Sharing Profits in the Sharing Economy

1. Motivation Model Results Conclusion Sharing Profits in the Sharing Economy Paolo Guasoni1, 2 Gu Wang3 Boston University1 Dublin City University2 Worcester Polytechnic Institute3 3rd Conference on Mathematical Economics and Finance, University of Manchester, September 2nd 2019

2. Motivation Model Results Conclusion Outline • Motivation: Monopolist Platforms: Uber, App Stores, Airbnb, and many others. • Model: Stackelberg Equilibrium: One Principal, Many Agents. • Results: Optimal Contract, Optimal Effort, Implied Profits. • Explicit Solution for Pareto Skill.

6. Motivation Model Results Conclusion Sharing Economy • Peer-to-peer markets: Myriad of individuals provide a service. • Uber drivers, Airbnb hosts, Youtubers, App developers. Providers can differ greatly in productivity (skill or popularity). • (Near) Monopoly: In each market, one dominant platform emerges. Competition among providers, not platforms. • Network effect: both customers and providers want to use the largest platform. • Platform takes large fraction of revenue generated by providers’.

11. Motivation Model Results Conclusion Optimal Taxation Analogies • Dominant platform similar to government: Taxes its own wage-earners. • Wage-earners differ in productivity. • Offers same contract to all providers: Either cannot observe productivity, or not allowed to discriminate. • Unlike government, the platform simply maximizes profits. • Optimal Taxation: Mirrlees (1971), Saez (2001), Golosov (2007), Mankiw et al. (2009). • Optimal Contracts with Hidden Type: Baron and Holmström (1980), Guesnierie and Laffont (1984), Darrough and Stoughton (1986), Alasseur et al. (2017), Cvitanic et al., (2018).

17. Motivation Model Results Conclusion Principal and Agents • One principal (platform) faces a population of agents (providers) who differ in skill. • An agent with skill t produces revenue of yt with y ∈ [0, ∞) units of effort. • The skill t of each agent remains hidden to the principal, who only knows its distribution, described by a density: Assumption The density m : [0, ∞) 7→ [0, ∞) is measurable, and R ∞ 0 m(t)dt < ∞ (i.e. the population is finite). Let M(t) = R ∞ t m(u)du.

21. Motivation Model Results Conclusion Contracts • The principal chooses a contract x 7→ c(x). • The revenue yt is split as net income: c(ty) for the agent and profit ty − c(ty) for the principal. Assumption The contract c : [0, ∞) 7→ R is upper semi-continuous and non-negative. C denotes the set of such contracts. • Nonnegativity: Incentive compatibility constraint. Can walk away with zero for zero effort. • Upper semicontinuity: cannot increase income by finite amount by infinitesimal change in effort. • Excludes penalties that can be avoided with negligible changes in effort, hence irrelevant in practice.

27. Motivation Model Results Conclusion Personal Cost • An agent with skill t who chooses effort y receives the reward uc (t, y) = c(ty) − f(t, y), • Net income c(ty) minus personal cost f(t, y). Assumption The cost f : [0, ∞) × [0, ∞) 7→ [0, ∞) satisfies: (i) f is strictly increasing and convex in y, and lim u→∞ fy (t, u) = ∞ for every t. (ii) f is continuously differentiable in y and t. (iii) f(t, 0) and fy are non-increasing in t. Denote by F the set of such cost functions. (i) Marginal cost of effort increasing and unbounded. (ii) Marginal cost of effort continuous. (iii) Marginal cost of effort lower (or constant) for higher skill.

36. Motivation Model Results Conclusion The Equilibrium • Agents with skill t choose the effort y(t) that maximizes individual reward: max y(t)≥0 (c(ty(t)) − f(t, y(t))) (1) • The principal chooses the contract s 7→ c(s) that maximizes profits: max c∈C P(c) := Z ∞ 0 (tyc (t) − c(tyc (t)))m(t)dt • Central tradeoff: Reducing c(s), profits would increase if effort stayed the same. But reducing c(s) also reduces effort. Lemma Given a contract c ∈ C, for any t ≥ 0 there exists some yc (t) ∈ [0, ∞] that maximizes (1), and such that yc (t) = min{y ≥ 0 : y maximizes (1)}. • Deterministic optimization problem. First-order condition?

