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14.121 Microeconomic Theory I (MIT) 14.121 Microeconomic Theory I (MIT)

Description

This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in the economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Topics include consumer and producer theory, markets and competition, general equilibrium, and tools of comparative statics and their application to price theory. Some topics of recent interest may also be covered. This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in the economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Topics include consumer and producer theory, markets and competition, general equilibrium, and tools of comparative statics and their application to price theory. Some topics of recent interest may also be covered.Subjects

microeconomic theory | microeconomic theory | demand theory | demand theory | producer theory; partial equilibrium | producer theory; partial equilibrium | competitive markets | competitive markets | general equilibrium | general equilibrium | externalities | externalities | Afriat's theorem | Afriat's theorem | pricing | pricing | robust comparative statics | robust comparative statics | utility theory | utility theory | properties of preferences | properties of preferences | choice as primitive | choice as primitive | revealed preference | revealed preference | classical demand theory | classical demand theory | Kuhn-Tucker necessary conditions | Kuhn-Tucker necessary conditions | implications of Walras?s law | implications of Walras?s law | indirect utility functions | indirect utility functions | theorem of the maximum (Berge?s theorem) | theorem of the maximum (Berge?s theorem) | expenditure minimization problem | expenditure minimization problem | Hicksian demands | Hicksian demands | compensated law of demand | compensated law of demand | Slutsky substitution | Slutsky substitution | price changes and welfare | price changes and welfare | compensating variation | compensating variation | and welfare from new goods | and welfare from new goods | price indexes | price indexes | bias in the U.S. consumer price index | bias in the U.S. consumer price index | integrability | integrability | demand aggregation | demand aggregation | aggregate demand and welfare | aggregate demand and welfare | Frisch demands | Frisch demands | and demand estimation | and demand estimation | increasing differences | increasing differences | producer theory applications | producer theory applications | the LeCh?telier principle | the LeCh?telier principle | Topkis? theorem | Topkis? theorem | Milgrom-Shannon monotonicity theorem | Milgrom-Shannon monotonicity theorem | monopoly pricing | monopoly pricing | monopoly and product quality | monopoly and product quality | nonlinear pricing | nonlinear pricing | and price discrimination | and price discrimination | simple models of externalities | simple models of externalities | government intervention | government intervention | Coase theorem | Coase theorem | Myerson-Sattherthwaite proposition | Myerson-Sattherthwaite proposition | missing markets | missing markets | price vs. quantity regulations | price vs. quantity regulations | Weitzman?s analysis | Weitzman?s analysis | uncertainty | uncertainty | common property externalities | common property externalities | optimization | optimization | equilibrium number of boats | equilibrium number of boats | welfare theorems | welfare theorems | uniqueness and determinacy | uniqueness and determinacy | price-taking assumption | price-taking assumption | Edgeworth box | Edgeworth box | welfare properties | welfare properties | Pareto efficiency | Pareto efficiency | Walrasian equilibrium with transfers | Walrasian equilibrium with transfers | Arrow-Debreu economy | Arrow-Debreu economy | separating hyperplanes | separating hyperplanes | Minkowski?s theorem | Minkowski?s theorem | Existence of Walrasian equilibrium | Existence of Walrasian equilibrium | Kakutani?s fixed point theorem | Kakutani?s fixed point theorem | Debreu-Gale-Kuhn-Nikaido lemma | Debreu-Gale-Kuhn-Nikaido lemma | additional properties of general equilibrium | additional properties of general equilibrium | Microfoundations | Microfoundations | core | core | core convergence | core convergence | general equilibrium with time and uncertainty | general equilibrium with time and uncertainty | Jensen?s inequality | Jensen?s inequality | and security market economy | and security market economy | arbitrage pricing theory | arbitrage pricing theory | and risk-neutral probabilities | and risk-neutral probabilities | Housing markets | Housing markets | competitive equilibrium | competitive equilibrium | one-sided matching house allocation problem | one-sided matching house allocation problem | serial dictatorship | serial dictatorship | two-sided matching | two-sided matching | marriage markets | marriage markets | existence of stable matchings | existence of stable matchings | incentives | incentives | housing markets core mechanism | housing markets core mechanismLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.01 Principles of Microeconomics (MIT) 14.01 Principles of Microeconomics (MIT)

Description

This introductory course teaches the fundamentals of microeconomics. Topics include consumer theory, producer theory, the behavior of firms, market equilibrium, monopoly, and the role of the government in the economy. 14.01 is a Humanities, Arts, and Social Sciences (HASS) elective and is offered both terms. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges. This introductory course teaches the fundamentals of microeconomics. Topics include consumer theory, producer theory, the behavior of firms, market equilibrium, monopoly, and the role of the government in the economy. 14.01 is a Humanities, Arts, and Social Sciences (HASS) elective and is offered both terms. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges.Subjects

