From 262f1803ac3871cd57a6ffafd973ff02192e18d9 Mon Sep 17 00:00:00 2001
From: kp992 <kpl992@outlook.com>
Date: Mon, 4 Mar 2024 11:49:44 +0530
Subject: [PATCH 1/5] Use np.interp instead of interpolation.interp

---
 lectures/BCG_complete_mkts.md   |  1 -
 lectures/BCG_incomplete_mkts.md | 53 +++++++++++++++++----------------
 lectures/coase.md               | 42 +++++++++++---------------
 lectures/lucas_model.md         | 33 +++++++++-----------
 lectures/opt_tax_recur.md       |  2 --
 5 files changed, 60 insertions(+), 71 deletions(-)

diff --git a/lectures/BCG_complete_mkts.md b/lectures/BCG_complete_mkts.md
index 0e77e277..dac8dea0 100644
--- a/lectures/BCG_complete_mkts.md
+++ b/lectures/BCG_complete_mkts.md
@@ -31,7 +31,6 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 tags: [hide-output]
 ---
 !pip install --upgrade quantecon
-!pip install interpolation
 !conda install -y -c plotly plotly plotly-orca
 ```
 
diff --git a/lectures/BCG_incomplete_mkts.md b/lectures/BCG_incomplete_mkts.md
index 78cb5578..90d3a53e 100644
--- a/lectures/BCG_incomplete_mkts.md
+++ b/lectures/BCG_incomplete_mkts.md
@@ -3,8 +3,10 @@ jupytext:
   text_representation:
     extension: .md
     format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.16.1
 kernelspec:
-  display_name: Python 3
+  display_name: Python 3 (ipykernel)
   language: python
   name: python3
 ---
@@ -26,12 +28,10 @@ kernelspec:
 
 In addition to what's in Anaconda, this lecture will need the following libraries:
 
-```{code-cell} ipython
----
-tags: [hide-output]
----
+```{code-cell} ipython3
+:tags: [hide-output]
+
 !pip install --upgrade quantecon
-!pip install interpolation
 !conda install -y -c plotly plotly plotly-orca
 ```
 
@@ -703,15 +703,14 @@ Parameters include:
 - $\beta$: Discount factor. Default value is 0.96.
 - bound: Bound for truncated normal distribution. Default value is 3.
 
-```{code-cell} ipython
+```{code-cell} ipython3
 import numpy as np
 from scipy.stats import truncnorm
 from scipy.integrate import quad
 from numba import njit
-from interpolation import interp
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 class BCG_incomplete_markets:
 
     # init method or constructor
@@ -784,7 +783,7 @@ class BCG_incomplete_markets:
         rv = truncnorm(ta, tb, loc=𝜇, scale=𝜎)
         𝜖_range = np.linspace(ta, tb, 1000000)
         pdf_range = rv.pdf(𝜖_range)
-        self.g = njit(lambda 𝜖: interp(𝜖_range, pdf_range, 𝜖))
+        self.g = njit(lambda 𝜖: np.interp(𝜖, 𝜖_range, pdf_range))
 
 
     #*************************************************************
@@ -1206,14 +1205,14 @@ Below we show some examples computed with the class `BCG_incomplete markets`.
 In the first example, we set up an instance of the BCG incomplete
 markets model with default parameter values.
 
-```{code-cell} python3
-:tags: ["hide-output"]
+```{code-cell} ipython3
+:tags: [hide-output]
 
 mdl = BCG_incomplete_markets()
 kss,bss,Vss,qss,pss,c10ss,c11ss,c20ss,c21ss,𝜃1ss = mdl.solve_eq(print_crit=False)
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 print(-kss+qss+pss*bss)
 print(Vss)
 print(𝜃1ss)
@@ -1229,7 +1228,7 @@ Thus, let’s see if the firm is actually maximizing its firm value given
 the equilibrium pricing function $q(k,b)$ for equity and
 $p(k,b)$ for  bonds.
 
