un_insure.ipynb

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   "source": [
    "# Optimal Unemployment Insurance"
   ]
  },
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   "source": [
    "## Overview\n",
    "\n",
    "This lecture describes a model of optimal unemployment\n",
    "insurance created by  Shavell and Weiss (1979) [[Shavell and Weiss, 1979](https://python-advanced.quantecon.org/zreferences.html#id9)].\n",
    "\n",
    "We  use recursive techniques of\n",
    "Hopenhayn and Nicolini (1997) [[Hopenhayn and Nicolini, 1997](https://python-advanced.quantecon.org/zreferences.html#id8)] to\n",
    "compute optimal insurance plans for  Shavell and\n",
    "Weiss’s model.\n",
    "\n",
    "Hopenhayn and Nicolini’s model is a generalization of Shavell and Weiss’s along dimensions that we’ll soon describe."
   ]
  },
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    "## Shavell and Weiss’s Model\n",
    "\n",
    "An unemployed worker orders stochastic processes of\n",
    "consumption and  search effort $ \\{c_t , a_t\\}_{t=0}^\\infty $\n",
    "according to\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo1'></a>\n",
    "$$\n",
    "E \\sum_{t=0}^\\infty \\beta^t \\left[ u(c_t) - a_t \\right] \\tag{40.1}\n",
    "$$\n",
    "\n",
    "where $ \\beta \\in (0,1) $ and $ u(c) $ is strictly increasing, twice differentiable,\n",
    "and strictly concave.\n",
    "\n",
    "We assume that $ u(0) $ is well defined.\n",
    "\n",
    "We require that $ c_t \\geq 0 $ and $ a_t \\geq 0 $.\n",
    "\n",
    "All jobs are alike and pay wage\n",
    "$ w >0 $ units of the consumption good each period forever.\n",
    "\n",
    "An unemployed\n",
    "worker searches with effort $ a $ and with probability\n",
    "$ p(a) $ receives a permanent job at the beginning\n",
    "of the next period.\n",
    "\n",
    "Furthermore, $ a=0 $ when the worker is\n",
    "employed.\n",
    "\n",
    "The probability\n",
    "of finding a job  is $ p(a) $.\n",
    "\n",
    "$ p $ is an increasing, strictly concave,\n",
    "and twice differentiable function of $ a $ that satisfies\n",
    "$ p(a)   \\in [0,1] $ for  $ a \\geq 0 $, $ p(0)=0 $.\n",
    "\n",
    ">**Note**\n",
    ">\n",
    ">When we compute examples below, we’ll use assume the same  $ p(a) $ function as\n",
    "[[Hopenhayn and Nicolini, 1997](https://python-advanced.quantecon.org/zreferences.html#id8)], namely, $ p(a) = 1 - \\exp(- r a) $, where $ r $ is a parameter\n",
    "that we’ll calibrate to hit the same target that [[Hopenhayn and Nicolini, 1997](https://python-advanced.quantecon.org/zreferences.html#id8)] did, namely,\n",
    "an empirical hazard rate of leaving unemployment.\n",
    "\n",
    "The consumption good is nonstorable.\n",
    "\n",
    "An unemployed worker has no savings and cannot borrow or lend.\n",
    "\n",
    "The unemployed worker’s only source of consumption\n",
    "smoothing over time and across states is an **insurance agency** or **planner**.\n",
    "\n",
    "Once a worker has found a job, he is beyond the planner’s grasp.\n",
    "\n",
    "- This is Shavell and Weiss’s assumption, but not Hopenhayn and Nicolini’s.  \n",
    "- Hopenhayn and Nicolini allow the unemployment insurance agency to\n",
    "  impose history-dependent taxes on previously unemployed workers.  \n",
    "- Since there is no incentive problem after the worker has found\n",
    "  a job, it is optimal for the agency to provide an employed worker with\n",
    "  a constant level of consumption.  \n",
    "- Hence, Hopenhayn and Nicolini’s insurance agency imposes\n",
    "  a permanent per-period history-dependent tax on a previously\n",
    "  unemployed but presently employed worker.  "
   ]
  },
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    "### Autarky\n",
    "\n",
    "As a benchmark, we first study the fate of an unemployed worker\n",
    "who has no access to unemployment insurance.\n",
    "\n",
    "Because employment\n",
    "is an absorbing state for the worker, we work backward from that\n",
    "state.\n",
    "\n",
    "Let $ V^e $ be the expected sum of discounted one-period utilities of an employed worker.\n",
    "\n",
    "Once the worker is employed, $ a=0 $, making  his period utility\n",
    "be $ u(c)-a = u(w) $ forever.\n",
    "\n",
    "Therefore,\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo2'></a>\n",
    "$$\n",
    "V^e = {u(w) \\over (1-\\beta)}  . \\tag{40.2}\n",
    "$$\n",
    "\n",
    "Now let $ V^u $ be the expected discounted present value of utility for an\n",
    "unemployed worker who chooses  consumption, effort  pair $ (c,a) $\n",
    "optimally.\n",
    "\n",
    "Value $ V^u $ satisfies the Bellman equation\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo3'></a>\n",
    "$$\n",
    "V^u = \\max_{a \\geq 0} \\biggl\\{ u(0) - a + \\beta \\left[\n",
    "   p(a) V^e + (1-p(a)) V^u \\right] \\biggr\\} . \\tag{40.3}\n",
    "$$\n",
    "\n",
    "The first-order condition for a maximum is\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo4'></a>\n",
    "$$\n",
    "\\beta p'(a) \\left[V^e - V^u \\right] \\leq 1 , \\tag{40.4}\n",
    "$$\n",
    "\n",
    "with equality if $ a>0 $.\n",
    "\n",
    "Since there is no state variable in this\n",
    "infinite horizon problem, there is a time-invariant optimal\n",
