Chapter 14 · Equilibrium, Liquidity, and the Allocation of Capital
Part IV · Risk, Robustness, and Equilibrium
Chapter at a glance
Modern Finance Laboratory · Module 14 · Week 14
Module 14 has four panels. The equilibrium panel lets the user set endowment growth states, beliefs, (𝛾, 𝛽), and watch (14.1) manufacture state prices, the riskless rate, and the equity premium—with a reverse mode that ingests observed prices and reports the implied (𝛾, 𝛽), preloaded with Chapter 1’s toy and with realistic consumption data (the premium puzzle as a slider experience). The Kyle panel runs the auction live: sliders for (Σ0 , 𝜎𝑢 ), simulated flows, the 𝜆 line forming through the scatter, the transfer table, and an “insider greed” override that lets the user trade more aggressively than (14.2) and watch profits fall. The spiral pa
Learning Outcome Statements
LOS 14.1 Define competitive equilibrium in a finite exchange economy and derive state prices from first-order conditions.
LOS 14.2 Give the stochastic discount factor its consumption identity, derive the consumption-based pricing formula, and quantify the equity premium puzzle.
LOS 14.3 Solve the Kyle model: derive the linear equilibrium, price impact, market depth, information revelation, and the transfer accounts.
LOS 14.4 Distinguish permanent (informational) from temporary (mechanical) price impact and connect each to its chapter of origin.
LOS 14.5 Explain funding versus market liquidity and the margin-spiral amplification mechanism.
LOS 14.6 Assemble the full Meridian agenda resolution and state what the mathematics settles and what remains judgment.
Laboratory · Module 14 (book §14.9)
Module 14: The Market Itself
Course Website · Week 14
Guided experiment (supports LOS 14.1–14.6). (i) Reverse-engineer Chapter 1’s market to (𝛾, 𝛽) = (1.41, 1.026); switch to consumption-scale volatility and report the 𝛾 the observed premium demands. (ii) Reproduce 𝜆 = 0.125 and the (+50, −50, 0) table; then use the override to verify that trading at 2𝛽 strictly reduces insider profit, and explain why in one sentence. (iii) Find the 𝑘 at which a 2% shock produces a 5% realized loss. (iv) Traverse the capstone panel and submit the one-page agenda summary in your own words—the course’s closing artifact.
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Exercises
Exercises are grouped A–E throughout the book: A concept checks, B computations, C proofs and extensions, D modeling and application, E laboratory. Starred exercises (∗) are on the advanced track.
Part A — Concept checks
Exercise 14.1. “Expensive states are hungry states.” State the equation this summarizes, and give one asset that is valuable because it pays in hungry states and one that is cheap because it pays in fed states, with the sign of each premium.
Exercise 14.2. In Kyle’s model, identify which player performs each of the book’s signature operations: conditional expectation (Chapter 3), optimal control (Chapter 10), and paying for immediacy (Chapter 14)—and explain why the market maker earns zero despite setting every price.
Exercise 14.3. Classify each cost of Meridian’s $600M transition as temporary or permanent impact, with the responsible model: the bid–offer concession for speed; the market’s inference from persistent one-sided flow; the price recovery in the week after completion.
Part B — Computations
Exercise 14.4. Reverse-engineer Chapter 1’s market in full: from 𝜓 = (0.476, 0.476), 𝑝 = (0.6, 0.4), growth (1.2, 0.9), derive 𝑚, 𝛾 = 1.41, and 𝛽 = 1.026; verify 𝑅 = 1.05 and the 3% premium; and recompute 𝛾 when the growth states are replaced by consumption-scale (1.028, 0.996) with the same premium target.
Exercise 14.5. A Lucas economy has two growth states, 1.045 and 0.985, with beliefs (0.7, 0.3), 𝛾 = 3, 𝛽 = 0.97. Compute the SDF states, the state prices, the riskless rate, then the price and expected return of the consumption claim, and finally the equity premium. √
Exercise 14.6. Verify Kyle’s accounts at Σ0 = 5, 𝜎𝑢 = 20: 𝛽 = 4, 𝜆 = 0.125, insider profit 50, posterior variance 12.5. Then double the noise and recompute all four, stating in one sentence who benefits from camouflage and why.
