diff --git a/.DS_Store b/.DS_Store new file mode 100644 index 00000000..eaee769c Binary files /dev/null and b/.DS_Store differ diff --git a/models/.DS_Store b/models/.DS_Store new file mode 100644 index 00000000..ac44b8e4 Binary files /dev/null and b/models/.DS_Store differ diff --git a/models/We-Would-Like-In-Econ-ARK/.DS_Store b/models/We-Would-Like-In-Econ-ARK/.DS_Store new file mode 100644 index 00000000..4a38c0bb Binary files /dev/null and b/models/We-Would-Like-In-Econ-ARK/.DS_Store differ diff --git a/models/We-Would-Like-In-Econ-ARK/Aiyagari1994QJE_updated.ipynb b/models/We-Would-Like-In-Econ-ARK/Aiyagari1994QJE_updated.ipynb new file mode 100644 index 00000000..b6e39e93 --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/Aiyagari1994QJE_updated.ipynb @@ -0,0 +1,218 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Uninsured Idiosyncratic Risk and Aggregate Saving\n", + "* Authors: S. Rao Aiyagari\n", + "* Source: The Quaterly Journal of Economics, Vol. 109, No. 3 (Aug., 1994), pp. 659-684\n", + "* Notebook by Zixuan Huang and Mingzuo Sun\n", + "* 2019 Fall" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Introduction of Heterogeneity\n", + "* The flaw of Representative Agent (RA) model: inconsistency with empirical studies. \n", + "> * Empricial studies suggest that individual consumption is much more variable than aggregate consumption, which is inconsistent with conclusions of RA model.\n", + "> * The behavior of wealth and portfolios is strongly at variance with the complete markets model implicit in RA models.\n", + "\n", + "* Thus due to incomplete markets, the existence of heterogeneity may be important especially with a context of borrowing constraints." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Abtract \n", + "* The paper modifies standard growth model to include precautionary saving motives and liquidity constraints. \n", + "* The paper presents the impact on the aggregate saving rate, the importance of asset trading to individuals and the relative inequality of wealth and income distributions. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Introduction\n", + "\n", + "### Goals \n", + "* To provide an exposition of models whose aggregate behavior is the result of market interaction among a large number of agents subject to idiosyncratic shocks. \n", + "> * This class of models contrasts with representative agent models where individual dynamics and uncertainty coincide with aggregate dynamics and uncertainty.
\n", + "> * This exposition is built around the standard growth model of Brock and Mirman(1972) modified to include a role for uninsured idiosyncratic risk and borrowing constraint.\n", + "* To use such a model to study the quantitative importance of individual risk for aggregate saving.
\n", + "\n", + "\n", + "### Key features\n", + "* Endogenous heterogeneity \n", + "* Aggregation\n", + "* Infinite horizons \n", + "* Borrowing constraint\n", + "* General equilibrium. i.e. interest rate is endogenously determined since in a steady state equilibrium the capital per capita must equal the per capita asset holdings of consumers, and the interest rate must equal the net marginal product of capital. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Related Literature\n", + "* Aiyagari model originates from Bewley model and Hugget model. These models all share the same key components as mentioned in the previous part. And they are used to study the following topics:\n", + "> * How much of observed wealth inequality can one explain through uninsurable earnings variation across agents?
\n", + "> * What is the fraction of aggregate savings due to the precautionary motive?
\n", + "> * What are the redistributional implications of various policies?\n", + "\n", + "* The evolution of those models\n", + "> * Bewley(1986): an exogenous interest rate $r$ is assumed.
\n", + "> * Huggett(1993): a pure exchange economy in which the interest rate is endogenized.
