\documentclass[10pt]{beamer}
%\documentclass[handout, 10pt]{beamer}

\mode<presentation>
\usetheme{Madrid}
%\definecolor{lightblue}{HTML}{3498DB}
\definecolor{lightblue}{HTML}{5DADE2}
\definecolor{foundersrock}{HTML}{3B7EA1}
\definecolor{goldenbear}{HTML}{FDB515}
\definecolor{berkeleyblue}{HTML}{003262}
%\definecolor{colblue2}{HTML}{0072CE} % Juan's blue
\usecolortheme[named=lightblue]{structure}
\setbeamercolor{button}{fg=white, bg=lightblue}

%\usecolortheme{seahorse}
%\usecolortheme{rose}
\usefonttheme[onlylarge]{structuresmallcapsserif}
%\usefonttheme[onlymath]{serif}

\definecolor{alertcolor}{rgb}{0.8,0,0} % dark red
%\definecolor{alertcolor}{HTML}{5DADE2} % light blue
\newcommand{\hl}[1]{\textcolor{darkred}{#1}}
\definecolor{beamer@darkred}{rgb}{0.8,0,0}
\setbeamercolor{alerted text}{fg=beamer@darkred}

\renewcommand{\baselinestretch}{1.3}   
\setbeamersize{text margin left=0.5cm}
\setbeamersize{text margin right=0.5cm}

\setbeamertemplate{itemize items}[boadilla]
\setbeamertemplate{enumerate items}[default]

\usenavigationsymbolstemplate{}

\renewcommand{\baselinestretch}{1.3}   
\setbeamersize{text margin left=0.5cm}
\setbeamersize{text margin right=0.5cm}

\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{times}
\usepackage[T1]{fontenc}
\usepackage{epstopdf}
\usepackage{hyperref}
\usepackage{appendixnumberbeamer}

\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{decorations.markings,decorations.pathmorphing,calc,intersections}

\title[Ramsey Model]{The Neoclassical Growth Model: \\ aka Ramsey-Cass-Koopmans Model}
\author[Steinsson]{Jón Steinsson}
\institute[]{University of California, Berkeley}
\date[]{Fall 2025}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}{Neoclassical Growth Model}
	\begin{itemize}
		\itemsep1em 
		\item Solow model makes many simplifying assumptions
		\item Now we relax exogenous savings rate assumption
		\item Households maximize utility / firms maximize profits
		\item Households and firms interact in competitive market
		\item Ramsey-Cass-Koopmans Model {\small (or Ramsy model for short)}
	\end{itemize}
\end{frame}

\begin{frame}{Technology}
	\begin{itemize}
		\item The same as in the Solow model
		\item Production function:
		\[ Y = F(K,AL) \]
		\item $A$ grows exogenously at rate $g$
		\item $L$ grows exogenously at rate $n$
		\item Evolution of capital stock:
		\[ \dot{K}(t) = Y(t) - C(t)L(t) \]
		\item No depreciation of capital (for simplicity)
		\item $C(t)$ denotes per capita consumption {\small (but $Y(t)$ is total output)}
	\end{itemize}
\end{frame}


\begin{frame}{Change of Variables} \label{tech}
	\begin{itemize}
		\itemsep1em 
		\item It is useful to rewrite 
		\[ \dot{K}(t) = Y(t) - C(t)L(t) \]
		in terms of capital, output, and consumption per effective unit of labor
		\item A few lines of algebra yield
		\[\dot{k}(t) = f(k(t)) - c(t) - (n+g)k(t) \] 
	\end{itemize}
	\vspace{20pt}
	\hyperlink{accdetails}{\beamergotobutton{Details}}
\end{frame}


\begin{frame}{Firms} \label{firms1}
	\begin{itemize}
		\item Large number of identical firms seek to maximize profits
		\item Hire workers and capital in competitive factor markets
		\item Can use $A$ for free
		\item Profit maximization implies that firms hire labor and capital to the point where their marginal product is equal to their price
		\item First order condition for capital:
		\[ \frac{\partial F(K,AL)}{\partial K} = r(t) \]
		which implies 
		\[ f'(k(t)) = r(t) \] 
	\end{itemize}
	\hyperlink{capitaldemand}{\beamergotobutton{Details}}
\end{frame}

