Semiparametric Identification of the Discount Factor and Payoff Function in Dynamic Discrete Choice Models

Semiparametric Identification of the Discount Factor and Payoff Function in Dynamic Discrete Choice Models

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Keywords: Dynamic discrete choice models; semiparametric identification; concavity; homogeneity of degree one; monotonicity, exchangeability.

Introduction

Dynamic discrete choice models are fundamental tools in empirical economics, providing a framework for analyzing forward-looking decision-making in contexts ranging from labor supply and human capital investment to firm entry/exit and technology adoption. While these models have proven valuable for policy analysis, a key identification challenge lies in disentangling the discount factor from other model primitives. Identifying the discount factor is essential because the discount factor governs how agents trade off current and future payoffs and directly affects both parameter estimates and counterfactual predictions.

The identification of structural parameters in dynamic discrete choice models has been extensively studied since the seminal work of rust87em. rust94hoe established a fundamental negative result, showing that the discount factor cannot be identified in these models without additional restrictions. Building on this insight, magnac02em showed that, without restrictions on preferences, the discount factor cannot be separately identified from current payoffs using only conditional choice probabilities.

Due to this non-identification result, most empirical studies of dynamic choice models assume that the annualized value of the discount factor is known and fix its value between 0.9 and 0.99 hendel2006, ryan2012, collardwexler2013demand,igami2020mergers, Miller2021. However, several recent studies that explicitly estimate the discount factor report values below the conventional lower bound of 0.9 yao2012determining,chung2014bonuses, GayleTomlin2018,DeGroote2019,kong2024nonparametric. This suggests that individuals and firms may be more impatient than typically assumed.\footnote{Reviewing past studies, frederick2002time find that discount factors vary considerably across different contexts and population samples.} Empirical studies that estimate the discount factor also reveal that optimal pricing, investment choices, and other forward-looking choices could be sensitive to the value of the discount factor. Accurately capturing individuals' actual time preferences, especially greater impatience, can therefore significantly change conclusions about the impact of counterfactual policy interventions, compared to models under the conventional discount factor value ChingOsborne2019.\footnote{fowlie2016market and igami2017estimating investigate the sensitivity of parameter estimates to the discount factor by conducting repeated estimations with a range of discount factors surrounding the value employed in their primary analysis. lau2024 proposes a sensitivity analysis framework for dynamic discrete choice models that examines how target parameters respond to variations in the discount factor.} Consequently, developing methodologies to credibly identify the discount factor is essential.

The identification of the discount factor has gained increasing attention in dynamic structural models. abbring20qe demonstrate that an exclusion restriction---requiring the payoff function to take the same value at two different action-state pairs---can identify the discount factor in single-agent dynamic discrete choice models up to a countable set of solutions to an infinite-order polynomial equation. abbring20wp sharpen the result of abbring20qe by showing that the cardinality of the identified set is no greater than that of the state space. Other studies impose assumptions such as linearity in parameters of the payoff function or the availability of a terminal action to achieve identification in single-agent dynamic models Bajari16qme, komarova18qe,chou24wp. However, these assumptions may be considered strong, lack clear economic justification, or significantly restrict the class of models. Moreover, the existing literature has exclusively focused on single-agent models and has not formally analyzed the identification of the discount factor in dynamic game models.

This paper makes two key contributions to the understanding of discount factor identification in dynamic discrete choice models. First, we demonstrate that standard nonparametric assumptions on period payoff---such as homogeneity, monotonicity, concavity, and zero cross-derivatives or complementarity across distinct state variables matzkin92em,matzkin94hoe---generate equality and inequality restrictions with substantial identifying power. Our identification strategy leverages nonparametric shape restrictions grounded in economic theory, which, when combined with the finite-set identification discussed above, enables researchers to achieve point identification or obtain an identified set containing only a small number of points without imposing arbitrary parametric assumptions on the payoff function. These restrictions also point or set identify the payoff function itself because the payoff function is identified once the discount factor is identified. Furthermore, as in abbring20qe, the cardinality of the identified set for $\beta$ is at most $\rho$ when the model exhibits $\rho$-finite dependence AltugMiller98restud, am11em.

Second, we analyze models of dynamic games. In addition to extending our identification results from the single-agent to the multi-agent setting, we show that nonparametric assumptions commonly used in empirical model of dynamic games---such as the irrelevance of other firms' lagged actions, the exchangeability of other firms' actions, and the independence of adjustment costs of changing states (e.g., entry costs) from other firms' actions in period payoff functions---provide equality restrictions that facilitate identification of the discount factor in multi-agent settings We also establish identification when discount factors vary heterogeneously across agents.

Our analysis builds on and extends several strands of literature. The conditional choice probability approach of hotzmiller93restud provides a foundational framework for analyzing dynamic discrete choice models without solving for the full solution. magnac02em characterize the identification of these models and show the importance of normalization. abbring20qe and abbring20wp pioneer a characterization of the identified set for the discount factor as the solution set to polynomial equations. In the context of dynamic games, am07em and pakes07rand develop tractable estimation methods in multi-agent settings. bajari07ecta introduce a computationally efficient two-step estimator, and pesendorfer08restud introduce an asymptotic least squares approach. aguirregabiriasuzuki14qme, noretstang14restud, am20joe, and Kalouptsidi21qe analyze the identification of counterfactuals, highlighting how normalizations affect what can be learned about preferences. In these studies, the discount factor is typically treated as known. We complement this literature by providing new identification results for the discount factor.

In static discrete choice and related econometric models, the literature has examined the identifying power of shape restrictions on utility functions derived from economic theory, such as monotonicity, concavity, and homogeneity of degree one. matzkin92em shows that binary threshold crossing and binary choice models can be identified without imposing parametric restrictions on either the utility function or the distribution of the unobservable term by employing such shape restrictions. matzkin93joe extends this approach to polychotomous choice models, while matzkin91em develops a semiparametric estimation method for these models under monotonicity and concavity assumptions. matzkin94hoe reviews the literature on nonparametric identification and estimation grounded in economically motivated restrictions. allen19em use a variant of Slutsky symmetry to nonparametrically identify latent utility models with additively separable unobservable heterogeneity. Furthermore, matzkin03em demonstrates that the homogeneity of degree one restriction enables identification of models with nonadditive unobservable heterogeneity.

In two-player binary choice games of complete information, shape restrictions such as strategic substitutability have been used for identification. berrytamer06book demonstrate that the payoff function and the distribution of unobservable heterogeneity are nonparametrically identified when the players' actions are strategic substitutes. foxlazzati17qe show that the payoff function can be nonparametrically identified when the econometrician knows the sign of the interaction effects. dunker18joe study a model with random coefficients and establish the identification of their joint distribution under strategic substitutability. To the best of our knowledge, shape restrictions on the payoff function have not yet been leveraged for identification in models of dynamic games with incomplete information.