-```{code-cell} python3
+```{code-cell} ipython3
 kgrid, bgrid, Vgrid, Qgrid, Pgrid = mdl.eq_valuation(c10ss, c11ss, c20ss, c21ss,N=30)
 
 print('Maximum valuation of the firm value in the (k,B) grid: {:.5f}'.format(Vgrid.max()))
@@ -1246,7 +1245,7 @@ Below we will plot the firm’s value as a function of $k,b$.
 We’ll also plot the equilibrium price functions $q(k,b)$ and
 $p(k,b)$.
 
-```{code-cell} python3
+```{code-cell} ipython3
 from IPython.display import Image
 import matplotlib.pyplot as plt
 from mpl_toolkits import mplot3d
@@ -1367,7 +1366,7 @@ $$
 
 The function also outputs agent 1’s bond holdings $\xi_1$.
 
-```{code-cell} python3
+```{code-cell} ipython3
 def off_eq_check(mdl,kss,bss,e=0.1):
     # Big K and big B
     k = kss
@@ -1603,7 +1602,7 @@ structure for the firm.
 We first check the case in which $b^{**} = b^* - e$ where
 $e = 0.1$:
 
-```{code-cell} python3
+```{code-cell} ipython3
 #====================== Experiment 1 ======================#
 Ve1,ke1,be1,pe1,qe1,c10e1,c11e1,c20e1,c21e1,𝜉1e1 = off_eq_check(mdl,
                                                                 kss,
@@ -1655,7 +1654,7 @@ Therefore, $(k^*,b^{*}-e)$ would not be an equilibrium.
 
 Next, we check for $b^{**} = b^* + e$.
 
-```{code-cell} python3
+```{code-cell} ipython3
 #====================== Experiment 2 ======================#
 Ve2,ke2,be2,pe2,qe2,c10e2,c11e2,c20e2,c21e2,𝜉1e2 = off_eq_check(mdl,
                                                                 kss,
@@ -1708,7 +1707,7 @@ want to hold corporate debt.
 
 For example,  $\xi^1 > 0$:
 
-```{code-cell} python3
+```{code-cell} ipython3
 print('Bond holdings of agent 1: {:.3f}'.format(𝜉1e2))
 ```
 
@@ -1726,7 +1725,7 @@ It is also interesting to look at the equilibrium price functions
 $q(k,b)$ and $p(k,b)$ faced by firms in our rational
 expectations equilibrium.
 
-```{code-cell} python3
+```{code-cell} ipython3
 # Equity Valuation
 fig = go.Figure(data=[go.Scatter3d(x=[kss],
                                    y=[bss],
@@ -1756,7 +1755,7 @@ Image(fig.to_image(format="png"))
 # code locally
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 # Bond Valuation
 fig = go.Figure(data=[go.Scatter3d(x=[kss],
                                    y=[bss],
@@ -1815,8 +1814,8 @@ $$
 The function `valuations_by_agent` is used in calculating these
 valuations.
 
-```{code-cell} python3
-:tags: ["hide-output"]
+```{code-cell} ipython3
+:tags: [hide-output]
 
 # Lists for storage
 wlist = []
@@ -1864,7 +1863,7 @@ for i in range(10):
     p2list.append(P2)
 ```
 
-```{code-cell} python3
+```{code-cell} ipython3
 # Plot
 fig, ax = plt.subplots(3,2,figsize=(12,12))
 ax[0,0].plot(wlist,klist)
@@ -1909,7 +1908,7 @@ Now let’s see how the two types of agents value bonds and equities,
 keeping in mind that the type that values the asset highest determines
 the equilibrium price (and thus the pertinent set of Big $C$’s).
 
-```{code-cell} python3
+```{code-cell} ipython3
 # Comparing the prices
 fig, ax = plt.subplots(1,3,figsize=(16,6))
 
@@ -1943,3 +1942,7 @@ Agents of type 2 value bonds more highly (they want more hedging).
 
