Input-output models are a key tool in mathematical economics, analyzing how different sectors of an economy interact. They show how changes in one industry can ripple through the entire system, helping economists understand complex economic relationships.

Open models treat final demand as external, while closed models incorporate it within the system. Both use matrix algebra to calculate how changes in demand affect output across industries. These models are vital for economic planning and impact analysis, despite some limitations.

Basic input-output model

• Fundamental framework in mathematical economics analyzes interdependencies between economic sectors
• Represents economy as system of linear equations describing flows of goods and services between industries
• Developed by Wassily Leontief, earned Nobel Prize in Economics for this work

Structure of input-output tables

• Consists of rows and columns representing industries or sectors in an economy
• Rows show distribution of industry's output across other sectors
• Columns depict inputs required by each industry from other sectors
• Includes intermediate consumption, final demand, and value added components
• Balances total output with total input for each sector

Leontief production function

• Assumes fixed proportions of inputs required to produce one unit of output
• Characterized by constant returns to scale and no substitution between inputs
• Expressed mathematically as $x_i = min(\frac{z_{i1}}{a_{i1}}, \frac{z_{i2}}{a_{i2}}, ..., \frac{z_{in}}{a_{in}})$
• Where $x_i$ represents output of industry i, $z_{ij}$ inputs from industry j, and $a_{ij}$ technical coefficients
• Simplifies production relationships but may not capture all real-world complexities

Technical coefficients matrix

• Represents direct input requirements per unit of output for each industry
• Calculated by dividing each input by the total output of the corresponding industry
• Expressed as $a_{ij} = \frac{z_{ij}}{x_j}$, where $a_{ij}$ technical coefficient, $z_{ij}$ input from i to j, $x_j$ total output of j
• Forms basis for analyzing inter-industry relationships and economic impacts
• Assumes stability over time, which may not hold in rapidly changing economies

Open input-output models

• Treats final demand as exogenous variable determined outside the model
• Allows analysis of how changes in final demand affect output levels across industries
• Widely used in economic impact studies and policy analysis

Exogenous final demand

• Includes components determined outside the production system (household consumption, government spending, exports)
• Represented as a vector $f$ in the input-output model
• Can be manipulated to simulate various economic scenarios or policy changes
• Allows for analysis of direct and indirect effects on industry outputs

Output determination

• Calculates total output required to meet given final demand
• Expressed as system of linear equations $x = Ax + f$
• Where $x$ output vector, $A$ technical coefficients matrix, $f$ final demand vector
• Solved through matrix algebra to find equilibrium output levels
• Provides insights into production requirements and inter-industry dependencies

Leontief inverse matrix

• Key component in solving open input-output models
• Calculated as $(I - A)^{-1}$, where $I$ identity matrix and $A$ technical coefficients matrix
• Also known as the total requirements matrix
• Captures both direct and indirect effects of changes in final demand
• Each element represents total output required from one industry to produce one unit of final demand in another

Closed input-output models

• Incorporates some components of final demand as endogenous variables within the model
• Typically includes household sector as part of the production system
• Allows for more comprehensive analysis of economic interdependencies and feedback effects

Endogenous final demand

• Treats certain final demand components as dependent on the economic system
• Often includes household consumption as function of income generated within the economy
• Expands technical coefficients matrix to include additional row and column for household sector
• Allows for analysis of induced effects alongside direct and indirect effects

Household sector inclusion

• Treats households as both producers (labor) and consumers in the economic system
• Adds row representing labor inputs to industries and column for household consumption
• Captures income-consumption feedback loop within the economy
• Increases complexity of model but provides more realistic representation of economic interactions

Multiplier effects

• Measures total impact on output resulting from change in exogenous final demand
• Includes direct, indirect, and induced effects in closed models
• Generally larger than multipliers in open models due to additional feedback effects
• Calculated using expanded Leontief inverse matrix $$(I - A^)^{-1}$, where $A^$ includes household sector
• Provides insights into full economic impacts of policy changes or external shocks

Mathematical formulation

• Utilizes linear algebra and matrix operations to represent and solve input-output models
• Allows for efficient computation and analysis of complex economic systems
• Forms basis for various extensions and applications in economic modeling

Matrix notation

• Represents entire input-output system using compact matrix equations
• Basic equation for open model $x = Ax + f$ or $(I - A)x = f$
• Closed model expands matrices to include endogenous sectors
• Facilitates manipulation and solution of large-scale economic systems
• Enables use of powerful matrix algebra techniques for analysis

System of linear equations

• Expresses input-output relationships as set of simultaneous linear equations
• Each equation represents balance between total output and sum of intermediate and final demands
• For n industries $x_i = \sum_{j=1}^n a_{ij}x_j + f_i$ for $i = 1, 2, ..., n$
• Solved simultaneously to determine equilibrium output levels
• Can be expanded to include additional constraints or variables as needed

