Oskar Morgenstern and Gerald L. Thompson · 1976
Morgenstern and Thompson’s study presents economic growth and contraction as problems of formal structure, optimization, and computability. Rather than treating “the economy” as a descriptive historical object, the work abstracts it into axiomatized systems whose productive possibilities, exchange relations, and temporal transformations can be compared mathematically. Its central ambition is to define when an economy can expand or contract, how optimal paths are characterized, and how such paths can be computed.
The text is alert to the danger that formal axioms may admit trivial or degenerate cases. Its theory therefore does not merely state assumptions; it tests their scope by identifying economies that technically satisfy them while lacking economic interest.
Observe that the following uninteresting economies satisfy Axioms (C1) through (C4) and Assumption (AO).
This kind of passage shows the authors’ methodological care: the axioms are not treated as self-validating. They must be refined, interpreted, or supplemented so that the class of admissible economies is mathematically tractable without becoming economically empty. Expansion and contraction are thus framed as properties of structured systems, not as loose metaphors for prosperity or decline.
A major part of the work translates the theory into optimization problems. The appearance of “central optimal strategies” suggests that the authors are concerned with strategies that remain optimal across related parameter values or model variants. The proof language is compact and algebraic, emphasizing equivalence of optimality under specified conditions.
Proof: Let $x^0$ be a central optimal strategy for $\alpha_i$ and let $y^0$ be a central optimal strategy for $\alpha_j$. Then Lemma 2-2 shows that $x^0$ is optimal for $\alpha_j$ and $y^0$ is optimal for $\alpha_i$.
The argument is characteristic of a theory seeking invariance: if strategies optimal for different indices can be exchanged while preserving optimality, then the structure of the model matters more than the accidental labeling of cases. The mathematical problem is to identify the conditions under which such transfer of optimality is legitimate.
The book also gives substantial attention to computational form. Expansion and contraction are not only defined axiomatically; they are rendered into problems that can be solved by tableau methods. The authors value reductions in dimensionality because they make the theory operational.
The initial condensed tableau for the problems (11) and (12) is shown in Figure 3-4. Note that it is $(m + 1) \times (n + 1)$, which is considerably smaller than the tableau of the method described in the previous section.
This emphasis on a “condensed tableau” indicates that the formal theory is designed with algorithmic economy in mind. The size of the tableau is not a mere technical detail: it marks the passage from abstract existence results to feasible computation. The work therefore belongs both to mathematical economics and to the tradition of linear programming and activity analysis.
The later examples appear to compare economies across time by following commodities and flows. Such examples make the abstract apparatus legible through concrete productive or trade quantities, while still preserving the formal comparative framework.
If we now compare the exportation of eggs in Figures 4-4 and 4-5, we see that at the first time period both economies export exactly the same number of eggs (336), even though the economy of Example 4-3 has less than half the number of chickens (28 compared to 64).
This passage suggests that the authors use numerical cases not as anecdotal illustrations but as controlled demonstrations of how two economies can coincide initially and then diverge under different structural or parametric assumptions. Expansion and contraction become observable through sequences of outputs, exports, and constraints rather than through aggregate intuition alone.
Overall, the work develops a rigorous mathematical vocabulary for economies whose scale changes over time. It combines axioms, optimality theorems, tableau algorithms, and numerical comparisons to show how growth and decline can be modeled as formal dynamic processes. Its importance lies in treating economic expansion and contraction as problems of structure and strategy: what assumptions define the system, what paths are optimal, how those paths can be computed, and how different economies can be compared when they share initial features but evolve differently.
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