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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 2}\\[1ex]
{\Large A Simple Macro Model, Equilibrium \& Welfare Theorems}\\
	\noindent \hspace{-0.3cm} \rule{12.8cm}{0.6pt} \vspace{-0.7cm}}
\author{Macroeconomics EC2B1\\[3ex]  Benjamin Moll \\\vspace{0.5cm}}
\institute{London School of Economics, Winter 2024}





\begin{document}

%%%%%%%%%%%%%%%%%%

{
\setbeamertemplate{footline}{}
\begin{frame}[noframenumbering]
\titlepage
\end{frame}
}





\begin{frame}{Today: your first modern macro model}
\bi
\bigskip
\item As already mentioned: model is very (ridiculously) simple, a baby version
\bigskip
\item ... but it will have key features of most modern macro models
\be
\smallskip
\item micro foundations
\smallskip
\item general equilibrium
\ee
\bigskip
\item To make point, will show you what richer modern macro model looks like
\bi
\medskip
\item Heterogeneous Agent New Keynesian (HANK) model from my work\\
\vspace{-1mm} {\scriptsize \url{https://benjaminmoll.com/HANK/}}
\medskip
\item one of the most complicated models I can think of...
\medskip
\item ... but will show you: structure is exactly the same
\ei
\ei
\end{frame}




\begin{frame}{Plan}
\be
\item Reminder: what we want a macro model to speak to
\bigskip
\item Overview of simple macro model
\bigskip
\item Optimal resource allocation: Robinson Crusoe's problem
\bigskip
\item Decentralized competitive equilibrium
\bigskip
\item First and second welfare theorems: the ``invisible hand"
\bigskip
\item What a rich modern macro model looks like: HANK model
\bigskip
\item Putting the simple macro model to use: long-run trends in hours worked
\ee
\end{frame}


\begin{frame}{Useful Readings (most on moodle)}
\vspace{-2mm}
\bi
\item Your EC2A1 lecture notes on ``The Market System" (Weeks 1 and 2)
\bi
\medskip
\item treatment very complementary though with some differences
\ei
\medskip
\item Varian, chapters 32 and 33
\bi
\medskip
\item ch. 32: treatment of first and second welfare theorems
\medskip
\item ch. 33: application to Robinson Crusoe economy
\ei
\bigskip
\item Chapter 9 of Kurlat, particularly chapter 9.2
\bi
\medskip
\item but our treatment is simpler: we use a static one-period model (like Varian and Williamson)
\medskip
\item ... whereas Kurlat uses more complicated two-period model
\ei
\medskip
\item Williamson, chapters 4 and 5
\medskip
\item (PhD-level treatment so only read if interested, also not on moodle) Chapter 10.B of MasColell-Whinston-Green ``Microeconomic Theory"
\ei
\end{frame}


\begin{frame}{Reminder: what we want a macro model to speak to}
\vspace{-4mm}
Data on macroeconomic aggregates like
\be
\medskip
\item GDP = aggregate output 
\medskip
\item aggregate hours worked
\medskip
\item aggregate consumption
\medskip
\item aggregate investment
\medskip
\item inflation
\medskip
\item unemployment
\medskip
\item ...
\ee
\bigskip
Today: build simple model that can speak to 1 to 3. Recall:
\bi
\bigskip
\item objective is \tr{not} to build one big model we use to address all issues
\bigskip
\item instead ``custom build" models for particular questions
\ei
\end{frame}



\begin{frame}{Reminder from EC1P1: GDP}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Ricardo_GDP_slide1.pdf}
\end{figure}
\end{frame}


\begin{frame}{Reminder from EC1P1: GDP}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Ricardo_GDP_slide2.pdf}
\end{figure}
\end{frame}


\begin{frame}{Reminder from EC1P1: GDP}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Ricardo_GDP_slide3.pdf}
\end{figure}
\end{frame}

\begin{frame}{Aggregate hours worked over time}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Kurlat_hours2.pdf}
\end{figure}
\end{frame}

\begin{frame}{Aggregate hours worked and GDP across countries}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Kurlat_hours1.pdf}
\end{figure}
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Overview of simple macro model}
\end{center}
\end{frame}


