Dynamic Competition in Negotiated Price Markets: Allen & Li (2025)

What this is. The paper’s core results, the dynamic structural model of price negotiation with search and switching frictions, and the maximum likelihood estimation procedure: enough to know what it found and how, without reading all 54 pages. To replicate or extend it, read the full source at the original.

TL;DR

Using anonymized Canadian mortgage contract data linked with credit bureau records (January 2014 to July 2019), Allen and Li document an “invest-and-harvest” pricing pattern: lenders charge loyal renewers 4 bps more than new borrowers, while switchers pay 7.9 bps less than stayers. To explain these patterns, they build a T-period dynamic game of price negotiation in which borrowers face search and switching frictions and lenders are forward-looking, competing aggressively ex ante to build a customer base (invest) and extracting rents later (harvest). Structural estimation yields average search costs of $386, switching costs of $462 (new) and $829 (renewers), and lender investment incentives of $843 per borrower. Counterfactual experiments show that dynamic competition attenuates the anticompetitive effects of market frictions relative to static model predictions, and that the mortgage stress test unintentionally distorts lender pricing strategies.

Core results

Magnitudes and significance are as reported. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Invest-and-harvest pricing documented: loyal renewers pay 4.0 bps more than new borrowers; switchers pay 7.9 bps less; only 9.55% of renewers switch	Table I, p. 569	Loyalty premium = 8.53 bps; switching rate at renewal = 9.55%
R2	OLS confirms invest-and-harvest: loyal renewal dummy +4.00 bps; switch renewal dummy -3.93 bps vs purchase baseline, controlling for borrower and contract characteristics	Table II col (1), p. 570	R-squared = 0.72; bond rates, FSA house prices, lender FE, year and region FE included
R3	Structural estimates: average per-borrower search cost $386 (1.3% of interest cost); average switching cost $462 (new), $829 (renewers); average lender investment incentive $843; lender annual discount factor 0.93	Table III, p. 587; Table IV, p. 588	Likelihood ratio test rejects static model (delta=0) at 0.1% level; LR statistic = 91.39
R4	Dynamic model predicts smaller consumer gains from removing frictions than static model: removing both frictions saves borrowers 1.4% at origination and 2.5% at renewal (dynamic); static model overpredicts at 3.0% and 4.6%	Table VIII Panel C, p. 597	Joint removal saves 58% (30%) more than sum of individual effects at origination (renewal): interaction effect
R5	Removing switching costs alone hurts new borrowers (+0.2% total cost) because lender investment incentives fall; renewers benefit (-0.6%); static model always predicts savings for both	Table VIII col (2), p. 597	Investment incentive V drops to $0.253 from $0.806 at origination when switching costs removed
R6	Investment incentive is driven by both frictions: average V = $843; increasing in switching cost; effect of switching cost on V is stronger when search costs are high	Table IV, p. 588; Table V col (2), p. 590	Interaction coefficient (next-period switching cost x next-period search cost) = 0.39 (SE 0.0038), R-squared = 0.96
R7	25-year FRMs benefit all players in the dynamic model: borrowers save $344 in total cost; lenders earn $423 more over 25 years; static model reverses sign, predicting lenders earn $1,163 less	Table IX, p. 602	Dynamic model: interest rate indifference spread = 1.79 bps; static = 10.14 bps; both far below observed 91 bps swap-adjusted spread
R8	Mortgage stress test at renewal unintentionally distorts pricing: unqualified borrowers (10.2%) pay 22 bps more and incur 7.8% higher total costs; switching rate falls from 20.2% to 3.0%; lender investment incentive at origination quadruples to $3,967	Table XI, p. 606; Table XII, p. 607	Anticipatory effect lowers origination rates 4.6% for affected borrowers but share of borrowers in financial distress (GDS>39%) rises from 4.69% to 5.81%

Overall (paper’s conclusion). Dynamic competition attenuates the anticompetitive effects of search and switching costs: each friction is less harmful when the other is also present (interaction effect), and lenders’ forward-looking investment incentives lower current prices more than a static model predicts. A static framework therefore systematically overpredicts consumer gains from removing frictions and misidentifies who benefits from policy changes. The invest-and-harvest pattern is consistent with Dube, Hitsch, and Rossi (2009) and Shcherbakov (2016), who find switching costs can lower equilibrium prices in dynamic frameworks. The paper extends Allen, Clark, and Houde (2019) (origination-only) to a full repeated-interaction setting. Identification uses an English auction approximation following Woodward and Hall (2012) and Allen, Clark, and Houde (2019). The framework for jointly modelling search and switching follows Honka (2014) but adds supply-side forward-looking responses and negotiated pricing. The stress-test results are analogous to Agarwal et al. (2023a), who document that HARP unintentionally strengthened incumbency advantages in U.S. mortgage refinancing.

