Pith. sign in

REVIEW 2 major objections 4 minor 58 references

The invisible hand as an emergent property: a gradient flow approach

T0 review · 2 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims that uncoordinated, myopic profit-seeking firms collectively follow a path that maximizes aggregate consumption in the long run.

desk verdict The baseline gradient-flow representation is a genuine contribution; the headline externality-efficiency claim is asserted, not proven, so treat the paper as promising but needing revision. read the letter →

arxiv 2508.14498 v1 pith:J7MWVTSE submitted 2025-08-20 econ.TH

Giorgio Fabbri , Davide Fiaschi , Cristiano Ricci This is my paper · ORCID
classification econ.TH MSC 35K5549Q2291B5091B62
keywords gradientflowWassersteinspaceemergentpropertiesmyopicfirmssectoralreallocationcompetitiveequilibriuminvisiblehandlabourimmobility
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to show that Adam Smith's invisible hand is a precise emergent property rather than a metaphor. Firms move between sectors only to chase currently higher profit rates and pay quadratic costs for doing so; the economy as a whole then behaves as if it were solving a sequence of global optimization problems. The central result is that this uncoordinated, myopic reallocation makes a functional of aggregate consumption rise monotonically over time, and the economy converges to the unique long-run competitive equilibrium that maximizes aggregate consumption and equalizes profit rates across sectors. The proof works by recasting the partial differential equation governing the distribution of firms across a circle of goods as a gradient flow in Wasserstein space, so that the aggregate law of motion is the steepest-ascent path of an explicitly identified potential. The authors also show that fixed reallocation costs can break efficiency and uniqueness, that positive intrasectoral externalities can still leave the decentralized outcome efficient, and that immobile labour causes a measurable consumption loss. An empirical application to EU firms finds convergence of sectoral profit rates but not labour productivity, consistent with capital reallocating faster than labour.

What carries the argument

The central object is the Wasserstein-2 gradient flow on the space of firm distributions over a circle of goods, realized through the Jordan-Kinderlehrer-Otto (JKO) scheme. The space's distance is the minimal quadratic reallocation cost between two distributions, which matches the firms' adjustment cost structure. The potential F(μ), explicitly identified for the model's profit function, is log aggregate consumption, and the PDE of firm reallocation is the steepest-ascent dynamics of F in this Wasserstein geometry.

What would settle it

Use firm-level panel data on sector switches to estimate the cost of moving from sector i to sector j as a function of economic distance between sectors. If the estimated cost is not approximately quadratic in distance, or if a fixed lumpy component dominates, the PDE is not the paper's gradient flow and the predicted monotone rise of aggregate consumption can fail. Alternatively, compute the paper's potential F from sector-level consumption data and test whether observed changes in sectoral profit rates follow its Wasserstein gradient; a rejection would falsify the central mechanism.

Watch

Extended reading notes

Core claim

The paper's central claim is that an economy in which firms move across sectors according to a myopic, profit-seeking rule with quadratic reallocation costs is exactly an optimizing system at the aggregate level. The density of firms over the circle of goods solves a continuity equation whose velocity is the local profit-rate gradient; Theorem 2.3 identifies this PDE as the Wasserstein-2 gradient flow of a potential F(μ) that is the natural logarithm of aggregate consumption. Consequently, along any trajectory F(μ) is nondecreasing (Corollary 2.4), and the firm distribution converges to the unique steady state that maximizes aggregate consumption and equalizes profit rates (Theorems 2.5 and

Load-bearing premise

Firms' reallocation costs are exactly quadratic in distance and firms use today's profit rate as a complete guide to where to move; if either premise fails, the economy's path is no longer a Wasserstein-2 gradient flow and the monotone-consumption and efficiency theorems need not follow.