 Taken together with our earlier plot of equity holdings, these graphs confirm our earlier conjecture that while both type
 of agents hold equities, only agents of type 2 holds bonds.
+
+```{code-cell} ipython3
+
+```
diff --git a/lectures/coase.md b/lectures/coase.md
index 41fe1c99..6bd33285 100644
--- a/lectures/coase.md
+++ b/lectures/coase.md
@@ -3,8 +3,10 @@ jupytext:
   text_representation:
     extension: .md
     format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.16.1
 kernelspec:
-  display_name: Python 3
+  display_name: Python 3 (ipykernel)
   language: python
   name: python3
 ---
@@ -24,14 +26,7 @@ kernelspec:
 :depth: 2
 ```
 
-In addition to what's in Anaconda, this lecture will need the following libraries:
-
-```{code-cell} ipython
----
-tags: [hide-output]
----
-!pip install interpolation
-```
++++
 
 ## Overview
 
@@ -57,12 +52,10 @@ Couldn't the associated within-firm planning be done more efficiently by the mar
 
 We'll use the following imports:
 
-```{code-cell} ipython
+```{code-cell} ipython3
 import numpy as np
 import matplotlib.pyplot as plt
-%matplotlib inline
 from scipy.optimize import fminbound
-from interpolation import interp
 ```
 
 ### Why Firms Exist
@@ -447,7 +440,7 @@ At each iterate, we will use continuous piecewise linear interpolation of functi
 
 To begin, here's a class to store primitives and a grid:
 
-```{code-cell} python3
+```{code-cell} ipython3
 class ProductionChain:
 
     def __init__(self,
@@ -464,7 +457,7 @@ Now let's implement and iterate with $T$ until convergence.
 Recalling that our initial condition must lie in $\mathcal P$, we set
 $p_0 = c$
 
-```{code-cell} python3
+```{code-cell} ipython3
 def compute_prices(pc, tol=1e-5, max_iter=5000):
     """
     Compute prices by iterating with T
@@ -481,7 +474,7 @@ def compute_prices(pc, tol=1e-5, max_iter=5000):
 
     while error > tol and i < max_iter:
         for j, s in enumerate(grid):
-            Tp = lambda t: delta * interp(grid, p, t) + c(s - t)
+            Tp = lambda t: delta * np.interp(t, grid, p) + c(s - t)
             new_p[j] = Tp(fminbound(Tp, 0, s))
         error = np.max(np.abs(p - new_p))
         p = new_p
@@ -492,14 +485,14 @@ def compute_prices(pc, tol=1e-5, max_iter=5000):
     else:
         print(f"Warning: iteration hit upper bound {max_iter}")
 
-    p_func = lambda x: interp(grid, p, x)
+    p_func = lambda x: np.interp(x, grid, p)
     return p_func
 ```
 
 The next function computes optimal choice of upstream boundary and range of
 task implemented for a firm face price function p_function and with downstream boundary $s$.
 
-```{code-cell} python3
+```{code-cell} ipython3
 def optimal_choices(pc, p_function, s):
     """
     Takes p_func as the true function, minimizes on [0,s]
@@ -524,7 +517,7 @@ their respective upstream boundary, treating the previous firm's upstream bounda
 
 In doing so, we start with firm 1, who has downstream boundary $s=1$.
 
-```{code-cell} python3
+```{code-cell} ipython3
 def compute_stages(pc, p_function):
     s = 1.0
     transaction_stages = [s]
@@ -539,7 +532,7 @@ Let's try this at the default parameters.
 The next figure shows the equilibrium price function, as well as the
 boundaries of firms as vertical lines
 
-```{code-cell} python3
+```{code-cell} ipython3
 pc = ProductionChain()
 p_star = compute_prices(pc)
 
@@ -558,7 +551,7 @@ plt.show()
 Here's the function $\ell^*$, which shows how large a firm with
 downstream boundary $s$ chooses to be
 
-```{code-cell} python3
+```{code-cell} ipython3
 ell_star = np.empty(pc.n)
 for i, s in enumerate(pc.grid):
     t, e = optimal_choices(pc, p_star, s)
@@ -590,9 +583,9 @@ Check your intuition by computing the number of firms at delta in (1.01, 1.05, 1
 ```{solution-start}  coa_ex1
 :class: dropdown
 ```
-Here is one solution 
+Here is one solution
 
-```{code-cell} python3
+```{code-cell} ipython3
 for delta in (1.01, 1.05, 1.1):
 
    pc = ProductionChain(delta=delta)
@@ -601,6 +594,7 @@ for delta in (1.01, 1.05, 1.1):
    num_firms = len(transaction_stages)
    print(f"When delta={delta} there are {num_firms} firms")
 ```
+
 ```{solution-end}
 ```
 
