Multi-Sector General Equilibrium Model

• Multi-sector GE models are formal frameworks that characterize equilibrium across multiple production sectors, households, and policy settings.
• They employ methodologies such as multifactor CES, nested CES, and networked input–output structures to simulate sectoral interactions and shock propagation.
• These models provide a basis for policy experiments and welfare analysis by assessing comparative statics, structural transformations, and dynamic spatial adjustments.

A multi-sector general equilibrium (GE) model is a formal mathematical structure describing the simultaneous equilibrium of multiple interacting production sectors, household consumers, factor markets, and government institutions within an economy. These models characterize the interdependencies among sectors via input-output linkages, substitution elasticities, and possibly spatial, network, or dynamic features. By imposing market-clearing, budget constraints, and rational agent optimization, multi-sector GE models are used to analyze comparative statics, policy counterfactuals, structural transformation, and the propagation of shocks across an interdependent economy.

1. Model Structures and Sectoral Technologies

Multi-sector GE models represent the economy as a finite or countable set of sectors (indexed by $j=1,\ldots,n$), each with a specific production technology and input requirements. The canonical sectoral production function can be specified by:

• Multifactor CES Technologies: Each sector $j$ produces output using a constant-returns multifactor CES (constant elasticity of substitution) production function over primary factors and intermediate inputs. The dual (unit-cost) representation is:

$$c_j(w) = A_j^{-1} \left( \sum_{i=0}^n s_{ij} w_i^{1-\sigma_j} \right)^{\frac{1}{1-\sigma_j}}$$

where $w_i$ are input prices, $s_{ij}$ are cost shares, $A_j$ is productivity, and $\sigma_j$ the sector-specific elasticity (Kim et al., 2016).

• Nested CES / Serial Nesting: Some models recursively nest binary CES processes, leading to cascaded cost functions parameterized by nest-specific substitution parameters (Nakano et al., 2019).
• Leontief and Cobb-Douglas Specializations: The Leontief ($\sigma_j = 0$) and Cobb-Douglas ($\sigma_j = 1$) cases are special limiting forms of the CES production function and are often used as benchmarks.
• Spatial and Networked Extensions: Models such as RHOMOLO include sectors distributed across multiple regions, with sectoral firm numbers, trade costs, and agglomeration/dispersion forces determining equilibrium location and production (Brandsma et al., 2014). Networked input-output models extend sectoral interactions across multiple economic systems using block-adjacency matrices encoding both intra- and inter-system dependencies (Trinh et al., 18 Dec 2024).
• Armington Aggregators: In trade-integrated models, aggregation over domestic and imported varieties employs Armington-type CES functions calibrated to match observed market shares and prices (Kim et al., 2017).

2. Agent Optimization and Preferences

Agents in multi-sector GE models—households, firms, governments—make decisions by solving optimization problems subject to constraints:

• Households: Typical preferences are represented as CES utilities over sectoral consumption bundles, e.g.

$$U(C_1, \ldots, C_n) = \left( \sum_{j=1}^n \mu_j^{1/(1-\lambda)} C_j^{\lambda/(\lambda-1)} \right)^{(\lambda-1)/\lambda}$$

subject to a budget constraint (Nakano et al., 2019). Local labor supplies can be further disaggregated by skill type as in RHOMOLO.

• Firms: Firms maximize profit, minimizing cost over input quantities and shadow prices, subject to production technology. Zero-profit (unit-cost) conditions characterize equilibrium prices, such as $p_j = c_j(w)$ for sector $j$ (Kim et al., 2016).
• Learning and Uncertainty: Some formulations endogenize firm beliefs about technological parameters (e.g., returns to scale), allowing for dynamic Bayesian updating based on observed outcomes, which influences subsequent input choices and equilibrium formation (Nasini et al., 7 Dec 2025).
• Government and R&D: Extended models include government budget constraints, taxation, public capital provision, and sectoral R&D with endogenous technological progress (Brandsma et al., 2014).

3. Equilibrium Conditions and Market Clearing

Equilibrium in multi-sector GE models is defined by a set of prices, quantities, and factor allocations that simultaneously clear all relevant markets:

• Goods Market Clearing:

$y_j = d_j + \sum_{k=1}^n b_{jk} y_k$

or $y = (I - B)^{-1} d$ for all sectors, where $B$ is the current input-output coefficient matrix, and $d$ is final demand (Kim et al., 2016).

• Labor Market Clearing: Aggregate labor employed across sectors equals total supply, possibly disaggregated by skill or region.
• Factor Markets: Primary factor prices adjust to clear supply and demand; often, a numéraire (e.g., the composite factor price $p_0 = 1$) is fixed for identification.
• Dynamic and Spatial Equilibrium: When the model is dynamic, law-of-motion for capital stock, human capital, and firm numbers are specified. In spatial models, equilibrium includes regional location choices and the trade-off between agglomeration and dispersion forces (Brandsma et al., 2014).
• Networked Equilibria: In inter-system models, the equilibrium is the solution to $x^* = (I_{dn} - \bar{A})^{-1} y$, where $\bar{A}$ is the block-adjacency matrix of sectoral coefficients across systems (Trinh et al., 18 Dec 2024).