\begin{frame}{Primitives of the simple macro model}
\vspace{-3mm}
\bi
\item \tr{\textbf{Preferences:}}
$$u(c) - v(h) $$
\vspace{-5mm}
\bi
\item $c$: consumption of coconuts, $u'(c)>0,u''(c)<0$
\smallskip
\item $h$: hours worked, $v'(h)>0,v''(h)>0$
\ei
\bigskip
\item \tr{\textbf{Technology:}}
$$y = f(n)$$
\vspace{-5mm}
\bi
\item $y$: output, i.e. production of coconuts
\smallskip
\item $n$: hours employed, $f'(n)>0, f''(n)<0$
\ei
\bigskip
\item \tr{\textbf{Resource constraint:}}
$$c = y, \qquad n = h $$
\ei
\end{frame}


\begin{frame}{Study two ways of organizing economy with these primitives}
\vspace{-2mm}
\be
\item \textbf{\tr{``Optimal resource allocation"}}
\bi
\medskip
\item Robinson has utility function $u(c) - v(h)$ and production function $y=f(n)$ and optimally chooses how much to consume and work
\medskip
\item no markets, prices with this organization, only physical quantities of coconuts and labor (the ``allocation")
\ei
\pause
\bigskip
\item \textbf{\tr{``Decentralized competitive equilibrium"}}
\bi
\medskip
\item representative household with utility function $u(c) - v(h)$ 
\medskip
\item representative firm with production function $y=f(n)$ (``Crusoe Inc.")
\medskip
\item (think of large number of identical households, large number of identical firms rather than literally one household and one firm)
\medskip
\item competitive coconut market: households buy, firms sell coconuts
\medskip
\item competitive labor market: households sell, firms buy labor
\ei
\ee
\end{frame}



\begin{frame}{Study two ways of organizing economy with these primitives}
Will show:
\bi
\bigskip
\item even though these sound very different: \tr{close connection}
\bigskip
\item there is a deep, much more general reason for this: \tr{welfare theorems}
\ei
\bigskip
Logic same as in EC2A1 but
\bi
\medskip
\item EC2A1: \tr{exchange economy} (or endowment economy) $\Rightarrow$ distribution of resources across different individuals, but no production
\medskip
\item here: \tr{production economy} (Robinson Crusoe economy) $\Rightarrow$ production but representative household so nothing to say about distribution
\ei
\end{frame}




\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\huge Optimal Resource Allocation:\\[1ex] Robinson Crusoe's Problem}
\end{center}
\end{frame}


\begin{frame}{Robinson Crusoe's Problem}
\bi
\item Robinson Crusoe is alone on an island, he lives of coconuts which he harvests, harvesting coconuts takes time
\bigskip
\item $\Rightarrow$ he chooses $c$ and $n$ to maximize his utility $u(c) -v(n)$ subject to the constraint that he can only eat the coconuts he harvested $c=f(n)$
$$\max_{c,n} \ u(c) - v(n) \qquad \mbox{s.t.} \qquad c = f(n) $$
\item Note: have used that $c=y$ and $n=h$ to drop $y$ and $h$
\bigskip
\pause
\item Optimality condition: $u'(c)f'(n) = v'(n)$ or
$$ \frac{v'(n)}{u'(c)} = f'(n) $$
\item Together with $c=f(n)$ this pins down optimal $(n^*,c^*)$
\ei
\end{frame}


\begin{frame}{Graphical representation of optimality condition}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/7.eps}
\end{figure}
\end{frame}

\begin{frame}{An increase in productivity from $A_0$ to $A_1>A_0$}
\vspace{-4mm}
Example: Robinson figures out that ladder $\Rightarrow$ more coconuts per hour worked
\begin{figure}[tph]
	\centering
\includegraphics[width=.72\textwidth]{figures/8.eps}
\end{figure}
\end{frame}


\begin{frame}{What will happen to consumption and hours worked?}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/9.eps}
\end{figure}
\end{frame}


\begin{frame}{Intuition: income and substitution effects}
\tr{\textbf{Substitution effect:}}
\bi
\medskip
\item $A \uparrow \Rightarrow$ Robinson's marginal product of labor $\uparrow$
\medskip
\item $\Rightarrow$ \tr{Robinson works more} (he substitutes leisure for consumption)
\ei
\bigskip
\tr{\textbf{Income effect:}}
\bi
\medskip
\item $A \uparrow \Rightarrow$ Robinson gets more output per hour worked (he is richer)
\medskip
\item $\Rightarrow$ \tr{Robinson works less} (he consumes more leisure because leisure is a normal good, i.e. a good whose consumption increases with income)
\ei
\end{frame}