@@ -636,7 +630,7 @@ associated with diminishing returns to management --- which induce convexity ---
 
 Here's one way to compute and graph value added across firms
 
-```{code-cell} python3
+```{code-cell} ipython3
 pc = ProductionChain()
 p_star = compute_prices(pc)
 stages = compute_stages(pc, p_star)
@@ -654,4 +648,4 @@ plt.show()
 ```
 
 ```{solution-end}
-```
\ No newline at end of file
+```
diff --git a/lectures/lucas_model.md b/lectures/lucas_model.md
index cc931e81..5d9f8496 100644
--- a/lectures/lucas_model.md
+++ b/lectures/lucas_model.md
@@ -3,8 +3,10 @@ jupytext:
   text_representation:
     extension: .md
     format_name: myst
+    format_version: 0.13
+    jupytext_version: 1.16.1
 kernelspec:
-  display_name: Python 3
+  display_name: Python 3 (ipykernel)
   language: python
   name: python3
 ---
@@ -27,14 +29,7 @@ kernelspec:
 :depth: 2
 ```
 
-In addition to what's in Anaconda, this lecture will need the following libraries:
-
-```{code-cell} ipython
----
-tags: [hide-output]
----
-!pip install interpolation
-```
++++
 
 ## Overview
 
@@ -52,13 +47,11 @@ Another difference to our [first asset pricing lecture](https://python-intro.qua
 
 Let's start with some imports:
 
-```{code-cell} ipython
+```{code-cell} ipython3
 import numpy as np
-from interpolation import interp
 from numba import njit, prange
 from scipy.stats import lognorm
 import matplotlib.pyplot as plt
-%matplotlib inline
 ```
 
 ## The Lucas Model
@@ -401,7 +394,7 @@ Utility will take the isoelastic form $u(c) = c^{1-\gamma}/(1-\gamma)$, where $\
 
 We will set up a `LucasTree` class to hold parameters of the model
 
-```{code-cell} python3
+```{code-cell} ipython3
 class LucasTree:
     """
     Class to store parameters of the Lucas tree model.
@@ -436,7 +429,7 @@ class LucasTree:
 The following function takes an instance of the `LucasTree` and generates a
 jitted version of the Lucas operator
 
-```{code-cell} python3
+```{code-cell} ipython3
 def operator_factory(tree, parallel_flag=True):
 
     """
@@ -458,7 +451,7 @@ def operator_factory(tree, parallel_flag=True):
         """
 
         # Turn f into a function
-        Af = lambda x: interp(grid, f, x)
+        Af = lambda x: np.interp(x, grid, f)
 
         Tf = np.empty_like(f)
         # Apply the T operator to f using Monte Carlo integration
@@ -474,7 +467,7 @@ def operator_factory(tree, parallel_flag=True):
 To solve the model, we write a function that iterates using the Lucas operator
 to find the fixed point.
 
-```{code-cell} python3
+```{code-cell} ipython3
 def solve_model(tree, tol=1e-6, max_iter=500):
     """
     Compute the equilibrium price function associated with Lucas
@@ -505,7 +498,7 @@ def solve_model(tree, tol=1e-6, max_iter=500):
 
 Solving the model and plotting the resulting price function
 
-```{code-cell} python3
+```{code-cell} ipython3
 tree = LucasTree()
 price_vals = solve_model(tree)
 
@@ -552,7 +545,8 @@ Replicate {ref}`the figure <mass_lt_cb>` to show how discount factors affect pri
 ```{solution-start} lucas_ex1
 :class: dropdown
 ```
-```{code-cell} python3
+
+```{code-cell} ipython3
 fig, ax = plt.subplots(figsize=(10, 6))
 
