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Search, Evaluate, and Read Evidence","\u002Fen\u002Facademic-writing\u002F02-reading-and-literature-matrix","en\u002Facademic-writing\u002F02-reading-and-literature-matrix",{"title":27,"path":28,"stem":29},"3. From Sources to Synthesis and Argument","\u002Fen\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs","en\u002Facademic-writing\u002F03-arguments-outlines-and-paragraphs",{"title":31,"path":32,"stem":33},"4. Literature Review and a Defensible Gap","\u002Fen\u002Facademic-writing\u002F04-writing-the-literature-review","en\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":35,"path":36,"stem":37},"5. Research Design, Evidence, and Core Sections","\u002Fen\u002Facademic-writing\u002F05-core-sections-and-evidence","en\u002Facademic-writing\u002F05-core-sections-and-evidence",{"title":39,"path":40,"stem":41},"6. Citation, Paraphrasing, Integrity, and AI","\u002Fen\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity","en\u002Facademic-writing\u002F06-citation-paraphrasing-and-integrity",{"title":43,"path":44,"stem":45},"7. Revision, Review, and Submission","\u002Fen\u002Facademic-writing\u002F07-revision-style-and-submission","en\u002Facademic-writing\u002F07-revision-style-and-submission",{"title":47,"path":48,"stem":49},"8. Research Workbook","\u002Fen\u002Facademic-writing\u002F08-literature-review-checklist-and-template","en\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":51,"path":52,"stem":53},"9. Research Workflow and Tools","\u002Fen\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management","en\u002Facademic-writing\u002F09-tools-for-academic-writing-and-literature-management",{"title":55,"path":56,"stem":57},"10. Worked Project — From Question to Defensible Conclusion","\u002Fen\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain","en\u002Facademic-writing\u002F10-worked-example-from-block-structure-to-question-chain",{"title":59,"path":60,"stem":61,"children":62,"page":249},"Accounting","\u002Fen\u002Faccounting","en\u002Faccounting",[63,69,87,93,193],{"title":64,"path":65,"stem":66,"children":67},"Accounting — From Evidence to Decisions","\u002Fen\u002Faccounting\u002F00-index","en\u002Faccounting\u002F00-index",[68],{"title":64,"path":65,"stem":66},{"title":70,"path":71,"stem":72,"children":73},"Accounting Appendix","\u002Fen\u002Faccounting\u002Fappendix","en\u002Faccounting\u002Fappendix\u002Findex",[74,75,79,83],{"title":70,"path":71,"stem":72},{"title":76,"path":77,"stem":78},"Worked Examples and Error Diagnosis","\u002Fen\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls","en\u002Faccounting\u002Fappendix\u002F24-examples-pitfalls",{"title":80,"path":81,"stem":82},"Accounting Glossary","\u002Fen\u002Faccounting\u002Fappendix\u002F26-glossary","en\u002Faccounting\u002Fappendix\u002F26-glossary",{"title":84,"path":85,"stem":86},"Reading and Evidence Map","\u002Fen\u002Faccounting\u002Fappendix\u002F27-reading-map","en\u002Faccounting\u002Fappendix\u002F27-reading-map",{"title":88,"path":89,"stem":90,"children":91},"Northstar Record-to-Decision Capstone","\u002Fen\u002Faccounting\u002Fcapstone","en\u002Faccounting\u002Fcapstone\u002Findex",[92],{"title":88,"path":89,"stem":90},{"title":94,"path":95,"stem":96,"children":97},"Financial Accounting","\u002Fen\u002Faccounting\u002Ffinancial-accounting","en\u002Faccounting\u002Ffinancial-accounting\u002Findex",[98,99,117,131,165,179],{"title":94,"path":95,"stem":96},{"title":100,"path":101,"stem":102,"children":103},"1. Foundations","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002Findex",[104,105,109,113],{"title":100,"path":101,"stem":102},{"title":106,"path":107,"stem":108},"Objectives and Qualitative Characteristics","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F01-objectives-qualitative-characteristics",{"title":110,"path":111,"stem":112},"Equation and Elements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F02-equation-elements",{"title":114,"path":115,"stem":116},"Accrual Basis, Estimates and Periods","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles","en\u002Faccounting\u002Ffinancial-accounting\u002F01-foundations\u002F03-accounting-bases-principles",{"title":118,"path":119,"stem":120,"children":121},"2. Recording Transactions","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002Findex",[122,123,127],{"title":118,"path":119,"stem":120},{"title":124,"path":125,"stem":126},"Double-Entry and Debit\u002FCredit","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F04-double-entry-dr-cr",{"title":128,"path":129,"stem":130},"Accounting Cycle","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle","en\u002Faccounting\u002Ffinancial-accounting\u002F02-recording-transactions\u002F05-accounting-cycle",{"title":132,"path":133,"stem":134,"children":135},"3. Measurement and Adjustments","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002Findex",[136,137,141,145,149,153,157,161],{"title":132,"path":133,"stem":134},{"title":138,"path":139,"stem":140},"Revenue Recognition","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F07-revenue-recognition",{"title":142,"path":143,"stem":144},"Inventory","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F08-inventory",{"title":146,"path":147,"stem":148},"Receivables and Expected Credit Losses","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F09-receivables",{"title":150,"path":151,"stem":152},"Property, Plant and Equipment","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F10-ppe",{"title":154,"path":155,"stem":156},"Intangible Assets","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F11-intangibles",{"title":158,"path":159,"stem":160},"Leases","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F12-leases",{"title":162,"path":163,"stem":164},"Current and Deferred Income Tax","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax","en\u002Faccounting\u002Ffinancial-accounting\u002F03-measurement-adjustments\u002F13-income-tax",{"title":166,"path":167,"stem":168,"children":169},"4. Reporting and Cash","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002Findex",[170,171,175],{"title":166,"path":167,"stem":168},{"title":172,"path":173,"stem":174},"Linked Financial Statements","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F06-financial-statements",{"title":176,"path":177,"stem":178},"Cash, Reconciliation and Internal Control","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control","en\u002Faccounting\u002Ffinancial-accounting\u002F04-reporting-cash\u002F14-cash-control",{"title":180,"path":181,"stem":182,"children":183},"5. Analysis and Comparison","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002Findex",[184,185,189],{"title":180,"path":181,"stem":182},{"title":186,"path":187,"stem":188},"Ratio Analysis","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F15-ratio-analysis",{"title":190,"path":191,"stem":192},"IFRS versus US GAAP","\u002Fen\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap","en\u002Faccounting\u002Ffinancial-accounting\u002F05-analysis-comparison\u002F25-ifrs-gaap",{"title":194,"path":195,"stem":196,"children":197},"Management Accounting","\u002Fen\u002Faccounting\u002Fmanagement-accounting","en\u002Faccounting\u002Fmanagement-accounting\u002Findex",[198,199,217,235],{"title":194,"path":195,"stem":196},{"title":200,"path":201,"stem":202,"children":203},"1. Cost Foundations","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002Findex",[204,205,209,213],{"title":200,"path":201,"stem":202},{"title":206,"path":207,"stem":208},"Cost Concepts and Behaviour","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F16-cost-concepts",{"title":210,"path":211,"stem":212},"Costing Systems","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F17-costing-systems",{"title":214,"path":215,"stem":216},"Variable and Absorption Costing","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption","en\u002Faccounting\u002Fmanagement-accounting\u002F01-cost-foundations\u002F19-variable-vs-absorption",{"title":218,"path":219,"stem":220,"children":221},"2. Planning and Control","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002Findex",[222,223,227,231],{"title":218,"path":219,"stem":220},{"title":224,"path":225,"stem":226},"Cost–Volume–Profit Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F18-cvp-analysis",{"title":228,"path":229,"stem":230},"Budgeting and Variance Analysis","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F20-budgeting-variances",{"title":232,"path":233,"stem":234},"Performance Measurement","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement","en\u002Faccounting\u002Fmanagement-accounting\u002F02-planning-control\u002F21-performance-measurement",{"title":236,"path":237,"stem":238,"children":239},"3. Decisions and Investment","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002Findex",[240,241,245],{"title":236,"path":237,"stem":238},{"title":242,"path":243,"stem":244},"Short-Term Decisions","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F22-short-term-decisions",{"title":246,"path":247,"stem":248},"Capital Budgeting","\u002Fen\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting","en\u002Faccounting\u002Fmanagement-accounting\u002F03-decisions-investment\u002F23-capital-budgeting",false,{"title":251,"path":252,"stem":253,"children":254,"page":249},"Business Analytics","\u002Fen\u002Fbusiness-analytics","en\u002Fbusiness-analytics",[255,261,279,305,335,365,383,400],{"title":256,"path":257,"stem":258,"children":259},"Business Analytics — From Data to Defensible Action","\u002Fen\u002Fbusiness-analytics\u002F00-index","en\u002Fbusiness-analytics\u002F00-index",[260],{"title":256,"path":257,"stem":258},{"title":262,"path":263,"stem":264,"children":265},"0. Decision and Data Foundations","\u002Fen\u002Fbusiness-analytics\u002F00-intro","en\u002Fbusiness-analytics\u002F00-intro\u002Findex",[266,267,271,275],{"title":262,"path":263,"stem":264},{"title":268,"path":269,"stem":270},"Decision Framing","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing","en\u002Fbusiness-analytics\u002F00-intro\u002F01-decision-framing",{"title":272,"path":273,"stem":274},"Data Contracts","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts","en\u002Fbusiness-analytics\u002F00-intro\u002F02-data-contracts",{"title":276,"path":277,"stem":278},"Reproducible Analytics Workflow","\u002Fen\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow","en\u002Fbusiness-analytics\u002F00-intro\u002F03-reproducible-workflow",{"title":280,"path":281,"stem":282,"children":283},"1. Descriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive","en\u002Fbusiness-analytics\u002F01-descriptive\u002Findex",[284,285,289,293,297,301],{"title":280,"path":281,"stem":282},{"title":286,"path":287,"stem":288},"Distributions and Exploratory Data Analysis","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda","en\u002Fbusiness-analytics\u002F01-descriptive\u002F04-distributions-eda",{"title":290,"path":291,"stem":292},"KPIs and Denominators","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators","en\u002Fbusiness-analytics\u002F01-descriptive\u002F05-kpis-denominators",{"title":294,"path":295,"stem":296},"Segments, Cohorts and Funnels","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels","en\u002Fbusiness-analytics\u002F01-descriptive\u002F06-segmentation-cohorts-funnels",{"title":298,"path":299,"stem":300},"Visual Evidence and Data Stories","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story","en\u002Fbusiness-analytics\u002F01-descriptive\u002F07-visualisation-story",{"title":302,"path":303,"stem":304},"Experiments and Causal Boundaries","\u002Fen\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary","en\u002Fbusiness-analytics\u002F01-descriptive\u002F08-experiments-causal-boundary",{"title":306,"path":307,"stem":308,"children":309},"2. Predictive Analytics","\u002Fen\u002Fbusiness-analytics\u002F02-predictive","en\u002Fbusiness-analytics\u002F02-predictive\u002Findex",[310,311,315,319,323,327,331],{"title":306,"path":307,"stem":308},{"title":312,"path":313,"stem":314},"Validation, Baselines and Leakage","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage","en\u002Fbusiness-analytics\u002F02-predictive\u002F09-validation-leakage",{"title":316,"path":317,"stem":318},"Regression and Forecasting","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting","en\u002Fbusiness-analytics\u002F02-predictive\u002F10-regression-forecasting",{"title":320,"path":321,"stem":322},"Classification and Calibration","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration","en\u002Fbusiness-analytics\u002F02-predictive\u002F11-classification-calibration",{"title":324,"path":325,"stem":326},"Trees and Ensembles","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles","en\u002Fbusiness-analytics\u002F02-predictive\u002F12-trees-ensembles",{"title":328,"path":329,"stem":330},"Decision Metrics and Thresholds","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds","en\u002Fbusiness-analytics\u002F02-predictive\u002F13-decision-metrics-thresholds",{"title":332,"path":333,"stem":334},"Explainability, Monitoring and Drift","\u002Fen\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift","en\u002Fbusiness-analytics\u002F02-predictive\u002F14-explainability-drift",{"title":336,"path":337,"stem":338,"children":339},"3. Prescriptive Analytics","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive","en\u002Fbusiness-analytics\u002F03-prescriptive\u002Findex",[340,341,345,349,353,357,361],{"title":336,"path":337,"stem":338},{"title":342,"path":343,"stem":344},"Decision-Making under Uncertainty","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F15-decision-uncertainty",{"title":346,"path":347,"stem":348},"Linear Optimisation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F16-linear-optimization",{"title":350,"path":351,"stem":352},"Inventory and Allocation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F17-inventory-allocation",{"title":354,"path":355,"stem":356},"Queueing and Simulation","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F18-queueing-simulation",{"title":358,"path":359,"stem":360},"Pricing and Revenue Management","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F19-pricing-experimentation",{"title":362,"path":363,"stem":364},"Causal Targeting and Policy Learning","\u002Fen\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting","en\u002Fbusiness-analytics\u002F03-prescriptive\u002F20-causal-targeting",{"title":366,"path":367,"stem":368,"children":369},"4. Deployment and Governance","\u002Fen\u002Fbusiness-analytics\u002F04-deployment","en\u002Fbusiness-analytics\u002F04-deployment\u002Findex",[370,371,375,379],{"title":366,"path":367,"stem":368},{"title":372,"path":373,"stem":374},"Data