\begin{frame}{Parametric example: functional forms from last lecture}
\vspace{-4mm}
\bi
\item Production function
$$f(n) = A n^\alpha, \quad 0<\alpha < 1,A>0$$
\item Utility function
$$u(c) = \log c, \quad v(n) = \theta \frac{n^{1+1/\varepsilon}}{1+1/\varepsilon}, \quad \theta, \varepsilon >0 \ \left(\mbox{note:} \ v'(n) = \theta n^{1/\varepsilon} \right)$$
\item Optimality condition
$$\frac{v'(n)}{u'(c)} = f'(n) \quad \Rightarrow \quad \frac{\theta n^{1/\varepsilon}}{1/c} = \alpha A n^{\alpha-1} $$
\item Using that $c = A n^\alpha$, solution is
\tr{$$n^* = \left(\frac{\alpha}{\theta} \right)^{\frac{\varepsilon}{1+\varepsilon}}, \qquad c^* = A\left(\frac{\alpha}{\theta} \right)^{\frac{\alpha\varepsilon}{1+\varepsilon}} $$}
\item How do these vary with $A,\theta,...$ and why?
\ei
\end{frame}


\begin{frame}{An increase in productivity in the parametric example}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/10.eps}
\end{figure}
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Competitive Equilibrium}
\end{center}
\end{frame}

\begin{frame}{Competitive equilibrium: Plan}
\be
\item Define ``competitive equilibrium" for a general, abstract economy
\bi
\medskip
\item only sketch this in lecture notes, see \tr{supplement} for details
\medskip
\item important thing is not the precise maths but to understand general structure
\ei
\bigskip
\item Apply analysis to Robinson Crusoe economy
\ee
\end{frame}




\begin{frame}{Preview of general definition of competitive equilibrium}
The following will be useful only in retrospect after you've seen next slides / supplement\\
\bigskip
A \tr{competitive equilibrium (CE)} are quantities and prices that satisfies \tr{three types of conditions}
\be
\bigskip
\item Households maximize taking prices as given
\bigskip
\item Firms maximize taking prices as given
\bigskip
\item All markets clear
\ee
\bigskip
\bigskip
Next: flesh this out in more detail
\end{frame}


\begin{frame}{Definition of CE for general, abstract economy (supplement)}
\vspace{-3mm}
\bi
\item $I$ consumers (households) indexed by \tr{$i=1,...,I$}
\medskip
\item $J$ producers (firms) indexed by \tr{$j=1,...,J$}
\medskip
\item $K$ factors of production (inputs) indexed by \tr{$k=1,...,K$}
\medskip
\item $L$ final goods indexed by \tr{$\ell = 1,...,L$}
\ei
\bigskip
\bi
\item \tr{\textbf{Preferences:}} household $i$'s utility over $L$ goods and $K$ factors
\medskip
\item \tr{\textbf{Technology:}} firm $j$'s production function for producing good $\ell$
\medskip
\item \tr{\textbf{Resource constraints (feasibility):}} 
\begin{align*}
\textrm{total demand of good $\ell$} &= \textrm{total supply of good $\ell$}, \quad \mbox{all} \ \ell = 1,...,L\\
\textrm{total demand of factor $k$} &= \textrm{total supply of factor $k$}, \quad \mbox{all} \ k = 1,...,K
\end{align*}
\ei
\end{frame}


\begin{frame}{Prices and price-taking (from EC1A1 notes)}
\vspace{-3mm}
Final bit of notation: prices
\bi
\smallskip
\item $p_{\ell}$: price of good $\ell=1,...,L$
\smallskip
\item $\tilde{p}_{k}$: price of factor $k=1,...,K$
\ei
\bigskip
\tr{\textbf{Assumption:}} both households and firms are \tr{price takers}. From EC1A1 notes:
\begin{figure}[tph]
	\centering
\includegraphics[width=.9\textwidth]{figures/EC1A1_price_taking.pdf}
\end{figure}
\end{frame}