Solving for equilibrium output

• Involves finding solution to system of linear equations
• For open models, solution given by $x = (I - A)^{-1}f$
• Closed models require solving expanded system including endogenous final demand
• May use direct matrix inversion or iterative methods for large systems
• Provides equilibrium output levels for given final demand and technical coefficients

Economic interpretation

• Translates mathematical results into meaningful economic insights
• Helps policymakers and analysts understand complex economic relationships
• Provides framework for assessing impacts of economic changes or policy interventions

Interdependence of industries

• Highlights how changes in one sector ripple through entire economy
• Quantifies strength of linkages between different industries
• Reveals key sectors with strong connections to rest of economy
• Helps identify potential bottlenecks or strategic industries for development
• Informs policy decisions on targeted economic interventions or support

Direct vs indirect effects

• Direct effects represent immediate impact of change in final demand on producing industry
• Indirect effects capture subsequent changes in output of supplying industries
• Induced effects (in closed models) include impacts from changes in household income and spending
• Total effect sum of direct, indirect, and induced effects
• Helps understand full scope of economic impacts beyond initial change

Backward and forward linkages

• Backward linkages measure industry's impact on its suppliers
• Calculated using column sums of Leontief inverse matrix
• Forward linkages represent industry's importance as supplier to other sectors
• Computed using row sums of output inverse (Ghosh inverse) matrix
• Identifies key industries with strong connections throughout economy
• Useful for targeting development strategies or assessing vulnerability to shocks

Applications and limitations

• Input-output models widely used in various economic analyses and planning
• Provide valuable insights but subject to certain assumptions and limitations
• Understanding both strengths and weaknesses crucial for appropriate application

Economic planning

• Used by governments and organizations for national and regional economic planning
• Helps identify key sectors for development or investment
• Allows simulation of different policy scenarios and their impacts
• Supports decision-making on infrastructure projects, industrial policies, and resource allocation
• Provides framework for coordinating different aspects of economic development

Impact analysis

• Assesses economic effects of changes in final demand, technology, or economic structure
• Used to evaluate impacts of major investments, policy changes, or external shocks
• Quantifies direct, indirect, and induced effects on output, employment, and income
• Supports cost-benefit analysis of public projects or private investments
• Helps estimate economic consequences of natural disasters or major events

Static vs dynamic models

• Traditional input-output models static, representing economy at single point in time
• Assume fixed technical coefficients and constant returns to scale
• Dynamic models incorporate changes in technology, prices, and economic structure over time
• May include investment functions, capital accumulation, and technological progress
• Static models simpler and require less data, but may not capture long-term economic changes
• Dynamic models more complex but can provide insights into economic growth and development paths

Extensions and variations

• Numerous extensions developed to address limitations of basic input-output models
• Adapt framework to specific analytical needs or incorporate additional economic factors
• Enhance applicability and relevance of input-output analysis in various contexts

Regional input-output models

• Focus on economic structure and interactions within specific geographic area
• Account for regional differences in production technologies and trade patterns
• May use location quotients or other techniques to estimate regional technical coefficients
• Often incorporate inter-regional trade flows and spillover effects
• Support regional development planning and impact assessments of localized events

Environmental input-output analysis

• Extends traditional model to include environmental impacts of economic activities
• Incorporates data on resource use, emissions, and waste generation by industries
• Allows analysis of environmental consequences of economic changes or policies
• Supports development of sustainable production strategies and environmental regulations
• Can be linked with life cycle assessment for comprehensive environmental impact studies

Social accounting matrices

• Expand input-output framework to include more detailed economic and social data
• Incorporate additional accounts for factors of production, institutions, and capital transactions
• Provide more comprehensive view of income distribution and economic structure
• Support analysis of poverty, inequality, and social impacts of economic policies
• Often used in developing countries for integrated economic and social planning

Empirical considerations

• Practical aspects of implementing and using input-output models in real-world contexts
• Addresses challenges in data collection, model maintenance, and result interpretation
• Crucial for ensuring reliability and relevance of input-output analysis in decision-making

Data collection and compilation

• Requires extensive data on inter-industry transactions and final demand components
• Often based on national accounts data, economic censuses, and surveys
• May involve reconciling data from multiple sources and resolving discrepancies
• Challenges include capturing informal sector activities and handling confidential business information
• Increasing use of big data and administrative records to supplement traditional data sources

Updating input-output tables

• Input-output tables typically produced with significant time lag due to data collection processes
• Various techniques used to update tables to more recent years (RAS method, commodity flow approach)
• Balancing methods ensure consistency between updated tables and current national accounts data
• Regular updates crucial for maintaining relevance of input-output models in rapidly changing economies
• Trade-off between frequency of updates and level of detail in input-output tables

Accuracy and reliability issues

• Input-output models subject to various sources of error and uncertainty
• Aggregation bias from grouping heterogeneous industries or products
• Assumption of fixed technical coefficients may not hold over time or for large changes
• Potential errors in data collection, estimation of missing values, and balancing procedures
• Sensitivity analysis and error propagation studies help assess robustness of results
• Important to communicate uncertainties and limitations when presenting input-output analysis results