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\title{\vspace{1.6cm} \\
	{\LARGE Lecture 1}\\[1ex]
Overview,  Labor Demand and Supply\\
	\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}{} 
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\titlepage
\end{frame}
}




\begin{frame}{Who are we?}
\vspace{-3mm}
\tr{\textbf{Who am I?}} \pause Macroeconomist. When I don't teach I work on:
\be
\medskip
\item Why are some countries so much poorer than others?
\medskip
\item How does micro heterogeneity affect macro economy / policy?
\ee
\bi
\medskip
\item How does your professor tick? Look at current working papers \\[-0.5ex]{\scriptsize \url{https://benjaminmoll.com/papers/}}
\bi
\medskip
{\footnotesize \item Asset-Price Redistribution
\medskip
\item Present Bias Amplifies the Household Balance-Sheet Channels of Macroecon Policy
\medskip
\item What if? The Economic Effects for Germany of a Stop of Energy Imports from Russia
\medskip
\item The Great Lockdown and the Big Stimulus: Tracing the Pandemic Possibility Frontier 
\item also see} {\scriptsize\url{https://benjaminmoll.com/research_agenda_2020/}}
\ei
\ei
\pause
\bigskip
\tr{\textbf{Who is Stephan and the class teachers?}} You've already met... \\
\bigskip
\pause
\tr{\textbf{Who are you?}} How many undergraduates, 2-year MSc? Econ background?

\end{frame}


\begin{frame}{This Course: \tr{Modern} Macroeconomics}
\bi
\item Modern = as practiced in current academic research
\bi
\medskip
\item $\neq$ what's in standard textbooks
\ei
\bigskip
\item Macro with \tr{micro foundations}
\bi
\medskip
\item utility and profit maximization just like in EC1A1
\medskip
\item ... but general equilibrium (= prices endogenous, clear markets)
\ei
\bigskip
%\item First time I'm teaching this course at LSE
%\bi
%\medskip
%\item updated version of course at Princeton some years ago
%\medskip
%\item ... but everything is flexible, feedback very useful
%\ei
%\bigskip
%\item Everyhing
\ei
\end{frame}



\begin{frame}{Admin}
\vspace{-4mm}
\be
\item Lectures: Monday 10:00-12:00, Old Theatre
\medskip
\item Office hours: SAL 1.19 (my office)
\bi
\item Weeks 3-11: Mon 12:40-13:40
\item Weeks 1\&2 only: Tue 15:00-16:00 
\ei
\medskip
\item Classes and class teacher office hours: see moodle
\medskip
\item Problem sets (as before)
\bi
\smallskip
\item one per week
\smallskip
\item handed out after lecture, discussed in class a week later
\smallskip
\item feedback on two: week 4 (PS \#3) and week 10 (PS \#9)
\ei
\medskip
\item Assessment
\bi
\medskip
\item big exam at end of year
\medskip
\item project
\ei
\medskip
\item Anything else?
\ee
\end{frame}


\begin{frame}{What this course is about}
\bi
\item Wouter den Haan's MT part: long-run issues
\bi
\medskip
\item growth, cross-country income differences etc
\ei
\bigskip
\item \tr{\textbf{This term: short-run issues }}
\bi
\medskip
\item business cycles
\medskip
\item economic crises: financial, energy, ...
\medskip
\item macroeconomic stabilization policies: monetary, fiscal, ...
\ei
\ei
\end{frame}

\begin{frame}{Topics we will cover (subject to change)}
\begin{figure}[tph]
	\centering
\includegraphics[width=.95\textwidth]{figures/topics}
\end{figure}
\end{frame}



\begin{frame}{Tools}
\vspace{-2mm}
As just mentioned: macro with \tr{micro foundations}
\bi
\medskip
\item utility and profit maximization just like in EC1A1
\ei
\bigskip
Reflected in tools:
\bi
\medskip
\item differential calculus
\medskip
\item heavy use of \tr{\textbf{constrained optimization using Lagrangean methods}}
\medskip
\item Need to know how to solve 
$$\max_{c_1,c_2} \ u(c_1,c_2) \quad \mbox{s.t.} \quad p_1 c_1 + p_2 c_2 \leq m $$
using
$$\mathcal{L} = u(c_1,c_2) + \lambda \left[m - p_1 c_1 - p_2 c_2\right] $$\\[1ex]
\item If you don't, refresh your memory using MA100, MA107 or EC1A1 notes
\medskip
\item Assignment 1 and class 1 will be about this
\ei
\end{frame}


