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Add notation guide to unemployment course
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content/courses/c1.md

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title: "Unemployment"
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date: 2025-01-07
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date: 2025-01-28
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url: /c1/
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aliases:
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- /t1.html
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- /uploads/7/0/2/0/70200055/t1b.pdf
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tags: ["Beveridge curve", "DMP model", "job rationing", "labor market tightness", "matching function", "matching model", "public employment", "Shimer puzzle", "unemployment", "unemployment insurance", "wage rigidity"]
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tags: ["Beveridge curve", "DMP model", "job rationing", "labor market tightness", "matching function", "matching model", "public employment", "Shimer puzzle", "unemployment fluctuations", "unemployment gap", "unemployment insurance", "wage rigidity"]
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author: "Pascal Michaillat"
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description: "This course presents a matching model of unemployment and uses it to study unemployment fluctuations, efficient unemployment, and labor market policies."
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summary: "This course presents a matching model of unemployment. It uses the model to study unemployment fluctuations; job rationing; efficient unemployment and unemployment gap; and labor market policies such as minimum wage, public employment, and unemployment insurance."
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This course presents several models of unemployment. These models explain why unemployment exist, and why unemployment fluctuate over the business cycle.
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The course also strives to justify all the assumptions introduced in the models.
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These models are extremely helpful to design effective policies to tackle unemployment. First, the models are useful to determine what is the socially efficient level of unemployment. This socially efficient level of unemployment is a key ingredient to good labor-market policies. Second, the models explain how various labor market policies should respond to fluctuations in unemployment.
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These models are extremely helpful to design effective policies to tackle unemployment. First, the models are useful to determine what is the socially efficient level of unemployment. This socially efficient level of unemployment is a key ingredient to good labor market policies. Second, the models explain how various labor market policies should respond to fluctuations in unemployment.
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##### Lecture videos
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##### Main readings
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+ [Michaillat (2012)](/1.pdf) – This paper first establishes that the DMP model does not have job rationing and then develops a matching model with job rationing. The job-rationing model departs from the DMP model by assuming that the production has diminishing returns to labor instead of constant returns to labor, and that the real wage is given by a rigid wage norm instead of Nash bargaining. In that model unemployment can be decomposed into rationing and frictional components. In recessions, the rationing component is large while the frictional component is small.
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+ [Crepon, Duflo, Gurgand, Rathelot, and Zamora (2013)](https://doi.org/10.1093/qje/qjt001) – This paper reports the results from a randomized experiment designed to evaluate the direct and indirect effects of job-placement assistance on the labor market outcomes of young educated jobseekers in France. It finds that jobseekers who are given job-placement assistance find jobs more rapidly, while jobseekers in the same labor market who are not assisted take longer to find a job. This finding suggests that jobs are somewhat rationed, and that assisted jobseekers move ahead of non-assisted jobseekers in job queues. The spillovers of job-placement assistance are particularly strong in bad times, when jobs are scarcer.
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+ [Crepon, Duflo, Gurgand, Rathelot, and Zamora (2013)](https://doi.org/10.1093/qje/qjt001) – This paper reports the results from a randomized experiment designed to evaluate the direct and indirect effects of job-placement assistance on the labor market outcomes of young educated job seekers in France. It finds that job seekers who are given job-placement assistance find jobs more rapidly, while job seekers in the same labor market who are not assisted take longer to find a job. This finding suggests that jobs are somewhat rationed, and that assisted job seekers move ahead of non-assisted job seekers in job queues. The spillovers of job-placement assistance are particularly strong in bad times, when jobs are scarcer.
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##### Additional readings
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+ [Robinson (1946)](https://tidsskrift.dk/nationaloekonomisktidsskrift/article/view/60263) – This essay discusses various definitions of full employment and explains why full employment does not mean zero unemployment.
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+ [Gokten, Heimberger, and Lichtenberger (2024)](https://doi.org/10.1016/j.euroecorev.2024.104725) – This paper uses the formula $u^\ast = \sqrt{uv}$ to compute the FERUs in selected European countries (Germany, Sweden, Austria, Finland, UK) between 1970 and 2022. It compares the FERUs and unemployment gaps in Europe to those in the United States.
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##### Practice material
