diff --git a/models/We-Would-Like-In-Econ-ARK/NKCross/Fig1_Bilbiie_2020.PNG b/models/We-Would-Like-In-Econ-ARK/NKCross/Fig1_Bilbiie_2020.PNG
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Binary files /dev/null and b/models/We-Would-Like-In-Econ-ARK/NKCross/Fig1_Bilbiie_2020.PNG differ
diff --git a/models/We-Would-Like-In-Econ-ARK/NKCross/NKCross.ipynb b/models/We-Would-Like-In-Econ-ARK/NKCross/NKCross.ipynb
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--- /dev/null
+++ b/models/We-Would-Like-In-Econ-ARK/NKCross/NKCross.ipynb
@@ -0,0 +1,184 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "d1e1394c-b184-48cb-aa98-978c8d770b18",
+   "metadata": {},
+   "source": [
+    "### New Keynesian Cross: Bilbiie 2020"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "12982804-4e21-4e49-adae-2c27cc037fa8",
+   "metadata": {},
+   "source": [
+    "### Motivation:\n",
+    "The idea of the Keynesian Cross and with it the Fiscal Multiplier which is $\\frac{1}{MPC}$ hinges on homogeneous agents. In case agents have different MPCs, the dynamics change.\n",
+    "\n",
+    "Bilbiie (2020) creates an analytical framework how to transfer the idea of the Keynesian Cross into new keynesian models with representative and heterogeneous agents (RANK, TANK and HANK). As a result, the exercise affords analytical insights into monetary, fiscal, and forward guidance multipliers."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d2418d70",
+   "metadata": {},
+   "source": [
+    "### New Keynesian Cross:\n",
+    "\n",
+    " \n",
+    "\\begin{align*}\n",
+    "\\text{PE curve: } C = C(Y,r) & \\text{ with } 0 < C_Y < 1; C_r <0 \\\\\n",
+    "\\omega & \\text{ MPC: slope of the PE}\\\\\n",
+    "\\Omega_D & \\text{ autonomous expenditure change: shift of the PE}\n",
+    "\\end{align*}\n",
+    "\n",
+    "Hence the multiplier is\n",
+    "$$\n",
+    "\\Omega = \\frac{\\Omega_D}{1-\\omega}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "59e2f621",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "from IPython.display import Image\n",
+    "Image(\"Fig1_Bilbiie_2020.png\")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "84436104",
+   "metadata": {},
+   "source": [
+    "### NK cross in RANK\n",
+    "\n",
+    "#### Proposition 1.\n",
+    "In RANK, the MPC $\\omega$, autonomous expenditure increase $\\Omega_D$ and multiplier $\\Omega$ (for an interest rate cut of persistence $p$) are:\n",
+    "\\begin{align*}\n",
+    "    &\\omega^* = \\frac{1 - \\beta}{1 - \\beta p}; & \\Omega^*_D = \\frac{\\sigma \\beta}{1 - \\beta p}; & & \\Omega^* = \\frac{\\sigma}{1 - p}\n",
+    "\\end{align*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48e873c5",
+   "metadata": {},
+   "source": [
+    "### NK cross in TANK\n",
+    "See: Bilbiie, 2008\n",
+    "1. H: Hand-to-Mouth: mass of $\\lambda$\n",
+    "2. S: Savers: mass of $1 - \\lambda$\n",
+    "3. $\\chi$: denotes the elasticity of H's consumption (and income) to aggregate income $y_t$\n",
+    "\\begin{align}\n",
+    "    c_t^H = y_t^H = \\chi y_t\\\\\n",
+    "    \\chi = 1 + \\phi\\Big(1 - \\frac{\\tau^D}{\\lambda}\\Big)\n",
+    "\\end{align}\n",
+    "\n",
+    "#### Proposition 2.\n",
+    "In TANK, the MPC $\\omega$, autonomous expenditure increase and multiplier $\\Omega$ (for an interest rate cut of persistence $p$) are:\n",
+    "\\begin{align*}\n",
+    "    &\\omega^* = \\frac{1 - \\beta(1 - \\lambda \\chi)}{1 - \\beta p(1 - \\lambda \\chi)}; & \\Omega^* = \\frac{\\sigma}{1 - p} \\frac{1 - \\lambda}{1 - \\lambda \\chi}\n",
+    "\\end{align*}\n",
+    "There is amplification $\\Big(\\frac{\\partial \\Omega}{\\partial \\lambda} > 0 \\Big)$ if and only oif income inequality is countercyclical:\n",
+    "\\begin{align*}\n",
+    "\\chi > 1\n",
+    "\\end{align*}\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "b0f4eda9",
+   "metadata": {},
+   "source": [
+    "### NK in an analytical HANK model\n",
+    "See: Bilbiie, 2018\n",
+    "1. Exogeneous stochastic change of status between constrained H and unconstrained S (idiosyncratic uncertainty)\n",
+    "2. Insurnace is full within type (after idiosyncratic uncertainty is revealed), bu limited across types\n",
+    "3. Different asset liquidity: bonds are liquid (can be used to self-insure, before idiosyncratic uncertainty is revealed), while stocks are completely illiquid (cannot be used to self-insure)\n",
+    "4. No bond trading (no equilibrium liquidity)\n",
+    "\n",
+    "Hence, mass of Hand-to-Mouth:\n",
+    "$$\n",
+    "\\lambda = \\frac{1 - s}{2 - s - h}\n",
+    "$$\n",
+    "And the multipliers are:\n",
+    "\\begin{align*}\n",
+    "    &\\omega = \\frac{1 - \\beta(1 - \\lambda \\chi)}{1 - \\delta \\beta p(1 - \\lambda \\chi)}; & \\Omega = \\frac{\\sigma}{1 - \\delta p} \\frac{1 - \\lambda}{1 - \\lambda \\chi}\n",
+    "\\end{align*}\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "24cd3339",
+   "metadata": {},
+   "source": [
+    "#### Additional Application: forward guidance puzzle\n",
+    "#### Proposition 4:\n",
+    "The multiplier of forward guidance FG (an interest rate cut in T periods) and the MPC in the analytical HANK model are:\n",
+    "\\begin{align*}\n",
+    "    &\\Omega^F_T = \\sigma \\frac{1 - \\lambda}{1 - \\lambda \\chi}delta^T; & \\omega^F_T = 1 - \\Big(\\beta(1 - \\lambda \\chi)\\Big)^{1+T}\n",
+    "\\end{align*}\n",
+    "The multiplier decreases with the horizon $\\frac{\\partial \\Omega_T^F}{\\partial T} < 0$, thus resolving the FG puzzle, if and only if there is discounting $\\delta < 1$. While in the compounding case, the multiplier increases with the horizon $\\frac{\\partial \\Omega_T^F}{\\partial T} > 0$ and the FG puzzle is aggravated."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5fff020b-2b55-4662-992c-36e2a7f7c814",
+   "metadata": {},
+   "source": [
+    "### References:\n",
+    "Bilbiie, Florin O. \"Limited asset markets participation, monetary policy and (inverted) aggregate demand logic.\" Journal of economic theory 140.1 (2008): 162-196.\n",
+    "\n",
+    "Bilbiie, Florin O. \"Monetary policy and heterogeneity: An analytical framework.\" (2018).\n",
+    "\n",
+    "Bilbiie, Florin O. \"The new Keynesian cross.\" Journal of Monetary Economics 114 (2020): 90-108."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5d7fee7c",
+   "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": "3.8.8"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}