Editorial extensions

If this is right

  • If the central claim is correct, the sequence of short-run competitive equilibria is not arbitrary: it has a variational structure, with aggregate consumption serving as a Lyapunov function.
  • The baseline economy is globally stable and Pareto efficient in the long run, so disequilibrium dynamics self-organize to a welfare maximum without coordination or rational expectations.
  • With fixed reallocation costs, multiple long-run equilibria and persistent profit-rate dispersion can occur; the empirical evidence suggests such costs are small at the sectoral level.
  • With immobile labour, the consumption loss relative to the mobile-labour benchmark is minimized when labour is allocated as in the mobile economy, tying labour-mobility frictions directly to efficiency.
  • The empirical finding of profit-rate convergence alongside labour-productivity divergence is exactly what the model predicts when capital reallocates faster than labour.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If reallocation costs scale with distance to a power other than two, the aggregate flow would live in a different Wasserstein space and the monotone quantity would change; the paper's proof does not cover this, but the JKO machinery suggests a testable family of models indexed by the cost exponent.
  • The result that myopic profit chasing remains efficient with positive externalities hints at a broader principle: whenever externalities are fully capitalized into current sectoral profit rates, decentralized reallocation can internalize them, and the paper's explicit example is one instance of that logic.
  • The paper leaves implicit a direct micro-level test of its key premise: reallocation flows should be proportional to the local profit-rate gradient, so firm-level data on sector switchers could be used to estimate the curvature of adjustment costs and thereby test the quadratic-cost assumption.
  • A natural extension, flagged by the authors, is joint capital and labour mobility with complementarities; that extension may produce agglomeration rather than equalization, reversing the paper's long-run efficiency conclusion.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper develops a general-equilibrium model with a continuum of firms distributed over a circle of goods. At each instant a short-run competitive equilibrium determines sectoral profit rates; firms then reallocate myopically under quadratic adjustment costs, generating a PDE for the evolution of the firm distribution (Section 2.2, Eq. (21)). The main theoretical claim is that this PDE is a Wasserstein-2 gradient flow of a functional F, so that the decentralized dynamic can be viewed as a sequence of global optimization problems (Theorem 2.3 and JKO scheme), that aggregate consumption increases monotonically (Corollary 2.4), and that the unique long-run equilibrium maximizes aggregate consumption and equalizes profit rates (Theorems 2.5 and 2.6). The paper then extends the model to non-symmetric preferences, intrasectoral externalities, fixed reallocation costs, and immobile labour, and reports empirical estimates using ORBIS data on EU firms.

Significance. If the central claims hold, the paper is a substantial contribution: it gives a rigorous optimal-transport formulation of Sonnenschein–Artzner–Simon disequilibrium dynamics, provides a formal 'invisible hand' result for myopic profit-seeking with reallocation costs, and extends the representation to several economically meaningful frictions. The baseline theorems are nontrivial and appear internally consistent: the proof strategy via Iacobelli et al.'s weighted ultrafast diffusion class and the concavity argument for Theorem 2.6 are credible. The paper also ships reproducible code and data on Zenodo, and the empirical application to 680 EU sectors is a useful first pass. However, the novelty emphasized in the abstract and in Section 2.5.1—efficiency with positive externalities—is asserted rather than proved, and is thus load-bearing for the paper's claimed contribution.

major comments (2)
  1. [§2.5.1, after Eq. (36)] The abstract's 'surprising result'—that the decentralized equilibrium remains efficient with positive (or mildly negative) externalities—is asserted with 'it can be shown' but no proof is supplied. Appendix B.1 gives the short-run equilibrium with externalities (Prop. B.1) and the fixed-cost example, but nowhere shows that the functional F in (34) is a monotone transform of aggregate consumption C in the externality case, nor that the stationary point (35) is the global maximizer of F or C. Condition (33) only ensures well-posedness. Since this claim is a headline contribution, please add a complete proof or explicitly weaken the claim.
  2. [§2.5.1, before Eq. (35)] The statements 'the rest of Theorem 2.3 still holds' and 'we can extend Theorem 2.5' to externalities are not backed by an argument in the appendix. The gradient-flow representation of Eq. (32), the JKO scheme, and global convergence to (35) require at least a proof sketch showing how condition (33) maps the equation into the Iacobelli et al. framework. Without this, the externality extension of the main theorems is only conjectural; this matters because the paper's abstract advertises the externality-efficiency result as one of its key findings.
minor comments (4)
  1. [§4.1.1, Eq. (54)] The inference that fixed costs are insignificant, based on the smoothness of the Nadaraya-Watson estimate in Figure 2, is not identified. A fixed cost creates an inaction region, but aggregation over five years and over 680 sectors could easily smooth this into a linear convergence pattern. Please either provide a formal test based on the implied plateau, or soften the abstract's claim of 'evidence suggesting the absence of significant fixed costs'.
  2. [§2.5.2, Eq. (39)] The fixed-cost equilibrium example uses a density that is unbounded at a point (it 'diverges at x=0'). The existence and uniqueness theorems in the paper assume bounded, strictly positive densities. Please clarify in what sense (39) is an admissible equilibrium, e.g., as a weak solution or after mollification.
  3. [Throughout] Several passages appear garbled, e.g., 'apart from' repeatedly appears where 'distribution' is presumably meant (Section 4.1.3), and the author affiliation line reads 'Giorgio F abbri'. Please proofread the final PDF carefully.
  4. [§2.4, after Eq. (28)] The statement that 'the speed of convergence appears to be decreasing in ρ' is vague and not derived from Eq. (28). Either give the precise parametric dependence or remove this remark.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the gradient-flow representation and the efficiency theorem are derived from the model's primitives rather than assumed.

