Material for the Dynare Summer School 2018

deterministic-examples/rbc_basic.mod

+// Endogenous variables: consumption and capital
+var c k;
+
+// Exogenous variable: technology level
+varexo A;
+
+// Parameters declaration and calibration
+parameters alpha beta gamma delta;
+
+alpha=0.5;
+beta=0.95;
+gamma=0.5;
+delta=0.02;
+
+// Equilibrium conditions
+model;
+c + k = A*k(-1)^alpha + (1-delta)*k(-1); // Resource constraint
+c^(-gamma) = beta*c(+1)^(-gamma)*(alpha*A(+1)*k^(alpha-1) + 1 - delta); // Euler equation
+end;
+
+// Steady state (analytically solved)
+initval;
+A = 1;
+k = ((1-beta*(1-delta))/(beta*alpha*A))^(1/(alpha-1));
+c = A*k^alpha-delta*k;
+end;
+
+// Check that this is indeed the steady state
+steady;
+
+// Declare a positive technological shock in period 1
+shocks;
+var A;
+periods 1;
+values 1.2;
+end;
+
+// Prepare the deterministic simulation of the model over 100 periods
+perfect_foresight_setup(periods=100);
+
+// Perform the simulation
+perfect_foresight_solver;
+
+// Display the path of consumption
+rplot c;

deterministic-examples/rbc_det1.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+steady;
+
+ik = varlist_indices('k',M_.endo_names);
+kstar = oo_.steady_state(ik);
+
+histval;
+ k(0) = kstar/2;
+end;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+rplot c;
+rplot k;

deterministic-examples/rbc_det2.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+steady;
+
+shocks;
+ var epsilon;
+ periods 1;
+ values -0.1;
+end;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+rplot c;
+rplot k;

deterministic-examples/rbc_det3.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+steady;
+
+shocks;
+ var epsilon;
+ periods 4, 5:8;
+ values 0.04, 0.01;
+end;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+rplot c;
+rplot k;

deterministic-examples/rbc_det4.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+initval;
+ epsilon = 0;
+end;
+
+steady;
+
+endval;
+ epsilon = (1-rho)*log(1.05);
+end;
+
+steady;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+rplot c;
+rplot k;

deterministic-examples/rbc_det5.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+initval;
+ epsilon = 0;
+end;
+
+steady;
+
+endval;
+ epsilon = (1-rho)*log(1.05);
+end;
+
+steady;
+
+shocks;
+ var epsilon;
+ periods 1:5;
+ values 0;
+end;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+rplot c;
+rplot k;

deterministic-examples/rbc_ep.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+steady;
+
+// Declare shocks as in a stochastic setup
+shocks;
+ var epsilon;
+ stderr 0.02;
+end;
+
+extended_path(periods=300);
+
+// Plot 20 first periods of consumption
+ic = varlist_indices('c',M_.endo_names);
+plot(oo_.endo_simul(ic, 1:21));

deterministic-examples/rbc_unexpected.mod

+var k, y, L, c, A, a;
+
+varexo epsilon;
+
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar;
+
+beta    =   0.9900;
+theta   =   0.3570;
+tau     =   2.0000;
+alpha   =   0.4500;
+psi     =  -0.1000;
+delta   =   0.0200;
+rho     =   0.8000;
+Astar   =   1.0000;
+
+model;
+ a = rho*a(-1) + epsilon;
+ A = Astar*exp(a);
+ y = A*(alpha*k(-1)^psi+(1-alpha)*L^psi)^(1/psi);
+ k = y-c + (1-delta)*k(-1);
+ (1-theta)/theta*c/(1-L) - (1-alpha)*(y/L)^(1-psi);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c = beta*(c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta);
+end;
+
+steady_state_model;
+ a = epsilon/(1-rho);
+ A = Astar*exp(a);
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ % Compute steady state of the endogenous variables.
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+end;
+
+steady;
+
+// Declare pre-announced shocks
+shocks;
+ var epsilon;
+ periods 5, 15;
+ values -0.1, -0.1;
+end;
+
+perfect_foresight_setup(periods=300);
+perfect_foresight_solver;
+
+// Declare unexpected shock (after first simulation!)
+oo_.exo_simul(11, 1) = 0.1; // Period 10 has index 11!
+
+saved_endo = oo_.endo_simul(:, 1:9); // Save periods 0 to 8
+saved_exo = oo_.exo_simul(1:9, :);
+oo_.endo_simul = oo_.endo_simul(:, 10:end); // Keep periods 9 to 301
+oo_.exo_simul = oo_.exo_simul(10:end, :);
+
+periods 291;
+perfect_foresight_solver;
+
+// Combine the two simulations
+oo_.endo_simul = [ saved_endo oo_.endo_simul ];
+oo_.exo_simul = [ saved_exo; oo_.exo_simul ];
+
+rplot epsilon;
+rplot c;