Products and Monitoring","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring","en\u002Fbusiness-analytics\u002F04-deployment\u002F21-data-products-monitoring",{"title":376,"path":377,"stem":378},"Governance, Fairness and Privacy","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy","en\u002Fbusiness-analytics\u002F04-deployment\u002F22-governance-fairness-privacy",{"title":380,"path":381,"stem":382},"Adoption and Business Value","\u002Fen\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value","en\u002Fbusiness-analytics\u002F04-deployment\u002F23-adoption-value",{"title":384,"path":385,"stem":386,"children":387},"Appendix and Revision Tools","\u002Fen\u002Fbusiness-analytics\u002Fappendix","en\u002Fbusiness-analytics\u002Fappendix\u002Findex",[388,389,393,397],{"title":384,"path":385,"stem":386},{"title":390,"path":391,"stem":392},"Formula and Decision Map","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map","en\u002Fbusiness-analytics\u002Fappendix\u002F24-formula-map",{"title":394,"path":395,"stem":396},"Worked Examples and Pitfalls","\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls","en\u002Fbusiness-analytics\u002Fappendix\u002F25-worked-examples-pitfalls",{"title":84,"path":398,"stem":399},"\u002Fen\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map","en\u002Fbusiness-analytics\u002Fappendix\u002F26-reading-map",{"title":401,"path":402,"stem":403,"children":404},"Capstone — The Evening Delivery Promise","\u002Fen\u002Fbusiness-analytics\u002Fcapstone","en\u002Fbusiness-analytics\u002Fcapstone\u002Findex",[405],{"title":401,"path":402,"stem":403},{"title":407,"path":408,"stem":409,"children":410,"page":249},"Financial Economic Time Series","\u002Fen\u002Ffinancial-economic-time-series","en\u002Ffinancial-economic-time-series",[411,417,423,429,435,441,447,453,459,465,471],{"title":412,"path":413,"stem":414,"children":415},"Financial and Economic Time Series — Course Guide","\u002Fen\u002Ffinancial-economic-time-series\u002F00-intro","en\u002Ffinancial-economic-time-series\u002F00-intro\u002Findex",[416],{"title":412,"path":413,"stem":414},{"title":418,"path":419,"stem":420,"children":421},"Three Versions of Time Series","\u002Fen\u002Ffinancial-economic-time-series\u002F01-bridge","en\u002Ffinancial-economic-time-series\u002F01-bridge\u002Findex",[422],{"title":418,"path":419,"stem":420},{"title":424,"path":425,"stem":426,"children":427},"Data, Clocks, and Transformations","\u002Fen\u002Ffinancial-economic-time-series\u002F02-data-transformations","en\u002Ffinancial-economic-time-series\u002F02-data-transformations\u002Findex",[428],{"title":424,"path":425,"stem":426},{"title":430,"path":431,"stem":432,"children":433},"Predictive Regressions and Persistent Predictors","\u002Fen\u002Ffinancial-economic-time-series\u002F03-predictive-regressions","en\u002Ffinancial-economic-time-series\u002F03-predictive-regressions\u002Findex",[434],{"title":430,"path":431,"stem":432},{"title":436,"path":437,"stem":438,"children":439},"Volatility, Tails, and Financial Risk","\u002Fen\u002Ffinancial-economic-time-series\u002F04-volatility-risk","en\u002Ffinancial-economic-time-series\u002F04-volatility-risk\u002Findex",[440],{"title":436,"path":437,"stem":438},{"title":442,"path":443,"stem":444,"children":445},"Unit Roots, Cointegration, and Error Correction","\u002Fen\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration","en\u002Ffinancial-economic-time-series\u002F05-unit-roots-cointegration\u002Findex",[446],{"title":442,"path":443,"stem":444},{"title":448,"path":449,"stem":450,"children":451},"VAR, Structural Identification, and Local Projections","\u002Fen\u002Ffinancial-economic-time-series\u002F06-var-identification","en\u002Ffinancial-economic-time-series\u002F06-var-identification\u002Findex",[452],{"title":448,"path":449,"stem":450},{"title":454,"path":455,"stem":456,"children":457},"State Space, Mixed Frequency, and Nowcasting","\u002Fen\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting","en\u002Ffinancial-economic-time-series\u002F07-state-space-nowcasting\u002Findex",[458],{"title":454,"path":455,"stem":456},{"title":460,"path":461,"stem":462,"children":463},"Forecast Evaluation for Decisions","\u002Fen\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation","en\u002Ffinancial-economic-time-series\u002F08-forecast-evaluation\u002Findex",[464],{"title":460,"path":461,"stem":462},{"title":466,"path":467,"stem":468,"children":469},"Integrated R Laboratory","\u002Fen\u002Ffinancial-economic-time-series\u002F09-r-laboratory","en\u002Ffinancial-economic-time-series\u002F09-r-laboratory\u002Findex",[470],{"title":466,"path":467,"stem":468},{"title":472,"path":473,"stem":474,"children":475},"Capstones, Data, and Reading Ladder","\u002Fen\u002Ffinancial-economic-time-series\u002F10-capstone-readings","en\u002Ffinancial-economic-time-series\u002F10-capstone-readings\u002Findex",[476],{"title":472,"path":473,"stem":474},{"title":478,"path":479,"stem":480,"children":481,"page":249},"Intro To Economics","\u002Fen\u002Fintro-to-economics","en\u002Fintro-to-economics",[482,486,490,494,498,502,506,510,514,518,522,526,530],{"title":483,"path":484,"stem":485},"Introduction to Economics — Decisions, Markets, and the Macroeconomy","\u002Fen\u002Fintro-to-economics\u002F00-intro","en\u002Fintro-to-economics\u002F00-intro",{"title":487,"path":488,"stem":489},"Chapter 1 — Choice, Opportunity Cost, and Trade","\u002Fen\u002Fintro-to-economics\u002F01-foundations","en\u002Fintro-to-economics\u002F01-foundations",{"title":491,"path":492,"stem":493},"Chapter 2 — Demand, Supply, Equilibrium, and Welfare","\u002Fen\u002Fintro-to-economics\u002F02-demand-and-supply","en\u002Fintro-to-economics\u002F02-demand-and-supply",{"title":495,"path":496,"stem":497},"Chapter 3 — Elasticity, Revenue, and Tax Incidence","\u002Fen\u002Fintro-to-economics\u002F03-elasticity","en\u002Fintro-to-economics\u002F03-elasticity",{"title":499,"path":500,"stem":501},"Chapter 4 — Firms, Market Power, and Market Failure","\u002Fen\u002Fintro-to-economics\u002F04-market-structures","en\u002Fintro-to-economics\u002F04-market-structures",{"title":503,"path":504,"stem":505},"Chapter 5 — GDP, Income, Wealth, and Welfare","\u002Fen\u002Fintro-to-economics\u002F05-gdp-and-wealth","en\u002Fintro-to-economics\u002F05-gdp-and-wealth",{"title":507,"path":508,"stem":509},"Chapter 6 — Inflation, Purchasing Power, and Labour Markets","\u002Fen\u002Fintro-to-economics\u002F06-inflation-and-unemployment","en\u002Fintro-to-economics\u002F06-inflation-and-unemployment",{"title":511,"path":512,"stem":513},"Chapter 7 — Productivity, Technology, and Economic Growth","\u002Fen\u002Fintro-to-economics\u002F07-economic-growth","en\u002Fintro-to-economics\u002F07-economic-growth",{"title":515,"path":516,"stem":517},"Chapter 8 — Money, Credit, and Banking","\u002Fen\u002Fintro-to-economics\u002F08-money-and-banking","en\u002Fintro-to-economics\u002F08-money-and-banking",{"title":519,"path":520,"stem":521},"Chapter 9 — Business Cycles, AD–AS, and Monetary Policy","\u002Fen\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as","en\u002Fintro-to-economics\u002F09-monetary-policy-and-ad-as",{"title":523,"path":524,"stem":525},"Chapter 10 — Fiscal Policy, Distribution, and Public Debt","\u002Fen\u002Fintro-to-economics\u002F10-fiscal-policy","en\u002Fintro-to-economics\u002F10-fiscal-policy",{"title":527,"path":528,"stem":529},"Chapter 11 — Trade, Capital Flows, and Exchange Rates","\u002Fen\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates","en\u002Fintro-to-economics\u002F11-open-economy-and-exchange-rates",{"title":531,"path":532,"stem":533},"Chapter 12 — Integrated Economic Analysis Studio","\u002Fen\u002Fintro-to-economics\u002F12-review-and-case-studies","en\u002Fintro-to-economics\u002F12-review-and-case-studies",{"title":535,"path":536,"stem":537,"children":538,"page":249},"Microeconometrics","\u002Fen\u002Fmicroeconometrics","en\u002Fmicroeconometrics",[539,557,563,581,603,629,651,673,691,709],{"title":540,"path":541,"stem":542,"children":543},"0. Causal Questions and Designs","\u002Fen\u002Fmicroeconometrics\u002F00-foundations","en\u002Fmicroeconometrics\u002F00-foundations\u002Findex",[544,545,549,553],{"title":540,"path":541,"stem":542},{"title":546,"path":547,"stem":548},"Causal Questions and Estimands","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands","en\u002Fmicroeconometrics\u002F00-foundations\u002F01-causal-question-estimands",{"title":550,"path":551,"stem":552},"Potential Outcomes and Experiments","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments","en\u002Fmicroeconometrics\u002F00-foundations\u002F02-potential-outcomes-experiments",{"title":554,"path":555,"stem":556},"Causal Diagrams and Controls","\u002Fen\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls","en\u002Fmicroeconometrics\u002F00-foundations\u002F03-dags-controls",{"title":558,"path":559,"stem":560,"children":561},"Microeconometrics — Designing Credible Counterfactuals","\u002Fen\u002Fmicroeconometrics\u002F00-index","en\u002Fmicroeconometrics\u002F00-index",[562],{"title":558,"path":559,"stem":560},{"title":564,"path":565,"stem":566,"children":567},"1. Regression and Inference","\u002Fen\u002Fmicroeconometrics\u002F01-regression","en\u002Fmicroeconometrics\u002F01-regression\u002Findex",[568,569,573,577],{"title":564,"path":565,"stem":566},{"title":570,"path":571,"stem":572},"OLS as a Projection","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection","en\u002Fmicroeconometrics\u002F01-regression\u002F04-ols-projection",{"title":574,"path":575,"stem":576},"FWL, Selection and Controls","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls","en\u002Fmicroeconometrics\u002F01-regression\u002F05-fwl-selection-controls",{"title":578,"path":579,"stem":580},"Inference and Clustering","\u002Fen\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering","en\u002Fmicroeconometrics\u002F01-regression\u002F06-inference-clustering",{"title":582,"path":583,"stem":584,"children":585},"2. Instruments and Panel Data","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002Findex",[586,587,591,595,599],{"title":582,"path":583,"stem":584},{"title":588,"path":589,"stem":590},"Instrumental Variables","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F07-iv-identification",{"title":592,"path":593,"stem":594},"Weak Instruments and LATE","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F08-weak-iv-late",{"title":596,"path":597,"stem":598},"Panel Fixed Effects","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F09-panel-fixed-effects",{"title":600,"path":601,"stem":602},"Dynamic Panels and Limits","\u002Fen\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel","en\u002Fmicroeconometrics\u002F02-iv-panel\u002F10-dynamic-panel",{"title":604,"path":605,"stem":606,"children":607},"3. Policy Evaluation Designs","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs","en\u002Fmicroeconometrics\u002F03-policy-designs\u002Findex",[608,609,613,617,621,625],{"title":604,"path":605,"stem":606},{"title":610,"path":611,"stem":612},"Difference-in-Differences","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F11-did-core",{"title":614,"path":615,"stem":616},"Staggered DID and Event Studies","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F12-staggered-event-studies",{"title":618,"path":619,"stem":620},"Regression Discontinuity","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F13-rdd",{"title":622,"path":623,"stem":624},"Matching, Weighting and Overlap","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F14-matching-weighting",{"title":626,"path":627,"stem":628},"Synthetic Control","\u002Fen\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control","en\u002Fmicroeconometrics\u002F03-policy-designs\u002F15-synthetic-control",{"title":630,"path":631,"stem":632,"children":633},"Module 4 — Choice and Limited Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002Findex",[634,635,639,643,647],{"title":630,"path":631,"stem":632},{"title":636,"path":637,"stem":638},"Binary Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F16-binary-choice",{"title":640,"path":641,"stem":642},"Multinomial and Ordered Choice","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F17-multinomial-choice",{"title":644,"path":645,"stem":646},"Count Outcomes","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F18-count-outcomes",{"title":648,"path":649,"stem":650},"Censoring, Truncation and Selection","\u002Fen\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection","en\u002Fmicroeconometrics\u002F04-outcomes-choice\u002F19-censoring-selection",{"title":652,"path":653,"stem":654,"children":655},"Module 5 — Modern Causal Analysis","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal","en\u002Fmicroeconometrics\u002F05-modern-causal\u002Findex",[656,657,661,665,669],{"title":652,"path":653,"stem":654},{"title":658,"path":659,"stem":660},"Double Machine Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F20-dml",{"title":662,"path":663,"stem":664},"Heterogeneity and Policy Learning","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F21-heterogeneity-policy",{"title":666,"path":667,"stem":668},"Sensitivity and Partial Identification","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F22-sensitivity-partial-id",{"title":670,"path":671,"stem":672},"External Validity and Transport","\u002Fen\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity","en\u002Fmicroeconometrics\u002F05-modern-causal\u002F23-external-validity",{"title":674,"path":675,"stem":676,"children":677},"Module 6 — From Data to Auditable Evidence","\u002Fen\u002Fmicroeconometrics\u002F06-workflow","en\u002Fmicroeconometrics\u002F06-workflow\u002Findex",[678,679,683,687],{"title":674,"path":675,"stem":676},{"title":680,"path":681,"stem":682},"Data Provenance","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance","en\u002Fmicroeconometrics\u002F06-workflow\u002F24-data-provenance",{"title":684,"path":685,"stem":686},"Reproducibility and Reporting","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting","en\u002Fmicroeconometrics\u002F06-workflow\u002F25-reproducibility-reporting",{"title":688,"path":689,"stem":690},"Paper and Evidence Audit","\u002Fen\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit","en\u002Fmicroeconometrics\u002F06-workflow\u002F26-paper-audit",{"title":692,"path":693,"stem":694,"children":695},"Appendix — Revision and Evidence Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix","en\u002Fmicroeconometrics\u002Fappendix\u002Findex",[696,697,701,705],{"title":692,"path":693,"stem":694},{"title":698,"path":699,"stem":700},"Formula and