    "search intensity $ a $ and an associated value of being unemployed $ V^u $.\n",
    "\n",
    "Let $ V_{\\rm aut} = V^u $ solve Bellman equation [(40.3)](#equation-eq-hugo3).\n",
    "\n",
    "Equations [(40.3)](#equation-eq-hugo3)\n",
    "and [(40.4)](#equation-eq-hugo4)\n",
    "form the basis for\n",
    "an iterative algorithm for computing $ V^u = V_{\\rm aut} $.\n",
    "\n",
    "- Let $ V^u_j $ be\n",
    "  the estimate of $ V_{\\rm aut} $ at the $ j $th iteration.  \n",
    "- Use this value\n",
    "  in equation [(40.4)](#equation-eq-hugo4)  and solve\n",
    "  for an estimate of effort  $ a_j $.  \n",
    "- Use this value in a version of equation\n",
    "  [(40.3)](#equation-eq-hugo3) with $ V^u_j $ on the right side\n",
    "  to compute $ V^u_{j+1} $.  \n",
    "- Iterate to convergence.  "
   ]
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    "### Full Information\n",
    "\n",
    "Another benchmark model helps set the stage for the model with private information that we ultimately want to study.\n",
    "\n",
    "We temporarily assume that an unemployment insurance agency has full information about the unemployed worker.\n",
    "\n",
    "We assume that the  insurance agency can control both the consumption and the search effort of an unemployed worker.\n",
    "\n",
    "The agency wants to design an unemployment insurance contract to give\n",
    "the unemployed worker expected discounted utility $ V > V_{\\rm aut} $.\n",
    "\n",
    "The agency, i.e., the  planner, wants to deliver value $ V $ efficiently,\n",
    "meaning in a  way that minimizes an expected  present value  discounted costs, using $ \\beta $ as the discount factor.\n",
    "\n",
    "We formulate the optimal insurance problem\n",
    "recursively.\n",
    "\n",
    "Let $ C(V) $ be the expected discounted cost of giving\n",
    "the worker expected discounted utility\n",
    "$ V $.\n",
    "\n",
    "The cost function is strictly convex because\n",
    "a higher $ V $ implies a lower marginal utility of the worker;\n",
    "that is, additional expected utils can be awarded to the worker\n",
    "only at an increasing marginal cost in terms of the consumption good.\n",
    "\n",
    "Given $ V $, the planner assigns first-period pair   $ (c,a) $ and promised\n",
    "continuation value $ V^u $ next period if   the worker  is  unlucky\n",
    "and does not find a job this period.\n",
    "\n",
    "The planner sets $ (c, a, V^u) $ as functions of $ V $ and to\n",
    "satisfy the following  Bellman equation for associated cost function $ C(V) $:\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo5'></a>\n",
    "$$\n",
    "C(V) = \\min_{c, a, V^u} \\biggl\\{ c  + \\beta [1 - p(a)] C(V^u) \\biggr\\} , \\tag{40.5}\n",
    "$$\n",
    "\n",
    "where  minimization is subject to the promise-keeping constraint\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo6'></a>\n",
    "$$\n",
    "V \\leq u(c) - a + \\beta\n",
    "\\left\\{ p(a) V^e + [1-p(a)] V^u \\right\\}. \\tag{40.6}\n",
    "$$\n",
    "\n",
    "Here $ V^e $ is given by equation [(40.2)](#equation-eq-hugo2), which reflects the\n",
    "assumption that once the worker is employed, he is beyond the\n",
    "reach of the unemployment insurance agency.\n",
    "\n",
    "The right side of  Bellman equation [(40.5)](#equation-eq-hugo5) is attained by\n",
    "policy functions $ c=c(V), a=a(V) $, and $ V^u=V^u(V) $.\n",
    "\n",
    "The promise-keeping constraint,\n",
    "equation  [(40.6)](#equation-eq-hugo6),\n",
    "asserts that the 3-tuple $ (c, a, V^u) $ attains\n",
    "at least $ V $.\n",
    "\n",
    "Let $ \\theta $ be a Lagrange multiplier\n",
    "on constraint [(40.6)](#equation-eq-hugo6).\n",
    "\n",
    "At an interior solution,  first-order\n",
    "conditions with\n",
    "respect to $ c, a $, and $ V^u $, respectively, are\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo7'></a>\n",
    "$$\n",
    "\\begin{aligned} \\theta & = {1 \\over u'(c)}\\,,   \\cr\n",
    "             C(V^u) & = \\theta \\left[ {1 \\over \\beta p'(a)} -\n",
    "                           (V^e - V^u) \\right]\\,,    \\cr\n",
    "             C'(V^u) & = \\theta\\,.\n",
    "\\end{aligned} \\tag{40.7}\n",
    "$$\n",
    "\n",
    "The envelope condition   $ C'(V) = \\theta $ and the third equation\n",
    "of [(40.7)](#equation-eq-hugo7)  imply that $ C'(V^u) =C'(V) $.\n",
    "\n",
    "Strict convexity of $ C $ then\n",
    "implies that $ V^u =V $.\n",
    "\n",
    "Applied repeatedly over time,\n",
    "$ V^u=V $ makes\n",
    "the continuation value remain constant during the entire\n",
    "spell of unemployment.\n",
    "\n",
    "The first equation of [(40.7)](#equation-eq-hugo7)\n",
    "determines $ c $, and the second equation of [(40.7)](#equation-eq-hugo7)  determines\n",
    "$ a $, both as functions of promised value $ V $.\n",
    "\n",
    "That $ V^u = V $ then\n",
    "implies that $ c $ and $ a $ are held constant during the unemployment\n",
    "spell.\n",
    "\n",
    "Thus, the unemployed worker’s consumption $ c $ and search effort $ a $ are both fully smoothed\n",
    "during the unemployment spell.\n",
    "\n",
    "But the worker’s consumption is not  smoothed across states of\n",
    "employment and unemployment unless $ V=V^e $."