Monotonicity restrictions have been widely used for identification in various nonlinear and nonseparable econometric models, though a comprehensive review is beyond the scope of this paper. matzkin13are and chetverikov18are provide excellent surveys. See also imbens94em, chesher03em, matzkin08em, imbens09em, shi18em, pakes24qe, and the references therein.

The remainder of the paper is organized as follows. Section 2 introduces the baseline model and assumptions. Section 3 analyzes identification in single-agent models. Section 4 provides examples of restrictions on per-period payoff functions that aid identification. Section 5 discusses finite dependence results. Section 6 provides numerical examples that illustrate the theoretical findings. Section 7 extends the analysis to dynamic game models. The Appendix contains the proofs of propositions and lemmas.

We use boldfaced letters to denote vectors and matrices. Let $\boldsymbol{I}_k$ denote the $k\times k$ identity matrix, and we suppress the subscript $k$ when no confusion arises. We use “$ \coloneqq$” to denote “equals by definition.” Let $\mathds{1}\{A\}$ denote the indicator function that takes the value one when $A$ is true and zero otherwise. For a set $\mathcal{S}$, let $|\mathcal{S}|$ denote its cardinality. We follow the convention that bold lowercase and uppercase letters denote vectors and matrices, respectively. For a matrix $\boldsymbol{A}$, let $\operatorname{Ker}(\boldsymbol{A})$ denote its null space. With a slight abuse of notation, we write $(\boldsymbol{a}_1,\ldots,\boldsymbol{a}_k)$ to denote the vector $(\boldsymbol{a}_1^{\top},\ldots,\boldsymbol{a}_k^{\top})^{\top}$ when no confusion arises.

Model and Assumptions

Framework and basic assumptions

We consider a stationary discrete-time infinite-horizon dynamic discrete choice model. Our presentation of the model follows abbring20qe. In each period, an agent observes state variables $(\boldsymbol{x},\boldsymbol{\varepsilon})$, where $\boldsymbol{x} \in \mathcal{X} = \{\boldsymbol{x}^1,\ldots,\boldsymbol{x}^J\}$ denotes variables observable to both the agent and the researcher, and $\boldsymbol{\varepsilon} = (\varepsilon_1,\ldots,\varepsilon_K)^{\top} \in \mathbb{R}^K$ denotes variables observable only to the agent. The agent then chooses the action $a$ from the set of alternatives $\mathcal{A} = \{1,2,\ldots, K\}$ to maximize the expected present discounted value of current and future payoffs. Let $\beta \in [0,1)$ denote the time discount factor.

Let $u_k(\boldsymbol{x},\boldsymbol{\varepsilon})$ be the per-period payoff function when choosing action $k \in \mathcal{A}$. Let $f_k(\boldsymbol{x}',\boldsymbol{\varepsilon}'|\boldsymbol{x},\boldsymbol{\varepsilon})$ denote the transition probability of $(\boldsymbol{x},\boldsymbol{\varepsilon})$ given action $k \in \mathcal{A}$. Let $V(\boldsymbol{x},\boldsymbol{\varepsilon})$ denote the agent's value function. By Bellman's principle of optimality, $V(\boldsymbol{x},\boldsymbol{\varepsilon})$ is the unique solution to Bellman's equation given by \[ V(\boldsymbol{x},\boldsymbol{\varepsilon}) = \max_{k \in \mathcal{A}} \left[ u_k(s) + \beta \int V(\boldsymbol{x},\boldsymbol{\varepsilon}) f_k(\boldsymbol{x}',\boldsymbol{\varepsilon}'|\boldsymbol{x},\boldsymbol{\varepsilon})\right] . \] We assume the per-period payoff function is additively separable as $u_k(\boldsymbol{x},\boldsymbol{\varepsilon}) = u_k(\boldsymbol{x}) + \varepsilon_k$ and the transition probablity factors as $f_k(\boldsymbol{x}',\boldsymbol{\varepsilon}'|\boldsymbol{x},\boldsymbol{\varepsilon}) = g(\boldsymbol{\varepsilon}'|\boldsymbol{x}') Q_k(\boldsymbol{x}'|\boldsymbol{x})$. Define the integrated value function (ex ante value function) as $V(\boldsymbol{x}) \coloneqq \int V(\boldsymbol{x},\boldsymbol{\varepsilon}) g(d\boldsymbol{\varepsilon}|\boldsymbol{x})$. Define the choice-specific value function for each $k \in \mathcal{A}$ as

The conditional choice probability (CCP) $p_k(\boldsymbol{x})$ is the probability that alternative $k$ is the optimal choice given the observable state $\boldsymbol{x}$: \[ p_k(\boldsymbol{x}) \coloneqq \int \mathds{1} \left\{ k = \operatorname*{arg\,max}_{\ell \in \mathcal{A}} \left[ v_\ell (\boldsymbol{x}) + \varepsilon_\ell\right] \right\} g(d\boldsymbol{\varepsilon}|\boldsymbol{x}). \] Define the CCP vector $\boldsymbol{p}(\boldsymbol{x}) \coloneqq (p_1(\boldsymbol{x}),\ldots,p_K(\boldsymbol{x}))^{\top}$. am11em show that for every $k \in \mathcal{A}$, there exists a function $\psi_k(\cdot)$ derived only from $g$ such that \footnote{am11em assume $g(\boldsymbol{\varepsilon}|\boldsymbol{x}) = g(\boldsymbol{\varepsilon})$, but their proof of Lemma 1 remains valid even if $\boldsymbol{g}$ depends on $\boldsymbol{x}$. $g$ also depends on $\boldsymbol{x}$ in Proposition 1 of hotzmiller93restud.}

From Lemma 3 of am11em, if $\varepsilon_1,\ldots,\varepsilon_K$ are independently drawn from type-I extreme value distribution, then $\psi_k(\boldsymbol{p}(\boldsymbol{x})) = \gamma - \ln (p_k(\boldsymbol{x}))$, where $\gamma$ is Euler's constant.\footnote{am11em also consider other distributions from the generalized extreme value (GEV) family; in the nested logit case, $\psi_k(\boldsymbol{p}) = \gamma\sigma\ln(p_k) + (1-\sigma)\ln\left(\sum_{k' \in \mathcal{K}} p_{k'}\right)$, where {$\mathcal{K}$ is the nest containing $k$, and} $\sigma$ captures the degree of correlation among alternatives within $\mathcal{K}$.} Substituting ((ref)) into ((ref)) gives

