Chapter 2 — Theories of Capital

Part I · The Aggregation Problem · Week 1

K→𝒢→Q→Ξ

An intellectual history and collision map: four traditions of capital theory, their formal commitments, and the eight requirements R1–R8 an adequate theory must meet. The chapter’s computational heart is the reswitching calculation — the Cambridge controversy’s proof that no scalar quantity of capital orders production techniques independently of prices.

Learning objectives

  • LOS 2.1 — Map the four traditions (classical, Austrian, neoclassical aggregate, Cambridge) onto their formal commitments, reach, and residual.
  • LOS 2.2 — Explain reswitching, reproduce the two-technique calculation, and derive its consequence for scalar capital.
  • LOS 2.6 — State the eight requirements R1–R8 and trace each to the tradition whose failure motivates it.

The laboratory module

Module 2 — The Aggregation Problem. Two instruments: a reswitching explorer (sliders for the dated-labor inputs, live plot of \(c_A(r)\) and \(c_B(r)\) with crossings marked) and a requirements navigator (the collision map rendered interactively).

The book’s reswitching example. One unit of output at date 0, two techniques, wage 1, with \(x = 1+r\):

\[c_A(r) = 7x^2, \qquad c_B(r) = 2x^3 + 6x.\]

They exchange places where \(c_A = c_B\); dividing by \(x\), \(\;2x^2 - 7x + 6 = 0 \Rightarrow x \in \{\tfrac{3}{2}, 2\}\), so the switches occur at \(r = 50\%\) and \(r = 100\%\).

\(r\) \(c_A\) \(c_B\) cost-minimizing
20% 10.08 10.656 A
75% 21.4375 21.219 B
150% 43.75 46.25 A

The choice of technique is not monotone in the interest rate — A wins low, B wins between the switch points, A returns high.

Guided experiments

  1. Reproduce the switch points at \(r = 0.5\) and \(r = 1.0\) from the default inputs.
  2. Find a no-reswitching configuration and state, in one sentence, what property of the cost polynomials changed.
  3. Requirements navigator — for two requirements of your choice, follow the chain → necessity via → formalized in and record one sentence per link.

📓 Notebook · 📊 Workbook · 🎞 Slides (.pptx) · Open the Laboratory →

ch02 · 8/8 PASS

Exercises

A · Concept checks

  • 2.1 For each tradition, state its one-sentence commitment and the requirement its failure motivates. Hint: the collision map’s rows are the requirements; the columns are the traditions.

B · Computations

  • 2.6 Prove reswitching in three lines from \((2.2)\): show \(c_B - c_A\) has two sign changes on \(r > 0\). Hint: the switch equation \(2x^2 - 7x + 6 = 0\) factors as \(2(x - \tfrac32)(x-2)\).

Statements and hints surfaced here; full solutions in the Instructor’s Manual.