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+ [Quiz on efficient unemployment and unemployment gap](/v11.pdf)
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##### Additional readings
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+ [Crepon and van den Berg (2016)](https://doi.org/10.1146/annurev-economics-080614-115738) – This survey assesses the effectiveness of various active labor market policies: policies designed to improve labor market outcomes of jobseekers. Policies covered include job-search assistance (to stimulate labor supply) and job subsidies (to stimulate labor demand). The survey highlights spillovers from such policies. Job-search assistance is not found to be very effective, which is unsurprising in a world with job rationing. Job subsidies are found to be moderately effective.
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+ [Crepon and van den Berg (2016)](https://doi.org/10.1146/annurev-economics-080614-115738) – This survey assesses the effectiveness of various active labor market policies: policies designed to improve labor market outcomes of job seekers. Policies covered include job-search assistance (to stimulate labor supply) and job subsidies (to stimulate labor demand). The survey highlights spillovers from such policies. Job-search assistance is not found to be very effective, which is unsurprising in a world with job rationing. Job subsidies are found to be moderately effective.
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+ [Neumark and Shirley (2022)](https://doi.org/10.1111/irel.12306) – This survey reviews evidence on the effects of the minimum wage on employment in the United States. Overall, the evidence points toward negative effects of minimum wages on employment of less-skilled workers.
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+ [Neumann, Fishback, and Kantor (2010)](https://www.jstor.org/stable/25654074) – This paper describes public employment in the United States during the First New Deal and Second New Deal. It then documents the effect of public employment on the private labor market, unearthing evidence of crowding out of private employment by public employment.
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+ [Auerbach and Gorodnichenko (2013)](https://www.nber.org/chapters/c12634) – This paper finds that in a large number of Organization for Economic Cooperation and Development (OECD) countries, government multipliers are larger in recessions than in expansions.
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## Unemployment insurance
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Fluctuations in unemployment raise another policy question: how should the generosity of unemployment insurance respond to unemployment fluctuations? This question was hotly debated during the Great Recession. Some argued that unemployment insurance should be reduced because it discouraged job search and would raise unemployment further. Other countered that unemployment insurance could be increased without raising unemployment much—as there were no jobs available for jobseekers.
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Fluctuations in unemployment raise another policy question: how should the generosity of unemployment insurance respond to unemployment fluctuations? This question was hotly debated during the Great Recession. Some argued that unemployment insurance should be reduced because it discouraged job search and would raise unemployment further. Other countered that unemployment insurance could be increased without raising unemployment much—as there were no jobs available for job seekers.
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This section weights the two sides of the argument. It discusses the different channels through which unemployment insurance affects the labor market. It then derives a formula for optimal unemployment insurance. The formula contrasts how generous unemployment insurance should be in good times and in bad times. The formula shows that the generosity of unemployment insurance should be adjusted over the business cycle; the adjustment depends on how unemployment insurance affects labor market tightness. The effect of unemployment insurance on tightness depends on the model: increasing unemployment insurance may raise tightness by alleviating the rat race for jobs or lower tightness by increasing wages through bargaining.
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##### Practice material
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+ [Quiz on unemployment insurance](/v13.pdf)
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+ [Problem set on labor market policies](/v4.pdf)
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+ [Problem set on labor market policies](/v4.pdf)
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## Notation guide
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+ $H>0$ – Labor force = number of workers who are able, willing, and seeking to work
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+ $U>0$ – Unemployment level = number of labor force participants who do not have a job
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+ $L>0$ – Employment level = number of labor force participants who do have a job
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+ $V>0$ – Vacancy level = number of job vacancies advertised by firms
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+ $u\in(0,1)$ – Unemployment rate = share of labor force without a job = $U / H$
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+ $v\in(0,1)$ – Vacancy rate = $V / H$
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+ $\theta>0$ – Labor market tightness = number of job vacancies per job seeker = $V/U$ = $v/u$
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+ $m(U,V)>0$ – Matching function
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+ $\eta\in(0,1)$ – Matching elasticity = $\partial \ln(m)/\partial \ln(U)$
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+ $f>0$ – Job-finding rate = $m(U,V)/U$
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+ $q>0$ – Vacancy-filling rate = $m(U,V)/V$
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+ $s>0$ – Job-separation rate