Let $\boldsymbol{v}_k$, $\boldsymbol{u}_k$, $\boldsymbol{p}_k$, $\boldsymbol{\psi}_k$, and $\boldsymbol{V}$ be $J\times 1$ vectors with the $j$-th elements $v_k(\boldsymbol{x}^j)$, $u_k(\boldsymbol{x}^j)$, $p_k(\boldsymbol{x}^j)$, $\psi_k(\boldsymbol{p}(\boldsymbol{x}^j))$, and $V(\boldsymbol{x}^j)$, respectively. Let $\boldsymbol{Q}_k$ be the $J \times J$ matrix with $(\ell,m)$-th entry $Q_k(\boldsymbol{x}^m|\boldsymbol{x}^\ell)$. Both $\boldsymbol{p}_k$ and $\boldsymbol{Q}_k$ for $k \in \mathcal{A}$ are directly identified from the data. Furthermore, under a distributional assumption on $g(\boldsymbol{\varepsilon}|\boldsymbol x)$, we can also identify $(\boldsymbol{\psi}_1,\ldots,\boldsymbol{\psi}_K)$ from $(\boldsymbol{p}_1,\ldots,\boldsymbol{p}_K)$. Therefore, we treat $\{\boldsymbol{p}_k, \boldsymbol{Q}_k, \boldsymbol{\psi}_k; k \in \mathcal{A}\}$ as known and analyze the identification of $(\{\boldsymbol{v}_k,\boldsymbol{u}_k\}_{k=1}^K, \boldsymbol{V}, \beta)$.

Let $\boldsymbol{Q}_k(\boldsymbol{x}^j)$ denote the $j$-th row of $\boldsymbol{Q}_k$. Stacking equation ((ref)) over $\boldsymbol{x} \in \mathcal{X}$ gives

Stacking equation ((ref)) over $\boldsymbol{x} \in \mathcal{X}$ yields

Equations ((ref)) and ((ref)) summarize the model's restrictions. Together, they provide $2JK$ equations in the $2JK+J+1$ unknowns $(\{\boldsymbol{v}_k,\boldsymbol{u}_k\}_{k=1}^K, \boldsymbol{V}, \beta )$. Thus, at least $J+1$ additional restrictions are required for identification.

We assume $\boldsymbol{u}_K=\boldsymbol{0}$ as in abbring20qe and abbring20wp. This normalization is common in empirical applications because the per-period payoff cannot be identified from the model and the observed conditional choice probability alone magnac02em, and because data on $\boldsymbol{u}_K$ or $\boldsymbol{V}$ are rarely available.\footnote{Kalouptsidi14aer, Kalouptsidi18res utilize external data on entry costs and scrap values to estimate the value function $\boldsymbol{V}$, thereby avoiding the need to assume $\boldsymbol{u}_K=\boldsymbol{0}$.} This assumption is not innocuous, however, as it can affect counterfactual predictions and other parameter estimates aguirregabiriasuzuki14qme, noretstang14restud, Kalouptsidi21qe.

Subtracting ((ref)) from ((ref)) for $k=K$ and using $\boldsymbol{u}_K=\boldsymbol{0}$ give $\boldsymbol{\psi}_K = (\boldsymbol{I} - \beta \boldsymbol{Q}_K ) \boldsymbol{V}$. Since $\boldsymbol{Q}_K$ is a stochastic matrix, its eigenvalues lie within the unit circle; therefore, given that $\beta < 1$, the matrix $\boldsymbol{I} - \beta \boldsymbol{Q}_K$ is invertible. Hence, $\boldsymbol{V}$ is identified as

Eliminating $\boldsymbol{v}_k$ from ((ref)) and ((ref)) and then substituting ((ref)) gives, for $k=1,\ldots,K-1$,\footnote{This equation corresponds to (3) in Kalouptsidi21qe except that we impose $\boldsymbol{\pi}_K=\boldsymbol{0}$.}

Therefore, the per-period payoff function is identified if $\beta$ were known, as shown by abbring20qe. berrytamer06book note that $\beta$ can be identified from ((ref)) if the value of $u_k(\widetilde{\boldsymbol{x}})$ is known for some $\widetilde{\boldsymbol{x}} \in \mathcal{X}$.

abbring20qe derive the identified set of $\beta$ using an exclusion restriction of the form $u_k(\boldsymbol{x}_a) = u_\ell(\boldsymbol{x}_b)$ for some known choices $k \in \mathcal{A}\setminus \{K\}, \ell \in \mathcal{A}$ and known states $\boldsymbol{x}_a,\boldsymbol{x}_b \in \mathcal{X}$, where either $k \neq \ell$, $\boldsymbol{x}_a \neq \boldsymbol{x}_b$, or both. Under this exclusion restriction, ((ref)) implies

where $(\boldsymbol{I} - \beta \boldsymbol{Q}_k)(\boldsymbol{x}_a)$ denotes the row of $\boldsymbol{I}-\beta\boldsymbol{Q}_k$ corresponding to $\boldsymbol{x}_a$, and similarly for $(\boldsymbol{I} - \beta \boldsymbol{Q}_\ell)(\boldsymbol{x}_b)$.\footnote{Equation ((ref)) is equivalent to equation (12) in abbring20qe; the equivalence follows from $\psi_K(\boldsymbol{x}_a)=(\boldsymbol{I} - \beta \boldsymbol{Q}_K)(\boldsymbol{x}_a)(\boldsymbol{I} - \beta\boldsymbol{Q}_K)^{-1}\boldsymbol{\psi}_K$.} abbring20qe note that $(\boldsymbol{I} - \beta\boldsymbol{Q}_K)^{-1}$ can be expressed as an infinite convergent power series in $\beta$ when $\beta\in [0, 1)$ and show that the solution set to ((ref)) is a closed discrete subset of $[0, 1)$. abbring20wp sharpen this result by expressing $( \boldsymbol{I} - \beta\boldsymbol{Q}_K )^{-1}$ as the ratio of two finite-order polynomials in $\beta$, thus bounding the cardinality of the identified set by $J$.