\begin{frame}{Firms} \label{firms2}
	\begin{itemize}
		\item First order condition for labor:
		\[\frac{\partial F(K,AL)}{\partial L} = W(t) \]
		\item We can rewrite this as
		\[ f(k(t)) - k(t)f'(k(t)) = w(t) \] 
		where $w(t) = W(t)/A(t)$ (wage per effective unit of labor)		  
	\end{itemize}
	\vspace{4mm}
	\hyperlink{labordemand}{\beamergotobutton{Details}}
\end{frame}

\begin{frame}{Representative Firm} \label{repfirm}
	\begin{itemize}
		\item One ``representative'' firm same as many small firms if
		\begin{itemize}
			\item Common production function
			\item Constant returns to scale
			\item Common factor prices
		\end{itemize}
		\item Easy to derive for Cobb-Douglas case:
		\[ Y = A K^{\alpha}L^{1-\alpha} \]
		\item Maximizing behavior implies \hyperlink{cobbdouglas}{\beamergotobutton{Details}}
		\[ Y = A \left(\frac{\alpha}{1-\alpha}\right)^{\alpha} \left( \frac{W}{r} \right)^{\alpha} L \]
		\item Since production is linear in L, firm has no optimal size \item Might as well be one large firm!  
	\end{itemize}
\end{frame}

\begin{frame}{Households}
	\begin{itemize}
		\itemsep1em 
		\item Large number $H$ of households
		\item Size of each household grows at rate $n$
		\item Each member supplies one unit of labor 
		\item Initial capital holdings $K(0)/H$
		\item Household rents capital to firms
		\item Key household choice:
		\begin{itemize}
			\item How to divide income between consumption and savings
		\end{itemize} 
	\end{itemize}
\end{frame}

\begin{frame}{Household Utility}
	\begin{itemize}
		\item Households maximize utility
		\[U = \int_{t=0}^{\infty} e^{-\rho t} u(C(t))\frac{L(t)}{H} dt \] 
		where
		\[ u(C(t)) = \frac{C(t)^{1-\theta}}{1-\theta} \]
		\item $\rho$ is discount rate
		\item $u(\cdot)$ gives each member's utility
		\item $L(t)/H$ is the number of household members
		\item $1/\theta$ is elasticity of intertemporal substitution  
	\end{itemize}
\end{frame}

\begin{frame}{Household Budget Constraint}
	\begin{itemize}
		\itemsep1em 
		\item Household's intertemporal budget constraint:
		\begin{itemize}
			\item Present value of consumption less than or equal to \\ present value of income plus initial wealth
		\end{itemize}
		\item Mathematically:
		\[ \int_{t=0}^{\infty} e^{-R(t)}C(t)\frac{L(t)}{H} dt \leq \frac{K(0)}{H} + \int_{t=0}^{\infty} e^{-R(t)} W(t) \frac{L(t)}{H} dt \]
		\item Discounting with time-varying interest rate:
		\[ R(t) = \int_{\tau =0}^{t} r(\tau) d\tau \]  
	\end{itemize}
\end{frame}


\begin{frame}{Household Budget Constraint}
	\begin{itemize}
		\itemsep1em 
		\item Why is this the budget constraint?
		\item This condition rules out Ponzi-schemes:
		\begin{itemize}
			\item Borrow today and consume
			\item Borrow more tomorrow to pay back principle and interest
			\item Do this forever
		\end{itemize}
		\item Intertemporal budget constraint implies that
		\[ \lim_{s\rightarrow \infty} e^{-R(s)} \frac{K(s)}{H} \geq 0 \]
		{\small (See Romer (2019, p. 54) for derivation)}
		\item Ponzi schemes violate this condition 
	\end{itemize}
\end{frame}

\begin{frame}{Change of Variables}
	\begin{itemize}
		\item Let's write household objective function in terms of consumption \\ and labor per effective worker
		\begin{align*}
			\frac{C(t)^{1-\theta}}{1-\theta} & = \frac{[A(t) c(t)]^{1-\theta}}{1-\theta} \\
			 & = \frac{[A(0)e^{gt}]^{1-\theta} c(t)^{1-\theta}}{1-\theta} \\
			 & = A(0)^{1-\theta}e^{(1-\theta)gt} \frac{c(t)^{1-\theta}}{1-\theta}
		\end{align*}
	\end{itemize}
\end{frame}