\begin{frame}{General definition of competitive equilibrium (CE)}
\tr{\textbf{Definition:}} a competitive equilibrium are quantities and prices $\{p_{\ell},\tilde{p}_k\}$ for $\ell=1,...,L$, $k=1,...,K$, $i=1,...,I$ and $j=1,...,J$ such that:
\be
\medskip
\item Utility maximization: \tr{taking as given prices $\{p_{\ell},\tilde{p}_k\}$}, households maximize utility subject to their budget constraints
\bigskip
\item Profit maximization: \tr{taking as given prices $\{p_{\ell},\tilde{p}_k\}$}, firms maximize profits
\medskip
\item Market clearing: demand = supply for each good and each factor
\begin{align*}
\textrm{total demand of good $\ell$} &= \textrm{total supply of good $\ell$}, \quad \mbox{all} \ \ell = 1,...,L\\
\textrm{total demand of factor $k$} &= \textrm{total supply of factor $k$}, \quad \mbox{all} \ k = 1,...,K
\end{align*}
\ee
\end{frame}




\begin{frame}{Useful facts about CEs: numeraire good and Walras' law}
\vspace{-4mm}
\bi
\item \tr{Numeraire good:} only relative prices $p_\ell/p_{\ell'}$ matter $\Rightarrow$ can always ``normalize" price of one good or factor to $1$, e.g. price of good $1$, $p_1 = 1$
\bi
\smallskip
\item only $N-1$ variable where $N=K+L = $ number of goods and factors
\ei
\medskip
\item \tr{Walras' law} in words: if we have found prices $\{p_{\ell},\tilde{p}_k\}$ such that all markets clear except one (i.e. demand = supply in all markets except one), then the remaining market must also clear (i.e. demand must also equal supply in that remaining market)
\bi
\smallskip
\item only $N-1$ equations which makes sense given $N-1$ variables
\ei
\medskip
\item Idea of proof (see Varian and MWG textbooks for formal proof):
\bi
\smallskip
\item add up all budget constraints and all firm profits
\smallskip
\item ... use all market clearing conditions except one
\smallskip
\item then the algebra implies the one remaining market clearing condition
\ei
\medskip
\item Useful implication: when computing a competitive equilibrium in practice we can drop one market clearing condition (and we will frequently do so)
\ei
\end{frame}




\begin{frame}{Now: apply analysis to Robinson Crusoe economy}
Special case with:
\bi
\medskip
\item $I=1$ consumer: representative household
\medskip
\item $J=1$ producer: representative firm
\medskip
\item $K=1$ factor of production: labor
\medskip
\item $L=1$ final good: coconuts
\ei
\bigskip
Prices:
\bi
\medskip
\item make coconuts the numeraire and normalize its price to 1
\medskip
\item only price is the wage (= price of labor)
\ei
\bigskip
Ownership: the representative household owns the representative firm and therefore receives its profits as dividends
\end{frame}


\begin{frame}{Competitive equilibrium in Robinson Crusoe economy}
\tr{\textbf{Definition:}} a competitive equilibrium in the Robinson Crusoe economy are quantities $(c,y,n,h)$ and a wage $w$ such that
\be
\medskip
\item Utility maximization: taking as given $w$ (and also $\Pi$), the representative household chooses $(c,h)$ to solve
$$\max_{c,h} \ u(c) - v(h) \quad \mbox{s.t.} \quad c = w h + \Pi$$
\item Profit maximization: taking as given $w$, the representative firm chooses $n$ and $y=f(n)$ to solve
$$\Pi = \max_n \ f(n) - wn $$
\smallskip
\item Market clearing: demand = supply for both coconuts and labor
\begin{align*}
\mbox{coconut market:} \qquad  c &= y \\[0.5ex]
\mbox{labor market:} \qquad n &= h
\end{align*}
\ee
\end{frame}



\begin{frame}{Comments on CE in Robinson Crusoe economy}
\vspace{-3mm}
Taken literally, the CE defined on previous slide has following features
\be
\smallskip
\item one household, one firm
\smallskip
\item household = firm's only employee
\smallskip
\item household = firm's only customer
\smallskip
\item firm is fully owned by household
\ee
\bigskip
\textbf{Question:} \tr{isn't this completely schizophrenic?}
\bi
\medskip
\item if it's literally just one household and one firm, why would they trade with each other in these competitive markets and even act as price takers?
\medskip
\item surely the household would just decide how much to work, tell the firm how much to produce, then eat it?
\medskip
\item why would Robinson articifically split himself in two and trade with himself?
\ei
\medskip
\textbf{Answer:} yes, this is 100\% correct, taken literally this definition of a competitive equilibrium is completely schizophrenic
\end{frame}