\n", + "> * Aiyagari(1994): a production sector is added to Huggett model." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Model\n", + "### The Individual's Problem\n", + "$$\n", + "\\begin{align}\n", + "max E_0[\\Sigma_{t=0}^\\infty \\beta^t U(c_t)]\n", + "\\end{align}\n", + "$$\n", + "subject to \n", + "$$\n", + "\\begin{align}\n", + "c_t+a_{t+1}&=wl_{t}+(1+r)a_t \\\\\n", + "c_t\\geq0&, a_t\\geq-b\n", + "\\end{align}\n", + "$$\n", + "where $b$ (if positive) is the limit on borrowing and $l_t$ is assumed to be i.i.d with bounded support given by $[l_{min},l_{max}]$, with $l_{min}>0$\n", + "\n", + "* If $r<0$, some limit on borrowing is required, otherwise the problem is not well posed and a solution does not exist. \n", + "* If $r>0$, a borrowing constraint, $a_{t} \\geq \\frac{-wl_{min}}{r}$, is implied by nonnegative consumption. Hence, if $b$ exceeds $\\frac{-wl_{min}}{r}$, the the borrowing limit $b$ wil never be binding, and $b$ should be replaced by $\\frac{wl_{min}}{r}$; In other words, the constraints may be specified as:\n", + "$$\n", + "\\begin{align}\n", + "a_t &\\geq -\\phi \\\\\n", + "\\phi &\\equiv min\\{b, \\frac{wl_{min}}{r}\\},\\ for\\ r>0; \\phi \\equiv b,\\ for\\ r \\leq 0\n", + "\\end{align}\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "* Define $\\hat{a_t}=a_t+\\phi$ and $z_t=wl_t+(1+r)\\hat{a_t}-r\\phi$, where $\\hat{a_t}$ and $z_t$ may be thought of as the total financial resources at data $t$ and total resources of the agent at date $t$ respectively, then the Bellman equation is as follows:\n", + "$$\n", + "\\begin{align}\n", + "V(z_t,b,w,r) \\equiv \\underset{\\hat{a_{t+1}}}{max}\\{U(z_t-\\hat{a_t})+\\beta \\int V(z_{t+1},b,w,r)dF(l_{t+1}) \\}\n", + "\\end{align}\n", + "$$\n", + "* Solving the Bellman equation, the paper derives the optimal asset demand rule: $\\hat{a_{t+1}}=A(z_t,b,w,r)$ and the transition law for total resources $z_t$ is $z_{t+1}=wl_{t+1}+(1+r)A(z_t,b,w,r)-r\\phi$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Endogenous Heteogeneity and Aggregation\n", + "* The distribution of $\\{z_t\\}$ and the value of $Ea_w$ (denoting long-run average assets) reflect the endogenous heterogeneity and the aggregation features mentioned above. $Ea_w$ represents the aggregate assets of the population consistent with the distribution of assets across population implied by individual optimal saving behavior.\n", + "* If earnings were certain (or, equivalently, markets complete), $Ea_w$ would equal ($-\\phi$) for all $r<\\lambda$, where $\\lambda$ denotes the rate of time preference. That is, per capita assets under certainty are at their lower permissible level since all agents are alike and everyone is constrained. However, in a steady state under incomplete markets, there is a distribution of agents with different total resources reflecting different histories of labor endowment shocks. \n", + "* Those with low total resources will continue to be liquidity constrained, whereas those with high total resources will accumulate assets beyond the constrained level. Aggregation then implies that per capita must necessarily exceed their level under certainty. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### General Equilibrium\n", + "* In the steady state, the capital desired by firms at the interest rate $r$ must be equal to the capital supplied by households at the interest rate $r$, which is time invariant.\n", + "* The effects of varying the borrowing limit $b$ is described: when borrowing is permitted individuals need not rely solely on holdings of capital to buffer earnings variations. Borrowing can also be used to buffer these shocks and, hence, leads to smaller holdings of capital. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Model Specification, Parameterization, and Consumption\n", + "\n", + "### Model specification and parameters\n", + "* Discount factor: $\\beta=0.96$\n", + "* Production function: Cobb Douglas with the capital share taken to be 0.36\n", + "* Depreciation rate: $\\delta=0.08$\n", + "* Utility function: CRRA with the relative risk aversion coefficient $\\mu \\in \\{1,3,5\\}$\n", + "* Labor endowment shocks: Markov charin specification with seven states to match the following first-order autoregressive representation for the logarithm of the labor endowment shock:\n", + "$$\n", + "\\begin{align}\n", + "\\log(l_t)=\\rho\\log(l_{t-1})+\\sigma(1-\\rho^2)^{1/2}\\epsilon_{t}, \\epsilon_t--Normal(0,1) \\\\\n", + "\\sigma \\in \\{0.2,0.4\\}, \\rho \\in \\{0,0.3,0.6,0.9\\}.