A Formal Model of Allocation in a Legitimation Economy
We formalize the structural dynamics documented above as a simple allocation model in which donors choose recipients based on three sources of utility. The model generates preferential attachment, edge persistence, and community clustering as equilibrium properties, and yields falsifiable comparative statics under AI-driven supply abundance.
Setup
Consider a bipartite network with N donors and M(θ) recipients, where θ ∈ [0, ∞) parameterizes the cost of producing cultural output. Higher θ corresponds to cheaper production (AI abundance), so M′(θ) > 0: as production costs fall, the pool of viable recipients expands. Each recipient i is characterized by an unobserved quality qi drawn i.i.d. from some distribution F and an observed endorsement stock ei,t equal to its cumulative weighted in-degree at time t. Each donor j belongs to a community cj ∈ {1, …, K}.
Donor utility
In each period, donor j allocates a grant to recipient i. The donor’s payoff from this allocation is:
rji,t ∈ {0, 1} is the relationship indicator—equal to 1 if donor j funded recipient i in period t−1. This term captures warm-glow and relational inertia.
s(i, cj) is the information signal—the quality of information donor j has about recipient i, which is higher when i is known within j’s community. We set s(i, cj) = s̄ if i has received funding from another donor in cj, and s(i, cj) = 0 otherwise.
λ(θ) is the search cost, which is increasing in M(θ): evaluating more candidates is costly. We assume λ′(θ) > 0.
The three β terms correspond directly to the three measured structural parameters. The status signal w generates preferential attachment; the relationship indicator r generates edge persistence; and the information signal s generates community clustering.
Attachment dynamics
For a new edge (donor j has no prior relationship with recipient i), the probability of selection is proportional to the utility from the status and information terms alone. Taking a logit specification over the M available recipients:
When w(e) = log(e), this reduces to the standard preferential attachment kernel Pr(i) ∝ eiα with α = β1. Our estimated α ≈ 1.07 implies β1 ≈ 1, meaning the marginal reputational return to endorsement stock is approximately proportional to the stock itself—a near-linear signal. The information term s introduces within-community bias, concentrating new edges within funding communities even conditional on endorsement stock.
Edge persistence
For an existing edge (rji,t = 1), the continuation probability is:
where σ is the logistic function. The key feature is that continuation depends on β2—the intrinsic value of the relationship—which is a donor-side parameter independent of the recipient’s production capacity. A supply shock (change in θ) affects persistence only through the search-cost term λ(θ), which enters negatively: higher search costs make continuation more attractive relative to switching, partially offsetting any supply-side disruption.
The observed persistence of 43–68% identifies the range of σ(β2 + ···), and the COVID null result (performing arts vs. museums, p = 0.55) is consistent with the prediction that edge persistence is driven by β2 (donor-side) rather than recipient-side production capacity.
Endorsement stock dynamics
The endorsement stock evolves as:
gji,t ∈ {0, 1} indicates whether donor j funds recipient i in period t.
dj is the centrality weight of donor j—grants from high-centrality donors contribute more to endorsement stock. This formalizes the gateway donor effect: dj is higher for donors with more and more diverse connections.
This generates a rich-get-richer dynamic: recipients with high ei,t attract more grants (equation 2), which further increases ei,t+1 (equation 4), which attracts still more grants. The near-linear kernel (α ≈ 1) means this process is just fast enough to produce heavy-tailed degree distributions without collapsing into a monopoly—consistent with the observed Gini above 0.85 but with a declining top-100 share.
Comparative statics under AI abundance
We now consider the effect of increasing θ (AI lowers production costs, expanding the recipient pool from M to M′ > M). Total donor attention is fixed: the number of grants per donor does not change, because philanthropic budgets are determined by endowment spending rules and board-level commitments, not by the supply of potential recipients.
With M′ > M recipients competing for fixed donor attention and a near-linear attachment kernel (α ≈ 1), the steady-state degree distribution develops heavier tails. Formally, the Gini coefficient of recipient funding G(M) is increasing in M when α ≥ 1. The new entrants dilute the lower tail of the distribution while the top recipients, whose endorsement stock ei is already high, continue to attract grants at rates proportional to eiα. The top-k share of total funding should increase with M.
The continuation probability (equation 3) depends on θ only through λ(θ), and the sign is ambiguous: higher M raises search costs, making continuation more attractive (∂Pr/∂λ < 0 in the outside option, so continuation becomes relatively preferred). Therefore ∂persistence/∂θ ≥ 0. This predicts the COVID null result: a shock that affected recipient production capacity (shuttered venues) but did not change donor-side utility (β2) should have no effect on edge persistence—confirmed empirically at p = 0.55.
For a new entrant with ei,0 ≈ 0, the probability of attracting a first grant (equation 2) is approximately 1/M absent community ties. As M increases, this probability falls. The gateway donor effect becomes more important: an initial grant from a high-centrality donor (dj large) provides a discrete jump in ei, making the recipient visible to the attachment kernel. The measured 6.2pp survival advantage of high-centrality first donors should therefore increase with θ, because the signal value of an endorsement rises when there are more alternatives.
The information signal s(i, cj) becomes more valuable when search costs λ(θ) are higher. As M increases, donors rely more heavily on community-based information channels to identify recipients, which increases the within-community edge share and raises network modularity Q. Formally, ∂Q/∂θ ≥ 0 when ∂λ/∂θ > 0. The measured excess modularity of 0.27–0.32 should be a lower bound on the modularity that would obtain under AI abundance.
Falsification conditions
The model is falsifiable. Each prediction identifies a specific empirical pattern that, if violated, would reject the legitimation economy framework for this domain:
If AI abundance leads to deconcentration—a declining Gini and falling top-k share even as M increases—then the attachment kernel is sublinear (α < 1) in the expanded regime, and endorsement stock is a weaker signal than the model assumes. The legitimation economy framework would not apply to this domain under abundance conditions.
If a supply shock that does not change donor-side utility (analogous to COVID venue closures) nonetheless disrupts edge persistence by more than 10pp, then β2 is not the dominant term in equation (3), and relationship inertia is weaker than the model requires. Persistence would be recipient-side, not donor-side, and the warm-glow mechanism would be rejected.
If AI abundance leads to declining modularity—donors diversifying across communities rather than retreating into them—then search costs are not increasing in M, likely because AI itself provides alternative information channels that substitute for community-based evaluation. This would imply that AI changes not only the supply side but also the information architecture of the legitimation economy, a qualitatively different regime.
The model is deliberately minimal. It does not specify the functional form of F(q), the community assignment process, or the endogenous formation of donor centrality. These omissions are intentional: the model’s purpose is to show that three standard economic mechanisms—status signaling, relational inertia, and community-based information—are jointly sufficient to generate the three structural parameters we measure, and to derive testable predictions under a single comparative static (θ increases). A richer model would accommodate heterogeneous donor budgets, endogenous community formation, and dynamic entry and exit; we leave these extensions to future work.