market | market | optimization | optimization | allocation | allocation | economic measurement | economic measurement | analysis | analysis | microeconomics | microeconomics | demand | demand | supply | supply | equilibrium | equilibrium | general equilibrium | general equilibrium | government interventions | government interventions | price elasticity of demand | price elasticity of demand | income elasticity of demand | income elasticity of demand | cross price elasticity of demand | cross price elasticity of demand | price elasticity of supply | price elasticity of supply | consumer behavior | consumer behavior | consumer preference | consumer preference | utility functions | utility functions | marginal rate of substitution | marginal rate of substitution | budget constraints | budget constraints | interior solutions | interior solutions | corner solutions | corner solutions | Engle curves | Engle curves | individual demand | individual demand | market demand | market demand | revealed preferences | revealed preferences | substitution effect | substitution effect | income effect | income effect | Giffen goods | Giffen goods | consumer surplus | consumer surplus | Irish potato famine | Irish potato famine | network externalities | network externalities | uncertainty | uncertainty | preference toward risk | preference toward risk | risk premium | risk premium | indifference curves | indifference curves | diversification | diversification | insurance | insurance | producer theory | producer theory | production functions | production functions | short run | short run | long run | long run | returns to scale | returns to scale | cost functions | cost functions | economies of scale | economies of scale | economies of scope | economies of scope | learning | learning | profit maximization | profit maximization | producer surplus | producer surplus | agricultural price support | agricultural price support | tax | tax | subsidy | subsidy | exchange economy | exchange economy | contract curves | contract curves | utility possibilities frontier | utility possibilities frontier | Edgeworth Box | Edgeworth Box | production possibilities frontier | production possibilities frontier | efficiency | efficiency | monopoly | monopoly | multiplant firm | multiplant firm | social cost | social cost | price regulation | price regulation | monopsony | monopsony | price discrimination | price discrimination | peak-load pricing | peak-load pricing | two-part tariffs | two-part tariffs | bundling | bundling | monopolistic competition | monopolistic competition | game theory | game theory | oligopoly | oligopoly | Cournot | Cournot | Stackelberg | Stackelberg | Bertrand | Bertrand | Prisoner's Dilemma | Prisoner's DilemmaLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.01 Principles of Microeconomics (MIT)

Description

This introductory course teaches the fundamentals of microeconomics. Topics include consumer theory, producer theory, the behavior of firms, market equilibrium, monopoly, and the role of the government in the economy. 14.01 is a Humanities, Arts, and Social Sciences (HASS) elective and is offered both terms. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges.Subjects

market | optimization | allocation | economic measurement | analysis | microeconomics | demand | supply | equilibrium | general equilibrium | government interventions | price elasticity of demand | income elasticity of demand | cross price elasticity of demand | price elasticity of supply | consumer behavior | consumer preference | utility functions | marginal rate of substitution | budget constraints | interior solutions | corner solutions | Engle curves | individual demand | market demand | revealed preferences | substitution effect | income effect | Giffen goods | consumer surplus | Irish potato famine | network externalities | uncertainty | preference toward risk | risk premium | indifference curves | diversification | insurance | producer theory | production functions | short run | long run | returns to scale | cost functions | economies of scale | economies of scope | learning | profit maximization | producer surplus | agricultural price support | tax | subsidy | exchange economy | contract curves | utility possibilities frontier | Edgeworth Box | production possibilities frontier | efficiency | monopoly | multiplant firm | social cost | price regulation | monopsony | price discrimination | peak-load pricing | two-part tariffs | bundling | monopolistic competition | game theory | oligopoly | Cournot | Stackelberg | Bertrand | Prisoner's DilemmaLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.121 Microeconomic Theory I (MIT)

Description

This half-semester course provides an introduction to microeconomic theory designed to meet the needs of students in the economics Ph.D. program. Some parts of the course are designed to teach material that all graduate students should know. Others are used to introduce methodologies. Topics include consumer and producer theory, markets and competition, general equilibrium, and tools of comparative statics and their application to price theory. Some topics of recent interest may also be covered.Subjects

microeconomic theory | demand theory | producer theory; partial equilibrium | competitive markets | general equilibrium | externalities | Afriat's theorem | pricing | robust comparative statics | utility theory | properties of preferences | choice as primitive | revealed preference | classical demand theory | Kuhn-Tucker necessary conditions | implications of Walras?s law | indirect utility functions | theorem of the maximum (Berge?s theorem) | expenditure minimization problem | Hicksian demands | compensated law of demand | Slutsky substitution | price changes and welfare | compensating variation | and welfare from new goods | price indexes | bias in the U.S. consumer price index | integrability | demand aggregation | aggregate demand and welfare | Frisch demands | and demand estimation | increasing differences | producer theory applications | the LeCh?telier principle | Topkis? theorem | Milgrom-Shannon monotonicity theorem | monopoly pricing | monopoly and product quality | nonlinear pricing | and price discrimination | simple models of externalities | government intervention | Coase theorem | Myerson-Sattherthwaite proposition | missing markets | price vs. quantity regulations | Weitzman?s analysis | uncertainty | common property externalities | optimization | equilibrium number of boats | welfare theorems | uniqueness and determinacy | price-taking assumption | Edgeworth box | welfare properties | Pareto efficiency | Walrasian equilibrium with transfers | Arrow-Debreu economy | separating hyperplanes | Minkowski?s theorem | Existence of Walrasian equilibrium | Kakutani?s fixed point theorem | Debreu-Gale-Kuhn-Nikaido lemma | additional properties of general equilibrium | Microfoundations | core | core convergence | general equilibrium with time and uncertainty | Jensen?s inequality | and security market economy | arbitrage pricing theory | and risk-neutral probabilities | Housing markets | competitive equilibrium | one-sided matching house allocation problem | serial dictatorship | two-sided matching | marriage markets | existence of stable matchings | incentives | housing markets core mechanismLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata15.818 Pricing (MIT) 15.818 Pricing (MIT)

