Corporate Investment Theory & Macroeconomics
Hayashi Marginal q & CapEx Adjustment-Cost Formula
Calculate Hayashi’s CRS Marginal/Average q Ratio V / K, Optimal Investment-to-Capital Rate I / K = delta + (q - 1) / phi, Optimal Gross CapEx, and Convex Installation Friction Cost.
Link observable Tobin’s Average Q to optimal CapEx intensity via Hayashi’s theorem!
James Tobin’s q theory states that firms should expand capital when marginal shadow value q exceeds replacement cost 1, yet marginal q is unobservable! Fumio Hayashi (1982, Econometrica) proved that when both operating profit and convex capital installation costs C(I, K) = (phi / 2) * ((I / K) - delta)^2 * K exhibit Constant Returns to Scale (CRS) in competitive markets, unobservable Marginal q equals observable Average Q = Market_Enterprise_Value / Replacement_Capital_Stock! Solving first-order optimality 1 + phi * ((I / K) - delta) = q yields the workhorse empirical corporate-finance investment equation: Optimal Gross CapEx Rate I / K = MAX(0, delta + (q - 1) / phi), where delta is economic depreciation and phi > 0 governs convex installation friction!
Verify on Replacement_Capital_Stock K = $500.0M (B2), Market_Enterprise_Value V = $650.0M (B3 -> Hayashi q = 650 / 500 = 1.3000 in E2), Depreciation_Rate delta = 0.10 (10.0%, B4), and Adjustment_Cost_Slope phi = 5.0 (B5): Optimal Investment Rate I / K equals 0.10 + (1.30 - 1) / 5.0 = 0.1600 (16.00% in E3); Optimal Gross CapEx equals 0.16 * $500M = $80.00M (E4); and Convex Installation Friction Cost equals (5.0 / 2) * (0.06)^2 * $500M = $4.50M (E5)!
| Hayashi q-Investment Metric (K=$500M, V=$650M -> q=1.30, delta=10%, phi=5.0) | Formula | Verified Output |
|---|---|---|
| CRS Marginal / Average q Ratio | Market_EV / Replacement_Capital_K | 1.3000 (1.0000 @ V=$500M) |
| Optimal CapEx Intensity I / K | MAX(0, delta + (q - 1) / phi) | 0.1600 (16.00%; 10.00% @ q=1) |
| Optimal Gross CapEx ($M) | (I / K) * K | $80.00M ($50.00M Maintenance @ q=1) |
| Convex Installation Friction ($M) | (phi / 2) * ((I/K) - delta)^2 * K | $4.50M ($0.00M @ q=1) |
Hayashi Optimal Investment-to-Capital Ratio (I / K) Formula
Clamp I / K at MAX(0, ...) when irreversible CapEx prevents negative gross investment (deep distress q < 1 - phi * delta) and validate K > 0, V > 0, 0 <= delta < 1, and phi > 0!
=LET(cp_k, B2, ev_m, B3, dl_r, B4, ph_c, B5, IF(OR(cp_k <= 0, ev_m <= 0, dl_r < 0, dl_r >= 1, ph_c <= 0), "Invalid Hayashi Input", LET(q_rt, ev_m / cp_k, ROUND(MAX(0, dl_r + (q_rt - 1) / ph_c), 4))))Steady-State Replacement Check (q = 1.00)
Set Market_EV B3 = $500.0M (q = 1.00): Optimal CapEx Rate equals depreciation delta = 10.00% ($50.00M maintenance CapEx) and convex installation friction drops to $0.00M!
=IF(ISTEXT(E3), E3, ROUND((B5 / 2) * ((E3 - B4) ^ 2) * B2, 2))Try the example
Use the original values or change them to test your understanding.