\begin{frame}{One of my main goals: train you to read newspaper properly}
\bi
\item Train you to critically read and evaluate discussions of macroeconomic policy in popular press, on TV, ...
\bigskip
\item Help you understand where people come from
\bi
\medskip
\item what's model in back of their minds (if any)?
\ei
\bigskip
\item Help you evaluate quality of argument
\bi
\medskip
\item is argument internally consistent?
\medskip
\item what empirical evidence is there and is it credible?
\medskip
\item you will find that there's huge variation in quality
\ei
\bigskip
\item ``Train your BS detector" so to speak
\ei
\end{frame}



\begin{frame}{Textbooks}
\vspace{-3mm}
Unfortunately, there exists no textbook with the proper level and focus for this course. Instead, it will rely on lecture notes and research journal articles\\
\bigskip
That being said, the following textbooks may be useful as background reading (given the extortionate prices of textbooks these days, I want to emphasize that these are not required):
\bi
\medskip
\item \tr{\textbf{Kurlat, Pablo, \emph{A Course in Modern Macroeconomics}}}\\
{\scriptsize \url{https://sites.google.com/view/pkurlat/a-course-in-modern-macroeconomics}}\\
\bi
\smallskip
\item if you do want to buy a book, buy this one (\$ 32.95) %used to be \$ 27.95 in 2022-23
\smallskip
\item but will also post relevant PDF excerpts on Moodle
\ei
\medskip
\item Jones, Chad, \emph{Macroeconomics} -- one of your books from EC1B1
\medskip
\item Williamson, Stephen, \emph{Macroeconomics}
\medskip
\item Romer, David, \emph{Advanced Macroeconomics}
\medskip
\item Abel, Andrew and Ben Bernanke and Dean Croushore, \emph{Macroeconomics}
\ei
\end{frame}


\begin{frame}{Finally, all of Wouter den Haan's tips still apply}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.6\textwidth]{figures/wouter_excerpt}
\end{figure}
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.5cm}
\begin{center}
{\huge Some thoughts on methodology:\\[1ex] what economists do and why}
\end{center}
\end{frame}

\begin{frame}{The interplay of models and data}
\vspace{-3mm}
\bi
\item Like in your previous courses, will make use of models and look at data
\bigskip
\item Example of model: firms choose labor demand to maximize profits
$$\max_{n} \ f(n) - w n $$
where $n$ is labor, $f(n)$ is output ($f$ is production function), $w$ is the wage
\bigskip
\item Example of data:
\begin{figure}[tph]
	\centering
\includegraphics[width=.8\textwidth]{figures/US_employment.png}
\end{figure}
\ei
\end{frame}



\begin{frame}{The interplay of models and data}
How to think about interplay of model and data?
\bi
\medskip
\item why not only data?
\medskip
\item why not only models?
\medskip
\ei
\bigskip
Here: quick, superficial treatment of important methodological issues
\bi
\medskip
\item part of much broader methodological debates (see references)
\medskip
\item my main goal here: get you thinking about these issues
\medskip
\item disclaimer: no single ``right" approach here, reflects my personal opinion
\ei
\end{frame}



\begin{frame}{The interplay of models and data}
Macroeconomics needs interplay of both theory and data
\be
\medskip
\item Theory without data
\bi
\medskip
\item ``smart kids playing in sandbox"
\ei
\bigskip
\item Data without theory
\bi
\medskip
\item flying blind
\medskip
\item ``Let the data speak" is fiction / wishful thinking
\medskip
\item there is no paper in economics without a (implicit) model
\medskip
\item policy questions are questions about counterfactuals (``what happens in world with policy relative to world without policy?")
\medskip
\item ... requires a model (either explicit or implicit)
\ei
\ee
\end{frame}


