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\title[OLG]{The Overlapping Generations Model}
\author[Steinsson]{Jón Steinsson}
\institute[]{University of California, Berkeley}
\date[]{Fall 2024}

\begin{document}

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


\begin{frame}{The Overlapping Generations Model}
	\begin{itemize}
		\itemsep1em 
		\item Neoclassical growth model has a representative agent
		\item No way to discuss implications of heterogeneity
		\item Life-cycle / generations are important examples of heterogeneity
		\item OLG model allows discussion these issues
		\item More generally, allows discussion of issues that arise with
		\begin{itemize}
			\item Heterogeneity
			\item Infinite number of agents
		\end{itemize}
		\item Seminal papers: Samuelson (1958), Diamond (1965)
	\end{itemize}
\end{frame}


\begin{frame}{Overlapping Generations Model}
	Specific issues we will discuss: \vspace{5pt}
	\begin{itemize}
		\itemsep1em 
		\item Dynamic efficiency (i.e., over-accumulation of capital)
		\item Social Security (i.e., old age pension systems)
		\item Public debt
		\item Money / Bubbles
	\end{itemize}
\end{frame}


\begin{frame}{Basic Structure}
	\begin{itemize}
		\itemsep1em 
		\item Two generations: Young and Old
		\item Each lives for two periods (discrete time)
		\item Young work, consume, save
		\item Old consume and dissave (do not work) 
		\item Common extensions:
		\begin{itemize}
			\item Many generations
			\item Perpetual youth model (Blanchard, 1985) 
		\end{itemize}
		\item Two generation version particularly simple because it precludes intertemporal trade (no one meets twice)
	\end{itemize}
\end{frame}


\begin{frame}{Basic Structure}
	\begin{itemize}
		\itemsep1em 
		\item $L_t$ individuals are born at time $t$
		\item Exogenous population growth at rate $n$:
		\[L_{t+1} = (1+n) L_t \]
		\item Each young agent supplies 1 unit of labor
		\item ``Youth'' need not be due to birth. Could be immigration \\ or the binding of a borrowing constraint. 
	\end{itemize}
\end{frame}


\begin{frame}{Production}
	\begin{itemize}
		\item Production function:
		\[ Y_t = F(K_t,A_t L_t) \]
		\item Exogenous productivity growth:
		\[A_{t+1} = (1+g)A_t \]
		\item Perfect competition in factor markets yields:
		\[r_t = f'(k_t) \hspace{40pt} w_t = f(k_t) - k_t f'(k_t) \] 
		{\footnotesize (See Ramsey model lecture for details)}
		 %\vspace{-10pt}
		\begin{itemize}
			\item $r_{t}$ is the return on savings held from period $t-1$ to $t$
			\item $w_t$ is the wage per effective unit of labor
		\end{itemize}   
	\end{itemize}
\end{frame}


\begin{frame}{Households}
	\begin{itemize}
		\item Preferences of households born at $t$:
		\[ U_t = \frac{C_{1t}^{1-\theta}}{1-\theta} + \frac{1}{1+\rho} \frac{C_{2t+1}^{1-\theta}}{1-\theta} \] \vspace{-10pt}
		\item Budget constraints:
		\[ C_{1t} + s_t = w_t A_t \]
		\[ C_{2t+1} = (1+r_{t+1})s_t\] \vspace{-10pt} 
		\begin{itemize}
			\item $s_t$ is savings of young at time $t$
			\item Old consume both interest and principle
			\item We are assuming no depreciation of capital (for simplicity)
		\end{itemize} 
	\end{itemize}
\end{frame}


\begin{frame}{Household Optimization}
	\begin{itemize}
		\item We can plug budget constraints into $U_t$ to get
		\[ U_t = \frac{(w_t A_t - s_t)^{1-\theta}}{1-\theta} + \frac{1}{1+\rho} \frac{((1+r_{t+1})s_t)^{1-\theta}}{1-\theta} \]
		\item Differentiating with respect to $s_t$ yields:
		\[ -(w_t A_t - s_t)^{-\theta} + \frac{1+r_{t+1}}{1+\rho} ((1+r_{t+1})s_t)^{-\theta} = 0 \]
		\item Rearranging and using budget constraints again:
		\[ C_{1t}^{-\theta} = \frac{1+r_{t+1}}{1+\rho} C_{2t+1}^{-\theta} \]   
		\item This is the consumption Euler equation {\small (same as Ramsey model)}
	\end{itemize}
\end{frame}


