name: bbl_dynasty # ───────────────────────────────────────────────────────────────────── # Benhabib-Bisin-Luo (2019) dynasty-level / cross-generational layer. # # Composes the 50 within-lifetime value/policy functions (one per # (tau, r_type) pair, defined in dolo-plus-draft.yaml) across # generations via independent intergenerational Markov chains # Pi_tau (10x10) and Pi_r (5x5). The lifetime map g(.; tau, r) is # treated as a black-box reference to the within-life solver. # # See dynasty-excerpt.md for the formal Bellman/algebra description. # ───────────────────────────────────────────────────────────────────── # unresolved (SPECULATIVE block-level): dolo-plus has no canonical # syntax for cross-generational composition. Matsya 2026-04-27 review # confirmed UNRESOLVED at the dolo-plus spec level for several related # items (calibration-override families, terminal-boundary blocks, per- # age overrides, period templates with state-dependent parameters); # cross-generational composition is one level above any of those, and # also has no canonical idiom in matsya's indexed corpus. The structure # below is therefore SPECULATIVE throughout — keyword names and sub-key # nesting may need renaming once a canonical idiom appears. # ═══════════════════════════════════════════════════════════════════ # WITHIN-LIFE REFERENCE # ═══════════════════════════════════════════════════════════════════ # The dynasty layer treats the solution of the within-lifetime # problem as a black box: given (tau, r_type) and initial wealth a, # it returns terminal wealth a_T = g(a; tau, r_type). The within- # life problem itself is fully specified in the sibling YAML. within_life: source: dolo-plus-draft.yaml family: bbl_consumption_savings description: | The within-life YAML defines a finite-horizon Bellman problem parameterized by (tau, r_type). Solving it produces: - optimal policy c*_t(a_t; tau, r_type) for t = 1..T - terminal-wealth function g(a_1; tau, r_type) = a_{T+1} where the second is the lifetime map this YAML composes across generations. instance_index: [tau, r_type] cardinality: {tau: 10, r_type: 5} # 50 instances total # ═══════════════════════════════════════════════════════════════════ # DYNASTY-LEVEL STATE AND DYNAMICS # ═══════════════════════════════════════════════════════════════════ generations: index: n # generation index n = 0, 1, 2, ... # ── Cross-generational state ──────────────────────────────────── # State at the start of generation n: initial wealth + types. # Both type variables are fixed within a life and resolve only # at birth (between generations), so they are dynasty-layer states # rather than within-life states. states: a: domain: '@in R+' description: | a^n = a^n_0 = newborn wealth of generation n. Equals terminal wealth of generation n-1 via the lifetime map: a^n = g(a^{n-1}; tau^{n-1}, r_type^{n-1}). tau: domain: '@in {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}' description: Earnings-decile type for generation n (paper Table 1). r_type: domain: '@in {1, 2, 3, 4, 5}' description: Rate-of-return-type index for generation n (paper Table 4). # ── Cross-generational stochastic process ─────────────────────── # Independence (paper §I): Pi[(tau,r) | (tau',r')] = Pi_tau[tau|tau'] * Pi_r[r|r']. # Both chains are 'drawn at the boundary between generations' — no # within-life resolution. exogenous: tau_chain: '@dist Markov(Pi_tau)' # 10-state intergenerational chain r_chain: '@dist Markov(Pi_r)' # 5-state intergenerational chain # unresolved: how dolo-plus expresses "two independent Markov chains" # in a single exogenous block — one per declaration is the natural # reading, but matsya did not surface a canonical example. # ── Transition: generation n → generation n+1 ─────────────────── # The lifetime map carries terminal wealth forward; the type chains # advance independently. transition: a[+1]: '= g(a; tau, r_type)' # lifetime map; symbolic reference # to within-life solver output tau[+1]: '~ Pi_tau[tau, :]' # next-gen earnings type r_type[+1]: '~ Pi_r[r_type, :]' # next-gen rate type # unresolved: dolo-plus syntax for "function-of-state" transitions # that invoke an external solver (g) is not documented. The notation # `= g(...)