\begin{frame}{Alternative interpretation: large number of households, firms}
\vspace{-3mm}
\bi
\item \tr{BUT} there is an alternative interpretation that is considerably less crazy:
\bi
\smallskip
\item there is a \tr{large number of households} $i=1,...,I$
\smallskip
\item there is a \tr{large number of firms} $j=1,...,J$
\smallskip
\item all households are \tr{identical}: have same utility function $u(c_i,h_i)$
\smallskip
\item all firms are \tr{identical}: have same production function $f(n_i)$
\smallskip 
\item resource constraints / market clearing conditions:
$$\sum_{i=1}^I c_i = \sum_{j=1}^J y_j, \qquad \sum_{j=1}^J n_j = \sum_{i=1}^I h_i, $$
\ei
\medskip
\item In this interpretation, trade and price-taking are much more sensible
\medskip
\item But equations same: clearly all households choose same $(c_i,h_i)$, all firms choose same $(y_j,n_j)$
\medskip
\item $\Rightarrow$ think of $(c,h,y,n)$ as variables in per capita terms
\ei
\end{frame}




\begin{frame}{Equations characterizing CE in Robinson Crusoe economy}
\tr{\textbf{Definition:}} a competitive equilibrium in the Robinson Crusoe economy are quantities $(c,y,n,h)$ and a wage $w$ such that
\be
\medskip
\item Utility maximization: taking as given $w$, the representative household chooses $(c,h)$ to solve
$$\max_{c,h} \ u(c) - v(h) \quad \mbox{s.t.} \quad c = w h + \Pi$$
\item Profit maximization: taking as given $w$, the representative firm chooses $n$ and $y=f(n)$ to solve
$$\Pi = \max_n \ f(n) - wn $$
\smallskip
\item Market clearing: demand = supply for both coconuts and labor
\begin{align*}
\mbox{coconut market:} \qquad  c &= y \\[0.5ex]
\mbox{labor market:} \qquad n &= h
\end{align*}
\ee
\end{frame}

\begin{frame}{Equations characterizing CE in Robinson Crusoe economy}
\tr{\textbf{Definition:}} a competitive equilibrium in the Robinson Crusoe economy are quantities $(c,y,n,h)$ and a wage $w$ such that
\be
\medskip
\item Utility maximization: taking as given $w$, the representative household chooses $(c,h)$ such that
$$\frac{v'(h)}{u'(c)} = w $$
\item Profit maximization: taking as given $w$, the representative firm chooses $n$ and $y=f(n)$  such that
$$f'(n) = w $$
\smallskip
\item Market clearing: demand = supply for both coconuts and labor
\begin{align*}
\mbox{coconut market:} \qquad  c &= y \\[0.5ex]
\mbox{labor market:} \qquad n &= h
\end{align*}
\ee
\end{frame}



\begin{frame}{Equations characterizing CE in Robinson Crusoe economy}
Summary: a CE is quantities $(c,y,n,h)$ and a wage $w$ such that
\begin{align}
\frac{v'(h)}{u'(c)} &= w\\
f'(n) &= w\\
y &= f(n)\\
c &= y \\
n &= h
\end{align}
Always want to check: same number of equations and unknowns? \\
\bigskip
Here: 5 equations (1)-(5) in 5 unknowns $(c,y,n,h,w)$ so this looks promising\\
\end{frame}



\begin{frame}{Equations characterizing CE in Robinson Crusoe economy}
\bi
\item Let's solve these equations as far as possible:
\bi
\medskip
\item equating (1) and (2) and using (4)
\begin{equation}\tag{$\ast$}\label{eq:opt1}
\frac{v'(n)}{u'(c)} = f'(n) 
\end{equation}
\item using (3) and (4)
\begin{equation}\tag{$\ast \ast$}\label{eq:opt2}
c = f(n) 
\end{equation}
\ei
\item So equilibrium allocation $(c,n)$ satisfies \eqref{eq:opt1} and \eqref{eq:opt2}
\bigskip
\pause
\item \tr{But these are same equations as in optimal resource allocation problem!}
\bigskip
\item So equilibrium allocation $(c,n)$ satisfies same two equations as optimal allocation $(c^*,n^*)$ $\Rightarrow$ \tr{CE allocation = optimal allocation!}
\bigskip
\item What's going on?
\ei
\end{frame}