\begin{frame}{Household Consumption Function}
	\begin{itemize}
		\item Combining the budget constraints:
		\[ C_{1t} + \frac{1}{1+r_{t+1}} C_{2t+1} = A_t w_t \]
		this is called the intertemporal budget constraint
		\item Rearranging Euler equation:
		\[ C_{2t+1} = \left( \frac{1+r_{t+1}}{1+\rho} \right)^{1/\theta} C_{1t} \] 
		\item Combining these two:
		\[ C_{1t} + \frac{(1+r_{t+1})^{(1-\theta)/\theta}}{(1+\rho)^{1/\theta}} C_{1t} = A_t w_t \]  
	\end{itemize}
\end{frame}


\begin{frame}{Consumption and Saving}
	\begin{itemize}
		\itemsep1em 
		\item Solving for $C_{1t}$ yields:
		\[ C_{1t} = \frac{(1+\rho)^{1/\theta}}{(1+\rho)^{1/\theta} + (1+r_{t+1})^{(1-\theta)\theta}} A_t w_t \] 
		\item Young spend some fraction of labor income on time 1 consumption
		\item Savings:
		\[ s_t = A_t w_t - C_{1t} = \frac{(1+r_{t+1})^{(1-\theta)/\theta}}{(1+\rho)^{1/\theta} + (1+r_{t+1})^{(1-\theta)\theta}} A_t w_t \] 
		\item Young save a complementary fraction of their labor income
	\end{itemize}
\end{frame}


\begin{frame}{Savings: Comparative Statics}
	\[ s_t = \frac{(1+r_{t+1})^{(1-\theta)/\theta}}{(1+\rho)^{1/\theta} + (1+r_{t+1})^{(1-\theta)/\theta}} A_t w_t \] \hspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Savings unambiguously increase in wage income \\ {\small (Both $C_{1t}$ and $C_{2t+1}$ are normal goods)}
		\item Effect of a change in $r_{t+1}$ is ambiguous
		\item Change in $r_{t+1}$ both and income effect and a substitution effect
		\begin{itemize}
			\item Increase in $r_{t+1}$ decreases price of $C_{2t+1}$ {\small (which increases savings)}
			\item Increase in $r_{t+1}$ increases feasible consumption set \\ (which decreases savings)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Savings: Comparative Statics}
	\[ s_t = \frac{(1+r_{t+1})^{(1-\theta)/\theta}}{(1+\rho)^{1/\theta} + (1+r_{t+1})^{(1-\theta)/\theta}} A_t w_t \] \hspace{-10pt}
	\begin{itemize}
		\itemsep1em 
		\item Savings increase in $r_{t+1}$ if $(1+r_{t+1})^{(1-\theta)/\theta}$ is increasing in $r_{t+1}$
		\[ \frac{d}{d r} (1+r)^{(1-\theta)/\theta} = \frac{1-\theta}{\theta} (1+r)^{(1-\theta)/\theta}  \]
		\item Savings increase in $r_{t+1}$ if $\theta < 1$, i.e., if IES > 1
		\item If IES > 1, substitution effect is strong and overwhelms income effect
		\item If IES = 1 (log utility) saving is unaffected by $r_{t+1}$
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Capital Stock}
	\begin{itemize}
		\item Savings of young at time $t$ become capital stock at time $t+1$:
		\[ K_{t+1} = s_t L_t \]
		\item Using notation from Romer (2019): $s_t = s(r_{t+1})A_t w_t$
		\[ K_{t+1} = s(r_{t+1})A_t w_t L_t \] 
		\item Dividing through by $A_{t+1}L_{t+1}$ yields:
		\[ k_{t+1} = \frac{s(r_{t+1})w_t}{(1+n)(1+g)} \]
		where $k_t = K_t/(A_tL_t)$ 
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Capital Stock}
	\begin{itemize}
		\itemsep1em 
		\item Plugging in for $w_t$ and $r_{t+1}$:
		\[ k_{t+1} = \frac{s(f'(k_{t+1}))[f(k_t)-k_tf'(k_t)]}{(1+n)(1+g)} \]
		\item Implicitly defines $k_{t+1}$ as a function of $k_t$
		\item Let's call this function the ``savings locus''
		\item Steady state when $k_{t+1} = k_t$
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Capital Stock}
	\[ k_{t+1} = \frac{s(f'(k_{t+1}))[f(k_t)-k_tf'(k_t)]}{(1+n)(1+g)} \] \vspace{-10pt}
	\begin{itemize}
		\item Let's start by considering special case:
		\begin{itemize}
			\item Logarithmic utility (i.e., $\theta = 1$)
			\item Cobb-Douglas production function ($y = k^{\alpha}$)
		\end{itemize}
		\item In this case:
		\[ s(r_{t+1}) = \frac{1}{2+\rho} \hspace{20pt} \text{and} \hspace{20pt} f(k)-kf'(k) = k^{\alpha}-\alpha k^{\alpha} = (1-\alpha)k^{\alpha}  \]
		\item So, we have:
		\[ k_{t+1} = \frac{(1-\alpha)}{(1+n)(1+g)(2+\rho)}k_{t}^{\alpha} \]
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Capital in Special Case}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75 \textheight]{FiguresTables/BlanchardFischer1989Figure32.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard and Fischer (1989)}
\end{frame}


