systems.builtin.deterministic.discrete.DiscreteSolowModel

systems.builtin.deterministic.discrete.DiscreteSolowModel(*args, **kwargs)

Discrete-time Solow-Swan neoclassical growth model.

Physical System (Economic System):

The Solow model describes how an economy's capital stock evolves over time
through the interplay of:
- **Investment:** Savings channeled into new capital
- **Depreciation:** Capital wears out and becomes obsolete
- **Population growth:** Dilutes per-capita capital
- **Technological progress:** Increases productivity

**The Central Equation (Continuous Time):**
    k̇ = s·f(k) - (δ + n)·k

where:
    k: Capital per worker (capital intensity)
    s: Savings rate (fraction of output saved)
    f(k): Production function (output per worker)
    δ: Depreciation rate
    n: Population growth rate

**Production Function (Cobb-Douglas):**
The standard form is:
    y = f(k) = A·k^α

where:
    y: Output per worker (GDP per capita)
    A: Total factor productivity (TFP), technology level
    k: Capital per worker
    α: Output elasticity of capital (0 < α < 1)

**Key Properties:**
1. **Positive but diminishing marginal product:**
   f'(k) > 0, f''(k) < 0
   - More capital → more output
   - But each additional unit contributes less (diminishing returns)

2. **Inada conditions:**
   f(0) = 0, f'(0) = ∞, f'(∞) = 0
   - No capital → no output
   - Initially very productive
   - Eventually saturates

3. **Constant returns to scale (with labor):**
   F(K, L) = A·K^α·L^(1-α)
   - Doubling both K and L doubles output

**Discrete-Time Dynamics:**
The discrete version models year-to-year changes:

    k[t+1] = (1 - δ)·k[t] + s·f(k[t])

Or with population growth:
    k[t+1] = [(1 - δ)·k[t] + s·f(k[t])] / (1 + n)

State Space:

State: x[t] = [k[t]]
    Capital per worker:
    - k: Capital intensity [$/worker] or [capital units per worker]
      * k ≥ 0: Non-negative by definition
      * k = 0: No capital (subsistence economy)
      * k → k*: Converges to steady state
      * Typical range: 0-500 for normalized units

**Physical/Economic Meaning:**
Capital per worker represents:
- Machines, equipment, tools available per person
- Infrastructure (roads, buildings) per capita
- Technology embodied in physical capital
- Higher k → higher productivity → higher income

Control: u[t] (optional, for policy intervention)
    - Can represent:
      * Temporary savings rate change: s → s + u[t]
      * Government investment/stimulus
      * Technology adoption rate change
      * Foreign aid or capital injection
    - Standard model: u = 0 (no intervention)

Output: y[t] = [k[t]] or [k[t], y[t], c[t]]
    Standard outputs:
    - k[t]: Capital per worker (state variable)
    - y[t]: Output per worker (GDP per capita)
    - c[t]: Consumption per worker

    Where:
        y[t] = A·k[t]^α
        c[t] = (1 - s)·y[t]

Dynamics (Economic Interpretation):

The discrete dynamics:
    k[t+1] = (1 - δ)·k[t] + s·A·k[t]^α

**Capital Evolution Has Three Components:**

1. **Undepreciated capital: (1 - δ)·k[t]**
   - Fraction (1 - δ) survives to next period
   - δ represents wearing out, obsolescence
   - If δ = 0.05: 5% of capital lost each year

2. **New investment: s·f(k[t])**
   - Savings rate s fraction of output is invested
   - f(k) = A·k^α is output per worker
   - Higher s → more investment → faster growth

3. **Population dilution (if n > 0):**
   - Growing population dilutes per-capita capital
   - k[t+1] must be divided by (1 + n)
   - Higher n → slower per-capita growth

**Steady State (Golden Rule):**
At steady state: k[t+1] = k[t] = k*

    (1 - δ)·k* + s·A·(k*)^α = k*
    s·A·(k*)^α = δ·k*
    k* = (s·A/δ)^(1/(1-α))

The steady-state capital is unique and globally stable (under standard
assumptions).

**Convergence to Steady State:**
- If k[0] < k*: Capital accumulates (growth)
- If k[0] > k*: Capital decumulates (decline)
- Convergence is monotonic (no oscillations for standard parameters)
- Speed of convergence: Faster when far from k*, slower when near

Parameters:

s : float, default=0.3
    Savings rate (fraction of output saved/invested)
    - Must satisfy: 0 < s < 1
    - Typical values: 0.15-0.40
    - Developed countries: ~0.20-0.25
    - Developing countries: ~0.30-0.40
    - China (historically): ~0.45-0.50

    **Economic Meaning:**
    - Higher s → more investment → higher steady-state k*
    - But also lower consumption (1-s)·y
    - Golden rule: s that maximizes steady-state consumption

delta : float, default=0.05
    Depreciation rate (fraction of capital lost per period)
    - Must satisfy: 0 < δ < 1
    - Typical annual values: 0.03-0.10
    - Physical capital: ~0.05-0.08
    - Structures: ~0.03
    - Equipment: ~0.10

    **Economic Meaning:**
    - Higher δ → more replacement needed → lower k*
    - Represents obsolescence, wear and tear

alpha : float, default=0.33
    Capital share of income (output elasticity of capital)
    - Must satisfy: 0 < α < 1
    - Typical values: 0.25-0.40
    - Empirically: ~0.33 (one-third)
    - Labor share: 1 - α ≈ 0.67 (two-thirds)

    **Economic Meaning:**
    - Measures importance of capital vs labor
    - α = 0.5: Equal importance (Cobb-Douglas symmetric)
    - α < 0.5: Labor more important
    - α > 0.5: Capital more important (rare)

A : float, default=1.0
    Total Factor Productivity (TFP) / technology level
    - Must satisfy: A > 0
    - Captures: Technology, institutions, efficiency
    - Grows over time via technological progress
    - Typical growth: 1-2% per year

    **Economic Meaning:**
    - Higher A → more output for same k
    - Represents "technological level"
    - Solow residual: Growth not explained by capital/labor

n : float, default=0.0
    Population/labor force growth rate
    - Can be positive, zero, or negative
    - Typical values: -0.01 to 0.03 (annually)
    - Developed countries: ~0.00-0.01
    - Developing countries: ~0.01-0.03

    **Economic Meaning:**
    - Higher n → dilutes per-capita capital
    - Must invest more just to maintain k
    - Effective depreciation: δ + n

dt : float, default=1.0
    Time period (typically 1 year for macroeconomic data)
    - Units: years
    - dt = 1: Annual model (most common)
    - dt = 0.25: Quarterly model

include_population_growth : bool, default=False
    If True, include population growth in dynamics
    If False, assume n = 0

use_technology_growth : bool, default=False
    If True, allow A to grow over time
    If False, A is constant

Equilibria:

**Steady State (k*):**
The unique non-zero equilibrium where capital per worker is constant:

    k* = (s·A/δ)^(1/(1-α))

At steady state:
- Capital accumulation exactly balances depreciation
- Investment = Depreciation: s·y* = δ·k*
- No growth in per-capita variables (k, y, c constant)
- But total output grows at rate n (population growth)

**Stability:**
The steady state is GLOBALLY ASYMPTOTICALLY STABLE:
- For ANY initial k[0] > 0, system converges to k*
- Convergence is monotonic (no oscillations)
- Speed: |λ| = (1 - δ) + s·α·A·(k*)^(α-1) < 1

**Output at Steady State:**
    y* = A·(k*)^α = A^(1/(1-α))·(s/δ)^(α/(1-α))

**Consumption at Steady State:**
    c* = (1 - s)·y*

**Golden Rule Capital Level:**
The capital level that maximizes steady-state consumption:
    k_gold = (α·A/δ)^(1/(1-α))

Golden rule savings rate:
    s_gold = α

If s = α, the economy is at the golden rule (max consumption).
If s > α, the economy is "over-saving" (dynamic inefficiency).
If s < α, the economy could save more to increase consumption.