Description

This course is designed to teach students how to price goods and services by providing a framework for understanding pricing strategies and tactics. Topics covered include economic value analysis, price elasticities, price customization, pricing complementary products, pricing in platform markets and anticipating competitive price responses. This course is designed to teach students how to price goods and services by providing a framework for understanding pricing strategies and tactics. Topics covered include economic value analysis, price elasticities, price customization, pricing complementary products, pricing in platform markets and anticipating competitive price responses.Subjects

pricing strategy | pricing strategy | economic value analysis | economic value analysis | price elasticities | price elasticities | price customization | price customization | pricing complementary products | pricing complementary products | pricing in platform markets | pricing in platform markets | competitive price responses | competitive price responsesLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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Subjects

rice | rice | planting | planting | ricefarming | ricefarming | ricefarmers | ricefarmers | asianamericans | asianamericansLicense

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See all metadata14.01SC Principles of Microeconomics (MIT) 14.01SC Principles of Microeconomics (MIT)

Description

Includes audio/video content: AV lectures. 14.01 Principles of Microeconomics is an introductory undergraduate course that teaches the fundamentals of microeconomics. This course introduces microeconomic concepts and analysis, supply and demand analysis, theories of the firm and individual behavior, competition and monopoly, and welfare economics. Students will also be introduced to the use of microeconomic applications to address problems in current economic policy throughout the semester. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmen Includes audio/video content: AV lectures. 14.01 Principles of Microeconomics is an introductory undergraduate course that teaches the fundamentals of microeconomics. This course introduces microeconomic concepts and analysis, supply and demand analysis, theories of the firm and individual behavior, competition and monopoly, and welfare economics. Students will also be introduced to the use of microeconomic applications to address problems in current economic policy throughout the semester. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmenSubjects

Microeconomics | Microeconomics | prices | prices | normative economics | normative economics | positive economics | positive economics | microeconomic applications | microeconomic applications | supply | supply | demand | demand | equilibrium | equilibrium | demand shift | demand shift | supply shift | supply shift | government interference | government interference | elasticity | elasticity | revenue | revenue | empirical economics | empirical economics | consumer theory | consumer theory | preference assumptions | preference assumptions | indifference curves | indifference curves | utility functions | utility functions | marginal utility | marginal utility | budget constraints | budget constraints | marginal rate of transformation | marginal rate of transformation | opportunity cost | opportunity cost | constrained utility maximization | constrained utility maximization | corner solutions | corner solutions | Engel curves | Engel curves | income effect | income effect | substitution effect | substitution effect | Giffin good | Giffin good | labor economics | labor economics | child labor | child labor | producer theory | producer theory | variable inputs | variable inputs | fixed inputs | fixed inputs | firm production functions | firm production functions | marginal rate of technical substitution | marginal rate of technical substitution | returns to scale | returns to scale | productivity | productivity | perfect competition | perfect competition | search theory | search theory | residual demand | residual demand | shutdown decisions | shutdown decisions | market equilibrium | market equilibrium | agency problem | agency problem | welfare economics | welfare economics | consumer surplus | consumer surplus | producer surplus | producer surplus | dead weight loss | dead weight loss | monopoly | monopoly | oligopoly | oligopoly | market power | market power | price discrimination | price discrimination | price regulation | price regulation | antitrust policy | antitrust policy | mergers | mergers | cartel | cartel | game theory | game theory | Nash equilibrium | Nash equilibrium | Cournot model | Cournot model | duopoly | duopoly | non-cooperative competition | non-cooperative competition | Bertrand competition | Bertrand competition | factor markets | factor markets | international trade | international trade | uncertainty | uncertainty | capital markets | capital markets | intertemporal choice | intertemporal choice | real interest rate | real interest rate | compounding | compounding | inflation | inflation | investment | investment | discount rate | discount rate | net present value | net present value | income distribution | income distribution | social welfare function | social welfare function | Utilitarianism | Utilitarianism | Raulsian criteria | Raulsian criteria | Nozickian | Nozickian | commodity egalitarianism | commodity egalitarianism | isowelfare curves | isowelfare curves | social insurance | social insurance | social security | social security | moral hazard | moral hazard | taxation | taxation | EITC | EITC | healthcare | healthcare | PPACA | PPACALicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata15.818 Pricing (MIT) 15.818 Pricing (MIT)

Description

This course, primarily discussion based, provides a framework for understanding pricing strategies and tactics. Topics covered include pricing in competitive markets, estimating demand, price discrimination, the role of price cues, anticipating competitive responses, pricing in business to business markets, and pricing of new products. Lectures and cases are the primary modes of learning. This course, primarily discussion based, provides a framework for understanding pricing strategies and tactics. Topics covered include pricing in competitive markets, estimating demand, price discrimination, the role of price cues, anticipating competitive responses, pricing in business to business markets, and pricing of new products. Lectures and cases are the primary modes of learning.Subjects

pricing | pricing | pricing strategy | pricing strategy | marketing | marketing | value-based pricing | value-based pricing | maximizing economic value | maximizing economic value | evc | evc | behavioral pricing | behavioral pricing | measuring demand | measuring demand | consumer behavior | consumer behavior | demand | demand | competition | competition | price cut | price cut | price war | price war | segmentation | segmentation | product line pricing | product line pricingLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.15J Networks (MIT) 14.15J Networks (MIT)