As in abbring20wp, we express $(\boldsymbol{I} - \beta\boldsymbol{Q}_K)^{-1}$ as the ratio of two finite-order polynomials. The adjoint matrix of a square matrix $\boldsymbol{A}$, denoted $\operatorname{adj}(\boldsymbol{A})$, is defined as the transpose of the cofactor matrix of $\boldsymbol{A}$. The cofactor matrix $\boldsymbol{C}$ has the same dimension as $\boldsymbol{A}$, and its $(i,j)$th element is $(-1)^{i+j} M_{ij}$, where $M_{ij}$ is the determinant of the submatrix obtained by removing the $i$th row and $j$th column from $\boldsymbol{A}$. The adjoint satisfies the identity magnus19book:

Hence, if $\boldsymbol{A}$ is invertible, $\boldsymbol{A}^{-1}$ is given by $\boldsymbol{A}^{-1} = \operatorname{adj}(\boldsymbol{A})/\det(\boldsymbol{A})$. Applying this to $( \boldsymbol{I} - \beta\boldsymbol{Q}_K )^{-1}$, we obtain

Each element of $\operatorname{adj}( \boldsymbol{I} - \beta\boldsymbol{Q}_K)$ is a polynomial of degree $J-1$ in $\beta$ because it is the determinant of a $(J-1) \times (J-1)$ submatrix of $\boldsymbol{I} - \beta\boldsymbol{Q}_K$. Substituting ((ref)) into ((ref)) and rearranging terms give

We collect equations ((ref)) for $k=1,\ldots,K-1$. Define the $J(K-1)$-dimensional vectors $\boldsymbol{U}$ and $\boldsymbol{\Psi}$, and the $J(K-1) \times J$ matrix $\boldsymbol{Q}(\beta)$ as \[ \boldsymbol{U} \coloneqq

,\quad \boldsymbol{\Psi} \coloneqq

, \quad \boldsymbol{Q}(\beta) \coloneqq

. \] Then, the model's restrictions are summarized by the following system of $J(K-1)$ equations:

Without additional assumptions or data, this system contains all available information about $\beta$. Note that both $\det\left( \boldsymbol{I} - \beta\boldsymbol{Q}_K\right)$ and the elements of $\boldsymbol{Q}(\beta)\operatorname{adj}( \boldsymbol{I} - \beta\boldsymbol{Q}_K)$ in ((ref)) are polynomials of degree $J$ in $\beta$. Therefore, if the payoff function $\boldsymbol{U}$ satisfies a linear restriction of the form $\boldsymbol{r}^{\top}\boldsymbol{U}=0$ for some known vector $\boldsymbol{r}$, then left-multiplying ((ref)) by $\boldsymbol{r}^{\top}$ gives a polynomial in $\beta$ of degree $J$ with known coefficients. Consequently, as discussed in Theorem 6 of abbring20wp, the identified set for $\beta$ consists of the roots of this degree-$J$ polynomial within the interval $[0,1)$.

Identification of $\beta$ by economic restrictions on per-period payoff

Economic theory often imposes restrictions on per-period payoffs such as homogeneity and monotonicity. These restrictions are expressed as linear constraints on the elements of $\boldsymbol{U}$, which represent the per-period payoffs across different actions and states. Consequently, economic theory provides a basis for deriving equality and inequality constraints on $\boldsymbol{U}$. In Section (ref), we show that homogeneity leads to equality constraints of the form $\boldsymbol{r}^{\top}\boldsymbol{U}=0$, while monotonicity and concavity imply inequality constraints of the form $\boldsymbol{r}^{\top}\boldsymbol{U} \geq 0$. Define $p \coloneqq J(K-1)$ as the length of $\boldsymbol{U}$.

The exclusion restriction used in abbring20qe and abbring20wp corresponds to Assumption (ref) with $q_1=1$, $c_1=0$, and a row vector $\boldsymbol{R}_1$ with entries $1$ and $-1$ in the positions corresponding to the two equalized elements, and zeros elsewhere. Left-multiplying both sides of ((ref)) by $\boldsymbol{R}_1$ gives the following proposition, which generalizes Theorem 6 of abbring20wp.

When $\boldsymbol{U}$ satisfies Assumption (ref), $\beta$ is identified as a solution to a system of $J$-degree polynomials in $\beta$ with coefficients identified from the data. Consequently, the identified set of $\beta$ contains at most $J$ elements. When the restriction is the exclusion restriction of the form $u_k(\boldsymbol{x}_a) = u_\ell(\boldsymbol{x}_b)$, equation (ref) corresponds to equation (41) in abbring20wp.

As we discuss in Section (ref), shape restrictions grounded in economic theory can provide multiple identifying restrictions. In such cases, we can reduce the degree of the resulting polynomial system by taking linear combinations of ((ref)) and eliminating higher-order terms in $\beta$ through Gauss-Jordan elimination. As a result, the degree of the polynomial may be reduced to $J-q_1+1$, provided that a suitable rank condition holds. Since $\beta \in [0,1)$, the number of admissible solutions is typically smaller than $J-q_1+1$.

Left-multiplying both sides of ((ref)) by $\boldsymbol{R}_2$ allows us to exploit the inequality restrictions on $\boldsymbol{U}$ imposed by Assumption (ref), as formalized in the following proposition. Note that $\det\left( \boldsymbol{I} - \beta\boldsymbol{Q}_K\right)>0$ because (i) the determinant of a matrix equals the product of its eigenvalues, and (ii) all eigenvalues of $\boldsymbol{I} - \beta\boldsymbol{Q}_K$ are positive Kalouptsidi21qe.

We can combine Assumptions (ref) and (ref) to further narrow the identified set of $\beta$. The following corollary summarizes this.

We rule out $\beta=1$ on economic grounds and because ((ref)) is not well-defined at $\beta=1$ due to the singularity of $\boldsymbol{I} - \boldsymbol{Q}_K$. Nevertheless, equations ((ref))--((ref)) remain valid at $\beta=1$ because all the terms on their left hand sides vanish at this value due to properties of the adjoint matrix.

Examples of Economic Restrictions on Per-Period Payoff Functions

In this section, we present several examples of per-period payoff functions that lead to the restrictions discussed in the previous section. In many applied economic models, payoff functions satisfy nonparametric restrictions such as monotonicity, concavity, and homogeneity of degree $\nu$; see matzkin92em for examples. These commonly used restrictions translate into equality and inequality constraints of the form $\boldsymbol{r}_1^{\top} \boldsymbol{U}=c_1$ or $\boldsymbol{r}_2^{\top} \boldsymbol{U}\geq c_2$.

