │ Factors │Households│ Gov │ RoW │
┌────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Activities  │          │ Domestic │         │          │     │Exp- │
│            │          │ sales    │         │          │     │orts │
├────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Commodities │ Intermed │          │         │Consumpt. │ Gov │     │
│            │ inputs   │          │         │          │     │     │
├────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Factors     │ Value    │          │         │          │     │     │
│(L, K)      │ added    │          │         │          │     │     │
├────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Households  │          │          │ Factor  │          │Trans│     │
│            │          │          │ income  │          │fers │     │
├────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Government  │ Indirect │          │         │ Direct   │     │     │
│            │ taxes    │          │         │ taxes    │     │     │
├────────────┼──────────┼──────────┼─────────┼──────────┼─────┼─────┤
│Rest of     │          │ Imports  │         │          │     │     │
│World       │          │          │         │          │     │     │
└────────────┴──────────┴──────────┴─────────┴──────────┴─────┴─────┘
     ↑
 RECEIPTS

Key property: Every row sum equals the corresponding column sum.

13.3 Data Sources

For GEM-E3:

Data Type	Primary Source	Secondary Sources
Economic structure	GTAP database	Eurostat, national accounts
Bilateral trade	GTAP	UN Comtrade
Energy balances	Eurostat, IEA	National statistics
Emissions	UNFCCC, IPCC	National inventories
Elasticities	Literature	Econometric estimates

GTAP Database (Global Trade Analysis Project):

Maintained by Purdue University
Global database covering 141 regions, 65 sectors
Used as starting point for most CGE models
Updated every few years (current: GTAP 11, base year 2017)

13.4 The Calibration Principle

Calibration assumption: The base year SAM represents an equilibrium.

Procedure:

Set all prices to 1 (numeraire normalization)
Choose functional forms (CES)
Specify elasticities from literature/estimation
Solve for share parameters such that model replicates SAM flows

Example: CES calibration

Given observed inputs $X_1 = 60$, $X_2 = 40$ and prices $P_1 = P_2 = 1$:

For CES: $Y = [\alpha X_1^{\rho} + (1-\alpha) X_2^{\rho}]^{1/\rho}$

At calibration: $$\alpha = \frac{X_1}{X_1 + X_2} = \frac{60}{100} = 0.6$$

(Exact formula depends on nesting structure and normalization.)

13.5 Replication Test

A properly calibrated model must replicate the base year exactly.

If you run the model with no policy shock:

All prices should remain at 1
All quantities should match the SAM
This is the "benchmark replication test"

If replication fails: There's a bug or inconsistency in the data/model.

13.6 SAM Balancing

Real data sources don't perfectly balance. SAM balancing adjusts raw data to ensure consistency.

Methods:

RAS method: Iteratively scale rows and columns
Cross-entropy: Minimize information distance from original data
Least squares: Minimize sum of squared adjustments

GEM-E3 constructs SAMs from multiple sources and balances them for each region.

13.7 Base Year and Projections

Base year: The year for which the model is calibrated (e.g., 2015, 2020)

Projections require additional assumptions:

GDP growth rates (from macroeconomic forecasts)
Population growth (from demographic projections)
World energy prices (from PROMETHEUS)
Policy changes (scenario-specific)

13.8 Uncertainty in Parameters

Key uncertain parameters:

Parameter	Source	Uncertainty Level
Armington elasticities	Literature	Medium
Capital-labor substitution	Econometrics	Medium
Energy-value added substitution	Literature	High
Learning rates	Historical	High
Intangible costs	Calibration	Very High

Sensitivity analysis is essential: Results should be tested against alternative parameter values.

13.9 Where This Appears in E3 Models

Model	Data/Calibration
GEM-E3	SAM from GTAP + Eurostat; calibrated CES functions
PRIMES	Energy balances from Eurostat; technology database
PROMETHEUS	Historical energy data from IEA, BP; econometric estimation

Chapter 14: Model Coupling

14.1 Why Couple Models?

No single model can do everything well:

Requirement	Best Approach
Economy-wide impacts	CGE (GEM-E3)
Technology detail	Partial equilibrium (PRIMES)
Uncertainty quantification	Stochastic (PROMETHEUS)

Solution: Couple models to get benefits of each.

14.2 Types of Model Linkage

Hard linking (full integration):

Models share same code/platform
Solve simultaneously
Computationally demanding
Ensures perfect consistency

Soft linking (iterative):

Models run separately
Exchange data between runs
Iterate until convergence
More flexible, easier to maintain

GEM-E3/PRIMES use soft linking — they iterate until energy prices and economic activity are consistent.

14.3 The E3-Modelling Coupling Architecture

┌──────────────────────────────────────────────────────────────┐
│                    E3-MODELLING SUITE                         │
└──────────────────────────────────────────────────────────────┘

         ┌─────────────────┐
         │   PROMETHEUS    │
         │  (World Energy) │
         └────────┬────────┘
                  │
                  │ World fuel prices
                  │ (Oil, Gas, Coal)
                  ▼
         ┌─────────────────┐
         │     GEM-E3      │◄──────────────┐
         │    (Economy)    │               │
         └────────┬────────┘               │
                  │                        │
                  │ GDP, sectoral          │
                  │ activity levels        │ Energy prices,
                  ▼                        │ system costs
         ┌─────────────────┐               │
         │     PRIMES      │───────────────┘
         │    (Energy)     │
         └─────────────────┘
                  │
                  │ Detailed energy
                  │ projections
                  ▼
         ┌─────────────────┐
         │    OUTPUTS      │
         │  (Integrated)   │
         └─────────────────┘

14.4 The Iteration Process

Step 1: PROMETHEUS runs

Generates world fossil fuel price trajectories
May include uncertainty bounds (P10, P50, P90)

Step 2: GEM-E3 initial run

Uses PROMETHEUS prices
Calculates GDP, sectoral output
First estimate of energy demand response

Step 3: PRIMES runs

Takes GDP trajectory from GEM-E3
Optimizes energy system
Calculates energy prices, system costs

Step 4: GEM-E3 re-runs

Uses energy prices from PRIMES
Updates economic projections
May change GDP, sectoral composition

Step 5: Iterate

Repeat steps 3-4 until convergence
Convergence = energy flows ↔ economic costs are consistent

Iteration 1:  GEM-E3(GDP₀) → PRIMES → Energy prices₁
Iteration 2:  GEM-E3(prices₁) → PRIMES → Energy prices₂
Iteration 3:  GEM-E3(prices₂) → PRIMES → Energy prices₃
...
Convergence: |prices_n - prices_{n-1}| < tolerance

14.5 What Gets Exchanged

GEM-E3 → PRIMES:

GDP by country and year
Sectoral value added (industry, services)
Population, households
Discount rates

PRIMES → GEM-E3:

Energy demand by sector and fuel
Energy prices (wholesale, retail)
Investment in energy sector
System costs

PROMETHEUS → GEM-E3/PRIMES:

World oil prices (€/barrel)
World gas prices (€/MWh)
World coal prices (€/ton)
Possibly: probability distributions

14.6 Ensuring Consistency

Challenges:

Different sectoral aggregations
Different time steps
Different geographic boundaries
Different price concepts

Solutions:

Mapping tables between classifications
Interpolation for time steps
Regional aggregation/disaggregation
Clear definition of price concepts

14.7 Advantages of the Coupled System

Feature	Standalone Model	Coupled System
Technology detail	Limited in CGE	Full from PRIMES
Economic feedback	None in PE	Full from GEM-E3
Price consistency	Assumed	Ensured by iteration
Uncertainty	Scenarios	PROMETHEUS distributions

14.8 Computational Considerations

Full model run:

PROMETHEUS: minutes
GEM-E3 (one year): seconds to minutes
PRIMES (one country-year): minutes
Full iteration: hours
All EU countries, all years: 1-2 days

Practical implications:

Scenario design matters (can't run everything)
Parallel computing helps
Results caching important

14.9 When to Use Which Model

Question	Model(s) to Use
GDP impact of carbon tax	GEM-E3
Optimal power mix in 2050	PRIMES
Oil price uncertainty	PROMETHEUS
Sectoral employment effects	GEM-E3
Technology investment needs	PRIMES
Carbon leakage from EU policy	GEM-E3
Complete policy assessment	All three (coupled)