Design Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map","en\u002Fmicroeconometrics\u002Fappendix\u002F27-formula-design-map",{"title":702,"path":703,"stem":704},"Ten Worked Microeconometric Pitfalls","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls","en\u002Fmicroeconometrics\u002Fappendix\u002F28-worked-pitfalls",{"title":706,"path":707,"stem":708},"Reading and Software Map","\u002Fen\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map","en\u002Fmicroeconometrics\u002Fappendix\u002F29-reading-software-map",{"title":710,"path":711,"stem":712,"children":713},"Capstone — Should Northbridge Expand Pathways?","\u002Fen\u002Fmicroeconometrics\u002Fcapstone","en\u002Fmicroeconometrics\u002Fcapstone\u002Findex",[714],{"title":710,"path":711,"stem":712},{"title":716,"path":717,"stem":718,"children":719,"page":249},"Microeconomics","\u002Fen\u002Fmicroeconomics","en\u002Fmicroeconomics",[720,726,732,738,744,750,756,762,768,774,780,786,792],{"title":721,"path":722,"stem":723,"children":724},"Advanced Microeconomics — Course Guide","\u002Fen\u002Fmicroeconomics\u002F00-intro","en\u002Fmicroeconomics\u002F00-intro\u002Findex",[725],{"title":721,"path":722,"stem":723},{"title":727,"path":728,"stem":729,"children":730},"Module 1 — Choice, Duality, and Revealed Preference","\u002Fen\u002Fmicroeconomics\u002F01-fundations","en\u002Fmicroeconomics\u002F01-fundations\u002Findex",[731],{"title":727,"path":728,"stem":729},{"title":733,"path":734,"stem":735,"children":736},"Module 2 — Comparative Statics and Welfare Measurement","\u002Fen\u002Fmicroeconomics\u002F02-comparative-statics","en\u002Fmicroeconomics\u002F02-comparative-statics\u002Findex",[737],{"title":733,"path":734,"stem":735},{"title":739,"path":740,"stem":741,"children":742},"Module 3 — Choice under Risk and Insurance","\u002Fen\u002Fmicroeconomics\u002F03-uncertainty","en\u002Fmicroeconomics\u002F03-uncertainty\u002Findex",[743],{"title":739,"path":740,"stem":741},{"title":745,"path":746,"stem":747,"children":748},"Module 4 — General Equilibrium and Welfare","\u002Fen\u002Fmicroeconomics\u002F04-general-equilibrium","en\u002Fmicroeconomics\u002F04-general-equilibrium\u002Findex",[749],{"title":745,"path":746,"stem":747},{"title":751,"path":752,"stem":753,"children":754},"Module 5 — Static, Dynamic, and Repeated Games","\u002Fen\u002Fmicroeconomics\u002F05-game-theory","en\u002Fmicroeconomics\u002F05-game-theory\u002Findex",[755],{"title":751,"path":752,"stem":753},{"title":757,"path":758,"stem":759,"children":760},"Module 6 — Oligopoly, Entry, and Algorithmic Pricing","\u002Fen\u002Fmicroeconomics\u002F06-oligopoly","en\u002Fmicroeconomics\u002F06-oligopoly\u002Findex",[761],{"title":757,"path":758,"stem":759},{"title":763,"path":764,"stem":765,"children":766},"Module 7 — Information Economics and Contracts","\u002Fen\u002Fmicroeconomics\u002F07-information-economics","en\u002Fmicroeconomics\u002F07-information-economics\u002Findex",[767],{"title":763,"path":764,"stem":765},{"title":769,"path":770,"stem":771,"children":772},"Module 8 — Mechanism Design and Auctions","\u002Fen\u002Fmicroeconomics\u002F08-mechanism-design","en\u002Fmicroeconomics\u002F08-mechanism-design\u002Findex",[773],{"title":769,"path":770,"stem":771},{"title":775,"path":776,"stem":777,"children":778},"Module 9 — Behavioural and Experimental Microeconomics","\u002Fen\u002Fmicroeconomics\u002F09-behavioural-economics","en\u002Fmicroeconomics\u002F09-behavioural-economics\u002Findex",[779],{"title":775,"path":776,"stem":777},{"title":781,"path":782,"stem":783,"children":784},"Module 10 — Externalities, Public Goods, and Collective Action","\u002Fen\u002Fmicroeconomics\u002F10-externalities-public-goods","en\u002Fmicroeconomics\u002F10-externalities-public-goods\u002Findex",[785],{"title":781,"path":782,"stem":783},{"title":787,"path":788,"stem":789,"children":790},"Module 11 — Matching and Market Design","\u002Fen\u002Fmicroeconomics\u002F11-market-design","en\u002Fmicroeconomics\u002F11-market-design\u002Findex",[791],{"title":787,"path":788,"stem":789},{"title":793,"path":794,"stem":795,"children":796},"Module 12 — Integrated Microeconomic Design Studio","\u002Fen\u002Fmicroeconomics\u002F12-review","en\u002Fmicroeconomics\u002F12-review\u002Findex",[797],{"title":793,"path":794,"stem":795},{"title":799,"path":800,"stem":801,"children":802},"Playground","\u002Fen\u002Fplayground","en\u002Fplayground\u002Findex",[803,804,808,812,816,820],{"title":799,"path":800,"stem":801},{"title":805,"path":806,"stem":807},"Typst Plugin Playground","\u002Fen\u002Fplayground\u002F01-typstex","en\u002Fplayground\u002F01-typstex",{"title":809,"path":810,"stem":811},"Citation Plugin Test","\u002Fen\u002Fplayground\u002F02-citation","en\u002Fplayground\u002F02-citation",{"title":813,"path":814,"stem":815},"Pyodide Playground","\u002Fen\u002Fplayground\u002F03-pyodide","en\u002Fplayground\u002F03-pyodide",{"title":817,"path":818,"stem":819},"WebR Playground","\u002Fen\u002Fplayground\u002F04-webr","en\u002Fplayground\u002F04-webr",{"title":821,"path":822,"stem":823},"Chart.js 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文献综述的构建与写作","\u002Fzh\u002Facademic-writing\u002F04-writing-the-literature-review","zh\u002Facademic-writing\u002F04-writing-the-literature-review",{"title":1056,"path":1057,"stem":1058},"8. 文献综述清单与模板","\u002Fzh\u002Facademic-writing\u002F08-literature-review-checklist-and-template","zh\u002Facademic-writing\u002F08-literature-review-checklist-and-template",{"title":1060,"path":1061,"stem":1062},"10. 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策略全流程","\u002Fzh\u002Fquant\u002F09-case-study","zh\u002Fquant\u002F09-case-study",{"title":1779,"path":1780,"stem":1781},"量化投资文献与软件图谱","\u002Fzh\u002Fquant\u002F10-reading-software-map","zh\u002Fquant\u002F10-reading-software-map",{"path":1555},{"id":1784,"title":727,"body":1785,"description":6819,"extension":6820,"features":6821,"hero":6821,"layout":6821,"locale":6821,"meta":6822,"navigation":6821,"path":728,"published":6823,"seo":6824,"stem":729,"__hash__":6825},"docs\u002Fen\u002Fmicroeconomics\u002F01-fundations\u002Findex.md",{"type":1786,"value":1787,"toc":6795},"minimark",[1788,1792,1797,1801,1805,1808,1827,1831,1839,1917,1920,2104,2111,2457,2467,2471,2482,2768,2771,2892,2895,2900,2903,3139,3142,3743,3765,3875,3878,4445,4449,4494,4497,4501,4504,4836,4839,5041,5044,5615,5624,5628,5696,5699,5703,5706,5729,5732,5737,5741,5757,5985,5989,6048,6064,6071,6075,6086,6090,6105,6342,6349,6685,6699,6703,6729,6733,6765,6769,6786],[1789,1790,727],"h1",{"id":1791},"module-1-choice-duality-and-revealed-preference",[1793,1794,1796],"h2",{"id":1795},"core-question","Core question",[1798,1799,1800],"p",{},"What can we infer about preferences or technology from observed choices, and what does an optimisation solution still leave unidentified?",[1793,1802,1804],{"id":1803},"learning-outcomes","Learning outcomes",[1798,1806,1807],{},"You will be able to:",[1809,1810,1811,1815,1818,1821,1824],"ul",{},[1812,1813,1814],"li",{},"state the assumptions needed for utility representation and well-behaved demand;",[1812,1816,1817],{},"derive Marshallian and Hicksian demand and use Roy's identity and Shephard's lemma;",[1812,1819,1820],{},"recognise corner solutions and verify demand properties;",[1812,1822,1823],{},"connect consumer and producer duality;",[1812,1825,1826],{},"test a small dataset for revealed-preference violations.",[1793,1828,1830],{"id":1829},"_1-preferences-are-rankings-not-measured-happiness","1. Preferences are rankings, not measured happiness",[1798,1832,1833,1834,1838],{},"A preference relation ",[1835,1836,1837],"code",{},"≽"," is usually assumed:",[1840,1841,1842,1858],"table",{},[1843,1844,1845],"thead",{},[1846,1847,1848,1852,1855],"tr",{},[1849,1850,1851],"th",{},"Assumption",[1849,1853,1854],{},"Meaning",[1849,1856,1857],{},"What it permits",[1859,1860,1861,1873,1884,1895,1906],"tbody",{},[1846,1862,1863,1867,1870],{},[1864,1865,1866],"td",{},"completeness",[1864,1868,1869],{},"any two bundles can be ranked",[1864,1871,1872],{},"a choice is defined",[1846,1874,1875,1878,1881],{},[1864,1876,1877],{},"transitivity",[1864,1879,1880],{},"rankings contain no cycle",[1864,1882,1883],{},"internally consistent ordering",[1846,1885,1886,1889,1892],{},[1864,1887,1888],{},"continuity",[1864,1890,1891],{},"nearby bundles do not cause arbitrary jumps",[1864,1893,1894],{},"continuous utility representation",[1846,1896,1897,1900,1903],{},[1864,1898,1899],{},"local nonsatiation",[1864,1901,1902],{},"every neighbourhood contains something better",[1864,1904,1905],{},"budget exhaustion at positive prices",[1846,1907,1908,1911,1914],{},[1864,1909,1910],{},"convexity",[1864,1912,1913],{},"mixtures are weakly preferred to extremes",[1864,1915,1916],{},"convex demand problem; diversification",[1798,1918,1919],{},"A utility function represents the ordering:",[1921,1922,1925],"span",{"className":1923},[1924],"katex-display",[1921,1926,1929,1996],{"className":1927},[1928],"katex",[1921,1930,1933],{"className":1931},[1932],"katex-mathml",[1934,1935,1938],"math",{"xmlns":1936,"display":1937},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML","block",[1939,1940,1941,1991],"semantics",{},[1942,1943,1944,1948,1952,1955,1959,1962,1964,1967,1971,1973,1976,1979,1981,1983,1985,1987],"mrow",{},[1945,1946,1947],"mi",{},"x",[1949,1950,1951],"mo",{},"⪰",[1945,1953,1954],{},"y",[1956,1957,1958],"mtext",{},"  ",[1949,1960,1961],{},"⟺",[1956,1963,1958],{},[1945,1965,1966],{},"u",[1949,1968,1970],{"stretchy":1969},"false","(",[1945,1972,1947],{},[1949,1974,1975],{"stretchy":1969},")",[1949,1977,1978],{},"≥",[1945,1980,1966],{},[1949,1982,1970],{"stretchy":1969},[1945,1984,1954],{},[1949,1986,1975],{"stretchy":1969},[1945,1988,1990],{"mathvariant":1989},"normal",".",[1992,1993,1995],"annotation",{"encoding":1994},"application\u002Fx-tex","x\\succeq y\\iff u(x)\\ge u(y).",[1921,1997,2001,2027,2053,2083],{"className":1998,"ariaHidden":2000},[1999],"katex-html","true",[1921,2002,2005,2010,2015,2020,2024],{"className":2003},[2004],"base",[1921,2006],{"className":2007,"style":2009},[2008],"strut","height:0.7719em;vertical-align:-0.136em;",[1921,2011,1947],{"className":2012},[2013,2014],"mord","mathnormal",[1921,2016],{"className":2017,"style":2019},[2018],"mspace","margin-right:0.2778em;",[1921,2021,1951],{"className":2022},[2023],"mrel",[1921,2025],{"className":2026,"style":2019},[2018],[1921,2028,2030,2034,2038,2041,2044,2047,2050],{"className":2029},[2004],[1921,2031],{"className":2032,"style":2033},[2008],"height:0.7194em;vertical-align:-0.1944em;",[1921,2035,1954],{"className":2036,"style":2037},[2013,2014],"margin-right:0.0359em;",[1921,2039],{"className":2040,"style":2019},[2018],[1921,2042],{"className":2043,"style":2019},[2018],[1921,2045,1961],{"className":2046},[2023],[1921,2048],{"className":2049,"style":2019},[2018],[1921,2051],{"className":2052,"style":2019},[2018],[1921,2054,2056,2060,2063,2067,2070,2074,2077,2080],{"className":2055},[2004],[1921,2057],{"className":2058,"style":2059},[2008],"height:1em;vertical-align:-0.25em;",[1921,2061,1966],{"className":2062},[2013,2014],[1921,2064,1970],{"className":2065},[2066],"mopen",[1921,2068,1947],{"className":2069},[2013,2014],[1921,2071,1975],{"className":2072},[2073],"mclose",[1921,2075],{"className":2076,"style":2019},[2018],[1921,2078,1978],{"className":2079},[2023],[1921,2081],{"className":2082,"style":2019},[2018],[1921,2084,2086,2089,2092,2095,2098,2101],{"className":2085},[2004],[1921,2087],{"className":2088,"style":2059},[2008],[1921,2090,1966],{"className":2091},[2013,2014],[1921,2093,1970],{"className":2094},[2066],[1921,2096,1954],{"className":2097,"style":2037},[2013,2014],[1921,2099,1975],{"className":2100},[2073],[1921,2102,1990],{"className":2103},[2013],[1798,2105,2106,2107,2110],{},"Any strictly increasing transformation ",[1835,2108,2109],{},"f(u)"," represents the same ordinal preferences. Marginal utility levels therefore depend on representation; the marginal rate of substitution does not:",[1921,2112,2114],{"className":2113},[1924],[1921,2115,2117,2189],{"className":2116},[1928],[1921,2118,2120],{"className":2119},[1932],[1934,2121,2122],{"xmlns":1936,"display":1937},[1939,2123,2124,2186],{},[1942,2125,2126,2129,2132,2142,2145,2184],{},[1945,2127,2128],{},"M",[1945,2130,2131],{},"R",[2133,2134,2135,2138],"msub",{},[1945,2136,2137],{},"S",[2139,2140,2141],"mn",{},"12",[1949,2143,2144],{},"=",[2146,2147,2148,2167],"mfrac",{},[1942,2149,2150,2153,2155,2158,2160],{},[1945,2151,2152],{"mathvariant":1989},"∂",[1945,2154,1966],{},[1945,2156,2157],{"mathvariant":1989},"\u002F",[1945,2159,2152],{"mathvariant":1989},[2133,2161,2162,2164],{},[1945,2163,1947],{},[2139,2165,2166],{},"1",[1942,2168,2169,2171,2173,2175,2177],{},[1945,2170,2152],{"mathvariant":1989},[1945,2172,1966],{},[1945,2174,2157],{"mathvariant":1989},[1945,2176,2152],{"mathvariant":1989},[2133,2178,2179,2181],{},[1945,2180,1947],{},[2139,2182,2183],{},"2",[1945,2185,1990],{"mathvariant":1989},[1992,2187,2188],{"encoding":1994},"MRS_{12}=\\frac{\\partial u\u002F\\partial x_1}{\\partial u\u002F\\partial