Description

Networks are ubiquitous in our modern society. The World Wide Web that links us to and enables information flows with the rest of the world is the most visible example. It is, however, only one of many networks within which we are situated. Our social life is organized around networks of friends and colleagues. These networks determine our information, influence our opinions, and shape our political attitudes. They also link us, often through important but weak ties, to everybody else in the United States and in the world. Economic and financial markets also look much more like networks than anonymous marketplaces. Firms interact with the same suppliers and customers and use Web-like supply chains. Financial linkages, both among banks and between consumers, companies and banks, also form a Networks are ubiquitous in our modern society. The World Wide Web that links us to and enables information flows with the rest of the world is the most visible example. It is, however, only one of many networks within which we are situated. Our social life is organized around networks of friends and colleagues. These networks determine our information, influence our opinions, and shape our political attitudes. They also link us, often through important but weak ties, to everybody else in the United States and in the world. Economic and financial markets also look much more like networks than anonymous marketplaces. Firms interact with the same suppliers and customers and use Web-like supply chains. Financial linkages, both among banks and between consumers, companies and banks, also form aSubjects

networks | networks | crowds | crowds | markets | markets | highly connected world | highly connected world | social networks | social networks | economic networks | economic networks | power networks | power networks | communication networks | communication networks | game theory | game theory | graph theory | graph theory | branching processes | branching processes | random graph models | random graph models | rich get richer phenomena | rich get richer phenomena | power laws | power laws | small worlds | small worlds | Erd?s-Renyi graphs | Erd?s-Renyi graphs | degree distributions | degree distributions | phase transitions | phase transitions | connectedness | connectedness | and giant component | and giant component | link analysis | link analysis | web search | web search | navigation | navigation | decentralized search | decentralized search | preferential attachment | preferential attachment | epidemics | epidemics | diffusion through networks | diffusion through networks | SIR | SIR | (susceptible | (susceptible | infected | infected | removed) | removed) | SIS | SIS | susceptible) | susceptible) | strategies | strategies | payoffs | payoffs | normal forms | normal forms | Nash equilibrium | Nash equilibrium | traffic networks | traffic networks | negative externalities | negative externalities | Braess' paradox | Braess' paradox | potential games | potential games | myopic behavior | myopic behavior | fictitious play | fictitious play | repeated games | repeated games | prisoner's dilemma | prisoner's dilemma | cooperation | cooperation | perfect information | perfect information | imperfect information | imperfect information | positive externalities | positive externalities | strategic complements | strategic complements | path dependence | path dependence | diffusion of innovation | diffusion of innovation | contagion pheonomena | contagion pheonomena | Bayes's rule | Bayes's rule | Bayesian Nash equilibrium | Bayesian Nash equilibrium | first price auctions | first price auctions | second price auctions | second price auctions | social learning | social learning | Bayesian learning | Bayesian learning | copying | copying | herding | herding | herd behavior | herd behavior | informational cascades | informational cascades | decisions | decisions | social choice | social choice | Condorcet jury theorem | Condorcet jury theorem | political economy | political economyLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata18.335J Introduction to Numerical Methods (MIT) 18.335J Introduction to Numerical Methods (MIT)

Description

This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB®. This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB®.Subjects

numerical linear algebra | numerical linear algebra | linear systems | linear systems | eigenvalue decomposition | eigenvalue decomposition | QR/SVD factorization | QR/SVD factorization | numerical algorithms | numerical algorithms | IEEE floating point standard | IEEE floating point standard | sparse matrices | sparse matrices | structured matrices | structured matrices | preconditioning | preconditioning | linear algebra software | linear algebra software | Matlab | MatlabLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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This subject examines the experiences of ordinary Chinese people as they lived through the tumultuous changes of the nineteenth and twentieth centuries. We look at personal narratives, primary sources, films alongside a textbook to think about how individual and family lives connect with the broader processes of change in modern China. In the readings and discussions, you should focus on how major political events have an impact on the characters' daily lives, and how the decisions they make cause large-scale social transformation. This subject examines the experiences of ordinary Chinese people as they lived through the tumultuous changes of the nineteenth and twentieth centuries. We look at personal narratives, primary sources, films alongside a textbook to think about how individual and family lives connect with the broader processes of change in modern China. In the readings and discussions, you should focus on how major political events have an impact on the characters' daily lives, and how the decisions they make cause large-scale social transformation.Subjects