\begin{frame}{Further reading for the interested (not required)}
Some relevant references (discussion in many much deeper than here):\\
\bi
\medskip
\item Nakamura-Steinsson ``Identification in Macroeconomics"
\medskip
\item Reis ``Is something really wrong with macroeconomics?"
\medskip
\item JEP Symposium ``Con out of Economics" {\tiny \url{https://www.aeaweb.org/issues/126}}\\
\medskip
\item Heckman-Urzua ``Comparing IV With Structural Models: What Simple IV Can and Cannot Identify"
\medskip
\item Card ``Model‐Based or Design‐Based? Competing Approaches in
`Empirical Micro'" {\scriptsize\url{https://davidcard.berkeley.edu/lectures/woytinsky.pdf}}\\[-.5ex] {\scriptsize\url{https://www.youtube.com/watch?v=S6xSEiB6E2s}}
\ei
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{3cm}
\begin{center}
{\Huge Modeling in (Macro)economics}
\end{center}
\end{frame}


\begin{frame}{Modeling in (Macro)economics}
\vspace{-3mm}
Objective is \tr{not} to build one big model we use to address all issues
\begin{itemize}
\smallskip
\item descriptive realism is not the objective
\smallskip
\item instead make modeling choices that are dependent on the issue
\smallskip
\item whether a model is ``good" is context dependent
\end{itemize}
\medskip
Approach to modeling in (macro)economics well summarized by
\begin{itemize}
\smallskip
\item ``All models are false; some are useful"\\[-1ex] {\tiny \url{https://en.wikipedia.org/wiki/All_models_are_wrong}}
\smallskip
\item ``If you want a model of the real world, look out the window" (kidding, but only half kidding) 
\smallskip
\item ``The map is not the territory" {\tiny \url{https://en.wikipedia.org/wiki/Map-territory_relation}}
\smallskip
\item Literary version of this: Borges ``On Exactitude in Science" 
\smallskip
\item Point in case: geographically accurate London tube map
{\tiny \url{https://kottke.org/plus/misc/images/tubegeo.gif}}\\
{\tiny \url{https://www.ifs.org.uk/publications/10282}} by Bandiera, Machin,...
\end{itemize}

\end{frame}




\begin{frame}{London tube map {\scriptsize(\url{https://kottke.org/plus/misc/images/tubegeo.gif})}}
\vspace{-5mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.85\textwidth]{figures/tubegeo.pdf}
\end{figure}
\end{frame}


\begin{frame}{My own version (same ideas)\\[-1ex] {\tiny \url{https://economicsobservatory.com/how-are-economic-models-adapting-to-rising-inequality-and-the-pandemic}}\vspace{-3mm}}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=0.62\textwidth]{figures/moll-rickard.png}
\end{figure}
\end{frame}



\begin{frame}{Crucial or Critical Assumptions}{}
\begin{figure}[tph]
	\centering
\includegraphics[width=.7\textwidth]{figures/solow}
\end{figure}
{\small Source: Solow (1956) ``A Contribution to the Theory of Economic Growth"}
\end{frame}

\begin{frame}{Modeling in (Macro)economics}{}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.65\textwidth]{figures/rodrik-commandments.pdf}
\end{figure}
\vspace{-3mm}
Source: Rodrik (2015) ``Economics Rules" \\[-1ex] {\scriptsize \url{http://www.centreformacroeconomics.ac.uk/pdf/Events/Slides/1510-RodrikD.pdf}}
\end{frame}

\begin{frame}{Navigating Among Models}
\vspace{-3mm}
\bi
\item Chapter 3 ``Navigating Among Models" of ``Economics Rules"
\item Responses to this tweet\\[-1ex] {\scriptsize \url{https://twitter.com/ben_moll/status/1312042597876142081}}
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/ben_tweet}
\end{figure}
\vspace{-3mm}
\item ... especially Beatrice Cherrier and Gianluca Violante's replies
\begin{figure}[tph]
	\centering
\includegraphics[width=.4\textwidth]{figures/beatrice_tweet}
\end{figure}
\vspace{-2mm}
{\scriptsize \url{https://twitter.com/Undercoverhist/status/1312053154515693568}}
\begin{figure}[tph]
	\centering
\includegraphics[width=.4\textwidth]{figures/gianluca_tweet}
\end{figure}
\ei
\end{frame}