We partition the state variable $\boldsymbol{x}$ as $\boldsymbol{x}=(\boldsymbol{w},\boldsymbol{z})$, where $\boldsymbol{z}$ may be empty, and write the payoff function as $u_k(\boldsymbol{w},\boldsymbol{z})$. Assume $\mathcal{X}=\mathcal{W}\times \mathcal{Z}$, where $\mathcal{W} = \{\boldsymbol{w}^1, \boldsymbol{w}^2, \dots, \boldsymbol{w}^{J_w}\}$ and $\mathcal{Z} = \{\boldsymbol{z}^1, \boldsymbol{z}^2, \dots, \boldsymbol{z}^{J_z}\}$. When $\boldsymbol{z}$ is empty, we let $J_z =1$. For simplicity, we assume that the domain of $\boldsymbol{w}$ does not depend on the value of $\boldsymbol{z}$, although the main results below remain valid even if the domain of $\boldsymbol{w}$ varies with $\boldsymbol{z}$. For a set $\mathcal{S}$, define $|\mathcal{S}|^+ \coloneqq \max\{|\mathcal{S}|,1\}$. In the following examples, we consider restrictions on the utility function of a single action $k$. If the same restriction holds across multiple actions, this increases the number of identifying restrictions.

Equality restrictions based on homogeneity assumptions

We first consider the case in which the payoff function $u_k(\boldsymbol{w},\boldsymbol{z})$ is homogeneous in $\boldsymbol{w}$. To define homogeneity formally, we assume that $u_k(\boldsymbol{w},\boldsymbol{z})$ is well-defined at $L$ points $(\widetilde{\boldsymbol{w}}, \lambda_2 \widetilde{\boldsymbol{w}}, \ldots, \lambda_L \widetilde{\boldsymbol{w}}) \in \mathcal{W}$ for some $\lambda_2,\ldots,\lambda_L \in \mathbb{R}^+\setminus\{1\}$.

Equality restrictions based on zero cross-difference assumptions

abbring20qe observe that excluding a variable from current utility yields multiple exclusion restrictions. Suppose $\boldsymbol{z}$ does not affect utilities for some $k \in \mathcal{A}\setminus \{K\}$: $u_k(\boldsymbol{w},\boldsymbol{z})= u_k(\boldsymbol{w})$ for all $(\boldsymbol{w},\boldsymbol{z}) \in \mathcal{W} \times \mathcal{Z}$. This conditon yields $|\mathcal{W}| (|\mathcal{Z}|-1)$ restrictions. If the restriction holds for multiple values of $k$, it may point identify $\beta$. Our zero cross-difference restriction may be viewed as a generalization of this condition, where the difference $u_k(\boldsymbol{w}_2,\boldsymbol{z}) - u_k(\boldsymbol{w}_1,\boldsymbol{z})$ is invariant to $\boldsymbol{z}$.

Inequality restrictions based on monotonicity, concavity, and complementarity/substitutability

Linear-in-parameter payoff function

In some models, the per-period payoff function is parameterized to be linear in parameters. In this case, the identifying constraint can be derived easily.

Suppose that the payoff function $\boldsymbol{U}$ can be expressed as, for a parameter vector $\boldsymbol{\theta}$ and a known $J \times \dim(\boldsymbol{\theta})$ matrix $\boldsymbol{H}$, \[ \boldsymbol{U} = \boldsymbol{H} \boldsymbol{\theta}. \] Let $d_H$ be the dimension of $\operatorname{Ker}(\boldsymbol{H}^{\top})$, and let $\boldsymbol{R}$ be a $d_H \times J$ matrix whose rows form a basis for $ \operatorname{Ker}(\boldsymbol{H}^{\top})$. Then, we have \[ \boldsymbol{R}\boldsymbol{U} = \boldsymbol{R}\boldsymbol{H} \boldsymbol{\theta} =\boldsymbol{0}. \] This provides $d_H$ identifying restrictions, with $d_H=J-\mathrm{rank}(\boldsymbol{H}^{\top}) \leq J - \dim(\boldsymbol{\theta})$.

Identification of $\beta$ by log differences in utilites

In some applications, the difference in log-payoffs, such as $\log(u_k(\boldsymbol{x}^i)) - \log(u_k(\boldsymbol{x}^j))$ and $\log(u_k(\boldsymbol{x}^k))-\log(u_k(\boldsymbol{x}^\ell))$, are related by a known function of $(\boldsymbol{x}^i,\boldsymbol{x}^j,\boldsymbol{x}^k,\boldsymbol{x}^\ell)$. This enables identification of $\beta$ by comparing these two differences.

For a $k$-vector $\boldsymbol{y}$, define $\log(\boldsymbol{y})$ as $(\log(y_1) ,\ldots, \log(y_k))^{\top}$. The following assumption, similar to Assumption (ref), is imposed on log differences in payoffs and includes the additional condition that the elements of $\boldsymbol{r}$ sum to 0, which is satisfied in Examples (ref) and (ref) below.

Rewriting ((ref)) and taking the logarithm elementwise give the system \[ \log (\boldsymbol{U}) = - \log \left(\det\left( \boldsymbol{I} - \beta\boldsymbol{Q}_K\right)\right) \boldsymbol{\iota}+ \log(\boldsymbol{G}(\beta)), \] where $\boldsymbol{\iota}$ is a $(p\times 1)$ vector of ones, and $\boldsymbol{G}(\beta) \coloneqq -\det\left( \boldsymbol{I} - \beta\boldsymbol{Q}_K\right) \boldsymbol{\Psi} + \boldsymbol{Q}(\beta)\operatorname{adj}( \boldsymbol{I} - \beta\boldsymbol{Q}_K) \boldsymbol{\psi}_K$.

Let $G_k(\beta)$ denote the $k$th element of $\boldsymbol{G}(\beta)$. Under Assumption (ref), we have $r_1\log(G_1(\beta)) + \cdots + r_p\log(G_p(\beta)) =c$. Taking the exponential of both sides gives the following proposition.

When some elements of $r_1,\ldots,r_p$ are non-integer, the resulting identifying polynomial in $\beta$ may be of non-integer order. The following examples satisfy Assumption (ref). Assume that $\mathcal{W}$ contains $\widetilde{\boldsymbol{w}}$ and $(\widetilde{\boldsymbol{w}}, \lambda_2 \widetilde{\boldsymbol{w}}, \ldots, \lambda_L \widetilde{\boldsymbol{w}})$ for some $\lambda_2,\ldots,\lambda_L \in \mathbb{R}^+\setminus\{1\}$.

Finite Dependence

AltugMiller98restud and am11em develop the concept of finite dependence and demonstrate that it can substantially reduce the computational cost of dynamic discrete choice models. According to their definition, a model exhibits finite dependence if two action sequences with different initial actions lead to the same state distribution after a finite number of periods.