   ]
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    "### Incentive Problem\n",
    "\n",
    "The preceding efficient insurance scheme assumes  that the insurance agency\n",
    "controls both $ c $ and $ a $.\n",
    "\n",
    "The insurance agency cannot simply  provide $ c $ and then allow the worker to choose $ a $.\n",
    "\n",
    "Here is why.\n",
    "\n",
    "The  agency delivers a value $ V^u $  higher than\n",
    "the autarky value $ V_{\\rm aut} $ by doing two things.\n",
    "\n",
    "It **increases** the unemployed worker’s consumption $ c $ and **decreases** his search\n",
    "effort $ a $.\n",
    "\n",
    "The prescribed\n",
    "search effort is **higher** than what the worker would choose\n",
    "if he were to be guaranteed consumption level $ c $ while he\n",
    "remains unemployed.\n",
    "\n",
    "This follows from the first two equations of [(40.7)](#equation-eq-hugo7) and the\n",
    "fact that the insurance scheme is costly, $ C(V^u)>0 $, which imply\n",
    "$ [ \\beta p'(a) ]^{-1} > (V^e - V^u) $.\n",
    "\n",
    "Now look at the worker’s\n",
    "first-order condition [(40.4)](#equation-eq-hugo4)  under autarky.\n",
    "\n",
    "It implies that if search effort $ a>0 $, then\n",
    "$ [\\beta p'(a)]^{-1} = [V^e - V^u] $, which is inconsistent\n",
    "with the  inequality\n",
    "$ [ \\beta p'(a) ]^{-1} > (V^e - V^u) $ that prevails when $ a >0 $ when the agency controls\n",
    "both $ a $ and $ c $.\n",
    "\n",
    "If he were free to choose $ a $, the worker would therefore want to\n",
    "fulfill [(40.4)](#equation-eq-hugo4), either at equality so long as $ a >0 $, or by setting\n",
    "$ a=0 $ otherwise.\n",
    "\n",
    "Starting from the  $ a $ associated with\n",
    "the full-information social insurance scheme in which the agency controls both $ c $ and $ a $,\n",
    "the worker  would establish the desired equality\n",
    "in [(40.4)](#equation-eq-hugo4) by *lowering* $ a $, thereby decreasing\n",
    "the term $ [ \\beta p'(a) ]^{-1} $ (which also lowers $ (V^e - V^u) $\n",
    "when the value of being\n",
    "unemployed $ V^u $ increases).\n",
    "\n",
    "If an equality can be established before\n",
    "$ a $ reaches zero, this would be the worker’s preferred search effort;\n",
    "otherwise the worker would find it optimal to accept the insurance\n",
    "payment, set $ a=0 $,  and  never work again.\n",
    "\n",
    "Thus, since the worker does not take the\n",
    "cost of the insurance scheme into account, he would choose a search\n",
    "effort below the socially optimal, full-information level.\n",
    "\n",
    "The full-information  contract thus\n",
    "relies on  the agency’s ability to control *both* the unemployed\n",
    "worker’s consumption *and* his search effort."
   ]
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    "## Private Information\n",
    "\n",
    "Following  [[Shavell and Weiss, 1979](https://python-advanced.quantecon.org/zreferences.html#id9)] and\n",
    "[[Hopenhayn and Nicolini, 1997](https://python-advanced.quantecon.org/zreferences.html#id8)], now assume that  the unemployment insurance agency cannot\n",
    "observe or control $ a $, though it can observe and control $ c $.\n",
    "\n",
    "The worker is free to choose $ a $, which puts expression [(40.4)](#equation-eq-hugo4), the worker’s first-order condition under autarky,\n",
    "back in the picture.\n",
    "\n",
    "- We are assuming that the worker’s\n",
    "  best response to the unemployment insurance arrangement is\n",
    "  completely characterized by the first-order condition [(40.4)](#equation-eq-hugo4),\n",
    "  an instance of the so-called first-order approach to incentive problems.  \n",
    "\n",
    "\n",
    "Given a contract, the individual will choose search effort according to\n",
    "first-order condition [(40.4)](#equation-eq-hugo4).\n",
    "\n",
    "This fact motivates the insurance agency\n",
    "to design an unemployment insurance contract that respects this restriction.\n",
    "\n",
    "Thus, the  contract design problem is now to minimize the right side of equation\n",
    "[(40.5)](#equation-eq-hugo5) subject to expression [(40.6)](#equation-eq-hugo6) and the incentive constraint [(40.4)](#equation-eq-hugo4).\n",
    "\n",
    "Since the restrictions [(40.4)](#equation-eq-hugo4) and [(40.6)](#equation-eq-hugo6) are not linear\n",
    "and generally do not define a convex set, it becomes challenging\n",
    "to provide conditions under which the solution to the dynamic\n",
    "programming problem results in a convex function $ C(V) $.\n",
    "\n",
    "- Sometimes this complication can be handled by convexifying\n",
    "  the constraint set by  introducing lotteries.  \n",
    "- A common finding is that optimal plans do not involve\n",
    "  lotteries, because convexity of the constraint set is a sufficient\n",
    "  but not necessary condition for convexity of the cost function.  \n",
    "- In order to characterize the optimal solution, we follow Hopenhayn and Nicolini (1997) [[Hopenhayn and Nicolini, 1997](https://python-advanced.quantecon.org/zreferences.html#id8)] by hopefully proceeding under the   assumption that $ C(V) $ is strictly convex.  \n",
    "\n",
    "\n",
    "Let $ \\eta $ be the multiplier on constraint [(40.4)](#equation-eq-hugo4), while\n",