\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.4cm}
\begin{center}
{\Huge Toward a Simple Macro Model:\\[1ex] Labor Supply and Demand}
\end{center}
\end{frame}


\begin{frame}{Toward a simple macro model}
\vspace{-3mm}
\bi
\item Next lecture: your first modern macro model
\medskip
\item Model will be extremely (ridiculously) simple, a baby version
\bi
\smallskip
\item a single ``representative" consumer
\smallskip
\item a single ``representative" firm
\smallskip
\item only one time period, i.e. model is static not dynamic
\smallskip
\item only one good
\smallskip
\item only one factor of production (labor)
\smallskip
\item no frictions, ``market imperfections"
\smallskip
\item model is sometimes called \textbf{\tr{``Robinson Crusoe economy"}}
\ei
\medskip
\item ... but it will have key features of most modern macro models
\be
\smallskip
\item micro foundations
\smallskip
\item general equilibrium
\ee
\medskip
\item Rest of course: build much richer, more satisfactory models
\ei
\end{frame}





\begin{frame}{Sketch of simple macro model (Robinson Crusoe economy)}
\vspace{-4mm}
\bi
\item \tr{\textbf{Preferences}} (i.e. utility function):
\vspace{-1mm}
$$U(c,h) $$
\vspace{-5mm}
\bi
\item $c$: consumption of coconuts, $U$ is increasing in $c$
\smallskip
\item $h$: hours worked, $U$ is decreasing in $h$
\ei
\medskip
\item \tr{\textbf{Technology}} (i.e. production function):
\vspace{-1mm}
$$y = f(n)$$
\vspace{-5mm}
\bi
\item $y$: output, i.e. production of coconuts
\smallskip
\item $n$: hours employed, $f$ is increasing in $n$
\ei
\medskip
\item \tr{\textbf{Resource constraints:}}
$$c = y, \qquad n= h $$\\
\pause
\item Next lecture: different ways of organizing this economy
\smallskip
\item One of them: \tr{Robinson Crusoe} is alone on an island, lives off coconuts, harvesting coconuts takes time. Hence the name.
\ei
\end{frame}



%\begin{frame}{Some thoughts on the representative agent assumption}
%\vspace{-3mm}
%\begin{itemize}
%\item Model assumes \tb{``representative household"} and \tb{``representative firm"} (jointly = ``representative agent")
%\bigskip
%\item When is this justified? If at least one of following 3 conditions is satisfied
%\begin{enumerate}
%\medskip
%\item all individuals in economy are identical
%\medskip
%\item particular assumptions on preferences (``homotheticity", ``Gorman aggregation")
%\medskip
%\item perfect markets
%\begin{itemize}
%\medskip
%\item representative firm $\Leftrightarrow$ perfect factor markets (capital, labor), equalize marginal products
%\medskip
%\item rep household $\Leftrightarrow$ perfect insurance markets, equalize marginal utilities
%\end{itemize}
%\end{enumerate}
%\bigskip
%\item Do we believe these conditions are satisfied? \textbf{\tr{No}}, but...
%\end{itemize}
%\end{frame}

\begin{frame}{Some thoughts on the representative agent assumption}
\vspace{-3mm}
\begin{itemize}
\item Model assumes \tr{``representative consumer"} and \tr{``representative firm"} (jointly = ``representative agent")
\medskip
\item When is this justified? Here: sketch, more in supplement (not examinable)
\medskip
\item \tr{\textbf{Firms}}: if at least one of following \tr{3} conditions is satisfied
\be
\smallskip
\item all firms in economy are identical
\medskip
\item particular assumptions on production functions (``homogeneity")
\medskip
\item perfect factor markets (capital, labor) $\Rightarrow$ equalize marginal products 
\bi
\item see supplement
\ei
\ee
\medskip
\item \tr{\textbf{Consumers}} (even harder): if one of following \tr{2} conditions is satisfied
\be
\medskip
\item all consumers in economy are identical
\medskip
\item particular assumptions on preferences (``Gorman aggregation")
\bi
\item see supplement
\ei
\ee
\medskip
\item Do we believe these conditions are satisfied? \textbf{\tr{No}}, but...
\end{itemize}
\end{frame}


\begin{frame}{Toward a simple macro model}
\vspace{-3mm}
Today: study key building blocks of simple macro model
\be
\bigskip
\item labor demand
\bigskip
\item labor supply
\ee
\bigskip
This should largely be a refresher of material you've already seen in EC1A1
\pause
\bi
\medskip
\item probably with different notation
\medskip
\item good point to mention: you need to be able to handle changing notation
\bi
\medskip
\item economists change notation all the time
\medskip
\item notation varies across textbooks, courses, even within lectures
\medskip
\item extreme example: rather than writing production as $n^\alpha$, write $\alpha^n$ 
\ei
\bigskip
\item my experience: if you can't handle changing notation, you probably don't understand the economics
\ei
\end{frame}