full rationale

Walking the derivation chain: the baseline model first solves the short-run competitive equilibrium (Prop. 2.1), derives the profit function (10), and inserts the myopic reallocation rule into the continuity equation (19)-(21). Theorem 2.3 uses external PDE results (Iacobelli et al. 2019) to represent that PDE as the Wasserstein gradient flow of the functional F in (24), with the JKO scheme (25)-(26). This is a representation theorem, not an assumption: F is recovered from the PDE's velocity field, and the identification of F with log aggregate consumption follows from the equilibrium price/profit formulas and (14)-(15); the monotonicity of aggregate consumption is then a standard gradient-flow consequence. Theorem 2.6 proves efficiency directly by a Hölder/Jensen inequality showing that the steady state (27) maximizes aggregate consumption; it does not use the gradient-flow functional in an inferential loop. In the externality extension, the claim 'it can be shown that μ* maximizes F ... first best efficiency holds' (Section 2.5.1) is asserted without an explicit proof in the displayed text, and the identification of F with aggregate consumption in the externality case is not fully demonstrated; however, this is an omitted-support/proof gap, not a circular reduction — the steady state is not defined as the maximizer of F, and Condition (33) is presented as a well-posedness restriction rather than as the efficiency conclusion. The empirical section calibrates parameters from the data and then uses those values in simulations; no fitted quantity is renamed as an out-of-sample prediction. There are no load-bearing self-citations. Hence no step reduces to its own inputs, and the appropriate score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central results rest on a mixture of standard optimal-transport mathematics (Wasserstein gradient flows, JKO scheme), specific economic functional forms (CES, Cobb-Douglas, quadratic adjustment costs), a strong behavioral assumption (myopia), and a technical positivity condition on initial conditions. The empirical parameters are estimated in-sample.

free parameters (3)
  • beta (labor share) = around 0.76-0.86, adjusted to around 0.24 in net terms with depreciation and intermediates (Section 4.1.1)
    Estimated from aggregate profit rate and labor share data; used to compute the equilibrium and calibrate the model.
  • epsilon (elasticity of substitution) = approximately 1.27 (Eq. 59)
    Recovered from regression (58) on the growth of value of production; governs substitutability among goods.
  • eta (intrasectoral externality) = approximately 0.04 (Eq. 60)
    Recovered from regression (58); measures how productivity responds to sector firm mass.
assumptions (6)
  • domain assumption Strictly positive initial firm distribution in H^2(S) (Theorem 2.2)
    Existence, uniqueness, and gradient flow representation are proven only for strictly positive initial densities, excluding economies with initially empty sectors.
  • domain assumption Myopic expectations: current profits are a perfect proxy for future relocation payoffs (Section 2.2, Problem (18))
    This makes the dynamic path a sequence of static optimizations; the efficiency result depends on it.
  • domain assumption Quadratic reallocation cost (Section 2.2, Eq. (17))
    The quadratic cost maps to the Wasserstein-2 distance and the JKO scheme; non-quadratic costs break the representation.
  • domain assumption CES preferences with elasticity epsilon and Cobb-Douglas production with labor share beta (Eqs. (1), (4))
    Functional forms used for closed-form equilibrium and explicit steady state.
  • standard math The validity of Iacobelli et al. (2019) theorems for the time-changed PDE (proof of Theorem 2.2)
    The paper relies on these theorems for existence and gradient flow identification but does not show the explicit variable transformation.
  • ad hoc to paper The space of goods is a circle (Eq. (16))
    Chosen to avoid boundary effects in the PDE; standard in this literature but not an economic primitive.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The invisible hand as an emergent property: a gradient flow approach." pith.science (2026). https://pith.science/paper/J7MWVTSE