 for β in (.95, 0.98):
@@ -566,5 +560,6 @@ ax.legend(loc='upper left')
 ax.set(xlabel='$y$', ylabel='price', xlim=(min(grid), max(grid)))
 plt.show()
 ```
+
 ```{solution-end}
-```
\ No newline at end of file
+```
diff --git a/lectures/opt_tax_recur.md b/lectures/opt_tax_recur.md
index 8c8815ec..929f48fc 100644
--- a/lectures/opt_tax_recur.md
+++ b/lectures/opt_tax_recur.md
@@ -31,7 +31,6 @@ In addition to what's in Anaconda, this lecture will need the following librarie
 tags: [hide-output]
 ---
 !pip install --upgrade quantecon
-!pip install interpolation
 ```
 
 ## Overview
@@ -73,7 +72,6 @@ import matplotlib.pyplot as plt
 from scipy.optimize import root
 from quantecon import MarkovChain
 from quantecon.optimize.nelder_mead import nelder_mead
-from interpolation import interp
 from numba import njit, prange, float64
 from numba.experimental import jitclass
 ```

From 5faa7b081a613272f9a6c642e2a5ba740376261e Mon Sep 17 00:00:00 2001
From: kp992 <kpl992@outlook.com>
Date: Sat, 9 Mar 2024 10:23:55 +0530
Subject: [PATCH 2/5] apply suggestions

---
 lectures/BCG_incomplete_mkts.md | 4 ----
 lectures/coase.md               | 2 --
 lectures/lucas_model.md         | 2 --
 3 files changed, 8 deletions(-)

diff --git a/lectures/BCG_incomplete_mkts.md b/lectures/BCG_incomplete_mkts.md
index 90d3a53e..91df9591 100644
--- a/lectures/BCG_incomplete_mkts.md
+++ b/lectures/BCG_incomplete_mkts.md
@@ -1942,7 +1942,3 @@ Agents of type 2 value bonds more highly (they want more hedging).
 
 Taken together with our earlier plot of equity holdings, these graphs confirm our earlier conjecture that while both type
 of agents hold equities, only agents of type 2 holds bonds.
-
-```{code-cell} ipython3
-
-```
diff --git a/lectures/coase.md b/lectures/coase.md
index 6bd33285..b34cec1a 100644
--- a/lectures/coase.md
+++ b/lectures/coase.md
@@ -26,8 +26,6 @@ kernelspec:
 :depth: 2
 ```
 
-+++
-
 ## Overview
 
 In 1937, Ronald Coase wrote a brilliant essay on the nature of the firm {cite}`coase1937nature`.
diff --git a/lectures/lucas_model.md b/lectures/lucas_model.md
index 5d9f8496..69900538 100644
--- a/lectures/lucas_model.md
+++ b/lectures/lucas_model.md
@@ -29,8 +29,6 @@ kernelspec:
 :depth: 2
 ```
 
-+++
-
 ## Overview
 
 As stated in an [earlier lecture](https://python-intro.quantecon.org/markov_asset.html), an asset is a claim on a stream of prospective payments.

From fb38c5562d7827ca8c69133c0f371445080b43c6 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Tue, 12 Mar 2024 10:06:08 +1100
Subject: [PATCH 3/5] re-enable build failures on warnings

---
 .github/workflows/ci.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 7934f3a5..787ab802 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -61,7 +61,7 @@ jobs:
         shell: bash -l {0}
         run: |
           rm -r _build/.doctrees
-          jb build lectures --path-output ./
+          jb build lectures --path-output ./ -nW --keep-going
       - name: Save Build as Artifact
         uses: actions/upload-artifact@v1
         with:

From cbd987617ec0742c7add7635eee713add851b157 Mon Sep 17 00:00:00 2001
From: mmcky <mamckay@gmail.com>
Date: Tue, 12 Mar 2024 10:30:14 +1100
Subject: [PATCH 4/5] MAINT: update ci.yml workflow versions and report upload