Theory / model

Consider a borrower seeking a mortgage with a fixed rate for $m$ years, amortizing over $T \times m$ years. The game has $T$ periods. In each period $t$, the home bank $h^t$ (the lender from the previous period, or a premortgage-relationship lender at $t=1$) moves first. The period is divided into two stages: (i) an initial quote stage, and (ii) a negotiation stage.

Borrower preferences (p. 574, eq. 1). The borrower chooses the lender $j$ from choice set $n^t$ that maximizes expected present value:

$$\max_{j \in n^t} \; v^t_j - p^t_j + \rho U^{t+1}_j, \tag{1}$$

where $v^t_j$ is the borrower’s valuation for lender $j$‘s mortgage, $p^t_j$ is the interest payment, $\rho$ is the borrower’s discount factor, and $U^{t+1}_j$ is the continuation value of being attached to lender $j$ next period. Products are homogeneous except for a switching disutility $\lambda^t$:

$$v^t_j = \begin{cases} \bar{v}^t, & j = h^t \\ \bar{v}^t - \lambda^t, & \text{otherwise.} \end{cases}$$

Lending costs (p. 575). Lender $j$‘s cost in period $t$ is:

$$c^t_j = \begin{cases} c^t, & j = h^t \text{ in initial quote stage} \\ c^t + \omega^t_j, & \text{otherwise,} \end{cases}$$

where $c^t \sim F(\cdot)$ is the common funding cost (observable to lenders but not the econometrician), and $\omega^t_j \sim G(\cdot)$ is a mean-zero IID idiosyncratic match-value component drawn at the negotiation stage.

Negotiation stage: English auction (p. 577, eqs. 2-5). With $n^t \geq 2$ lenders in the choice set, lenders compete in a descending procurement auction. Lender $j$ stays in the auction so long as the present value of winning exceeds the present value of losing (p. 577, eq. 2):

$$\bar{b}^t - (c^t + \omega^t_j) + \delta W^{t+1}_j \geq \delta L^{t+1}_j, \tag{2}$$

where $\delta$ is the lender discount factor, $W^{t+1}_j$ is the continuation value of winning, and $L^{t+1}_j$ is the continuation value of losing. The weakly dominant strategy is to bid one’s reservation value. Lender $j$‘s equilibrium drop-out bid (p. 577, eq. 3) is:

$$b^t_j(c^t, \omega^t_j) = c^t + \omega^t_j - \delta(W^{t+1} - L^{t+1}). \tag{3}$$

The term $V^{t+1} \equiv W^{t+1} - L^{t+1}$ is the net continuation value (investment incentive): lenders bid below cost in period $t$ to secure the incumbency advantage in period $t+1$. The equilibrium price given the state vector $s^t$ is (p. 578, eq. 5):

$$p^{t*}(s^t) = \begin{cases} c^t - \delta V^{t+1} + \omega^t_{(2)} + \lambda, & \omega_{h^t} - \lambda = \omega^t_{(1)} \\ c^t - \delta V^{t+1} + \omega^t_{(2)}, & \omega_{h^t} - \lambda \leq \omega^t_{(2)}, \end{cases} \tag{5}$$

where $\omega_{h^t}$ is the home bank’s idiosyncratic match value and $\omega^t_{(k)}$ denotes the $k$th-order statistic among the rival banks’ adjusted costs. The home bank charges more when it ranks first in expected utility (first line), capturing the switching cost $\lambda$ as a rent.