\begin{frame}{What's going on? Recall utility and profit maximization}
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/2.eps}\includegraphics[width=.5\textwidth]{figures/6.eps}
\end{figure}
\end{frame}



\begin{frame}{Competitive equilibrium in Robinson Crusoe economy}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/11.eps}
\end{figure}
\end{frame}


\begin{frame}{A remarkable fact about the competitive equilibrium}
\vspace{-4mm}
\bi
\item Have studied equilibrium in which households and firms maximize taking as given prices and trade in competitive markets
\bigskip
\item Somehow magically the resulting competitive equilibrium allocation = optimal allocation!
\bigskip
\item So even though both households and firms act purely in their own self-interest and only do what's individually optimal for them (maximize utility and profits), this generates an allocation that is jointly optimal!
\bigskip
\item Also has a convenient practical implication for us: instead of solving for CE can just solve for optimal allocation (which is much easier).
\bigskip
\item Natural questions:
\bi
\smallskip
\item is this true more generally? 
\smallskip
\item if so, when does this hold? Always? Sometimes? Under reasonable conditions? Under crazy conditions?
\ei
\ei
\end{frame}




\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Welfare Theorems: the ``Invisible Hand"}
\end{center}
\end{frame}

\begin{frame}{Pareto efficiency (or Pareto optimality)}
\vspace{-2mm}
\bi
\item Consider general economy from earlier with many individuals $i=1,...,I$ 
\medskip
\item \tr{\textbf{Definition:}} an allocation is Pareto efficient (PE) if it is impossible to find another feasible allocation that improves everyone's welfare
\bi
\smallskip
\item more precisely: there is no feasible allocation that makes one individual better off without making another worse off
\smallskip
\item feasible allocation = allocation that respects resource constraints
\ei
\medskip
\item If PE, government intervention cannot make everyone better off
\medskip
\item But note: while Pareto efficiency is desirable it is also a very weak requirement (a single person consuming everything may be PE)
\bi
\smallskip
\item ``A society can be Pareto optimal and still perfectly disgusting" (Sen)
\smallskip
\item even if PE, may still want to intervene for distributional reasons
\ei
\medskip
\pause
\item To find Pareto efficient allocations: solve problem of \tr{fictitious ``social planner"} who maximizes weighted sum of individual utilities s.t. feasibility
\ei
\end{frame}


\begin{frame}{Pareto efficiency in Robinson Crusoe economy}
\vspace{-3mm}
\bi
\item One individual or everyone identical (and gets same): no dist'nal conflict
\medskip
\item PE = allocation that attains highest utility of rep household s.t. feasibility
\medskip
\item \tr{\textbf{Definition:}} a Pareto efficient allocation in the Robinson Crusoe economy are quantities $(c,y,n,h)$ that maximize household utility
$$u(c) - v(h) $$
subject to being technologically feasible
$$y = f(n), \qquad c = y, \qquad n = h.$$
\item Can interpret as problem of fictitious \tr{benevolent social planner}
\medskip
\pause
\item But exact same problem we solved earlier! max $u(c) - v(n)$ s.t. $c = f(n)$\\
\medskip
\item So when we studied ``optimal resource allocation" or ``Robinson Crusoe's problem" we actually studied the Pareto efficient allocation in this economy\\
\medskip
\item \tr{In this economy, CE allocation = PE allocation}. True more generally?
\ei
\end{frame}




\begin{frame}{First and second welfare theorems}
Very roughly:
\bi
\bigskip
\item \tr{\textbf{1st welfare theorem:}} (under some strong assumptions) CE allocations are PE, i.e. CE $\Rightarrow$ PE
\bigskip
\item \tr{\textbf{2nd welfare theorem:}} (under even stronger assumptions*) PE allocations are CE, i.e. CE $\Leftarrow$ PE
\bigskip
\item Together: CE $\Leftrightarrow$ PE
\ei
\bigskip
*even stronger assumptions = same assumptions as 1st welfare theorem + ``convexity assumptions", e.g. utility and production functions are concave\\
\bigskip
For more precise treatment, see references in beginning of slides in particular Varian and MWG
\end{frame}


