The rationality of adaptive decision-making and the feasibility of optimal growth planning

Rationality and rational expectations are foundational in economics but represent an idealized behavioral norm. Even with advances in information technology that improve probabilistic forecasting, the coordinated harmony implied by rational expectations for future events remains fictional for real decision processes. In practice, intertemporal choices in firms and households follow adaptive, budget-control procedures: formulate a plan, implement it, compare outcomes to plan, and revise iteratively. Such behavior is sequentially rational given available information and is well aligned with information technology advances. This paper studies households’ intertemporal consumption/saving planning and compares rational (optimal) versus adaptive (budget-controlled) decision principles. The study: (1) documents the universality of budget-controlled decision-making; (2) sets the Ramsey-type optimal growth model as the rational benchmark; (3) builds an adaptive, budget-controlled planning model using replicator dynamics and phase-diagram analysis; and (4) evaluates social welfare via numerical simulations comparing adaptive paths to the optimal growth path.

The paper situates practical decision-making within management accounting and budgetary control traditions, emphasizing Plan-Do-Study-Act (PDSA) cycles (Deming). Empirical evidence shows widespread adoption of planning, budgeting, and forecasting (PBF) procedures across firms (Deloitte 2014; ACCA & KPMG 2015) and applications in government (Results-Based Management; UNDP 2002) and households. Household budgeting and self-control literature highlight plan–actual monitoring to manage intertemporal trade-offs (Thaler & Shefrin 1981; Heath & Soll 1996; Hernandez et al. 2014; Jonker 2016). Surveys show broad use of budgets, savings accounts, and retirement planning (Hilgert et al. 2003; Federal Reserve 2019), with knowledge and experience improving financial behaviors. As the rational benchmark, the paper reviews Ramsey–Cass–Koopmans optimal growth models (Ramsey 1928; Cass 1965; Koopmans 1963; Uzawa 1964, 1965), extensions to overlapping generations (Diamond 1965) and endogenous growth (Romer 1986). It contrasts these with adaptive and boundedly rational perspectives from psychology and behavioral economics, noting context-dependent preferences and adaptive strategies (Payne & Bettman 1988; Knoll 2010; Rosati & Stevens 2009). Adaptive learning in macroeconomics models expectation formation updated with observed data and studies stability under adaptive learning (Evans & McGough 2020). The paper adopts replicator dynamics as an alternative adaptive mechanism, reinterpreted for socioeconomic settings beyond random matching via agent-based learning dynamics (Deguchi 2004), and previously applied to technology/industry evolution (Safarzynska & van den Bergh 2011; Cantner et al. 2019; Sakaki 2004, 2018). This literature motivates modeling intertemporal planning as sequential, adaptive, and information-updating decisions.

