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@ -14,15 +14,17 @@ Use: When a user asks about TU Q101 equity premium experiments or wants
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# TU Q101 MVP: toy equity premium tension
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_Status: work in progress. This page records early MVP designs and will be updated after the first notebooks run._
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_Status: MVP A implemented (offline, single cell). MVP B is design only and will be added later._
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> This page sketches toy experiments for TU Q101.
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> The goal is to encode the equity premium puzzle as a tension between
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> simple models and simple observed targets, not to resolve real markets.
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> TU Q101 encodes the equity premium puzzle as a tension between
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> simple consumption based models and stylised target bands for
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> long run equity premia.
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> The goal is to expose where the puzzle appears in tiny toy worlds,
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> not to resolve real markets.
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**Navigation**
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- [← Back to Experiments index](../README.md)
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- [← Back to Experiments index](../README.md)
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- [← Back to Event Horizon (WFGY 3.0)](../../EventHorizon/README.md)
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---
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@ -31,68 +33,158 @@ _Status: work in progress. This page records early MVP designs and will be updat
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TU Q101 looks at the equity premium puzzle as a structured mismatch between
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- predicted premia from simple asset pricing models, and
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- empirical target bands for long run equity premia.
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- predicted premia from simple asset pricing models, and
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- empirical style targets for long run equity premia.
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The MVP here uses small simulated economies with:
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The MVP here uses small simulated economies with
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- consumption processes,
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- risk free and risky assets,
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- consumption processes,
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- risk free and risky assets,
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- representative agents with utility functions,
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and tracks tension observables when the model cannot match observed premia without extreme parameters.
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Users can understand the result just by reading this page.
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Running the Colab notebook is only required if you want to reproduce the numbers or modify parameters.
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---
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## 1. Experiment A: simple consumption based asset pricing model
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### 1.1 Research question
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If we build a minimal consumption based asset pricing model, can we define a scalar observable T_premium that
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If we build a minimal consumption based asset pricing model, can we define a scalar observable `T_premium` that
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- is small when predicted equity premia and volatility sit inside a plausible band,
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- and increases as the model requires extreme risk aversion or unrealistic parameters.
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- is small when predicted equity premia and volatility stay inside a plausible band,
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- and grows when the model only matches targets by pushing risk aversion or other parameters into unrealistic regions.
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### 1.2 Setup
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### 1.2 Notebook and Colab entry
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The notebook will:
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**Notebook**
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- Simulate a simple consumption growth process over many periods.
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- Define two assets:
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- `TensionUniverse/Experiments/Q101_MVP/Q101_A.ipynb`
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- a risk free asset with fixed return,
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- a risky asset with return tied to consumption growth.
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**One click Colab**
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- Use a basic power utility or similar to compute implied prices.
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- From these, derive
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[](https://colab.research.google.com/github/onestardao/WFGY/blob/main/TensionUniverse/Experiments/Q101_MVP/Q101_A.ipynb)
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- the implied equity premium,
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- volatility of returns,
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- required risk aversion parameter to match a chosen target premium.
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The notebook is a single cell script.
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All runs are fully offline. No API key is needed.
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Define T_premium using:
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### 1.3 Model and tension observable
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- the squared deviation between implied and target equity premium,
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- penalties for risk aversion parameters outside a reasonable range,
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- penalties when volatility patterns are inconsistent with empirical targets.
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The notebook simulates three tiny consumption based worlds and computes a scalar tension observable `T_premium` for each world.
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### 1.3 Expected pattern
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Core ingredients:
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We expect:
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- A lognormal consumption growth process.
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- Two assets:
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- a risk free asset with return `R_f`,
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- a risky asset whose payoff is correlated with consumption.
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- A representative agent with power utility and risk aversion `gamma`.
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- Standard consumption based pricing:
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- stochastic discount factor `m = beta * g^{-gamma}`,
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- prices from expectations of `m` and payoffs,
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- equity premium `E[R_e] - R_f` and volatility `vol`.
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- toy configurations that generate plausible premia with reasonable parameters to have lower T_premium,
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- toy configurations that only match observed premia with extreme risk aversion to have higher T_premium.
