Abstract
We prove the existence of a continuous-time Radner equilibrium with multiple agents and transaction costs. The agents are incentivized to trade towards a targeted number of shares throughout the trading period and seek to maximize their expected wealth minus a penalty for deviating from their targets. Their wealth is further reduced by transaction costs that are proportional to the number of stock shares traded.
Summary
This paper provides the first existence result for a continuous-time Radner equilibrium with proportional (not quadratic) transaction costs and an arbitrary finite number of agents I >= 2. Each agent has a target number of shares a-tilde_i to acquire by time T, trades via a TWAP-like strategy (time-weighted target trajectory gamma(t)), and faces proportional costs lambda on total shares traded.
The key structural insight is that in equilibrium, each agent trades monotonically (always buying or always selling) during an initial interval [0, tau_i] and then stops. The main challenge is determining the ORDER in which agents stop trading. This is solved by a “rank-based ordering” (Section 4): agents are re-indexed as (1), (2), …, (I) so that tau^{(1)} ⇐ tau^{(2)} ⇐ … ⇐ tau^{(I-1)} = tau^{(I)} < T. The last two agents must stop simultaneously (for market clearing).
The equilibrium stock price drift mu_t is piecewise continuous, changing form each time an agent stops trading (equation 4.7). On each interval [tau^{(j)}, tau^{(j+1)}), the drift depends on the remaining active agents’ targets, weighted by the trajectory function F(t) = integral_t^T kappa(u)(gamma(u) - gamma(t)) du.
Key Contributions
- First equilibrium existence with proportional costs for I >= 3 agents (Theorem 2.2 / Theorem 4.1)
- Rank-based ordering construction for stop-trade times via backward induction (Section 4)
- Transaction cost level lambda directly impacts equilibrium stock price drift (unlike N=2 case from Noh-Weston)
- Monotonic trading property: agents always buy or always sell (Proposition 5.3)
- Stop-trade times are F_0-measurable (determined at time 0 from the targets)
- Equilibrium stock price given explicitly by equation 4.9
Methodology
The construction proceeds by backward induction on the rank j from I-1 down to 1. At each step, the pair of agents with the most extreme relative targets is identified — they are the last two to stop trading (at time tau^{(I-1)} = tau^{(I)}). The stop-trade times are defined implicitly via equation 4.3: tau^{(j)} = inf{t: |first-order-condition process Y^{(j)}_t| ⇐ lambda}. The candidate equilibrium is then verified: Proposition 5.3 proves monotonicity of the optimal strategies theta^{(j)}, Proposition 5.5 expresses Y^{(j)} as a conditional expectation of the first-order conditions, and Section 5 verifies that the proposed strategies indeed form an equilibrium.
Key Findings
- With proportional costs, agents optimally choose a stop-trade time — unlike quadratic costs where trading is continuous throughout
- Having more than two agents leads to a richer equilibrium stock price drift that changes form as agents exit (Figure 2)
- Agents with targets closest to the “average” stop trading first; those with extreme targets trade longest
- The last two agents (the most extreme buyer and seller) must stop simultaneously for market clearing
- The 20-agent numerical example (Section 3) illustrates the rich structure: trading intervals, non-monotone drift, and ordered stopping
Important References
- Asset Pricing with General Transaction Costs — Gonon, Muhle-Karbe, Shi (2020), also treats proportional costs in equilibrium but only for N=2
- Equilibrium Returns with Transaction Costs — Herdegen, Muhle-Karbe (2018), the quadratic cost equilibrium framework
- Equilibrium under TWAP trading with quadratic transaction costs — Noh (2020), the N=2 targeted trading precursor