\begin{frame}{Change of Variables}
	\begin{itemize}
		\item Overall utility function becomes:
		\begin{align*}
			U & = \int_{t=0}^{\infty} e^{-\rho t} \frac{C(t)^{1-\theta}}{1-\theta} \frac{L(t)}{H} dt \\
			& = \int_{t=0}^{\infty} e^{-\rho t} \left[ A(0)^{1-\theta}e^{(1-\theta)gt} \frac{c(t)^{1-\theta}}{1-\theta} \right] \frac{L(0)e^{nt}}{H} dt \\
			& = B \int_{t=0}^{\infty} e^{-\beta t} \frac{c(t)^{1-\theta}}{1-\theta} dt 
		\end{align*}
		where 
		\[ B = A(0)^{1-\theta}\frac{L(0)}{H} \hspace{30pt} \beta = \rho - n - (1-\theta) g \] 
		\item A similar change of variables for the budget constraint yields:
		\[\int_{t=0}^{\infty} e^{-R(t)} c(t) e^{(n+g)t}dt \leq k(0) + \int_{t=0}^{\infty} e^{-R(t)} w(t) e^{(n+g)t}dt \]  
	\end{itemize}
	
\end{frame}


\begin{frame}{Household's Problem}
	\begin{itemize}
		\itemsep1em 
		\item Maximize:
		\[ U = B \int_{t=0}^{\infty} e^{-\beta t} \frac{c(t)^{1-\theta}}{1-\theta} dt \] 
		\item Subject to:
		\[\int_{t=0}^{\infty} e^{-R(t)} c(t) e^{(n+g)t}dt \leq k(0) + \int_{t=0}^{\infty} e^{-R(t)} w(t) e^{(n+g)t}dt \] \vspace{0pt}
		\item Household's choice variable is the path for $c(t)$ 
	\end{itemize}
\end{frame}


\begin{frame}{Household Optimization}
	\begin{itemize}
		\itemsep2em 
		\item We form a Lagrangian:
		\begin{align*}
			\mathcal{L} & = B \int_{t=0}^{\infty} e^{-\beta t} \frac{c(t)^{1-\theta}}{1-\theta} dt \\ 
			 & + \lambda \left[ k(0) + \int_{t=0}^{\infty} e^{-R(t)} w(t) e^{(n+g)t}dt - \int_{t=0}^{\infty} e^{-R(t)} c(t) e^{(n+g)t}dt \right]
		\end{align*}
		\item Differentiate this with respect to $c(t)$:
		\[B e^{-\beta t} c(t)^{-\theta} = \lambda e^{-R(t)} e^{(n+g) t} \] 
	\end{itemize}
\end{frame}


\begin{frame}{Household Optimization}
	\[B e^{-\beta t} c(t)^{-\theta} = \lambda e^{-R(t)} e^{(n+g) t} \] \vspace{-20pt}
	\begin{itemize}
		\item Let's take logs:
		\[ \log B - \beta t - \theta \log c(t) = \log \lambda - \int_{\tau=0}^{t} r(\tau) d\tau + (n+g)t \] 
		\item Differentiate both sides with respect to time
		\[ - \beta - \theta \frac{\dot{c}(t)}{c(t)} = -r(t) + (n+g) \]
		\item Rearrange and use definition of $\beta$:
		\[ \frac{\dot{c}(t)}{c(t)} = \frac{r(t) - \rho - \theta g}{\theta} \]  
	\end{itemize}
\end{frame}

\begin{frame}{Consumption Euler Equation}
	\[ \frac{\dot{c}(t)}{c(t)} = \frac{r(t) - \rho - \theta g}{\theta} \] 
	\begin{itemize}
		\item And for per capita consumption:
		\begin{align*}
			\frac{\dot{C}(t)}{C(t)} & = \frac{\dot{A}(t)}{A(t)} + \frac{\dot{c}(t)}{c(t)} \\
			& = g + \frac{r(t) - \rho - \theta g}{\theta} \\
			& = \frac{r(t) - \rho}{\theta} 
		\end{align*}
		\item Consumption growth depends on $r(t) - \rho$ and nothing else
	\end{itemize}
\end{frame}