Convergence Dynamics:

**Conditional Convergence:**
Countries with same parameters (s, δ, α, A) converge to same k*.
- Poor countries (low k) grow faster initially
- Rich countries (high k) grow slower
- Convergence rate: β = (1 - α)(δ + n)
- Half-life: t_half = ln(2)/β

**Absolute Convergence:**
Empirically NOT observed - poor countries don't always catch up.
Reasons:
- Different savings rates s
- Different technology levels A
- Different institutional quality
- Different human capital

**Speed of Convergence:**
Empirical estimate: β ≈ 0.02 (2% per year)
- Half-life: ~35 years
- Slow convergence due to diminishing returns

Economic Insights:

**The Fundamental Question:**
Why are some countries rich and others poor?

Solow model answers:
1. **Capital accumulation:** Rich countries have more k
2. **Savings rates:** Higher s → higher k*
3. **Technology:** Higher A → higher productivity
4. **Population:** Lower n → easier to maintain high k

**The Growth Puzzle:**
Solow model predicts:
- Long-run growth ONLY from technological progress (A growing)
- Capital accumulation alone cannot sustain growth (diminishing returns)
- Saving more raises level but not long-run growth rate

**Policy Implications:**
1. **Increase savings rate s:**
   - Raises steady-state income
   - Temporary growth boost during transition
   - Reduces consumption in short run

2. **Reduce depreciation δ:**
   - Better infrastructure maintenance
   - Raises k* and y*

3. **Improve technology A:**
   - Education, R&D, technology adoption
   - Permanent growth effects
   - Most important for long run

4. **Control population n:**
   - Lower n → higher k* per capita
   - But total output growth slower

Control Objectives (Policy):

**1. Maximize Steady-State Consumption:**
   Goal: Choose s to maximize c* = (1 - s)·y*
   Solution: Golden rule s* = α

**2. Convergence Acceleration:**
   Goal: Reach k* faster via temporary policy
   Method: Increase s temporarily, then reduce
   Challenge: Political economy (delayed gratification)

**3. Poverty Trap Escape:**
   Goal: Overcome low-level equilibrium trap
   Method: Large temporary investment (foreign aid, stimulus)
   Extension: Add threshold effects, multiple equilibria

**4. Optimal Savings Path:**
   Goal: Maximize discounted utility ∫e^(-ρt)·u(c[t])dt
   Solution: Ramsey-Cass-Koopmans model (extends Solow)

**5. Technological Catch-Up:**
   Goal: Model technology diffusion from frontier
   Extension: A[t+1] = A[t] + φ(A_frontier - A[t])

State Constraints:

**1. Non-negativity: k[t] ≥ 0**
   - Capital cannot be negative
   - Economic necessity
   - k = 0 is absorbing state (poverty trap if no external help)

**2. Feasibility: k[t] ≤ k_max**
   - Some maximum possible capital
   - Typically not binding (grows without bound theoretically)

**3. Savings rate bounds: 0 < s < 1**
   - Cannot save more than 100% of output
   - Cannot save negative amount
   - Policy variable, not state constraint

**4. Non-negative output: y[t] ≥ 0**
   - Automatic if k ≥ 0 and A > 0

Numerical Considerations:

**Stability:**
The discrete Solow model is always stable (under standard parameters):
- Eigenvalue at k*: λ = (1 - δ) + s·α·A·(k*)^(α-1)
- For typical parameters: 0 < λ < 1
- Monotonic convergence (no oscillations)

**Accuracy:**
Discrete vs continuous models differ for large dt:
- Annual data (dt = 1): Discrete appropriate
- Continuous time limit as dt → 0

**Numerical Precision:**
No special considerations - well-conditioned problem.

Example Usage:

>>> # Create Solow model with US-like parameters
>>> model = DiscreteSolowModel(
...     s=0.25,      # 25% savings rate
...     delta=0.05,  # 5% annual depreciation
...     alpha=0.33,  # 1/3 capital share
...     A=1.0,       # Normalized technology
...     n=0.01,      # 1% population growth
...     dt=1.0,      # Annual periods
...     include_population_growth=True
... )
>>>
>>> # Compute steady state
>>> k_star = model.compute_steady_state()
>>> y_star = model.compute_output(k_star)
>>> c_star = model.compute_consumption(k_star)
>>>
>>> print(f"Steady-state capital: k* = {k_star:.2f}")
>>> print(f"Steady-state output: y* = {y_star:.2f}")
>>> print(f"Steady-state consumption: c* = {c_star:.2f}")
>>>
>>> # Check golden rule
>>> k_gold = model.compute_golden_rule_capital()
>>> s_gold = model.alpha
>>> print(f"