China; rice; bowl; Chinese; East Asia; ordinary people; nineteenth century; twentieth century; personal narratives; primary sources; films; textbook; individual; family; lives; change; modern; readings; discussions; political events; daily; decisions; large-scale; social; transformation. | China; rice; bowl; Chinese; East Asia; ordinary people; nineteenth century; twentieth century; personal narratives; primary sources; films; textbook; individual; family; lives; change; modern; readings; discussions; political events; daily; decisions; large-scale; social; transformation. | China | China | rice | rice | bowl | bowl | Chinese | Chinese | East Asia | East Asia | ordinary people | ordinary people | nineteenth century | nineteenth century | twentieth century | twentieth century | personal narratives | personal narratives | primary sources | primary sources | films | films | textbook | textbook | individual | individual | family | family | lives | lives | change | change | modern | modern | readings | readings | discussions | discussions | political events | political events | daily | daily | decisions | decisions | large-scale | large-scale | social | social | transformation | transformation | 21F.191 | 21F.191 | 21F.991 | 21F.991License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.452 Macroeconomic Theory II (MIT) 14.452 Macroeconomic Theory II (MIT)

Description

This is the second course in the four-quarter graduate sequence in macroeconomics. Its purpose is to introduce the basic models macroeconomists use to study fluctuations. The course is organized around nine topics/sections: Fluctuations and Facts; The basic model: the consumption/saving choice; Allowing for a labor/leisure choice (the RBC model); Allowing for non trivial investment decisions; Allowing for two goods; Introducing money; Introducing price setting; Introducing staggering of price decisions; and Applications to fiscal and monetary policy. This is the second course in the four-quarter graduate sequence in macroeconomics. Its purpose is to introduce the basic models macroeconomists use to study fluctuations. The course is organized around nine topics/sections: Fluctuations and Facts; The basic model: the consumption/saving choice; Allowing for a labor/leisure choice (the RBC model); Allowing for non trivial investment decisions; Allowing for two goods; Introducing money; Introducing price setting; Introducing staggering of price decisions; and Applications to fiscal and monetary policy.Subjects

Economics | Economics | Macroeconomics | Macroeconomics | macroeconomics | macroeconomics | fluctuations | fluctuations | consumption | consumption | saving | saving | choice | choice | labor | labor | leisure | leisure | RBC model | RBC model | non trivial investment decisions | non trivial investment decisions | money | money | price setting | price setting | staggering price decisions | staggering price decisions | fiscal policy | fiscal policy | monetary policy. | monetary policy. | monetary policy | monetary policyLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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Subjects

newzealand | newzealand | landscape | landscape | paisaje | paisaje | wanganui | wanganui | durie | durie | nuevazelanda | nuevazelanda | whanganuiriver | whanganuiriver | duriehill | duriehill | williamaprice | williamaprice | williamarcherprice | williamarcherpriceLicense

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See all metadata14.462 Advanced Macroeconomics II (MIT) 14.462 Advanced Macroeconomics II (MIT)

Description

14.462 is the second semester of the second-year Ph.D. macroeconomics sequence. The course is intended to introduce the students, not only to particular areas of current research, but also to some very useful analytical tools. It covers a selection of topics that varies from year to year. Recent topics include: Growth and Fluctuations Heterogeneity and Incomplete Markets Optimal Fiscal Policy Time Inconsistency Reputation Coordination Games and Macroeconomic Complementarities Information 14.462 is the second semester of the second-year Ph.D. macroeconomics sequence. The course is intended to introduce the students, not only to particular areas of current research, but also to some very useful analytical tools. It covers a selection of topics that varies from year to year. Recent topics include: Growth and Fluctuations Heterogeneity and Incomplete Markets Optimal Fiscal Policy Time Inconsistency Reputation Coordination Games and Macroeconomic Complementarities InformationSubjects

macroeconomics research; analytical tools; analysis; endogenous growth; coordintation; incomplete markets; technolgy; distribution; employment; intellectual property rights; bounded rationality; demographics; complementarities; amplification; recursive equilibria; uncertainty; morris; shin; global games; policy; price; aggregation; social learning; dynamic adjustment; business cycle; heterogeneous agents; savings; utility; aiyagari; steady state; krusell; smith; idiosyncratic investment risk | macroeconomics research; analytical tools; analysis; endogenous growth; coordintation; incomplete markets; technolgy; distribution; employment; intellectual property rights; bounded rationality; demographics; complementarities; amplification; recursive equilibria; uncertainty; morris; shin; global games; policy; price; aggregation; social learning; dynamic adjustment; business cycle; heterogeneous agents; savings; utility; aiyagari; steady state; krusell; smith; idiosyncratic investment risk | macroeconomics research | macroeconomics research | analytical tools | analytical tools | analysis | analysis | endogenous growth | endogenous growth | coordintation | coordintation | incomplete markets | incomplete markets | technolgy | technolgy | distribution | distribution | employment | employment | intellectual property rights | intellectual property rights | bounded rationality | bounded rationality | demographics | demographics | complementarities | complementarities | amplification | amplification | recursive equilibria | recursive equilibria | uncertainty | uncertainty | morris | morris | shin | shin | global games | global games | policy | policy | price | price | aggregation | aggregation | social learning | social learning | dynamic adjustment | dynamic adjustment | business cycle | business cycle | heterogeneous agents | heterogeneous agents | savings | savings | utility | utility | aiyagari | aiyagari | steady state | steady state | krusell | krusell | smith | smith | idiosyncratic investment risk | idiosyncratic investment risk | growth | growth | fluctuations | fluctuations | heterogeneity | heterogeneity | optimal fiscal policy | optimal fiscal policy | time inconsistency | time inconsistency | reputation | reputation | information | information | coordination games | coordination gamesLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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This course focuses on alternative ways in which the issues of growth, restructuring, innovation, knowledge, learning, and accounting and measurements can be examined, covering both industrialized and emerging countries. We give special emphasis to recent transformations in regional economies throughout the world and to the implications these changes have for the theories and research methods used in spatial economic analyses. Readings will relate mainly to the United States, but we cover pertinent material on foreign countries in lectures. This course focuses on alternative ways in which the issues of growth, restructuring, innovation, knowledge, learning, and accounting and measurements can be examined, covering both industrialized and emerging countries. We give special emphasis to recent transformations in regional economies throughout the world and to the implications these changes have for the theories and research methods used in spatial economic analyses. Readings will relate mainly to the United States, but we cover pertinent material on foreign countries in lectures.Subjects