14.10 Related Models in the Ecosystem

The E3-Modelling suite can link with other models for extended analysis:

E3ME (Cambridge Econometrics)

Type: Macroeconometric model (not CGE)
Approach: Based on historical relationships, not optimization
Key difference: Allows for under-utilized resources, path-dependent dynamics
Use case: Alternative to GEM-E3 for macro drivers; sometimes used when Keynesian features desired
Linkage: Can substitute for GEM-E3 in providing macro drivers to PRIMES

GAINS (IIASA)

Type: Air pollution and greenhouse gas model
Focus: Multi-pollutant, multi-effect analysis
Key feature: Detailed emission control technology costs
Use case: Air quality co-benefits; pollution control costs
Linkage: Can receive energy scenarios from PRIMES; provides air pollutant costs

Model	Developer	Linkage to E3 Suite
E3ME	Cambridge Econometrics	Alternative macro driver for PRIMES
GAINS	IIASA	Air pollutant costs; co-benefits analysis
GTAP	Purdue	Database source for GEM-E3 calibration
IEA WEO	IEA	Benchmark for PROMETHEUS validation

Why multiple models exist: Different methodological approaches have different strengths. CGE models (GEM-E3) assume equilibrium; econometric models (E3ME) capture historical dynamics. Using multiple approaches provides robustness checks for policy conclusions.

End of Part V: Integration

Continue to Appendices for reference materials.

Appendices

Appendix A: Glossary

Term	Definition
AEEI	Autonomous Energy Efficiency Improvement — exogenous efficiency gains over time
Armington elasticity	Substitution elasticity between domestic and imported goods
Calibration	Process of determining model parameters to replicate base year data
Carbon leakage	Increase in emissions outside a policy region due to the policy
CES	Constant Elasticity of Substitution — a production/utility function
CGE	Computable General Equilibrium — economy-wide model with optimization
Complementarity	Mathematical condition where either price=0 or market clears
Discount rate	Rate used to convert future values to present values
Dominant firm	Market structure where one large firm sets price, others follow
E3ME	Energy-Environment-Economy Model for Europe — macroeconometric alternative to CGE
EPEC	Equilibrium Problem with Equilibrium Constraints
ETS	Emissions Trading System (cap-and-trade)
Efficiency wage	Above-market wage paid to prevent shirking; explains unemployment
Equivalent Variation	Welfare measure: money to give before price change to reach new utility
Externality	Cost or benefit not reflected in market prices
GAINS	Greenhouse Gas and Air Pollution Interactions and Synergies model (IIASA)
GTAP	Global Trade Analysis Project — database for CGE models
Hotelling rule	Price of exhaustible resource rises at rate of interest
Learning curve	Relationship between cumulative production and unit cost
LCOE	Levelized Cost of Electricity
LP	Linear Programming
MAC	Marginal Abatement Cost — cost of reducing one more unit of emissions
MCP	Mixed Complementarity Problem — mathematical formulation used to solve CGE equilibrium
MILP	Mixed-Integer Linear Programming
Monte Carlo	Simulation method using random sampling
Nested CES	Hierarchical structure of CES functions for multiple inputs
NLP	Nonlinear Programming
Numeraire	Good whose price is normalized to 1
OPEC	Organization of Petroleum Exporting Countries — oil cartel
Partial equilibrium	Analysis of one market holding others constant
Ramsey-Boiteux pricing	Setting prices to recover fixed costs while minimizing welfare loss
SAM	Social Accounting Matrix — balanced data of economic flows
Shadow price	Marginal value of relaxing a constraint
Shapiro-Stiglitz	Efficiency wage model explaining involuntary unemployment
Sobol indices	Global sensitivity analysis measures
TFP	Total Factor Productivity
Time slice	Representative period used to approximate temporal variation
URR	Ultimate Recoverable Resources
Walrasian equilibrium	Price vector where all markets clear simultaneously
Welfare	Economic well-being; often measured as utility or equivalent variation

Appendix B: Mathematical Notation

Sets and Indices

Symbol	Meaning
$i, j$	Goods/sectors
$f$	Fuels
$r, s$	Regions
$t$	Time periods
$h$	Hours/time slices

Variables

Symbol	Meaning	Units
$Y$	Output	€ or physical
$K$	Capital	€
$L$	Labor	persons or hours
$E$	Energy	GJ, MWh
$P$	Price	€/unit
$W$	Wage	€/hour
$r$	Interest rate	%
$U$	Utility	index
$C$	Consumption	€
$I$	Investment	€
$X$	Exports	€
$M$	Imports	€
$EMI$	Emissions	Mt CO₂
$\tau$	Carbon price	€/tCO₂

Parameters

Symbol	Meaning
$\sigma$	Elasticity of substitution
$\alpha, \beta$	Share parameters
$\rho$	CES substitution parameter ($\rho = (\sigma-1)/\sigma$)
$\delta$	Depreciation rate
$\epsilon$	Emission coefficient
$\nu$	Weibull heterogeneity parameter
$\mu$	Intangible cost

Appendix C: Key Equations Cheat Sheet

Production

CES Production Function: $$Y = A \left[ \alpha K^{\rho} + (1-\alpha) L^{\rho} \right]^{1/\rho}, \quad \sigma = \frac{1}{1-\rho}$$ Output Y is produced by combining capital K and labor L, where σ controls how easily one can substitute for the other.

Cost Minimization (FOC): $$\frac{K}{L} = \left( \frac{\alpha}{1-\alpha} \cdot \frac{W}{R} \right)^{\sigma}$$ Firms use more capital relative to labor when wages W rise relative to the cost of capital R; σ determines how much they switch.

Demand

Price Elasticity: $$\varepsilon = \frac{\partial Q / Q}{\partial P / P} = \frac{\partial \ln Q}{\partial \ln P}$$ If price rises by 1%, quantity demanded falls by ε%. Inelastic (|ε|<1) means demand barely budges; elastic (|ε|>1) means big response.

Discrete Choice (Logit): $$P_i = \frac{e^{V_i / \mu}}{\sum_j e^{V_j / \mu}}$$ Probability of choosing option i depends on its "attractiveness" V_i relative to all alternatives. Same as softmax in ML.

Equilibrium

Market Clearing: $$Q^{supply}(P^) = Q^{demand}(P^)$$ The equilibrium price P is where buyers want exactly what sellers offer — no excess supply or demand.*

Complementarity: $$0 \leq P \perp (S - D) \geq 0$$ Either price is zero (free good) OR supply equals demand. Can't have both positive price AND excess supply.

Trade

Armington Composite: $$X = \left[ \delta D^{\rho} + (1-\delta) M^{\rho} \right]^{1/\rho}$$ Domestic goods D and imports M are imperfect substitutes. Prevents unrealistic "all-or-nothing" trade swings.

Environment

Emissions: $$EMI_f = \epsilon_f \cdot E_f$$ Emissions from fuel f = emission factor × energy consumed. Coal has high ε, gas has lower ε, renewables have zero.

Effective Fuel Price (with carbon tax): $$P_f^{eff} = P_f + \tau \cdot \epsilon_f$$ Carbon tax τ raises dirty fuels more than clean ones, making low-carbon alternatives relatively cheaper.

Learning

Experience Curve: $$C_t = C_0 \cdot \left( \frac{Q_t}{Q_0} \right)^{-b}, \quad LR = 1 - 2^{-b}$$ Costs fall as cumulative production grows. Learning rate LR = % cost drop per doubling of capacity (e.g., 20% for solar PV).

Capital Dynamics

Stock Update: $$K_{t+1} = (1 - \delta) K_t + I_t$$ Next year's capital = what survives depreciation + new investment. Drives the dynamics in recursive models.

NPV

Net Present Value: $$NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t}$$ Future cash flows discounted to today. Positive NPV = profitable investment. Used for technology choice in PRIMES.

Resource Economics

Hotelling Rule: $$\frac{dP}{dt} = r \cdot P$$ Price of exhaustible resources rises at the interest rate. If not, arbitrage: everyone would extract now (or wait).

Resource Rent Growth: $$\frac{d(P - MC)}{dt} = r \cdot (P - MC)$$ The profit margin (price minus extraction cost) grows at rate r. Drives long-run oil/gas price dynamics in PROMETHEUS.