\begin{frame}{Evolution of Capital in Special Case}
	\begin{itemize}
		\itemsep1em 
		\item In this special case: \vspace{5pt}
		\begin{itemize}
			\itemsep1em 
			\item There is a single steady state (with positive capital)
			\item The steady state is locally stable
		\end{itemize} 
		\item What is it that makes the steady state locally stable?
		\[ \left. \frac{d k_{t+1}}{d k_{t}}\right|_{ss} < 1 \] 
	\end{itemize}
\end{frame}

\begin{frame}{Evolution of Capital More Generally}
	\[ k_{t+1} = \frac{s(f'(k_{t+1}))[f(k_t)-k_tf'(k_t)]}{(1+n)(1+g)} \]
	\begin{itemize}
		\itemsep1em 
		\item More generally, the savings locus can take many different shapes
		\item This can lead to various types of pathologies
		\begin{itemize}
			\item No steady state with positive capital
			\item Multiple steady states with positive capital
			\item Multiple equilibria
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Evolution of Capital}
	\begin{figure}
		\centering
		\includegraphics[height = 0.75 \textheight]{FiguresTables/BlanchardFischer1989Figure31.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard and Fischer (1989)}
\end{frame}


\begin{frame}{Evolution of Capital}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/Romer2019Figure212.pdf}
	\end{figure}
	\vspace{-6mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Evolution of Capital}
	\[ k_{t+1} = \frac{s(f'(k_{t+1}))[f(k_t)-k_tf'(k_t)]}{(1+n)(1+g)} \] \vspace{-10pt}
	\begin{itemize}
		\item We can rewrite this as follows:
		\[ k_{t+1} = \frac{1}{(1+n)(1+g)}
		\underbrace{s(r_{t+1})}_{\text{savings rate}}
		\underbrace{\frac{f(k_t)-k_tf'(k_t)}{f(k_t)}}_{\text{labor share}}
		\underbrace{f(k_t)}_{\text{output per person}} \]
		\item $f(k)$ concave (diminishing returns)
		\item With log utility $s(r)$ constant, with Cobb-Douglas labor share constant
		\item Multiple steady states: need sharply rising savings rate or labor share
	\end{itemize}
\end{frame}


\begin{frame}{Welfare in the OLG Model}
	\begin{itemize}
		\itemsep1em 
		\item Common in macro to compare market outcome to outcome from ``planner's problem''
		\item Conceptually simple in a model with a representative agent \\ {\small (planner will maximize that agent's welfare)}
		\item Not as simple in model with heterogeneous agents such as OLG model
		\item How should planner weight the welfare of different generations?
		\item However, Pareto optimality is still unambiguous 
	\end{itemize}
\end{frame}


