The structure and existence of solutions of the problem of consumption with satiation in continuous time
Bibliographic record
Abstract
With the help of the method of Lagrange multipliers and KKT theory, we investigate the structure and existence of optimal solutions of the continuous-time model of consumption with satiation.We show that the differential equations have no solutions in the C 1 class but that solutions exist in a wider space of functions, namely, the space of functions of bounded variation with non-negative Borel measures as controls.We prove our theorems with no additional assumptions about the structure of the control Borel measures.We prove the conjecture made in the earlier literature, that there are only three types of solutions: I-shaped solutions, with a gulp of consumption at the end of the interval and no consumption at the beginning or in the interior; U-shaped solutions, with consumption in the entire interior of the interval and gulps at the beginning and the end; and intermediate (J-shaped) solutions, with an initial interval of abstinence followed by a terminal interval of distributed consumption at rates and a gulp at the end.We also establish the criteria that permit determination of the solution type using the problem's parameters.When the solution structure is known, we reduce the problem of the existence of a solution to algebraic equations and discuss the solvability of these equations.We construct explicit solutions for logarithmic utility and CRRA utility. IntroductionRecently, Baucells and Sarin [1, 2] described a new and interesting discrete-time model of consumer behavior: the satiation model.The psychological justifications of the model are explained in plain language in [3].In [4,5] this model was extended to continuous time, and the solutions were constructed for CRRA utilities.The satiation model presented in [4, 5] is mathematically described as an optimal control problem (Problem 1) in this paper.In this problem the control (consumption c(t)) enters the evolution equation ( Eq (2)) in this paper linearly and may be unbounded, hence the classical Pontryagin Maximum Principle (see [6]) is not directly applicable.For problems of this type,
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.003 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".