Jones Semi-Endogenous R&D Growth & Spillover Formula

Resolve the R&D scale-effect puzzle: why doubling scientists raises TFP LEVEL, not permanent g_A*!

In Paul Romer’s (1990) knife-edge knowledge spillover phi = 1, doubling researchers L_A permanently doubles TFP growth — contradicted by post-1950 data where scientist counts surged 5x while TFP growth stayed near ~2%/yr! Charles Jones (1995) generalized idea creation to dA/dt = delta * (L_A^lambda) * (A^phi) with research duplication exponent 0 < lambda <= 1 ("stepping on toes") and diminishing spillover phi < 1 ("fishing out" relative to Romer’s phi = 1). Along a balanced growth path with researcher growth rate n > 0, steady-state TFP growth is g_A* = (lambda * n) / (1 - phi), while raising R&D labor share from s_R0 to s_R1 creates a permanent TFP LEVEL multiplier (s_R1 / s_R0)^(lambda / (1 - phi))!

Verify with researcher growth n = 0.015 (B2, 1.5%/yr), duplication exponent lambda = 0.80 (B3), knowledge spillover phi = 0.40 (B4), and an R&D subsidy doubling R&D share from s_R0 = 0.02 (B5) to s_R1 = 0.04 (B6): steady-state TFP growth is (0.80 * 0.015) / (1 - 0.40) = 0.0200 (2.00%/yr!), impact initial TFP growth jumps temporarily to 0.0200 * 2^0.80 = 0.0348 (3.48%/yr), and long-run income per worker permanently rises by 2^(0.80 / 0.60) = 2^(4/3) = 2.5198x (+151.98%)!

Jones Balanced-Growth Idea Growth Rate g_A* = (lambda * n) / (1 - phi) Formula

Validate n >= 0, 0 < lambda <= 1, and semi-endogenous stability phi < 1 (at phi >= 1, growth explodes super-exponentially whenever n > 0!).

=LET(n_gr, B2, lam_r, B3, phi_s, B4, IF(OR(n_gr < 0, lam_r <= 0, lam_r > 1, phi_s >= 1), "Require 0 < lambda <= 1 and phi < 1", ROUND((lam_r * n_gr) / (1 - phi_s), 4)))

Long-Run Steady-State TFP Level Multiplier from R&D Policy Shift Formula

Evaluate POWER(s_R1 / s_R0, lambda / (1 - phi)) across positive R&D labor shares 0 < s_R0, s_R1 < 1.

=LET(lam_r, B3, phi_s, B4, sr_0, B5, sr_1, B6, IF(OR(lam_r <= 0, lam_r > 1, phi_s >= 1, sr_0 <= 0, sr_0 >= 1, sr_1 <= 0, sr_1 >= 1), "Invalid R&D Share Input", ROUND(POWER(sr_1 / sr_0, lam_r / (1 - phi_s)), 4)))

Further reading

• Arte Sheets Lesson — Solow-Swan Steady-State Capital per Effective Worker Formula
• Microsoft Support — POWER function