deterministic-examples/rbcii.mod

+var k, y, L, c, i, A, a, mu;
+varexo epsilon;
+parameters beta, theta, tau, alpha, psi, delta, rho, Astar, sigma;
+
+beta    =  0.990;
+theta   =  0.357;
+tau     =  2.000;
+alpha   =  0.450;
+psi     = -2.500;
+delta   =  0.020;
+rho     =  0.998;
+Astar   =  1.000;
+sigma   =  0.100;
+
+model;
+ a = rho*a(-1) + sigma*epsilon;
+ A = Astar*exp(a);
+ (c^theta*(1-L)^(1-theta))^(1-tau)/c - mu = beta*((c(+1)^theta*(1-L(+1))^(1-theta))^(1-tau)/c(+1)*(alpha*(y(+1)/k)^(1-psi)+1-delta)-mu(+1)*(1-delta));
+ ((1-theta)/theta)*(c/(1-L)) - (1-alpha)*(y/L)^(1-psi);
+ y = A*(alpha*(k(-1)^psi)+(1-alpha)*(L^psi))^(1/psi);
+ k = y-c+(1-delta)*k(-1);
+ i = k-(1-delta)*k(-1);
+
+[ mcp = 'i > 0' ]
+ mu = 0;
+end;
+
+steady_state_model;
+ a=0;
+ mu=0;
+ A=Astar;
+    
+ // Steady state ratios 
+ Output_per_unit_of_Capital=((1/beta-1+delta)/alpha)^(1/(1-psi));
+ Consumption_per_unit_of_Capital=Output_per_unit_of_Capital-delta;
+ Labour_per_unit_of_Capital=(((Output_per_unit_of_Capital/A)^psi-alpha)/(1-alpha))^(1/psi);
+ Output_per_unit_of_Labour=Output_per_unit_of_Capital/Labour_per_unit_of_Capital;
+ Consumption_per_unit_of_Labour=Consumption_per_unit_of_Capital/Labour_per_unit_of_Capital;
+
+ L=1/(1+Consumption_per_unit_of_Labour/((1-alpha)*theta/(1-theta)*Output_per_unit_of_Labour^(1-psi)));
+ c=Consumption_per_unit_of_Labour*L;
+ k=L/Labour_per_unit_of_Capital;
+ y=Output_per_unit_of_Capital*k;
+ i=delta*k;
+end;
+
+shocks;
+ var epsilon;
+ periods 10;
+ values -1;
+end;
+
+steady;
+
+perfect_foresight_setup(periods=400);
+perfect_foresight_solver(lmmcp, maxit=200);
+
+n = 40;
+
+figure(2);
+subplot(3,2,1); plot(1:n,A(1:n)); title('A');
+subplot(3,2,2); plot(2:n,y(2:n)); title('y');
+subplot(3,2,3); plot(2:n,L(2:n)); title('L');
+subplot(3,2,4); plot(1:n,k(1:n)); title('k');
+subplot(3,2,5); plot(2:n,c(2:n)); title('c');
+subplot(3,2,6); plot(2:n, y(2:n)-c(2:n)); title('i');
+

deterministic.tex

+\documentclass{beamer}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{lmodern}
+\usepackage{amsmath}
+\usepackage[copyright]{ccicons}
+
+\usepackage{pstricks}
+
+\usetheme{Boadilla}
+
+\title{Deterministic Models}
+\subtitle{Perfect foresight, nonlinearities and occasionally binding constraints}
+\author{Sébastien Villemot}
+\pgfdeclareimage[height=0.6cm]{logo}{cepremap}
+\institute[CEPREMAP]{\pgfuseimage{logo}}
+\date{June 12, 2018}
+
+\AtBeginSection[]
+{
+  \begin{frame}
+    \frametitle{Outline}
+    \tableofcontents[currentsection]
+  \end{frame}
+}
+
+\AtBeginSubsection[]
+{
+  \begin{frame}
+    \frametitle{Outline}
+    \tableofcontents[currentsection,currentsubsection]
+  \end{frame}
+}
+
+\begin{document}
+
+\begin{frame}
+  \titlepage
+\end{frame}
+
+\begin{frame}
+  \frametitle{Introduction}
+  \begin{itemize}
+  \item Perfect foresight = agents perfectly anticipate all future shocks
+  \item Concretely, at period 1:
+    \begin{itemize}
+    \item agents learn the value of all future shocks;
+    \item since there is shared knowledge of the model and of future shocks,
+      agents can compute their optimal plans for all future periods;
+    \item optimal plans are not adjusted in periods 2 and later \\
+      $\Rightarrow$ the model behaves as if it was deterministic.
+    \end{itemize}
+  \item Cost of this approach: the effect of future uncertainty is not taken
+    into account (\textit{e.g.} no precautionary motive)
+  \item Advantage: numerical solution can be computed exactly (up to rounding
+    errors), contrarily to perturbation or global solution methods for rational
+    expectations models
+  \item In particular, nonlinearities fully taken into account (\textit{e.g.}
+    occasionally binding constraints)
+  \end{itemize}
+\end{frame}
+
+\begin{frame}
+  \frametitle{Outline}
+  \tableofcontents
+\end{frame}
+
+\section{Presentation of the problem}
+
+\begin{frame}
+  \frametitle{The (deterministic) neoclassical growth model}
+  \begin{equation*}
+    \max_{\{c_t\}_{t=1}^\infty}\sum_{t=1}^\infty \beta^{t-1}\frac{c_t^{1-\sigma}}{1-\sigma}
+  \end{equation*}
+  s.t.
+  \begin{equation*}
+    c_t+k_t = A_t k_{t-1}^\alpha + (1-\delta)k_{t-1}
+  \end{equation*}
+  First order conditions:
+  \begin{align*}
+    c_t^{-\sigma} &= \beta c_{t+1}^{-\sigma}\left(\alpha A_{t+1}k_t^{\alpha-1}+1-\delta\right)\\
+    c_t+k_t &= A_t k_{t-1}^\alpha + (1-\delta)k_{t-1}
+  \end{align*}
+  Steady state:
+  \begin{align*}
+    \bar k &= \left(\frac{1-\beta(1-\delta)}{\beta\alpha\bar A}\right)^{\frac{1}{\alpha-1}}\\
+    \bar c &= \bar A \bar k^\alpha -\delta\bar k
+  \end{align*}
+
+  Note the absence of stochastic elements! \\
+  No expectancy term, no probability distribution
+\end{frame}
+