Golden rule capital: k_gold = {k_gold:.2f}“) >>> print(f”Golden rule savings rate: s_gold = {s_gold:.2f}“) >>> print(f”Current vs optimal: {‘Over-saving’ if model.s > s_gold else ‘Under-saving’}“) >>> >>> # Simulate convergence from low initial capital (poor country) >>> k0_poor = np.array([10.0]) # Low starting capital >>> result_poor = model.simulate( … x0=k0_poor, … u_sequence=None, … n_steps=100 # 100 years … ) >>> >>> # Simulate from high initial capital (rich country) >>> k0_rich = np.array([100.0]) # High starting capital >>> result_rich = model.simulate( … x0=k0_rich, … u_sequence=None, … n_steps=100 … ) >>> >>> # Plot convergence >>> import plotly.graph_objects as go >>> from plotly.subplots import make_subplots >>> >>> fig = make_subplots( … rows=2, cols=2, … subplot_titles=[‘Capital per Worker’, ‘Output per Worker’, … ‘Growth Rate’, ‘Consumption per Worker’] … ) >>> >>> t = result_poor[‘time_steps’] >>> >>> # Capital >>> fig.add_trace(go.Scatter(x=t, y=result_poor[‘states’][:, 0], … name=‘Poor’, line=dict(color=‘blue’)), row=1, col=1) >>> fig.add_trace(go.Scatter(x=t, y=result_rich[‘states’][:, 0], … name=‘Rich’, line=dict(color=‘red’)), row=1, col=1) >>> fig.add_hline(y=k_star, line_dash=‘dash’, line_color=‘green’, row=1, col=1) >>> >>> # Output >>> y_poor = model.compute_output(result_poor[‘states’][:, 0]) >>> y_rich = model.compute_output(result_rich[‘states’][:, 0]) >>> fig.add_trace(go.Scatter(x=t, y=y_poor, showlegend=False, … line=dict(color=‘blue’)), row=1, col=2) >>> fig.add_trace(go.Scatter(x=t, y=y_rich, showlegend=False, … line=dict(color=‘red’)), row=1, col=2) >>> fig.add_hline(y=y_star, line_dash=‘dash’, line_color=‘green’, row=1, col=2) >>> >>> # Growth rate >>> growth_poor = np.diff(result_poor[‘states’][:, 0]) / result_poor[‘states’][:-1, 0] >>> growth_rich = np.diff(result_rich[‘states’][:, 0]) / result_rich[‘states’][:-1, 0] >>> fig.add_trace(go.Scatter(x=t[:-1], y=growth_poor100, showlegend=False, … line=dict(color=‘blue’)), row=2, col=1) >>> fig.add_trace(go.Scatter(x=t[:-1], y=growth_rich100, showlegend=False, … line=dict(color=‘red’)), row=2, col=1) >>> >>> # Consumption >>> c_poor = model.compute_consumption(result_poor[‘states’][:, 0]) >>> c_rich = model.compute_consumption(result_rich[‘states’][:, 0]) >>> fig.add_trace(go.Scatter(x=t, y=c_poor, showlegend=False, … line=dict(color=‘blue’)), row=2, col=2) >>> fig.add_trace(go.Scatter(x=t, y=c_rich, showlegend=False, … line=dict(color=‘red’)), row=2, col=2) >>> >>> fig.update_xaxes(title_text=‘Time [years]’) >>> fig.update_yaxes(title_text=‘k’, row=1, col=1) >>> fig.update_yaxes(title_text=‘y’, row=1, col=2) >>> fig.update_yaxes(title_text=‘Growth [%]’, row=2, col=1) >>> fig.update_yaxes(title_text=‘c’, row=2, col=2) >>> fig.update_layout(height=800, width=1000, title_text=‘Solow Model: Convergence Dynamics’) >>> fig.show() >>> >>> # Phase diagram (Solow diagram) >>> fig_phase = model.plot_solow_diagram() >>> fig_phase.show() >>> >>> # Sensitivity analysis: Vary savings rate >>> s_values = np.linspace(0.1, 0.5, 20) >>> k_stars = [] >>> y_stars = [] >>> c_stars = [] >>> >>> for s_test in s_values: … model_temp = DiscreteSolowModel( … s=s_test, delta=0.05, alpha=0.33, A=1.0 … ) … k_ss = model_temp.compute_steady_state() … y_ss = model_temp.compute_output(k_ss) … c_ss = model_temp.compute_consumption(k_ss) … k_stars.append(k_ss) … y_stars.append(y_ss) … c_stars.append(c_ss) >>> >>> fig_sens = go.Figure() >>> fig_sens.add_trace(go.Scatter(x=s_values, y=k_stars, name=‘k’)) >>> fig_sens.add_trace(go.Scatter(x=s_values, y=y_stars, name=’y’)) >>> fig_sens.add_trace(go.Scatter(x=s_values, y=c_stars, name=’c*‘)) >>> fig_sens.add_vline(x=model.alpha, line_dash=’dash’, … annotation_text=‘Golden Rule’) >>> fig_sens.update_layout( … title=‘Steady State vs Savings Rate’, … xaxis_title=‘Savings Rate s’, … yaxis_title=‘Steady-State Value’ … ) >>> fig_sens.show() >>> >>> # Simulate policy experiment: Increase savings temporarily >>> k0 = np.array([20.0]) # Below steady state >>> >>> # Baseline: constant s = 0.25 >>> result_baseline = model.simulate(x0=k0, u_sequence=None, n_steps=50) >>> >>> # Policy: Increase s to 0.35 for 10 years, then back to 0.25 >>> def policy_intervention(x, k_time): … if 10 <= k_time < 20: … # Increase savings rate temporarily … model_temp = DiscreteSolowModel( … s=0.35, delta=model.delta, alpha=model.alpha, A=model.A … ) … # Compute additional investment … extra_savings = 0.35 - 0.25 … y_current = model.compute_output(x[0]) … # This requires modifying the system… … # For now, return zero (standard model doesn’t have control) … return np.array([]) … return np.array([]) >>> >>> # For policy experiments, better to simulate with modified parameters >>> # Create model with higher savings for intervention period >>> model_high_s = DiscreteSolowModel(s=0.35, delta=0.05, alpha=0.33, A=1.0) >>> >>> # Simulate in segments >>> # Years 0-10: Normal savings >>> result_seg1 = model.simulate(x0=k0, u_sequence=None, n_steps=10) >>> # Years 10-20: High savings >>> k_10 = result_seg1[‘states’][-1, :] >>> result_seg2 = model_high_s.simulate(x0=k_10, u_sequence=None, n_steps=10) >>> # Years 20-50: Back to normal >>> k_20 = result_seg2[‘states’][-1, :] >>> result_seg3 = model.simulate(x0=k_20, u_sequence=None, n_steps=30) >>> >>> # Combine results >>> k_policy = np.concatenate([ … result_seg1[‘states’][:, 0], … result_seg2[‘states’][1:, 0], … result_seg3[‘states’][1:, 0] … ]) >>> >>> fig_policy = go.Figure() >>> fig_policy.add_trace(go.Scatter( … x=np.arange(51), … y=result_baseline[‘states’][:, 0], … name=‘Baseline (s=0.25)’, … line=dict(color=‘blue’) … )) >>> fig_policy.add_trace(go.Scatter( … x=np.arange(51), … y=k_policy, … name=‘Policy (s=0.35 for years 10-20)’, … line=dict(color=‘red’) … )) >>> fig_policy.add_vrect(x0=10, x1=20, fillcolor=‘gray’, opacity=0.2, … annotation_text=‘High Savings’) >>> fig_policy.update_layout( … title=‘Policy Experiment: Temporary Savings Increase’, … xaxis_title=‘Time [years]’, … yaxis_title=‘Capital per Worker k’ … ) >>> fig_policy.show()

Economic Paradoxes and Puzzles:

**1. Solow Paradox (Computer Age):**
"You can see the computer age everywhere but in the productivity statistics"
- Massive IT investment in 1980s-90s
- But productivity growth didn't accelerate much
- Possible explanations: Mismeasurement, adjustment lags, diminishing returns

**2. Productivity Puzzle:**
Why does α ≈ 0.33 (Kaldor fact)?
- Empirically robust across countries and time
- Suggests deep structural feature of production
- Labor share stable despite technological change