Pricing

Ramsey-Boiteux Rule: $$\frac{P_i - MC_i}{P_i} = \frac{k}{\varepsilon_i}$$ To recover fixed costs, charge higher markups to less price-sensitive customers. Used for electricity grid tariffs.

Labor Market

Efficiency Wage (Shapiro-Stiglitz): $$w = w^* \cdot \left(1 + \frac{e}{b + \rho/q}\right)$$ Firms pay above market-clearing wage to discourage shirking. Creates equilibrium unemployment — key feature of GEM-E3.

Appendix D: Bridging Notes — From Physics & ML to E3 Modeling

For readers with a background in computational physics and/or machine learning.

D.1 Statistical Physics → Economic Equilibrium

E3 Concept	Physics Analogue	Connection
Walrasian equilibrium	Thermodynamic equilibrium	Both are stable states where no agent/particle has incentive to change; system minimizes a "potential"
Market clearing	Detailed balance	Flows in = flows out; conservation at each node
CES aggregation	Generalized mean / partition function	$Y = [\sum \alpha_i X_i^\rho]^{1/\rho}$ interpolates between sum (ρ→1) and min (ρ→-∞), like temperature controlling ensemble behavior
Elasticity of substitution (σ)	Inverse "temperature"	High σ = agents easily switch (high T, flat distribution); Low σ = locked in (low T, peaked distribution)
Shadow price (λ)	Lagrange multiplier	Identical math — the marginal "force" enforcing a constraint
Utility maximization	Free energy minimization	Agents maximize U subject to budget ↔ systems minimize F subject to constraints
SAM balance	Conservation laws	Row sums = column sums, like current conservation or mass balance

D.2 Computational Physics → CGE Computation

E3 Technique	Physics Analogue	Connection
Monte Carlo (PROMETHEUS)	MC in condensed matter	Same idea — sample parameter space, propagate to outputs, build distributions
Solving coupled nonlinear systems	Self-consistent field methods	CGE solves for prices where all markets clear simultaneously; iterative like SCF
Sensitivity analysis (Sobol)	Parameter sweeps	Which inputs drive output variance? Same question, same methods
GAMS/PATH solver	Newton-Raphson, conjugate gradient	Iterative solution of F(x)=0; PATH uses pivoting for complementarity

D.3 Machine Learning → E3 Modeling

E3 Concept	ML Analogue	Connection
Calibration	Training / fitting	Adjust parameters so model replicates observed data
Elasticities	Hyperparameters	Control model behavior; often from literature, not estimated on the SAM
CES production function	Parametric model family	Like choosing between L1/L2/Huber loss — functional form matters
Discrete choice (logit)	Softmax	$P_i = e^{V_i}/\sum e^{V_j}$ — identical to softmax in classification
Intangible costs (μ)	Regularization / prior	Calibrated to match observed behavior; captures "non-financial" factors
Counterfactual simulation	What-if analysis	Change inputs (policy), observe outputs (welfare, emissions)
Shadow prices	Dual variables in constrained optimization	scipy.optimize returns these too

D.4 Key Intuition Shortcuts

"Zero profit" doesn't mean firms make no money — it means no excess returns above opportunity cost (competitive equilibrium)
"Elasticity" is just the log-log slope: $\varepsilon = d\ln Y / d\ln X$
"Armington" = "domestic and imported goods are imperfect substitutes" — prevents unrealistic all-or-nothing trade swings
"Recursive dynamic" = solve year-by-year, updating capital stocks — no perfect foresight (unlike intertemporal optimization)
"Complementarity" = "either the constraint binds OR the shadow price is zero" — like KKT conditions you know from physics optimization

End of E3 Modeling Concepts Primer

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GEM-E3.md

GEM-E3: Technical Distillation

General Equilibrium Model for Energy-Economy-Environment

1. Model Identity

Attribute	Value
Type	Computable General Equilibrium (CGE)
Mathematical Class	Mixed Complementarity Problem (MCP)
Temporal Structure	Recursive Dynamic
Spatial Coverage	Multi-regional (EU-28 + major world regions)
Time Horizon	Typically 2020–2070, 5-year steps
Resolution	31 economic sectors (8 energy, 23 non-energy)
Primary Output	GDP, prices, welfare (EV), emissions by region/sector
Role in E3 Suite	Macro-economic feedback; economy-wide policy impacts

2. Theoretical Foundations

2.1 Walrasian General Equilibrium

GEM-E3 is grounded in Arrow-Debreu general equilibrium theory:

Core Axiom: There exists a price vector p* such that all markets clear simultaneously:

$$ z_i(p^*) = 0 \quad \forall i \in {goods, factors, emissions} $$

where $z_i$ is excess demand in market $i$.

Implications:

All agents are price-takers
No excess supply or demand in equilibrium
Prices carry all information needed for coordination

2.2 Microfoundations

Households: Maximize utility subject to budget constraint

Inter-temporal: choose consumption vs. savings
Intra-temporal: choose leisure vs. labor, and consumption bundle

Firms: Maximize profit (equivalently, minimize cost) subject to technology

Represented by nested CES production functions
Zero economic profit in equilibrium (constant returns to scale)

Government: Collects taxes, provides transfers, purchases goods

Budget constraint enforced (or deficit financed)

2.3 Deviation from Standard CGE: Labor Market

Unlike Walrasian models assuming full employment, GEM-E3 incorporates involuntary unemployment via the Shapiro-Stiglitz efficiency wage model:

$$ w = w^* \cdot \left(1 + \frac{e}{b + \rho/q}\right) $$

where:

$w$ = efficiency wage
$w^*$ = market-clearing wage
$e$ = effort level required
$b$ = unemployment benefits
$\rho$ = discount rate
$q$ = probability of being caught shirking

Result: Labor market does not clear; unemployment rate is endogenous.

3. Inputs

3.1 Data Requirements

Input Category	Source	Description
Social Accounting Matrix (SAM)	GTAP, Eurostat	Base year flows (€bn)
Bilateral Trade Flows	GTAP	Imports/exports by origin-destination
Energy Balances	IEA, Eurostat	Physical energy flows (PJ)
Emission Coefficients	IPCC, national inventories	CO₂, CH₄, N₂O, SO₂, NOx, PM
Elasticities	Econometric literature	Substitution, trade, labor supply

3.2 Exogenous Drivers

Variable	Typical Source	Role
Population growth	UN projections	Labor supply
Autonomous productivity growth	Historical trends	TFP assumptions
World fossil fuel prices	PROMETHEUS model	Import prices
Policy instruments	Scenario definition	Carbon tax, ETS cap, subsidies

3.3 Key Elasticity Parameters

Elasticity	Symbol	Typical Range	Sensitivity
Capital-Labor substitution	$\sigma_{KL}$	0.4–1.0	High
Energy-Value Added substitution	$\sigma_{E,VA}$	0.1–0.5	Very High
Armington (domestic vs. import)	$\sigma_A$	2.0–8.0	High
Intertemporal (consumption smoothing)	$\sigma_C$	0.5–1.0	Medium

4. Outputs

4.1 Macroeconomic Indicators

Output	Unit	Description
GDP	€bn (real)	By region and aggregate
Consumption	€bn	Private, public
Investment	€bn	By sector
Employment	Million persons	By sector, with unemployment rate
Trade balance	€bn	Exports - Imports by region

4.2 Sectoral Results

Output	Detail
Output	By 31 sectors × regions
Prices	Producer, consumer prices by good
Factor use	Labor, capital by sector
Energy use	Physical units by fuel and sector

4.3 Environmental Indicators

Output	Unit
CO₂ emissions	Mt CO₂
Other GHGs	Mt CO₂-eq
Air pollutants	Kt SO₂, NOx, PM
Carbon price	€/tCO₂ (shadow price of constraint)

4.4 Welfare Measures

Measure	Definition
Equivalent Variation (EV)	Income change at old prices giving same utility as new equilibrium
GDP loss/gain	% deviation from baseline
Double dividend	Whether environmental tax revenue recycling improves welfare

5. Mathematical Formalization

5.1 The Mixed Complementarity Problem (MCP)

GEM-E3 is solved as an MCP, which generalizes the Karush-Kuhn-Tucker (KKT) conditions:

For each market $i$: $$ \begin{aligned} p_i &\geq 0 \ z_i(p) &\geq 0 \ p_i \cdot z_i(p) &= 0 \end{aligned} $$

Interpretation: Either the price is zero (free good) OR the market clears. The complementarity condition prevents both excess supply and positive price.