11.481 | 11.481 | 1.284 | 1.284 | ESD.192 | ESD.192 | regional growth | regional growth | political economy | political economy | spatial economic analysis | spatial economic analysis | regional economic growth | regional economic growth | economics | economics | regional theories | regional theories | regional planning | regional planning | regional and urban economics | regional and urban economics | neoclassical | neoclassical | dispersal economies | dispersal economies | regional accounting | regional accounting | social accounting matrices | social accounting matrices | underground economy | underground economy | price indices | price indices | shift share analyses | shift share analyses | energy | energy | determinants of growth | determinants of growthLicense

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See all metadata18.338J Infinite Random Matrix Theory (MIT) 18.338J Infinite Random Matrix Theory (MIT)

Description

In this course on the mathematics of infinite random matrices, students will learn about the tools such as the Stieltjes transform and Free Probability used to characterize infinite random matrices. In this course on the mathematics of infinite random matrices, students will learn about the tools such as the Stieltjes transform and Free Probability used to characterize infinite random matrices.Subjects

Infinite Random Matrices | Infinite Random Matrices | The Hermite Ensemble | The Hermite Ensemble | Wigner's Semi-Circle Law; | Wigner's Semi-Circle Law; | The Laguerre Ensemble | The Laguerre Ensemble | Marcenko-Pastur Theorem | Marcenko-Pastur Theorem | The Jacobi Ensemble | The Jacobi Ensemble | McKay's Random Graph Theorem | McKay's Random Graph Theorem | The ?Semi-Circular? Element | The ?Semi-Circular? Element | Central Limit Theorem | Central Limit Theorem | Free Cumulants in Free Probability | Free Cumulants in Free Probability | Non-Crossing Partitionsm | Non-Crossing Partitionsm | Free Cumulants | Free Cumulants | The Semi-Circular and ?Free Poisson? distributions | The Semi-Circular and ?Free Poisson? distributions | Additive Free Convolution | Additive Free Convolution | The R-Transform and the Marcenko-Pastur Theorem | The R-Transform and the Marcenko-Pastur Theorem | Multiplicative Free Convolution | Multiplicative Free Convolution | The S-Transform | The S-Transform | Non-Crossing Partitions | Non-Crossing Partitions | Orthogonal Polynomials and the Classical Matrix Ensembles | Orthogonal Polynomials and the Classical Matrix Ensembles | Tracy Widom Distribution | Tracy Widom Distribution | Eigenvalue Spectrum Fluctuations | Eigenvalue Spectrum Fluctuations | Free Probability and Fluctuations | Free Probability and Fluctuations | Zonal Polynomials and Random Matrices | Zonal Polynomials and Random Matrices | Symmetric Group Representations and Free Probability | Symmetric Group Representations and Free Probability | 18.338 | 18.338 | 16.394 | 16.394 | Wigner's Semi-Circle Law | Wigner's Semi-Circle LawLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications. This course is an introduction to the basics of random matrix theory, motivated by engineering and scientific applications.Subjects

Random matrix theory | Random matrix theory | Matrix Jacobians | Matrix Jacobians | Wishart Matrices | Wishart Matrices | Wigner's Semi-Circular laws | Wigner's Semi-Circular laws | Matrix beta ensembles | Matrix beta ensembles | free probability | free probability | spherical coordinates | spherical coordinates | wedging | wedging | Plucker coordinates | Plucker coordinates | matrix factorizations | matrix factorizations | householder transformations | householder transformations | Stiefel manifold | Stiefel manifold | Cauchey-Binet theorem | Cauchey-Binet theorem | Telatar's paper | Telatar's paper | level densities | level densities | orthogonal polynomials | orthogonal polynomials | matrix integrals | matrix integrals | hypergeometric functions | hypergeometric functions | wireless communictions | wireless communictions | eigenvalue density | eigenvalue density | sample covariance matrices | sample covariance matrices | Marcenko-Pastur theorem | Marcenko-Pastur theorem | wireless communications | wireless communicationsLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications. Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.Subjects