\begin{frame}{Consumption Euler Equation}
	\[ \frac{\dot{C}(t)}{C(t)} = \frac{r(t) - \rho}{\theta} \] 
	\begin{itemize}
		\itemsep1em 
		\item Consumption growth depends on $r(t) - \rho$ and nothing else. \item Why?
		\begin{itemize}
			\item Why nothing else?
			\item Why higher growth with higher $r(t)$?
		\end{itemize} \pause
		\item Embodies desire to smooth consumption \\ {\small (due to diminishing marginal utility)}
	\end{itemize}
\end{frame}


\begin{frame}{Dynamic Optimization in Continuous Time}
	\begin{itemize}
		\item Our derivation was somewhat heuristic
		\item More rigorous to use Hamiltonians / Maximum Principle
		\item Barro R.J. and X. Sala-I-Martin (2004): \textit{Economic Growth} \\ (Appendix A.3)
		\item Obstfeld, M. (1992): ``Dynamic Optimization in Continuous-Time Economic Models (A Guide for the Perplexed)
		\item Acemoglu, D. (2009): \textit{Introduction to Modern Economic Growth} (Chapter 7)
	\end{itemize}
\end{frame}

\begin{frame}{Interest Rate in General Equilibrium}
	\begin{itemize}
		\itemsep1em 
		\item Household takes $r(t)$ as given:
		\[ \frac{\dot{c}(t)}{c(t)} = \frac{r(t) - \rho - \theta g}{\theta} \]
		\item But in general equilibrium:
		\[ r(t) = f'(k(t)) \]
		\item So we have that
		\[ \frac{\dot{c}(t)}{c(t)} = \frac{f'(k(t)) - \rho - \theta g}{\theta} \]
	\end{itemize}
\end{frame}


\begin{frame}{Dynamic System}
	\begin{itemize}
		\itemsep1em 
		\item Two endogenous variables $c(t)$, $k(t)$
		\item Two dynamic equations:
		\[ \frac{\dot{c}(t)}{c(t)} = \frac{f'(k(t)) - \rho - \theta g}{\theta} \]
		\[\dot{k}(t) = f(k(t)) - c(t) - (n+g)k(t) \]
		\item Two boundary conditions:
		\begin{itemize}
			\item k(0) given (initial condition)
			\item Intertemporal budget constraint with equality (terminal condition)
		\end{itemize}
		\item {\small It is the fact that dynamic system has a terminal condition (rather than full set of initial conditions) that makes the system ``forward looking''.}
	\end{itemize}
\end{frame}


\begin{frame}{Dynamics of $c(t)$}
	\[ \frac{\dot{c}(t)}{c(t)} = \frac{f'(k(t)) - \rho - \theta g}{\theta} \] \vspace{-0pt}
	\begin{itemize}
		\itemsep1em 
		\item $\dot{c}(t) = 0$ if $f'(k(t)) = \rho + \theta g$
		\begin{itemize}
			\item Recall that $f'(k)$ is decreasing in $k$
			\item Let $k^{*}$ denote $k$ for which $f'(k) = \rho + \theta g$
			\item Then $\dot{c}(t) = 0$ if $k(t) = k^{*}$
		\end{itemize}
		\item $\dot{c}(t) > 0$ if $f'(k(t)) > \rho + \theta g$ or equivalently $k(t) <  k^{*}$
		\item $\dot{c}(t) < 0$ if $f'(k(t)) < \rho + \theta g$ or equivalently $k(t) >  k^{*}$
	\end{itemize}
\end{frame}