\begin{frame}{First welfare theorem: the ``invisible hand"}
\vspace{-3mm}
Focus on first welfare theorem and unpack assumptions a bit
\bi
\item \tr{\textbf{1st welfare theorem:}} \tr{if}
\be
\item perfect competition (individuals and firms are price takers)
\item individuals and firms are rational
\item no externalities
\item perfect information
\ee
\smallskip
\tr{then} CE allocations are Pareto efficient
\bigskip
\item How to think about this result and the ``invisible hand"?
\bigskip
\item Does it perhaps mean that free markets always deliver socially optimal outcomes and government intervention is never desirable?
\ei
\end{frame}



\begin{frame}{Limitations and uses of the first welfare theorem}
\vspace{-3mm}
\bi
\item Does 1st welfare theorem mean that free markets always deliver socially optimal outcomes and government intervention is never desirable?
\medskip
\item Definitely not. Instead \tr{1st welfare theorem is all about its assumptions}.
\bi
\smallskip
\item violations of these are called ``market failures" or ``frictions"
\smallskip
\item in real world and also in modern macro, market failures and frictions are everywhere
\smallskip
\item ``The reason that the invisible hand often seems invisible is that it is often not there" (Stiglitz)
\ei
\medskip
\item ... but 1st welfare theorem is nevertheless an important benchmark
\medskip
\item In my view, perhaps the most important use of 1st welfare theorem: force you to think about rationale for policy intervention
\bi
\medskip
\item what market failures, frictions, externalities? (or about distribution?)
\medskip
\item once identified, what policies can tackle these? (targeting principle)
\ei
\ei
\end{frame}


\begin{frame}{Idea of proof of 1st welfare theorem: CE $\Rightarrow$ PE}
\bigskip
\includegraphics[width=.44\textwidth]{figures/11.eps} \includegraphics[width=.1\textwidth]{figures/arrow.pdf}\includegraphics[width=.44\textwidth]{figures/7.eps}
Idea: if indifference curves and production possibilities are tangent to price line, then they are also tangent to each other and hence allocation is optimal
\end{frame}


\begin{frame}{Idea of proof of 2nd welfare theorem: PE $\Rightarrow$ CE}
\bigskip
\includegraphics[width=.44\textwidth]{figures/7.eps} \includegraphics[width=.1\textwidth]{figures/arrow.pdf}\includegraphics[width=.44\textwidth]{figures/11.eps}
Idea: if indifference curves and production possibilities are tangent to each other, then can find a price line that separates them in a tangent fashion
\bi
\medskip
\item in general (see MWG): a ``separating hyperplane"
\ei
\end{frame}



\begin{frame}{Link to EC2A1: logic exactly same as in exchange economy}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.7\textwidth]{figures/EC2A1_welfare_theorem} 
\end{figure}
\vspace{-2mm}
\bi
\item EC2A1: \tr{exchange economies} (or endowment economies)
\medskip
\item Here: \tr{production economies}, e.g. Robinson Crusoe economy
\medskip
\item Different uses but same logic, welfare theorems are very general
\ei
\end{frame}




\begin{frame}{Another use of welfare theorems: a shortcut for finding CE}
\bi
\item In many applications, planning problem is much easier to solve than CE
\bi
\medskip
\item think of Robinson Crusoe economy
\ei
\bigskip
\item In such cases: if welfare theorems hold, can use planning problem as a shortcut to find CE allocation
\bi
\medskip
\item will often see this in macro, and will sometimes do this in this course
\ei
\bigskip
\item Obviously only works if assumptions of welfare theorems hold, i.e. \tr{if there are no market failures / frictions}
\bi
\medskip
\item technically need PE $\Rightarrow$ CE so need 2nd welfare theorem assumptions (recall = same as 1st welfare theorem + ``convexity assumptions")
\medskip
\item in practice assumptions of 1st welfare theorem will fail so check those
\ei
\ei
\end{frame}




\begin{frame}{Equilibrium vs Planning Problem: Key Differences}
\bi
\item Even though the competitive equilibrium and planning problem may lead to the same allocation, they are very different animals
\bigskip
\item Some key differences to remember when you solve these in practice
\be
\medskip
\item equilibrium features \tr{prices}, the planning problem does not
\medskip
\item equation for resource constraint plays mathematically different roles:
\bi
\smallskip
\item in competitive equilibrium: market clearing condition
\smallskip
\item in planning problem: constraint on planner's maximization problem
\ei
\ee
\bigskip
\item A typical mistake that students make: write planning problem as planner maximizing utility subject to budget constraint (which features prices)
\bi
\medskip
\item Please don't do this. If in exam: zero points on that subquestion.
\ei
\ei
\end{frame}