For brevity, we focus on a single action ($K$) $\rho$-period dependence: for some current action-state pairs $(k,\boldsymbol{x})$, the state distribution at time $\rho + 1$ periods into the future is independent of the current action or state if action $K$ is taken in each of the subsequent $\rho$ periods. In this section, we show that this finite dependence reduces the degree of the identifying polynomial to $\rho$. As a result, the cardinality of the identified set of $\beta$ is no larger than $\rho$.

The literature considers three alternative versions of finite dependence, each differing in how the current action-state pairs are specified:

We introduce the following assumption, which assumption holds if $\{\boldsymbol{Q}_k: k \in \mathcal{A}\}$ satisfies any of the three versions above. For example, setting $\boldsymbol{x}_a=\boldsymbol{x}_b$ in ((ref)) gives the first specification.

The following proposition shows that, under finite dependence, the payoff difference becomes a polynomial of degree $\rho$ in $\beta$. The proof is provided in the Appendix. Define $g(\beta;k,\boldsymbol{x}) \coloneqq -\psi_{k}(\boldsymbol{x}) - \beta \boldsymbol{Q}_{k}(\boldsymbol{x})(\boldsymbol{I}+\beta\boldsymbol{Q}_K + \cdots + \beta^{\rho-1}\boldsymbol{Q}_K^{\rho-1})$.

As in Section (ref), we derive restrictions on $\beta$ implied by equality and inequality constraints on the per-period payoff function. For brevity, we consider a single restriction of the form $\boldsymbol{r}^{\top} \boldsymbol{U}=c$ or $\boldsymbol{r}^{\top} \boldsymbol{U} \geq c$ for a vector $\boldsymbol{r}$. Extending the results to multiple restrictions is straightforward but introduces notational complexity. We first consider the equality case.

Example (ref) satisfies Assumption (ref) with $M=2$, $\alpha_1 = 1/(\lambda_3^\nu-1)$, $\alpha_2 = - 1/(\lambda_2^\nu-1)$, $c=0$, $k_{11}=k_{12}=k_{21}=k_{22}=k$, $\boldsymbol{x}_{11} = (\lambda_3 \widetilde{\boldsymbol{w}}, \boldsymbol{z})$, $\boldsymbol{x}_{12} = \boldsymbol{x}_{22} =(\widetilde{\boldsymbol{w}}, \boldsymbol{z})$, and $\boldsymbol{x}_{21} = (\lambda_2 \widetilde{\boldsymbol{w}}, \boldsymbol{z})$ (see ((ref))) when $\{\boldsymbol{Q}_k: k \in \mathcal{A}\}$ and these action-state pairs satisfy Assumption (ref).

The following corollary follows directly from Proposition (ref) and shows that the cardinality of the identified set of $\beta$ is no greater than $\rho$ if Assumption (ref) holds. For example, in a renewal model where action $K$ resets a state variable to $0$, the identifying equation becomes linear in $\beta$, and $\beta$ is point identified. Note that $\beta=1$ does not necessarily solve this identifying equation.

We can also incorporate inequality restrictions in models with finite dependence.

Numerical Example 1: Dynamic Entry Model

This section uses a dynamic entry model to demonstrate the application of the identifying restrictions discussed above. The firm decides whether to operate ($a = 1$) or not ($a = 2$) after observing $(\boldsymbol{x},\varepsilon)$, where $\varepsilon$ represents unobserved heterogeneity and $\boldsymbol{x} = (w,z,y)$ is the observed state variable. Here, $w$ and $z$ are observable exogenous shocks, and $y$ indicates the firm's past action, indicating its market presence in the previous year.

Model

The state at time $t$ is $(w_{t},z_t,y_t,\varepsilon_t)$, where $y_t=a_{t-1}$, and $\varepsilon_t$ follows a Type-I extreme value distribution. The firm's per-period payoff function (net of $\varepsilon_t$) is

where $\theta_1 + \exp(z) (\theta_2 + \theta_3 w)$ represents variable profit, and $(1-y) \theta_4$ captures the entry cost. The parameter are set as $\theta_1=1$, $\theta_2 = 0.5$, $\theta_3 = 1.0$, and $\theta_4 = 1.0$. The discount factor is set to $\beta = 0.95$.

This model satisfies the homogeneity assumption in Example (ref) with $u_1^w(\boldsymbol{w},\boldsymbol{z}) = \exp(z)(\theta_2 + \theta_3 w)$ and $u_1^z(\boldsymbol{z})=\theta_1 + (1-y)\theta_4$ because $\exp(z)(\theta_2 + \theta_3 w)$ is homogeneous of degree 1 in $w$. It also satisfies zero cross-difference assumption in Section (ref) as $u_1(w, z,y=1;\boldsymbol{\theta}) - u_1(w, z,y=0;\boldsymbol{\theta}) = \theta_4$ for all $(w, z)$.

The exogenous shocks $(w_t,z_t)$ follow two independent $AR(1)$ processes, where $w_{t}$ evolves according to $w_{t} = \gamma_1^{w} w_{t-1} + e^w_{t}$ with $\gamma_1^{w} = 0.5$ and $e^w_{t} \sim \text{i.i.d.} N(0,1)$. The productivity shock $z_t$ follows the process

with parameters $(\gamma_a^z,\gamma_1^z)=(1,0.5)$, and $e^z_{t} \sim \text{i.i.d.} N(0,1)$, independent of $e^w_t$. $\gamma_a^z$ captures the effect of the lagged action $a_{t-1}$ on the transition of $z_t$. Because $\gamma_a^z \neq 0$, the model does not exhibit finite dependence.

We apply the method by tauchen1986finite to discretize these processes into a finite state space. Let $J_w$ and $J_z$ denote the number of discrete grid points for $w_t$ and $z_t$, respectively.\footnote{To discretize $w_t$, we set the endpoints of the grid at the $0.5/J_w$ and $1 - 0.5/J_w$ quantiles of its stationary distribution and place equispaced points between these endpoints. The stationary distribution of $w_t$ is centered at 0 with variance $\sigma_w^2 / (1 - (\gamma_1^w)^2)$. Discretizing $z_t$ is more involved because the center of its stationary distribution depends on the equilibrium conditional choice probability through the lagged action term $\gamma_a^z a_{t-1}$. For simplicity, we center the distribution at $(0.5 \gamma_a^z) / (1 - \gamma_1^z)$.} We set $J_z=3$ and $J_w=3$, so the resulting state space $\mathcal{X}$ has cardinality $J = 2 J_z J_w = 18$. The true value of the discount factor is set to $\beta=0.95$.

