Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Structural Models and Economic Computation/IV-09
Dynamic Choice, Aggregation, and Policy Capacity · Chapter IV-09

Corporate Control and Policy-Invariant Aggregation

Corporate groups can coordinate financing, pricing, or investment across legal entities. Aggregates built from entity-level balance sheets may conceal control links that govern joint adjustment to policy. A policy-invariant state must preserve the directions through which the relevant corporate organization changes equilibrium responses.

Conceptual map

  1. IV-09.01Economic organization
  2. IV-09.02Joint adjustment
  3. IV-09.03Comparative statics
  4. IV-09.04Identification limits
  5. IV-09.05Empirical operationalization

1. Legal-entity totals can hide response-relevant organization

An aggregate is policy invariant only relative to a policy-to-target response. Corporate-control records matter when hidden ownership or coordination directions change that response.

Let \(x\in \mathbb{R}^{2}\) collect activity controlled by two corporate groups and let the reported legal-entity total be \(h(x)=x_{1}+x_{2}\). Consider equilibrium outcome under scalar policy \(p\),

\[y(p,x)=x_{1}+x_{2}+p(x_{1}-x_{2}).\](1)

At \(p=0\), baseline outcome equals the reported total. Its policy derivative is \(Bx=x_{1}-x_{2}\), which varies along the hidden direction of \(A\)=[1,1].

2. Response rank gives the local repair dimension

Proposition 1 · Corporate-state repair in the example

The restricted response derivative \(B\)=[1,\(-1\)] has rank one on \(\operatorname{ker}(A)\)=span{(1,\(-1\))}. One additional linear corporate record is necessary and sufficient for exact local response recovery.

Proof. The hidden direction maps to two under \(B\), so the deficit is one. Record \(r=x_{1}-x_{2}\) directly supplies the target derivative. ∎

States (1,0) and (0,1) share total one and baseline outcome one. Their policy derivatives are +1 and \(-1\). The aggregate alone therefore cannot preserve the response even in this two-state comparison.

3. A real control network requires explicit construction

In an ownership network, the state must specify edge direction, voting or economic weights, consolidation rules, treatment of cycles, and the map from direct ownership to ultimate control. The equilibrium model then states how group financing, internal markets, or coordinated pricing enters policy response.

A group identifier can serve as a record only if it is observed at the decision date under a reproducible entity-resolution rule. Later ownership information creates look-ahead bias in historical policy experiments.

4. Response heterogeneity and control interventions differ

Failure case · Control proxy association interpreted as a control effect

Correlation between a control index and monetary response can identify heterogeneity under a maintained response model. The effect of changing control requires an intervention, structural transition, or design that specifies how ownership changes other primitives.

At ownership ties or cycles, ultimate-control labels can be set valued. Near-zero singular values make repair dimension threshold dependent. A finite policy move additionally requires curvature control beyond the derivative in (1).

5. Implementation, exercises, and sources

Version entity identifiers and ownership dates, construct the control map, aggregate under both legal and ultimate-control definitions, and differentiate the equilibrium at a declared policy baseline. Report hidden-response singular values, metric, rank threshold, added records, and finite-move residuals.

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Exercises

  1. Reproduce the baseline and derivative comparison for the two states.
  2. Add record \(x_{1}\) and construct a decoder for the response derivative.
  3. Build a three-entity ownership cycle and compare two declared tie-breaking rules.
Partial solutions

1. Both totals equal one; substituting into \(\partial y/\partial p=x_{1}-x_{2}\) gives opposite signs. 2. From total \(s\) and \(x_{1}\), recover \(x_{2}=s-x_{1}\) and response \(2x_{1}-s\).

  1. Oliver E. Williamson, Markets and Hierarchies.Organizational control and firm boundaries.
  2. Jean Tirole, The Theory of Corporate Finance, chapters on control rights.Control, financing, and organizational incentives.
  3. Chae-Yeon Xon (2026), “Too Big to Aggregate: Corporate Control and Policy-Invariant Aggregation.”Related monetary-response application.

6. Audit checkpoint

State entity and control units, ownership date and direction, cycle and tie rules, legal aggregate, control state, policy, equilibrium map, derivative metric, hidden-response rank and threshold, feasible records, causal estimand, and finite-policy domain.

7. Scope boundary

The chapter gives a local aggregation example and network-data contract. Corporate-control causality and full monetary transmission require separate empirical and structural designs.

Prerequisites