\begin{frame}{Welfare in OLG Model}
	\begin{itemize}
		\itemsep1em 
		\item Is market outcome Pareto optimal in OLG model?
		\item Turns out this is not necessarily the case
		\item Economy may accumulate ``too much'' capital 
		\item If so, it is possible to make everyone better off
	\end{itemize}
\end{frame}


\begin{frame}{Golden Rule Capital}
	\begin{itemize}
		\itemsep1em 
		\item Let's consider log-utility, Cobb-Douglas production case
		\item Let's also assume $g = 0$ for simplicity and focus on steady state
		\item Golden Rule capital stock:
		\begin{itemize}
			\item Capital stock that yields the highest steady state consumption \\ per effective unit of labor
		\end{itemize}
		\item Never makes sense to have more capital than Golden Rule capital
		\begin{itemize}
			\item In this case, less capital would give more consumption
			\item ``the economy staggers under the weight of the need to maintain the \\ per capita capital stock constant.'' (Blanchard and Fischer, 1989)
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Resource Constraint}
	\begin{itemize}
		\item Economy's resource constraint:
		\[ K_t + F(K_t,A_tL_t) = K_{t+1} + C_{1t}L_t + C_{2t}L_{t-1} \] 
		\item Divide through by $A_tL_t$
		\[ k_t + f(k_t) = (1+n)k_{t+1} + A_t^{-1} c_t\]
		where $c_t = C_{1t} + (1+n)^{-1}C_{2t}$ {\footnotesize(weighted average of young and old consumption)}
		\item In steady state with $g = 0$:
		\[ A^{-1} c = f(k) - n k  \] 
	\end{itemize}
\end{frame}



\begin{frame}{Golden Rule Capital}
	\begin{itemize}
		\itemsep1em 
		\item In steady state with $g = 0$
		\[ A^{-1} c = f(k) - nk \]
		\item $c$ is maximized when 
		\[ f'(k_{GK}) = n\]  
		which implicitly gives the Golden Rule capital stock
	\end{itemize}
\end{frame}


\begin{frame}{OLG Market Steady State}
	\begin{itemize}
		\item OLG savings locus:
		\[ k_{t+1} = \frac{(1-\alpha)}{(1+n)(1+g)(2+\rho)}k_{t+1}^{\alpha} \]
		\item With $g = 0$ and in steady state:
		\[ k^* = \frac{(1-\alpha)}{(1+n)(2+\rho)}k^{*\alpha} \]
		which simplifies to
		\[ k^* = \left[ \frac{(1-\alpha)}{(1+n)(2+\rho)} \right]^{1/(1-\alpha)} \]  
	\end{itemize}
\end{frame}


\begin{frame}{OLG Market Steady State}
	\begin{itemize}
		\itemsep1em 
		\item If
		\[ k^* = \left[ \frac{(1-\alpha)}{(1+n)(2+\rho)} \right]^{1/(1-\alpha)} \]
		then
		\[ f'(k^*) = \alpha k^{*\alpha-1} = \frac{\alpha}{1-\alpha}(1+n)(2+\rho)  \]
		\item We have ignored depreciation. If $f(k) = k^{\alpha} - \delta k$:
		\[ f'(k^*) = \frac{\alpha}{1-\alpha}(1+n)(2+\rho) - \delta  \] 
		\item Recall that $r = f'(k)$. So, we have
		\[ r^* = \frac{\alpha}{1-\alpha}(1+n)(2+\rho) - \delta \] 
	\end{itemize}
\end{frame}


\begin{frame}{Dynamic Inefficiency}
	\begin{itemize}
		\itemsep1em 
		\item If 
		\[ r^* < n \]
		economy has more capital than Golden Rule capital
		\item This outcome is Pareto inefficient
		\item Economy is said to be dynamically inefficient
		\item Suppose in some period $t_0$, social planner cuts capital to $k_{GK}$
		\begin{itemize}
			\item In period $t_0$: More resources available for consumption due to cut
			\item In periods $t > t_0$: More resources available for consumption because \\ $n k$ falls more than $f(k)$ 
		\end{itemize}
		\item This policy change can thus make everyone better off
	\end{itemize}
\end{frame}