5.2 Production Technology: Nested CES

Output $Y$ is produced using a nested Constant Elasticity of Substitution (CES) structure:

Top nest (Output): $$ Y = A \left[ \alpha \cdot VA^{\rho} + (1-\alpha) \cdot M^{\rho} \right]^{1/\rho} $$

where $\sigma = 1/(1-\rho)$ is the elasticity of substitution.

Value Added nest: $$ VA = \left[ \beta \cdot K^{\rho_{KL}} + (1-\beta) \cdot L^{\rho_{KL}} \right]^{1/\rho_{KL}} $$

Energy nest (within Materials or separate): $$ E = \left[ \sum_f \gamma_f \cdot E_f^{\rho_E} \right]^{1/\rho_E} $$

where $f \in {coal, oil, gas, electricity}$.

Nesting Structure (typical):

Output (σ ≈ 0)
├── Value Added (σ_KL ≈ 0.5)
│   ├── Capital
│   └── Labor
└── Intermediate Bundle (σ_M ≈ 0.3)
    ├── Energy Bundle (σ_E ≈ 0.5–1.0)
    │   ├── Electricity
    │   └── Fossil Fuels (σ_F ≈ 1.0)
    │       ├── Coal
    │       ├── Oil
    │       └── Gas
    └── Non-Energy Materials (σ_NE ≈ 0.2)

5.3 Consumer Optimization

Utility function (CES over time and goods):

$$ U = \sum_{t=0}^{T} \left(\frac{1}{1+\rho}\right)^t \cdot u(C_t, L_t) $$

where instantaneous utility: $$ u(C, L) = \left[ \alpha C^{\rho_u} + (1-\alpha) L^{\rho_u} \right]^{1/\rho_u} $$

Budget constraint: $$ \sum_t \frac{P_t \cdot C_t}{(1+r)^t} = \sum_t \frac{w_t \cdot \bar{L} + r_t \cdot K_t + Transfers_t}{(1+r)^t} $$

5.4 International Trade: Armington Specification

Domestic goods and imports are imperfect substitutes:

$$ X = \left[ \delta \cdot D^{\rho_A} + (1-\delta) \cdot M^{\rho_A} \right]^{1/\rho_A} $$

Import aggregation (by origin): $$ M = \left[ \sum_r \phi_r \cdot M_r^{\rho_M} \right]^{1/\rho_M} $$

This prevents "bang-bang" specialization and allows for intra-industry trade.

5.5 Emissions and Environmental Policy

Emissions function: $$ EMI_{f,s,r} = \epsilon_f \cdot E_{f,s,r} $$

where $\epsilon_f$ is the emission coefficient (tCO₂/TJ) for fuel $f$.

Carbon pricing (tax or ETS):

The carbon price $\tau$ enters the effective fuel price: $$ P_f^{eff} = P_f + \tau \cdot \epsilon_f $$

Cap-and-trade: $$ \sum_{s,r} EMI_{s,r} \leq \bar{EMI} \quad (\lambda = \text{permit price}) $$

The shadow price $\lambda$ on this constraint is the equilibrium carbon price.

5.6 Recursive Dynamics

Unlike intertemporal models with perfect foresight, GEM-E3 is recursive dynamic:

Solve static equilibrium for year $t$
Update capital stocks: $$ K_{t+1} = (1-\delta) K_t + I_t $$
Solve year $t+1$

Savings rule: Adaptive expectations or fixed savings rate (not Ramsey optimal).

6. Solution Method

6.1 GAMS/MPSGE Implementation

GEM-E3 is implemented in GAMS using the MPSGE (Mathematical Programming System for General Equilibrium) subsystem:

$MODEL:GEM_E3

$SECTORS:
    Y(s,r)      ! Production by sector and region
    C(r)        ! Consumption aggregate
    
$COMMODITIES:
    P(s,r)      ! Goods prices
    W(r)        ! Wage rate
    R(r)        ! Return to capital
    PCO2        ! Carbon permit price

$CONSUMERS:
    HH(r)       ! Household by region

$PROD:Y(s,r) s:0 va:sigma_va(s) e:sigma_e(s)
    O:P(s,r)    Q:output0(s,r)
    I:P(i,r)    Q:input0(i,s,r)
    I:W(r)      Q:labor0(s,r)     va:
    I:R(r)      Q:capital0(s,r)   va:
    I:P(e,r)    Q:energy0(e,s,r)  e:  A:GOVT T:tau(e)

6.2 Solver

PATH solver for MCP
Convergence criterion: excess demand < tolerance (typically 10⁻⁶)
Typical solve time: seconds to minutes per year

7. Calibration

7.1 The Calibration Principle

Assumption: The base year SAM represents an equilibrium.

Procedure:

Set all prices to 1 (numeraire normalization)
Choose functional forms (CES)
Solve for share parameters ($\alpha$, $\beta$, $\gamma$) such that model replicates SAM quantities at unit prices

Example (CES calibration):

Given observed inputs $X_1^0, X_2^0$ and prices $P_1 = P_2 = 1$: $$ \alpha = \frac{X_1^0}{X_1^0 + X_2^0} $$

7.2 Replication Test

A properly calibrated model must exactly replicate the base year when solved with no policy shock.

8. Key Assumptions and Limitations

Assumption	Implication
Representative agent per region	No within-region inequality
Constant returns to scale	Zero pure profits
Perfect competition	Price = marginal cost
No money, no financial sector	Real economy only
Recursive dynamics	No perfect foresight, myopic investment
Armington trade	No firm heterogeneity in trade

Known limitations:

Elasticities are uncertain → sensitivity analysis essential
Comparative statics, not transition dynamics
Aggregation hides sectoral heterogeneity

8.1 Closure Rules

CGE models have more potential endogenous variables than equations. To make the system "square" (solvable), some variables must be fixed exogenously. This choice is called a closure rule.

Key closure choices:

Closure	Options	Policy Implication
Government budget	Fixed deficit vs. endogenous taxes	Does carbon tax revenue offset other taxes?
Foreign sector	Fixed current account vs. flexible exchange rate	Does energy price shock affect trade balance or borrowing?
Investment-savings	Savings-driven vs. investment-driven	Who adjusts when investment demand changes?

Why closures matter: The same policy shock with different closures yields different GDP and employment results. There is no universally "correct" closure—it depends on the policy question and time horizon.

Worked Example: Same Policy, Different Closures

Consider a €50/tCO₂ carbon tax generating €10bn in revenue:

Closure	Mechanism	Illustrative GDP Impact
Lump-sum recycling	Revenue returned as equal transfers to households	GDP: -0.5%
Labor tax cut	Revenue used to cut payroll taxes	GDP: -0.2% (or even +0.1% with efficiency wage effects)
Capital tax cut	Revenue used to cut corporate taxes	GDP: -0.3%
Deficit reduction	Revenue reduces government borrowing	GDP: -0.6%

These numbers are illustrative—actual results depend on model calibration and other assumptions. The key point is that the same carbon tax can look costly or nearly costless depending on what you assume happens to the revenue.

Why this matters: When reading GEM-E3 results, always ask: "What closure was used?" Results presented without specifying the closure are incomplete.

GEM-E3's typical approach: Closures are scenario-dependent, but commonly:

Government budget balanced via lump-sum transfers or tax adjustment
Current account may be fixed or allowed to adjust depending on research question
Savings rate follows adaptive expectations (recursive dynamic structure)

9. Connections to Other Models

Model	Interface
PRIMES	Receives: GDP, sectoral activity. Sends: energy prices, costs
PROMETHEUS	Receives: world fuel prices as exogenous boundary conditions
GTAP	Uses GTAP database for SAM construction
JRC-GEM-E3	JRC version used for EU Commission impact assessments

10. Key References

Capros, P. et al. (2013). "GEM-E3 Model Documentation." E3MLab Technical Report.
Böhringer, C., Rutherford, T. (2008). "Combining bottom-up and top-down." Energy Economics.
Sue Wing, I. (2004). "Computable General Equilibrium Models." MIT Technical Note 6.
Armington, P. (1969). "A Theory of Demand for Products Distinguished by Place of Production." IMF Staff Papers.

