Buchanan Club Goods Optimal Membership Size Formula

Balance cost-sharing economies against crowding externalities in excludable club goods!

Between pure private goods and pure non-rival Samuelson public goods sit James M. Buchanan’s (1965, Economica) "Club Goods"—swimming pools, cloud GPU clusters, golf clubs, and toll bridges—where Fixed Facility Cost F is shared across N excludable members (lowering per-member dues F / N), while each entrant imposes linear crowding cost c_cong * N per member! Maximizing Representative Member Net Surplus u(N) = B_0 - F / N - c_cong * N sets Marginal Cost-Sharing Saving F / N^2 equal to Marginal Congestion Cost c_cong, yielding Buchanan’s exact Optimal Club Size N* = SQRT(F / c_cong)! At N*, Per-Member Dues F / N* = SQRT(F * c_cong) EXACTLY equal Marginal Congestion Externality c_cong * N*, requiring Gross Member Benefit B_0 >= 2 * SQRT(F * c_cong) for club viability!

Verify on Annual_Facility_Fixed_Cost F = $90,000 (B2), Uncongested_Member_Valuation B_0 = $5,000 (B3), and Crowding_Cost_Slope c_cong = $25/member (B4): Optimal Club Membership is N* = SQRT(90,000 / 25) = 60 members (E2); Optimal Annual Dues equal 90,000 / 60 = $1,500.00 (matching Congestion Cost 25 * 60 = $1,500.00 in E3!); Per-Member Net Surplus is 5,000 - 1,500 - 1,500 = $2,000.00 (E4); and Aggregate Club Welfare reaches 60 * 2,000 = $120,000.00 (E5)! Test doubling facility cost to F = $360,000 (4x): N* doubles to 120 members and dues double to $3,000—or test B_0 = $2,500 (< $3,000 minimum threshold) to trigger the guarded "Unviable Club (Cost > Benefit)" flag!

Viability-Guarded Optimal Club Membership & Net Surplus Formula

Check F > 0, B_0 > 0, c_cong > 0, and B_0 >= 2 * SQRT(F * c_cong) so the club never organizes at negative member utility!

=LET(fx_c, B2, bn_g, B3, cg_s, B4, IF(OR(fx_c <= 0, bn_g <= 0, cg_s <= 0), "Invalid Club Input", LET(du_op, SQRT(fx_c * cg_s), IF(bn_g < 2 * du_op, "Unviable Club (Cost > Benefit)", ROUND(bn_g - 2 * du_op, 2)))))

Aggregate Club Welfare Formula

Notice that an unpriced open-access commons drives entry until average net benefit hits zero at N_open = (B_0 + SQRT(B_0^2 - 4*F*c_cong)) / (2*c_cong), dissipating 100% of the $120,000 club surplus!

=IF(ISTEXT(E4), E4, ROUND(SQRT(B2 / B4) * E4, 2))

Further reading

• Arte Sheets Lesson — Samuelson Public Good Sum-of-MRS & Lindahl Benefit Prices
• Arte Sheets Lesson — Tiebout Sorting & Median Voter Public Spending Formula