\begin{frame}{Dynamic Inefficiency}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/Romer2019Figure213.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Romer (2019)}
\end{frame}


\begin{frame}{Dynamic Inefficiency: Intuition}
	\begin{itemize}
		\itemsep1em 
		\item Only technology available to households to transfer resources from when they are young to when they are old is capital accumulation
		\item At the margin, the return on this technology is 
		\[ r = f'(k) \] 
		\item If households are patient enough, they will accumulate capital to the point where
		\[ r < n \] 
		\item They have no private reason to pay any attention to $n$
	\end{itemize}
\end{frame}

\begin{frame}{Dynamic Inefficiency: Intuition}
	\begin{itemize}
		\itemsep1em 
		\item Society (the government) has another technology for transferring resources from the young to the old
		\item The government can simply:
		\begin{itemize}
			\item Take $d$ units from each young
			\item Give $(1+n)d$ units to each old
		\end{itemize} 
		\item Notice that the ``return'' on this technology is $n$ \\ {\small (because the old generation is less populous than the young)}
		\item Must be repeated forever to be a Pareto improvement
		\item If $r < n$, this ``government technology'' is better than what is available \\ to people ``in the market'' (i.e., through saving or bilateral trade)
	\end{itemize}
\end{frame}


\begin{frame}{Dynamic Inefficiency}
	\begin{itemize}
		\itemsep1em 
		\item With growth in output per person ($g \neq 0$) we get
		\begin{itemize}
			\item Economy is dynamically efficient if
			\[ r^* > g + n \]
			\item Economy is dynamically inefficient if
			\[ r^* < g + n \]
		\end{itemize}
		\item This suggests a way to test dynamic efficiency
		\item Complication: Which interest rate to use?	\\ {\small (More on this later.)}	
	\end{itemize}
\end{frame}

\begin{frame}{Why Inefficiency?}
	\begin{itemize}
		\itemsep1em 
		\item It may seem puzzling that the market equilibrium in inefficient
		\item What is the failure of the First Welfare Theorem?
		\begin{itemize}
			\item All markets are competitive
			\item All agents are rational
			\item Property rights are well defined and costlessly enforced
		\end{itemize}
		\item Isn't this enough?
	\end{itemize}
\end{frame}


\begin{frame}{Problem with Infinity}
	\begin{itemize}
		\itemsep1em 
		\item Things can get complicated when there are an infinite number of agents
		\item Consider ``government technology'' discussed above:
		\begin{itemize}
			\item Take 1 from each young and give $1+n$ to each old \\ {\footnotesize (Recall that the young generation is more populous)}
			\item Do this again next period, and so on
			\item If return to saving is less than $n$, this makes everyone better off
		\end{itemize}
		\item This scheme only works if there are infinite number of generations
		\item FWT holds with infinite agents if present value of endowments is finite \\ {\small (which does not hold if economy is dynamically inefficient)}
	\end{itemize}
\end{frame}


\begin{frame}{Public Debt}
	\begin{itemize}
		\itemsep0.5em 
		\item When $r < n$, government can issue debt at no cost
		\item Suppose government borrows $B$ from each young person
		\item Next period it owes $(1+r)B$ to each old. 
		\item Suppose it again borrows $B$ from each young
		\item Since there are $(1+n)$ young for each old, it borrows $(1+n)B$ \\ for each $(1+r)B$ that it owes
		\item System is self-financing as long as $r < n$!!
		\item With growth, relevant issue is perhaps debt-to-GDP ratio. \\ Relevant condition is then $r < g$ 
	\end{itemize}
\end{frame}


\begin{frame}{More Public Debt, Anyone?}
	\begin{figure}
		\centering
		\includegraphics[height = 0.7 \textheight]{FiguresTables/Blanchard2019Figure1.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard (2019)}
\end{frame}


\begin{frame}{Should We Issue More Public Debt?}
	\begin{itemize}
		\itemsep0.5em
		\item Looks like $r<g$ much of the time
		\item So, looks like public debt is a ``free lunch''
		\item Does this mean we should issue more?
		\item Well, public debt ``crowds out'' private capital
		\item But with $r<g$, isn't there overaccumulation of capital?
		\item Not so fast! Relevant $r$ for dynamic efficiency is not necessarily \\ the same as for debt sustainability
	\end{itemize}
\end{frame}