4. Calibration, Estimation, and Solution Methods

Empirical application of multi-sector GE models requires careful calibration of cost shares, substitution elasticities, productivity parameters, and agent preferences:

• Two-Point Calibration and Regression: Parameters such as CES elasticities and productivity growths are estimated by regressing the change in factor-wise cost shares against input price growths over temporally distant input-output tables (Kim et al., 2016, Kim et al., 2017, Nakano et al., 2019).
• Armington Elasticities: Computed by matching observed changes in market shares and prices across two states (Kim et al., 2017).
• Spatial Parameters: Spatial models use gravity-type estimates or transport survey data to calibrate trade costs and regional interdependence (Brandsma et al., 2014).
• Solution Algorithms: Models are typically solved by fixed-point iteration, Newton-type solvers, or block-recursive algorithms for the large nonlinear system of equilibrium conditions. Dynamic spatial models solve a sequence of static equilibria linked by capital and human capital evolution (Brandsma et al., 2014).
• Agent-Based Behavioral Metrics: Extensions utilize micro-level behavioral parameters to capture sectoral price versus quantity adjustment response to shocks; these are empirically estimated by minimizing residual differences between predicted and observed IO data (Hurt et al., 15 May 2025).

5. Policy Experiments, Comparative Statics, and Welfare Analysis

Multi-sector GE frameworks support a wide array of counterfactual analyses and welfare computations:

• Productivity and Tariff Shocks: Simulation of sector-specific productivity changes or trade policy interventions, such as tariff elimination, is performed by re-solving the model under new exogenous parameter values and comparing resulting equilibria (Kim et al., 2016, Kim et al., 2017).
• Structural Transformation: Dynamic models extrapolate sectoral and aggregate welfare outcomes under hypothetical productivity trajectories using restoring methods on serially nested IO data (Nakano et al., 2019).
• Welfare Metrics: Social cost saved, equivalent variation, and compensating variation are computed from post-shock output, prices, and utility levels.
• Resilience and Recovery: Agent-based extensions to input-output models evaluate sectoral and regional resilience and recovery strategies to exogenous demand shocks, extracting behavioral metrics for price versus quantity adjustments (Hurt et al., 15 May 2025).

6. Extensions: Network, Dynamic, and Spatial GE Models

Recent research has expanded the canonical multi-sector GE framework in several directions:

• Networked Input-Output Models: The system is represented as a matrix-weighted graph of multiple interacting economic systems, allowing distributed algorithms for equilibrium computation and consensus-based stability analysis (Trinh et al., 18 Dec 2024).
• Mean-Field, Stochastic, and Environmental Extensions: Heterogeneous agents interact via common external variables (e.g., pollution), with equilibrium characterized by forward-backward stochastic differential equations and the mean-field game apparatus (Lavigne et al., 21 Jan 2025).
• Learning and Bayesian Updating: Firm technology learning dynamics, especially under uncertainty about returns to scale, change equilibrium trajectories. Path-independent belief updating often yields faster and less biased convergence compared to long-memory approaches (Nasini et al., 7 Dec 2025).
• Spatial Models: RHOMOLO and related frameworks combine multi-sector GE with spatial economics, modeling entry and exit of firms, transport costs, R&D spillovers, and regionally differentiated labor and capital dynamics (Brandsma et al., 2014).

7. Comparative Insights and Limitations

Multi-sector GE models provide tractable and rigorous foundations for analyzing inter-industry dependencies, resource allocation, and the effects of exogenous shocks. CES-based structures allow flexible substitution patterns, while networked and spatial GE models accommodate distributed and location-dependent phenomena. However, key limitations include assumptions of constant technology matrices, the absence of endogenous innovation beyond prescribed processes, and potential empirical challenges in calibrating high-dimensional parameter spaces and behavioral rules.

A central implication is that sectoral shock propagation, welfare impacts, and policy effectiveness depend crucially on the microstructure of substitution elasticities, input-output topology, spatial clustering, and agent learning dynamics. Ongoing extensions integrating agent-based behavior, stochastic feedbacks, and network structures are expanding the analytic reach and empirical relevance of multi-sector general equilibrium modeling in contemporary economic research (Brandsma et al., 2014, Trinh et al., 18 Dec 2024, Lavigne et al., 21 Jan 2025, Kim et al., 2016, Nasini et al., 7 Dec 2025, Hurt et al., 15 May 2025, Nakano et al., 2019).