Discrete-time filters | Discrete-time filters | convolution | convolution | Fourier transform | Fourier transform | owpass and highpass filters | owpass and highpass filters | Sampling rate change operations | Sampling rate change operations | upsampling and downsampling | upsampling and downsampling | ractional sampling | ractional sampling | interpolation | interpolation | Filter Banks | Filter Banks | time domain (Haar example) and frequency domain | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | conditions for alias cancellation and no distortion | perfect reconstruction | perfect reconstruction | halfband filters and possible factorizations | halfband filters and possible factorizations | Modulation and polyphase representations | Modulation and polyphase representations | Noble identities | Noble identities | block Toeplitz matrices and block z-transforms | block Toeplitz matrices and block z-transforms | polyphase examples | polyphase examples | Matlab wavelet toolbox | Matlab wavelet toolbox | Orthogonal filter banks | Orthogonal filter banks | paraunitary matrices | paraunitary matrices | orthogonality condition (Condition O) in the time domain | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | modulation domain and polyphase domain | Maxflat filters | Maxflat filters | Daubechies and Meyer formulas | Daubechies and Meyer formulas | Spectral factorization | Spectral factorization | Multiresolution Analysis (MRA) | Multiresolution Analysis (MRA) | requirements for MRA | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | Refinement equation | iterative and recursive solution techniques | iterative and recursive solution techniques | infinite product formula | infinite product formula | filter bank approach for computing scaling functions and wavelets | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | Orthogonal wavelet bases | connection to orthogonal filters | connection to orthogonal filters | orthogonality in the frequency domain | orthogonality in the frequency domain | Biorthogonal wavelet bases | Biorthogonal wavelet bases | Mallat pyramid algorithm | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | Accuracy of wavelet approximations (Condition A) | vanishing moments | vanishing moments | polynomial cancellation in filter banks | polynomial cancellation in filter banks | Smoothness of wavelet bases | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | convergence of the cascade algorithm (Condition E) | splines | splines | Bases vs. frames | Bases vs. frames | Signal and image processing | Signal and image processing | finite length signals | finite length signals | boundary filters and boundary wavelets | boundary filters and boundary wavelets | wavelet compression algorithms | wavelet compression algorithms | Lifting | Lifting | ladder structure for filter banks | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | lifting form of refinement equationSec | Wavelets and subdivision | Wavelets and subdivision | nonuniform grids | nonuniform grids | multiresolution for triangular meshes | multiresolution for triangular meshes | representation and compression of surfaces | representation and compression of surfaces | Numerical solution of PDEs | Numerical solution of PDEs | Galerkin approximation | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | convergence | Subdivision wavelets for integral equations | Subdivision wavelets for integral equations | Compression and convergence estimates | Compression and convergence estimates | M-band wavelets | M-band wavelets | DFT filter banks and cosine modulated filter banks | DFT filter banks and cosine modulated filter banks | Multiwavelets | MultiwaveletsLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadataView of the station at Helensville View of the station at Helensville

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Subjects

train | train | tren | tren | railway | railway | helensville | helensville | williamaprice | williamaprice | williamarcherprice | williamarcherpriceLicense

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See all metadataWork in progress on two Clark-Sulzer engines Work in progress on two Clark-Sulzer engines

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shadow | shadow | abstract | abstract | industry | industry | metal | metal | wall | wall | shirt | shirt | hair | hair | kneel | kneel | landscape | landscape | shoe | shoe | daylight | daylight | workers | workers | construction | construction | support | support | industrial | industrial | pattern | pattern | ship | ship | shine | shine | darkness | darkness | hole | hole | timber | timber | steel | steel | parts | parts | pipe | pipe | working | working | platform | platform | engine | engine | structures | structures | progress | progress | plate | plate | vessel | vessel | social | social | row | row | tools | tools | hose | hose | pole | pole | beam | beam | climbing | climbing | maritime | maritime | bolt | bolt | cylinder | cylinder | overalls | overalls | ladder | ladder | unusual | unusual | curve | curve | shipyard | shipyard | airvent | airvent | development | development | attentive | attentive | lid | lid | lever | lever | fascinating | fascinating | digitalimage | digitalimage | sunderland | sunderland | bending | bending | manufacture | manufacture | shipbuilding | shipbuilding | southwick | southwick | blackandwhitephotograph | blackandwhitephotograph | marineengineering | marineengineering | bedplate | bedplate | americastar | americastar | may1963 | may1963 | 6cylinderengine | 6cylinderengine | 8cylinderengine | 8cylinderengine | refrigeratedcargovessel | refrigeratedcargovessel | bartramsonsltd | bartramsonsltd | williamdoxfordsonsltd | williamdoxfordsonsltd | georgeclarkltd | georgeclarkltd | southwickengineworks | southwickengineworks | workersofsunderland | workersofsunderland | clarksulzer | clarksulzer | ‘mauricedelmas’ | ‘mauricedelmas’ | twamref1361323 | twamref1361323 | mauricedelmas | mauricedelmasLicense

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See all metadata18.335J Introduction to Numerical Methods (MIT) 18.335J Introduction to Numerical Methods (MIT)