\begin{frame}{Vocabulary: ``partial equilibrium" and ``general equilibrium"}
\vspace{-3mm}
Will frequently hear macroeconomists use following two words
\bi
\medskip
\item \tr{``general equilibrium"} = all prices endogenously determined in equilibrium 
\medskip
\item \tr{``partial equilibrium"} = problem of individual household or firm in isolation taking as given prices
\ei
\bigskip
Let's apply vocabulary:
\bi
\medskip
\item Today: labor demand and supply in \tr{partial equilibrium}
\medskip
\item Next lecture: labor demand and supply in \tr{general equilibrium}
\ei
\bigskip
Note:  ``partial equilibrium" = \tr{misnomer}
\bi
\medskip
\item Nothing ``partial" about it, should really be \tr{``no equilibrium"}
\medskip
\item This is specific to macro. Other fields use ``partial equilibrium" differently: equilibrium in a single market.
\medskip
\item But very common language in macro $\Rightarrow$ will adopt even though misnomer
\ei
\end{frame}




\begin{frame}{Labor demand}
\vspace{-3mm}
\bi
\item Representative firm has production function
$$y = f(n) $$
\item Assumptions: $f'(n)>0$ and $f''(n)<0$ for all $n$
\vspace{-1mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.55\textwidth]{figures/1.eps}
\end{figure}
\vspace{-4mm}
\item Example: $f(n) = A n^\alpha$ with $0<\alpha < 1,A>0$
\ei
\end{frame}



\begin{frame}{Labor demand}
\bi
\item Representative firm maximizes profits
$$\Pi = \max_n \ f(n) - wn $$
\item Solution: labor demand function
$$n^* = n^d(w) $$
\item Optimality condition
$$f'(n) = w $$
\ei
\end{frame}


\begin{frame}{Labor demand: graphical representation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/2b.eps}
\end{figure}
\end{frame}



\begin{frame}{Labor demand: more minimal graphical representation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/2.eps}
\end{figure}
\end{frame}


\begin{frame}{Labor demand: alternative graphical representation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/3.eps}
\end{figure}
\end{frame}



\begin{frame}{Labor demand: parametric example}
\bi
\item Example: $f(n) = A n^\alpha$ with $0<\alpha < 1,A>0$
\bigskip
\item Representative firm maximizes profits
$$\Pi = \max_n \ f(n) - wn $$
\item $\Rightarrow$ optimal labor demand is
$$n^* = n^d(w) = \left(\frac{\alpha A}{w}\right)^{\frac{1}{1-\alpha}} $$
which is
\bi
\medskip
\item decreasing in wage $w$ (as a demand curve typically is)
\medskip
\item increasing in productivity $A$
\ei
\ei
\end{frame}





\begin{frame}{Labor supply}
\bi
\item Derive labor supply from utility maximization of representative household
\bigskip
\item Good treatment in \tr{Kurlat, chapter 7.2} (see moodle)
\bigskip
\item Two \tr{equivalent} formulations
\be
\bigskip
\item in terms of hours worked $h$
\bigskip
\item in terms of leisure $\ell$
\ee
\bigskip
\item My guess: you've probably seen the latter $\Rightarrow$ start with that one
\ei
\end{frame}


\begin{frame}{Derivation of labor supply in terms of leisure $\ell$}
\vspace{-2mm}
\bi
\item Preferences (role of $\~$ will become clear momentarily)
$$u(c) + \tilde{v}(\ell) $$
where $c$ = consumption, $\ell$ = leisure, $\tilde{v}(\ell)$ = utility from leisure
\bigskip
\item Assumptions
\bi
\medskip
\item $u'(c)>0$ and $\tilde{v}'(\ell)>0$ 
\medskip
\item $u''(c)<0$ and $\tilde{v}''(\ell)<0$
\ei
\bigskip
\item Example 
$$u(c) = \log c, \quad \tilde{v}(\ell) = \theta \log \ell, \quad \theta > 0 $$
\item More general version would feature non-separable utility function
$$\tilde{U}(c,\ell) $$
but this doesn't add very much so work with simpler separable version
\ei
\end{frame}