Initial quote stage (p. 578-580, eqs. 6-12). Given the home bank’s initial offer $p^t_0$, the borrower’s expected gain from searching $l$ lenders (relative to accepting) is:

$$\Delta^t_l = \begin{cases} 0, & l = 1 \\ p^t_0 - \lambda - (c^t - \delta V^{t+1} + E[\omega^t_{(2)} | n^t = l]), & l = 2, 3, \ldots, N, \end{cases} \tag{6}$$

and the expected marginal benefit of adding lender $l$ to the choice set is $\kappa^t_l \equiv \Delta^t_l - \Delta^t_{l-1}$ (eq. 7). The borrower chooses choice set size to maximize net expected benefit of searching (eq. 8):

$$n^t = \text{argmax}_l \; \Delta^t_l - (l-1)\kappa^t, \quad l = 1, 2, \ldots, N. \tag{8}$$

The home bank anticipates search probabilities and sets the optimal initial quote $p^{t*}_0$ to maximize expected profit (eq. 11-12):

$$\underbrace{p^{t*}_0 - c^t + \delta W^{t+1}}_{\text{Profit from } p_0} = \underbrace{E[\pi^{t*}_h | n^t = 2]}_{\text{Auction profit}} + \underbrace{\Delta^{t*}_2}_{\text{Rent from } H(\cdot)}, \tag{12}$$

showing that the optimal initial price equals the expected auction profit plus the rents the home bank can extract from search frictions.

Continuation values (p. 581-582, eqs. 13-15). The investment incentive $V^t = W^t - L^t$ is determined by the search cost distribution $H(\cdot)$, the idiosyncratic cost distribution $G(\cdot)$, the switching cost $\lambda$, and the number of available lenders $N$:

$$V^t = [1 - H(\Delta^{t*}_2)]\bigl(\Delta^{t*}_2 + E[\max\{\omega_{-h^t} - (\omega_{h^t} - \lambda), 0\} | n^t = 2]\bigr) + \sum_{l=2}^N \Pr(n^t = l) E[\max\{\omega_{-h^t} - (\omega_{h^t} - \lambda), 0\} | n^t = l] - \sum_{l=2}^N \Pr(n^t = l) \frac{l-1}{N-1} E[\max\{\omega^t_{-j} - \omega^t_j, 0\} | n^t = l]. \tag{15}$$

Since $V^t$ depends only on the distributions $H(\cdot)$, $G(\cdot)$, $\lambda$, and $N$, which are assumed time-invariant, $V^t$ does not depend on future continuation values, greatly simplifying the solution.

Equilibrium is a Markov perfect equilibrium: (i) the home bank sets $p^{t*}_0$ to maximize expected profit; (ii) the borrower sets $n^t$ to maximize net search benefit; (iii) lenders in the choice set bid $b^t_j(\cdot)$ as in eq. (3).

Method

The model is estimated by maximum likelihood on a sample of 34,554 Canadian mortgage contracts. The method builds on blp-demand (parametric demand) and search-bargaining-otc (price as second-order statistic from an English auction). The structural approach is required because search decisions are unobserved: only the final contract rate, the home bank’s identity, and the switching decision are observed.

Parametric specification (p. 583-584). Per-unit common cost $c^t_i$ is drawn from a Normal distribution $N(\mathbf{x}_i^t \boldsymbol{\beta}, \sigma_c^2)$, where $\mathbf{x}_i^t$ includes credit score, outstanding balance, bond rate, amortization, income, house price, and year/region fixed effects. The idiosyncratic cost for lender $j$ in the negotiation stage is $M_i^t \omega_{i,j}$, where $\omega_{i,j} \sim \text{T1EV}(\gamma \sigma_\omega, \sigma_\omega)$ (Type 1 Extreme Value).

Search costs follow an exponential distribution with a mean determined by the borrower’s age, credit score, and FSA-level income (p. 584):

$$H_i(\kappa) = 1 - \exp\!\left(-\frac{\kappa}{\alpha_i}\right), \quad \alpha_i = \exp(\alpha_0 + \alpha_{\text{credit}} \, \text{Credit}_i + \alpha_{\text{inc}} \, \text{Income}_i + \alpha_{\text{age}} \, \text{Age}_i).$$

Switching costs are a linear function of borrower type, origination amount, age, credit score, and income (p. 584):

$$\lambda_i = \lambda_0 + \lambda_{\text{new}} + M_i^1 \times (\lambda_{\text{credit}} \, \text{Credit}_i + \lambda_{\text{inc}} \, \text{Income}_i + \lambda_{\text{age}} \, \text{Age}_i).$$

The vector $(\sigma_c, \sigma_\omega, \delta, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\lambda})$ is estimated by maximizing the likelihood of observed switching decisions and interest rates given the equilibrium of the model. The likelihood ratio test rejects the static model ($\delta = 0$) at the 0.1% significance level (LR statistic = 91.39; Table III, p. 587).