\n", + "\\end{align}\n", + "$$\n", + "* Borrowing limit: $b=0$" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Computation\n", + "* Approximate the asset demand as a function of total resources (for each of the seven possible current labor endowment shocks).\n", + "* Approximate the steady state: start with some value of $r$ (say, $r_1$), then compute the asset demand function corresponding to $r_1$, then simulate the Markov chain for the labor endowment shock using a random number generator and obtain a series of 10,000 draws. These are used with the asset demand function to obtain a simulated series of assets. The sample mean of this is taken to be $Ea$.Then calculate $r_2$ such that $K(r_2)$ equals $Ea$. If $r_2$ exceeds the rate of time preference, it is replaced by the rate of time preference. Define $r_3=(r_1+r_2)/2$ and calculate $Ea$ corresponding to $r_3$. If $Ea$ exceeds $K(r_3)$, then $r_2$ is replaced by $r_3$, and use bisection again. If $Ea$ is less than $K(r_3)$, then $r_1$ is replaced by $r_3$ then use bisection again. Typically this yields an excellent approximation to the steady state within ten iterations. \n", + "* Once the steady state is approximated, calculate the mean, median, standard deviation, skewness, serial correlation coefficient for labor income, asset holdings, gross saving, consumtion, etc. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Results\n", + "\n", + "### Aggregate Saving\n", + "* The differences between the saving rates with an without insurance are quite small for moderate and empirically plausible values of $\\sigma$, $\\rho$, and $\\mu$. However for high values of $\\sigma$, $\\rho$, and $\\mu$, the presence of idiosyncratic risk can raise the saving rate quite significantly by up to seven percentage points. \n", + "\n", + "\n", + "### Variabilities \n", + "* Consumption varies about 50-70 percent as much as income. Saving and assets are much more volatile than income: Saving varies about three times as much as income, and assets vary about twice as much as income. \n", + "* Risk aversion tends to reduce the variabilities of all these variabilities. \n", + "\n", + "### Importance of Asset Trading\n", + "* If $\\mu=3$,$\\sigma=4$ and $\\rho=0.6$, then, by optimally accumulating and depleting assets, consumption variability is cut in half, yielding a welfare benefit of about 14 percent of per capita consumption.\n", + "\n", + "### Cross-Section Distributions and Inequality Measures\n", + "* Since long-run distrubutions for an individual coincide with cross-section distrbutions for the population, results for variabilities of individual consumption, income, and assets have immediate implications for cross-section distributions. \n", + "* There is much less dispersion across households in consumption compared with income and much greater dispersion in wealth compared with income. \n", + "* Skewness coefficients reveal another aspect of inequality. All of the cross-section distributions are positively skewed (median$<$mean). However, the degree of skewness is less than in the data. " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Conclusions and Comments\n", + "The Aiyagari model extends the Bewley model and the Hugget model to a context with a production sector. Deviating from the representative agent model in which complete market is implicitly assumed, it tries to study the aggregate saving behavior of the economy with the agents facing uninsured idiosyncratic risk. With an empirically plausible set of parameters, it finds that the aggregate saving rate does increase compared to the case with a complete market, however, the change here caused by the precautionary saving motive is mild. Also, the results of the model qualitatively match the real data in terms of the ranking of the fluctuations of some economic variables. However, in terms of approaching the real inequalities of income and wealth shown by the data, the model does not perform very well. Also, in this model, the joint distribution of income and wealth is not treated as a state variable, which neglects the distribution effect Krusell and Smith(1998) try to address." + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/models/We-Would-Like-In-Econ-ARK/OFPSun/.DS_Store b/models/We-Would-Like-In-Econ-ARK/OFPSun/.DS_Store new file mode 100644 index 00000000..b580260e Binary files /dev/null and b/models/We-Would-Like-In-Econ-ARK/OFPSun/.DS_Store differ diff --git a/models/We-Would-Like-In-Econ-ARK/OFPSun/Bassetto-2014-Quantitative_Economics.pdf b/models/We-Would-Like-In-Econ-ARK/OFPSun/Bassetto-2014-Quantitative_Economics.pdf