6.10.3 The Duffie–Gârleanu–Pedersen Search-and-Bargaining Model

Basic Prerequisites: 02 · OTC Market Structure and Stochastic Calculus & Itô’s Lemma.

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1. Intuition & Practical Objective

Duffie, Gârleanu & Pedersen (2005) is the anchor model of OTC economics. It is the first model to derive dealer bid and ask prices from search and bargaining rather than from inventory (Ho–Stoll) or adverse selection (Glosten–Milgrom). Its answer: the OTC spread is the Nash split of the gains from trade, and the split depends on the search intensities $\lambda$ (how fast investors find each other) and $\rho$ (how fast they find a dealer) and on the bargaining power $z$.

The practical objective: be able to compute the equilibrium prices $A,B,P$ from the primitives, and to understand which knob tightens the spread. The headline comparative statics are exactly the ones a market-structure practitioner cares about:

1. Better investor search ($\lambda \uparrow$) → tighter spread, unconditionally: the customer always has the credible threat of trading with another investor, so the dealer must concede.
2. Better dealer access ($\rho \uparrow$) → tighter spread IF dealers compete ($z<1$) - “sequential competition” between dealers drives the spread to zero.
3. Better dealer access ($\rho \uparrow$) → WIDER spread IF the dealer is a monopolist ($z=1$) - because easier access makes the investor’s alternative worse, since the dealer has hoovered up all the counterparties.

The one-sentence result. “The OTC bid-ask is $A-B=\delta z/D$ - a $z$-fraction of the fundamental surplus $\delta /r$ - and it converges to zero under fast search only when dealers lack total bargaining power; a monopolist dealer’s spread actually widens with more accessibility.”

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2. Mathematical Ground Truth & Derivations

2.1 The setup: types, search, and meeting

A unit continuum of risk-neutral investors. Each holds at most one unit of a long-lived asset (total supply $s$). An investor is of intrinsic type low or high; type switches are Poisson: low$\to$high at ${\lambda }_u$, high$\to$low at ${\lambda }_d$. In the steady state the fraction of high types is ${\lambda }_u/({\lambda }_u+{\lambda }_d)$ - high-type owners want to keep the asset, high-type non-owners want to buy it.

Investors meet each other at intensity $\lambda$ (a search friction) and meet dealers at intensity $\rho$ (dealer accessibility). Dealers can recycle positions instantly in a frictionless interdealer market at price $M$, so they bear no inventory risk - a deliberate contrast with inventory models.

Conservation gives two identities that solve the four-type system with only one unknown:

$${\mu }_{lo}+{\mu }_{ho}=s,{\mu }_{ho}+{\mu }_{hn}=\frac{{\lambda }_u}{{\lambda }_u+{\lambda }_d}.$$

Under Condition 1 ($s<{\lambda }_u/({\lambda }_u+{\lambda }_d)$), which holds in the paper’s example ($0.8<0.9091$), the market has more high-type non-owners than low-type owners (${\mu }_{lo}<{\mu }_{hn}$), so a dealer sells to every non-owner he meets and the interdealer price is the buyer/taker side, $M=A=H$. The steady-state mass ${\mu }_{lo}$ then solves a quadratic:

$$2\lambda x^2+(2\lambda a+\rho +{\lambda }_u+{\lambda }_d)x-{\lambda }_{d}s=0,a\equiv \frac{{\lambda }_u}{{\lambda }_u+{\lambda }_d}-s.$$

2.2 Value functions and bargaining (the HJB)

Let $V_{\sigma }$ be an investor’s continuation value by type. In steady state the Hamilton–Jacobi–Bellman equations are (DGP eq 10):

$${\dot{V}}_{lo}=r{V}_{lo}-{\lambda }_u(V_{ho}-V_{lo})-2\lambda {\mu }_{hn}(P+V_{ln}-V_{lo})-\rho (B+V_{ln}-V_{lo})-(1-\delta ),$$ $${\dot{V}}_{ln}=r{V}_{ln}-{\lambda }_u(V_{hn}-V_{ln}),{\dot{V}}_{ho}=r{V}_{ho}-{\lambda }_d(V_{lo}-V_{ho})-1,$$ $${\dot{V}}_{hn}=r{V}_{hn}-{\lambda }_d(V_{ln}-V_{hn})-2\lambda {\mu }_{ho}(V_{ho}-V_{hn}-P)-\rho (V_{ho}-V_{hn}-A).$$

