Rent-Funded Solow–Zeira UBI

• Rent-Funded Solow–Zeira UBI is a framework that defines the minimum AI productivity threshold for captured capital rents to sustainably fund UBI without new taxes.
• The model adapts the dynamic Solow–Zeira framework by integrating AI automation parameters and a continuum of tasks to quantify optimal policy levers.
• Comparative statics show that public revenue share, oversight costs, and market structure significantly influence the threshold for UBI financing.

The Rent-Funded Solow–Zeira UBI framework defines the minimum threshold at which AI capital rents can sustainably finance a universal basic income (UBI), without recourse to new taxes or job creation. It adapts the dynamic Solow–Zeira model—incorporating a continuum of automatable tasks, a constant net saving rate $s$, and sub-unitary task elasticity $\sigma$—to analyze how advances in AI productivity ($\gamma$) translate into public revenue streams that could fund society-wide transfers. The central insight is an explicit closed-form condition for when public capture of rents from AI-automated production crosses the threshold necessary to sustain a UBI equal to a given fraction $(\lambda)$ of GDP, isolating the functional dependence on economic parameters, technological progress, and policy levers (Nayebi, 24 May 2025).

1. The Solow–Zeira Model with AI Automation

The foundational model aggregates output via a CES (constant elasticity of substitution) production function with a continuum of tasks $i \in [0,1]$, each producing $X_{it}$: $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$ where $A_t$ is Hicks-neutral total factor productivity, $\rho = (\sigma-1)/\sigma < 0$, and $\sigma \in (0,1)$ is the elasticity of substitution across tasks.

The Zeira "knife-edge" microtechnology partitions tasks into a fixed automated fraction $\sigma$0, with the remainder performed by labor: $\sigma$1 Aggregating, this yields: $\sigma$2 AI capability is parameterized by $\sigma$3, denoting productivity relative to baseline automation. Under AI advancement, the automated share is rescaled: $\sigma$4 resulting in an AI-augmented aggregate output: $\sigma$5

Capital accumulates as $\sigma$6, with the long-run Solow ratio $\sigma$7 for small growth rate $\sigma$8.

2. UBI Financing Condition and Capability Threshold

UBI is modeled as a fixed per-period transfer $\sigma$9, with $\gamma$0 its GDP share. The government captures a fraction $\gamma$1 of gross AI-capital income but faces oversight costs $\gamma$2. The net public rent extraction is $\gamma$3, where $\gamma$4 is the marginal product of capital. The feasibility constraint for rent-funded UBI is

$\gamma$5

Solving for the AI capability threshold $\gamma$6: $\gamma$7 This condition gives the minimum relative AI productivity necessary for AI capital rents, net of regulatory costs and given existing automation, to fund a desired UBI share.

3. Comparative Statics and Elasticity Analysis

The threshold $\gamma$8 is sensitive to several economic parameters. Its logarithm

$\gamma$9

allows the following comparative static results:

• Public revenue share ($(\lambda)$0): $(\lambda)$1. Higher $(\lambda)$2 lowers the threshold.
• Oversight cost ($(\lambda)$3): $(\lambda)$4. Higher costs raise $(\lambda)$5 proportionally.
• Net saving rate ($(\lambda)$6): Since $(\lambda)$7 and $(\lambda)$8, $(\lambda)$9; higher $i \in [0,1]$0 increases the threshold.
• Task elasticity ($i \in [0,1]$1): Effects are mixed, involving multiple terms, and must be simulated numerically.

These comparative statics clarify the principal levers for lowering the required AI capability for sustainable, rent-financed UBI.

4. Market Structure: Oligopoly versus Competition

The analysis extends to imperfectly competitive AI capital markets. In an $i \in [0,1]$2-firm Cournot oligopoly, the conduct parameter $i \in [0,1]$3 and market demand elasticity $i \in [0,1]$4 yield a pure-profit share of output $i \in [0,1]$5. If government captures all pure profit plus $i \in [0,1]$6 of conventional AI rents, the UBI constraint becomes: $i \in [0,1]$7 The closed-form for $i \in [0,1]$8 is lowered; monopolistic or oligopolistic markups directly reduce the capability threshold compared to perfect competition, where $i \in [0,1]$9. Under perfect competition, the most stringent requirement applies, but even then, only a modest order-of-magnitude productivity gain is necessary.

5. Calibration to Empirical Baseline

Calibrating to U.S. data (c. 2024):

• Target UBI share $X_{it}$0
• Automated task share $X_{it}$1
• Net saving rate $X_{it}$2, depreciation $X_{it}$3, output growth $X_{it}$4
• Solow ratio $X_{it}$5
• Public revenue share $X_{it}$6
• Oversight cost $X_{it}$7
• Task elasticity $X_{it}$8, hence $X_{it}$9
• Hicks-neutral productivity $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$0

Substituting into the threshold formula: $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$1 Thus, AI must be approximately 5–6 times more productive than pre-AI automation to fund an 11% of GDP UBI through captured rents.

6. Policy Levers and Diminishing Returns

The model quantifies the impact of policy interventions:

• Increasing $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$2: Doubling from 15% to 33% halves the capability bar to $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$3. Above 50%, additional gains shrink rapidly.
• Lowering $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$4: Reducing oversight costs decreases the threshold proportionally; values of $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$5 raise the requirement to $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$6, delaying feasible implementation.
• Market structure: Moderate concentration (oligopoly) sharply reduces the required $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$7 compared to perfect competition. Further concentration yields diminishing additional rent capture.
• Net saving rate: Raising $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$8 increases the capability threshold; thus, excessively promoting saving is counterproductive.

Combining moderate profit taxation (30–40%), non-competitive AI sector structure, and efficient regulation can reduce the threshold from $$Y_t = A_t \left( \int_0^1 X_{it}^\rho\, di \right)^{1/\rho}$$9 to as low as $A_t$0, significantly easing the technological requirements (Nayebi, 24 May 2025).

7. Broader Implications and Limitations

The closed-form rent-funded Solow–Zeira UBI threshold provides a robust benchmark for evaluating the fiscal potential of AI-driven economic transitions. It delineates the interplay between automation, technological progress, and policy interventions in sustaining societal transfers. Notably, the analysis is pessimistic, assuming no new jobs or tasks are created; any endogenous labor demand would further relax the AI capability requirements.

A plausible implication is that managing the distribution of economic rents—via both taxation and strategic market regulation—serves as a primary policy tool to realize the social potential of AI progress. However, the model abstracts from transitional dynamics, sectoral heterogeneity, and broader general equilibrium feedbacks, which are subjects for future research and refinement (Nayebi, 24 May 2025).