\begin{frame}{Derivation of labor supply in terms of leisure $\ell$}
\vspace{-3mm}
Representative household solves
\begin{align*}
\max_{c,\ell} \ u(c) + \tilde{v}(\ell) \quad \mbox{s.t.} \quad c = w(1-\ell) + \Pi
\end{align*}
Explanation:
\bi
\medskip
\item household's time endowment = 1
\medskip
\item hours worked $h = 1 - \ell$
\medskip
\item note: $1$ is just a normalization to save on notation, could equally have assumed time endowment = 24 hours or some abstract parameter $\bar{\ell}$
\bigskip
\item $w(1-\ell) = wh =$ is labor income
\bigskip
\item $\Pi$ = other unearned income, e.g. dividend income from owning stocks
\ei
\bigskip
\tr{\textbf{Solution:}} labor supply function
$$\tr{h^* = h^s(w) = 1 - \ell^*} $$
\end{frame}


\begin{frame}{Labor supply in terms of leisure $\ell$: graphical representation}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/4.eps}
\end{figure}
\end{frame}


\begin{frame}{Derivation of optimality condition}
You should already know this but I will do derivation once (and never again)\\
\bigskip
Two equivalent strategies:
\be
\bigskip
\item Direct substitution (brute force)
\uncover<2->{
$$\max_{\ell} \ u(\underbrace{w(1-\ell) + \Pi}_{c}) + \tilde{v}(\ell) \quad \Rightarrow \quad - u'(c)w + \tilde{v}'(\ell) = 0 $$}
\vspace{-3mm}
\item Lagrangean
\uncover<3->{
$$\mathcal{L} = u(c) + \tilde{v}(\ell) + \lambda\left[w(1-\ell) + \Pi - c\right] $$}
\uncover<4->{
$$
\begin{rcases}
c: \quad u'(c) &= \lambda\\
\ell: \quad \tilde{v}'(\ell) &= \lambda w
\end{rcases}
\quad \Rightarrow \quad  \frac{\tilde{v}'(\ell)}{u'(c)} = w $$
}
\ee
\end{frame}

\begin{frame}{Derivation of labor supply in terms of \tr{hours worked $h$}}
\vspace{-4mm}
\bi
\item Preferences
$$u(c) - v(h) $$
where $c$ = consumption, $h$ = hours worked, \tr{$v(h)$ = disutility from work}
\bigskip
\item Assumptions
\bi
\medskip
\item $u'(c)>0$ and $v'(h)>0$ 
\medskip
\item $u''(c)<0$ and $v''(h)>0$, i.e. disutility is convex
\ei
\bigskip
\item Link to leisure formulation: set $v(h) = - \tilde{v}(1-h) \Rightarrow $ the two are equivalent 
\bigskip
\item Example 1: $u(c) = \log c, \ v(h) = - \theta\log(1-h),\theta >0$
\bigskip
\item Example 2:
\vspace{-2mm}
$$u(c) = \log c, \quad v(h) = \theta \frac{h^{1+1/\varepsilon}}{1+1/\varepsilon}, \quad \theta, \varepsilon >0 \ \left(\mbox{note:} \ v'(h) = \theta h^{1/\varepsilon} \right)$$
\item As before, could also work with non-separable $U(c,h)$
\ei
\end{frame}


\begin{frame}{Derivation of labor supply in terms of hours worked $h$}
Representative household solves
\begin{align*}
\max_{c,h} \ u(c) - v(h) \quad \mbox{s.t.} \quad c = w h + \Pi
\end{align*}
\bi
\item $wh =$ is labor income
\bigskip
\item $\Pi$ = other unearned income, e.g. dividend income from owning stocks
\ei
\bigskip
\tr{\textbf{Solution:}} labor supply function
$$\tr{h^* = h^s(w)} $$
\end{frame}



\begin{frame}{Graphical representation in terms of hours worked $h$}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/5.eps}
\end{figure}
\end{frame}



\begin{frame}{Graphical representation in terms of hours worked $h$}
\vspace{-4mm}
\begin{figure}[tph]
	\centering
\includegraphics[width=.78\textwidth]{figures/6.eps}
\end{figure}
\end{frame}


\begin{frame}{Note: same graph as before but with flipped $x$-axis}
\begin{figure}[tph]
	\centering
\includegraphics[width=.5\textwidth]{figures/4.eps}\includegraphics[width=.5\textwidth]{figures/6.eps}
\end{figure}
\end{frame}



