From 354e78438fd8beb9c5d5fe69662088c6b84dd315 Mon Sep 17 00:00:00 2001 From: Adrian Torchiana Date: Fri, 3 Dec 2021 09:53:11 -0800 Subject: [PATCH 1/2] WIP Lifecycle consumption + savings model (extended cake eating problem) --- .gitignore | 2 + python/consumption_savings.py | 378 ++++++++++++++++++++++++++++++++++ 2 files changed, 380 insertions(+) create mode 100644 python/consumption_savings.py diff --git a/.gitignore b/.gitignore index ac7cea3..957ed67 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,4 @@ tags *~ +*png +*csv diff --git a/python/consumption_savings.py b/python/consumption_savings.py new file mode 100644 index 0000000..afebf8c --- /dev/null +++ b/python/consumption_savings.py @@ -0,0 +1,378 @@ +import os + +import matplotlib.pyplot as plt +import numpy as np +import pandas as pd +from scipy.interpolate import RegularGridInterpolator + + +# State variable: (wealth, income, age) +# Action variable: how much to consume (implicitly, how much to save) +# No durables like houses, cars, etc +# Savings earn a constant interest rate, no randomness (no equities for simplicity) +# Reward: utility(consumption), pick something simple like CRRA +# Income is stochastic: you might randomly get a raise (or get fired) + + +# Very simple lifecycle model: the agent lives from age MIN_AGE to MAX_AGE and then dies +MIN_AGE = 30 +RETIREMENT_AGE = 50 +MAX_AGE = 60 +AGE_GRID = np.arange(MIN_AGE, MAX_AGE + 1) + +DISCOUNT_FACTOR = 0.95 + +# The agent puts their savings in a bank and earns yearly interest +# TODO Make this random? But would have to be careful calculating E[log(wealth)] in continuation values +INTEREST_RATE = 0.01 + +# At age MIN_AGE, the agent earns an income of INITIAL_INCOME +# Every year, they receive a raise with probability RAISE_PROBABILITY +# The agent retires (stops earning income) when they reach RETIREMENT_AGE +INITIAL_INCOME = 1 +RAISE_PROBABILITY = 0.5 + +# If the agent gets a raise, it'll be randomly distributed +MIN_RAISE, MAX_RAISE = (0.0, 0.05) + +# The agent begins their career at MIN_AGE with a little gift from their parents +INITIAL_WEALTH = INITIAL_INCOME + +CONSUMPTION_FRACTION_GRID = np.arange(0.01, 1.0, 0.01) + +INCOME_GRID = np.hstack([np.zeros((1, )), INITIAL_INCOME * (1 + MAX_RAISE) ** np.arange(RETIREMENT_AGE - MIN_AGE + 1)]) + +# TODO Calculate max possible wealth, choose a better grid +WEALTH_GRID = np.arange(0.001 * INITIAL_INCOME, 0.3 * (RETIREMENT_AGE - MIN_AGE) * np.max(INCOME_GRID), 0.1 * INITIAL_INCOME) + +# Ways of extending this problem and making it more interesting: +# Investment: allow the agent to choose between putting savings in a bank (safe, low interest rate) and in equities (higher expected return, but some volatility) +# See how optimal investment changes with age -- expect agent to put more of their savings in equities when young +# Home ownership +# Non-deterministic lifespan :-) +# Probability of unemployment + +PLOT_DIR = "plots" + +def reward_function(consumption): + + # The agent's action is how much to consume in the current period + # They enjoy logarithmic utility from consumption flows + return np.log(consumption) + + +def is_terminal_state(wealth, income, age): + + # The problem ends (the agent dies) when they reach the max age + return age >= MAX_AGE + + +def simulate_next_state(wealth, income, age, consumption): + + next_age = age + 1 + next_wealth = (wealth + income - consumption) * (1 + INTEREST_RATE) + + if next_age >= RETIREMENT_AGE: + next_income = 0 + return next_wealth, next_income, next_age + + next_income = income + + agent_receives_raise = np.random.uniform() < RAISE_PROBABILITY + if agent_receives_raise: + rate = np.random.uniform(low=MIN_RAISE, high=MAX_RAISE) + next_income = income * (1 + rate) + + return next_wealth, next_income, next_age + + +def get_feature_vector(wealth, income, age): + + # Sutton and Barto 9.4: "the vector x(s) is called a feature vector representing state s" + + # TODO Could include interactions, higher order terms, etc + # TODO Partial dependence plots for approximate value fn + age_scaled = age / 100 + return np.array([ + 1.0, + np.log(wealth), + income, + age_scaled, + np.log(wealth) * income, + np.log(wealth) * age_scaled, + income * age_scaled, + ]) + + +def feature_vector_size(): + + return get_feature_vector(INITIAL_WEALTH, INITIAL_INCOME, MIN_AGE).size + + +def discounted_value_starting_from_idx(rewards, idx): + + return np.sum(np.array(rewards[idx:]) * (DISCOUNT_FACTOR ** np.arange(len(rewards[idx:])))) + + +def calculate_approx_value_function(policy, n_episodes=500, learning_rate=0.000_001): + + # Return approximate value function (or weights for approx value function?) for agent following this policy + # Follows Sutton and Barto 9.3 Gradient Monte Carlo for Estimating v_pi + + policy_interpolator = RegularGridInterpolator(points=(WEALTH_GRID, INCOME_GRID, AGE_GRID), values=policy, method="linear", bounds_error=False, fill_value=None) + + # The shape of the coefficient vector needs to be consistent with get_feature_vector + value_fn_coefficients = np.ones((feature_vector_size(), )) + + dfs = [] + + for episode_idx in range(n_episodes): + + reached_terminal_state = False + next_wealth, next_income, next_age = (INITIAL_WEALTH, INITIAL_INCOME, MIN_AGE) + + rewards, predicted_values, feature_vectors = ([], [], []) + consumption_fractions = [] + + while not reached_terminal_state: + + wealth, income, age = (next_wealth, next_income, next_age) + + # This tells us whether the _current_ state is terminal (as opposed to next period's state) + reached_terminal_state = is_terminal_state(wealth, income, age) + + feature_vector = get_feature_vector(wealth, income, age) + feature_vectors.append(feature_vector) + + predicted_value = np.dot(feature_vector, value_fn_coefficients) + predicted_values.append(predicted_value) + + consumption_fraction = policy_interpolator(np.array([wealth, income, age]))[0] + + # Convention: the policy tells us the _fraction_ of wealth + income that the agent will consume in the current period + consumption = consumption_fraction * (wealth + income) + consumption_fractions.append(consumption_fraction) + + reward = reward_function(consumption) + rewards.append(reward) + + discounted_predicted_continuation_value = 0 + if not reached_terminal_state: + next_wealth, next_income, next_age = simulate_next_state(wealth, income, age, consumption) + + actual_values = [] + + for idx, predicted_value in enumerate(predicted_values): + feature_vector = feature_vectors[idx] + actual_value = discounted_value_starting_from_idx(rewards, idx) + actual_values.append(actual_value) + value_fn_coefficients = value_fn_coefficients + learning_rate * (actual_value - predicted_value) * feature_vector + + # Messy, these colnames need to be consistent with get_feature_vector + columns = [ + "constant", + "log_wealth", + "income", + "age_scaled", + "log_wealth_income", + "log_wealth_age", + "income_age", + ] + df = pd.DataFrame(feature_vectors, columns=columns) + df["age"] = df["age_scaled"] * 100 + df["wealth"] = np.exp(df["log_wealth"]) + df["actual_value"] = actual_values + df["reward"] = rewards + df["consumption"] = consumption_fractions * (df["wealth"] + df["income"]) + df["consumption_fractions"] = consumption_fractions + df["episode_idx"] = episode_idx + dfs.append(df) + # print(f"Done running episode {episode_idx}, coefficients {np.round(value_fn_coefficients, 2)}") + + df = pd.concat(dfs) + + return value_fn_coefficients, df + + +def update_policy(value_fn_coefficients, old_policy): + + # Policy is input is (wealth, income, age) + # TODO Could try solving with first order condition instead of brute force, compare results + + policy = np.zeros_like(old_policy) + + for wealth_idx, wealth in enumerate(WEALTH_GRID): + print(f"Updating policy, wealth idx {wealth_idx} of {WEALTH_GRID.size - 1}") + for income_idx, income in enumerate(INCOME_GRID): + for age_idx, age in enumerate(AGE_GRID): + + # Shortcut: optimal policy in final year is to consume everything + if age >= MAX_AGE: + policy[wealth_idx, income_idx, age_idx] = 1.0 + continue + + # Another shortcut: for states that aren't reachable (positive income after retirement), fill things in quickly + if age >= RETIREMENT_AGE and income_idx > 0: + policy[wealth_idx, income_idx, age_idx] = policy[wealth_idx, 0, age_idx] + continue + + # Candidate values for consumption + consumption = CONSUMPTION_FRACTION_GRID * (income + wealth) + + reward = reward_function(consumption) + + next_wealth = (wealth + income - consumption) * (1 + INTEREST_RATE) + next_age = (age + 1) * np.ones_like(next_wealth) + + if next_age[0] >= RETIREMENT_AGE: + next_income = np.zeros_like(next_wealth) + next_state = np.stack([next_wealth, next_income, next_age]) + features = np.apply_along_axis(lambda x: get_feature_vector(*x), 0, next_state) + continuation_value = np.dot(value_fn_coefficients, features) + + else: + # The next state is random (because income is random) + # We need to calculate an expected value + expected_next_income = RAISE_PROBABILITY * income * (1 + (MIN_RAISE + MAX_RAISE) / 2) + (1 - RAISE_PROBABILITY) * income + expected_next_income = expected_next_income * np.ones_like(next_wealth) + + expected_next_state = np.stack([next_wealth, expected_next_income, next_age]) + + features = np.apply_along_axis(lambda x: get_feature_vector(*x), 0, expected_next_state) + continuation_value = np.dot(value_fn_coefficients, features) + + value = reward + DISCOUNT_FACTOR * continuation_value + + best_consumption_fraction = CONSUMPTION_FRACTION_GRID[np.argmax(value)] + + policy[wealth_idx, income_idx, age_idx] = best_consumption_fraction + + print("Policy fn sanity check") + correct_policy_penultimate_period = 1 / (1 + DISCOUNT_FACTOR) # TODO Compare to policy[:, 0, AGE_GRID.size - 2] + + # TODO Another sanity check: plot correct value function in final period v(w) = ln(w) versus approximate value fn + + # import pdb; pdb.set_trace() + + return policy + + +def main(n_policy_updates=5): + + # These are _fractions_ of available wealth+income to consume in the current period + # State is wealth, income, age + # Note that the grid includes states that are not reachable! For example, the grid includes + # states where the agent is above RETIREMENT_AGE but still earns income + # TODO Could have a separate grid for before and after retirement age (fewer points to loop over) + # The optimal policy after retiring is very similar to the cake eating problem! + policy = np.min(CONSUMPTION_FRACTION_GRID) * np.ones((WEALTH_GRID.size, INCOME_GRID.size, AGE_GRID.size)) + + for policy_update_idx in range(n_policy_updates): + + value_fn_coefficients, df = calculate_approx_value_function(policy) + policy = update_policy(value_fn_coefficients, policy) + + # TODO Run sanity checks on policy (check last and before last period, for example) + + # TODO Put all of these plots in separate functions + fig, ax = plt.subplots(figsize=(10, 8)) + + plt.scatter(df["age"], df["wealth"], alpha=0.1) + + plt.xlabel("age") + plt.ylabel("wealth (at beginning of period)") + + outdir = os.path.join(os.path.dirname(os.path.abspath(__file__)), PLOT_DIR) + + outfile = f"simulations_age_and_wealth_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + + plt.scatter(df["age"], df["income"], alpha=0.1) + + plt.xlabel("age") + plt.ylabel("income") + + outfile = f"simulations_age_and_income_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + + plt.scatter(df["age"], df["consumption"], alpha=0.1) + + plt.xlabel("age") + plt.ylabel("consumption") + + outfile = f"simulations_age_and_consumption_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + + plt.scatter(df["age"], df["consumption_fractions"], alpha=0.1) + + plt.xlabel("age") + plt.ylabel("consumption fraction") + + outfile = f"simulations_age_and_consumption_fraction_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + + for wealth_idx in [0, 5, 10]: + for income_idx in [0, 5, 10]: + label = f"wealth = {np.round(WEALTH_GRID[wealth_idx], 2)}, income = {np.round(INCOME_GRID[income_idx], 2)}" + plt.plot(AGE_GRID, policy[wealth_idx, income_idx, :], label=label) + + plt.xlabel("age (part of the state variable)") + plt.ylabel("fraction of wealth + income consumed in current period (action)") + ax.legend() + + outfile = f"policy_fraction_consumed_as_of_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + + for wealth_idx in [0, 5, 10]: + for income_idx in [0, 5, 10]: + label = f"wealth = {np.round(WEALTH_GRID[wealth_idx], 2)}, income = {np.round(INCOME_GRID[income_idx], 2)}" + wealth = WEALTH_GRID[wealth_idx] + income = INCOME_GRID[income_idx] + amount_consumed = (wealth + income) * policy[wealth_idx, income_idx, :] + plt.plot(AGE_GRID, amount_consumed, label=label) + + plt.xlabel("age (part of the state variable)") + plt.ylabel("amount consumed in current period (action)") + ax.legend() + + outfile = f"policy_amount_consumed_as_of_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + fig, ax = plt.subplots(figsize=(10, 8)) + for age_idx in [0, 5, 10]: + for income_idx in [0, 5]: + age = AGE_GRID[age_idx] + income = INCOME_GRID[income_idx] + state = np.stack([WEALTH_GRID, income * np.ones_like(WEALTH_GRID), age * np.ones_like(WEALTH_GRID)]) + features = np.apply_along_axis(lambda x: get_feature_vector(*x), 0, state) + approx_value = np.dot(value_fn_coefficients, features) + + label = f"age = {age}, income = {np.round(income, 2)}" + plt.plot(WEALTH_GRID, approx_value, label=label) + + plt.xlabel("wealth (part of the state variable)") + plt.ylabel("approx value fn") + ax.legend() + outfile = f"approx_value_as_of_iteration_{policy_update_idx}.png" + plt.savefig(os.path.join(outdir, outfile)) + plt.close() + + +if __name__ == "__main__": + main() From c42c707cd96a2eff665b2e9f94e0073b49d1fd20 Mon Sep 17 00:00:00 2001 From: Adrian Torchiana Date: Fri, 3 Dec 2021 09:54:49 -0800 Subject: [PATCH 2/2] Add some plots --- ...age_and_consumption_fraction_iteration_4.png | Bin 0 -> 23507 bytes ...ulations_age_and_consumption_iteration_4.png | Bin 0 -> 42524 bytes .../simulations_age_and_income_iteration_4.png | Bin 0 -> 50387 bytes .../simulations_age_and_wealth_iteration_4.png | Bin 0 -> 63530 bytes 4 files changed, 0 insertions(+), 0 deletions(-) create mode 100644 python/plots/simulations_age_and_consumption_fraction_iteration_4.png create mode 100644 python/plots/simulations_age_and_consumption_iteration_4.png create mode 100644 python/plots/simulations_age_and_income_iteration_4.png create mode 100644 python/plots/simulations_age_and_wealth_iteration_4.png diff --git a/python/plots/simulations_age_and_consumption_fraction_iteration_4.png 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