\n",
" \n",
" \n",
" \n",
"

"
]
},
{
"cell_type": "markdown",
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"source": [
"# Optimal Taxation without State-Contingent Debt"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Contents\n",
"\n",
"- [Optimal Taxation without State-Contingent Debt](#Optimal-Taxation-without-State-Contingent-Debt) \n",
" - [Overview](#Overview) \n",
" - [Competitive Equilibrium with Distorting Taxes](#Competitive-Equilibrium-with-Distorting-Taxes) \n",
" - [Recursive Version of AMSS Model](#Recursive-Version-of-AMSS-Model) \n",
" - [Examples](#Examples) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**Software Requirement:**\n",
"\n",
"This lecture requires the use of some older software versions to run. If\n",
"you would like to execute this lecture please download the following\n",
"amss_environment.yml\n",
"file. This specifies the software required and an environment can be\n",
"created using [conda](https://docs.conda.io/en/latest/):\n",
"\n",
"Open a terminal:"
]
},
{
"cell_type": "markdown",
"metadata": {
"hide-output": false
},
"source": [
"```bash\n",
"conda env create --file amss_environment.yml\n",
"conda activate amss\n",
"```\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In addition to what’s in Anaconda, this lecture will need the following libraries:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"hide-output": true
},
"outputs": [],
"source": [
"!pip install --upgrade quantecon"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Overview\n",
"\n",
"Let’s start with following imports:"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"hide-output": false
},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"from scipy.optimize import root, fmin_slsqp\n",
"from scipy.interpolate import UnivariateSpline\n",
"from quantecon import MarkovChain"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In [an earlier lecture](https://python-programming.quantecon.org/opt_tax_recur.html), we described a model of\n",
"optimal taxation with state-contingent debt due to\n",
"Robert E. Lucas, Jr., and Nancy Stokey [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983).\n",
"\n",
"Aiyagari, Marcet, Sargent, and Seppälä [[AMSSeppala02]](https://python-programming.quantecon.org/zreferences.html#aiyagari2002optimal) (hereafter, AMSS)\n",
"studied optimal taxation in a model without state-contingent debt.\n",
"\n",
"In this lecture, we\n",
"\n",
"- describe assumptions and equilibrium concepts \n",
"- solve the model \n",
"- implement the model numerically \n",
"- conduct some policy experiments \n",
"- compare outcomes with those in a corresponding complete-markets model \n",
"\n",
"\n",
"We begin with an introduction to the model."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Competitive Equilibrium with Distorting Taxes\n",
"\n",
"Many but not all features of the economy are identical to those of [the Lucas-Stokey economy](https://python-programming.quantecon.org/opt_tax_recur.html).\n",
"\n",
"Let’s start with things that are identical.\n",
"\n",
"For $ t \\geq 0 $, a history of the state is represented by $ s^t = [s_t, s_{t-1}, \\ldots, s_0] $.\n",
"\n",
"Government purchases $ g(s) $ are an exact time-invariant function of $ s $.\n",
"\n",
"Let $ c_t(s^t) $, $ \\ell_t(s^t) $, and $ n_t(s^t) $ denote consumption,\n",
"leisure, and labor supply, respectively, at history $ s^t $ at time $ t $.\n",
"\n",
"Each period a representative household is endowed with one unit of time that can be divided between leisure\n",
"$ \\ell_t $ and labor $ n_t $:\n",
"\n",
"\n",
"\n",
"$$\n",
"n_t(s^t) + \\ell_t(s^t) = 1 \\tag{1}\n",
"$$\n",
"\n",
"Output equals $ n_t(s^t) $ and can be divided between consumption $ c_t(s^t) $ and $ g(s_t) $\n",
"\n",
"\n",
"\n",
"$$\n",
"c_t(s^t) + g(s_t) = n_t(s^t) \\tag{2}\n",
"$$\n",
"\n",
"Output is not storable.\n",
"\n",
"The technology pins down a pre-tax wage rate to unity for all $ t, s^t $.\n",
"\n",
"A representative household’s preferences over $ \\{c_t(s^t), \\ell_t(s^t)\\}_{t=0}^\\infty $ are ordered by\n",
"\n",
"\n",
"\n",
"$$\n",
"\\sum_{t=0}^\\infty \\sum_{s^t} \\beta^t \\pi_t(s^t) u[c_t(s^t), \\ell_t(s^t)] \\tag{3}\n",
"$$\n",
"\n",
"where\n",
"\n",
"- $ \\pi_t(s^t) $ is a joint probability distribution over the sequence $ s^t $, and \n",
"- the utility function $ u $ is increasing, strictly concave, and three times continuously differentiable in both arguments. \n",
"\n",
"\n",
"The government imposes a flat rate tax $ \\tau_t(s^t) $ on labor income at time $ t $, history $ s^t $.\n",
"\n",
"Lucas and Stokey assumed that there are complete markets in one-period Arrow securities; also see [smoothing models](https://python-programming.quantecon.org/smoothing.html).\n",
"\n",
"It is at this point that AMSS [[AMSSeppala02]](https://python-programming.quantecon.org/zreferences.html#aiyagari2002optimal) modify the Lucas and Stokey economy.\n",
"\n",
"AMSS allow the government to issue only one-period risk-free debt each period.\n",
"\n",
"Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in [[Bar79]](https://python-programming.quantecon.org/zreferences.html#barro1979) than they do in [[LS83]](https://python-programming.quantecon.org/zreferences.html#lucasstokey1983)."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Risk-free One-Period Debt Only\n",
"\n",
"In period $ t $ and history $ s^t $, let\n",
"\n",
"- $ b_{t+1}(s^t) $ be the amount of the time $ t+1 $ consumption good that at time $ t $ the government promised to pay \n",
"- $ R_t(s^t) $ be the gross interest rate on risk-free one-period debt between periods $ t $ and $ t+1 $ \n",
"- $ T_t(s^t) $ be a non-negative lump-sum transfer to the representative household **[1]** In an allocation that solves the Ramsey problem and that levies distorting\n",
"taxes on labor, why would the government ever want to hand revenues back\n",
"to the private sector? It would not in an economy with state-contingent debt, since\n",
"any such allocation could be improved by lowering distortionary taxes\n",
"rather than handing out lump-sum transfers. But, without state-contingent\n",
"debt there can be circumstances when a government would like to make\n",
"lump-sum transfers to the private sector.\n",
"\n",
"

**[2]** From the first-order conditions for the Ramsey\n",
"problem, there exists another realization $ \\tilde s^t $ with\n",
"the same history up until the previous period, i.e., $ \\tilde s^{t-1}=\n",
"s^{t-1} $, but where the multiplier on constraint [(11)](#equation-amss-46) takes a positive value, so\n",
"$ \\gamma_t(\\tilde s^t)>0 $."
]
}
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"title": "Optimal Taxation with State-Contingent Debt"
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"title": "Optimal Taxation without State-Contingent Debt"
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