\begin{frame}{Should We Issue More Public Debt?}
	Blanchard (2019): \vspace{5pt}
	\begin{itemize}
		\itemsep1em 
		\item Two types of welfare effects of more debt:
		\begin{itemize}
			\item Lower capital accumulation
			\item Induced changes in returns to labor and capital
		\end{itemize}
		\item Relevant interest rate for first of these:
		\begin{itemize}
			\item Safe rate because safe rate is the ``risk adjusted'' rate of return on capital
		\end{itemize}
		\item Relevant interest rate for second of these:
		\begin{itemize}
			\item Average (risky) marginal return on capital
		\end{itemize}
		\item Welfare effects of more debt ambiguous
	\end{itemize}
\end{frame}


\begin{frame}{OLG Economy without Capital/Production}
	Consider the following simpler setting: \vspace{5pt}
	\begin{itemize}
		\itemsep0.5em 
		\item Two generation OLG model: young and old
		\item Population growth: $L_{t} = (1+n)^t$
		\item No production / \textbf{No capital} 
		\item Each young individual endowed with 1 unit of consumption good
		\item Old receive no endowment
		\item Consumption good is perishable 
		\item Individuals have standard utility function $U(C_{1t},C_{2t+1})$
	\end{itemize}
\end{frame}


\begin{frame}{Society's Consumption Possibilities}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BlanchardFischer1989Figure41.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard and Fischer (1989)}
\end{frame}


\begin{frame}{Individual's Lifetime C Possibilities}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BlanchardFischer1989Figure42.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard and Fischer (1989)}
\end{frame}


\begin{frame}{Barter Equilibrium}
	\begin{itemize}
		\itemsep0.75em 
		\item Given this set of possibilities, individual would choose an ``interior'' point (e.g., C on last slide)
		\item However, this is not attainable through bilateral trade
		\item Initial old have nothing to offer
		\item Initial young would like to exchange goods today for goods next period, but next period's young not yet born 
		\item No trade possible!!
		\item ``Market outcome'' is A on last slide, which is highly Pareto inefficient 
	\end{itemize}
\end{frame}


\begin{frame}{Shadow Interest Rate}
	\begin{itemize}
		\itemsep1em 
		\item Intertemporal trade not possible. So no actual interest rate
		\item But we can define a ``shadow interest rate''
		\item I.e., interest rate that would make young happy not to trade
		\item For ``normal preferences'', this interest rate would be -100\% \\ {\small (i.e., if $U'(C) \rightarrow \infty$ as $C \rightarrow 0$)}
		\item So, this simple case is clearly a case of 
		\[ r < n + g\] 
	\end{itemize}
\end{frame}


\begin{frame}{Pay-as-You-Go Government Pension System}
	\begin{itemize}
		\itemsep1em 
		\item Suppose the government transferred an amount $d < 1$ \\ from young to old from period $t$ onward
		\item Initial old obviously much better off
		\item Young and all future generations also better off
		\begin{itemize}
			\item No longer destitute in old age. 
		\end{itemize}
		\item For moderate $d$, an increase in $d$ is a Pareto improvement
		\begin{itemize}
			\item Marginal cost: $U'(1-d)$
			\item Marginal benefit $(1+n) U'((1+n)d)(1+\rho)^{-1}$
		\end{itemize}
		\item Increase in $d$ is a Pareto improvement as long as
		\[ (1+n) \frac{U'((1+n)d)}{(1+\rho)} > U'(1-d) \hspace{10pt} => \hspace{10pt} 1+n > 1+r \] 
	\end{itemize}
	{\footnotesize (Recall that $(1+r)^{-1} = U'(C_{t+1})/(U'(C_t)(1+\rho))$)} 
\end{frame}