@misc{pith2026250814498,
  author       = {Pith},
  title        = {Pith review of: The invisible hand as an emergent property: a gradient flow approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J7MWVTSE}},
  note         = {Machine review of arXiv:2508.14498}
}
read the original abstract

We develop a general equilibrium model in which, at each instant, a short-run competitive equilibrium arises. Heterogeneity in factor allocation generates differential profit rates across sectors, prompting firms to move between them under myopic profit-seeking behaviour, subject to quadratic reallocation costs. The aggregate dynamics of the economy can be formalised as a gradient flow in a Wasserstein space, starting from a partial differential equation that describes the reallocation of firms across sectors. Two key emergent properties arise: (i) decentralised and uncoordinated decisions can be reinterpreted as the solution to a sequence of global optimisation problems, involving a function of aggregate consumption, which increases monotonically along the dynamic path; (ii) the long-run competitive equilibrium is efficient, as the distribution of firms maximises aggregate consumption and profit rates are equalised across sectors. We extend the baseline model to incorporate non-symmetric preferences, intrasectoral externalities, a fixed cost of reallocation, and labour immobility. These extensions reveal conditions under which the efficiency and uniqueness of the long-run equilibrium may fail, but also highlight the surprising result that the decentralised equilibrium can remain efficient even in the presence of externalities. Finally, using a large sample of EU firms from the period 2018-2023, we empirically document convergence in sectoral profit rates, but not in labour productivity, pointing to a certain degree of labour immobility. We also find evidence suggesting the absence of significant fixed costs of reallocation at the sectoral level, the presence of positive but limited intrasectoral externalities, and a moderate degree of substitutability among goods.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

58 extracted references · 57 canonical work pages

  1. [1]

    and Arkolakis, C

    Allen, T. and Arkolakis, C. (2014). Trade and the topography of the spatial economy. The Quarterly Journal of Economics , 129(3):1085--1140

    work page 2014

  2. [2]

    Ambrosio, L., Bru \'e , E., and Semola, D. (2021). Lectures on Optimal Transport . Springer, Cham

    work page 2021

  3. [3]

    Ambrosio, L., Gigli, N., and Savar \'e , G. (2005). Gradient flows: in metric spaces and in the space of probability measures . Springer Science & Business Media

    work page 2005

  4. [4]

    Arrow, K. J. (1986). Rationality of self and others in an economic system. Journal of business , pages S385--S399

    work page 1986

  5. [5]

    J., Block, H

    Arrow, K. J., Block, H. D., and Hurwicz, L. (1959). On the stability of the competitive equilibrium, ii. Econometrica: Journal of the Econometric Society , pages 82--109

    work page 1959

  6. [6]

    P., and Sonnenschein, H

    Artzner, P., Simon, C. P., and Sonnenschein, H. (1986). Convergence of myopic firms to long-run equilibrium via the method of characteristics. In Models of Economic Dynamics: Proceedings of a Workshop held at the IMA, University of Minnesota, Minneapolis, USA, October 24--28, 1983 , pages 157--183. Springer

    work page 1986

  7. [7]

    and Etro, F

    Bertoletti, P. and Etro, F. (2017). Monopolistic competition when income matters. The Economic Journal , 127(603):1217--1243

    work page 2017

  8. [8]

    Blanchard, O. (1997). The medium run. Brookings Papers on Economic Activity , 28(2):89--158

    work page 1997

Show all 58 references
  1. [9]

    Blanchard, O. (2025). Convergence? thoughts about the evolution of macroeconomics over the last 40 years

    2025

  2. [10]

    A., Esposito, A., Skrzeczkowski, J., and Wu, J

    Carrillo, J. A., Esposito, A., Skrzeczkowski, J., and Wu, J. S.-H. (2024). Nonlocal particle approximation for linear and fast diffusion equations. arXiv preprint. arXiv:2408.02345