Analytical framework: two decision principles are compared. 1) Rational (optimal growth) benchmark - Economy: single closed economy; production Y = F(K, L), with constant returns and Inada conditions. Per-capita variables: y = f(k), c, s. Labor grows at n > 0, technology constant. - Capital accumulation (continuous time): k̇ = f(k) − c − n k (Eq. 1). - Planner/household maximizes discounted utility ∫₀^∞ u(c_t) e^{-ρ t} dt subject to k̇ and initial k₀ > 0; u'(c) > 0, u''(c) < 0, u'(0) = ∞. - Hamiltonian H_t = u(c_t) e^{-ρ t} + λ_t (f(k_t) − c_t − n k_t), FOCs yield Keynes–Ramsey rule linking consumption growth to the interest–time preference gap: ċ/c = (1/σ)[f'(k) − n − ρ] for CES utility (Eqs. 2, 15), plus TVC lim λ_t k_t = 0. - Phase diagram: unique saddle-path to steady state E at Modified Golden Rule k** (f'(k**) = n + ρ), with c lower than Golden Rule k* (f'(k*) = n). 2) Adaptive, budget-controlled planning via PDSA and replicator dynamics - Decision process mapped to PDSA: Plan/Do: Firms produce f(k_t) from k_t; households consume planned c_t; firms plan next-period capital k_{t+1} and output f(k_{t+1}). Capital update (discrete): k_{t+1} = k_t + f(k_t) − c_t − n k_t (Eq. 5). Study: Households compare current output f(k_t) with discounted next planned output f(k_{t+1})/(1+r). Using replicator dynamics, they update the allocation share x_{t+1} of output toward current vs next period: x_{t+1} = x_t f(k_t)/[x_t f(k_t) + (1−x_t) f(k_{t+1})/(1+r)] (Eq. 6). Act: Revise next consumption plan c_{t+1} using the change in allocation shares and output differences: Δc_t = A_c[(x_{t+1}−x_t) f(k_t) + x_t (f(k_{t+1}) − f(k_t))]; c_{t+1} = c_t + Δc_t (Eq. 7). Parameters are set so that x correlates with the average propensity to consume. - Continuous-time approximation yields dynamic system (Eqs. 8–10): ẋ = x(1−x)[f(k) − e^{−ρ}(f(k) + f'(k)k)] ≈ x f(k) + x f'(k) k (approximation for analysis), ċ = function of x and k capturing adaptive revision (Eq. 9), k̇ = f(k) − c − n k (Eq. 10). - Phase-diagram analysis: Steady-state loci include k̇ = 0 (same as optimal), an ẋ = 0 vertical at k = k** (>0), and ċ = 0 verticals at k = k* and k = k*2 with k* < k*2 < k** (for k > 0), plus behavior for k < 0 (with k^#). Trajectories partition into four qualitative path types depending on initial (k_0, c_0, x_0). Paths may converge to regions (11) 0 < k_t < k** (k_t>0) on k̇=0 or (12) k**(k_t>0) ≤ k_t ≤ k**(k_t<0) on k̇=0, with possibilities to approach Modified Golden Rule and mitigate dynamic inefficiency by avoiding overaccumulation. Numerical evaluation - Production: Cobb–Douglas f(k) = k^α with α = 0.5 (Eq. 13). Parameters: n = 0.01, ρ = 0.01, initial k_0 = 100 (mature, low-growth economy). Golden Rule: k* = 2500, c* = 25; Modified Golden Rule: k## = 625, c## = 18.75. - Preferences: CES utility u(c) = (c^{1−σ} − 1)/(1 − σ) with σ = 1/1.4 (inverse of intertemporal elasticity per Yagihashi & Katano 2020). Social welfare W = ∫₀^∞ u(c_t) e^{-ρ t} dt (Eq. 14). Optimal path computed from Keynes–Ramsey differential Eq. (15). Adaptive (PDSA) paths simulated in Mathematica 12 for varying initial c_0 in [0,10], with an initial production allocation plan parameter x_0 ≈ 0.99.

- Optimal growth vs adaptive paths: • The optimal growth model yields a unique saddle-path to the steady state at the Modified Golden Rule. Practical implementation is unstable and requires perfect foresight. • Budget-controlled (PDSA/replicator) decision-making generates a diversity of stable trajectories around the optimal path, enabling sequential plan revision without perfect foresight. - Phase-diagram insights: Multiple steady-state regions exist; four qualitative adaptive path types emerge. Some paths mitigate dynamic inefficiency by avoiding overaccumulation and can approach the Modified Golden Rule region. - Numerical welfare comparisons (k_0 = 100; α = 0.5; n = ρ = 0.01; σ = 1/1.4): • Optimal benchmark: W_opt ≈ 357.615; long-run k_{1000} ≈ 625.006; c_{1000} ≈ 18.7499. • Adaptive PDSA paths vary with c_0: - For c_0 ∈ [0, 2.376]: paths converge to c ≈ 0 (types ②–1); welfare ratios W_PDSA/W_opt ≈ 0.014–0.256. - For c_0 ∈ [2.377, 2.4102]: transient overaccumulation (type ①); welfare ratios ≈ 0.848–0.971. - For c_0 ∈ [2.4103, 9]: uniform convergence to region around Modified Golden Rule (type ③), approaching optimal levels; best matches near c_0 ≈ 3.46–3.65. With c_0 equal to the optimal path’s 3.94697, steady-state k and c are slightly lower but welfare is high: W_PDSA/W_opt ≈ 0.9988. Peak ratios reach ≈ 0.999 for c_0 ∈ [3.1, 3.9]; ≈ 0.99 for c_0 ∈ [2.5, 5.1]; and ≥ 0.9 for c_0 ∈ [2.4103, 8.2]. - For c_0 ∈ (9, 10]: stagnation/decline (type ⑧); welfare ratios ≈ 0.846–0.622. • Manageability metric: The share of initial plans yielding welfare ≥ 0.9 of optimal is ≈ 0.58 of the feasible c_0 range [0,10]; ≈ 0.26 for ≥ 0.99; ≈ 0.08 for ≈ 0.999. - Administrative stability: The redundancy of many near-optimal paths enhances operational robustness relative to the saddle-path fragility of the optimal control solution.