41. Motivation Model Results Conclusion Not So Fast • First-order condition for agents: c0 (s(t)) = 1 t fy (t, y(t)) • Optimal effort equalizes marginal cost with marginal gain. • Intuitively but deceptive. • Moot for s where c(s) is not differentiable. • Ambiguous for t where y(t) is discontinuous. • Neither case can be excluded. • Find a priori restrictions on optimal contracts.

48. Motivation Model Results Conclusion Positive Sharing Rates (1) Theorem Let ˜ C = {c ∈ C : non-decreasing, right-continuous, 0 ≤ c(x) ≤ x for all x ≥ 0}. Then sup c∈C P(c) = sup c∈C̃ P(c). (2) • Intuitively natural, but requires some care. • c(x) ≥ 0 excludes average penalties. • c(x) ≤ x excludes average subsidies. • Nondecreasing property excludes marginal penalties. • Right-continuity from nondecreasing and upper-semicontinuity. • Does not exclude marginal subsidies.

55. Motivation Model Results Conclusion Positive Sharing Rates (2) • If you replace c by its increasing envelope, it can only improve. • If you truncate c(x) below x, it can only improve. Lemma For any contract c ∈ C, define its increasing envelope c∗ (x) = sup 0≤y≤x c(y). Then c∗ ∈ C and for any t ≥ 0 the optimal efforts yc (t) = yc∗ (t) coincide, as do the profits P(c) = P(c∗ ). Lemma For any non-decreasing c ∈ C, the contract c̃(x) = min(c(x), x) ∈ C and P(c̃) ≥ P(c).

59. Motivation Model Results Conclusion Revenue Restrictions Lemma Given c ∈ C̃, yc (t) is finite for every t, and lim t→0 yc (0) = 0. Furthermore, for 0 < t1 < t2, if yc (t1) > 0, then t1yc (t1) < t2yc (t2). • Any revenue s(t) = tyc (t) for c ∈ C must be in the admissible set: S = {s ≥ 0 : s(0) = 0, lim t→0 s(t) t = 0, s increasing, strictly increasing at t such that s(t) > 0}. • Idea: for each admissible revenue, construct a “nice” contract that induces such revenue.

62. Motivation Model Results Conclusion Canonical Contracts Definition For s ∈ S, define the function cs : R+ → R+ as cs (x) = R x 0 cs x (z)dz, where cs x (z) = 1 tz fy tz, z tz and tz = inf{t : s(t) z}, (3) with the convention that if {t : s(t) z} = ∅, then tz = ∞. • Canonical contract is constructed so that the first-order condition holds... • ... with the right-derivative... • ... and with skill as right-continuous inverse of revenue. • Because canonical contracts generate all revenues that can be generated by potentially optimal contracts, they are enough for the agents. • Are they enough for the principal?

68. Motivation Model Results Conclusion Canonical Profits Proposition The canonical contract cs defined above: (i) Induces the revenue s(t) (i.e., s(t) = tycs (t)) up to countably many t. (ii) Satisfies P(cs ) ≥ P(c) for any contract c ∈ ˜ C that satisfies (i), and cs ∈ C̃. • Profits from the canonical contract are maximal. • Proof technical. Requires to deal with discontinuity points of s(t). Theorem If the contract cŝ , corresponding to ŝ, is optimal, then cŝ x (z) ≤ 1 for every z ŝ (t), where t = sup{t ≥ 0 : M(t) 0}. • No marginal subsidies in canonical contracts.