new file mode 100644 index 00000000..b5cc1a02 Binary files /dev/null and b/models/We-Would-Like-In-Econ-ARK/OFPSun/Bassetto-2014-Quantitative_Economics.pdf differ diff --git a/models/We-Would-Like-In-Econ-ARK/OFPSun/OptimalFiscalPolicy.ipynb b/models/We-Would-Like-In-Econ-ARK/OFPSun/OptimalFiscalPolicy.ipynb new file mode 100644 index 00000000..eaf89c3a --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/OFPSun/OptimalFiscalPolicy.ipynb @@ -0,0 +1,189 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Optimal Fiscal Policy with Heterogeneous Agents\n", + "## Journal: Quantitative Economics\n", + "## Author: Marco Bassetto\n", + "## Notebook by Mingzuo Sun\n", + "## Sep 15, 2019\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Goal of the paper\n", + "This paper is designed to study the relationship between the intertemporal behavior of taxes and wealth distribution in a Heterogeneous-Agent framework. While optimal-taxation problem appears quite often in the literature of representative agent model, it is much less often discussed in a heterogeneous-agent context, one of the reasons of which is that Representative Agent ('RA') model has much stronger welfare implications since the maximization of the social welfare and the maximization of the representative agent's welfare are equivalent. However, in Heterogeneous Agent ('HA') model, with a given social welfare function, we can still do that by studying how the government can manipulate the intertemporal prices to favor some groups over others." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Overview\n", + "\n", + "This paper tries to study the optimal fiscal policy with heterogeneous agents on both of the normative side and positive side. On the normative side, the Pareto frontier on which there is a trade-off between 'equity and efficiency' is being traced in this paper. In addition, it also tries to find some positive implications since an ideal political system will select policies close to the Pareto frontier. Particularly when there are large fiscal shocks occurring like a war, the welfare differences among possible fiscal policies will differ a lot, which in turn could make the main argument of the paper more evident." + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Model\n", + "\n", + "#### Setup of the general model\n", + "\n", + "\n", + "##### Endowment and technology\n", + "\n", + "Exogenous stream of public spending which has no effect on utility function.\n", + "\n", + "$\\{g_t\\}_{t=0}^\\infty$: stochastic process with a finite range G.\n", + "\n", + "Randomization of taxation plan: $h_0$, revealed after the choice of consumption and tax plans at time 0. （Note: Here tax rates are randomized in individuals but not in periods. Stiglitz(1981) made the argument that randomization will capture the feature of optimal taxation under a series of conditions related to the consumer's utility functions and the average tax rate. And randomization here can still preserve ex ante equity.)\n", + "\n", + "No storage and only one consumption good. \n", + "\n", + "CRS technology with labor being the only input, productivity $w^{i}$.\n", + "\n", + "Each agent is endowed with 1 unit of time.\n", + "\n", + "Government can levy proportional taxes on the labor income of each agent in the economy.\n", + "\n", + "Complete market for both privately issued and publicly issued securities. No defalt is allowed.\n", + "\n", + "$b^i(s^t)$: the amount of government-issued contingent claims payable at time $t$.\n", + "\n", + "$\\eta^i(s^t)$: entitlement to the state-contingent goods owned by agent $i$ as of time 0. $$\\sum_{n=1}^{N} \\eta^i(s^t) \\equiv 0$$\n", + "##### Preferences\n", + "###### Agent\n", + "$Agent=\\{1,...,i,...,N \\}$.\n", + "\n", + "$U^i\\equiv E_0 \\sum_{t=0}^{\\infty} \\beta^t u^i(c^i(s^t),u^i(s^t))$\n", + "\n", + "The $ith$ agent has the Arrow-Debreu budget constraint\n", + "\n", + " $\\sum _ { t = 0 } ^ { \\infty } \\sum _ { s^t \\in S^t } ^ { n } p \\left( s ^ { t } \\right) \\left[ c ^ { i } \\left( s ^ { t } \\right) - \\eta ^ { i } \\left( s ^ { t } \\right) - b ^ { i } \\left( s ^ { t } \\right) - \\left( 1 - \\tau \\left( s ^ { t } \\right) \\right) w ^ { i } \\left( 1 - x ^ { i } \\left( s ^ { t } \\right) \\right) \\right] = 0 $\n", + "\n", + "where $p(s^t)$ is the time-0 price of an Arrow-Debreu security that pays 1 at date $t$ in state $s^t$.\n", + "\n", + "###### Government\n", + "\n", + "The preference of the goverment(social welfare function)\n", + "\n", + "$W \\equiv \\sum_{n=1}^{N} \\alpha^iU^i$,$\\alpha^i$ is the Pareto weight of the $i$th agent.