\begin{frame}{Two exercises and a question}
\bi
\item Exercise 1: derive optimality condition in terms of hours worked
\bigskip
\item Exercise 2: derive optimal labor supply in example 2
$$u(c) = \log c, \quad v(h) = \theta \frac{h^{1+1/\varepsilon}}{1+1/\varepsilon}, \quad \theta, \varepsilon >0 $$
\item \tr{\textbf{Question:}} suppose $w$ increases, what will happen to labor supply? Why?
\ei
\end{frame}


\begin{frame}{Next time}
\bi
\item Combine labor supply and labor demand
\bigskip
\item $\Rightarrow$ your first (baby, ridiculously simple) modern macro model
\ei
\end{frame}



\begin{frame}
\renewcommand{\baselinestretch}{1}
\vspace{2.3cm}
\begin{center}
{\Huge Appendix on \\[1ex] Income and Substitution Effects}
\end{center}
\end{frame}


\begin{frame}{A way of thinking about income and substitution effects}
\vspace{-3mm}
\bi
\item Note: only some people find this useful (those who ``think in equations")
\medskip
\item If you don't, please just ignore. The standard graphical analysis is just fine.\\
\medskip
\item This definition of income and substitution effects is also non-standard (compared to textbook definition) but it's helpful nevertheless
\bi
\smallskip
\item on my to-do list: work out exact relation to standard definition
\ei
\bigskip
\pause
\item Recall: representative household solves
\begin{align*}
\max_{c,h} \ u(c) - v(h) \quad \mbox{s.t.} \quad c = w h + \Pi
\end{align*}
\item Optimal $h^*(w)$ solves optimality condition and budget constraint
\begin{align*}
\frac{v'(h)}{u'(c)} &= w\\
c &= wh + \Pi
\end{align*}
\ei
\end{frame}



\begin{frame}{A way of thinking about income and substitution effects}
\vspace{-2mm}
\bi
\item Let's do something funny: separate notation for $w$ in the two equations
\begin{align*}
\frac{v'(h)}{u'(c)} &= \tg{w^S} \\
c &= \tr{w^I} h + \Pi
\end{align*}
where \tg{$S$} stands for \tg{``substitution"} and \tr{$I$} for \tr{``income"} (explain momentarily)
\medskip
\item Obviously when $\tg{w^S} = \tr{w^I} = w \Rightarrow$ obtain $h^*(w)$ again
\medskip
\item But can still solve system even when $\tg{w^S} \neq \tr{w^I} \Rightarrow$ solution $h^*(\tg{w^S},\tr{w^I})$
\medskip
\item \tr{\textbf{Key observation:}}
\bi
\medskip
\item \tg{Substitution effect} = change in $h^*$ from changing $\tg{w^S}$ but not $\tr{w^I}$, i.e. effect of price change working through \tg{first-order condition}
\medskip
\item \tr{Income effect} = change in $h^*$ from changing $\tr{w^I}$ but not $\tg{w^S}$, i.e. effect of price change working through \tr{budget constraint}
\ei
\ei
\end{frame}


\begin{frame}{This is much more general}
\bi
\item Example: intertemporal choice (Lecture 4 and EC1A1)
$$\max_{c_1,c_2} \ u(c_1) + \beta u(c_2) \quad \mbox{s.t.} \quad c_1 + \frac{c_2}{1+r} = y_1 + \frac{y_2}{1+r} $$
\smallskip
\item Following logic of previous slide, write first-order condition and budget constraint as
\begin{align*}
u'(c_1) &= \beta (1+ \tg{r^S}) u'(c_2) \\
c_1 + \frac{c_2}{1+\tr{r^I}} &= y_1 + \frac{y_2}{1+\tr{r^I}}
\end{align*}
\item \tg{Substitution effect} = change in $(c_1^*,c_2^*)$ from changing $\tg{r^S}$ but not $\tr{r^I}$, i.e. effect of price change working through \tg{first-order condition}
\bigskip
\item \tr{Income effect} = change in $(c_1^*,c_2^*)$ from changing $\tr{r^I}$ but not $\tg{r^S}$, i.e. effect of price change working through \tr{budget constraint}
\ei
\end{frame}






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\end{document}

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