75. Motivation Model Results Conclusion Integral Representation Lemma Let cs be the canonical contract in Definition 9 for some admissible revenue s ∈ S, and let y(t) = s(t) t . Then the corresponding utility ucs (t) satisfies ucs (t) − ucs (0) = Z t 0 y(v) v fy (v, y(v)) + ft (v, y(v)) dv. (4) • Obtains the agents’ utility in terms of cost function and implied effort. • Critical to obtain optimal contract.

78. Motivation Model Results Conclusion Optimal Contracts Theorem Let f be independent of t, yfy (y) be convex, and tm(t) M(t) increasing. Define g(t, s) = s − f s t m(t) − M(t) s t2 fy s t t, s ≥ 0. (5) (i) The revenue ŝ(t) defined as ŝ(t) = ( 0 t = 0, inf{s ≥ 0 : s maximizes g(t, ·)} t 0, (6) is optimal and finite for every t 0. (ii) If f is twice differentiable, then for t 0, ŝ(t) = 0 if gs(t, 0) ≤ 0. Otherwise, ŝ(t) = inf{s : gs(t, s) = 0}, where gs(t, s) = 1 − 1 t fy s t m(t) − s t fyy s t + fy s t M(t) t2 . (7)

81. Motivation Model Results Conclusion Solution Approach • Maximize g(s, t) to find optimal revenue ŝ(t). • Use derivatives if cost f smooth, otherwise do it by hand. • Find optimal contract as canonical contract generated by ŝ. • Does it work?

85. Motivation Model Results Conclusion Pareto Skill, Linear-Power costs • Concrete model. Minimal skill t. • Pareto distribution for skill: m(t) = αtα t−α−1 with α 1, so that M(t) = (t/t)−α . • Common in optimal taxation literature. • Linear-power costs: f(y) = ayλ + by for a, b 0 and λ 1, such that λ λ−1 α. • b controls small costs, a large costs. Conditions on λ ensure that total revenue is finite (well-posedness). • Optimal revenue: ŝ(t) =    t αt−(α+1)b aλ(α+λ) 1 λ−1 t (1+α)b α ∨ t, 0 t ≤ (1+α)b α ∨ t. (8)

91. Motivation Model Results Conclusion Power Costs • Consider first b = 0, i.e., f(y) = ayλ . • Optimal contract affine: cŝ x (ŝ(t)) = α α + λ for t ≥ t. (9) • Principal collects profit: ŝ(t) − cŝ (ŝ(t)) = 1 − α λ(α + λ) ŝ(t) + λ λ + α (ŝ(t) − ŝ(t)) (10) • Fees follow a two-tiered structure. Higher fees below t, lower above. • Everyone participates. (s(t) 0 for all t ≥ t.) • Fraction of total revenue to the principal: Γ = R ∞ t ŝ(t) − cŝ (ŝ(t)) m(t)dt R ∞ t ŝ(t)m(t)dt = 1 − 1 λ . (11) • Fraction of revenue independent of distribution of skill. (Total amount does depend.)

98. Motivation Model Results Conclusion Linear-Power Costs: Two Regimes � α � - � λ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.1 0.2 0.3 0.4 0.5 0.6 • Principal’s share (Γ) against linear costs (b in f(y) = by + ayλ ). • As linear costs increase from 0 to αt (1+α) , the principal’s share shifts from 1 − 1/λ (low-cost regime) to 1/α (high-cost regime). • High costs: Principal’s share insensitive to personal costs. Depends on skill distribution.

101. Motivation Model Results Conclusion Conclusion • Monopolistic platform (principal). Population of providers diverse in skill (agents). • Agents choose effort to maximize utility. • Principal chooses contract to maximize profits. • From a priori restrictions on optimal contracts, to a priori restrictions on optimal revenues. • Canonical contracts. • Characterization of optimal contract from costs and distribution. • Explicit formula for Pareto skill and linear-power costs. • Two regimes. Low costs: principal’s share independent of skill distribution. High costs: principal’s share insensitive to costs.

109. Motivation Model Results Conclusion Thank You! Questions? https://papers.ssrn.com/abstract=3278274

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