    "$ \\theta $ continues to denote the multiplier on constraint [(40.6)](#equation-eq-hugo6).\n",
    "\n",
    "But now we replace the weak inequality in [(40.6)](#equation-eq-hugo6) by an equality.\n",
    "\n",
    "- We do this because the unemployment insurance agency cannot award a higher utility than\n",
    "  $ V $ because that might violate an incentive-compatibility constraint\n",
    "  for exerting the proper search effort in earlier periods.  \n",
    "\n",
    "\n",
    "At an interior solution,  first-order conditions with\n",
    "respect to $ c, a $, and $ V^u $, respectively, are\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo8'></a>\n",
    "$$\n",
    "\\begin{aligned} \\theta & = {1 \\over u'(c)}\\,,   \\cr\n",
    " C(V^u)  & = \\theta \\left[ {1 \\over \\beta p'(a)} - (V^e - V^u) \\right]\n",
    "            \\,-\\, \\eta {p''(a) \\over p'(a)} (V^e - V^u)                  \\cr\n",
    "         & = \\,- \\eta {p''(a) \\over p'(a)} (V^e - V^u) \\,,   \\cr\n",
    " C'(V^u) & = \\theta \\,-\\, \\eta {p'(a) \\over 1-p(a)}\\, ,\n",
    "\\end{aligned} \\tag{40.8}\n",
    "$$\n",
    "\n",
    "where the second equality in the second equation in [(40.8)](#equation-eq-hugo8)  follows from strict equality\n",
    "of the incentive constraint [(40.4)](#equation-eq-hugo4) when $ a>0 $.\n",
    "\n",
    "As long as the\n",
    "insurance scheme is associated with costs, so that $ C(V^u)>0 $, the first-order\n",
    "condition in the second equation of [(40.8)](#equation-eq-hugo8) implies that the multiplier $ \\eta $ is strictly\n",
    "positive.\n",
    "\n",
    "The first-order condition in the second equation of the third equality in [(40.8)](#equation-eq-hugo8)  and the\n",
    "envelope condition $ C'(V) = \\theta $ together allow us to conclude that\n",
    "$ C'(V^u) < C'(V) $.\n",
    "\n",
    "Convexity of $ C $ then implies that $ V^u < V $.\n",
    "\n",
    "After we have also used  the first equation of [(40.8)](#equation-eq-hugo8), it follows that\n",
    "in order to provide  the proper incentives, the consumption\n",
    "of the unemployed worker must decrease as the duration of the unemployment\n",
    "spell lengthens.\n",
    "\n",
    "It also follows from [(40.4)](#equation-eq-hugo4) at equality that\n",
    "search effort $ a $ rises as $ V^u $ falls, i.e., it rises with the duration\n",
    "of unemployment.\n",
    "\n",
    "The  of benefits on the duration of unemployment is  designed to provide the worker an\n",
    "incentive to search.\n",
    "\n",
    "To understand this, from  the third equation of [(40.8)](#equation-eq-hugo8), notice how\n",
    "the conclusion that consumption falls with the duration of\n",
    "unemployment depends on the assumption that more search effort\n",
    "raises the prospect of finding a job, i.e., that $ p'(a) > 0 $.\n",
    "\n",
    "If\n",
    "$ p'(a) =0 $, then  the third equation of [(40.8)](#equation-eq-hugo8) and the strict convexity of $ C $ imply that\n",
    "$ V^u =V $.\n",
    "\n",
    "Thus, when $ p'(a) =0 $, there is no reason for the\n",
    "planner to make consumption fall with the duration of\n",
    "unemployment."
   ]
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    "### Computational Details\n",
    "\n",
    "It is useful to note that there\n",
    "are natural lower and upper bounds to the set\n",
    "of continuation values $ V^u $.\n",
    "\n",
    "The lower bound is\n",
    "the expected lifetime utility in autarky,\n",
    "$ V_{\\rm aut} $.\n",
    "\n",
    "To compute an upper bound,\n",
    "represent condition [(40.4)](#equation-eq-hugo4) as\n",
    "\n",
    "$$\n",
    "V^u \\geq V^e  - [\\beta  p'(a)]^{-1},\n",
    "$$\n",
    "\n",
    "with equality if $ a > 0 $.\n",
    "\n",
    "If there is zero search effort, then $ V^u \\geq V^e -[\\beta p'(0)]^{-1} $.\n",
    "\n",
    "Therefore, to rule out zero search effort we require\n",
    "\n",
    "$$\n",
    "V^u < V^e - [\\beta p'(0)]^{-1} .\n",
    "$$\n",
    "\n",
    "(Remember that $ p''(a) < 0 $.)\n",
    "\n",
    "This step gives  our upper bound\n",
    "for $ V^u $.\n",
    "\n",
    "To formulate the Bellman equation numerically,\n",
    "we suggest using the constraints to eliminate $ c $ and $ a $ as choice\n",
    "variables, thereby reducing the Bellman equation to\n",
    "a minimization over the one choice variable $ V^u $.\n",
    "\n",
    "First express the promise-keeping constraint [(40.6)](#equation-eq-hugo6)  at equality as\n",
    "\n",
    "$$\n",
    "u(c) = V + a - \\beta \\{p(a) V^e +[1-p(a)] V^u \\}\n",
    "$$\n",
    "\n",
    "so that  consumption is\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo21'></a>\n",
    "$$\n",
    "c = u^{-1}\\left(\n",
    "     V+a -\\beta [p(a)V^e + (1-p(a))V^u] \\right). \\tag{40.9}\n",
    "$$\n",
    "\n",
    "Similarly, solving the inequality [(40.4)](#equation-eq-hugo4)  for $ a $  leads to\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo22a'></a>\n",
    "$$\n",
    "a = \\max\\left\\{0, p'^{-1} \\left({ 1 \\over \\beta (V^e - V^u)  } \\right) \\right\\}. \\tag{40.10}\n",
    "$$\n",
    "\n",
    "When we specialize [(40.10)](#equation-eq-hugo22a) to the   functional\n",