\begin{frame}{Dynamics of $c(t)$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure21.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Dynamics of $k(t)$}
		\[\dot{k}(t) = \underbrace{f(k(t)) - c(t)}_{\text{Investment}} - \underbrace{(n+g)k(t)}_{\text{breakeven investment}} \]
	\begin{itemize}
		\item $\dot{k}(t) = 0$ if investment is equal to breakeven investment
		\item Rearranging, we get that $\dot{k}(t) = 0$ when 
		\[c = f(k) - (n+g)k \]
		\item Let's think about this as a function of $k$
		\begin{itemize}
			\item $f(0) = 0$ and $(n+g)0 = 0$
			\item $f'(k)$ is very high for small $k$
			\item $f(k)$ is concave (slope goes to $-(n+g)$ when $k \rightarrow \infty$) 
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Dynamics of $k(t)$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure22.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Dynamics of $c(t)$ and $k(t)$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure23.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{$k^{*}$ is below Golden Rule $k$}
	\begin{itemize}
		\item Steady state $k$: $f'(k^{*}) = \rho + \theta g$
		\item Golden Rule $k$: $f'(k_{GR}) = n + g$
		\item Since $f''(k) < 0$, $k^{*} < k_{GR}$ if
		\[ \rho + \theta g > n + g \]
		\[ \rho - n - (1-\theta) g > 0 \]
		\[ \beta > 0\] 
		\item We assume $\beta > 0$ since otherwise household utility is infinite
	\end{itemize}
\end{frame}

\begin{frame}{Which Path Will Economy Take?}
	\begin{itemize}
		\item Phase diagram has many paths
		\item All of them satisfy the two dynamic equations
		\item Which one of these paths will the economy take? \pause 
		\item[]
		\item Answer determined by boundary conditions
		\begin{itemize}
			\item Boundary condition \#1: Initial condition for $k$
			\item But there is no initial condition for $c$!!!
			\item $c(0)$ is a choice of the household \pause 
		\end{itemize}
		\item[]
		\item So, how do we determine $c(0)$? \pause 
		\begin{itemize}
			\item Boundary condition \#2: Intertemporal budget constraint holds with equality
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Determination of $c(0)$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure24.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Determination of $c(0)$}
	\begin{itemize}
		\itemsep1em 
		\item Paths starting at A, B, and C eventually hit $k = 0$. For consumption Euler equation to continue to hold, $k$ must keep falling, which it can't. 
		\item Path starting at D, eventually rises above Golden Rule level of consumption. At this point $f'(k) < n + g$, so $e^{-R(s)}e^{(n+g)s}$ is rising in $s$ and $k(s)$ is rising in $s$. So:
		\[ \lim_{s\rightarrow \infty} e^{-R(s)}e^{(n+g)s} k(s) = \infty  \]
		In other words, net present value of future wealth is exploding, which means household should consumer more. (Violates intertemporal budget constraint.)
	\end{itemize}
\end{frame}


\begin{frame}{Saddle Path}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure25.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Long-Run Growth in Ramsey Model}
	\begin{itemize}
		\itemsep1em 
		\item In the long run, $k^{*}$, $y^{*}$, $c^{*}$ are constant
		\item Level variables grow at rate $n+g$
		\item Per capita variables grow at rate $g$
		\item Long-run growth rate independent of many model parameters. \\ E.g., Production function parameters, preference parameters ($\rho$ and $\theta$)
	\end{itemize}
\end{frame}


\begin{frame}{Steady State}
	\begin{itemize}
		\itemsep1em 
		\item Dynamic equations:
		\[ \frac{\dot{c}(t)}{c(t)} = \frac{f'(k(t)) - \rho - \theta g}{\theta} \]
		\[\dot{k}(t) = f(k(t)) - c(t) - (n+g)k(t) \]
		\item To calculate steady state, we set $\dot{c}(t) =0$ and $\dot{k}(t) = 0$:
		\[f'(k^{*}) = \rho + \theta g \]
		\[c^{*} = f(k^{*}) - (n+g)k^{*} \] 
		\item Level of steady state \textbf{is} a function of $\rho$, $\theta$, and production function
	\end{itemize}
\end{frame}


\begin{frame}{Unanticipated Fall in Discount Rate}
	\begin{itemize}
		\itemsep1em 
		\item Consider an unanticipated fall in the discount rate $\rho$
		\item Conceptually similar to an increase in the savings rate \\ in the Solow model
		\item Long-run effect depends on shift in steady state: 
		\[f'(k^{*}) = \rho + \theta g \]
		\[c^{*} = f(k^{*}) - (n+g)k^{*} \]
		\item Shift in $\rho$ only affects first of these equation
		\item Since $f''(k) < 0$, $k^{*}_{\text{NEW}} > k^{*}_{\text{OLD}}$
	\end{itemize}
\end{frame}