` is symbolic; an implementation would need to wire the # within-life policy function into this transition explicitly. # ── Lifetime map (symbolic; not computed in this YAML) ────────── lifetime_map: name: g signature: '(a_0: R+, tau: int, r_type: int) -> R+' description: | g(a_0; tau, r_type) is the terminal wealth of a household born with initial wealth a_0 and types (tau, r_type), obtained by forward-simulating the optimal within-life policy: a_{t+1} = (1 + r) * (a_t - c*_t(a_t; tau, r_type)) + w_t(tau) for t = 1, ..., T, with a_1 = a_0; return a_T. Where r is the value indexed by r_type (see by_r_type in dolo-plus-draft.yaml::calibration_family). Symbolic only at this layer; computation lives in a downstream solver that ingests dolo-plus-draft.yaml. properties: # paper §I Proposition mu_eq_sigma: description: | When mu = sigma, g is affine in a_0: g(a_0; tau, r) = alpha(tau, r) * a_0 + beta_g(tau, r). Savings rate alpha(tau, r) is independent of a_0. mu_lt_sigma: description: | When mu < sigma (paper's empirical case: mu=0.5993 < sigma=2), g is strictly convex in a_0: d^2 g / da_0^2 > 0. Savings rate is increasing in wealth; the rich save proportionally more. # ═══════════════════════════════════════════════════════════════════ # INTERGENERATIONAL MARKOV CHAINS # ═══════════════════════════════════════════════════════════════════ # Reproduced here for self-containment of the dynasty layer; the # authoritative source is dolo-plus-draft.yaml::calibration_family # .population. Pi_tau: description: | 10 x 10 transition matrix for tau (earnings decile across generations), from Chetty et al. (2014) reduced to a 10-state chain (paper §IIB). Online Appendix B.2 documents the construction procedure but does not tabulate the matrix. Reconstruction requires either Chetty et al.'s online_data_tables.xls or the BBL replication package (https://doi.org/10.3886/E113112V1). source_reference: dolo-plus-draft.yaml::calibration_family.population.Pi_tau # unresolved: full matrix entries (not in online Appendix B.2). Pi_r: description: | 5 x 5 transition matrix for r. Paper Table 4 (diagonal) plus online Appendix C.1 (full matrix; downloaded 2026-04-27). Row-stochastic; rows 1-4 are symmetric around the diagonal (decay per paper footnote 13); row 5 has constant off-diagonals 0.2448. Full matrix transcribed in the within-life YAML. source_reference: dolo-plus-draft.yaml::calibration_family.population.Pi_r matrix: - [0.0338, 0.5013, 0.2600, 0.1349, 0.0700] - [0.2876, 0.2676, 0.2876, 0.1129, 0.0443] - [0.1158, 0.3163, 0.1360, 0.3163, 0.1158] - [0.0446, 0.1136, 0.2894, 0.2630, 0.2894] - [0.2448, 0.2448, 0.2448, 0.2448, 0.0208] # ═══════════════════════════════════════════════════════════════════ # STATIONARY DISTRIBUTION (annotation only, not computed here) # ═══════════════════════════════════════════════════════════════════ # Paper §I Proposition characterizes the stationary distribution of # {a^n}_n. This block records the result for downstream solvers # without computing anything. See dynasty-excerpt.md → "Stationary # distribution and Pareto tail" for the full statement. stationary_distribution: regime: mu_eq_sigma: description: | Linear stochastic recurrence equation a^{n+1} = alpha * a^n + beta_g. Stationary distribution exists under standard conditions on the joint distribution of (alpha, beta_g) (paper footnote 9; see Grey 1994 / Hay-Rastegar-Roitershtein 2011 / Benhabib-Bisin-Zhu 2011). tail: 'Pr(a > a_underbar) ~ Q * a_underbar^(-gamma) asymptotically' tail_index_implicit_eq: | gamma solves: lim_{N->inf} E[ prod_{n=0}^{N-1} alpha(tau^{-n}, r^{-n})^gamma ]^(1/N) = 1 independence_of_earnings: description: | The constant term beta_g does NOT affect the asymptotic Pareto exponent gamma — only the multiplicative term alpha does. This is the paper's mechanism for generating a thick wealth tail despite a relatively thin earnings distribution. mu_lt_sigma: description: | Convex map; stationary distribution may not exist. When it does, the tail is at least as thick as Pareto: Pr(a > a_underbar) >= Q * a_underbar^(-gamma). Paper's quantitative analysis (Table 5) confirms existence at estimated parameters and matches empirical wealth shares. computational_status: description: | Determining gamma numerically requires simulating the dynasty process at calibrated parameters and fitting the empirical stationary tail. Out of scope for this formalization layer (per AGENTS.md: this repo describes models, does not solve them). # ═══════════════════════════════════════════════════════════════════ # OUT OF SCOPE FOR THIS BASELINE DYNASTY YAML # ═══════════════════════════════════════════════════════════════════ # - Section IIID wealth-dependent r extension. In that variant, # r^n becomes a function of a^n via a wealth-percentile indexing, # so r_chain becomes a state-conditioned Markov chain. Belongs in # a separate dynasty variant YAML pair. See dynasty-excerpt.md # → "Out-of-scope" and bellman-excerpt.md → Open Issue #6. # - Section V transitional-dynamics exercise (non-stationary initial # distribution starting from SCF 1962-63). Same dynasty operator # as the baseline; would specify a different initial distribution. # Out of scope here, in scope for any future replication exercise. # - Numerical computation of the lifetime map g and of the # stationary distribution. Out of scope per AGENTS.md "Common # next tasks" — this repo produces formal specifications, not # solvers. The dynasty YAML pair describes the model; downstream # tooling (econ-ARK or otherwise) would compute it.