    2024 arXiv

  3. [11]

    and Ji, Y

    De Grauwe, P. and Ji, Y. (2019). Behavioural macroeconomics: Theory and policy . Oxford University Press

    2019

  4. [12]

    Dixit, A. K. and Stiglitz, J. E. (1977). Monopolistic competition and optimum product diversity. The American economic review , 67(3):297--308

    1977

  5. [13]

    and Roventini, A

    Dosi, G. and Roventini, A. (2025). Agent-based Macroeconomics: The Schumpeter Meeting Keynes Models . Elements in Evolutionary Economics. Cambridge University Press

    2025

  6. [14]

    Durlauf, W. B. A. S. N. (1997). The Economy As A Complex Evolving System II . Santa Fe Institute Studies in the Sciences of Complexity Lecture Notes. Westview Press

    1997

  7. [15]

    Edgeworth, F. Y. (1881). Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences . C. Kegan Paul & Co., London

  8. [16]

    Efron, B. (2011). Tweedie’s formula and selection bias. Journal of the American Statistical Association , 106(494):1608--1618

    2011

  9. [17]

    Eisfeldt, A. L. and Rampini, A. A. (2006). Capital reallocation and liquidity. Journal of monetary Economics , 53(3):369--399

    2006

  10. [18]

    Foster, L., Haltiwanger, J., and Syverson, C. (2008). Reallocation, firm turnover, and efficiency: Selection on productivity or profitability? American Economic Review , 98(1):394--425

    2008

  11. [19]

    and Thisse, J.-F

    Fujita, M. and Thisse, J.-F. (1996). Economics of agglomeration. Journal of the Japanese and international economies , 10(4):339--378

    1996

  12. [20]

    and Thisse, J.-F

    Fujita, M. and Thisse, J.-F. (2002). Economics of Agglomeration: Cities, Industrial Location, and Regional Growth . Cambridge University Press

    2002

  13. [21]

    Griliches, Z. (1992). The search for r&d spillovers. Scandinavian Journal of Economics , 94(Supplement):S29--S47

    1992

  14. [22]

    Grossman, G. M. and Helpman, E. (1991). Innovation and Growth in the Global Economy . MIT Press, Cambridge, MA

    1991

  15. [23]

    Gupta, S., Kangur, A., Papageorgiou, C., and Wane, A. (2014). Efficiency-adjusted public capital and growth. World development , 57:164--178

    2014

  16. [24]

    Hadamard, J. (1923). Lectures on Cauchy's Problem in Linear Partial Differential Equations . Courier Corporation

    1923

  17. [25]

    Hayek, F. A. (1945). The use of knowledge in society. American Economic Review , 35(4):519--530

    1945

  18. [26]

    Horiguchi, S. A. and Kobayashi, T. J. (2023). Cellular gradient flow structure linking single-cell-level rules and population-level dynamics. Physical Review Research , 5(2):L022052

    2023

  19. [27]

    Howitt, P. (2006). Coordination issues in long-run growth. Handbook of Computational Economics , 2:1605--1624

    2006

  20. [28]

    L., and Volzone, B

    Huang, Y., Mainini, E., V \'a zquez, J. L., and Volzone, B. (2024). Nonlinear aggregation-diffusion equations with riesz potentials. Journal of Functional Analysis , 287(2):110465

    2024

  21. [29]

    S., and Santambrogio, F

    Iacobelli, M., Patacchini, F. S., and Santambrogio, F. (2019). Weighted ultrafast diffusion equations: from well-posedness to long-time behaviour. Archive for Rational Mechanics and Analysis , 232:1165--1206

    2019

  22. [30]

    Jordan, R., Kinderlehrer, D., and Otto, F. (1998). The variational formulation of the fokker–planck equation. SIAM Journal on Mathematical Analysis , 29(1):1--17

    1998

  23. [31]

    and Rob, R

    Jovanovic, B. and Rob, R. (1989). The growth and diffusion of knowledge. Review of Economic Studies , 56(4):569--582

    1989

  24. [32]

    Kirman, A. (2016). Ants and nonoptimal self-organization: Lessons for macroeconomics. Macroeconomic Dynamics , 20(2):601--621

    2016

  25. [33]