x_2}.",[1921,2190,2192,2276],{"className":2191,"ariaHidden":2000},[1999],[1921,2193,2195,2199,2203,2207,2267,2270,2273],{"className":2194},[2004],[1921,2196],{"className":2197,"style":2198},[2008],"height:0.8333em;vertical-align:-0.15em;",[1921,2200,2128],{"className":2201,"style":2202},[2013,2014],"margin-right:0.109em;",[1921,2204,2131],{"className":2205,"style":2206},[2013,2014],"margin-right:0.0077em;",[1921,2208,2210,2214],{"className":2209},[2013],[1921,2211,2137],{"className":2212,"style":2213},[2013,2014],"margin-right:0.0576em;",[1921,2215,2218],{"className":2216},[2217],"msupsub",[1921,2219,2223,2258],{"className":2220},[2221,2222],"vlist-t","vlist-t2",[1921,2224,2227,2253],{"className":2225},[2226],"vlist-r",[1921,2228,2232],{"className":2229,"style":2231},[2230],"vlist","height:0.3011em;",[1921,2233,2235,2240],{"style":2234},"top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;",[1921,2236],{"className":2237,"style":2239},[2238],"pstrut","height:2.7em;",[1921,2241,2247],{"className":2242},[2243,2244,2245,2246],"sizing","reset-size6","size3","mtight",[1921,2248,2250],{"className":2249},[2013,2246],[1921,2251,2141],{"className":2252},[2013,2246],[1921,2254,2257],{"className":2255},[2256],"vlist-s","​",[1921,2259,2261],{"className":2260},[2226],[1921,2262,2265],{"className":2263,"style":2264},[2230],"height:0.15em;",[1921,2266],{},[1921,2268],{"className":2269,"style":2019},[2018],[1921,2271,2144],{"className":2272},[2023],[1921,2274],{"className":2275,"style":2019},[2018],[1921,2277,2279,2283,2454],{"className":2278},[2004],[1921,2280],{"className":2281,"style":2282},[2008],"height:2.363em;vertical-align:-0.936em;",[1921,2284,2286,2290,2451],{"className":2285},[2013],[1921,2287],{"className":2288},[2066,2289],"nulldelimiter",[1921,2291,2293],{"className":2292},[2146],[1921,2294,2296,2442],{"className":2295},[2221,2222],[1921,2297,2299,2439],{"className":2298},[2226],[1921,2300,2303,2367,2378],{"className":2301,"style":2302},[2230],"height:1.427em;",[1921,2304,2306,2310],{"style":2305},"top:-2.314em;",[1921,2307],{"className":2308,"style":2309},[2238],"height:3em;",[1921,2311,2313,2317,2320,2323,2326],{"className":2312},[2013],[1921,2314,2152],{"className":2315,"style":2316},[2013],"margin-right:0.0556em;",[1921,2318,1966],{"className":2319},[2013,2014],[1921,2321,2157],{"className":2322},[2013],[1921,2324,2152],{"className":2325,"style":2316},[2013],[1921,2327,2329,2332],{"className":2328},[2013],[1921,2330,1947],{"className":2331},[2013,2014],[1921,2333,2335],{"className":2334},[2217],[1921,2336,2338,2359],{"className":2337},[2221,2222],[1921,2339,2341,2356],{"className":2340},[2226],[1921,2342,2344],{"className":2343,"style":2231},[2230],[1921,2345,2347,2350],{"style":2346},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[1921,2348],{"className":2349,"style":2239},[2238],[1921,2351,2353],{"className":2352},[2243,2244,2245,2246],[1921,2354,2183],{"className":2355},[2013,2246],[1921,2357,2257],{"className":2358},[2256],[1921,2360,2362],{"className":2361},[2226],[1921,2363,2365],{"className":2364,"style":2264},[2230],[1921,2366],{},[1921,2368,2370,2373],{"style":2369},"top:-3.23em;",[1921,2371],{"className":2372,"style":2309},[2238],[1921,2374],{"className":2375,"style":2377},[2376],"frac-line","border-bottom-width:0.04em;",[1921,2379,2381,2384],{"style":2380},"top:-3.677em;",[1921,2382],{"className":2383,"style":2309},[2238],[1921,2385,2387,2390,2393,2396,2399],{"className":2386},[2013],[1921,2388,2152],{"className":2389,"style":2316},[2013],[1921,2391,1966],{"className":2392},[2013,2014],[1921,2394,2157],{"className":2395},[2013],[1921,2397,2152],{"className":2398,"style":2316},[2013],[1921,2400,2402,2405],{"className":2401},[2013],[1921,2403,1947],{"className":2404},[2013,2014],[1921,2406,2408],{"className":2407},[2217],[1921,2409,2411,2431],{"className":2410},[2221,2222],[1921,2412,2414,2428],{"className":2413},[2226],[1921,2415,2417],{"className":2416,"style":2231},[2230],[1921,2418,2419,2422],{"style":2346},[1921,2420],{"className":2421,"style":2239},[2238],[1921,2423,2425],{"className":2424},[2243,2244,2245,2246],[1921,2426,2166],{"className":2427},[2013,2246],[1921,2429,2257],{"className":2430},[2256],[1921,2432,2434],{"className":2433},[2226],[1921,2435,2437],{"className":2436,"style":2264},[2230],[1921,2438],{},[1921,2440,2257],{"className":2441},[2256],[1921,2443,2445],{"className":2444},[2226],[1921,2446,2449],{"className":2447,"style":2448},[2230],"height:0.936em;",[1921,2450],{},[1921,2452],{"className":2453},[2073,2289],[1921,2455,1990],{"className":2456},[2013],[2458,2459,2461,2462,2466],"warning",{"title":2460},"Curvature needs care","Convex preferences mean the ",[2463,2464,2465],"strong",{},"upper contour sets"," are convex; they do not require the chosen utility representation to be concave. A monotone transformation can change second derivatives without changing choice.",[1793,2468,2470],{"id":2469},"_2-utility-maximisation","2. Utility maximisation",[1798,2472,2473,2474,2477,2478,2481],{},"At prices ",[1835,2475,2476],{},"p ≫ 0"," and income ",[1835,2479,2480],{},"m",", Marshallian demand solves:",[1921,2483,2485],{"className":2484},[1924],[1921,2486,2488,2572],{"className":2487},[1928],[1921,2489,2491],{"className":2490},[1932],[1934,2492,2493],{"xmlns":1936,"display":1937},[1939,2494,2495,2569],{},[1942,2496,2497,2499,2501,2503,2506,2508,2510,2513,2516,2519,2538,2541,2543,2545,2547,2549,2552,2554,2557,2559,2562,2564,2567],{},[1945,2498,1947],{},[1949,2500,1970],{"stretchy":1969},[1945,2502,1798],{},[1949,2504,2505],{"separator":2000},",",[1945,2507,2480],{},[1949,2509,1975],{"stretchy":1969},[1949,2511,2512],{},"∈",[1945,2514,2515],{},"arg",[1949,2517,2518],{},"⁡",[2520,2521,2522,2529],"munder",{},[1942,2523,2524,2527],{},[1945,2525,2526],{},"max",[1949,2528,2518],{},[1942,2530,2531,2533,2535],{},[1945,2532,1947],{},[1949,2534,1978],{},[2139,2536,2537],{},"0",[1949,2539,2540],{"stretchy":1969},"{",[1945,2542,1966],{},[1949,2544,1970],{"stretchy":1969},[1945,2546,1947],{},[1949,2548,1975],{"stretchy":1969},[1949,2550,2551],{},":",[1945,2553,1798],{},[1949,2555,2556],{},"⋅",[1945,2558,1947],{},[1949,2560,2561],{},"≤",[1945,2563,2480],{},[1949,2565,2566],{"stretchy":1969},"}",[1945,2568,1990],{"mathvariant":1989},[1992,2570,2571],{"encoding":1994},"x(p,m)\\in\\arg\\max_{x\\ge0}\\{u(x):p\\cdot x\\le m\\}.",[1921,2573,2575,2613,2714,2735,2753],{"className":2574,"ariaHidden":2000},[1999],[1921,2576,2578,2581,2584,2587,2590,2594,2598,2601,2604,2607,2610],{"className":2577},[2004],[1921,2579],{"className":2580,"style":2059},[2008],[1921,2582,1947],{"className":2583},[2013,2014],[1921,2585,1970],{"className":2586},[2066],[1921,2588,1798],{"className":2589},[2013,2014],[1921,2591,2505],{"className":2592},[2593],"mpunct",[1921,2595],{"className":2596,"style":2597},[2018],"margin-right:0.1667em;",[1921,2599,2480],{"className":2600},[2013,2014],[1921,2602,1975],{"className":2603},[2073],[1921,2605],{"className":2606,"style":2019},[2018],[1921,2608,2512],{"className":2609},[2023],[1921,2611],{"className":2612,"style":2019},[2018],[1921,2614,2616,2620,2629,2632,2690,2693,2696,2699,2702,2705,2708,2711],{"className":2615},[2004],[1921,2617],{"className":2618,"style":2619},[2008],"height:1.5623em;vertical-align:-0.8123em;",[1921,2621,2624,2625],{"className":2622},[2623],"mop","ar",[1921,2626,2628],{"style":2627},"margin-right:0.0139em;","g",[1921,2630],{"className":2631,"style":2597},[2018],[1921,2633,2636],{"className":2634},[2623,2635],"op-limits",[1921,2637,2639,2681],{"className":2638},[2221,2222],[1921,2640,2642,2678],{"className":2641},[2226],[1921,2643,2646,2667],{"className":2644,"style":2645},[2230],"height:0.4306em;",[1921,2647,2649,2652],{"style":2648},"top:-2.3829em;margin-left:0em;",[1921,2650],{"className":2651,"style":2309},[2238],[1921,2653,2655],{"className":2654},[2243,2244,2245,2246],[1921,2656,2658,2661,2664],{"className":2657},[2013,2246],[1921,2659,1947],{"className":2660},[2013,2014,2246],[1921,2662,1978],{"className":2663},[2023,2246],[1921,2665,2537],{"className":2666},[2013,2246],[1921,2668,2670,2673],{"style":2669},"top:-3em;",[1921,2671],{"className":2672,"style":2309},[2238],[1921,2674,2675],{},[1921,2676,2526],{"className":2677},[2623],[1921,2679,2257],{"className":2680},[2256],[1921,2682,2684],{"className":2683},[2226],[1921,2685,2688],{"className":2686,"style":2687},[2230],"height:0.8123em;",[1921,2689],{},[1921,2691,2540],{"className":2692},[2066],[1921,2694,1966],{"className":2695},[2013,2014],[1921,2697,1970],{"className":2698},[2066],[1921,2700,1947],{"className":2701},[2013,2014],[1921,2703,1975],{"className":2704},[2073],[1921,2706],{"className":2707,"style":2019},[2018],[1921,2709,2551],{"className":2710},[2023],[1921,2712],{"className":2713,"style":2019},[2018],[1921,2715,2717,2721,2724,2728,2732],{"className":2716},[2004],[1921,2718],{"className":2719,"style":2720},[2008],"height:0.6389em;vertical-align:-0.1944em;",[1921,2722,1798],{"className":2723},[2013,2014],[1921,2725],{"className":2726,"style":2727},[2018],"margin-right:0.2222em;",[1921,2729,2556],{"className":2730},[2731],"mbin",[1921,2733],{"className":2734,"style":2727},[2018],[1921,2736,2738,2741,2744,2747,2750],{"className":2737},[2004],[1921,2739],{"className":2740,"style":2009},[2008],[1921,2742,1947],{"className":2743},[2013,2014],[1921,2745],{"className":2746,"style":2019},[2018],[1921,2748,2561],{"className":2749},[2023],[1921,2751],{"className":2752,"style":2019},[2018],[1921,2754,2756,2759,2762,2765],{"className":2755},[2004],[1921,2757],{"className":2758,"style":2059},[2008],[1921,2760,2480],{"className":2761},[2013,2014],[1921,2763,2566],{"className":2764},[2073],[1921,2766,1990],{"className":2767},[2013],[1798,2769,2770],{},"For a regular interior solution,",[1921,2772,2774],{"className":2773},[1924],[1921,2775,2777,2815],{"className":2776},[1928],[1921,2778,2780],{"className":2779},[1932],[1934,2781,2782],{"xmlns":1936,"display":1937},[1939,2783,2784,2812],{},[1942,2785,2786,2789,2791,2793,2801,2803,2805,2808,2810],{},[1945,2787,2788],{"mathvariant":1989},"∇",[1945,2790,1966],{},[1949,2792,1970],{"stretchy":1969},[2794,2795,2796,2798],"msup",{},[1945,2797,1947],{},[1949,2799,2800],{},"∗",[1949,2802,1975],{"stretchy":1969},[1949,2804,2144],{},[1945,2806,2807],{},"λ",[1945,2809,1798],{},[1945,2811,1990],{"mathvariant":1989},[1992,2813,2814],{"encoding":1994},"\\nabla u(x^*)=\\lambda p.",[1921,2816,2818,2876],{"className":2817,"ariaHidden":2000},[1999],[1921,2819,2821,2824,2827,2830,2833,2864,2867,2870,2873],{"className":2820},[2004],[1921,2822],{"className":2823,"style":2059},[2008],[1921,2825,2788],{"className":2826},[2013],[1921,2828,1966],{"className":2829},[2013,2014],[1921,2831,1970],{"className":2832},[2066],[1921,2834,2836,2839],{"className":2835},[2013],[1921,2837,1947],{"className":2838},[2013,2014],[1921,2840,2842],{"className":2841},[2217],[1921,2843,2845],{"className":2844},[2221],[1921,2846,2848],{"className":2847},[2226],[1921,2849,2852],{"className":2850,"style":2851},[2230],"height:0.7387em;",[1921,2853,2855,2858],{"style":2854},"top:-3.113em;margin-right:0.05em;",[1921,2856],{"className":2857,"style":2239},[2238],[1921,2859,2861],{"className":2860},[2243,2244,2245,2246],[1921,2862,2800],{"className":2863},[2731,2246],[1921,2865,1975],{"className":2866},[2073],[1921,2868],{"className":2869,"style":2019},[2018],[1921,2871,2144],{"className":2872},[2023],[1921,2874],{"className":2875,"style":2019},[2018],[1921,2877,2879,2883,2886,2889],{"className":2878},[2004],[1921,2880],{"className":2881,"style":2882},[2008],"height:0.8889em;vertical-align:-0.1944em;",[1921,2884,2807],{"className":2885},[2013,2014],[1921,2887,1798],{"className":2888},[2013,2014],[1921,2890,1990],{"className":2891},[2013],[1798,2893,2894],{},"The condition says marginal utility per pound is equalised. It is not enough on its own: verify feasibility, curvature, and whether a non-negativity constraint binds.",[2896,2897,2899],"h3",{"id":2898},"worked-example-cobbdouglas-demand","Worked example: Cobb–Douglas demand",[1798,2901,2902],{},"Let",[1921,2904,2906],{"className":2905},[1924],[1921,2907,2909,2972],{"className":2908},[1928],[1921,2910,2912],{"className":2911},[1932],[1934,2913,2914],{"xmlns":1936,"display":1937},[1939,2915,2916,2969],{},[1942,2917,2918,2920,2922,2924,2926,2928,2930,2932,2939,2952,2954,2957,2959,2962,2964,2966],{},[1945,2919,1966],{},[1949,2921,1970],{"stretchy":1969},[1945,2923,1947],{},[1949,2925,2505],{"separator":2000},[1945,2927,1954],{},[1949,2929,1975],{"stretchy":1969},[1949,2931,2144],{},[2794,2933,2934,2936],{},[1945,2935,1947],{},[1945,2937,2938],{},"α",[2794,2940,2941,2943],{},[1945,2942,1954],{},[1942,2944,2945,2947,2950],{},[2139,2946,2166],{},[1949,2948,2949],{},"−",[1945,2951,2938],{},[1949,2953,2505],{"separator":2000},[2018,2955],{"width":2956},"2em",[2139,2958,2537],{},[1949,2960,2961],{},"\u003C",[1945,2963,2938],{},[1949,2965,2961],{},[2139,2967,2968],{},"1.",[1992,2970,2971],{"encoding":1994},"u(x,y)=x^\\alpha y^{1-\\alpha},\\qquad 0\u003C\\alpha\u003C1.",[1921,2973,2975,3011,3110,3129],{"className":2974,"ariaHidden":2000},[1999],[1921,2976,2978,2981,2984,2987,2990,2993,2996,2999,3002,3005,3008],{"className":2977},[2004],[1921,2979],{"className":2980,"style":2059},[2008],[1921,2982,1966],{"className":2983},[2013,2014],[1921,2985,1970],{"className":2986},[2066],[1921,2988,1947],{"className":2989},[2013,2014],[1921,2991,2505],{"className":2992},[2593],[1921,2994],{"className":2995,"style":2597},[2018],[1921,2997,1954],{"className":2998,"style":2037},[2013,2014],[1921,3000,1975],{"className":3001},[2073],[1921,3003],{"className":3004,"style":2019},[2018],[1921,3006,2144],{"className":3007},[2023],[1921,3009],{"className":3010,"style":2019},[2018],[1921,3012,3014,3018,3049,3088,3091,3095,3098,3101,3104,3107],{"className":3013},[2004],[1921,3015],{"className":3016,"style":3017},[2008],"height:1.0585em;vertical-align:-0.1944em;",[1921,3019,3021,3024],{"className":3020},[2013],[1921,3022,1947],{"className":3023},[2013,2014],[1921,3025,3027],{"className":3026},[2217],[1921,3028,3030],{"className":3029},[2221],[1921,3031,3033],{"className":3032},[2226],[1921,3034,3037],{"className":3035,"style":3036},[2230],"height:0.7144em;",[1921,3038,3039,3042],{"style":2854},[1921,3040],{"className":3041,"style":2239},[2238],[1921,3043,3045],{"className":3044},[2243,2244,2245,2246],[1921,3046,2938],{"className":3047,"style":3048},[2013,2014,2246],"margin-right:0.0037em;",[1921,3050,3052,3055],{"className":3051},[2013],[1921,3053,1954],{"className":3054,"style":2037},[2013,2014],[1921,3056,3058],{"className":3057},[2217],[1921,3059,3061],{"className":3060},[2221],[1921,3062,3064],{"className":3063},[2226],[1921,3065,3068],{"className":3066,"style":3067},[2230],"height:0.8641em;",[1921,3069,3070,3073],{"style":2854},[1921,3071],{"className":3072,"style":2239},[2238],[1921,3074,3076],{"className":3075},[2243,2244,2245,2246],[1921,3077,3079,3082,3085],{"className":3078},[2013,2246],[1921,3080,2166],{"className":3081},[2013,2246],[1921,3083,2949],{"className":3084},[2731,2246],[1921,3086,2938],{"className":3087,"style":3048},[2013,2014,2246],[1921,3089,2505],{"className":3090},[2593],[1921,3092],{"className":3093,"style":3094},[2018],"margin-right:2em;",[1921,3096],{"className":3097,"style":2597},[2018],[1921,3099,2537],{"className":3100},[2013],[1921,3102],{"className":3103,"style":2019},[2018],[1921,3105,2961],{"className":3106},[2023],[1921,3108],{"className":3109,"style":2019},[2018],[1921,3111,3113,3117,3120,3123,3126],{"className":3112},[2004],[1921,3114],{"className":3115,"style":3116},[2008],"height:0.5782em;vertical-align:-0.0391em;",[1921,3118,2938],{"className":3119,"style":3048},[2013,2014],[1921,3121],{"className":3122,"style":2019},[2018],[1921,3124,2961],{"className":3125},[2023],[1921,3127],{"className":3128,"style":2019},[2018],[1921,3130,3132,3136],{"className":3131},[2004],[1921,3133],{"className":3134,"style":3135},[2008],"height:0.6444em;",[1921,3137,2968],{"className":3138},[2013],[1798,3140,3141],{},"The tangency condition and budget exhaustion