\begin{frame}{Two kinds of Government Pension Systems}
	\begin{enumerate}
		\itemsep1em 
		\item Fully Funded
		\begin{itemize}
			\item Government forces young to save (buy capital)
			\item No effect on capital accumulation if people are fully rational \\ (and forced saving is not too large)
			\item Increases capital accumulation if people are myopic
		\end{itemize}
		\item Pay-as-You-Go
		\begin{itemize}
			\item Government taxes young and gives proceeds to current old
			\item Reduces capital accumulation if people are fully rational
			\item Welfare improving even with rational agents if economy is \\ dynamically inefficient ($r<n+g$)
		\end{itemize}
	\end{enumerate}
	\vspace{15pt}
	{\small (See Blanchard and Fischer (1989, ch. 3.2))}
\end{frame}


\begin{frame}{Intergenerational Risk Sharing}
	\begin{itemize}
		\itemsep1em 
		\item We have ignored risk up until now
		\item Risk introduces another source of inefficiency in OLG models
		\item Efficient intergenerational risk sharing is not possible
		\item Suppose there is a shock at time $t$:
		\begin{itemize}
			\item Efficient to smooth the shock over infinite future
			\item This will not happen in an OLG model
		\end{itemize}
		\item Gov. pension system can help bring about efficient risk sharing
		\item Ball and Mankiw (2007) take a ``first stab'' at this
	\end{itemize}
\end{frame}


\begin{frame}{Pure Fiat Money}
	\begin{itemize}
		\itemsep0.7em 
		\item Consider again the simple barter economy 
		\item Suppose at $t=0$ government gives old H units of (completely divisible) inherently useless green pieces of paper
		\item Let's call these pieces of paper money 
		\item Suppose the old and every future generation believe they will be able to exchange goods for money at price $P_t$ in period $t$
		\item $P_t$ is the price level in this economy
		\item If this is an equilibrium, individuals can trade:
		\begin{itemize}
			\item Buy money for goods when young
			\item Sell money for goods when old
		\end{itemize}
	\end{itemize}
\end{frame}


\begin{frame}{Household Problem}
	\begin{itemize}
		\item Maximize 
		\[ U_t = \frac{C_{1t}^{1-\theta}}{1-\theta} + \frac{1}{1+\rho} \frac{C_{2t+1}^{1-\theta}}{1-\theta} \]
		subject to 
		\[ P_t(1-C_{1t}) = M^d_t \]
		\[ P_{t+1} C_{2t+1} = M^d_t \] 
		\item Plugging constraints into objective, differentiating, setting result to zero, and rearranging yields:
		\[ \frac{M_t^d}{P_t} = \frac{1}{1+(1+\rho)^{1/\theta}\Pi_{t+1}^{(\theta-1)/\theta}} \hspace{20pt} \text{where} \hspace{20pt} \Pi_{t+1} = \frac{P_{t+1}}{P_t} \] 
		\item This is the money demand function, also the savings function 
	\end{itemize}
\end{frame}


\begin{frame}{Money Demand}
	\[ \frac{M_t^d}{P_t} = \frac{1}{1+(1+\rho)^{1/\theta}\Pi_{t+1}^{(\theta-1)/\theta}} \hspace{20pt} \] \vspace{-5pt}
	\begin{itemize}
		\itemsep1em 
		\item $\Pi_{t+1}$ is the (inverse of the) rate of return on money
		\item Effect of an increase in $\Pi_{t+1}$ on money demand ambiguous
		\begin{itemize}
			\item If $\theta > 1$, higher $\Pi_{t+1}$ leads to lower money demand \\ (substitution effect dominates)
			\item If $\theta < 1$, higher $\Pi_{t+1}$ leads to higher money demand \\ (income effect dominates)
		\end{itemize}  
		\item Let's denote money demand function:
		\[ \frac{M_t^d}{P_t} = L(\Pi_{t+1}) \] 
	\end{itemize}
\end{frame}


\begin{frame}{Equilibrium with Money}
	\begin{itemize}
		\item Money demand equal to money supply:
		\[ (1+n)^t M_t^d = H \]
		\item Also true in period $t+1$
		\[ (1+n)^t M_t^d = (1+n)^{t+1} M_{t+1}^d \]
		\item Dividing by $P_t$ on both sides:
		\[ \frac{M_t^d}{P_t} = (1+n) \frac{P_{t+1}}{P_t} \frac{M_{t+1}^d}{P_{t+1}} \]
		\item Plugging in for money demand:
		\[ L(\Pi_{t+1}) = (1+n) \Pi_{t+1} L(\Pi_{t+2}) \]    
	\end{itemize}
\end{frame}