Description

This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB®. This course offers an advanced introduction to numerical linear algebra. Topics include direct and iterative methods for linear systems, eigenvalue decompositions and QR/SVD factorizations, stability and accuracy of numerical algorithms, the IEEE floating point standard, sparse and structured matrices, preconditioning, linear algebra software. Problem sets require some knowledge of MATLAB®.Subjects

numerical linear algebra | numerical linear algebra | linear systems | linear systems | eigenvalue decomposition | eigenvalue decomposition | QR/SVD factorization | QR/SVD factorization | numerical algorithms | numerical algorithms | IEEE floating point standard | IEEE floating point standard | sparse matrices | sparse matrices | structured matrices | structured matrices | preconditioning | preconditioning | linear algebra software | linear algebra software | Matlab | MatlabLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata15.433 Investments (MIT) 15.433 Investments (MIT)

Description

The focus of this course is on financial theory and empirical evidence for making investment decisions. Topics include: portfolio theory; equilibrium models of security prices (including the capital asset pricing model and the arbitrage pricing theory); the empirical behavior of security prices; market efficiency; performance evaluation; and behavioral finance. The focus of this course is on financial theory and empirical evidence for making investment decisions. Topics include: portfolio theory; equilibrium models of security prices (including the capital asset pricing model and the arbitrage pricing theory); the empirical behavior of security prices; market efficiency; performance evaluation; and behavioral finance.Subjects

Financial theory | Financial theory | empirical evidence | empirical evidence | investment decisions | investment decisions | portfolio theory | portfolio theory | equilibrium models of security prices | equilibrium models of security prices | capital asset pricing model | capital asset pricing model | arbitrage pricing theory | arbitrage pricing theory | empirical behavior of security prices | empirical behavior of security prices | market efficiency | performance evaluation | market efficiency | performance evaluation | market efficiency | market efficiency | performance evaluation | performance evaluation | behavioral finance | behavioral financeLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htmSite sourced from

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This course is designed to teach students how to price goods and services by providing a framework for understanding pricing strategies and tactics. Topics covered include economic value analysis, price elasticities, price customization, pricing complementary products, pricing in platform markets and anticipating competitive price responses.Subjects

pricing strategy | economic value analysis | price elasticities | price customization | pricing complementary products | pricing in platform markets | competitive price responsesLicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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See all metadata14.01SC Principles of Microeconomics (MIT)

Description

14.01 Principles of Microeconomics is an introductory undergraduate course that teaches the fundamentals of microeconomics. This course introduces microeconomic concepts and analysis, supply and demand analysis, theories of the firm and individual behavior, competition and monopoly, and welfare economics. Students will also be introduced to the use of microeconomic applications to address problems in current economic policy throughout the semester. This course is a core subject in MIT's undergraduate Energy Studies Minor. This Institute-wide program complements the deep expertise obtained in any major with a broad understanding of the interlinked realms of science, technology, and social sciences as they relate to energy and associated environmental challenges.Subjects

Microeconomics | prices | normative economics | positive economics | microeconomic applications | supply | demand | equilibrium | demand shift | supply shift | government interference | elasticity | revenue | empirical economics | consumer theory | preference assumptions | indifference curves | utility functions | marginal utility | budget constraints | marginal rate of transformation | opportunity cost | constrained utility maximization | corner solutions | Engel curves | income effect | substitution effect | Giffin good | labor economics | child labor | producer theory | variable inputs | fixed inputs | firm production functions | marginal rate of technical substitution | returns to scale | productivity | perfect competition | search theory | residual demand | shutdown decisions | market equilibrium | agency problem | welfare economics | consumer surplus | producer surplus | dead weight loss | monopoly | oligopoly | market power | price discrimination | price regulation | antitrust policy | mergers | cartel | game theory | Nash equilibrium | Cournot model | duopoly | non-cooperative competition | Bertrand competition | factor markets | international trade | uncertainty | capital markets | intertemporal choice | real interest rate | compounding | inflation | investment | discount rate | net present value | income distribution | social welfare function | Utilitarianism | Raulsian criteria | Nozickian | commodity egalitarianism | isowelfare curves | social insurance | social security | moral hazard | taxation | EITC | healthcare | PPACALicense

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htmSite sourced from

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Linear algebra is the study of linear equations, vector spaces, linear maps and Euclidean spaces. The subject covers all topics in a first year college in a linear algebra course. Linear Algebra finds applications in virtually every area of mathematics, including Multivariate Calculus, Differential Equations, and Probability Theory. The subject will mainly develop the theory of Linear Algebra, and will focus on the computational aspects. Linear Algebra as the structure underlying in the study of the Euclidean Geometry is developed and explained with a interesting figure description of the movements in the space. The mathematical formulas are also written with different color in order to make easier the compression of the subject. Linear algebra is the study of linear equations, vector spaces, linear maps and Euclidean spaces. The subject covers all topics in a first year college in a linear algebra course. Linear Algebra finds applications in virtually every area of mathematics, including Multivariate Calculus, Differential Equations, and Probability Theory. The subject will mainly develop the theory of Linear Algebra, and will focus on the computational aspects. Linear Algebra as the structure underlying in the study of the Euclidean Geometry is developed and explained with a interesting figure description of the movements in the space. The mathematical formulas are also written with different color in order to make easier the compression of the subject.Subjects

Espacio Euclídeo | Espacio Euclídeo | Álgebra Lineal | Álgebra Lineal | Álgebra | Álgebra | Ecuaciones Lineales | Ecuaciones Lineales | Matrices | MatricesLicense

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