give:",[1921,3143,3145],{"className":3144},[1924],[1921,3146,3148,3254],{"className":3147},[1928],[1921,3149,3151],{"className":3150},[1932],[1934,3152,3153],{"xmlns":1936,"display":1937},[1939,3154,3155,3251],{},[1942,3156,3157,3159,3161,3167,3169,3175,3177,3179,3181,3183,3197,3199,3201,3203,3205,3211,3213,3219,3221,3223,3225,3227,3249],{},[1945,3158,1947],{},[1949,3160,1970],{"stretchy":1969},[2133,3162,3163,3165],{},[1945,3164,1798],{},[1945,3166,1947],{},[1949,3168,2505],{"separator":2000},[2133,3170,3171,3173],{},[1945,3172,1798],{},[1945,3174,1954],{},[1949,3176,2505],{"separator":2000},[1945,3178,2480],{},[1949,3180,1975],{"stretchy":1969},[1949,3182,2144],{},[2146,3184,3185,3191],{},[1942,3186,3187,3189],{},[1945,3188,2938],{},[1945,3190,2480],{},[2133,3192,3193,3195],{},[1945,3194,1798],{},[1945,3196,1947],{},[1949,3198,2505],{"separator":2000},[2018,3200],{"width":2956},[1945,3202,1954],{},[1949,3204,1970],{"stretchy":1969},[2133,3206,3207,3209],{},[1945,3208,1798],{},[1945,3210,1947],{},[1949,3212,2505],{"separator":2000},[2133,3214,3215,3217],{},[1945,3216,1798],{},[1945,3218,1954],{},[1949,3220,2505],{"separator":2000},[1945,3222,2480],{},[1949,3224,1975],{"stretchy":1969},[1949,3226,2144],{},[2146,3228,3229,3243],{},[1942,3230,3231,3233,3235,3237,3239,3241],{},[1949,3232,1970],{"stretchy":1969},[2139,3234,2166],{},[1949,3236,2949],{},[1945,3238,2938],{},[1949,3240,1975],{"stretchy":1969},[1945,3242,2480],{},[2133,3244,3245,3247],{},[1945,3246,1798],{},[1945,3248,1954],{},[1945,3250,1990],{"mathvariant":1989},[1992,3252,3253],{"encoding":1994},"x(p_x,p_y,m)=\\frac{\\alpha m}{p_x},\\qquad\ny(p_x,p_y,m)=\\frac{(1-\\alpha)m}{p_y}.",[1921,3255,3257,3379,3612],{"className":3256,"ariaHidden":2000},[1999],[1921,3258,3260,3264,3267,3270,3311,3314,3317,3358,3361,3364,3367,3370,3373,3376],{"className":3259},[2004],[1921,3261],{"className":3262,"style":3263},[2008],"height:1.0361em;vertical-align:-0.2861em;",[1921,3265,1947],{"className":3266},[2013,2014],[1921,3268,1970],{"className":3269},[2066],[1921,3271,3273,3276],{"className":3272},[2013],[1921,3274,1798],{"className":3275},[2013,2014],[1921,3277,3279],{"className":3278},[2217],[1921,3280,3282,3303],{"className":3281},[2221,2222],[1921,3283,3285,3300],{"className":3284},[2226],[1921,3286,3289],{"className":3287,"style":3288},[2230],"height:0.1514em;",[1921,3290,3291,3294],{"style":2346},[1921,3292],{"className":3293,"style":2239},[2238],[1921,3295,3297],{"className":3296},[2243,2244,2245,2246],[1921,3298,1947],{"className":3299},[2013,2014,2246],[1921,3301,2257],{"className":3302},[2256],[1921,3304,3306],{"className":3305},[2226],[1921,3307,3309],{"className":3308,"style":2264},[2230],[1921,3310],{},[1921,3312,2505],{"className":3313},[2593],[1921,3315],{"className":3316,"style":2597},[2018],[1921,3318,3320,3323],{"className":3319},[2013],[1921,3321,1798],{"className":3322},[2013,2014],[1921,3324,3326],{"className":3325},[2217],[1921,3327,3329,3349],{"className":3328},[2221,2222],[1921,3330,3332,3346],{"className":3331},[2226],[1921,3333,3335],{"className":3334,"style":3288},[2230],[1921,3336,3337,3340],{"style":2346},[1921,3338],{"className":3339,"style":2239},[2238],[1921,3341,3343],{"className":3342},[2243,2244,2245,2246],[1921,3344,1954],{"className":3345,"style":2037},[2013,2014,2246],[1921,3347,2257],{"className":3348},[2256],[1921,3350,3352],{"className":3351},[2226],[1921,3353,3356],{"className":3354,"style":3355},[2230],"height:0.2861em;",[1921,3357],{},[1921,3359,2505],{"className":3360},[2593],[1921,3362],{"className":3363,"style":2597},[2018],[1921,3365,2480],{"className":3366},[2013,2014],[1921,3368,1975],{"className":3369},[2073],[1921,3371],{"className":3372,"style":2019},[2018],[1921,3374,2144],{"className":3375},[2023],[1921,3377],{"className":3378,"style":2019},[2018],[1921,3380,3382,3386,3490,3493,3496,3499,3502,3505,3545,3548,3551,3591,3594,3597,3600,3603,3606,3609],{"className":3381},[2004],[1921,3383],{"className":3384,"style":3385},[2008],"height:1.988em;vertical-align:-0.8804em;",[1921,3387,3389,3392,3487],{"className":3388},[2013],[1921,3390],{"className":3391},[2066,2289],[1921,3393,3395],{"className":3394},[2146],[1921,3396,3398,3478],{"className":3397},[2221,2222],[1921,3399,3401,3475],{"className":3400},[2226],[1921,3402,3405,3453,3461],{"className":3403,"style":3404},[2230],"height:1.1076em;",[1921,3406,3407,3410],{"style":2305},[1921,3408],{"className":3409,"style":2309},[2238],[1921,3411,3413],{"className":3412},[2013],[1921,3414,3416,3419],{"className":3415},[2013],[1921,3417,1798],{"className":3418},[2013,2014],[1921,3420,3422],{"className":3421},[2217],[1921,3423,3425,3445],{"className":3424},[2221,2222],[1921,3426,3428,3442],{"className":3427},[2226],[1921,3429,3431],{"className":3430,"style":3288},[2230],[1921,3432,3433,3436],{"style":2346},[1921,3434],{"className":3435,"style":2239},[2238],[1921,3437,3439],{"className":3438},[2243,2244,2245,2246],[1921,3440,1947],{"className":3441},[2013,2014,2246],[1921,3443,2257],{"className":3444},[2256],[1921,3446,3448],{"className":3447},[2226],[1921,3449,3451],{"className":3450,"style":2264},[2230],[1921,3452],{},[1921,3454,3455,3458],{"style":2369},[1921,3456],{"className":3457,"style":2309},[2238],[1921,3459],{"className":3460,"style":2377},[2376],[1921,3462,3463,3466],{"style":2380},[1921,3464],{"className":3465,"style":2309},[2238],[1921,3467,3469,3472],{"className":3468},[2013],[1921,3470,2938],{"className":3471,"style":3048},[2013,2014],[1921,3473,2480],{"className":3474},[2013,2014],[1921,3476,2257],{"className":3477},[2256],[1921,3479,3481],{"className":3480},[2226],[1921,3482,3485],{"className":3483,"style":3484},[2230],"height:0.8804em;",[1921,3486],{},[1921,3488],{"className":3489},[2073,2289],[1921,3491,2505],{"className":3492},[2593],[1921,3494],{"className":3495,"style":3094},[2018],[1921,3497],{"className":3498,"style":2597},[2018],[1921,3500,1954],{"className":3501,"style":2037},[2013,2014],[1921,3503,1970],{"className":3504},[2066],[1921,3506,3508,3511],{"className":3507},[2013],[1921,3509,1798],{"className":3510},[2013,2014],[1921,3512,3514],{"className":3513},[2217],[1921,3515,3517,3537],{"className":3516},[2221,2222],[1921,3518,3520,3534],{"className":3519},[2226],[1921,3521,3523],{"className":3522,"style":3288},[2230],[1921,3524,3525,3528],{"style":2346},[1921,3526],{"className":3527,"style":2239},[2238],[1921,3529,3531],{"className":3530},[2243,2244,2245,2246],[1921,3532,1947],{"className":3533},[2013,2014,2246],[1921,3535,2257],{"className":3536},[2256],[1921,3538,3540],{"className":3539},[2226],[1921,3541,3543],{"className":3542,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Check:",[1921,3766,3768],{"className":3767},[1924],[1921,3769,3771,3809],{"className":3770},[1928],[1921,3772,3774],{"className":3773},[1932],[1934,3775,3776],{"xmlns":1936,"display":1937},[1939,3777,3778,3806],{},[1942,3779,3780,3782,3784,3787,3789,3792,3795,3797,3799,3801,3803],{},[2139,3781,2183],{},[1949,3783,1970],{"stretchy":1969},[2139,3785,3786],{},"20",[1949,3788,1975],{"stretchy":1969},[1949,3790,3791],{},"+",[2139,3793,3794],{},"5",[1949,3796,1970],{"stretchy":1969},[2139,3798,2141],{},[1949,3800,1975],{"stretchy":1969},[1949,3802,2144],{},[2139,3804,3805],{},"100.",[1992,3807,3808],{"encoding":1994},"2(20)+5(12)=100.",[1921,3810,3812,3839,3866],{"className":3811,"ariaHidden":2000},[1999],[1921,3813,3815,3818,3821,3824,3827,3830,3833,3836],{"className":3814},[2004],[1921,3816],{"className":3817,"style":2059},[2008],[1921,3819,2183],{"className":3820},[2013],[1921,3822,1970],{"className":3823},[2066],[1921,3825,3786],{"className":3826},[2013],[1921,3828,1975],{"className":3829},[2073],[1921,3831],{"className":3832,"style":2727},[2018],[1921,3834,3791],{"className":3835},[2731],[1921,3837],{"className":3838,"style":2727},[2018],[1921,3840,3842,3845,3848,3851,3854,3857,3860,3863],{"className":3841},[2004],[1921,3843],{"className":3844,"style":2059},[2008],[1921,3846,3794],{"className":3847},[2013],[1921,3849,1970],{"className":3850},[2066],[1921,3852,2141],{"className":3853},[2013],[1921,3855,1975],{"className":3856},[2073],[1921,3858],{"className":3859,"style":2019},[2018],[1921,3861,2144],{"className":3862},[2023],[1921,3864],{"className":3865,"style":2019},[2018],[1921,3867,3869,3872],{"className":3868},[2004],[1921,3870],{"className":3871,"style":3135},[2008],[1921,3873,3805],{"className":3874},[2013],[1798,3876,3877],{},"The indirect utility function is:",[1921,3879,3881],{"className":3880},[1924],[1921,3882,3884,3987],{"className":3883},[1928],[1921,3885,3887],{"className":3886},[1932],[1934,3888,3889],{"xmlns":1936,"display":1937},[1939,3890,3891,3984],{},[1942,3892,3893,3896,3898,3904,3906,3912,3914,3916,3918,3920,3944,3982],{},[1945,3894,3895],{},"v",[1949,3897,1970],{"stretchy":1969},[2133,3899,3900,3902],{},[1945,3901,1798],{},[1945,3903,1947],{},[1949,3905,2505],{"separator":2000},[2133,3907,3908,3910],{},[1945,3909,1798],{},[1945,3911,1954],{},[1949,3913,2505],{"separator":2000},[1945,3915,2480],{},[1949,3917,1975],{"stretchy":1969},[1949,3919,2144],{},[2794,3921,3922,3942],{},[1942,3923,3924,3926,3940],{},[1949,3925,1970],{"fence":2000},[2146,3927,3928,3934],{},[1942,3929,3930,3932],{},[1945,3931,2938],{},[1945,3933,2480],{},[2133,3935,3936,3938],{},[1945,3937,1798],{},[1945,3939,1947],{},[1949,3941,1975],{"fence":2000},[1945,3943,2938],{},[2794,3945,3946,3974],{},[1942,3947,3948,3950,3972],{},[1949,3949,1970],{"fence":2000},[2146,3951,3952,3966],{},[1942,3953,3954,3956,3958,3960,3962,3964],{},[1949,3955,1970],{"stretchy":1969},[2139,3957,2166],{},[1949,3959,2949],{},[1945,3961,2938],{},[1949,3963,1975],{"stretchy":1969},[1945,3965,2480],{},[2133,3967,3968,3970],{},[1945,3969,1798],{},[1945,3971,1954],{},[1949,3973,1975],{"fence":2000},[1942,3975,3976,3978,3980],{},[2139,3977,2166],{},[1949,3979,2949],{},[1945,3981,2938],{},[1945,3983,1990],{"mathvariant":1989},[1992,3985,3986],{"encoding":1994},"v(p_x,p_y,m)=\n\\left(\\frac{\\alpha m}{p_x}\\right)^\\alpha\n\\left(\\frac{(1-\\alpha)m}{p_y}\\right)^{1-\\alpha}.",[1921,3988,3990,4109],{"className":3989,"ariaHidden":2000},[1999],[1921,3991,3993,3996,3999,4002,4042,4045,4048,4088,4091,4094,4097,4100,4103,4106],{"className":3992},[2004],[1921,3994],{"className":3995,"style":3263},[2008],[1921,3997,3895],{"className":3998,"style":2037},[2013,2014],[1921,4000,1970],{"className":4001},[2066],[1921,4003,4005,4008],{"className":4004},[2013],[1921,4006,1798],{"className":4007},[2013,2014],[1921,4009,4011],{"className":4010},[2217],[1921,4012,4014,4034],{"className":4013},[2221,2222],[1921,4015,4017,4031],{"className":4016},[2226],[1921,4018,4020],{"className":4019,"style":3288},[2230],[1921,4021,4022,4025],{"style":2346},[1921,4023],{"className":4024,"style":2239},[2238],[1921,4026,4028],{"className":4027},[2243,2244,2245,2246],[1921,4029,1947],{"className":4030},[2013,2014,2246],[1921,4032,2257],{"className":4033},[2256],[1921,4035,4037],{"className":4036},[2226],[1921,4038,4040],{"className":4039,"style":2264},[2230],[1921,4041],{},[1921,4043,2505],{"className":4044},[2593],[1921,4046],{"className":4047,"style":2597},[2018],[1921,4049,4051,4054],{"className":4050},[2013],[1921,4052,1798],{"className":4053},[2013,2014],[1921,4055,4057],{"className":4056},[2217],[1921,4058,4060,4080],{"className":4059},[2221,2222],[1921,4061,4063,4077],{"className":4062},[2226],[1921,4064,4066],{"className":4065,"style":3288},[2230],[1921,4067,4068,4071],{"style":2346},[1921,4069],{"className":4070,"style":2239},[2238],[1921,4072,4074],{"className":4073},[2243,2244,2245,2246],[1921,4075,1954],{"className":4076,"style":2037},[2013,2014,2246],[1921,4078,2257],{"className":4079},[2256],[1921,4081,4083],{"className":4082},[2226],[1921,4084,4086],{"className":4085,"style":3355},[2230],[1921,4087],{},[1921,4089,2505],{"className":4090},[2593],[1921,4092],{"className":4093,"style":2597},[2018],[1921,4095,2480],{"className":4096},[2013,2014],[1921,4098,1975],{"className":4099},[2073],[1921,4101],{"className":4102,"style":2019},[2018],[1921,4104,2144],{"className":4105},[2023],[1921,4107],{"className":4108,"style":2019},[2018],[1921,4110,4112,4116,4265,4268,4439,4442],{"className":4111},[2004],[1921,4113],{"className":4114,"style":4115},[2008],"height:2.6261em;vertical-align:-0.9721em;",[1921,4117,4120,4240],{"className":4118},[4119],"minner",[1921,4121,4123,4132,4234],{"className":4122},[4119],[1921,4124,4128],{"className":4125,"style":4127},[2066,4126],"delimcenter","top:0em;",[1921,4129,1970],{"className":4130},[4131,2245],"delimsizing",[1921,4133,4135,4138,4231],{"className":4134},[2013],[1921,4136],{"className":4137},[2066,2289],[1921,4139,4141],{"className":4140},[2146],[1921,4142,4144,4223],{"className":4143},[2221,2222],[1921,4145,4147,4220],{"className":4146},[2226],[1921,4148,4150,4198,4206],{"className":4149,"style":3404},[2230],[1921,4151,4152,4155],{"style":2305},[1921,4153],{"className":4154,"style":2309},[2238],[1921,4156,4158],{"className":4157},[2013],[1921,4159,4161,4164],{"className":4160},[2013],[1921,4162,1798],{"className":4163},[2013,2014],[1921,4165,4167],{"className":4166},[2217],[1921,4168,4170,4190],{"className":4169},[2221,2222],[1921,4171,4173,4187],{"className":4172},[2226],[1921,4174,4176],{"className":4175,"style":3288},[2230],[1921,4177,4178,4181],{"style":2346},[1921,4179],{"className":4180,"style":2239},[2238],[1921,4182,4184],{"className":4183},[2243,2244,2245,2246],[1921,4185,1947],{"className":4186},[2013,2014,2246],[1921,4188,2257],{"className":4189},[2256],[1921,4191,4193],{"className":4192},[2226],[1921,4194,4196],{"className":4195,"style":2264},[2230],[1921,4197],{},[1921,4199,4200,4203],{"style":2369},[1921,4201],{"className":4202,"style":2309},[2238],[1921,4204],{"className":4205,"style":2377},[2376],[1921,4207,4208,4211],{"style":2380},[1921,4209],{"className":4210,"style":2309},[2238],[1921,4212,4214,4217],{"className":4213},[2013],[1921,4215,2938],{"className":4216,"style":3048},[2013,2014],[1921,4218,2480],{"className":4219},[2013,2014],[1921,4221,2257],{"className":4222},[2256],[1921,4224,4226],{"className":4225},[2226],[1921,4227,4229],{"className":4228,"style":3484},[2230],[1921,4230],{},[1921,4232],{"className":4233},[2073,2289],[1921,4235,4237],{"className":4236,"style":4127},[2073,4126],[1921,4238,1975],{"className":4239},[4131,2245],[1921,4241,4243],{"className":4242},[2217],[1921,4244,4246],{"className":4245},[2221],[1921,4247,4249],{"className":4248},[2226],[1921,4250,4253],{"className":4251,"style":4252},[2230],"height:1.5043em;",[1921,4254,4256,4259],{"style":4255},"top:-3.9029em;margin-right:0.05em;",[1921,4257],{"className":4258,"style":2239},[2238],[1921,4260,4262],{"className":4261},[2243,2244,2245,2246],[1921,4263,2938],{"className":4264,"style":3048},[2013,2014,2246],[1921,4266],{"className":4267,"style":2597},[2018],[1921,4269,4271,4406],{"className":4270},[4119],[1921,4272,4274,4280,4400],{"className":4273},[4119],[1921,4275,4277],{"className":4276,"style":4127},[2066,4126],[1921,4278,1970],{"className":4279},[4131,2245],[1921,4281,4283,4286,4397],{"className":4282},[2013],[1921,4284],{"className":4285},[2066,2289],[1921,4287,4289],{"className":4288},[2146],[1921,4290,4292,4389