\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.4cm}
\begin{center}
{\huge A much richer modern macro model:\\[1ex] HANK}
\end{center}
\end{frame}


\begin{frame}{A richer modern macro model}
\vspace{-4mm}
\bi
\item I picked one that I know well: a Heterogeneous Agent New Keynesian (HANK) model from my own work 
 {\scriptsize\url{https://benjaminmoll.com/HANK/}}
\begin{figure}[tph]
	\centering
\includegraphics[width= .45\textwidth]{figures/HANK_abstract.pdf}
\end{figure}
\vspace{-1mm}
\item ... but could have also picked lots of other papers / models
\smallskip
\item See Carlin-Soskice chapter on moodle for good accessible discussion of the heterogeneous agent and HANK literatures
\ei
\end{frame}


\begin{frame}{A richer modern macro model}
\vspace{-3mm}
\bi
\item A very complicated model with lots of bells \& whistles 
\bi
\medskip
\item lots of heterogeneity and inequality
\medskip
\item lots of assumptions that invalidate welfare theorems (borrowing constraints, incomplete insurance markets, monopolistic comp., ...)
\ei
\ei
\includegraphics[width=0.32\textwidth]{figures/liq_dist.eps} 
\includegraphics[width=0.32\textwidth]{figures/ill_dist.eps}
\includegraphics[width=0.32\textwidth]{figures/mpc500_quarterly_function_av.eps}
\bi
\item \tr{Still the model's structure is exactly the same as our baby model}
\ei
\end{frame}



\begin{frame}{Primitives of the HANK economy}
\vspace{-4mm}
\bi
\item \tr{\textbf{Preferences:}} \uncover<2->{
\vspace{-3mm}
\begin{figure}[tph]
	\centering
\includegraphics[width= .42\textwidth]{figures/HANK_preferences.pdf}
\end{figure}
}
\item \tr{\textbf{Technology:}} \uncover<2->{
\vspace{-3mm}
\begin{figure}[tph]
	\centering
\includegraphics[width= .42\textwidth]{figures/HANK_technology.pdf}
\end{figure}
}
\item \tr{\textbf{Resource constraints (feasibility):}}
\uncover<2->{
\begin{figure}[tph]
	\centering
\includegraphics[width= .42\textwidth]{figures/HANK_feasibility.pdf}
\end{figure}
}
\ei
\end{frame}

\begin{frame}{Equilibrium in the HANK model}
\begin{figure}[tph]
	\centering
\includegraphics[width= \textwidth]{figures/HANK_equilibrium.pdf}
\end{figure}
\begin{figure}[tph]
	\centering
\includegraphics[width= \textwidth]{figures/HANK_household_max.pdf}
\end{figure}
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\huge Long-run trends in hours worked}
\end{center}
\end{frame}




\begin{frame}{Aggregate hours worked over time}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Kurlat_hours2.pdf}
\end{figure}
\end{frame}

\begin{frame}{Aggregate hours worked and GDP across countries}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/Kurlat_hours1.pdf}
\end{figure}
\end{frame}


\begin{frame}{Can our parametric example reproduce these trends?}
\vspace{-4mm}
$$n^* = \left(\frac{\alpha}{\theta} \right)^{\frac{\varepsilon}{1+\varepsilon}}, \qquad c^* = A\left(\frac{\alpha}{\theta} \right)^{\frac{\alpha\varepsilon}{1+\varepsilon}} $$
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.65\textwidth]{figures/10.eps}
\end{figure}
\end{frame}


\begin{frame}{What alternative model specification can?}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.77\textwidth]{figures/12.eps}
\end{figure}
\end{frame}



\begin{frame}{Interesting recent book: ``Friday is the new Saturday"}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=\textwidth]{figures/gomes_praise}
\end{figure}
\end{frame}





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\end{document}

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