**3. Growth Miracle Puzzle:**
Some countries grow much faster than Solow predicts:
- East Asian tigers: 7-10% growth for decades
- China: 10% growth for 30+ years
- Solow: Convergence only, not sustained super-growth
- Answer: Rapid A growth, human capital, institutions

**4. Poverty Trap:**
Basic Solow predicts convergence, but some countries stay poor:
- Extensions: Threshold effects, multiple equilibria
- S-shaped production function
- Institutional quality matters

Common Pitfalls:

1. **Confusing levels with growth rates:**
   Higher s → higher k* (level effect)
   But NOT higher long-run growth rate
   Growth sustained only by technological progress

2. **Ignoring diminishing returns:**
   Cannot grow forever by capital accumulation alone
   α < 1 ensures diminishing returns

3. **Misinterpreting steady state:**
   k* constant doesn't mean economy stopped growing
   Total output Y = y·L grows at rate n (population)
   Per-capita y constant, but total Y growing

4. **Assuming instant convergence:**
   Convergence takes decades (half-life ~35 years)
   Short-run dynamics matter for policy

5. **Forgetting population dilution:**
   With n > 0, investment must first offset dilution
   Effective depreciation: δ + n

6. **Using wrong α:**
   α should be capital share, empirically ~0.33
   α = 0.5 gives equal capital/labor importance (unrealistic)

Extensions:

1. **Human capital:**
   k → (k_physical, k_human)
   Education and skills accumulation

2. **Technology diffusion:**
   A[t+1] = A[t] + g·A[t] + φ(A_frontier - A[t])
   Catch-up dynamics

3. **Multiple sectors:**
   Agriculture, manufacturing, services
   Structural transformation

4. **Open economy:**
   Include international trade, capital flows
   Foreign investment, remittances

5. **Endogenous growth:**
   Make technology growth endogenous
   R&D, learning-by-doing, spillovers

6. **Resource constraints:**
   Add natural resources (oil, land)
   Environmental limits to growth

Methods

Name	Description
compute_consumption	Compute consumption per worker c = (1-s)·y.
compute_convergence_speed	Compute speed of convergence β.
compute_golden_rule_capital	Compute golden rule capital level k_gold.
compute_investment	Compute investment per worker i = s·y.
compute_output	Compute output per worker y = A·k^α.
compute_steady_state	Compute steady-state capital per worker k*.
define_system	Define discrete-time Solow growth model.
plot_solow_diagram	Plot classic Solow diagram (phase diagram).
setup_equilibria	Set up steady-state equilibrium.

compute_consumption

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_consumption(k)

Compute consumption per worker c = (1-s)·y.

Parameters

Name	Type	Description	Default
k	np.ndarray	Capital per worker	required

Returns

Name	Type	Description
	np.ndarray	Consumption per worker

compute_convergence_speed

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_convergence_speed(
)

Compute speed of convergence β.

Returns

Name	Type	Description
	float	Convergence coefficient β

Notes

Linearization around k* gives: k[t+1] - k* ≈ λ·(k[t] - k*)

where λ = 1 - β and: β = (1 - α)(δ + n)

Half-life: t_half = ln(2)/β

Examples

>>> model = DiscreteSolowModel()
>>> beta = model.compute_convergence_speed()
>>> print(f"Convergence rate: {beta:.4f}")
>>> print(f"Half-life: {np.log(2)/beta:.1f} years")

compute_golden_rule_capital

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_golden_rule_capital(
)

Compute golden rule capital level k_gold.

The golden rule maximizes steady-state consumption.

Returns

Name	Type	Description
	float	Golden rule capital

Notes

At golden rule: MPK = δ + n α·A·k_gold^(α-1) = δ + n k_gold = (α·A/(δ + n))^(1/(1-α))

Examples

>>> model = DiscreteSolowModel()
>>> k_gold = model.compute_golden_rule_capital()
>>> k_star = model.compute_steady_state()
>>> print(f"Current k*: {k_star:.2f}")
>>> print(f"Golden rule k_gold: {k_gold:.2f}")

compute_investment

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_investment(k)

Compute investment per worker i = s·y.

Parameters

Name	Type	Description	Default
k	np.ndarray	Capital per worker	required

Returns

Name	Type	Description
	np.ndarray	Investment per worker

compute_output

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_output(k)

Compute output per worker y = A·k^α.

Parameters

Name	Type	Description	Default
k	np.ndarray	Capital per worker	required

Returns

Name	Type	Description
	np.ndarray	Output per worker

compute_steady_state

systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_steady_state()

Compute steady-state capital per worker k*.

Returns

Name	Type	Description
	float	Steady-state capital k*

Notes

At steady state: k[t+1] = k[t] = k (1 - δ)·k + s·A·(k)^α = k·(1 + n) s·A·(k)^α = (δ + n)·k k* = (s·A/(δ + n))^(1/(1-α))

Examples

>>> model = DiscreteSolowModel(s=0.25, delta=0.05, alpha=0.33)
>>> k_star = model.compute_steady_state()
>>> print(f"Steady-state capital: {k_star:.2f}")

define_system

systems.builtin.deterministic.discrete.DiscreteSolowModel.define_system(
    s=0.3,
    delta=0.05,
    alpha=0.33,
    A=1.0,
    n=0.0,
    dt=1.0,
    include_population_growth=False,
    use_technology_growth=False,
    g_A=0.0,
)

Define discrete-time Solow growth model.

Parameters

Name	Type	Description	Default
s	float	Savings rate (0 < s < 1)	`0.3`
delta	float	Depreciation rate (0 < δ < 1)	`0.05`
alpha	float	Capital share / output elasticity (0 < α < 1)	`0.33`
A	float	Technology level / TFP (A > 0)	`1.0`
n	float	Population growth rate	`0.0`
dt	float	Time period [years]	`1.0`
include_population_growth	bool	If True, include population growth in dynamics	`False`
use_technology_growth	bool	If True, allow A to grow at rate g_A	`False`
g_A	float	Technology growth rate (if use_technology_growth=True)	`0.0`

plot_solow_diagram

systems.builtin.deterministic.discrete.DiscreteSolowModel.plot_solow_diagram(
    k_range=None,
)

Plot classic Solow diagram (phase diagram).

Shows: - Investment curve: s·f(k) - Depreciation line: δ·k (or (δ+n)·k) - Steady state at intersection

Parameters

Name	Type	Description	Default
k_range	Optional[tuple]	Range of k values to plot	`None`

Returns

Name	Type	Description
	go.Figure	Solow diagram

Examples

>>> model = DiscreteSolowModel()
>>> fig = model.plot_solow_diagram()
>>> fig.show()

setup_equilibria

systems.builtin.deterministic.discrete.DiscreteSolowModel.setup_equilibria()

Set up steady-state equilibrium.