Identifying assumptions

While the payoff function ((ref)) is parametric, it satisfies various nonparametric and semiparametric identifying assumptions discussed in Section (ref) as follows. In the following Sections (ref)--(ref), let the values of $w$ and $z$ be ordered as $w^{J_w} > w^{J_w-1} > \cdots > w^1$ and $z^{J_z} > z^{J_z-1} > \cdots > z^1$.

Homogeneity in $w$

We first investigate how $\beta$ is identified by the assumption that $u_1(w,z,y)$ is additively separable with a homogeneous function of degree 1 in $w$. This assumption implies the following restrictions:

for $\ell=1,\ldots,J_w-2$ and for all $(z,y) \in \mathcal{Z} \times \{1,2\}$. We impose these restrictions on $\boldsymbol{U}$ using a matrix \(\boldsymbol{R}_{\text{homo}}\), whose rows correspond to the constraints specified in ((ref)).

Zero cross-difference in $y$ and $(w,z)$

If the entry cost is independent of $w$ and $z$, then

for all $(w, z), (w', z') \in \mathcal{W}\times \mathcal{Z}$. We impose these restrictions on $\boldsymbol{U}$ using a matrix \(\boldsymbol{R}_{\text{zero}}\), whose rows correspond to the restrictions in ((ref)).

Monotonicity in $z$

The monotonicity assumption in $z$ implies that

for $\ell=1,\ldots,J_z-1$ and all $(w,y) \in \mathcal{W} \times \{1,2\}$. We construct a matrix $\boldsymbol{R}_{\text{mono}}$ whose rows correspond to the restrictions in ((ref)), such that the monotonicity constraint is expressed as $\boldsymbol{R}_{\text{mono}} \boldsymbol{U} \geq \boldsymbol{0}$.

Concavity in $z$

The concavity assumption in $z$ implies that \[ \frac{u_1(w, z^{\ell+2}, y) - u_1(w, z^{\ell+1}, y)}{z^{\ell+2} - z^{\ell+1}} - \frac{u_1(w, z^{\ell+1}, y) - u_1(w, z^{\ell}, y)}{z^{\ell+1} - z^{\ell}} \geq 0, \] for $\ell=1,\ldots,J_z-2$ and all $(w,y) \in \mathcal{W} \times \{1,2\}$. We impose these restrictions using a matrix $\boldsymbol{R}_{\text{concav}}$ as $\boldsymbol{R}_{\text{concav}} \boldsymbol{U} \geq \boldsymbol{0}$.

Complementarity between $w$ and $z$

The complementarity assumption between $w$ and $z$ implies that \[ u_1(w^{\ell+1}, z^{m+1}) - u_1(w^{\ell}, z^{m+1}) - u_1(w^{\ell+1}, z^{m+1}) + u_1(w^{\ell}, z^{m}) \geq 0, \] for all $(\ell,m)\in \{1,\ldots,J_w-1\}\times \{1,\ldots,J_z-1\}$. We express this condition via a matrix $\boldsymbol{R}_{\text{comp}}$ as $\boldsymbol{R}_{\text{comp}} \boldsymbol{U} \geq \boldsymbol{0}$.

Linearlity in parameters

Because this model is linear in parameters, the per-period payoff function can be written as \[ \boldsymbol{U} = \boldsymbol{u}_1(\boldsymbol{\theta}) = \boldsymbol{H}\boldsymbol{\theta}, \quad \boldsymbol{H} =

_{(w, z, y) \in \mathcal{X} }, \quad \boldsymbol{\theta} =

^{\top}. \] In this example, the matrix $\boldsymbol{H}$ is $18 \times 4$ and has rank 4. Let the rows of $\boldsymbol{R}$ form a basis for $\operatorname{Ker}(\boldsymbol{H}^{\top})$. It then follows that $\boldsymbol{R}\boldsymbol{U} = \boldsymbol{R}\boldsymbol{H} \boldsymbol{\theta} =\boldsymbol{0}$. The linear-in-parameter assumption imposes $18-4=14$ restrictions on $\beta$ .

Identification of $\beta$

We now investigate how these nonparametric shape restrictions contribute to the identification of $\beta$ without relying on the parametric form in ((ref)). Recall that the true value of $\beta$ is $0.95$.

Figures (ref) and (ref) plot the identifying polynomials in $\beta$ implied by various assumptions in Section (ref) over the ranges $\beta \in [0.85,1.05]$ and $\beta \in [0.0,1.2]$, respectively. The panels titled “Homogeneity in $w$” and “Zero Cross-Difference” plot the polynomials defined in ((ref)) formed by applying the homogeneity assumption ((ref)) and the zero cross-difference assumption ((ref)), respectively. These assumptions impose six and eight restrictions, respectively. The resulting polynomials are of order 18. All curves intersect the horizontal axis at $\beta=0.95$. This suggests that either the homogeneity assumption or the zero cross-difference assumption alone is sufficient to achieve point identification of the discount factor without explicitly specifying the parametric form of the payoff function. In addition, all curves intersect the horizontal axis at $\beta=1$, which is consistent with Proposition (ref).

The panels titled “Monotonicity in $z$,” “Concavity in $z$,” and “Complementarity between $w$ and $z$” depict the polynomials in $\beta$ for which equation ((ref)) holds with equality. The yellow regions indicate the sets of $\beta$ values that satisfy the corresponding inequalities. In Figure (ref), the monotonicity restriction implies $\beta \in [0.1, 0.95]$, the concavity restriction implies $\beta \in [0.69, 0.95]$, and the complementarity assumption yields the bound $\beta \in [0.04,0.95]$, ruling out discount factor values above $0.95$.

The panel titled “Linearity in Parameters” shows the polynomials in $\beta$ implied by the linear-in-parameter restriction $\boldsymbol{U}=\boldsymbol{H}\boldsymbol{\theta}$. All curves intersect the horizontal axis at the true value $\beta=0.95$, which illustrates the strong identification power of the linearity assumption.

Finite dependence case

When $\gamma_a^z=0$ in ((ref)), $z_t$ becomes exogenous, and the model exhibits $1$-dependence. Then, Proposition (ref) implies that the identifying equations ((ref))--((ref)) are linear in $\beta$. We confirm this implication.

Figure (ref) displays the identified set of $\beta$ implied by the assumptions in Section (ref) when $\gamma_a^z=0$. Consistent with Corollaries (ref) and (ref), all plotted lines are linear in $\beta$ and intersect the horizontal axis at $\beta=0.95$. In this model, the inequality restrictions imply $\beta \geq 0.95$. Notably, the zero cross-difference assumption alone identifies $\beta=0.95$ even without imposing any functional form restrictions on the payoff function in terms of $w$ and $z$.

