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…uction function Add a lecture on Houthakker (1955), "The Pareto Distribution and the Cobb-Douglas Production Function in Activity Analysis", with Levhari (1968) as an extension. The lecture proves Houthakker's efficiency theorem, derives his Cobb-Douglas function from a Pareto distribution of input coefficients and explains which property of that distribution delivers it, and computes the aggregate production function of finitely many cells as a linear program whose dual is a minimization over prices. It shows substitution through entry and exit of cells, gives a general inversion formula for Levhari's inverse problem, and corrects Levhari's description of the CES distribution: his text and Figure 1 swap the cases sigma < 1/2 and 1/2 < sigma < 1, and his density formula (8) is off by a factor of beta^(-2/rho). Exercises cover distributions that scale jointly but not separately, an identity linking the capacity distribution to labor's share and the elasticity of substitution, and the Pareto tails of Levhari's distributions. Place the lecture after von_neumann_model in the Linear Programming part, add short return links in von_neumann_model and wealth_dynamics, and add eight bibliography entries. Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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🟡 Changes recommended
There are unresolved moderate issues: the optimization code does not guard against non-convergence (risking silently incorrect results) and the figure construction triggers a very large number of linear programs on a mesh, making execution time potentially unreliable in builds.
2 open issues
What changed in this PR
Adds a new Jupyter Book lecture on Houthakker’s (1955) aggregation result (with Levhari (1968) extensions), integrating theory, proofs, computational illustrations, and exercises into the Linear Programming sequence.
Changes:
- Add new lecture
houthakker_aggregationcovering aggregation via activity analysis, LP duality, and Levhari’s inverse problem (with numerical examples and exercises) - Insert the lecture into the book TOC after
von_neumann_modeland add short cross-links from related lectures - Add bibliography entries needed to support the new lecture’s citations
| File | Description |
|---|---|
| lectures/wealth_dynamics.md | Adds a short navigational link connecting Lorenz/Pareto material to the new Houthakker aggregation lecture. |
| lectures/von_neumann_model.md | Adds a forward/reference link pointing readers to the new Houthakker aggregation lecture as a related application. |
| lectures/houthakker_aggregation.md | Introduces a full new lecture with derivations, LP-duality formulation, numerical illustrations, figures, and exercises on Houthakker aggregation and Levhari’s extension. |
| lectures/_toc.yml | Updates the Jupyter Book table of contents to place the new lecture after von_neumann_model in the Linear Programming section. |
| lectures/_static/quant-econ.bib | Adds new bibliography entries cited by the newly added lecture. |
Review effort: Balanced
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…ragraph Apply the QuantEcon style guide to the new lecture. Printed outputs are unchanged. - Shorten the six figure captions to at most six words (qe-fig-004) and write p_1, F_N and phi as math so they render; move the tails figure's "dashed lines" note into the text. - Turn the "three features" and "four things" lists, whose items held two to four sentences, into one-sentence paragraphs (qe-writing-001); this also drops the bold run-in heads, which were not definitions (qe-writing-005). - Italicize "separately", which is emphasis, not a definition. - Split the long overview bullet and the Johansen, Hildenbrand and Jones sentences, and unwrap the two-sentence parenthetical on Levhari's notation. - Add lw=2 to the share and elasticity plots (qe-fig-008). - Note that the 100-cell isoquants lie inside the Cobb-Douglas ones by luck: across samples of 100 cells the gap has a standard deviation of about 9 percent (400 samples, three price vectors). Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>
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thanks @thomassargent30 |
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✅ Translation sync completed (zh-cn)Target repo: QuantEcon/lecture-python.zh-cn
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Add a lecture on Houthakker (1955), "The Pareto Distribution and the Cobb-Douglas Production Function in Activity Analysis", with Levhari (1968) as an extension.
The lecture proves Houthakker's efficiency theorem, derives his Cobb-Douglas function from a Pareto distribution of input coefficients and explains which property of that distribution delivers it, and computes the aggregate production function of finitely many cells as a linear program whose dual is a minimization over prices. It shows substitution through entry and exit of cells, gives a general inversion formula for Levhari's inverse problem, and corrects Levhari's description of the CES distribution: his text and Figure 1 swap the cases sigma < 1/2 and 1/2 < sigma < 1, and his density formula (8) is off by a factor of beta^(-2/rho). Exercises cover distributions that scale jointly but not separately, an identity linking the capacity distribution to labor's share and the elasticity of substitution, and the Pareto tails of Levhari's distributions.
Place the lecture after von_neumann_model in the Linear Programming part, add short return links in von_neumann_model and wealth_dynamics, and add eight bibliography entries.