# systems.builtin.deterministic.discrete.DiscreteSolowModel { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel(*args, **kwargs) ``` Discrete-time Solow-Swan neoclassical growth model. ## Physical System (Economic System): {.doc-section .doc-section-----physical-system-economic-system} The Solow model describes how an economy's capital stock evolves over time through the interplay of: - **Investment:** Savings channeled into new capital - **Depreciation:** Capital wears out and becomes obsolete - **Population growth:** Dilutes per-capita capital - **Technological progress:** Increases productivity **The Central Equation (Continuous Time):** k̇ = s·f(k) - (δ + n)·k where: k: Capital per worker (capital intensity) s: Savings rate (fraction of output saved) f(k): Production function (output per worker) δ: Depreciation rate n: Population growth rate **Production Function (Cobb-Douglas):** The standard form is: y = f(k) = A·k^α where: y: Output per worker (GDP per capita) A: Total factor productivity (TFP), technology level k: Capital per worker α: Output elasticity of capital (0 < α < 1) **Key Properties:** 1. **Positive but diminishing marginal product:** f'(k) > 0, f''(k) < 0 - More capital → more output - But each additional unit contributes less (diminishing returns) 2. **Inada conditions:** f(0) = 0, f'(0) = ∞, f'(∞) = 0 - No capital → no output - Initially very productive - Eventually saturates 3. **Constant returns to scale (with labor):** F(K, L) = A·K^α·L^(1-α) - Doubling both K and L doubles output **Discrete-Time Dynamics:** The discrete version models year-to-year changes: k[t+1] = (1 - δ)·k[t] + s·f(k[t]) Or with population growth: k[t+1] = [(1 - δ)·k[t] + s·f(k[t])] / (1 + n) ## State Space: {.doc-section .doc-section-----state-space} State: x[t] = [k[t]] Capital per worker: - k: Capital intensity [$/worker] or [capital units per worker] * k ≥ 0: Non-negative by definition * k = 0: No capital (subsistence economy) * k → k*: Converges to steady state * Typical range: 0-500 for normalized units **Physical/Economic Meaning:** Capital per worker represents: - Machines, equipment, tools available per person - Infrastructure (roads, buildings) per capita - Technology embodied in physical capital - Higher k → higher productivity → higher income Control: u[t] (optional, for policy intervention) - Can represent: * Temporary savings rate change: s → s + u[t] * Government investment/stimulus * Technology adoption rate change * Foreign aid or capital injection - Standard model: u = 0 (no intervention) Output: y[t] = [k[t]] or [k[t], y[t], c[t]] Standard outputs: - k[t]: Capital per worker (state variable) - y[t]: Output per worker (GDP per capita) - c[t]: Consumption per worker Where: y[t] = A·k[t]^α c[t] = (1 - s)·y[t] ## Dynamics (Economic Interpretation): {.doc-section .doc-section-----dynamics-economic-interpretation} The discrete dynamics: k[t+1] = (1 - δ)·k[t] + s·A·k[t]^α **Capital Evolution Has Three Components:** 1. **Undepreciated capital: (1 - δ)·k[t]** - Fraction (1 - δ) survives to next period - δ represents wearing out, obsolescence - If δ = 0.05: 5% of capital lost each year 2. **New investment: s·f(k[t])** - Savings rate s fraction of output is invested - f(k) = A·k^α is output per worker - Higher s → more investment → faster growth 3. **Population dilution (if n > 0):** - Growing population dilutes per-capita capital - k[t+1] must be divided by (1 + n) - Higher n → slower per-capita growth **Steady State (Golden Rule):** At steady state: k[t+1] = k[t] = k* (1 - δ)·k* + s·A·(k*)^α = k* s·A·(k*)^α = δ·k* k* = (s·A/δ)^(1/(1-α)) The steady-state capital is unique and globally stable (under standard assumptions). **Convergence to Steady State:** - If k[0] < k*: Capital accumulates (growth) - If k[0] > k*: Capital decumulates (decline) - Convergence is monotonic (no oscillations for standard parameters) - Speed of convergence: Faster when far from k*, slower when near ## Parameters: {.doc-section .doc-section-----parameters} s : float, default=0.3 Savings rate (fraction of output saved/invested) - Must satisfy: 0 < s < 1 - Typical values: 0.15-0.40 - Developed countries: ~0.20-0.25 - Developing countries: ~0.30-0.40 - China (historically): ~0.45-0.50 **Economic Meaning:** - Higher s → more investment → higher steady-state k* - But also lower consumption (1-s)·y - Golden rule: s that maximizes steady-state consumption delta : float, default=0.05 Depreciation rate (fraction of capital lost per period) - Must satisfy: 0 < δ < 1 - Typical annual values: 0.03-0.10 - Physical capital: ~0.05-0.08 - Structures: ~0.03 - Equipment: ~0.10 **Economic Meaning:** - Higher δ → more replacement needed → lower k* - Represents obsolescence, wear and tear alpha : float, default=0.33 Capital share of income (output elasticity of capital) - Must satisfy: 0 < α < 1 - Typical values: 0.25-0.40 - Empirically: ~0.33 (one-third) - Labor share: 1 - α ≈ 0.67 (two-thirds) **Economic Meaning:** - Measures importance of capital vs labor - α = 0.5: Equal importance (Cobb-Douglas symmetric) - α < 0.5: Labor more important - α > 0.5: Capital more important (rare) A : float, default=1.0 Total Factor Productivity (TFP) / technology level - Must satisfy: A > 0 - Captures: Technology, institutions, efficiency - Grows over time via technological progress - Typical growth: 1-2% per year **Economic Meaning:** - Higher A → more output for same k - Represents "technological level" - Solow residual: Growth not explained by capital/labor n : float, default=0.0 Population/labor force growth rate - Can be positive, zero, or negative - Typical values: -0.01 to 0.03 (annually) - Developed countries: ~0.00-0.01 - Developing countries: ~0.01-0.03 **Economic Meaning:** - Higher n → dilutes per-capita capital - Must invest more just to maintain k - Effective depreciation: δ + n dt : float, default=1.0 Time period (typically 1 year for macroeconomic data) - Units: years - dt = 1: Annual model (most common) - dt = 0.25: Quarterly model include_population_growth : bool, default=False If True, include population growth in dynamics If False, assume n = 0 use_technology_growth : bool, default=False If True, allow A to grow over time If False, A is constant ## Equilibria: {.doc-section .doc-section-----equilibria} **Steady State (k*):** The unique non-zero equilibrium where capital per worker is constant: k* = (s·A/δ)^(1/(1-α)) At steady state: - Capital accumulation exactly balances depreciation - Investment = Depreciation: s·y* = δ·k* - No growth in per-capita variables (k, y, c constant) - But total output grows at rate n (population growth) **Stability:** The steady state is GLOBALLY ASYMPTOTICALLY STABLE: - For ANY initial k[0] > 0, system converges to k* - Convergence is monotonic (no oscillations) - Speed: |λ| = (1 - δ) + s·α·A·(k*)^(α-1) < 1 **Output at Steady State:** y* = A·(k*)^α = A^(1/(1-α))·(s/δ)^(α/(1-α)) **Consumption at Steady State:** c* = (1 - s)·y* **Golden