\begin{frame}{Unanticipated Fall in $\rho$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/Romer2019Figure26.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Shape of the Saddle Path}
	\begin{itemize}
		\itemsep1em 
		\item The saddle path gives c(k) (called the policy function)
		\item What is the shape of this path?
		\item Consider different values of $\theta$
		\item Recall that $1/\theta$ is the intertemporal elasticity of substitution
		\item High $\theta$ (low $1/\theta$) implies strong desire to smooth consumption \\ Household will try to shift consumption from the future \\ Saddle path will be close to $\dot{k}(t) = 0$ locus 
	\end{itemize}
\end{frame}


\begin{frame}{Saddle Paths with Different in $\theta$}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/BarroSalaimartin2004Figure22.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004)}
\end{frame}


\begin{frame}{Savings Rate in the Ramsey Model}
	\begin{itemize}
		\itemsep1em
		\item Savings rate constant in Solow model. What about Ramsey model?
		\item In general it is complicated
		\item Intuitively two forces:
		\begin{itemize}
			\itemsep0.5em 
			\item Substitution effect: As $k$ rises, $f'(k)$ falls, and $r$ falls. Incentive to save fall.
			\item Income effect: When households are below steady state, their income \\ is below future income. They would like to smooth consumption by consuming a lot relative to income. As income rises, this force weakens and savings rise.  
		\end{itemize}
		\item Which force wins depends on parameters. 
	\end{itemize}
\end{frame}


\begin{frame}{Savings Rate in the Ramsey Model}
	\begin{itemize}
		\itemsep1em 
		\item Cobb-Douglas case ($y = k^{\alpha}$): savings rate is monotonic
		\item If $\alpha (g + n) = (\rho + \theta g)/\theta$, savings rate is constant
		\item If $\alpha (g + n) > (\rho + \theta g)/\theta$, savings rate is increasing \\ {\small (e.g., small value of $1/\theta$ -- weak substitution effect) }
		\item If $\alpha (g + n) < (\rho + \theta g)/\theta$, savings rate is decreasing \\ {\small (e.g., large value of $1/\theta$ -- strong substitution effect) }
		\item See Barro and Sala-I-Martin (2004, sec. 2.6.4) for details
		\begin{itemize}
			\item They argue that $\alpha (g + n) \approx (\rho + \theta g)/\theta$ is not unreasonable.
			\item Empirically they argue savings rate rise a slight bit with development 
		\end{itemize}
	\end{itemize}
\end{frame}

\begin{frame}{Savings Rate in Ramsey Model}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75  \textheight]{FiguresTables/BarroSalaimartin2004Figure23.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Barro and Sala-I-Martin (2004)}
\end{frame}



\begin{frame}{Welfare in the Ramsey Model}
	\begin{itemize}
		\itemsep1em 
		\item Is the market outcome in the Ramsey model a ''good'' outcome?
		\item First Welfare Theorem holds in the Ramsay model:
		\begin{itemize}
			\item Markets are competitive and complete
			\item Households are rational and optimize
			\item Property rights exist over all objects of value
		\end{itemize}
		\item This means decentralized outcome is Pareto efficient \\ {\footnotesize (Actually there are some technicalities which we will come back to in OLG lecture)}
		\item Easy to verify by solving the ``planner's problem''
	\end{itemize}
\end{frame}


\begin{frame}{Ramsey versus Solow}
	\begin{itemize}
		\itemsep1em 
		\item Basic message of Solow model carries over to Ramsey model
		\begin{itemize}
			\item Balanced growth in steady state
			\item Savings rate $(y-c)/y$ constant in steady state
			\item Level variables grow at rate $n+g$
			\item Per capita growth due to exogenous growth in $A$
			\item Growth higher below steady state, lower above steady state
		\end{itemize}
		\item Differences in Ramsey:
		\begin{itemize}
			\item Savings rate not necessarily constant (but constant not unreasonable)
			\item $n$ does not affect $k^{*}$. ($f'(k^{*}) = \rho - \theta g$)
			\begin{itemize}
				\item Sensitive to having $L(t)/H$ term in utility function
				\item Without this term $f'(k^{*}) = \rho + n - \theta g$
				\item Depends on nature of altruism toward offspring
			\end{itemize}
		\end{itemize}
	\end{itemize}
\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% APPENDIX
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\appendix