The research asks whether adaptive, budget-controlled decision-making can rival or supplant rational, optimal planning for intertemporal consumption/saving. The findings show that while the rational benchmark ensures uniqueness and optimality, it hinges on perfect foresight and saddle-path stability, making it operationally fragile. In contrast, the adaptive PDSA/replicator approach, by iteratively updating plans from realized outcomes, produces many feasible trajectories that gravitate toward the Modified Golden Rule region, often with welfare levels indistinguishable from optimal for a wide set of initial conditions. This addresses the research question affirmatively: sequentially rational, adaptive planning can yield sufficiently efficient outcomes without omniscience. The significance is twofold. First, it provides a realistic micro-to-macro mechanism (PDSA cycles coupled to replicator dynamics) that aligns with observed budgeting practices in firms and households, thereby improving external validity. Second, it offers practical policy and management implications: planners can implement robust, trial-and-error budgeting that mitigates dynamic inefficiency (e.g., avoiding prolonged overaccumulation) and achieves high welfare without tracking a knife-edge saddle path. The approach leverages contemporary data and computational tools, suggesting a pathway to integrate behavioral and information-rich adaptive control into macroeconomic planning.

The paper contributes a practical alternative to dynamic optimization for intertemporal planning by formalizing budget-controlled, sequentially rational decisions through PDSA and replicator dynamics, and benchmarking against the Ramsey optimal growth model. It shows theoretically and numerically that myriad adaptive paths cluster around the optimal growth path, providing administrative stability and sufficient efficiency. Social welfare under adaptive planning can exceed 0.9 of optimal for a wide range of initial plans (≈58% of the tested range), and in many cases reaches ≈0.99–0.999. Future research directions include: incorporating explicit uncertainty and expectation formation (e.g., adaptive learning under stochastic environments, real options); extending to endogenous technological change; integrating mutation/recombination mechanisms to capture innovation dynamics; and applying the framework to policy domains and heterogeneous-agent settings using richer data and machine learning to enhance the Plan and Study stages.

- Domain dependence of PDSA: While widely adopted in business and budgeting, strict PDSA applications can face methodological and practical challenges in healthcare and other public-service domains (e.g., low adherence to iterative cycles, heavy resource needs, unclear links to outcomes). - Expectations and uncertainty: The replicator dynamics model here is largely deterministic and does not explicitly model stochastic shocks or forward-looking expectations, limiting its treatment of risk and time inconsistency relative to, e.g., Merton-type portfolio problems or recursive preferences. - Technology and innovation: The baseline analysis abstracts from technological progress; without mutation/recombination mechanisms, replicator dynamics capture selection but not the generative process of innovation. Endogenous growth extensions and innovation dynamics (with mutation/recombination) are needed for long-run realism. - Dynamic inefficiency: Without a transversality-like criterion, some adaptive paths can exhibit overaccumulation and dynamic inefficiency, though certain paths mitigate this via feedback and plan revision.