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PRIMES.md

PRIMES: Technical Distillation

Price-Induced Market Equilibrium System

1. Model Identity

Attribute	Value
Type	Partial Equilibrium (Energy Sector)
Mathematical Class	EPEC (Equilibrium Problem with Equilibrium Constraints) / Hybrid
Temporal Structure	Mixed: Intertemporal optimization (supply) + Dynamic simulation (demand)
Spatial Coverage	EU Member States + UK, candidate countries
Time Horizon	Typically 2020–2070, 5-year steps
Resolution	High technological detail (hundreds of technologies)
Primary Output	Energy balances, technology mix, energy prices, investment
Role in E3 Suite	Detailed energy sector projections; technology pathways

2. Theoretical Foundations

2.1 Partial Equilibrium Framework

Unlike CGE, PRIMES models only the energy sector in detail, taking the rest of the economy as exogenous:

Exogenous from GEM-E3:

GDP growth by country
Sectoral value added (industry, services)
Population, household numbers

Endogenous within PRIMES:

Energy demand by fuel and sector
Technology choices (stock turnover)
Energy supply, transformation, prices
Infrastructure investment

2.2 Hybrid Modeling Philosophy

PRIMES is "hybrid" because it combines:

Engineering/Technology Models (bottom-up)
- Explicit representation of technologies
- Physical constraints (efficiency, capacity, ramping)
Economic/Behavioral Models (top-down)
- Price-responsive demand
- Consumer heterogeneity
- Market equilibrium conditions

2.3 Theoretical Basis by Module

Module	Theoretical Approach
Demand (Residential, Tertiary, Industry)	Discrete choice theory (McFadden)
Transport	Nested logit models, generalized cost
Power Generation	Least-cost optimization (LP/MILP)
Refining, Gas Supply	Market equilibrium, Stackelberg games
Pricing	Ramsey-Boiteux (regulated), marginal cost (wholesale)

2.4 Why Transitions Are Slow: Two Distinct Mechanisms

PRIMES produces gradual transitions for two fundamentally different reasons. Understanding the distinction matters for policy analysis:

1. Capital Inertia (Stock Dynamics)

Existing equipment has remaining economic/physical lifetime
A power plant built in 2020 runs until 2050+ regardless of new policy
Early retirement is possible but costly (stranded assets)
Policy implication: Even with strong price signals, old stock persists. Carbon pricing affects new investment decisions, not existing capacity.

2. Adoption Heterogeneity (Discrete Choice)

Even for new purchases, not everyone adopts the lowest-cost option simultaneously
People differ in preferences, constraints, information, risk tolerance
Captured via Weibull/logit distributions over generalized costs
Policy implication: Market shares shift gradually even for new purchases. A carbon tax doesn't make everyone buy EVs—it shifts the distribution of choices.

Mechanism	What It Affects	Time Scale	Policy Lever
Inertia	Existing stock retirement	Decades (equipment lifetimes)	Early retirement incentives, stranded asset compensation
Heterogeneity	New purchase decisions	Years (adoption diffusion)	Price signals, information campaigns, infrastructure

These mechanisms stack: slow turnover of existing stock + gradual adoption of new technologies = multi-decade transitions.

3. Inputs

3.1 Macroeconomic Drivers (from GEM-E3)

Input	Source	Description
GDP	GEM-E3	Real GDP by country, scenario
Sectoral output	GEM-E3	Value added by NACE sector
Population	Eurostat projections	Persons, households
Discount rates	Scenario assumption	Social vs. private (4-12%)

3.2 Energy System Data

Input	Source	Description
Energy balances	Eurostat, IEA	Base year physical flows
Installed capacity	ENTSO-E, Platts	Power plants, refineries
Technology database	IEA, IRENA, internal	Costs, efficiencies, lifetimes
Fuel prices	PROMETHEUS	World oil, gas, coal prices
CO₂ prices	Policy scenario	ETS price trajectory

3.3 Technology Parameters

Parameter	Examples	Uncertainty
Investment cost	€/kW for wind, solar, nuclear	Medium-High
Fixed O&M	€/kW/year	Low
Variable O&M	€/MWh	Low
Efficiency	% (thermal plants)	Low
Capacity factor	% (renewables)	Medium
Lifetime	Years	Low
Learning rate	% cost reduction per doubling	High

3.4 Policy Inputs

Policy Type	Implementation
Carbon price	Tax or ETS cap + price floor
Renewable targets	Constraint on RES share
Efficiency standards	Minimum efficiency for appliances, vehicles
Phase-out dates	Nuclear, coal exit years
Subsidies	Feed-in tariffs, investment support

4. Outputs

4.1 Energy Balances

Output	Detail
Final energy demand	By sector × fuel × country × year
Primary energy supply	By source (domestic, imports)
Transformation	Power, heat, refining losses
Energy intensity	Energy/GDP, energy/capita

4.2 Technology Stocks

Output	Detail
Installed capacity	GW by technology × country × year
Generation mix	TWh by technology
Fleet composition	Million vehicles by type
Building stock	m² by efficiency class
Industrial processes	Technology shares

4.3 Prices

Output	Unit
Wholesale electricity price	€/MWh
Retail electricity price	€/kWh (by sector)
Natural gas price	€/MWh
Oil product prices	€/liter
Heat prices	€/GJ

4.4 Investment and Costs

Output	Description
Investment expenditure	€bn/year by sector
Total system cost	NPV of all costs
Marginal abatement cost	€/tCO₂ avoided
Stranded assets	Value of early retirements

4.5 Emissions

Output	Coverage
CO₂ by sector and fuel	Mt CO₂/year
Other GHGs	CH₄ from gas, N₂O from transport
Air pollutants	SO₂, NOx, PM (linked to fuel use)

5. Mathematical Formalization

5.1 Overall Structure: EPEC

PRIMES solves an Equilibrium Problem with Equilibrium Constraints (EPEC):

Multiple agents (producers, consumers) each solve their own optimization problem, and the model finds market-clearing prices:

$$ \text{Find } (p^_, q^_) \text{ such that } \sum_i q_i^{demand}(p^_) = \sum_j q_j^{supply}(p^_) $$

where each agent's quantity is the solution to their own optimization.

5.2 Demand Side: Discrete Choice Models

5.2.1 Useful Energy Demand

First, calculate demand for energy services (not fuels):

$$ UE_{s,r,t} = f(VA_{s,r,t}, P_{UE}, AEEI_t) $$

where:

$UE$ = useful energy (e.g., process heat in GJ)
$VA$ = sectoral value added (from GEM-E3)
$P_{UE}$ = price of useful energy
$AEEI$ = autonomous energy efficiency improvement

5.2.2 Technology Choice: Weibull Distribution

Consumers don't all choose the lowest-cost option. Market shares follow a Weibull distribution:

$$ S_i = \frac{e^{-\nu \cdot C_i}}{\sum_j e^{-\nu \cdot C_j}} $$

where:

$S_i$ = market share of technology $i$
$C_i$ = generalized cost (not just financial)
$\nu$ = parameter controlling heterogeneity

Generalized cost includes: $$ C_i = \underbrace{CAPEX_i + OPEX_i}{\text{engineering cost}} + \underbrace{\mu_i}{\text{intangible cost (risk, hassle)}} $$

The intangible cost $\mu_i$ captures:

Risk aversion toward new technologies
Hidden costs (installation complexity)
Behavioral inertia

5.2.3 Nested Logit (Transport)

Transport mode and vehicle choice uses nested logit:

$$ P(\text{mode } m | \text{trip type } t) = \frac{e^{V_m / \lambda}}{\sum_{m'} e^{V_{m'} / \lambda}} $$

where $V_m$ is the systematic utility: $$ V_m = \beta_0 + \beta_1 \cdot \text{cost}_m + \beta_2 \cdot \text{time}_m + \beta_3 \cdot \text{comfort}_m $$

5.3 Supply Side: Cost Minimization

5.3.1 Power Sector Objective

Minimize total discounted system cost:

$$ \min \sum_{t=0}^{T} \frac{1}{(1+r)^t} \left[ \sum_i (IC_i \cdot \Delta K_{i,t} + FC_i \cdot K_{i,t} + VC_i \cdot G_{i,t}) \right] $$

where:

$IC_i$ = investment cost (€/kW)
$FC_i$ = fixed O&M (€/kW/year)
$VC_i$ = variable cost (€/MWh)
$K_{i,t}$ = capacity of technology $i$ in year $t$
$G_{i,t}$ = generation (MWh)

5.3.2 Key Constraints

Demand satisfaction: $$ \sum_i G_{i,t,h} = D_{t,h} \quad \forall t, h $$

Capacity constraint: $$ G_{i,t,h} \leq CF_{i,h} \cdot K_{i,t} \quad \forall i, t, h $$

where $CF_{i,h}$ is the capacity factor in hour $h$.