    "form for $ p(a) $ used by Hopenhayn and Nicolini, we obtain\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo22'></a>\n",
    "$$\n",
    "a = \\max\\left\\{0, {\\log[r \\beta (V^e - V^u)] \\over r } \\right\\}. \\tag{40.11}\n",
    "$$\n",
    "\n",
    "Formulas [(40.9)](#equation-eq-hugo21)  and [(40.11)](#equation-eq-hugo22)  express $ (c,a) $ as functions\n",
    "of $ V $ and  the continuation value $ V^u $.\n",
    "\n",
    "Using these functions\n",
    "allows us to write the Bellman equation in $ C(V) $ as\n",
    "\n",
    "\n",
    "<a id='equation-eq-hugo23'></a>\n",
    "$$\n",
    "C(V)  = \\min_{V^u} \\left\\{ c + \\beta [1 - p(a)] C(V^u) \\right\\} \\tag{40.12}\n",
    "$$\n",
    "\n",
    "where $ c $ and $ a $ are given by equations [(40.9)](#equation-eq-hugo21)  and [(40.11)](#equation-eq-hugo22)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6cb73c5",
   "metadata": {},
   "source": [
    "### Python Computations\n",
    "\n",
    "We’ll approximate the planner’s optimal cost function  with cubic splines.\n",
    "\n",
    "To do this, we’ll load some useful modules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "781762a5",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import scipy as sp\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d90dc73b",
   "metadata": {},
   "source": [
    "We first create a class to set up a particular parametrization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d845a9ec",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "class params_instance:\n",
    "\n",
    "    def __init__(self,\n",
    "                r,\n",
    "                β = 0.999,\n",
    "                σ = 0.500,\n",
    "                w = 100,\n",
    "                n_grid = 50):\n",
    "\n",
    "        self.β,self.σ,self.w,self.r = β,σ,w,r\n",
    "        self.n_grid = n_grid\n",
    "        uw = self.w**(1-self.σ)/(1-self.σ)  #Utility from consuming all wage\n",
    "        self.Ve = uw/(1-β)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d38d30ac",
   "metadata": {},
   "source": [
    "### Parameter Values\n",
    "\n",
    "For  the other parameters appearing in  the above Python code, we’ll calibrate parameter $ r $\n",
    "that pins down the function $ p(a) = 1 - \\exp(- r a) $ to match an observerd hazard rate –  the probability that an unemployed worker finds a job each  –  in US data.\n",
    "\n",
    "In particular, we seek an $ r $ so that in  autarky `p(a(r)) = 0.1`, where `a` is the optimal search effort.\n",
    "\n",
    "First, we create some helper functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "051e1ef8",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "# The probability of finding a job given search effort, a and parameter r.\n",
    "def p(a,r):\n",
    "    return 1-np.exp(-r*a)\n",
    "\n",
    "def invp_prime(x,r):\n",
    "    return -np.log(x/r)/r\n",
    "\n",
    "def p_prime(a,r):\n",
    "    return r*np.exp(-r*a)\n",
    "\n",
    "# The utiliy function\n",
    "def u(self,c):\n",
    "    return (c**(1-self.σ))/(1-self.σ)\n",
    "\n",
    "def u_inv(self,x):\n",
    "    return ((1-self.σ)*x)**(1/(1-self.σ))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e574c593",
   "metadata": {},
   "source": [
    "Recall that under  autarky the value for an unemployed worker\n",
    "satisfies  the Bellman equation\n",
    "\n",
    "\n",
    "<a id='equation-eq-yad1'></a>\n",
    "$$\n",
    "V^u = \\max_{a} \\{u(0) - a + \\beta\\left[p_{r}(a)V^e + (1-p_{r}(a))V^u\\right]\\} \\tag{40.13}\n",
    "$$\n",
    "\n",
    "At the optimal choice of $ a $, we have first-order necessary condition:\n",
    "\n",
    "\n",
    "<a id='equation-eq-yad2'></a>\n",
    "$$\n",
    "\\beta p_{r}'(a)[V^e - V^u] \\leq 1 \\tag{40.14}\n",
    "$$\n",
    "\n",
    "with equality when a >0.\n",
    "\n",
    "Given a value of parameter  $ \\bar{r} $, we can solve the autarky problem as follows:\n",
    "\n",
    "1. Guess $ V^u \\in \\mathbb{R}^{+} $  \n",
    "1. Given $ V^u $, use the FOC [(40.14)](#equation-eq-yad2) to calculate the implied optimal search effort $ a $  \n",
    "1. Evaluate the difference between the LHS and RHS of the Bellman equation [(40.13)](#equation-eq-yad1)  \n",
    "1. Update guess for $ V^u $ accordingly, then return  to 2) and repeat until the Bellman equation is satisfied.  \n",
    "\n",
    "\n",
    "For a given $ r $ and guess $ V^u $,\n",
    "the function `Vu_error` calculates the error in the Bellman equation under the optimal search intensity.\n",
    "\n",
    "We’ll  soon use this as an input to computing $ V^u $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "ae89881c",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "# The error in the Bellman equation that requires equality at\n",
    "# the optimal choices.\n",
    "def Vu_error(self,Vu,r):\n",
    "    β= self.β\n",
    "    Ve = self.Ve\n",
    "\n",
    "    a = invp_prime(1/(β*(Ve-Vu)),r)\n",
    "    error = u(self,0) -a + β*(p(a,r)*Ve + (1-p(a,r))*Vu) - Vu\n",
    "    return error"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "af389579",
   "metadata": {},
   "source": [
    "Since the calibration exercise is to match the hazard rate under autarky to the data, we must find a parameter  $ r $ to match `p(a,r) = 0.1`.\n",
    "\n",