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For each world the notebook:
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The point is to show an explicit tension between model structure and target bands, even in a simple setup.
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1. Sweeps `gamma` over a grid from 0.5 to 20.
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2. For every `gamma`, computes the implied equity premium and volatility.
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3. Picks `gamma*` that gets closest to a target long run equity premium.
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### 1.4 How to reproduce
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Target band and plausible parameter region (stylised):
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After `Q101_A.ipynb` is created:
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- Target long run equity premium: **6.0%**.
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- Plausible risk aversion range for the representative agent: **gamma in [1.0, 5.0]**.
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- Premium mismatch is scaled at **2.0% per unit** in the tension score.
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1. Open the notebook.
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2. Read the header comments describing the consumption process, assets and parameter ranges.
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3. Run the simulation and compute T_premium across parameter sweeps.
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4. Inspect the table of results and plots.
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The scalar observable `T_premium` combines three components at `gamma*`:
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1. Premium mismatch
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- absolute deviation between implied and target premium, scaled by 2% per unit.
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2. Risk aversion penalty
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- zero if `gamma*` is in [1, 5],
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- linear penalty when `gamma*` lies outside this plausible band.
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3. Volatility penalty
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- linear penalty when the realised volatility falls outside a simple per world target interval.
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Weights are chosen so that premium mismatch and risk aversion dominate, with volatility as a weaker regulariser.
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Configured scenarios:
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- `no_puzzle`
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High volatility world with strongly correlated risky asset. A moderate `gamma` can support a six percent equity premium without stress.
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- `realistic_puzzle`
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Low volatility consumption world that is closer to real data. We still demand a six percent equity premium. The model tends to require very large `gamma` values to get close.
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- `anemic_asset`
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Low volatility risky asset. Consumption volatility is low and the risky asset itself is not very volatile. Asking for six percent premium here is almost impossible without extreme parameters.
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### 1.4 First reference run
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All simulations complete in a single cell.
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The notebook prints a summary table sorted by `T_premium` (lower means closer to a plausible world):
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- Target premium: **6.0%** for all worlds.
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- Plausible `gamma` band: **[1.0, 5.0]**.
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Key outputs from the reference run:
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- **no_puzzle**
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- `gamma* ≈ 3.46` (inside plausible band)
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- model premium ≈ **5.83%** (difference −0.17%)
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- volatility ≈ **0.22**
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- `T_premium ≈ 0.035`
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- **realistic_puzzle**
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- `gamma* = 20.0` (hitting the right edge of the grid)
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- model premium ≈ **4.53%** (difference −1.47%)
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- volatility ≈ **0.28**
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- `T_premium ≈ 1.52`
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- **anemic_asset**
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- `gamma* = 20.0`
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- model premium ≈ **1.28%** (difference −4.72%)
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- volatility ≈ **0.11**
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- `T_premium ≈ 2.14`
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Quick interpretation:
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- In the **no_puzzle** world, a moderate risk aversion around three and a half already delivers a premium very close to six percent, with reasonable volatility. Tension is almost zero.
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- In the **realistic_puzzle** world, matching a six percent premium demands extreme risk aversion. Even at `gamma = 20` the model only reaches about four and a half percent. Tension is clearly elevated.
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- In the **anemic_asset** world, the risky asset is too quiet. The model cannot get anywhere near six percent premium even at the most extreme `gamma` on the grid. Tension is highest.
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This behaviour is also visible in the plots saved by the notebook:
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The premium versus `gamma` plot shows that only the high volatility world passes through the six percent target band for plausible `gamma` values.
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The low volatility and anemic asset worlds stay far below the target even when `gamma` is pushed to extremes.
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The `T_premium` bar chart summarises the story in a single view:
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- `no_puzzle` sits near zero tension,
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- `realistic_puzzle` has moderate tension,
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- `anemic_asset` has the largest tension.
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### 1.5 How to reproduce
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To reproduce or modify the experiment:
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1. Open `Q101_A.ipynb` in Colab using the badge above.
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2. Read the header block that explains the three worlds, target band and tension definition.