Ramping constraints: $$ |G_{i,t,h+1} - G_{i,t,h}| \leq RR_i \cdot K_{i,t} $$

Reserve margin: $$ \sum_i AvailCap_{i,t} \geq (1 + \xi) \cdot PeakDemand_t $$

Renewable portfolio standard: $$ \frac{\sum_{i \in RES} G_{i,t}}{\sum_i G_{i,t}} \geq \theta_t $$

5.3.3 Investment Decision

New capacity is built when the Net Present Value is positive:

$$ NPV_i = \sum_{t=0}^{L_i} \frac{R_{i,t} - C_{i,t}}{(1+r)^t} - IC_i > 0 $$

where $R_{i,t}$ is expected revenue (price × generation).

5.4 Pricing Mechanisms

5.4.1 Wholesale Electricity: Marginal Cost

The wholesale price equals the shadow price of the demand constraint:

$$ P_{wholesale,h} = \lambda_h^{demand} $$

This equals the marginal cost of the most expensive unit dispatched.

5.4.2 Retail Pricing: Ramsey-Boiteux

For regulated infrastructure (grids), prices are set to recover fixed costs while minimizing welfare loss:

$$ \frac{P_i - MC_i}{P_i} = \frac{k}{\epsilon_i} $$

where:

$\epsilon_i$ = price elasticity of demand for consumer class $i$
$k$ = constant ensuring cost recovery

Interpretation: Charge higher markups to consumers with lower elasticity (they'll reduce demand less).

5.4.3 Retail Price Decomposition

$$ P_{retail} = P_{wholesale} + T_{network} + T_{policy} + Taxes $$

where:

$T_{network}$ = grid tariffs
$T_{policy}$ = RES support, capacity payments
$Taxes$ = VAT, energy taxes

5.5 Temporal Resolution

PRIMES represents intra-annual variation through:

Load duration curve: Hours ranked by demand level Time slices: Typically 12-24 representative periods per year

$$ \text{Periods} = {Peak, Shoulder, OffPeak} \times {Winter, Summer, Intermediate} \times {Day, Night} $$

This captures:

Diurnal patterns (solar availability)
Seasonal patterns (heating/cooling demand)
Peak vs. baseload operation

6. Solution Method

6.1 Iterative Market Equilibrium

EPEC is the conceptual umbrella—multiple agents each solving their own optimization, with market clearing linking them. Operationally, PRIMES implements this as modular equilibrium iteration with embedded optimizations:

PRIMES doesn't solve a single monolithic optimization. It iterates between:

Demand modules calculate quantities given prices
Supply modules calculate prices given quantities
Check convergence; if not, update and repeat

Initialize prices P⁰

For iteration k:
    1. Demand modules: Q^k = D(P^{k-1})
    2. Supply modules: P^k = S(Q^k)
    3. If ||P^k - P^{k-1}|| < ε: STOP
    4. Else: k = k+1

6.2 GAMS Implementation

PRIMES is implemented in GAMS with:

Demand modules as simulation/econometric equations
Supply modules as LP/MILP optimization
Master loop controlling iteration

6.3 Computational Complexity

Component	Method	Solve Time
Demand modules	Simulation	Fast (seconds)
Power module	LP/MILP	Medium (minutes)
Full model iteration	Market equilibrium	Hours
All EU countries + years	Full scenario	1-2 days

7. Calibration

7.1 Base Year Calibration

The model must replicate Eurostat energy balances for the base year:

Adjust technology shares to match observed generation mix
Calibrate demand functions to observed consumption
Set intangible costs $\mu_i$ to match observed technology adoption rates

7.2 Validation

Metric	Validation Source
Historical backcasting	Replicate 2010-2020 with historical inputs
Cross-model comparison	Compare with TIMES, MESSAGE
Expert review	Sense-check technology pathways

8. Key Assumptions and Limitations

Assumption	Implication
Exogenous GDP	No feedback from energy costs to macroeconomy (handled by GEM-E3 coupling)
Perfect foresight (supply)	Investors know future prices
Myopic consumers (demand)	Households don't optimize intertemporally
No market power	Perfect competition in most markets
Representative consumers	Within each class, homogeneous behavior

Known limitations:

Intangible costs are calibrated, not estimated
Hourly resolution approximated by time slices
Technology learning rates uncertain
Behavioral change beyond technology adoption not fully captured

9. Connections to Other Models

Model	Interface	Direction
GEM-E3	GDP, sectoral activity	Receives
GEM-E3	Energy prices, system costs	Sends
PROMETHEUS	World fuel prices	Receives
E3ME	Alternative macro driver	Can substitute for GEM-E3
GAINS	Air pollutant costs	Can link

Coupling with GEM-E3

The coupling iteration:

GEM-E3: GDP_t → PRIMES
PRIMES: Energy system → Energy prices_t
Energy prices_t → GEM-E3
GEM-E3: GDP_{t+1} → ...

Convergence when $|GDP_{iteration} - GDP_{iteration-1}| < \epsilon$

10. Comparison with Alternative Energy Models

Feature	PRIMES	TIMES/MARKAL	OSeMOSYS
Equilibrium	Partial	None (cost min)	None (cost min)
Demand	Behavioral	Elastic	Fixed
Pricing	Endogenous	Shadow prices	Shadow prices
Scope	EU focus	Flexible	Flexible
Openness	Proprietary	Commercial	Open source

11. Key References

E3Modelling (2018). "The PRIMES Model 2018." Technical documentation.
Capros, P. et al. (2014). "Description of models and scenarios used to assess European decarbonisation pathways." Energy Strategy Reviews.
McFadden, D. (1974). "Conditional logit analysis of qualitative choice behavior." Frontiers in Econometrics.
Ramsey, F. (1927). "A Contribution to the Theory of Taxation." Economic Journal.

Raw

PROMETHEUS.md

PROMETHEUS: Technical Distillation

Stochastic World Energy Model

1. Model Identity

Attribute	Value
Type	Stochastic Simulation Model
Mathematical Class	Non-linear equation system + Monte Carlo
Temporal Structure	Dynamic simulation (annual steps)
Spatial Coverage	Global (world regions)
Time Horizon	Typically 2020–2070, annual steps
Resolution	Aggregate (major fuels × world regions)
Primary Output	Probability distributions of world fuel prices
Role in E3 Suite	Boundary condition generator; uncertainty quantification

2. Theoretical Foundations

2.1 The Problem of Uncertainty

Energy system projections face deep uncertainty in:

Fossil fuel resource availability
Technology development (costs, performance)
Economic growth in emerging economies
Policy evolution globally

PROMETHEUS addresses this by:

Modeling key parameters as probability distributions rather than point estimates
Running Monte Carlo simulations to propagate uncertainty
Producing probabilistic forecasts rather than single scenarios

2.1.1 Structure Shapes Uncertainty

PROMETHEUS is not just a Monte Carlo wrapper that passes random inputs through a black box. The value lies in how structural features of energy markets interact with parameter uncertainty:

Structural Feature	How It Shapes Uncertainty
Depletion dynamics	Uncertainty in URR compounds over time—small differences in reserves lead to divergent long-run price paths
OPEC behavior	Strategic responses amplify or dampen shocks—a demand shock triggers supply response that feeds back
Demand lags	Partial adjustment means prices overshoot and then correct—volatility patterns emerge from structure
Learning curves	Uncertain learning rates create path dependence—early deployment affects long-run costs

The insight isn't the distribution itself but how structural features propagate and transform input uncertainty into output uncertainty.