],{"className":4291},[2221,2222],[1921,4293,4295,4386],{"className":4294},[2226],[1921,4296,4298,4346,4354],{"className":4297,"style":2302},[2230],[1921,4299,4300,4303],{"style":2305},[1921,4301],{"className":4302,"style":2309},[2238],[1921,4304,4306],{"className":4305},[2013],[1921,4307,4309,4312],{"className":4308},[2013],[1921,4310,1798],{"className":4311},[2013,2014],[1921,4313,4315],{"className":4314},[2217],[1921,4316,4318,4338],{"className":4317},[2221,2222],[1921,4319,4321,4335],{"className":4320},[2226],[1921,4322,4324],{"className":4323,"style":3288},[2230],[1921,4325,4326,4329],{"style":2346},[1921,4327],{"className":4328,"style":2239},[2238],[1921,4330,4332],{"className":4331},[2243,2244,2245,2246],[1921,4333,1954],{"className":4334,"style":2037},[2013,2014,2246],[1921,4336,2257],{"className":4337},[2256],[1921,4339,4341],{"className":4340},[2226],[1921,4342,4344],{"className":4343,"style":3355},[2230],[1921,4345],{},[1921,4347,4348,4351],{"style":2369},[1921,4349],{"className":4350,"style":2309},[2238],[1921,4352],{"className":4353,"style":2377},[2376],[1921,4355,4356,4359],{"style":2380},[1921,4357],{"className":4358,"style":2309},[2238],[1921,4360,4362,4365,4368,4371,4374,4377,4380,4383],{"className":4361},[2013],[1921,4363,1970],{"className":4364},[2066],[1921,4366,2166],{"className":4367},[2013],[1921,4369],{"className":4370,"style":2727},[2018],[1921,4372,2949],{"className":4373},[2731],[1921,4375],{"className":4376,"style":2727},[2018],[1921,4378,2938],{"className":4379,"style":3048},[2013,2014],[1921,4381,1975],{"className":4382},[2073],[1921,4384,2480],{"className":4385},[2013,2014],[1921,4387,2257],{"className":4388},[2256],[1921,4390,4392],{"className":4391},[2226],[1921,4393,4395],{"className":4394,"style":3734},[2230],[1921,4396],{},[1921,4398],{"className":4399},[2073,2289],[1921,4401,4403],{"className":4402,"style":4127},[2073,4126],[1921,4404,1975],{"className":4405},[4131,2245],[1921,4407,4409],{"className":4408},[2217],[1921,4410,4412],{"className":4411},[2221],[1921,4413,4415],{"className":4414},[2226],[1921,4416,4419],{"className":4417,"style":4418},[2230],"height:1.654em;",[1921,4420,4421,4424],{"style":4255},[1921,4422],{"className":4423,"style":2239},[2238],[1921,4425,4427],{"className":4426},[2243,2244,2245,2246],[1921,4428,4430,4433,4436],{"className":4429},[2013,2246],[1921,4431,2166],{"className":4432},[2013,2246],[1921,4434,2949],{"className":4435},[2731,2246],[1921,4437,2938],{"className":4438,"style":3048},[2013,2014,2246],[1921,4440],{"className":4441,"style":2597},[2018],[1921,4443,1990],{"className":4444},[2013],[2896,4446,4448],{"id":4447},"when-tangency-fails","When tangency fails",[1809,4450,4451,4466,4475],{},[1812,4452,4453,4454,4457,4458,4461,4462,4465],{},"For ",[1835,4455,4456],{},"u(x,y)=ax+by",", compare ",[1835,4459,4460],{},"a\u002Fp_x"," with ",[1835,4463,4464],{},"b\u002Fp_y","; usually buy one good only.",[1812,4467,4453,4468,4471,4472,1990],{},[1835,4469,4470],{},"u(x,y)=min{x,y}",", choose the kink ",[1835,4473,4474],{},"x=y",[1812,4476,4453,4477,4480,4481,4483,4484,4487,4488,4490,4491,4493],{},[1835,4478,4479],{},"u(x,y)=v(x)+y",", an interior ",[1835,4482,1947],{}," satisfies ",[1835,4485,4486],{},"v'(x)=p_x\u002Fp_y","; ",[2463,4489,1947],{},", not the numeraire ",[1835,4492,1954],{},", has zero income effect once income is high enough.",[1798,4495,4496],{},"KKT conditions handle these cases without pretending all optima are interior.",[1793,4498,4500],{"id":4499},"_3-expenditure-minimisation-and-duality","3. Expenditure minimisation and duality",[1798,4502,4503],{},"Hicksian demand holds utility fixed:",[1921,4505,4507],{"className":4506},[1924],[1921,4508,4510,4594],{"className":4509},[1928],[1921,4511,4513],{"className":4512},[1932],[1934,4514,4515],{"xmlns":1936,"display":1937},[1939,4516,4517,4591],{},[1942,4518,4519,4522,4524,4526,4528,4536,4538,4540,4542,4544,4561,4563,4565,4567,4569,4571,4573,4575,4577,4579,4581,4587,4589],{},[1945,4520,4521],{},"h",[1949,4523,1970],{"stretchy":1969},[1945,4525,1798],{},[1949,4527,2505],{"separator":2000},[4529,4530,4531,4533],"mover",{"accent":2000},[1945,4532,1966],{},[1949,4534,4535],{},"ˉ",[1949,4537,1975],{"stretchy":1969},[1949,4539,2512],{},[1945,4541,2515],{},[1949,4543,2518],{},[2520,4545,4546,4553],{},[1942,4547,4548,4551],{},[1945,4549,4550],{},"min",[1949,4552,2518],{},[1942,4554,4555,4557,4559],{},[1945,4556,1947],{},[1949,4558,1978],{},[2139,4560,2537],{},[1949,4562,2540],{"stretchy":1969},[1945,4564,1798],{},[1949,4566,2556],{},[1945,4568,1947],{},[1949,4570,2551],{},[1945,4572,1966],{},[1949,4574,1970],{"stretchy":1969},[1945,4576,1947],{},[1949,4578,1975],{"stretchy":1969},[1949,4580,1978],{},[4529,4582,4583,4585],{"accent":2000},[1945,4584,1966],{},[1949,4586,4535],{},[1949,4588,2566],{"stretchy":1969},[1945,4590,1990],{"mathvariant":1989},[1992,4592,4593],{"encoding":1994},"h(p,\\bar u)\\in\\arg\\min_{x\\ge0}\\{p\\cdot x:u(x)\\ge\\bar u\\}.",[1921,4595,4597,4665,4748,4766,4793],{"className":4596,"ariaHidden":2000},[1999],[1921,4598,4600,4603,4606,4609,4612,4615,4618,4653,4656,4659,4662],{"className":4599},[2004],[1921,4601],{"className":4602,"style":2059},[2008],[1921,4604,4521],{"className":4605},[2013,2014],[1921,4607,1970],{"className":4608},[2066],[1921,4610,1798],{"className":4611},[2013,2014],[1921,4613,2505],{"className":4614},[2593],[1921,4616],{"className":4617,"style":2597},[2018],[1921,4619,4622],{"className":4620},[2013,4621],"accent",[1921,4623,4625],{"className":4624},[2221],[1921,4626,4628],{"className":4627},[2226],[1921,4629,4632,4640],{"className":4630,"style":4631},[2230],"height:0.5678em;",[1921,4633,4634,4637],{"style":2669},[1921,4635],{"className":4636,"style":2309},[2238],[1921,4638,1966],{"className":4639},[2013,2014],[1921,4641,4642,4645],{"style":2669},[1921,4643],{"className":4644,"style":2309},[2238],[1921,4646,4650],{"className":4647,"style":4649},[4648],"accent-body","left:-0.2222em;",[1921,4651,4535],{"className":4652},[2013],[1921,4654,1975],{"className":4655},[2073],[1921,4657],{"className":4658,"style":2019},[2018],[1921,4660,2512],{"className":4661},[2023],[1921,4663],{"className":4664,"style":2019},[2018],[1921,4666,4668,4671,4676,4679,4733,4736,4739,4742,4745],{"className":4667},[2004],[1921,4669],{"className":4670,"style":2619},[2008],[1921,4672,2624,4674],{"className":4673},[2623],[1921,4675,2628],{"style":2627},[1921,4677],{"className":4678,"style":2597},[2018],[1921,4680,4682],{"className":4681},[2623,2635],[1921,4683,4685,4725],{"className":4684},[2221,2222],[1921,4686,4688,4722],{"className":4687},[2226],[1921,4689,4692,4712],{"className":4690,"style":4691},[2230],"height:0.6679em;",[1921,4693,4694,4697],{"style":2648},[1921,4695],{"className":4696,"style":2309},[2238],[1921,4698,4700],{"className":4699},[2243,2244,2245,2246],[1921,4701,4703,4706,4709],{"className":4702},[2013,2246],[1921,4704,1947],{"className":4705},[2013,2014,2246],[1921,4707,1978],{"className":4708},[2023,2246],[1921,4710,2537],{"className":4711},[2013,2246],[1921,4713,4714,4717],{"style":2669},[1921,4715],{"className":4716,"style":2309},[2238],[1921,4718,4719],{},[1921,4720,4550],{"className":4721},[2623],[1921,4723,2257],{"className":4724},[2256],[1921,4726,4728],{"className":4727},[2226],[1921,4729,4731],{"className":4730,"style":2687},[2230],[1921,4732],{},[1921,4734,2540],{"className":4735},[2066],[1921,4737,1798],{"className":4738},[2013,2014],[1921,4740],{"className":4741,"style":2727},[2018],[1921,4743,2556],{"className":4744},[2731],[1921,4746],{"className":4747,"style":2727},[2018],[1921,4749,4751,4754,4757,4760,4763],{"className":4750},[2004],[1921,4752],{"className":4753,"style":2645},[2008],[1921,4755,1947],{"className":4756},[2013,2014],[1921,4758],{"className":4759,"style":2019},[2018],[1921,4761,2551],{"className":4762},[2023],[1921,4764],{"className":4765,"style":2019},[2018],[1921,4767,4769,4772,4775,4778,4781,4784,4787,4790],{"className":4768},[2004],[1921,4770],{"className":4771,"style":2059},[2008],[1921,4773,1966],{"className":4774},[2013,2014],[1921,4776,1970],{"className":4777},[2066],[1921,4779,1947],{"className":4780},[2013,2014],[1921,4782,1975],{"className":4783},[2073],[1921,4785],{"className":4786,"style":2019},[2018],[1921,4788,1978],{"className":4789},[2023],[1921,4791],{"className":4792,"style":2019},[2018],[1921,4794,4796,4799,4830,4833],{"className":4795},[2004],[1921,4797],{"className":4798,"style":2059},[2008],[1921,4800,4802],{"className":4801},[2013,4621],[1921,4803,4805],{"className":4804},[2221],[1921,4806,4808],{"className":4807},[2226],[1921,4809,4811,4819],{"className":4810,"style":4631},[2230],[1921,4812,4813,4816],{"style":2669},[1921,4814],{"className":4815,"style":2309},[2238],[1921,4817,1966],{"className":4818},[2013,2014],[1921,4820,4821,4824],{"style":2669},[1921,4822],{"className":4823,"style":2309},[2238],[1921,4825,4827],{"className":4826,"style":4649},[4648],[1921,4828,4535],{"className":4829},[2013],[1921,4831,2566],{"className":4832},[2073],[1921,4834,1990],{"className":4835},[2013],[1798,4837,4838],{},"The expenditure function is the minimum value:",[1921,4840,4842],{"className":4841},[1924],[1921,4843,4845,4898],{"className":4844},[1928],[1921,4846,4848],{"className":4847},[1932],[1934,4849,4850],{"xmlns":1936,"display":1937},[1939,4851,4852,4895],{},[1942,4853,4854,4857,4859,4861,4863,4869,4871,4873,4875,4877,4879,4881,4883,4885,4891,4893],{},[1945,4855,4856],{},"e",[1949,4858,1970],{"stretchy":1969},[1945,4860,1798],{},[1949,4862,2505],{"separator":2000},[4529,4864,4865,4867],{"accent":2000},[1945,4866,1966],{},[1949,4868,4535],{},[1949,4870,1975],{"stretchy":1969},[1949,4872,2144],{},[1945,4874,1798],{},[1949,4876,2556],{},[1945,4878,4521],{},[1949,4880,1970],{"stretchy":1969},[1945,4882,1798],{},[1949,4884,2505],{"separator":2000},[4529,4886,4887,4889],{"accent":2000},[1945,4888,1966],{},[1949,4890,4535],{},[1949,4892,1975],{"stretchy":1969},[1945,4894,1990],{"mathvariant":1989},[1992,4896,4897],{"encoding":1994},"e(p,\\bar u)=p\\cdot h(p,\\bar u).",[1921,4899,4901,4965,4983],{"className":4900,"ariaHidden":2000},[1999],[1921,4902,4904,4907,4910,4913,4916,4919,4922,4953,4956,4959,4962],{"className":4903},[2004],[1921,4905],{"className":4906,"style":2059},[2008],[1921,4908,4856],{"className":4909},[2013,2014],[1921,4911,1970],{"className":4912},[2066],[1921,4914,1798],{"className":4915},[2013,2014],[1921,4917,2505],{"className":4918},[2593],[1921,4920],{"className":4921,"style":2597},[2018],[1921,4923,4925],{"className":4924},[2013,4621],[1921,4926,4928],{"className":4927},[2221],[1921,4929,4931],{"className":4930},[2226],[1921,4932,4934,4942],{"className":4933,"style":4631},[2230],[1921,4935,4936,4939],{"style":2669},[1921,4937],{"className":4938,"style":2309},[2238],[1921,4940,1966],{"className":4941},[2013,2014],[1921,4943,4944,4947],{"style":2669},[1921,4945],{"className":4946,"style":2309},[2238],[1921,4948,4950],{"className":4949,"style":4649},[4648],[1921,4951,4535],{"className":4952},[2013],[1921,4954,1975],{"className":4955},[2073],[1921,4957],{"className":4958,"style":2019},[2018],[1921,4960,2144],{"className":4961},[2023],[1921,4963],{"className":4964,"style":2019},[2018],[1921,4966,4968,4971,4974,4977,4980],{"className":4967},[2004],[1921,4969],{"className":4970,"style":2720},[2008],[1921,4972,1798],{"className":4973},[2013,2014],[1921,4975],{"className":4976,"style":2727},[2018],[1921,4978,2556],{"className":4979},[2731],[1921,4981],{"className":4982,"style":2727},[2018],[1921,4984,4986,4989,4992,4995,4998,5001,5004,5035,5038],{"className":4985},[2004],[1921,4987],{"className":4988,"style":2059},[2008],[1921,4990,4521],{"className":4991},[2013,2014],[1921,4993,1970],{"className":4994},[2066],[1921,4996,1798],{"className":4997},[2013,2014],[1921,4999,2505],{"className":5000},[2593],[1921,5002],{"className":5003,"style":2597},[2018],[1921,5005,5007],{"className":5006},[2013,4621],[1921,5008,5010],{"className":5009},[2221],[1921,5011,5013],{"className":5012},[2226],[1921,5014,5016,5024],{"className":5015,"style":4631},[2230],[1921,5017,5018,5021],{"style":2669},[1921,5019],{"className":5020,"style":2309},[2238],[1921,5022,1966],{"className":5023},[2013,2014],[1921,5025,5026,5029],{"style":2669},[1921,5027],{"className":5028,"style":2309},[2238],[1921,5030,5032],{"className":5031,"style":4649},[4648],[1921,5033,4535],{"className":5034},[2013],[1921,5036,1975],{"className":5037},[2073],[1921,5039,1990],{"className":5040},[2013],[1798,5042,5043],{},"For Cobb–Douglas utility,",[1921,5045,5047],{"className":5046},[1924],[1921,5048,5050,5158],{"className":5049},[1928],[1921,5051,5053],{"className":5052},[1932],[1934,5054,5055],{"xmlns":1936,"display":1937},[1939,5056,5057,5155],{},[1942,5058,5059,5061,5063,5069,5071,5077,5079,5085,5087,5089,5153],{},[1945,5060,4856],{},[1949,5062,1970],{"stretchy":1969},[2133,5064,5065,5067],{},[1945,5066,1798],{},[1945,5068,1947],{},[1949,5070,2505],{"separator":2000},[2133,5072,5073,5075],{},[1945,5074,1798],{},[1945,5076,1954],{},[1949,5078,2505],{"separator":2000},[4529,5080,5081,5083],{"accent":2000},[1945,5082,1966],{},[1949,5084,4535],{},[1949,5086,1975],{"stretchy":1969},[1949,5088,2144],{},[2146,5090,5091,5125],{},[1942,5092,5093,5099,5102,5111],{},[4529,5094,5095,5097],{"accent":2000},[1945,5096,1966],{},[1949,5098,4535],{},[1956,5100,5101],{}," ",[5103,5104,5105,5107,5109],"msubsup",{},[1945,5106,1798],{},[1945,5108,1947],{},[1945,5110,2938],{},[5103,5112,5113,5115,5117],{},[1945,5114,1798],{},[1945,5116,1954],{},[1942,5118,5119,5121,5123],{},[2139,5120,2166],{},[1949,5122,2949],{},[1945,5124,2938],{},[1942,5126,5127,5133,5135,5137,5139,5141],{},[2794,5128,5129,5131],{},[1945,5130,2938],{},[1945,5132,2938],{},[1949,5134,1970],{"stretchy":1969},[2139,5136,2166],{},[1949,5138,2949],{},[1945,5140,2938],{},[2794,5142,5143,5145],{},[1949,5144,1975],{"stretchy":1969},[1942,5146,5147,5149,5151],{},[2139,5148,2166],{},[1949,5150,2949],{},[1945,5152,2938],{},[1949,5154,2505],{"separator":2000},[1992,5156,5157],{"encoding":1994},"e(p_x,p_y,\\bar u)=\n\\frac{\\bar u\\,p_x^\\alpha