Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Structural Models and Economic Computation/IV-10
Dynamic Choice, Aggregation, and Policy Capacity · Chapter IV-10

Heterogeneous States, Transition Economies, and Additive Summaries

A distribution of households, firms, or cases is often the true dynamic state. Moments or counts provide useful summaries when their transitions close under the policy and equilibrium law. Fixed-price transitions and equilibrium feedback must be separated, because a statistic sufficient for logistics can fail once prices or incentives respond to the distribution.

Conceptual map

  1. IV-10.01Distribution states
  2. IV-10.02Fixed prices and equilibrium
  3. IV-10.03Resolution
  4. IV-10.04Moment closure
  5. IV-10.05Transition and logistics examples

1. A population distribution is a dynamic state

Counts or moments close only when every microdistribution sharing them generates the same next summary under the declared policy and equilibrium environment.

Let \(\mu _{t}\) be a row probability vector on \(K\) finite types and let row-stochastic \(P\) govern a fixed-policy transition, so \(\mu _{t+1}=\mu _{t}P\). A linear summary is \(m_{t}=C\mu _{t}^{\top}\).

Proposition 1 · Linear moment closure

A linear map \(L\) satisfies \(m_{t+1}=Lm_{t}\) for every signed state vector exactly when \(\operatorname{ker}(C)\subseteq \operatorname{ker}(CP^{\top})\), equivalently \(CP^{\top}=LC\).

Proof. Apply the linear factorization theorem to current summary \(C\) and next summary \(CP^{\top}\). Restricting to the probability simplex gives the same sufficient condition and can admit additional affine formulations when total mass is fixed. ∎

2. Within-cell composition changes next counts

States one and two form group \(G\); state three forms group \(B\). Let transition rows be (0.9,0,0.1), (0,0.5,0.5), and (0,0,1). Distributions \(\mu ^{a}\)=(1,0,0) and \(\mu ^{b}\)=(0,1,0) both report (\(G,B\))=(1,0). Their next \(B\) masses are 0.1 and 0.5.

Adding the mass in state two makes the three-state distribution recoverable from total mass and the two group summaries in this example. A lower-dimensional repair may suffice for a narrower target or horizon.

3. Fixed-price closure need not survive equilibrium feedback

If transition matrix is \(P(p,u)\) and equilibrium price solves \(p=\phi(\mu,u)\), a summary must preserve both the price map and the conditional transition. Two distributions can generate equal fixed-price next moments and different equilibrium prices, which then alter choices and transitions.

Logistical closure holding prices fixed is useful for forecasting that controlled environment. A structural policy state additionally carries every distribution feature entering prices, selection, or policy assignment.

4. Simplex faces and inaccessible types

Failure case · Fixed transition closure extended to endogenous prices

A factorization of \(CP^{\top}\) through \(C\) at one fixed price says nothing about a price rule that varies inside a summary fiber.

Zero-mass types lie on simplex faces and can become positive only if transition inflows permit. Reducible transitions create invariant faces. Finite-horizon closure can hold for one target even when one-step full-distribution closure fails; the horizon and target belong in the test.

5. Implementation, exercises, and sources

Fix row or column orientation, verify stochasticity, construct \(C\), and test the factorization with exact arithmetic or a declared rank tolerance. Search summary fibers for pairs with different next moments and repeat after enabling equilibrium feedback.

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Exercises

  1. Reproduce the next \(B\) masses in Section 2.
  2. Find an augmented summary that closes the displayed transition.
  3. Construct a price map depending on within-\(G\) composition and show fixed-price closure can fail in equilibrium.
Partial solutions

1. Multiply each point-mass distribution by its transition row. 2. Retain the mass of state two in addition to group \(B\) and total mass. 3. Let price equal the state-two share and make a transition probability depend on price.

  1. John G. Kemeny and J. Laurie Snell, Finite Markov Chains.Lumpability and aggregation.
  2. Lars Peter Hansen and Thomas J. Sargent, Recursive Models of Dynamic Linear Economies.State closure and recursive equilibrium.
  3. Chae-Yeon Xon (2026), “The Polar Express: Dynamic Aggregation under Policy Holonomy.”Related dynamic-aggregation application.

6. Audit checkpoint

State the distribution simplex, transition orientation, policy and price environment, summary map, total-mass convention, closure target and horizon, kernel residual, simplex faces, reducibility, equilibrium feedback, and counterexample fiber.

7. Scope boundary

The chapter covers finite distribution transitions and linear closure diagnostics. Infinite-dimensional heterogeneous-agent computation requires further approximation theory.

Prerequisites