2.2 Theoretical Basis

2.2.1 Supply: Hotelling-style Resource Economics

Oil and gas supply follows principles from exhaustible resource theory:

$$ \frac{\partial P}{\partial t} = r \cdot P - \frac{\partial MC}{\partial Q} $$

where:

$P$ = resource price
$r$ = discount rate
$MC$ = marginal extraction cost
$Q$ = cumulative extraction

Key insight: As resources deplete, extraction moves to more expensive sources, driving up prices.

2.2.2 Demand: Econometric Relationships

Demand is modeled using reduced-form econometric equations:

$$ \ln(D_{f,r,t}) = \alpha + \beta_1 \ln(GDP_{r,t}) + \beta_2 \ln(P_{f,t}) + \beta_3 \ln(D_{f,r,t-1}) + \epsilon $$

Characteristics:

Log-log specification (constant elasticities)
Lagged dependent variable (partial adjustment)
Estimated from historical data (1970-present)

2.2.3 Technology: Learning Curves with Uncertainty

Technology costs follow experience curves:

$$ C_t = C_0 \cdot \left(\frac{Q_t}{Q_0}\right)^{-b} $$

where $b = -\ln(1-LR)/\ln(2)$ and $LR$ is the learning rate.

PROMETHEUS innovation: The learning rate $LR$ is itself a random variable:

$$ LR \sim \mathcal{N}(\mu_{LR}, \sigma_{LR}^2) $$

3. Inputs

3.1 Stochastic Parameters (Distributions)

Parameter	Distribution	Uncertainty Source
Oil URR (Ultimate Recoverable Resources)	Triangular/Lognormal	Geological estimates vary
Gas URR	Triangular/Lognormal	Same
Coal resources	Triangular	Less uncertain
GDP growth (non-OECD)	Normal	Economic forecasts diverge
Technology learning rates	Normal/Triangular	Historical variance
Demand elasticities	Normal	Econometric estimation error

3.2 Deterministic Inputs

Input	Source	Description
Historical data	IEA, BP	Prices, production, consumption (1970-present)
Proven reserves	OPEC, EIA	Current known resources
Econometric estimates	Literature	Central elasticity values
Policy assumptions	Scenario	Global climate policy strength

3.3 Scenario Dimensions

PROMETHEUS typically explores scenarios along:

Resource availability: Pessimistic → Optimistic
Demand growth: Low → High
Technology progress: Slow → Fast
Climate policy: Weak → Strong

4. Outputs

4.1 Primary Outputs: Price Distributions

Output	Format	Use
Oil price	$/barrel, probability distribution	PRIMES, GEM-E3 input
Gas price (regional)	$/MMBtu, probability distribution	PRIMES input
Coal price	$/ton, probability distribution	PRIMES input

Output format:

Mean, median, standard deviation
Percentiles (P10, P50, P90)
Full empirical distribution from Monte Carlo

4.2 Secondary Outputs

Output	Description
Global primary energy demand	By fuel and region
Production by region	Oil, gas, coal extraction
Trade flows	Net exports/imports by region
Depletion rates	Remaining resources over time

4.3 Uncertainty Quantification

Metric	Description
Variance decomposition	Which input uncertainties drive output variance
Sensitivity indices	Sobol indices for global sensitivity
Confidence intervals	90% CI for price projections

5. Mathematical Formalization

5.1 Overall Model Structure

PROMETHEUS consists of:

Supply module for each fossil fuel
Demand module for each region
Market clearing condition
Monte Carlo wrapper for uncertainty propagation

5.2 Supply Functions

5.2.1 Oil and Gas Supply (Non-linear Cost Curves)

Supply is derived from a cost-resource curve:

$$ C(R) = C_0 + \gamma \cdot \left(\frac{R}{URR}\right)^\eta $$

where:

$C(R)$ = marginal cost when cumulative extraction is $R$
$C_0$ = initial (lowest) extraction cost
$URR$ = ultimate recoverable resources (uncertain)
$\gamma, \eta$ = curve shape parameters

Production decision: Extract where marginal revenue ≥ marginal cost: $$ Q_t = \min\left{Q^{capacity}_t, Q : P_t = C'(R_t + Q)\right} $$

5.2.2 OPEC Behavior

OPEC is modeled as a dominant firm with competitive fringe:

$$ Q^{OPEC}_t = D_t(P_t) - S^{fringe}_t(P_t) $$

OPEC sets price to maximize discounted profits: $$ \max_{P_t} \sum_{t} \frac{(P_t - MC^{OPEC}_t) \cdot Q^{OPEC}_t}{(1+r)^t} $$

Subject to: market share constraints, capacity limits, depletion dynamics.

5.2.3 Coal Supply

Coal supply is more competitive (no cartel), modeled as:

$$ S^{coal}_t = f(P^{coal}_t, Capacity_t, TransportCosts) $$

5.3 Demand Functions

5.3.1 Regional Demand (Econometric)

For region $r$ and fuel $f$:

$$ \ln D_{f,r,t} = \alpha_{f,r} + \beta^{GDP}_{f,r} \ln GDP_{r,t} + \beta^{P}_{f,r} \ln P_{f,t} + \beta^{lag}_{f,r} \ln D_{f,r,t-1} $$

Estimated parameters:

Fuel	Income elasticity ($\beta^{GDP}$)	Price elasticity ($\beta^P$)
Oil	0.4–0.8	-0.1 to -0.3
Gas	0.6–1.0	-0.2 to -0.5
Coal	0.3–0.6	-0.1 to -0.2

5.3.2 Inter-fuel Substitution

Long-run substitution between fuels:

$$ \frac{D_{oil}}{D_{gas}} = \kappa \cdot \left(\frac{P_{gas}}{P_{oil}}\right)^{\sigma_{sub}} $$

where $\sigma_{sub}$ is the substitution elasticity (typically 0.5–2.0).

5.4 Market Clearing

For each fuel $f$ and time $t$:

$$ \sum_r D_{f,r,t}(P_{f,t}) = \sum_r S_{f,r,t}(P_{f,t}) $$

Solution: Find $P^*_{f,t}$ that clears the market.

5.5 Monte Carlo Implementation

5.5.1 Sampling

For each Monte Carlo iteration $m = 1, \ldots, M$:

Draw random parameters: $$ \theta^{(m)} \sim \pi(\theta) $$

where $\theta = (URR^{oil}, URR^{gas}, LR^{renewables}, \beta^{GDP}, \ldots)$

Solve model: Find equilibrium prices $P^{(m)}_t$ for all $t$
Store results: $P^{(m)}_t$ becomes one sample in the output distribution

5.5.2 Number of Iterations

Typically $M = 1000$ to $10000$ iterations, chosen to achieve: $$ SE(\hat{\mu}) = \frac{\hat{\sigma}}{\sqrt{M}} < \epsilon $$

where $\epsilon$ is the desired precision (e.g., $1/barrel for oil price).

5.5.3 Latin Hypercube Sampling (LHS)

To improve efficiency, PROMETHEUS uses Latin Hypercube Sampling rather than simple random sampling:

Divide each parameter's range into $M$ equal-probability intervals
Sample exactly once from each interval
Randomly pair samples across parameters

Advantage: Better coverage of parameter space with fewer iterations.

5.6 Solving the Stochastic System

For each Monte Carlo draw:

Input: θ^(m) = {URR, elasticities, growth rates, ...}

For t = t_0 to T:
    1. Update GDP: GDP_t = GDP_{t-1} × (1 + g_t)
    2. Update resources: R_t = R_{t-1} + Q_{t-1}
    3. Solve market clearing:
       Find P_t such that D(P_t) = S(P_t)
    4. Compute quantities: Q_t = D(P_t)

Output: {P_t}_{t=t_0}^T for this iteration

5.7 Uncertainty Quantification

5.7.1 Output Statistics

From $M$ price trajectories ${P^{(m)}_t}$:

Mean: $\hat{\mu}_t = \frac{1}{M} \sum_m P^{(m)}_t$

Variance: $\hat{\sigma}^2_t = \frac{1}{M-1} \sum_m (P^{(m)}_t - \hat{\mu}_t)^2$

Percentiles: $P_{10,t}, P_{50,t}, P_{90,t}$ from empirical distribution

5.7.2 Sensitivity Analysis: Sobol Indices

To identify which uncertain parameters drive output variance:

First-order index: $$ S_i = \frac{V[\mathbb{E}[Y | X_i]]}{V[Y]} $$

Measures the fraction of output variance explained by parameter $X_i$ alone.