content/courses/c2.md

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tags: ["Beveridge curve", "business cycles", "economic slack", "fiscal policy", "inequality", "Keynesian unemployment", "Kuhnian model", "matching function", "matching model", "monetary policy"]
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tags: ["Beveridge curve", "business cycles", "economic slack", "fiscal policy", "frictional unemployment", "inequality", "Keynesian unemployment", "Kuhnian model", "matching function", "matching model", "monetary policy", "unemployment gap"]
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author: "Pascal Michaillat"
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description: "This graduate course presents matching models of economic slack and uses them to study business-cycle fluctuations, monetary policy, and fiscal policy."
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summary: "This graduate course presents various matching models of economic slack. It uses them to study business-cycle fluctuations; Keynesian, classical, and frictional unemployment; optimal monetary policy and the zero lower bound; and optimal government spending."

content/courses/c3.md

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tags: ["Bellman equation", "Benveniste-Scheinkman equation", "differential equations", "dynamic programming", "dynamical systems", "Hamiltonian", "HJB equation", "optimal control", "optimization", "phase diagrams"]
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author: "Pascal Michaillat"
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description: "This graduate course covers basic mathematical methods for macroeconomics: dynamic programming, optimal control, and differential equations."
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summary: "This graduate course covers basic mathematical methods for macroeconomics: dynamic programming, optimal control, and differential equations."
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summary: "This graduate course covers basic mathematical methods for macroeconomics: dynamic programming, optimal control, and differential equations. The methods are used to study dynamical macroeconomic systems in discrete time and continuous time."
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cover:
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image: "/c3s.png"
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alt: "Phase diagram depicting a saddle path"
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## Introduction
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This course covers basic mathematical methods to study dynamical systems, in discrete time and in continuous time. Dynamical systems are systems that involve more than one time-period; they are prevalent in macroeconomics. We first discuss dynamic programming, which is a method to solve dynamic optimization problems in discrete time. We then turn to optimal control, which is a method to solve dynamic optimization problems in continuous time. Finally, we show how to solve differential equations, which are used to describe continuous-time dynamical systems.
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This course covers basic mathematical methods to study dynamical systems in discrete time and in continuous time. Dynamical systems are systems that involve more than one time-period; they are prevalent in macroeconomics. We first discuss dynamic programming, which is a method to solve dynamic optimization problems in discrete time. We then turn to optimal control, which is a method to solve dynamic optimization problems in continuous time. Finally, we show how to solve differential equations, which are used to describe continuous-time dynamical systems.
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content/papers/1.md

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tags: ["business cycles", "DMP model", "frictional unemployment", "job rationing", "job vacancies", "matching model", "recessions", "rationing unemployment", "Shimer puzzle","unemployment", "wage rigidity"]
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tags: ["DMP model", "frictional unemployment", "job rationing", "job vacancies", "matching model", "recessions", "rationing unemployment", "Shimer puzzle","unemployment fluctuations", "wage rigidity"]
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author: "Pascal Michaillat"
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description: "This paper proposes a matching model of the labor market with job rationing: unemployment persists even without matching frictions. Published in AER, 2012."
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summary: "This paper proposes a matching model of the labor market with job rationing: unemployment does not disappear in the absence of matching frictions. In recessions, job rationing drives the rise of unemployment, whereas matching frictions contribute little to it."

content/papers/2.md

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tags: ["business cycles", "crowding out", "fiscal multiplier", "fiscal policy", "job rationing", "matching model", "New Keynesian model", "public employment", "state dependence", "unemployment"]
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tags: ["crowding out", "fiscal multiplier", "fiscal policy", "job rationing", "labor market tightness", "matching model", "New Keynesian model", "public employment", "state dependence", "unemployment fluctuations"]
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author: "Pascal Michaillat"
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description: "This paper develops a New Keynesian model in which the government multiplier doubles when unemployment rises from 5% to 8%. Published in AEJ Macro, 2014."
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summary: "This paper develops a New Keynesian model in which the government multiplier doubles when the unemployment rate rises from 5% to 8%. The multiplier is so countercyclical because in bad times, on the labor market, job rationing dwarfs matching frictions."

content/papers/4.md

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tags: ["Baily-Chetty formula", "job creation", "job rationing", "job search", "matching model", "moral hazard", "rat race", "sufficient statistics", "unemployment", "unemployment insurance"]
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tags: ["Baily-Chetty formula", "job creation", "job rationing", "job search", "labor market tightness", "matching model", "moral hazard", "rat race", "sufficient statistics", "unemployment insurance"]
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author: ["Camille Landais","Pascal Michaillat","Emmanuel Saez"]
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description: "This paper develops a theory of optimal unemployment insurance in matching models. Published in AEJ Policy, 2018."
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summary: "This paper develops a theory of optimal unemployment insurance in matching models. It derives a sufficient-statistic formula for optimal unemployment insurance, which is useful to determine the optimal cyclicality of unemployment insurance."

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tags: ["Baily-Chetty formula", "job rationing", "labor market tightness", "matching model", "recessions", "state dependence", "sufficient statistics", "unemployment benefits", "unemployment", "unemployment insurance"]
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tags: ["Baily-Chetty formula", "job rationing", "labor market tightness", "macro elasticity of unemployment", "matching model", "micro elasticity of unemployment", "recessions", "state dependence", "unemployment benefits", "unemployment insurance"]
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author: ["Camille Landais","Pascal Michaillat","Emmanuel Saez"]
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description: "This paper studies how the generosity of unemployment insurance should vary over the business cycle in the United States. Published in AEJ Policy, 2018."
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summary: "This paper explores how the optimal generosity of unemployment insurance varies over the business cycle in the United States. It finds that the optimal replacement rate is countercyclical, just like the actual replacement rate."

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