    "The function below `r_error` calculates, for a given guess of $ r $ the difference between the model implied equilibrium hazard rate and 0.1.\n",
    "\n",
    "We’ll use this to compute   a calibrated $ r^* $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d782e859",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "# The error of our p(a^*) relative to our calibration target\n",
    "def r_error(self,r):\n",
    "    β = self.β\n",
    "    Ve = self.Ve\n",
    "\n",
    "    Vu_star = sp.optimize.fsolve(Vu_error_Λ,15000,args = (r))\n",
    "    a_star = invp_prime(1/(β*(Ve-Vu_star)),r) # Assuming a>0\n",
    "    return    p(a_star,r) - 0.1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f9fb0905",
   "metadata": {},
   "source": [
    "Now, let us create an instance of the model with our parametrization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "449913fa",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "params = params_instance(r = 1e-2)\n",
    "# Create some lambda functions useful for fsolve function\n",
    "Vu_error_Λ =  lambda Vu,r: Vu_error(params,Vu,r)\n",
    "r_error_Λ =  lambda r: r_error(params,r)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8bd0095e",
   "metadata": {},
   "source": [
    "We want to compute an  $ r $ that is consistent with the hazard rate 0.1 in autarky.\n",
    "\n",
    "To do so, we will use a bisection strategy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "9d6a3b4d",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "r_calibrated = sp.optimize.brentq(r_error_Λ,1e-10,1-1e-10)\n",
    "print(f\"Parameter to match 0.1 hazard rate: r = {r_calibrated}\")\n",
    "\n",
    "Vu_aut = sp.optimize.fsolve(Vu_error_Λ,15000,args = (r_calibrated))[0]\n",
    "a_aut = invp_prime(1/(params.β*(params.Ve-Vu_aut)),r_calibrated)\n",
    "\n",
    "print(f\"Check p at r: {p(a_aut,r_calibrated)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5e23bea0",
   "metadata": {},
   "source": [
    "Now that we have calibrated our the parameter $ r $, we can continue with solving the  model with private information."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "59916ede",
   "metadata": {},
   "source": [
    "### Computation under Private Information\n",
    "\n",
    "Our approach to solving the full model follows ideas of  Judd (1998) [[Judd, 1998](https://python-advanced.quantecon.org/zreferences.html#id169)], who uses a polynomial to approximate the value function and a numerical optimizer to perform the optimization at each iteration.\n",
    "\n",
    ">**Note**\n",
    ">\n",
    ">For further details of the Judd (1998) [[Judd, 1998](https://python-advanced.quantecon.org/zreferences.html#id169)] method, see [[Ljungqvist and Sargent, 2018](https://python-advanced.quantecon.org/zreferences.html#id176)], Section 5.7.\n",
    "\n",
    "We will use cubic splines to interpolate across a pre-set grid of points to approximate the value function.\n",
    "\n",
    "Our strategy involves finding a function $ C(V) $ – the expected cost of giving the worker value $ V $ – that satisfies the Bellman equation:\n",
    "\n",
    "\n",
    "<a id='equation-eq-yad3'></a>\n",
    "$$\n",
    "C(V) = \\min_{c,a,V^u} \\{c + \\beta\\left[1-p(a)\\right]C(V^u)\\} \\tag{40.15}\n",
    "$$\n",
    "\n",
    "Notice that in equations [(40.9)](#equation-eq-hugo21) and [(40.11)](#equation-eq-hugo22), we have analytical solutions of $ c $ and $ a $ in terms of  promised value $ V $ and $ V^u $ (and other parameters).\n",
    "\n",
    "We can substitute these equations for $ c $ and $ a $ and obtain the functional equation [(40.12)](#equation-eq-hugo23)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "44462ecb",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def calc_c(self,Vu,V,a):\n",
    "    '''\n",
    "    Calculates the optimal consumption choice coming from the constraint of the insurer's problem\n",
    "    (which is also a Bellman equation)\n",
    "    '''\n",
    "    β,Ve,r = self.β,self.Ve,self.r\n",
    "\n",
    "    c = u_inv(self,V + a - β*(p(a,r)*Ve + (1-p(a,r))*Vu))\n",
    "    return c\n",
    "\n",
    "def calc_a(self,Vu):\n",
    "    '''\n",
    "    Calculates the optimal effort choice coming from the worker's effort optimality condition.\n",
    "    '''\n",
    "\n",
    "    r,β,Ve = self.r,self.β,self.Ve\n",
    "\n",
    "    a_temp = np.log(r*β*(Ve - Vu))/r\n",
    "    a = max(0,a_temp)\n",
    "    return a"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11ab31ec",
   "metadata": {},
   "source": [
    "With these analytical solutions for optimal $ c $ and $ a $ in hand, we can reduce the minimization to  [(40.12)](#equation-eq-hugo23)  in the single variable\n",
    "$ V^u $.\n",
    "\n",
    "With this in hand, we have our algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11f939cf",
   "metadata": {},
   "source": [
    "### Algorithm\n",
    "\n",
    "1. Fix a set of grid points $ grid_V $ for $ V $   and $ Vu_{grid} $ for $ V^u $  \n",
    "1. Guess a function $ C_0(V) $ that is evaluated at a grid $ grid_V $.  \n",
    "1. For each point in $ grid_V $ find the $ V^u $ that minimizes the expression on  right side  of [(40.12)](#equation-eq-hugo23). We find the minimum by evaluating the right side of [(40.12)](#equation-eq-hugo23) at each point in $ Vu_{grid} $  and then finding the minimum using cubic splines.  \n",