{  \setbeamercolor{background canvas}{bg=lightblue}
	\begin{frame}
		\addtocounter{framenumber}{-1}
		\thispagestyle{empty}
		
		\begin{center}
			{\Large \color{white} Appendix}
		\end{center}
		
	\end{frame}
}

\begin{frame}{Capital Accumulation -- Change of Variables} \label{accdetails}
	Notice that:
	\[\dot{k}(t) = \frac{d}{dt} \left( \frac{K(t)}{A(t)L(t)} \right) = \frac{\dot{K}(t)}{A(t)L(t)} - \frac{K(t)}{[A(t)L(t)]^2} [A(t)\dot{L}(t) + \dot{A}(t)L(t)] \] 
	\[\dot{k}(t) = \frac{\dot{K}(t)}{A(t)L(t)} - (n+g)k \]
	Dividing capital accumulation equation by $A(t)L(t)$ we get
	\[ \frac{\dot{K}(t)}{A(t)L(t)} = \frac{Y(t)}{A(t)L(t)} - \frac{C(t)L(t)}{A(t)L(t)} \]
	\[ \dot{k}(t) + (n+g)k = y(t) - c(t) \] 
	\[\dot{k}(t) = f(k(t)) - c(t) - (n+g)k(t) \] 
	\hyperlink{tech}{\beamerreturnbutton{Back}}
\end{frame}


\begin{frame}{Capital Demand -- Details} \label{capitaldemand}
	\begin{itemize}
		\item Recall that $f(k) \equiv F(k,1)$.
		\item $F(\cdot,\cdot)$ is homogeneous of degree one
		\item This implies that $F_1(\cdot,\cdot)$ is homogeneous of degree zero \\ {\small (where the subscript denotes a derivative with respect to the first argument)}
		\[ f'(k) = F_1(k,1) = F_1(K/AL,1) = F_1(K,AL) = F_K(K,AL) \] 
	\end{itemize}
	\vspace{4mm}
	\hyperlink{firms1}{\beamerreturnbutton{Back}}
\end{frame}

\begin{frame}{Labor Demand -- Details} \label{labordemand}
	\begin{itemize}
		\item Notice that 
		\[ \frac{\partial F(K,AL)}{\partial L} = A\frac{\partial F(K,AL)}{\partial AL} = A F_2(K,AL) \] 
		\item Since $F(K,AL)$ is homogeneous of degree 1 we have that
		\begin{align*}
			F(K,AL) & = F_1(K,AL) K + F_2(K,AL) AL \\
			AL F(k,1) & = F_1(K,AL) K + F_2(K,AL) AL \\
			f(k) & = f'(k) k + F_2(K,AL) \\
			F_2(K,AL) & = f(k) - f'(k) k
		\end{align*}
	\end{itemize}
	\vspace{4mm}
	\hyperlink{firms2}{\beamerreturnbutton{Back}}
\end{frame}

\begin{frame}{Cobb-Douglas Firm} \label{cobbdouglas}
	\begin{itemize}
		\item Labor demand:
		\[ W = (1-\alpha)AK^{\alpha}L^{-\alpha}\]
		\item Capital demand:
		\[ r = \alpha A K^{\alpha-1}L^{1-\alpha}\]
		\item Divide one by the other
		\[\frac{K}{L} = \frac{\alpha}{1-\alpha} \frac{W}{r} \]
		\item Plug this into production function for K:
		\[ Y = A K^{\alpha} L^{1-\alpha} = A \left( \frac{\alpha}{1-\alpha} \right)^{\alpha} \left( \frac{W}{r} \right)^{\alpha} L^{\alpha} L^{1-\alpha} \] 
	\end{itemize}
	\vspace{4mm}
	\hyperlink{repfirm}{\beamerreturnbutton{Back}}
\end{frame}


\end{document}