---
 .github/workflows/ci.yml | 16 +++++++++++-----
 1 file changed, 11 insertions(+), 5 deletions(-)

diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml
index 787ab802..f072929a 100644
--- a/.github/workflows/ci.yml
+++ b/.github/workflows/ci.yml
@@ -5,9 +5,9 @@ jobs:
     runs-on: ubuntu-latest
     steps:
       - name: Checkout
-        uses: actions/checkout@v2
+        uses: actions/checkout@v4
       - name: Setup Anaconda
-        uses: conda-incubator/setup-miniconda@v2
+        uses: conda-incubator/setup-miniconda@v3
         with:
           auto-update-conda: true
           auto-activate-base: true
@@ -35,7 +35,7 @@ jobs:
         shell: bash -l {0}
         run: pip list
       - name: Download "build" folder (cache)
-        uses: dawidd6/action-download-artifact@v2
+        uses: dawidd6/action-download-artifact@v3
         with:
           workflow: cache.yml
           branch: main
@@ -62,13 +62,19 @@ jobs:
         run: |
           rm -r _build/.doctrees
           jb build lectures --path-output ./ -nW --keep-going
+      - name: Upload Execution Reports (Download Notebooks)
+        uses: actions/upload-artifact@v4
+        if: failure()
+        with:
+          name: execution-reports
+          path: _build/jupyter/reports
       - name: Save Build as Artifact
-        uses: actions/upload-artifact@v1
+        uses: actions/upload-artifact@v4
         with:
           name: _build
           path: _build
       - name: Preview Deploy to Netlify
-        uses: nwtgck/actions-netlify@v1.1
+        uses: nwtgck/actions-netlify@v2
         with:
           publish-dir: '_build/html/'
           production-branch: master

From 794a3763f09544baf088003d27a58240379095a6 Mon Sep 17 00:00:00 2001
From: kp992 <kpl992@outlook.com>
Date: Tue, 12 Mar 2024 07:29:36 +0530
Subject: [PATCH 5/5] Fix interp issues

---
 .../opt_tax_recur/recursive_allocation.py            | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/lectures/_static/lecture_specific/opt_tax_recur/recursive_allocation.py b/lectures/_static/lecture_specific/opt_tax_recur/recursive_allocation.py
index 2ce9387e..c197f11c 100644
--- a/lectures/_static/lecture_specific/opt_tax_recur/recursive_allocation.py
+++ b/lectures/_static/lecture_specific/opt_tax_recur/recursive_allocation.py
@@ -131,8 +131,8 @@ def simulate(self, b0, s0, T, sHist=None):
         # Time 1 onward
         for t in range(1, T):
             s, x = sHist[t], xHist[t-1]
-            cHist[t] = interp(self.x_grid, self.c1[:, s], x)
-            nHist[t] = interp(self.x_grid, self.n1[:, s], x)
+            cHist[t] = np.interp(x, self.x_grid, self.c1[:, s])
+            nHist[t] = np.interp(x, self.x_grid, self.n1[:, s])
 
             τHist[t] = self.τ(cHist[t], nHist[t])
 
@@ -140,8 +140,8 @@ def simulate(self, b0, s0, T, sHist=None):
 
             c, n = np.empty((2, self.S))
             for sprime in range(self.S):
-                c[sprime] = interp(x_grid, self.c1[:, sprime], x)
-                n[sprime] = interp(x_grid, self.n1[:, sprime], x)
+                c[sprime] = np.interp(x, x_grid, self.c1[:, sprime])
+                n[sprime] = np.interp(x, x_grid, self.n1[:, sprime])
             Euc = π[sHist[t-1]] @ Uc(c, 1-n)
             RHist[t-1] = Uc(cHist[t-1], 1-nHist[t-1]) / (self.pref.β * Euc)
 
@@ -150,7 +150,7 @@ def simulate(self, b0, s0, T, sHist=None):
             
             if t < T-1:
                 sprime = sHist[t+1]
-                xHist[t] = interp(self.x_grid, self.xprime1[:, s, sprime], x)
+                xHist[t] = np.interp(x, self.x_grid, self.xprime1[:, s, sprime])
 
         return [cHist, nHist, Bhist, τHist, gHist, yHist, xHist, RHist]
 
@@ -209,7 +209,7 @@ def obj_V(z_sub, x, s, V, pref, π, g, x_grid, b0=None):
     # prepare Vprime vector
     Vprime = np.empty(S)
     for sprime in range(S):
-        Vprime[sprime] = interp(x_grid, V[:, sprime], xprime[sprime])
+        Vprime[sprime] = np.interp(xprime[sprime], x_grid, V[:, sprime])
 
     # compute the objective value
     obj = U(c, l) + β * π[s] @ Vprime