    "1. Evaluating the minimum across all points in $ grid_V $ gives you another function $ C_1(V) $.  \n",
    "1. If $ C_0(V) $ and $ C_1(V) $ are sufficiently different, then repeat steps 3-4 again. Otherwise, we are done.  \n",
    "1. Thus, the iterations are $ C_{j+1}(V) = \\min_{c,a, V^u} \\{c - \\beta [1 - p(a) ] C_j(V)\\} $.  \n",
    "\n",
    "\n",
    "The function `iterate_C` below executes step 3 in the above algorithm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a34bf1f8",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "# Operator iterate_C that calculates the next iteration of the cost function.\n",
    "def iterate_C(self,C_old,Vu_grid):\n",
    "\n",
    "    '''\n",
    "    We solve the model by minimising the value function across a grid of possible promised values.\n",
    "    '''\n",
    "    β,r,n_grid = self.β,self.r,self.n_grid\n",
    "\n",
    "    C_new = np.zeros(n_grid)\n",
    "    cons_star = np.zeros(n_grid)\n",
    "    a_star = np.zeros(n_grid)\n",
    "    V_star = np.zeros(n_grid)\n",
    "\n",
    "    C_new2 = np.zeros(n_grid)\n",
    "    V_star2 = np.zeros(n_grid)\n",
    "\n",
    "    for V_i in range(n_grid):\n",
    "        C_Vi_temp = np.zeros(n_grid)\n",
    "        cons_Vi_temp = np.zeros(n_grid)\n",
    "        a_Vi_temp = np.zeros(n_grid)\n",
    "\n",
    "        for Vu_i in range(n_grid):\n",
    "            a_i = calc_a(self,Vu_grid[Vu_i])\n",
    "            c_i = calc_c(self,Vu_grid[Vu_i],Vu_grid[V_i],a_i)\n",
    "\n",
    "            C_Vi_temp[Vu_i] = c_i + β*(1-p(a_i,r))*C_old[Vu_i]\n",
    "            cons_Vi_temp[Vu_i] = c_i\n",
    "            a_Vi_temp[Vu_i] = a_i\n",
    "\n",
    "        # Interpolate across the grid to get better approximation of the minimum\n",
    "        C_Vi_temp_interp = sp.interpolate.interp1d(Vu_grid,C_Vi_temp, kind = 'cubic')\n",
    "        cons_Vi_temp_interp = sp.interpolate.interp1d(Vu_grid,cons_Vi_temp, kind = 'cubic')\n",
    "        a_Vi_temp_interp = sp.interpolate.interp1d(Vu_grid,a_Vi_temp, kind = 'cubic')\n",
    "\n",
    "        res = sp.optimize.minimize_scalar(C_Vi_temp_interp,method='bounded',bounds = (Vu_min,Vu_max))\n",
    "        V_star[V_i] = res.x\n",
    "        C_new[V_i] = res.fun\n",
    "\n",
    "        # Save the associated consumpton and search policy functions as well\n",
    "        cons_star[V_i] = cons_Vi_temp_interp(V_star[V_i])\n",
    "        a_star[V_i] = a_Vi_temp_interp(V_star[V_i])\n",
    "\n",
    "    return C_new,V_star,cons_star,a_star"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3f4d2bb",
   "metadata": {},
   "source": [
    "The following code executes steps 4 and 5 in the Algorithm  until convergence to a function $ C^*(V) $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "642633e0",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def solve_incomplete_info_model(self,Vu_grid,Vu_aut,tol = 1e-6,max_iter = 10000):\n",
    "    iter = 0\n",
    "    error = 1\n",
    "\n",
    "    C_init = np.ones(self.n_grid)*0\n",
    "    C_old = np.copy(C_init)\n",
    "\n",
    "    while iter<max_iter and error >tol:\n",
    "        C_new,V_new,cons_star,a_star = iterate_C(self,C_old,Vu_grid)\n",
    "        error = np.max(np.abs(C_new - C_old))\n",
    "\n",
    "        #Only print the iterations every 50 steps\n",
    "        if iter % 50 ==0:\n",
    "            print(f\"Iteration: {iter}, error:{error}\")\n",
    "        C_old = np.copy(C_new)\n",
    "        iter+=1\n",
    "\n",
    "    return C_new,V_new,cons_star,a_star"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c5b4a84",
   "metadata": {},
   "source": [
    "## Outcomes\n",
    "\n",
    "Using the above functions, we create another instance of the parameters with our calibrated parameter $ r $."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5d3e4ed1",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "##? Create another instance with the correct r now\n",
    "params = params_instance(r = r_calibrated)\n",
    "\n",
    "#Set up grid\n",
    "Vu_min = Vu_aut\n",
    "Vu_max = params.Ve - 1/(params.β*p_prime(0,params.r))\n",
    "Vu_grid = np.linspace(Vu_min,Vu_max,params.n_grid)\n",
    "\n",
    "#Solve model\n",
    "C_star,V_star,cons_star,a_star = solve_incomplete_info_model(params,Vu_grid,Vu_aut,tol = 1e-6,max_iter = 10000) #,cons_star,a_star\n",
    "\n",
    "# Since we have the policy functions in grid form, we will interpolate them to be able to\n",
    "# evaluate any promised value\n",
    "cons_star_interp = sp.interpolate.interp1d(Vu_grid,cons_star)\n",
    "a_star_interp = sp.interpolate.interp1d(Vu_grid,a_star)\n",
    "V_star_interp = sp.interpolate.interp1d(Vu_grid,V_star)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "65d99a51",
   "metadata": {},
   "source": [
    "### Replacement Ratios and Continuation Values\n",
    "\n",
    "Let’s graph the  replacement ratio ($ c/w $) and  search effort $ a $ as  functions of the duration of unemployment.\n",
    "\n",
    "We’ll do this for three levels of $ V_0 $, the lowest being the autarky value $ V_{\\rm aut} $.\n",
    "\n",
    "We accomplish this by using the optimal policy functions `V_star`, `cons_star` and `a_star` computed above as well the following  iterative procedure:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "647b0179",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "# Replacement ratio and effort as a function of unemployment duration\n",