Total-effect index: $$ S_{T,i} = 1 - \frac{V[\mathbb{E}[Y | X_{-i}]]}{V[Y]} $$

Includes all interactions involving $X_i$.

Typical findings:

Parameter	First-order $S_i$
Oil URR	0.2–0.4
GDP growth	0.1–0.3
OPEC behavior	0.1–0.2
Demand elasticity	0.05–0.15

6. Solution Method

6.1 Implementation

PROMETHEUS is typically implemented in:

GAMS for the core market model
R or Python for Monte Carlo wrapper and post-processing
Excel/VBA for parameter input management

6.2 Computational Requirements

Component	Time
Single deterministic run	Seconds
Full Monte Carlo (5000 iterations)	Minutes to hours
Sensitivity analysis	Hours (requires many runs)

6.3 Solver

Market clearing equations solved using:

Newton-Raphson for single markets
PATH solver for simultaneous multi-fuel equilibrium

7. Calibration

7.1 Historical Calibration

Backcast test: Run model for 1990–2020 with historical GDP
Compare simulated prices to actual IEA/EIA prices
Adjust parameters until historical fit is acceptable

7.2 Parameter Estimation

Parameter	Estimation Method
Demand elasticities	Time-series regression
Supply cost curves	Engineering cost data + resource estimates
OPEC behavior	Calibration to historical market share

7.3 Distribution Specification

Challenge: Specify probability distributions for uncertain parameters.

Approaches:

Expert elicitation: Ask experts for min/max/most likely
Historical variance: Use observed volatility
Literature review: Combine multiple estimates

8. Key Assumptions and Limitations

Assumption	Implication
Reduced-form demand	No structural technology modeling
OPEC as rational actor	May not capture political decisions
Independent parameters	Correlation structure simplified
No feedback from prices to GDP	Handled by GEM-E3 coupling
Annual resolution	No intra-year price volatility

Known limitations:

Black swan events (wars, revolutions) not modeled
Correlation between parameters may be underestimated
Technology disruptions (shale revolution) hard to predict
Political factors (sanctions, OPEC+ decisions) exogenous

9. Connections to Other Models

Model	Interface	Frequency
PRIMES	Oil, gas, coal prices	Annual, probabilistic
GEM-E3	World fuel prices	Annual, probabilistic
IEA WEO	Comparison/validation	External benchmark

9.1 How Outputs Are Used

Deterministic runs (PRIMES/GEM-E3):

Use P50 (median) price trajectory as central case
Run additional scenarios at P10, P90 for sensitivity

Probabilistic analysis:

Feed distribution into stochastic version of PRIMES (if available)
Report results as ranges, not point estimates

10. Comparison with Alternative Models

Feature	PROMETHEUS	IEA WEO	EIA IEO	OPEC WOO
Stochastic	✅ Full MC	❌ Scenarios	❌ Scenarios	❌ Scenarios
Behavioral OPEC	✅	Partial	❌	✅
Learning curves	✅ Uncertain	✅ Fixed	✅ Fixed	Limited
Open source	❌	❌	❌	❌

PROMETHEUS advantage: Explicit uncertainty quantification vs. scenario-based approaches.

11. Practical Usage Guidance

11.1 Interpreting Outputs

Don't:

Treat mean as a forecast
Ignore the distribution width
Say "PROMETHEUS predicts X"

Do:

Report ranges (e.g., "oil prices in 2040: $60–120/barrel with 80% probability")
Use for stress-testing policies under price uncertainty
Identify which uncertainties matter most (via sensitivity analysis)

11.1.1 Communicating P10/P50/P90 to Policymakers

Percentile	Meaning	Use In Policy Discussion
P50	Median—50% chance above, 50% below	Central planning assumption; "our baseline"
P10	10th percentile—90% chance of being higher	Downside stress test; "even if prices are low..."
P90	90th percentile—90% chance of being lower	Upside stress test; "even if prices are high..."

Framing suggestions:

❌ "Oil will cost $80/barrel in 2040"
✅ "There's an 80% chance oil prices fall between $50 and $120/barrel in 2040, with $80 as the median estimate"
✅ "Our policy remains cost-effective even at P90 prices"
✅ "The key uncertainty is not the price level but whether OPEC accommodates or restricts supply"

Note: P10/P90 bounds the 80% confidence interval, not 90%. For 90% CI, use P5/P95.

11.2 Typical Results

Horizon	Oil Price Range (P10–P90)	Key Driver of Uncertainty
2030	$50–90/barrel	OPEC behavior, demand growth
2050	$40–150/barrel	URR, technology, climate policy
2070	$30–200/barrel	Deep uncertainty dominates

12. Key References

E3Modelling (2018). "The PROMETHEUS Model." Technical documentation.
Hotelling, H. (1931). "The Economics of Exhaustible Resources." Journal of Political Economy.
Kaufmann, R.K. (1995). "Econometric models of energy demand." Energy Economics.
Sobol, I.M. (2001). "Global sensitivity indices for nonlinear mathematical models." Mathematics and Computers in Simulation.
McKay, M.D., Beckman, R.J., Conover, W.J. (1979). "A Comparison of Three Methods for Selecting Values of Input Variables." Technometrics.

davidefiocco/EnergyMarketsPrimer.md

E3 Modeling Concepts Primer

Foundations for Understanding GEM-E3, PRIMES, and PROMETHEUS

Table of Contents

Part I: Economic Foundations

Part II: Mathematical Methods

Part III: Energy Systems

Part IV: Trade and Environment

Part V: Integration

Appendices

Part I: Economic Foundations

Chapter 1: Markets and Demand

1.1 What Is a Market?

1.2 Demand

1.3 Price Elasticity

1.4 Where This Appears in E3 Models

Chapter 2: Production and Supply

2.1 What Is a Production Function?

2.2 Key Concepts

2.3 The CES Production Function

2.4 Nested CES Production

2.5 Elasticity Values Matter!

2.6 Technical Change

2.7 Cost Functions and Input Demands

2.8 From Costs to Supply

2.9 Where This Appears in E3 Models

Chapter 3: Equilibrium Concepts

3.1 Market Equilibrium

3.2 What Is Economic Equilibrium? (General Concept)

3.3 Partial Equilibrium vs. General Equilibrium

3.4 Walrasian General Equilibrium

3.5 The Auctioneer Metaphor

3.6 Existence and Uniqueness

3.7 Comparative Statics

3.8 Dynamic Extensions

3.9 Labor Markets: Beyond Perfect Competition

3.10 Where This Appears in E3 Models

Chapter 4: Welfare Economics

4.1 What Is Welfare?

4.2 Utility

4.3 Consumer and Producer Surplus

4.4 Pareto Efficiency

4.5 Measuring Welfare Changes

4.6 Social Welfare Functions

4.7 Efficiency vs. Equity Trade-offs

4.8 The Double Dividend Hypothesis

4.9 Where This Appears in E3 Models

Part II: Mathematical Methods

Chapter 5: Optimization Primer

5.1 Why Optimization?

5.2 The Basic Optimization Problem

5.3 Constrained Optimization and Lagrange Multipliers

5.4 Types of Optimization Problems

5.5 Shadow Prices (Dual Variables)

5.6 Complementarity Problems

5.7 Solving Optimization Problems

5.8 Where This Appears in E3 Models

Chapter 6: Discrete Choice Models

6.1 The Problem: Heterogeneous Choices

6.2 Random Utility Models

6.3 The Logit Model

6.4 The Weibull Distribution (in PRIMES)

6.5 Nested Logit (for Transport)

6.6 Calibration of Intangible Costs

6.7 Why This Matters for Policy

6.8 Where This Appears in E3 Models

Chapter 7: Stochastic Methods

7.1 The Challenge of Uncertainty

7.2 Types of Uncertainty

7.3 Probability Distributions

7.4 Monte Carlo Simulation

7.5 Latin Hypercube Sampling (LHS)

7.6 Sensitivity Analysis

7.7 Interpreting Stochastic Results

7.8 Where This Appears in E3 Models

Part III: Energy Systems

Chapter 8: Energy Fundamentals

8.1 Why Energy Matters for Economic Modeling

8.2 Energy Forms and Conversions

8.3 Energy Units