Identification (p. 584-585). Search and switching frictions are separately identified because they have different effects on the correlation between the number of lenders ($N$) and switching probability. Search costs reduce pass-through from $N$ to the number of quotes $n$; switching costs do not. Variation in $N$ across local markets (FSA-level) and variation in borrower characteristics that differentially predict search versus switching costs enable separation of the two frictions. The lenders’ discount factor $\delta$ is identified by the relationship between amortization period and price (p. 585-586): longer amortization implies larger outstanding balance at renewal and hence stronger investment incentives.

Empirical specifications

The paper combines descriptive reduced-form evidence with structural estimation.

Descriptive evidence (R1, R2). OLS regressions of mortgage rates and switching decisions on borrower/contract characteristics (Table II, p. 570), with bond rates, FSA house prices, transaction volume, lender fixed effects, and year and region fixed effects as controls. The estimating equation for rates is:

$$\text{Rate}_{it} = \alpha + \mathbf{x}_{it} \boldsymbol{\gamma} + \text{BondRate}_t + \text{HousePrice}_{fsa} + \text{LenderFE} + \text{Year} + \text{RegionFE} + \varepsilon_{it},$$

with standard errors clustered at the FSA level. For switching probability, the linear probability model replaces rate as the outcome.

Structural estimation and model fit (R3, R6). Maximum likelihood estimation over 34,554 observations. Parameters estimated: $(\sigma_c, \sigma_\omega, \delta, \boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\lambda})$. Model fit is assessed by simulating 1,000 samples of 34,554 borrowers from the benchmark and single-friction models and comparing the distribution of predicted switching probabilities and interest rates to the data (Figure 4, p. 595). The benchmark dual-friction model reproduces data patterns; single-friction models fail (Table VII, p. 593).

Counterfactual exercises (R4, R5, R7, R8). Simulate 100,000 borrowers from the estimated dynamic model under alternative market structures (no switching cost; no search cost; no frictions; 25-year FRM; mortgage stress test). For each, solve the equilibrium home bank offer, borrower search decision, and auction outcome; compute total financing cost (interest plus search/switching costs incurred). Panel C of Table VIII (p. 597) sums origination and renewal costs to obtain the lifetime comparison.

Reduced-form validation of mechanisms (R6). OLS of the investment incentive on model-estimated search costs, switching costs, and their interaction (Table V, p. 590), using the full final sample of 34,554 contracts. Column (3) shows interest rate is decreasing in the estimated investment incentive (-7.74 per unit, SE 1.09), consistent with the forward-looking pricing mechanism.

Datasets used

Dataset	Role in paper	Wiki page
TransUnion credit bureau data (Canada)	Monthly credit bureau records for Canadian population: borrower characteristics (age, credit score, address at FSA level, nonmortgage debt), mortgage identity, switching activities, financial inquiries	no page yet
OSFI / federally regulated lender administrative data	Contract-level mortgage information: lender identity, loan size, funding date, monthly payment, outstanding balance, mortgage rate, amortization, LTV, total debt-servicing ratio	no page yet
2016 FSA-level demographic data	Population and average household income at the forward sortation area (FSA) level	no page yet
Teranet quarterly FSA-level house price index	Local house price controls	no page yet

Sample: January 2014 to July 2019 (cross-section of new borrowers and first-time renewers). Final estimation sample: 34,554 contracts (17,277 purchase, 15,627 loyal renewal, 1,650 switch renewal). Restricted to insured FRM-5Y contracts, excluding broker transactions, movers, and contracts without matching administrative data.

When to read the full paper

Use the original if you are: studying mortgage market competition, search and switching frictions in negotiated-price markets, or the welfare implications of market frictions in the presence of forward-looking firms; extending the framework to markets with endogenous refinancing (U.S.) or broker intermediation; or designing counterfactual macroprudential policy experiments using a structural dynamic model. The locators above point to the exact tables and figures for each result.

Attribution and rights

Source: peer-reviewed, The Journal of Finance 80(1). This distillation was extracted by an LLM on 2026-06-06 and is not human-verified or independently reproduced. The CC BY-NC 4.0 licence permits noncommercial reproduction; the verbatim PDF is not hosted in this batch.

Citation. Allen, Jason, and Shaoteng Li. “Dynamic Competition in Negotiated Price Markets.” The Journal of Finance 80, no. 1 (February 2025): 561-614. DOI: 10.1111/jofi.13408. © 2024 The Author(s). Licensed under CC BY-NC 4.0. This page is an extract-only distillation by the Institute for Automated Research; it does not reproduce the full text and is for noncommercial research purposes only.