    "T_max = 52\n",
    "Vu_t = np.empty((T_max,3))\n",
    "cons_t = np.empty((T_max-1,3))\n",
    "a_t = np.empty((T_max-1,3))\n",
    "\n",
    "# Calculate the replacement ratios depending on different initial\n",
    "# promised values\n",
    "Vu_0_hold = np.array([Vu_aut,16942,17000])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "3f348e1b",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "for i,Vu_0, in enumerate(Vu_0_hold):\n",
    "    Vu_t[0,i] = Vu_0\n",
    "    for t in range(1,T_max):\n",
    "        cons_t[t-1,i] = cons_star_interp(Vu_t[t-1,i])\n",
    "        a_t[t-1,i] = a_star_interp(Vu_t[t-1,i])\n",
    "        Vu_t[t,i] = V_star_interp(Vu_t[t-1,i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "5e968be0",
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "fontSize = 10\n",
    "plt.rc('font', size=fontSize)          # controls default text sizes\n",
    "plt.rc('axes', titlesize=fontSize)     # fontsize of the axes title\n",
    "plt.rc('axes', labelsize=fontSize)    # fontsize of the x and y labels\n",
    "plt.rc('xtick', labelsize=fontSize)    # fontsize of the tick labels\n",
    "plt.rc('ytick', labelsize=fontSize)    # fontsize of the tick labels\n",
    "plt.rc('legend', fontsize=fontSize)    # legend fontsize\n",
    "\n",
    "f1 = plt.figure(figsize = (8,8))\n",
    "plt.subplot(2,1,1)\n",
    "plt.plot(range(T_max-1),cons_t[:,0]/params.w,label = '$V^u_0$ = 16759 (aut)',color = 'red')\n",
    "plt.plot(range(T_max-1),cons_t[:,1]/params.w,label = '$V^u_0$ = 16942',color = 'blue')\n",
    "plt.plot(range(T_max-1),cons_t[:,2]/params.w,label = '$V^u_0$ = 17000',color = 'green')\n",
    "plt.ylabel(\"Replacement ratio (c/w)\")\n",
    "plt.legend()\n",
    "plt.title(\"Optimal replacement ratio\")\n",
    "\n",
    "plt.subplot(2,1,2)\n",
    "plt.plot(range(T_max-1),a_t[:,0],color = 'red')\n",
    "plt.plot(range(T_max-1),a_t[:,1],color = 'blue')\n",
    "plt.plot(range(T_max-1),a_t[:,2],color = 'green')\n",
    "plt.ylim(0,320)\n",
    "plt.ylabel(\"Optimal search effort (a)\")\n",
    "plt.xlabel(\"Duration of unemployment\")\n",
    "plt.title(\"Optimal search effort\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98b79b33",
   "metadata": {},
   "source": [
    "For an initial promised value $ V^u = V_{\\rm aut} $, the planner chooses the autarky level of $ 0 $ for the replacement ratio and instructs the worker to search at the autarky search intensity, regardless of the duration of unemployment\n",
    "\n",
    "But for $ V^u > V_{\\rm aut} $, the planner makes the replacement ratio  decline and search effort increase  with the duration of unemployment."
   ]
  },
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   "cell_type": "markdown",
   "id": "4e3f62c8",
   "metadata": {},
   "source": [
    "### Interpretations\n",
    "\n",
    "The  downward slope of the replacement ratio when $ V^u > V_{\\rm aut} $ is a consequence of the\n",
    "planner’s limited information about the worker’s search effort.\n",
    "\n",
    "By providing the worker with a duration-dependent schedule of replacement ratios, the planner induces the worker in effect to reveal\n",
    "his/her search effort to the planner.\n",
    "\n",
    "We saw earlier that with full  information, the\n",
    "planner would smooth consumption\n",
    "over an unemployment spell by\n",
    "keeping the  replacement ratio constant.\n",
    "\n",
    "With private information, the planner can’t observe the worker’s search effort\n",
    "and therefore makes   the replacement ratio  fall.\n",
    "\n",
    "Evidently, search\n",
    "effort rise as the duration of unemployment increases, especially\n",
    "early in an unemployment spell.\n",
    "\n",
    "There is a **carrot-and-stick**\n",
    "aspect to the replacement rate and search effort  schedules:\n",
    "\n",
    "- the **carrot** occurs in the forms of high compensation and low search\n",
    "  effort early in an unemployment spell.  \n",
    "- the **stick** occurs in the low compensation and high effort  later in\n",
    "  the spell.  \n",
    "\n",
    "\n",
    "We shall encounter a related  carrot-and-stick feature in our other lectures about dynamic programming squared.\n",
    "\n",
    "The planner offers declining benefits and induces increased search\n",
    "effort as the duration of an unemployment spell rises in order to provide an\n",
    "unemployed worker with proper incentives, not to punish an unlucky worker\n",
    "who has been unemployed for a long time.\n",
    "\n",
    "The planner believes that a worker who has been unemployed a long time is unlucky, not that he has\n",
    "done anything wrong (e.g.,that he has not lived up to the contract).\n",
    "\n",
    "Indeed, the\n",
    "contract is designed to induce the unemployed workers to search in\n",
    "the way the planner expects.\n",
    "\n",
    "The falling consumption and rising\n",
    "search effort of the unlucky ones with long unemployment spells are\n",
    "simply  costs that have to be paid in order to  provide\n",
    "proper incentives."
   ]
  }
 ],
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