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3. Run the single cell to:
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- simulate all three worlds,
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- sweep `gamma`,
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- compute `T_premium` and print the summary table,
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- regenerate the two plots and save them as
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- `Q101A_premium_vs_gamma.png`
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- `Q101A_T_premium.png`.
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4. Adjust parameters if you want to explore variants, for example different target premia or different volatility ranges.
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---
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@ -100,44 +192,34 @@ After `Q101_A.ipynb` is created:
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### 2.1 Research question
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Can we use a language model to evaluate narrative explanations for the equity premium
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and compare them against the toy numerical model, defining a narrative tension observable T_narrative.
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Can we use a language model to evaluate narrative explanations for the equity premium and compare them against the toy numerical model, defining a narrative tension observable `T_narrative`.
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### 2.2 Setup
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### 2.2 Setup (design only)
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The notebook will:
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The planned notebook will:
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- Generate a small set of narrative hypotheses, for example
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- rare disasters,
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- habit formation,
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- institutional constraints.
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- For each configuration of the toy numerical model, produce a short summary describing its mechanism.
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- Ask a language model to judge which narrative hypothesis best fits the numerical summary.
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- Generate a small set of narrative hypotheses, for example
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- rare disasters,
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- habit formation,
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- institutional constraints.
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- For each configuration of the toy numerical model, produce a short summary describing its mechanism.
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- Ask a language model to judge which narrative hypothesis best fits the numerical summary.
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- Extract consistency scores.
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Define T_narrative as a function of:
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`T_narrative` will be defined as a function of:
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- mismatch between assigned narrative and true simulated mechanism,
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- low consistency scores.
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- mismatch between assigned narrative and true simulated mechanism,
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- low consistency scores when the narrative does not fit the numerical story.
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### 2.3 Expected pattern
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We expect:
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- lower T_narrative when numerical mechanisms and selected narratives match,
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- higher T_narrative when narratives and numerical structure disagree.
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- lower `T_narrative` when numerical mechanisms and selected narratives match,
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- higher `T_narrative` when narratives and numerical structure disagree.
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This bridges TU Q101 with narrative level explanations at the effective layer.
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### 2.4 How to reproduce
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Once `Q101_B.ipynb` exists:
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- open the notebook,
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- inspect the narrative set and prompt format,
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- run the evaluation and compare T_narrative across scenarios.
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This bridges TU Q101 with narrative level explanations at the effective layer.
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Implementation of `Q101_B.ipynb` is deferred to a later iteration.
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---
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TU Q101 treats the equity premium puzzle as a tension between
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- simple asset pricing models,
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- empirical target bands,
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- simple asset pricing models,
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- empirical style target bands,
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- and narrative explanations.
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This MVP page provides:
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This MVP page currently provides:
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- a minimal numerical experiment with T_premium,
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- a minimal narrative experiment with T_narrative.
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- a minimal numerical experiment with `T_premium` that shows where a toy model world behaves plausibly and where it runs into the equity premium puzzle,
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- a planned narrative experiment with `T_narrative` that will connect numerical mechanisms to verbal stories.
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Both are meant as starting points, not as final economic models.
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Both are starting points at the effective layer, not final economic models.
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For the broader project:
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- [Experiments index](../README.md)
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- [Experiments index](../README.md)
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- [Event Horizon (WFGY 3.0)](../../EventHorizon/README.md)
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---
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@ -167,6 +249,6 @@ For the broader project:
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The design of this MVP follows:
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- [TU Effective Layer Charter](../../Charters/TU_EFFECTIVE_LAYER_CHARTER.md)
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- [TU Encoding and Fairness Charter](../../Charters/TU_ENCODING_AND_FAIRNESS_CHARTER.md)
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- [TU Effective Layer Charter](../../Charters/TU_EFFECTIVE_LAYER_CHARTER.md)
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- [TU Encoding and Fairness Charter](../../Charters/TU_ENCODING_AND_FAIRNESS_CHARTER.md)
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- [TU Tension Scale Charter](../../Charters/TU_TENSION_SCALE_CHARTER.md)
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