Rule Capital Level:** The capital level that maximizes steady-state consumption: k_gold = (α·A/δ)^(1/(1-α)) Golden rule savings rate: s_gold = α If s = α, the economy is at the golden rule (max consumption). If s > α, the economy is "over-saving" (dynamic inefficiency). If s < α, the economy could save more to increase consumption. ## Convergence Dynamics: {.doc-section .doc-section-----convergence-dynamics} **Conditional Convergence:** Countries with same parameters (s, δ, α, A) converge to same k*. - Poor countries (low k) grow faster initially - Rich countries (high k) grow slower - Convergence rate: β = (1 - α)(δ + n) - Half-life: t_half = ln(2)/β **Absolute Convergence:** Empirically NOT observed - poor countries don't always catch up. Reasons: - Different savings rates s - Different technology levels A - Different institutional quality - Different human capital **Speed of Convergence:** Empirical estimate: β ≈ 0.02 (2% per year) - Half-life: ~35 years - Slow convergence due to diminishing returns ## Economic Insights: {.doc-section .doc-section-----economic-insights} **The Fundamental Question:** Why are some countries rich and others poor? Solow model answers: 1. **Capital accumulation:** Rich countries have more k 2. **Savings rates:** Higher s → higher k* 3. **Technology:** Higher A → higher productivity 4. **Population:** Lower n → easier to maintain high k **The Growth Puzzle:** Solow model predicts: - Long-run growth ONLY from technological progress (A growing) - Capital accumulation alone cannot sustain growth (diminishing returns) - Saving more raises level but not long-run growth rate **Policy Implications:** 1. **Increase savings rate s:** - Raises steady-state income - Temporary growth boost during transition - Reduces consumption in short run 2. **Reduce depreciation δ:** - Better infrastructure maintenance - Raises k* and y* 3. **Improve technology A:** - Education, R&D, technology adoption - Permanent growth effects - Most important for long run 4. **Control population n:** - Lower n → higher k* per capita - But total output growth slower ## Control Objectives (Policy): {.doc-section .doc-section-----control-objectives-policy} **1. Maximize Steady-State Consumption:** Goal: Choose s to maximize c* = (1 - s)·y* Solution: Golden rule s* = α **2. Convergence Acceleration:** Goal: Reach k* faster via temporary policy Method: Increase s temporarily, then reduce Challenge: Political economy (delayed gratification) **3. Poverty Trap Escape:** Goal: Overcome low-level equilibrium trap Method: Large temporary investment (foreign aid, stimulus) Extension: Add threshold effects, multiple equilibria **4. Optimal Savings Path:** Goal: Maximize discounted utility ∫e^(-ρt)·u(c[t])dt Solution: Ramsey-Cass-Koopmans model (extends Solow) **5. Technological Catch-Up:** Goal: Model technology diffusion from frontier Extension: A[t+1] = A[t] + φ(A_frontier - A[t]) ## State Constraints: {.doc-section .doc-section-----state-constraints} **1. Non-negativity: k[t] ≥ 0** - Capital cannot be negative - Economic necessity - k = 0 is absorbing state (poverty trap if no external help) **2. Feasibility: k[t] ≤ k_max** - Some maximum possible capital - Typically not binding (grows without bound theoretically) **3. Savings rate bounds: 0 < s < 1** - Cannot save more than 100% of output - Cannot save negative amount - Policy variable, not state constraint **4. Non-negative output: y[t] ≥ 0** - Automatic if k ≥ 0 and A > 0 ## Numerical Considerations: {.doc-section .doc-section-----numerical-considerations} **Stability:** The discrete Solow model is always stable (under standard parameters): - Eigenvalue at k*: λ = (1 - δ) + s·α·A·(k*)^(α-1) - For typical parameters: 0 < λ < 1 - Monotonic convergence (no oscillations) **Accuracy:** Discrete vs continuous models differ for large dt: - Annual data (dt = 1): Discrete appropriate - Continuous time limit as dt → 0 **Numerical Precision:** No special considerations - well-conditioned problem. ## Example Usage: {.doc-section .doc-section-----example-usage} >>> # Create Solow model with US-like parameters >>> model = DiscreteSolowModel( ... s=0.25, # 25% savings rate ... delta=0.05, # 5% annual depreciation ... alpha=0.33, # 1/3 capital share ... A=1.0, # Normalized technology ... n=0.01, # 1% population growth ... dt=1.0, # Annual periods ... include_population_growth=True ... ) >>> >>> # Compute steady state >>> k_star = model.compute_steady_state() >>> y_star = model.compute_output(k_star) >>> c_star = model.compute_consumption(k_star) >>> >>> print(f"Steady-state capital: k* = {k_star:.2f}") >>> print(f"Steady-state output: y* = {y_star:.2f}") >>> print(f"Steady-state consumption: c* = {c_star:.2f}") >>> >>> # Check golden rule >>> k_gold = model.compute_golden_rule_capital() >>> s_gold = model.alpha >>> print(f" Golden rule capital: k_gold = {k_gold:.2f}") >>> print(f"Golden rule savings rate: s_gold = {s_gold:.2f}") >>> print(f"Current vs optimal: {'Over-saving' if model.s > s_gold else 'Under-saving'}") >>> >>> # Simulate convergence from low initial capital (poor country) >>> k0_poor = np.array([10.0]) # Low starting capital >>> result_poor = model.simulate( ... x0=k0_poor, ... u_sequence=None, ... n_steps=100 # 100 years ... ) >>> >>> # Simulate from high initial capital (rich country) >>> k0_rich = np.array([100.0]) # High starting capital >>> result_rich = model.simulate( ... x0=k0_rich, ... u_sequence=None, ... n_steps=100 ... ) >>> >>> # Plot convergence >>> import plotly.graph_objects as go >>> from plotly.subplots import make_subplots >>> >>> fig = make_subplots( ... rows=2, cols=2, ... subplot_titles=['Capital per Worker', 'Output per Worker', ... 'Growth Rate', 'Consumption per Worker'] ... ) >>> >>> t = result_poor['time_steps'] >>> >>> # Capital >>> fig.add_trace(go.Scatter(x=t, y=result_poor['states'][:, 0], ... name='Poor', line=dict(color='blue')), row=1, col=1) >>> fig.add_trace(go.Scatter(x=t, y=result_rich['states'][:, 0], ... name='Rich', line=dict(color='red')), row=1, col=1) >>> fig.add_hline(y=k_star, line_dash='dash', line_color='green', row=1, col=1) >>> >>> # Output >>> y_poor = model.compute_output(result_poor['states'][:, 0]) >>> y_rich = model.compute_output(result_rich['states'][:, 0]) >>> fig.add_trace(go.Scatter(x=t, y=y_poor, showlegend=False, ... line=dict(color='blue')), row=1, col=2) >>> fig.add_trace(go.Scatter(x=t, y=y_rich, showlegend=False, ... line=dict(color='red')), row=1, col=2) >>> fig.add_hline(y=y_star, line_dash='dash', line_color='green', row=1, col=2) >>> >>> # Growth rate >>> growth_poor = np.diff(result_poor['states'][:, 0]) / result_poor['states'][:-1, 0] >>> growth_rich = np.diff(result_rich['states'][:, 0]) / result_rich['states'][:-1, 0] >>> fig.add_trace(go.Scatter(x=t[:-1], y=growth_poor*100, showlegend=False, ... line=dict(color='blue')), row=2, col=1) >>> fig.add_trace(go.Scatter(x=t[:-1], y=growth_rich*100, showlegend=False, ... line=dict(color='red')), row=2, col=1) >>> >>> # Consumption >>> c_poor = model.compute_consumption(result_poor['states'][:, 0]) >>> c_rich = model.compute_consumption(result_rich['states'][:, 0]) >>> fig.add_trace(go.Scatter(x=t, y=c_poor, showlegend=False, ... line=dict(color='blue')), row=2, col=2) >>> fig.add_trace(go.Scatter(x=t, y=c_rich, showlegend=False, ... line=dict(color='red')), row=2, col=2) >>> >>> fig.update_xaxes(title_text='Time [years]') >>> fig.update_yaxes(title_text='k', row=1, col=1) >>> fig.update_yaxes(title_text='y', row=1, col=2) >>> fig.update_yaxes(title_text='Growth [%]', row=2, col=1) >>> fig.update_yaxes(title_text='c', row=2, col=2) >>> fig.update_layout(height=800, width=1000, title_text='Solow Model: Convergence Dynamics') >>> fig.show() >>> >>> # Phase diagram (Solow diagram) >>> fig_phase = model.plot_solow_diagram() >>> fig_phase.show() >>> >>> # Sensitivity analysis: Vary savings rate >>> s_values = np.linspace(0.1, 0.5, 20) >>> k_stars = [] >>> y_stars = [] >>> c_stars = [] >>> >>> for s_test in s_values: ... model_temp = DiscreteSolowModel( ... s=s_test, delta=0.05, alpha=0.33, A=1.0 ... ) ... k_ss = model_temp.compute_steady_state() ... y_ss = model_temp.compute_output(k_ss) ... c_ss = model_temp.compute_consumption(k_ss) ... k_stars.append(k_ss) ... y_stars.append(y_ss) ... c_stars.append(c_ss) >>> >>> fig_sens = go.Figure() >>> fig_sens.add_trace(go.Scatter(x=s_values, y=k_stars, name='k*')) >>> fig_sens.add_trace(go.Scatter(x=s_values, y=y_stars, name='y*')) >>> fig_sens.add_trace(go.Scatter(x=s_values, y=c_stars, name='c*')) >>> fig_sens.add_vline(x=model.alpha, line_dash='dash', ... annotation_text='Golden Rule') >>> fig_sens.update_layout( ... title='Steady State vs Savings Rate', ... xaxis_title='Savings Rate s', ... yaxis_title='Steady-State Value' ... ) >>> fig_sens.show() >>> >>> # Simulate policy experiment: Increase savings temporarily >>> k0 = np.array([20.0]) # Below steady state >>> >>> # Baseline: constant s = 0.25 >>> result_baseline = model.simulate(x0=k0, u_sequence=None, n_steps=50) >>> >>> # Policy: Increase s to 0.35 for 10 years, then back to 0.25 >>> def policy_intervention(x, k_time): ... if 10 <= k_time < 20: ... # Increase savings rate temporarily ... model_temp = DiscreteSolowModel( ... s=0.35, delta=model.delta, alpha=model.alpha, A=model.A ... ) ... # Compute additional investment ... extra_savings = 0.35 - 0.25 ... y_current = model.compute_output(x[0]) ... # This requires modifying the system... ... # For now, return zero (standard model doesn't have control) ... return np.array([]) ... return np.array([]) >>> >>> # For policy experiments, better to simulate with modified parameters >>> # Create model with higher savings for intervention period >>> model_high_s = DiscreteSolowModel(s=0.35, delta=0.05, alpha=0.33, A=1.0) >>> >>> # Simulate in segments >>> # Years 0-10: Normal savings >>> result_seg1 = model.simulate(x0=k0, u_sequence=None, n_steps=10) >>> # Years 10-20: High savings >>> k_10 = result_seg1['states'][-1, :] >>> result_seg2 = model_high_s.simulate(x0=k_10, u_sequence=None, n_steps=10) >>> # Years 20-50: Back to normal >>> k_20 = result_seg2['states'][-1, :] >>> result_seg3 = model.simulate(x0=k_20, u_sequence=None, n_steps=30) >>> >>> # Combine results >>> k_policy = np.concatenate([ ... result_seg1['states'][:, 0], ... result_seg2['states'][1:, 0], ... result_seg3['states'][1:, 0] ... ]) >>> >>> fig_policy = go.Figure() >>> fig_policy.add_trace(go.Scatter( ... x=np.arange(51), ... y=result_baseline['states'][:, 0], ... name='Baseline (s=0.25)', ... line=dict(color='blue') ... )) >>> fig_policy.add_trace(go.Scatter( ... x=np.arange(51), ... y=k_policy, ... name='Policy (s=0.35 for years 10-20)', ... line=dict(color='red') ... )) >>> fig_policy.add_vrect(x0=10, x1=20, fillcolor='gray', opacity=0.2, ... annotation_text='High Savings') >>> fig_policy.update_layout( ... title='Policy Experiment: Temporary Savings Increase', ... xaxis_title='Time [years]', ... yaxis_title='Capital per Worker k' ... ) >>> fig_policy.show() ## Economic Paradoxes and Puzzles: {.doc-section .doc-section-----economic-paradoxes-and-puzzles} **1. Solow Paradox (Computer Age):** "You can see the computer age everywhere but in the productivity statistics" - Massive IT investment in 1980s-90s - But productivity growth didn't accelerate much - Possible explanations: Mismeasurement, adjustment lags, diminishing returns **2. Productivity Puzzle:** Why does α ≈ 0.33 (Kaldor fact)? - Empirically robust across countries and time - Suggests deep structural feature of production - Labor share stable despite technological change **3. Growth Miracle Puzzle:** Some countries grow much faster than Solow predicts: - East Asian tigers: 7-10% growth for decades - China: 10% growth for 30+ years - Solow: Convergence only, not sustained super-growth - Answer: Rapid A growth, human capital, institutions **4. Poverty Trap:** Basic Solow predicts convergence, but some countries stay poor: - Extensions: Threshold effects, multiple equilibria - S-shaped production function - Institutional quality matters ## Common Pitfalls: {.doc-section .doc-section-----common-pitfalls} 1. **Confusing levels with growth rates:** Higher s → higher k* (level effect) But NOT higher long-run growth rate Growth sustained only by technological progress 2. **Ignoring diminishing returns:** Cannot grow forever by capital accumulation alone α < 1 ensures diminishing returns 3. **Misinterpreting steady state:** k* constant doesn't mean economy stopped growing Total output Y = y·L grows at rate n (population) Per-capita y constant, but total Y growing 4. **Assuming instant convergence:** Convergence takes decades (half-life ~35 years) Short-run dynamics matter for policy 5. **Forgetting population dilution:** With n > 0, investment must first offset dilution Effective depreciation: δ + n 6. **Using wrong α:** α should be capital share, empirically ~0.33 α = 0.5 gives equal capital/labor importance (unrealistic) ## Extensions: {.doc-section .doc-section-----extensions} 1. **Human capital:** k → (k_physical, k_human) Education and skills accumulation 2. **Technology diffusion:** A[t+1] = A[t] + g·A[t] + φ(A_frontier - A[t]) Catch-up dynamics 3. **Multiple sectors:** Agriculture, manufacturing, services Structural transformation 4. **Open economy:** Include international trade, capital flows Foreign investment, remittances 5. **Endogenous growth:** Make technology growth endogenous R&D, learning-by-doing, spillovers 6. **Resource constraints:** Add natural resources (oil, land) Environmental limits to growth ## See Also: {.doc-section .doc-section-----see-also} LogisticMap : Similar mathematical structure (but very different interpretation) ## Methods | Name | Description | | --- | --- | | [compute_consumption](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_consumption) | Compute consumption per worker c = (1-s)·y. | | [compute_convergence_speed](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_convergence_speed) | Compute speed of convergence β. | | [compute_golden_rule_capital](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_golden_rule_capital) | Compute golden rule capital level k_gold. | | [compute_investment](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_investment) | Compute investment per worker i = s·y. | | [compute_output](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_output) | Compute output per worker y = A·k^α. | | [compute_steady_state](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_steady_state) | Compute steady-state capital per worker k*. | | [define_system](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.define_system) | Define discrete-time Solow growth model. | | [plot_solow_diagram](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.plot_solow_diagram) | Plot classic Solow diagram (phase diagram). | | [setup_equilibria](#cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.setup_equilibria) | Set up steady-state equilibrium. | ### compute_consumption { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_consumption } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_consumption(k) ``` Compute consumption per worker c = (1-s)·y. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|------------|--------------------|------------| | k | np.ndarray | Capital per worker | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|------------------------| | | np.ndarray | Consumption per worker | ### compute_convergence_speed { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_convergence_speed } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_convergence_speed( ) ``` Compute speed of convergence β. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------------| | | float | Convergence coefficient β | #### Notes {.doc-section .doc-section-notes} Linearization around k* gives: k[t+1] - k* ≈ λ·(k[t] - k*) where λ = 1 - β and: β = (1 - α)(δ + n) Half-life: t_half = ln(2)/β #### Examples {.doc-section .doc-section-examples} ```python >>> model = DiscreteSolowModel() >>> beta = model.compute_convergence_speed() >>> print(f"Convergence rate: {beta:.4f}") >>> print(f"Half-life: {np.log(2)/beta:.1f} years") ``` ### compute_golden_rule_capital { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_golden_rule_capital } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_golden_rule_capital( ) ``` Compute golden rule capital level k_gold. The golden rule maximizes steady-state consumption. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|---------------------| | | float | Golden rule capital | #### Notes {.doc-section .doc-section-notes} At golden rule: MPK = δ + n α·A·k_gold^(α-1) = δ + n k_gold = (α·A/(δ + n))^(1/(1-α)) #### Examples {.doc-section .doc-section-examples} ```python >>> model = DiscreteSolowModel() >>> k_gold = model.compute_golden_rule_capital() >>> k_star = model.compute_steady_state() >>> print(f"Current k*: {k_star:.2f}") >>> print(f"Golden rule k_gold: {k_gold:.2f}") ``` ### compute_investment { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_investment } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_investment(k) ``` Compute investment per worker i = s·y. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|------------|--------------------|------------| | k | np.ndarray | Capital per worker | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-----------------------| | | np.ndarray | Investment per worker | ### compute_output { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_output } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_output(k) ``` Compute output per worker y = A·k^α. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |--------|------------|--------------------|------------| | k | np.ndarray | Capital per worker | _required_ | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|------------|-------------------| | | np.ndarray | Output per worker | ### compute_steady_state { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_steady_state } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.compute_steady_state() ``` Compute steady-state capital per worker k*. #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|--------|-------------------------| | | float | Steady-state capital k* | #### Notes {.doc-section .doc-section-notes} At steady state: k[t+1] = k[t] = k* (1 - δ)·k* + s·A·(k*)^α = k*·(1 + n) s·A·(k*)^α = (δ + n)·k* k* = (s·A/(δ + n))^(1/(1-α)) #### Examples {.doc-section .doc-section-examples} ```python >>> model = DiscreteSolowModel(s=0.25, delta=0.05, alpha=0.33) >>> k_star = model.compute_steady_state() >>> print(f"Steady-state capital: {k_star:.2f}") ``` ### define_system { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.define_system } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.define_system( s=0.3, delta=0.05, alpha=0.33, A=1.0, n=0.0, dt=1.0, include_population_growth=False, use_technology_growth=False, g_A=0.0, ) ``` Define discrete-time Solow growth model. #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------------------------|--------|--------------------------------------------------------|-----------| | s | float | Savings rate (0 < s < 1) | `0.3` | | delta | float | Depreciation rate (0 < δ < 1) | `0.05` | | alpha | float | Capital share / output elasticity (0 < α < 1) | `0.33` | | A | float | Technology level / TFP (A > 0) | `1.0` | | n | float | Population growth rate | `0.0` | | dt | float | Time period [years] | `1.0` | | include_population_growth | bool | If True, include population growth in dynamics | `False` | | use_technology_growth | bool | If True, allow A to grow at rate g_A | `False` | | g_A | float | Technology growth rate (if use_technology_growth=True) | `0.0` | ### plot_solow_diagram { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.plot_solow_diagram } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.plot_solow_diagram( k_range=None, ) ``` Plot classic Solow diagram (phase diagram). Shows: - Investment curve: s·f(k) - Depreciation line: δ·k (or (δ+n)·k) - Steady state at intersection #### Parameters {.doc-section .doc-section-parameters} | Name | Type | Description | Default | |---------|-------------------|---------------------------|-----------| | k_range | Optional\[tuple\] | Range of k values to plot | `None` | #### Returns {.doc-section .doc-section-returns} | Name | Type | Description | |--------|-----------|---------------| | | go.Figure | Solow diagram | #### Examples {.doc-section .doc-section-examples} ```python >>> model = DiscreteSolowModel() >>> fig = model.plot_solow_diagram() >>> fig.show() ``` ### setup_equilibria { #cdesym.systems.builtin.deterministic.discrete.DiscreteSolowModel.setup_equilibria } ```python systems.builtin.deterministic.discrete.DiscreteSolowModel.setup_equilibria() ``` Set up steady-state equilibrium.