Bargaining. When a low owner meets a high non-owner, the gains from trade are $L=V_{lo}-V_{ln}$ (the seller’s) and $H=V_{ho}-V_{hn}$ (the buyer’s). Nash bargaining with seller power $q$ splits them (DGP eq 11):

$$\boxed{P=(1-q)L+qH}$$

When an investor meets a dealer, the dealer’s outside option is the interdealer price $M$ and his power is $z$ (DGP eqs 12–13):

$$\boxed{A=zH+(1-z)M,B=zL+(1-z)M}⟹A-B=z(H-L).$$

2.3 The closed-form equilibrium (Theorem 2)

Solving the linear value-function system gives the prices directly (DGP eqs 14–16), with

$$D=r+{\lambda }_d+2\lambda {\mu }_{lo}(1-q)+{\lambda }_u+2\lambda {\mu }_{hn}q+\rho (1-z),$$ $$A=\frac{1}{r}-\frac{\delta }{r}\frac{{\lambda }_d+2\lambda {\mu }_{lo}(1-q)}{D},B=\frac{1}{r}-\frac{\delta }{r}\frac{zr+{\lambda }_d+2\lambda {\mu }_{lo}(1-q)}{D},P=\frac{1}{r}-\frac{\delta }{r}\frac{(1-q)r+{\lambda }_d+2\lambda {\mu }_{lo}(1-q)}{D}.$$

Each price is the present value $1/r$ of the dividend stream, reduced by an illiquidity discount $(\delta /r)\times (\text{ratio})$. The spread is

$$A-B=\frac{\delta z}{D}.$$

Monopolist case ($z=1$): the spread becomes $A-B=\delta /D$ with the full denominator $D$ (the $2\lambda \mu$ investor-search terms do not vanish), so it still falls in the investor search intensity $\lambda$ - but it is increasing in the dealer-accessibility $\rho$ - the counterintuitive core of the paper (faster dealer contact widens the monopolist’s spread). Competitive case ($z<1$): as $\rho \to \infty$, $D\to \infty$ and $A-B\to 0$ (sequential competition). Fast investors: as $\lambda \to \infty$, $D\to \infty$ regardless of $z$, so $A-B\to 0$ (Theorem 3, part 1). The equilibrium converges to Walrasian $P^{\ast }=1/r$.

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3. Computational Implementation - the equilibrium engine

Stdlib only. We solve the steady-state masses from the quadratic, then compute Theorem-2 prices and verify the three comparative statics plus the spread identity. Ran and verified.

Read the two tables. As $\lambda$ grows $P\to 1/r=20$ and the spread $\to 0$: fast investor search is unconditionally good for customers. As $\rho$ grows, the competing-dealer spread ($z=0.8$) falls from $0.1757$ to $0.0039$ while the monopolist spread ($z=1.0$) rises from $0.2196$ to $0.2506$ - the paper’s signature result, reproduced exactly. The spread identity $A-B=\delta z/D$ confirms the closed form to ten decimals.

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4. Failure Modes & First-Principles Breakdowns

1. Bargaining power $z$ is the master parameter, and it is unobservable. Every comparative static flips on whether dealers compete or monopolise. Calibrating $z$ from data is hard; treating $z$ as a free knob without justification is the classic misuse.
2. The model abstracts from inventory and information. DGP dealers bear no inventory risk (interdealer market is frictionless) and agents are symmetrically informed. Real dealers do both - this model must be combined with inventory and adverse-selection views.
3. Risk-neutral, single asset, no capital constraints. There is no balance sheet: dealers never run out of capital. The capacity failure - the one that caused 2008 - is outside this model and requires 04 · Dealer Capacity.
4. Instant trade assumption. Meeting ⇒ immediate trade. With asymmetric information or strategic delay, trade can be delayed (the model deliberately abstracts from this).
5. Steady state. The closed forms hold at the stationary masses; transitional dynamics (a crisis) require the full $\dot{\mu }$ system.

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5. References

• Duffie, Gârleanu & Pedersen (2005), Over-the-counter markets, Econometrica 73(6), 1815–1847
• Duffie, Gârleanu & Pedersen (2003), Valuation in dynamic bargaining markets
• Duffie (2012), Dark Markets
• Rubinstein & Wolinsky (1985), Equilibrium in a market with sequential bargaining, Econometrica

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6. Connected Graph Bridges

• Back: 02 · OTC Market Structure
• Forward: 04 · Dealer Capacity & Balance Sheets · Index Hub
• Sibling: Ho–Stoll Dealer Model · The Glosten–Milgrom Model