p_y^{1-\\alpha}}\n{\\alpha^\\alpha(1-\\alpha)^{1-\\alpha}},",[1921,5159,5161,5308],{"className":5160,"ariaHidden":2000},[1999],[1921,5162,5164,5167,5170,5173,5213,5216,5219,5259,5262,5265,5296,5299,5302,5305],{"className":5163},[2004],[1921,5165],{"className":5166,"style":3263},[2008],[1921,5168,4856],{"className":5169},[2013,2014],[1921,5171,1970],{"className":5172},[2066],[1921,5174,5176,5179],{"className":5175},[2013],[1921,5177,1798],{"className":5178},[2013,2014],[1921,5180,5182],{"className":5181},[2217],[1921,5183,5185,5205],{"className":5184},[2221,2222],[1921,5186,5188,5202],{"className":5187},[2226],[1921,5189,5191],{"className":5190,"style":3288},[2230],[1921,5192,5193,5196],{"style":2346},[1921,5194],{"className":5195,"style":2239},[2238],[1921,5197,5199],{"className":5198},[2243,2244,2245,2246],[1921,5200,1947],{"className":5201},[2013,2014,2246],[1921,5203,2257],{"className":5204},[2256],[1921,5206,5208],{"className":5207},[2226],[1921,5209,5211],{"className":5210,"style":2264},[2230],[1921,5212],{},[1921,5214,2505],{"className":5215},[2593],[1921,5217],{"className":5218,"style":2597},[2018],[1921,5220,5222,5225],{"className":5221},[2013],[1921,5223,1798],{"className":5224},[2013,2014],[1921,5226,5228],{"className":5227},[2217],[1921,5229,5231,5251],{"className":5230},[2221,2222],[1921,5232,5234,5248],{"className":5233},[2226],[1921,5235,5237],{"className":5236,"style":3288},[2230],[1921,5238,5239,5242],{"style":2346},[1921,5240],{"className":5241,"style":2239},[2238],[1921,5243,5245],{"className":5244},[2243,2244,2245,2246],[1921,5246,1954],{"className":5247,"style":2037},[2013,2014,2246],[1921,5249,2257],{"className":5250},[2256],[1921,5252,5254],{"className":5253},[2226],[1921,5255,5257],{"className":5256,"style":3355},[2230],[1921,5258],{},[1921,5260,2505],{"className":5261},[2593],[1921,5263],{"className":5264,"style":2597},[2018],[1921,5266,5268],{"className":5267},[2013,4621],[1921,5269,5271],{"className":5270},[2221],[1921,5272,5274],{"className":5273},[2226],[1921,5275,5277,5285],{"className":5276,"style":4631},[2230],[1921,5278,5279,5282],{"style":2669},[1921,5280],{"className":5281,"style":2309},[2238],[1921,5283,1966],{"className":5284},[2013,2014],[1921,5286,5287,5290],{"style":2669},[1921,5288],{"className":5289,"style":2309},[2238],[1921,5291,5293],{"className":5292,"style":4649},[4648],[1921,5294,4535],{"className":5295},[2013],[1921,5297,1975],{"className":5298},[2073],[1921,5300],{"className":5301,"style":2019},[2018],[1921,5303,2144],{"className":5304},[2023],[1921,5306],{"className":5307,"style":2019},[2018],[1921,5309,5311,5315,5612],{"className":5310},[2004],[1921,5312],{"className":5313,"style":5314},[2008],"height:2.5232em;vertical-align:-0.936em;",[1921,5316,5318,5321,5609],{"className":5317},[2013],[1921,5319],{"className":5320},[2066,2289],[1921,5322,5324],{"className":5323},[2146],[1921,5325,5327,5601],{"className":5326},[2221,2222],[1921,5328,5330,5598],{"className":5329},[2226],[1921,5331,5334,5430,5438],{"className":5332,"style":5333},[2230],"height:1.5872em;",[1921,5335,5336,5339],{"style":2305},[1921,5337],{"className":5338,"style":2309},[2238],[1921,5340,5342,5373,5376,5379,5382,5385,5388,5391],{"className":5341},[2013],[1921,5343,5345,5348],{"className":5344},[2013],[1921,5346,2938],{"className":5347,"style":3048},[2013,2014],[1921,5349,5351],{"className":5350},[2217],[1921,5352,5354],{"className":5353},[2221],[1921,5355,5357],{"className":5356},[2226],[1921,5358,5361],{"className":5359,"style":5360},[2230],"height:0.5904em;",[1921,5362,5364,5367],{"style":5363},"top:-2.989em;margin-right:0.05em;",[1921,5365],{"className":5366,"style":2239},[2238],[1921,5368,5370],{"className":5369},[2243,2244,2245,2246],[1921,5371,2938],{"className":5372,"style":3048},[2013,2014,2246],[1921,5374,1970],{"className":5375},[2066],[1921,5377,2166],{"className":5378},[2013],[1921,5380],{"className":5381,"style":2727},[2018],[1921,5383,2949],{"className":5384},[2731],[1921,5386],{"className":5387,"style":2727},[2018],[1921,5389,2938],{"className":5390,"style":3048},[2013,2014],[1921,5392,5394,5397],{"className":5393},[2073],[1921,5395,1975],{"className":5396},[2073],[1921,5398,5400],{"className":5399},[2217],[1921,5401,5403],{"className":5402},[2221],[1921,5404,5406],{"className":5405},[2226],[1921,5407,5410],{"className":5408,"style":5409},[2230],"height:0.7401em;",[1921,5411,5412,5415],{"style":5363},[1921,5413],{"className":5414,"style":2239},[2238],[1921,5416,5418],{"className":5417},[2243,2244,2245,2246],[1921,5419,5421,5424,5427],{"className":5420},[2013,2246],[1921,5422,2166],{"className":5423},[2013,2246],[1921,5425,2949],{"className":5426},[2731,2246],[1921,5428,2938],{"className":5429,"style":3048},[2013,2014,2246],[1921,5431,5432,5435],{"style":2369},[1921,5433],{"className":5434,"style":2309},[2238],[1921,5436],{"className":5437,"style":2377},[2376],[1921,5439,5441,5444],{"style":5440},"top:-3.7731em;",[1921,5442],{"className":5443,"style":2309},[2238],[1921,5445,5447,5478,5481,5536],{"className":5446},[2013],[1921,5448,5450],{"className":5449},[2013,4621],[1921,5451,5453],{"className":5452},[2221],[1921,5454,5456],{"className":5455},[2226],[1921,5457,5459,5467],{"className":5458,"style":4631},[2230],[1921,5460,5461,5464],{"style":2669},[1921,5462],{"className":5463,"style":2309},[2238],[1921,5465,1966],{"className":5466},[2013,2014],[1921,5468,5469,5472],{"style":2669},[1921,5470],{"className":5471,"style":2309},[2238],[1921,5473,5475],{"className":5474,"style":4649},[4648],[1921,5476,4535],{"className":5477},[2013],[1921,5479],{"className":5480,"style":2597},[2018],[1921,5482,5484,5487],{"className":5483},[2013],[1921,5485,1798],{"className":5486},[2013,2014],[1921,5488,5490],{"className":5489},[2217],[1921,5491,5493,5527],{"className":5492},[2221,2222],[1921,5494,5496,5524],{"className":5495},[2226],[1921,5497,5500,5512],{"className":5498,"style":5499},[2230],"height:0.6644em;",[1921,5501,5503,5506],{"style":5502},"top:-2.453em;margin-left:0em;margin-right:0.05em;",[1921,5504],{"className":5505,"style":2239},[2238],[1921,5507,5509],{"className":5508},[2243,2244,2245,2246],[1921,5510,1947],{"className":5511},[2013,2014,2246],[1921,5513,5515,5518],{"style":5514},"top:-3.063em;margin-right:0.05em;",[1921,5516],{"className":5517,"style":2239},[2238],[1921,5519,5521],{"className":5520},[2243,2244,2245,2246],[1921,5522,2938],{"className":5523,"style":3048},[2013,2014,2246],[1921,5525,2257],{"className":5526},[2256],[1921,5528,5530],{"className":5529},[2226],[1921,5531,5534],{"className":5532,"style":5533},[2230],"height:0.247em;",[1921,5535],{},[1921,5537,5539,5542],{"className":5538},[2013],[1921,5540,1798],{"className":5541},[2013,2014],[1921,5543,5545],{"className":5544},[2217],[1921,5546,5548,5589],{"className":5547},[2221,2222],[1921,5549,5551,5586],{"className":5550},[2226],[1921,5552,5555,5566],{"className":5553,"style":5554},[2230],"height:0.8141em;",[1921,5556,5557,5560],{"style":5502},[1921,5558],{"className":5559,"style":2239},[2238],[1921,5561,5563],{"className":5562},[2243,2244,2245,2246],[1921,5564,1954],{"className":5565,"style":2037},[2013,2014,2246],[1921,5567,5568,5571],{"style":5514},[1921,5569],{"className":5570,"style":2239},[2238],[1921,5572,5574],{"className":5573},[2243,2244,2245,2246],[1921,5575,5577,5580,5583],{"className":5576},[2013,2246],[1921,5578,2166],{"className":5579},[2013,2246],[1921,5581,2949],{"className":5582},[2731,2246],[1921,5584,2938],{"className":5585,"style":3048},[2013,2014,2246],[1921,5587,2257],{"className":5588},[2256],[1921,5590,5592],{"className":5591},[2226],[1921,5593,5596],{"className":5594,"style":5595},[2230],"height:0.3831em;",[1921,5597],{},[1921,5599,2257],{"className":5600},[2256],[1921,5602,5604],{"className":5603},[2226],[1921,5605,5607],{"className":5606,"style":2448},[2230],[1921,5608],{},[1921,5610],{"className":5611},[2073,2289],[1921,5613,2505],{"className":5614},[2593],[1798,5616,5617,5618,5620,5621,1990],{},"and expenditure shares remain ",[1835,5619,2938],{}," and ",[1835,5622,5623],{},"1−α",[2896,5625,5627],{"id":5626},"the-four-links","The four links",[1840,5629,5630,5643],{},[1843,5631,5632],{},[1846,5633,5634,5637,5640],{},[1849,5635,5636],{},"Result",[1849,5638,5639],{},"Formula",[1849,5641,5642],{},"Interpretation",[1859,5644,5645,5658,5671,5684],{},[1846,5646,5647,5650,5655],{},[1864,5648,5649],{},"Shephard's lemma",[1864,5651,5652],{},[1835,5653,5654],{},"∂e\u002F∂p_i = h_i",[1864,5656,5657],{},"price derivative of required spending is compensated demand",[1846,5659,5660,5663,5668],{},[1864,5661,5662],{},"Roy's identity",[1864,5664,5665],{},[1835,5666,5667],{},"x_i = −(∂v\u002F∂p_i)\u002F(∂v\u002F∂m)",[1864,5669,5670],{},"recover ordinary demand from indirect utility",[1846,5672,5673,5676,5681],{},[1864,5674,5675],{},"inverse relation",[1864,5677,5678],{},[1835,5679,5680],{},"e(p,v(p,m))=m",[1864,5682,5683],{},"spending to recover achieved utility equals income",[1846,5685,5686,5688,5693],{},[1864,5687,5675],{},[1864,5689,5690],{},[1835,5691,5692],{},"v(p,e(p,u))=u",[1864,5694,5695],{},"minimum spending attains target utility",[1798,5697,5698],{},"These are envelope results: the derivative of the optimised value ignores the first-order effect of re-optimising the choice.",[1793,5700,5702],{"id":5701},"_4-properties-that-catch-algebra-mistakes","4. Properties that catch algebra mistakes",[1798,5704,5705],{},"Marshallian demand satisfies, under local nonsatiation:",[1809,5707,5708,5718,5726],{},[1812,5709,5710,5713,5714,5717],{},[2463,5711,5712],{},"Walras' law:"," ",[1835,5715,5716],{},"p·x(p,m)=m",";",[1812,5719,5720,5713,5723,5717],{},[2463,5721,5722],{},"degree-zero homogeneity:",[1835,5724,5725],{},"x(tp,tm)=x(p,m)",[1812,5727,5728],{},"adding-up and aggregation restrictions on demand derivatives.",[1798,5730,5731],{},"The expenditure function is increasing in utility, homogeneous of degree one and concave in prices. Hicksian demand obeys the compensated law of demand; its substitution matrix is symmetric and negative semidefinite under regularity.",[5733,5734,5736],"tip",{"title":5735},"Verification routine","After deriving a demand, check units, budget exhaustion, homogeneity, signs, limiting cases, and corners. A formula that fails one check is not an economic solution even if the differentiation was correct.",[1793,5738,5740],{"id":5739},"_5-revealed-preference-learn-from-budgets-not-utility-surveys","5. Revealed preference: learn from budgets, not utility surveys",[1798,5742,5743,5744,5747,5748,5751,5752,5754,5755,2551],{},"If bundle ",[1835,5745,5746],{},"xᵗ"," is chosen when ",[1835,5749,5750],{},"xˢ"," was affordable, then ",[1835,5753,5746],{}," is directly revealed at least as good as ",[1835,5756,5750],{},[1921,5758,5760],{"className":5759},[1924],[1921,5761,5763,5809],{"className":5762},[1928],[1921,5764,5766],{"className":5765},[1932],[1934,5767,5768],{"xmlns":1936,"display":1937},[1939,5769,5770,5806],{},[1942,5771,5772,5779,5781,5787,5789,5795,5797,5804],{},[2794,5773,5774,5776],{},[1945,5775,1798],{},[1945,5777,5778],{},"t",[1949,5780,2556],{},[2794,5782,5783,5785],{},[1945,5784,1947],{},[1945,5786,5778],{},[1949,5788,1978],{},[2794,5790,5791,5793],{},[1945,5792,1798],{},[1945,5794,5778],{},[1949,5796,2556],{},[2794,5798,5799,5801],{},[1945,5800,1947],{},[1945,5802,5803],{},"s",[1945,5805,1990],{"mathvariant":1989},[1992,5807,5808],{"encoding":1994},"p^t\\cdot x^t\\ge p^t\\cdot x^s.",[1921,5810,5812,5858,5903,5947],{"className":5811,"ariaHidden":2000},[1999],[1921,5813,5815,5819,5849,5852,5855],{"className":5814},[2004],[1921,5816],{"className":5817,"style":5818},[2008],"height:1.038em;vertical-align:-0.1944em;",[1921,5820,5822,5825],{"className":5821},[2013],[1921,5823,1798],{"className":5824},[2013,2014],[1921,5826,5828],{"className":5827},[2217],[1921,5829,5831],{"className":5830},[2221],[1921,5832,5834],{"className":5833},[2226],[1921,5835,5838],{"className":5836,"style":5837},[2230],"height:0.8436em;",[1921,5839,5840,5843],{"style":2854},[1921,5841],{"className":5842,"style":2239},[2238],[1921,5844,5846],{"className":5845},[2243,2244,2245,2246],[1921,5847,5778],{"className":5848},[2013,2014,2246],[1921,5850],{"className":5851,"style":2727},[2018],[1921,5853,2556],{"className":5854},[2731],[1921,5856],{"className":5857,"style":2727},[2018],[1921,5859,5861,5865,5894,5897,5900],{"className":5860},[2004],[1921,5862],{"className":5863,"style":5864},[2008],"height:0.9795em;vertical-align:-0.136em;",[1921,5866,5868,5871],{"className":5867},[2013],[1921,5869,1947],{"className":5870},[2013,2014],[1921,5872,5874],{"className":5873},[2217],[1921,5875,5877],{"className":5876},[2221],[1921,5878,5880],{"className":5879},[2226],[1921,5881,5883],{"className":5882,"style":5837},[2230],[1921,5884,5885,5888],{"style":2854},[1921,5886],{"className":5887,"style":2239},[2238],[1921,5889,5891],{"className":5890},[2243,2244,2245,2246],[1921,5892,5778],{"className":5893},[2013,2014,2246],[1921,5895],{"className":5896,"style":2019},[2018],[1921,5898,1978],{"className":5899},[2023],[1921,5901],{"className":5902,"style":2019},[2018],[1921,5904,5906,5909,5938,5941,5944],{"className":5905},[2004],[1921,5907],{"className":5908,"style":5818},[2008],[1921,5910,5912,5915],{"className":5911},[2013],[1921,5913,1798],{"className":5914},[2013,2014],[1921,5916,5918],{"className":5917},[2217],[1921,5919,5921],{"className":5920},[2221],[1921,5922,5924],{"className":5923},[2226],[1921,5925,5927],{"className":5926,"style":5837},[2230],[1921,5928,5929,5932],{"style":2854},[1921,5930],{"className":5931,"style":2239},[2238],[1921,5933,5935],{"className":5934},[2243,2244,2245,2246],[1921,5936,5778],{"className":5937},[2013,2014,2246],[1921,5939],{"className":5940,"style":2727},[2018],[1921,5942,2556],{"className":5943},[2731],[1921,5945],{"className":5946,"style":2727},[2018],[1921,5948,5950,5953,5982],{"className":5949},[2004],[1921,5951],{"className":5952,"style":3036},[2008],[1921,5954,5956,5959],{"className":5955},[2013],[1921,5957,1947],{"className":5958},[2013,2014],[1921,5960,5962],{"className":5961},[2217],[1921,5963,5965],{"className":5964},[2221],[1921,5966,5968],{"className":5967},[2226],[1921,5969,5971],{"className":5970,"style":3036},[2230],[1921,5972,5973,5976],{"style":2854},[1921,5974],{"className":5975,"style":2239},[2238],[1921,5977,5979],{"className":5978},[2243,2244,2245,2246],[1921,5980,5803],{"className":5981},[2013,2014,2246],[1921,5983,1990],{"className":5984},[2013],[2896,5986,5988],{"id":5987},"a-two-observation-violation","A two-observation violation",[1840,5990,5991,6008],{},[1843,5992,5993],{},[1846,5994,5995,5998,6001,6004],{},[1849,5996,5997],{},"Observation",[1849,5999,6000],{},"Prices",[1849,6002,6003],{},"Chosen bundle",[1849,6005,6007],{"align":6006},"right","Cost of other bundle",[1859,6009,6010,6031],{},[1846,6011,6012,6014,6019,6025],{},[1864,6013,2166],{},[1864,6015,6016],{},[1835,6017,6018],{},"(2,1)",[1864,6020,6021,6024],{},[1835,6022,6023],{},"A=(2,1)",", cost 5",[1864,6026,6027,6030],{"align":6006},[1835,6028,6029],{},"B=(1,2)"," costs 4",[1846,6032,6033,6035,6040,6044],{},[1864,6034,2183],{},[1864,6036,6037],{},[1835,6038,6039],{},"(1,2)",[1864,6041,6042,6024],{},[1835,6043,6029],{},[1864,6045,6046,6030],{"align":6006},[1835,6047,6023],{},[1798,6049,6050,6051,6054,6055,6058,6059,6054,6061,6063],{},"At observation 1, ",[1835,6052,6053],{},"A"," is chosen although ",[1835,6056,6057],{},"B"," is cheaper; at observation 2, ",[1835,6060,6057],{},[1835,6062,6053],{}," is cheaper. Both