\begin{frame}{Equilibrium with Money}
	\[ L(\Pi_t) = (1+n) \Pi_{t} L(\Pi_{t+1}) \] \vspace{-10pt}
	\begin{itemize}
		\item Consider a steady state where
		\[ \Pi_{t} = \Pi_{t+1} = \bar{\Pi} \]
		\item Then we have that
		\[ L(\bar{\Pi}) = (1+n) \bar{\Pi} L(\bar{\Pi}) \]  
		\item This simplifies to 
		\[ \bar{\Pi} = (1+n)^{-1} \] 
	\end{itemize}
\end{frame}


\begin{frame}{Equilibrium with Money}
	\begin{itemize}
		\itemsep1em 
		\item This means that there is an equilibrium of the model with a constant inflation rate equal to $(1+n)^{-1}$
		\item Return on holding money is $\Pi^{-1}$
		\item In equilibrium with constant inflation rate, return on holding money is 
		\[ \bar{\Pi}^{-1} = (1+n) \]
		\item This is the ``golden rule'' return on assets in this economy
		\item Money allows economy to reach efficient equilibrium 
	\end{itemize}
\end{frame}


\begin{frame}{Consumption Possibilities with Money}
	\begin{figure}
		\centering
		\includegraphics[height = 0.8 \textheight]{FiguresTables/BlanchardFischer1989Figure41.pdf}
	\end{figure}
	\vspace{-4mm}
	{\scriptsize Source: Blanchard and Fischer (1989)}
\end{frame}


\begin{frame}{Fiat Money in OLG Model}
	\begin{itemize}
		\itemsep1em 
		\item Money is intrinsically worthless in this model
		\item Yet, it is valued in equilibrium
		\item Valued because everyone believes it will continue to be valued
		\item Not just valued, it allows economy to reach Pareto efficient outcome!
	\end{itemize}
\end{frame}


\begin{frame}{Fiat Money and Time Horizon}
	\begin{itemize}
		\itemsep1em 
		\item For money to be valued, economy must go on forever
		\item If world ends at time $T$, money will not be valued in period $T$
		\item If money not valued in period $T$, also not valued in period $T-1$
		\item[]
		\item Many other equilibria including one were money is not valued
		\item If people don't believe money will be valued tomorrow, \\ it will not be valued today
		\item Lots of equilibria in between
	\end{itemize}
\end{frame}


\begin{frame}{Fragility of Monetary Equilibrium}
	\begin{itemize}
		\itemsep1em 
		\item In simple economy $r < n$
		\item In economy with assets with $r > n$, there is no monetary equilibrium (Blanchard and Fischer, 1989, ch. 4.1)
		\item Monetary equilibrium only exists when economy is \\ dynamically inefficient
		\item Money plays the same role as government pension system
	\end{itemize}
\end{frame}


\begin{frame}{Money and OLG Model}
	\begin{itemize}
		\itemsep1em 
		\item In OLG model, money is only valued if it is not dominated \\ in rate of return
		\item In reality, money is dominated in rate of return 
		\item In OLG model, money is a store of value
		\item In reality, money is a unit of account (and medium of exchange)
		\item OLG model doesn't capture some crucial features of money
	\end{itemize}
\end{frame}


\begin{frame}{Rational Bubbles}
	\begin{itemize}
		\itemsep1em 
		\item In OLG model, money can be valued even though it pays no dividends
		\item Example of a ``rational bubble''
		\item Bubble: Asset that has a higher price than discounted value \\ of future dividends
		\item Bubbles cannot arise in Ramsey model
		\item Bubbles can arise in OLG model \\ {\small (Tirole, 1985; Blanchard and Fischer, 1989, ch. 5)}
		\item Bubbles can arise in some other settings as well \\ {\small (Santos and Woodford, 1997)}
	\end{itemize}
\end{frame}


\end{document}