    Krugman, P. (1991). Increasing returns and economic geography. Journal of political economy , 99(3):483--499

    1991

  26. [34]

    Krugman, P. (1993). On the number and location of cities. European economic review , 37(2-3):293--298

    1993

  27. [35]

    Krugman, P. (1994). Complex landscapes in economic geography. American Economic Review , 84(2):412--416

    1994

  28. [36]

    and Wolpin, K

    Lee, D. and Wolpin, K. I. (2006). Intersectoral labor mobility and the growth of the service sector. Econometrica , 74(1):1--46

    2006

  29. [37]

    and Magenes, E

    Lions, J.-L. and Magenes, E. (1972). Non-homogeneous boundary value problems and applications , volume 2. Springer, Berlin

    1972

  30. [38]

    Lunardi, A. (1995). Analytic semigroups and optimal regularity in parabolic problems . Birkhauser

    1995

  31. [39]

    and Phillips, G

    Maksimovic, V. and Phillips, G. (2001). The market for corporate assets: Who engages in mergers and asset sales and are there efficiency gains? The Journal of Finance , 56(6):2019--2065

    2001

  32. [40]

    Mas-Colell, A. (1986). Notes on price and quantity t \^a tonnement dynamics. In Models of Economic Dynamics: Proceedings of a Workshop held at the IMA, University of Minnesota, Minneapolis, USA, October 24--28, 1983 , pages 49--68. Springer

    1986

  33. [41]

    D., Green, J

    Mas-Colell, A., Whinston, M. D., Green, J. R., et al. (1995). Microeconomic theory , volume 1. Oxford university press New York

    1995

  34. [42]

    Melitz, M. J. (2003). The impact of trade on intra-industry reallocations and aggregate industry productivity. econometrica , 71(6):1695--1725

    2003

  35. [43]

    Moll, B. (2024). The trouble with rational expectations in heterogeneous agent models: A challenge for macroeconomics

    2024

  36. [44]

    Nelson, R. R. and Winter, S. G. (1985). An evolutionary theory of economic change . harvard university press

    1985

  37. [45]

    Osotimehin, S. (2019). Aggregate productivity and the allocation of resources over the business cycle. Review of Economic Dynamics , 32:180--205

    2019

  38. [46]

    Parenti, M., Ushchev, P., and Thisse, J.-F. (2017). Toward a theory of monopolistic competition. Journal of Economic theory , 167:86--115

    2017

  39. [47]

    Pavliotis, G. A. (2014). Stochastic Processes and Applications , volume 60 of Texts in Applied Mathematics . Springer

    2014

  40. [48]

    Romer, P. M. (1986). Increasing returns and long-run growth. Journal of Political Economy , 94(5):1002--1037

    1986

  41. [49]

    Romer, P. M. (1990). Endogenous technological change. Journal of Political Economy , 98(5):71--102

    1990

  42. [50]

    Santambrogio, F. (2015). Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , volume 87 of Progress in Nonlinear Differential Equations and Their Applications . Springer, Cham

    2015

  43. [51]

    Santambrogio, F. (2017). Euclidean, metric, and Wasserstein gradient flows: an overview . Bulletin of Mathematical Sciences , 7(2):157--227

    2017

  44. [52]

    Simon, H. A. (1997). An empirically based microeconomics . Cambridge University Press

    1997

  45. [53]

    Solow, R. M. (2000). Toward a macroeconomics of the medium run. Journal of Economic perspectives , 14(1):151--158

    2000

  46. [54]

    Sonnenschein, H. (1972). Market excess demand functions. Econometrica , 40(3):549--563

    1972

  47. [55]

    Sonnenschein, H. (1981). Price dynamics and the disappearance of short-run profits: An example. Journal of Mathematical Economics , 8(2):201--204

    1981

  48. [56]

    Sonnenschein, H. (1982). Price dynamics based on the adjustment of firms. The American Economic Review , 72(5):1088--1096

    1982

  49. [57]

    Stiglitz, J. E. (2018). Where modern macroeconomics went wrong. Oxford Review of Economic Policy , 34(1-2):70--106

    2018

  50. [58]

    Walras, L. (1874). \'E l \'e ments d' \'e conomie politique pure: ou, Th \'e orie de la richesse sociale . L. Corbaz, Lausanne

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.