Exercise 14.7. With feedback 𝑘: derive the amplification 1/(1 − 𝑘) from the geometric loop; compute realized losses from a 2% shock at 𝑘 = 0.2, 0.4, 0.6; and find the 𝑘 at which Meridian’s October −6.8% month is consistent with a −4% fundamental shock. 316 14 Equilibrium, Liquidity, and the Allocation of Capital
Part C — Proofs and extensions
Exercise 14.8. Extend Theorem 14.1 to 𝐼 investors with common beliefs and CRRA utilities of common 𝛾: show that equilibrium prices are those of a representative agent with the aggregate endowment (aggregate the first-order conditions), and identify exactly where common curvature is used—the door to heterogeneous-agent theory.
Exercise 14.9. Derive the consumption CAPM in the lognormal-CRRA case: with ln 𝑚 = ln 𝛽 − 𝛾 Δ ln 𝑐 jointly normal with returns, prove E[𝑅] − 𝑅 𝑓 ≈ 𝛾 Cov(Δ ln 𝑐, 𝑅) from E[𝑚𝑅] = 1 (expand with the Gaussian MGF), recovering the wall of Figure 14.1(b).
Exercise 14.10. Complete Theorem 14.2: verify second-order conditions on both sides; show that within linear strategies the equilibrium is unique; and prove the half-revelation identity Var(𝑉 |𝑦) = Σ0 /2 holds for any (Σ0 , 𝜎𝑢 )—the insider’s rationing exactly offsets the signal quality.
Exercise 14.11. ∗ In Kyle’s model, let the insider’s signal be noisy: 𝑠 = 𝑉 + 𝜀 with 𝜀 ∼ N(0, Σ 𝜀 ) independent. Re-derive the linear equilibrium (the insider first filters, then trades on E[𝑉 |𝑠]—two Chapter 12 projections composed), and show 𝜆 falls with Σ 𝜀 : worse-informed insiders make deeper markets.
Part D — Modeling and application
Exercise 14.12. Write the desk’s camouflage addendum to Chapter 10’s execution instruction: how Kyle’s model changes the $600M plan (slicing, randomization, venue choice, pre-announcement of mechanical intent); which parameter each tactic attacks (𝜆 via perceived Σ0 , versus 𝜂); and the one measurement the desk should collect to estimate the trade’s realized permanent-versus-temporary split.
Exercise 14.13. (The capstone memo.) Write the two-page memorandum the CIO circulates after the meeting: the five resolutions with their numbers and the theorem each rests on; the standing methodological rules adopted this year (model-resolution escalation, de-smoothing, robust limits, explicit (𝛾, 𝜌)); the three known boundaries of the toolkit (spanning, liquidity under stress, parameter learning) with the volume of the trilogy that addresses each; and the closing paragraph on what the committee now does differently at 7:55 on a Tuesday morning.
Part E — Laboratory
Exercise 14.14. (Laboratory Module 14; supports LOS 14.1–14.6.) Perform the fourpart guided experiment of Section 14.9. Submit: (a) the two (𝛾, 𝛽) readouts; (b) the Kyle 14.11 Notes and Sources 317 table with the greed-override result and its sentence; (c) the spiral 𝑘; (d) the one-page agenda summary.
Exercise 14.15. (Course Website, Week 14, Notebook 14.) Run the full-course regression test: execute the notebook that recomputes, from stated seeds and inputs, every number in Figure 14.3(b)—the collar values, the de-smoothed volatility and nowcast, the implied preferences, the platform thresholds, and the robust limit—and submit the passing log. Three sentences on which single input, if changed, moves the most resolutions, and what that says about where the committee’s scrutiny belongs.
Full solutions are distributed to instructors in the Instructor’s Manual, Chapter 14; they are not posted here. Problem-set files are on the Course Website, Week 14.