\n", + "\n", + "Government budget constraint\n", + "\n", + "$\\sum _ { t = 0 } ^ { \\infty } \\sum _ { s ^ { t } \\in S ^ { t } } p \\left( s ^ { t } \\right) \\left[ g _ { t } + \\sum _ { i = 1 } ^ { N } \\left( b ^ { i } \\left( s ^ { t } \\right) - \\tau \\left( s ^ { t } \\right) w ^ { i } \\left( 1 - x ^ { i } \\left( s ^ { t } \\right) \\right) \\right) \\right] = 0$\n", + "#### Market clear condition\n", + "\n", + " $\\sum _ { i = 1 } ^ { N } c ^ { i } \\left( s ^ { t } \\right) + g _ { t } = \\sum _ { i = 1 } ^ { N } w ^ { i } \\left( 1 - x ^ { i } \\left( s ^ { t } \\right) \\right) + \\sum _ { i = 1 } ^ { N } \\eta ^ { i } \\left( s ^ { t } \\right) \\quad \\forall t \\geq 0 \\quad \\forall s ^ { t } \\in S ^ { t } $\n", + "\n", + "\n", + "#### The two class economy\n", + "\n", + "The general framework is specialized into a particular case in which there are two types of agents in the economy M agents of type 1 and (by normalization) one agent of type 2. Type-1 agents are 'rentiers'. Their productivity $w^1=0$ is 0, so they always choose $x^1_t= 1$.They have no labor income and live only out of their assets.Type-2 agents\n", + "are identified as the 'taxpayers', as they are the only ones who have labor income and therefore pay taxes. We normalize their productivity to be $w^2 = 1$.\n", + "We assume the agents have the utility functions\n", + "\n", + "$u^1(c_t^1,x_t^1)=\\frac{(c_t^1)^{(1-\\gamma)} \\quad -1}{1-\\gamma}$ \n", + "\n", + "$u^2(c_t^2,x_t^2)=\\frac{(c_t^2)^{(1-\\gamma)} \\quad -1}{1-\\gamma}+\\xi\\frac{(x_t^2)^{(1-\\sigma)} \\quad -1}{1-\\sigma} $ \n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Basic results\n", + "\n", + "Different tax rates in different periods can distort intertemporal prices, which is an important redistribution tool of the government.\n", + "\n", + "For the rentiers, the intertemporal change of tax rates will not have direct effect but indirect effect(general equilibrium effect) on them.\n", + "\n", + "By taxing labor in odd periods and subsidizing it in even periods, the government generates an artificial scarcity of goods in odd periods, which is beneficial to the rentiers but very costly to the tax payers.\n", + "\n", + "A government that draws its main support from the people who pay taxes should optimally run larger deficits and wait for the end of the war to levy the taxes necessary to repay the defense expenses and vice versa.\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Novelties of the paper\n", + "\n", + "Using a simple model, this paper tries to find a link between the way of financing large public project and political inclination with efficiency considerations. By doing that, it manages to reveal some features of the optimal distortionary fiscal policy in a 2-agent model. Besides the theoretical work, it also tries to validate its result through some empirical evidence. By a simulation of the economies of France and Britain during their war time, this paper shows that the patterns of fiscal policy suggested by the model coincide with the real patterns in history, so it also has some political implications." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.6.8" + }, + "latex_envs": { + "LaTeX_envs_menu_present": true, + "autoclose": false, + "autocomplete": true, + "bibliofile": "biblio.bib", + "cite_by": "apalike", + "current_citInitial": 1, + "eqLabelWithNumbers": true, + "eqNumInitial": 1, + "hotkeys": { + "equation": "Ctrl-E", + "itemize": "Ctrl-I" + }, + "labels_anchors": false, + "latex_user_defs": false, + "report_style_numbering": false, + "user_envs_cfg": false + }, + "toc": { + "base_numbering": 1, + "nav_menu": {}, + "number_sections": true, + "sideBar": true, + "skip_h1_title": false, + "title_cell": "Table of Contents", + "title_sidebar": "Contents", + "toc_cell": false, + "toc_position": { + "height": "calc(100% - 180px)", + "left": "10px", + "top": "150px", + "width": "165px" + }, + "toc_section_display": true, + "toc_window_display": true + } + }, + "nbformat": 4, + "nbformat_minor": 2 +} diff --git a/models/We-Would-Like-In-Econ-ARK/OFPSun/placeholder b/models/We-Would-Like-In-Econ-ARK/OFPSun/placeholder new file mode 100644 index 00000000..8b137891 --- /dev/null +++ b/models/We-Would-Like-In-Econ-ARK/OFPSun/placeholder @@ -0,0 +1 @@ + diff --git a/models/We-Would-Like-In-Econ-ARK/Table2.png b/models/We-Would-Like-In-Econ-ARK/Table2.png new file mode 100644 index 00000000..01dba93d Binary files /dev/null and b/models/We-Would-Like-In-Econ-ARK/Table2.png differ