strict comparisons cannot come from one locally nonsatiated, stable preference ordering. This violates WARP.",[1798,6065,6066,6067,6070],{},"For many observations, GARP rules out an indirect revealed-preference cycle with at least one strict step. ",[2463,6068,6069],{},"Afriat's theorem"," says that a finite dataset satisfies GARP if and only if it can be rationalised by a continuous, monotone, concave utility function.",[2896,6072,6074],{"id":6073},"recent-extension","Recent extension",[1798,6076,6077,6078,6085],{},"Chambers and Echenique show how finite choice data can support bounds on counterfactual welfare and whether an allocation could be Pareto efficient for some preferences consistent with the data (",[6079,6080,6084],"a",{"href":6081,"rel":6082},"https:\u002F\u002Fdoi.org\u002F10.3982\u002FTE5845",[6083],"nofollow","2025","). The key boundary is incompleteness: observed budgets rarely rank every counterfactual bundle.",[1793,6087,6089],{"id":6088},"_6-producer-choice-uses-the-same-dual-logic","6. Producer choice uses the same dual logic",[1798,6091,6092,6093,6096,6097,6100,6101,6104],{},"For technology ",[1835,6094,6095],{},"f(z)",", output price ",[1835,6098,6099],{},"q",", and input prices ",[1835,6102,6103],{},"w",", profit maximisation is:",[1921,6106,6108],{"className":6107},[1924],[1921,6109,6111,6180],{"className":6110},[1928],[1921,6112,6114],{"className":6113},[1932],[1934,6115,6116],{"xmlns":1936,"display":1937},[1939,6117,6118,6177],{},[1942,6119,6120,6123,6125,6127,6129,6131,6133,6135,6152,6154,6156,6159,6161,6163,6165,6167,6169,6171,6173,6175],{},[1945,6121,6122],{},"π",[1949,6124,1970],{"stretchy":1969},[1945,6126,6099],{},[1949,6128,2505],{"separator":2000},[1945,6130,6103],{},[1949,6132,1975],{"stretchy":1969},[1949,6134,2144],{},[2520,6136,6137,6143],{},[1942,6138,6139,6141],{},[1945,6140,2526],{},[1949,6142,2518],{},[1942,6144,6145,6148,6150],{},[1945,6146,6147],{},"z",[1949,6149,1978],{},[2139,6151,2537],{},[1949,6153,2540],{"stretchy":1969},[1945,6155,6099],{},[1945,6157,6158],{},"f",[1949,6160,1970],{"stretchy":1969},[1945,6162,6147],{},[1949,6164,1975],{"stretchy":1969},[1949,6166,2949],{},[1945,6168,6103],{},[1949,6170,2556],{},[1945,6172,6147],{},[1949,6174,2566],{"stretchy":1969},[1945,6176,1990],{"mathvariant":1989},[1992,6178,6179],{"encoding":1994},"\\pi(q,w)=\\max_{z\\ge0}\\{qf(z)-w\\cdot z\\}.",[1921,6181,6183,6220,6308,6327],{"className":6182,"ariaHidden":2000},[1999],[1921,6184,6186,6189,6192,6195,6198,6201,6204,6208,6211,6214,6217],{"className":6185},[2004],[1921,6187],{"className":6188,"style":2059},[2008],[1921,6190,6122],{"className":6191,"style":2037},[2013,2014],[1921,6193,1970],{"className":6194},[2066],[1921,6196,6099],{"className":6197,"style":2037},[2013,2014],[1921,6199,2505],{"className":6200},[2593],[1921,6202],{"className":6203,"style":2597},[2018],[1921,6205,6103],{"className":6206,"style":6207},[2013,2014],"margin-right:0.0269em;",[1921,6209,1975],{"className":6210},[2073],[1921,6212],{"className":6213,"style":2019},[2018],[1921,6215,2144],{"className":6216},[2023],[1921,6218],{"className":6219,"style":2019},[2018],[1921,6221,6223,6226,6280,6283,6286,6290,6293,6296,6299,6302,6305],{"className":6222},[2004],[1921,6224],{"className":6225,"style":2619},[2008],[1921,6227,6229],{"className":6228},[2623,2635],[1921,6230,6232,6272],{"className":6231},[2221,2222],[1921,6233,6235,6269],{"className":6234},[2226],[1921,6236,6238,6259],{"className":6237,"style":2645},[2230],[1921,6239,6240,6243],{"style":2648},[1921,6241],{"className":6242,"style":2309},[2238],[1921,6244,6246],{"className":6245},[2243,2244,2245,2246],[1921,6247,6249,6253,6256],{"className":6248},[2013,2246],[1921,6250,6147],{"className":6251,"style":6252},[2013,2014,2246],"margin-right:0.044em;",[1921,6254,1978],{"className":6255},[2023,2246],[1921,6257,2537],{"className":6258},[2013,2246],[1921,6260,6261,6264],{"style":2669},[1921,6262],{"className":6263,"style":2309},[2238],[1921,6265,6266],{},[1921,6267,2526],{"className":6268},[2623],[1921,6270,2257],{"className":6271},[2256],[1921,6273,6275],{"className":6274},[2226],[1921,6276,6278],{"className":6277,"style":2687},[2230],[1921,6279],{},[1921,6281,2540],{"className":6282},[2066],[1921,6284,6099],{"className":6285,"style":2037},[2013,2014],[1921,6287,6158],{"className":6288,"style":6289},[2013,2014],"margin-right:0.1076em;",[1921,6291,1970],{"className":6292},[2066],[1921,6294,6147],{"className":6295,"style":6252},[2013,2014],[1921,6297,1975],{"className":6298},[2073],[1921,6300],{"className":6301,"style":2727},[2018],[1921,6303,2949],{"className":6304},[2731],[1921,6306],{"className":6307,"style":2727},[2018],[1921,6309,6311,6315,6318,6321,6324],{"className":6310},[2004],[1921,6312],{"className":6313,"style":6314},[2008],"height:0.4445em;",[1921,6316,6103],{"className":6317,"style":6207},[2013,2014],[1921,6319],{"className":6320,"style":2727},[2018],[1921,6322,2556],{"className":6323},[2731],[1921,6325],{"className":6326,"style":2727},[2018],[1921,6328,6330,6333,6336,6339],{"className":6329},[2004],[1921,6331],{"className":6332,"style":2059},[2008],[1921,6334,6147],{"className":6335,"style":6252},[2013,2014],[1921,6337,2566],{"className":6338},[2073],[1921,6340,1990],{"className":6341},[2013],[1798,6343,6344,6345,6348],{},"Cost minimisation for target output ",[1835,6346,6347],{},"\\bar y"," is:",[1921,6350,6352],{"className":6351},[1924],[1921,6353,6355,6432],{"className":6354},[1928],[1921,6356,6358],{"className":6357},[1932],[1934,6359,6360],{"xmlns":1936,"display":1937},[1939,6361,6362,6429],{},[1942,6363,6364,6367,6369,6371,6373,6379,6381,6383,6399,6401,6403,6405,6407,6409,6411,6413,6415,6417,6419,6425,6427],{},[1945,6365,6366],{},"c",[1949,6368,1970],{"stretchy":1969},[1945,6370,6103],{},[1949,6372,2505],{"separator":2000},[4529,6374,6375,6377],{"accent":2000},[1945,6376,1954],{},[1949,6378,4535],{},[1949,6380,1975],{"stretchy":1969},[1949,6382,2144],{},[2520,6384,6385,6391],{},[1942,6386,6387,6389],{},[1945,6388,4550],{},[1949,6390,2518],{},[1942,6392,6393,6395,6397],{},[1945,6394,6147],{},[1949,6396,1978],{},[2139,6398,2537],{},[1949,6400,2540],{"stretchy":1969},[1945,6402,6103],{},[1949,6404,2556],{},[1945,6406,6147],{},[1949,6408,2551],{},[1945,6410,6158],{},[1949,6412,1970],{"stretchy":1969},[1945,6414,6147],{},[1949,6416,1975],{"stretchy":1969},[1949,6418,1978],{},[4529,6420,6421,6423],{"accent":2000},[1945,6422,1954],{},[1949,6424,4535],{},[1949,6426,2566],{"stretchy":1969},[1945,6428,1990],{"mathvariant":1989},[1992,6430,6431],{"encoding":1994},"c(w,\\bar y)=\\min_{z\\ge0}\\{w\\cdot z:f(z)\\ge\\bar y\\}.",[1921,6433,6435,6512,6586,6604,6631],{"className":6434,"ariaHidden":2000},[1999],[1921,6436,6438,6441,6444,6447,6450,6453,6456,6500,6503,6506,6509],{"className":6437},[2004],[1921,6439],{"className":6440,"style":2059},[2008],[1921,6442,6366],{"className":6443},[2013,2014],[1921,6445,1970],{"className":6446},[2066],[1921,6448,6103],{"className":6449,"style":6207},[2013,2014],[1921,6451,2505],{"className":6452},[2593],[1921,6454],{"className":6455,"style":2597},[2018],[1921,6457,6459],{"className":6458},[2013,4621],[1921,6460,6462,6491],{"className":6461},[2221,2222],[1921,6463,6465,6488],{"className":6464},[2226],[1921,6466,6468,6476],{"className":6467,"style":4631},[2230],[1921,6469,6470,6473],{"style":2669},[1921,6471],{"className":6472,"style":2309},[2238],[1921,6474,1954],{"className":6475,"style":2037},[2013,2014],[1921,6477,6478,6481],{"style":2669},[1921,6479],{"className":6480,"style":2309},[2238],[1921,6482,6485],{"className":6483,"style":6484},[4648],"left:-0.1944em;",[1921,6486,4535],{"className":6487},[2013],[1921,6489,2257],{"className":6490},[2256],[1921,6492,6494],{"className":6493},[2226],[1921,6495,6498],{"className":6496,"style":6497},[2230],"height:0.1944em;",[1921,6499],{},[1921,6501,1975],{"className":6502},[2073],[1921,6504],{"className":6505,"style":2019},[2018],[1921,6507,2144],{"className":6508},[2023],[1921,6510],{"className":6511,"style":2019},[2018],[1921,6513,6515,6518,6571,6574,6577,6580,6583],{"className":6514},[2004],[1921,6516],{"className":6517,"style":2619},[2008],[1921,6519,6521],{"className":6520},[2623,2635],[1921,6522,6524,6563],{"className":6523},[2221,2222],[1921,6525,6527,6560],{"className":6526},[2226],[1921,6528,6530,6550],{"className":6529,"style":4691},[2230],[1921,6531,6532,6535],{"style":2648},[1921,6533],{"className":6534,"style":2309},[2238],[1921,6536,6538],{"className":6537},[2243,2244,2245,2246],[1921,6539,6541,6544,6547],{"className":6540},[2013,2246],[1921,6542,6147],{"className":6543,"style":6252},[2013,2014,2246],[1921,6545,1978],{"className":6546},[2023,2246],[1921,6548,2537],{"className":6549},[2013,2246],[1921,6551,6552,6555],{"style":2669},[1921,6553],{"className":6554,"style":2309},[2238],[1921,6556,6557],{},[1921,6558,4550],{"className":6559},[2623],[1921,6561,2257],{"className":6562},[2256],[1921,6564,6566],{"className":6565},[2226],[1921,6567,6569],{"className":6568,"style":2687},[2230],[1921,6570],{},[1921,6572,2540],{"className":6573},[2066],[1921,6575,6103],{"className":6576,"style":6207},[2013,2014],[1921,6578],{"className":6579,"style":2727},[2018],[1921,6581,2556],{"className":6582},[2731],[1921,6584],{"className":6585,"style":2727},[2018],[1921,6587,6589,6592,6595,6598,6601],{"className":6588},[2004],[1921,6590],{"className":6591,"style":2645},[2008],[1921,6593,6147],{"className":6594,"style":6252},[2013,2014],[1921,6596],{"className":6597,"style":2019},[2018],[1921,6599,2551],{"className":6600},[2023],[1921,6602],{"className":6603,"style":2019},[2018],[1921,6605,6607,6610,6613,6616,6619,6622,6625,6628],{"className":6606},[2004],[1921,6608],{"className":6609,"style":2059},[2008],[1921,6611,6158],{"className":6612,"style":6289},[2013,2014],[1921,6614,1970],{"className":6615},[2066],[1921,6617,6147],{"className":6618,"style":6252},[2013,2014],[1921,6620,1975],{"className":6621},[2073],[1921,6623],{"className":6624,"style":2019},[2018],[1921,6626,1978],{"className":6627},[2023],[1921,6629],{"className":6630,"style":2019},[2018],[1921,6632,6634,6637,6679,6682],{"className":6633},[2004],[1921,6635],{"className":6636,"style":2059},[2008],[1921,6638,6640],{"className":6639},[2013,4621],[1921,6641,6643,6671],{"className":6642},[2221,2222],[1921,6644,6646,6668],{"className":6645},[2226],[1921,6647,6649,6657],{"className":6648,"style":4631},[2230],[1921,6650,6651,6654],{"style":2669},[1921,6652],{"className":6653,"style":2309},[2238],[1921,6655,1954],{"className":6656,"style":2037},[2013,2014],[1921,6658,6659,6662],{"style":2669},[1921,6660],{"className":6661,"style":2309},[2238],[1921,6663,6665],{"className":6664,"style":6484},[4648],[1921,6666,4535],{"className":6667},[2013],[1921,6669,2257],{"className":6670},[2256],[1921,6672,6674],{"className":6673},[2226],[1921,6675,6677],{"className":6676,"style":6497},[2230],[1921,6678],{},[1921,6680,2566],{"className":6681},[2073],[1921,6683,1990],{"className":6684},[2013],[1809,6686,6687,6690,6696],{},[1812,6688,6689],{},"Hotelling's lemma recovers net supply from derivatives of profit.",[1812,6691,6692,6693,1990],{},"Shephard's lemma recovers conditional input demand: ",[1835,6694,6695],{},"∂c\u002F∂w_i=z_i^c",[1812,6697,6698],{},"Profit is convex and homogeneous of degree one in the price vector; cost is concave and homogeneous of degree one in input prices.",[2896,6700,6702],{"id":6701},"worked-cost-problem","Worked cost problem",[1798,6704,6705,6706,3749,6709,6712,6713,6716,6717,6720,6721,6724,6725,6728],{},"If ",[1835,6707,6708],{},"f(L,K)=√(LK)",[1835,6710,6711],{},"w=r=1",", and required output is 10, minimise ",[1835,6714,6715],{},"L+K"," subject to ",[1835,6718,6719],{},"LK≥100",". The cost-minimising bundle is ",[1835,6722,6723],{},"L=K=10",", so ",[1835,6726,6727],{},"c=20",". If labour becomes relatively expensive, conditional demand substitutes toward capital; the output target remains fixed.",[1793,6730,6732],{"id":6731},"practice","Practice",[6734,6735,6736,6746,6749,6756,6759],"ol",{},[1812,6737,6738,6739,6742,6743,1990],{},"Derive Marshallian demand and ",[1835,6740,6741],{},"v(p,m)"," for ",[1835,6744,6745],{},"u=x^{1\u002F3}y^{2\u002F3}",[1812,6747,6748],{},"Derive its expenditure function and verify Shephard's lemma.",[1812,6750,6751,6752,6755],{},"Solve ",[1835,6753,6754],{},"u=2x+y"," at three price ratios and identify all corners\u002Fties.",[1812,6757,6758],{},"Construct three observations that pass WARP pairwise but fail GARP through a longer cycle.",[1812,6760,4453,6761,6764],{},[1835,6762,6763],{},"f(L,K)=L^{1\u002F3}K^{1\u002F3}",", derive the conditional input ratio.",[1793,6766,6768],{"id":6767},"quick-check","Quick check",[1809,6770,6771,6774,6777,6780,6783],{},[1812,6772,6773],{},"Utility represents rankings; its numerical scale is usually arbitrary.",[1812,6775,6776],{},"Tangency is a candidate, not a universal solution rule.",[1812,6778,6779],{},"Marshallian demand holds income fixed; Hicksian demand holds utility fixed.",[1812,6781,6782],{},"Dual value functions recover behaviour through envelope theorems.",[1812,6784,6785],{},"Revealed preference disciplines—but does not fully identify—unobserved welfare.",[1798,6787,6788,5713,6791,1990],{},[2463,6789,6790],{},"Next:",[6079,6792,6794],{"href":6793},"..\u002F02-comparative-statics\u002F","decompose responses and measure welfare changes",{"title":10,"searchDepth":6796,"depth":6796,"links":6797},2,[6798,6799,6800,6801,6806,6809,6810,6814,6817,6818],{"id":1795,"depth":6796,"text":1796},{"id":1803,"depth":6796,"text":1804},{"id":1829,"depth":6796,"text":1830},{"id":2469,"depth":6796,"text":2470,"children":6802},[6803,6805],{"id":2898,"depth":6804,"text":2899},3,{"id":4447,"depth":6804,"text":4448},{"id":4499,"depth":6796,"text":4500,"children":6807},[6808],{"id":5626,"depth":6804,"text":5627},{"id":5701,"depth":6796,"text":5702},{"id":5739,"depth":6796,"text":5740,"children":6811},[6812,6813],{"id":5987,"depth":6804,"text":5988},{"id":6073,"depth":6804,"text":6074},{"id":6088,"depth":6796,"text":6089,"children":6815},[6816],{"id":6701,"depth":6804,"text":6702},{"id":6731,"depth":6796,"text":6732},{"id":6767,"depth":6796,"text":6768},"Consumer and producer choice, value functions, envelope results, demand properties, and revealed-preference tests.","md",null,{},true,{"title":727,"description":6819},"icii5R2iQRalH5H-atqyyb3FSf30Eb1cb4Z1AlPPRGY",[6827,6829],{"title":721,"path":722,"stem":723,"description":6828,"children":-1},"A rigorous course from individual choice and duality to games, information, mechanisms, externalities, and market design.",{"title":733,"path":734,"stem":735,"description":6830,"children":-1},"Implicit derivatives, Slutsky decomposition, compensating and equivalent variation, surplus, incidence, and empirical welfare.",1785754732132]