Models of Dynamic Discrete Games

Framework and basic assumptions

We consider the model of dynamic discrete games studied by am07em. There are $N$ firms operating in the market. Each firm $i$ selects an action $a_{it}$ in period $t$ from the set $\mathcal{A}_i = \{1,\ldots, K\}$. The joint action space is $\mathcal{A} \coloneqq \times_{i=1}^N \mathcal{A}_i$. Define $\boldsymbol{a}_t \coloneqq (a_{1t}, \ldots, a_{Nt})$. Let $\boldsymbol{d}_t$ denote demand shifters, and define the observable state variables as $\boldsymbol{x}_t \coloneqq (\boldsymbol{d}_t,\boldsymbol{x}_{1t}, \ldots, \boldsymbol{x}_{Nt}) \in \mathcal{X}$. Firm $i$'s private information shock is $\boldsymbol{\varepsilon}_{it} \coloneqq (\varepsilon_{it}(1), \ldots, \varepsilon_{it}(K)) \in \mathbb{R}^K$, which is i.i.d.\ over time and independent across firms, with a density $g_i(\boldsymbol{\varepsilon}_{it})$ that is absolutely continuous with respect to the Lebesgue measure. Firm $i$'s per-period payoff is $\widetilde\pi_i(\boldsymbol{a}_{t}, \boldsymbol{x}_t,\boldsymbol{\varepsilon}_{it})$. Set $m_a \coloneqq |\mathcal{A}|$ and $m_x \coloneqq |\mathcal{X}|$, with $\mathcal{X} = \{\boldsymbol{x}^1,\ldots,\boldsymbol{x}^{m_x}\}$. Define $\boldsymbol{a}_{-it}$ as the action profile of all firms except $i$, with corresponding action space $\mathcal{A}_{-i} \coloneqq \times_{j \neq i} \mathcal{A}_j$. The joint private information vector is $\boldsymbol{\varepsilon}_t \coloneqq(\boldsymbol{\varepsilon}_{1t}, \ldots, \boldsymbol{\varepsilon}_{Nt})$. The sequence $\{\boldsymbol{x}_t, \boldsymbol{\varepsilon}_t\}$ follows a controlled Markov process with transition probability $p(\boldsymbol{x}_{t+1}, \boldsymbol{\varepsilon}_{t+1}|\boldsymbol{x}_t, \boldsymbol{\varepsilon}_t, \boldsymbol{a}_t)$.

Each firm maximizes the expected discounted sum of current and future payoffs, \[ E \left\{ \sum_{s=t}^\infty \beta_i^{s-t} \, \widetilde\pi_i(\boldsymbol{a}_s,\boldsymbol{x}_s,\boldsymbol{\varepsilon}_{is}) \middle|\boldsymbol{x}_t, \boldsymbol{\varepsilon}_{it} \right\}, \] where the discount factor $\beta_i \in [0,1)$ may differ across firms. We adopt the following standard assumptions. Let $Q(\boldsymbol{x}_{t+1}|\boldsymbol{x}_t, \boldsymbol{a}_t)$ denote the transition probability of $\boldsymbol{x}_t$.

We assume firms follow stationary Markov strategies and hence omit the time superscript henceforth. Let $\boldsymbol{x}'$ and $\boldsymbol{\varepsilon}'$ denote the next-period state variables. Let $\sigma = \{\sigma_i(\boldsymbol{x},\boldsymbol{\varepsilon}_i)\}_{i=1}^N$ denote the strategy profile, where each firm $i$'s strategy function is $\sigma_i: \mathcal{X} \times \mathbb{R}^K \to \mathcal{A}_i$. am07em establish the existence of Markov perfect equilibrium (MPE) strategies in this model. Let $\boldsymbol{P}^* \coloneqq \{[P_i^*(a|\boldsymbol{x})]_{a \in \mathcal{A}_i, \boldsymbol{x}\in\mathcal{X}}\}_{i=1}^N$ denote the equilibrium choice probabilities corresponding to an MPE strategy profile $\sigma^*$. Define

as the equilibrium conditional choice probability of all firms other than firm $i$, where $a_{-i}[j]$ is the $j$th firm's element in $\boldsymbol{a}_{-i}$. Under equilibrium choice probabilities $\boldsymbol{P}^*$, firm $i$'s expected payoff and expected transition probability are given by

and \[ Q_i^*(\boldsymbol{x}'|\boldsymbol{x},a_i) \coloneqq \sum_{\boldsymbol{a}_{-i} \in \mathcal{A}_{-i}} P_{-i}^*(\boldsymbol{a}_{-i}|\boldsymbol{x}) Q(\boldsymbol{x}'|\boldsymbol{x},a_i,\boldsymbol{a}_{-i}). \]

As shown by am07em, the equilibrium choice probability satisfies \[ P^*_i(a_i|\boldsymbol x) = \int \mathds{1}\left\{a_i=\operatorname*{arg\,max}_{k \in \mathcal{A}_i} \left\{ v_i^{*}(k,\boldsymbol{x}) + \varepsilon_i(k) \right\}\right\} g_i(\boldsymbol{\varepsilon}_i) d\boldsymbol{\varepsilon}_i, \] where the equilibrium choice-specific value function $v_i^{*}(k,\boldsymbol{x})$ is defined as

and the integrated value function $V_i^*$ satisfies the integrated Bellman equation

Implications of the model

This section derives the equations summarizing the model's restrictions. Following am07em and pesendorfer08restud, we assume the data are generated by a single Markov perfect equilibrium, though the model may admit multiple equilibria.

We now express $\pi_i^*(a_i,\boldsymbol{x})$ in ((ref)) in terms of the payoff function and equilibrium choice probabilities. Define $\boldsymbol{P}_{-i}^{*}(\boldsymbol{x}) \coloneqq[P_{-i}^*(\boldsymbol{a}_{-i}|\boldsymbol{x})]_{\boldsymbol{a}_{-i} \in \mathcal{A}_{-i}}$ as the $K^{N-1} \times 1$ vector of conditional choice probabilities of firms other than firm $i$. Similarliy, define the $K^{N-1} \times 1$ vector $\boldsymbol{\pi}_i^{-i}(a_i,\boldsymbol{x}) \coloneqq [\pi_i(a_i,\boldsymbol{a}_{-i},\boldsymbol{x})]_{\boldsymbol{a}_{-i} \in \mathcal{A}_{-i}}$. Then, from ((ref)), \[ \pi_i^*(a_i,\boldsymbol{x}) = \boldsymbol{P}_{-i}^{*}(\boldsymbol{x})^{\top} \boldsymbol{\pi}_i^{-i}(a_i,\boldsymbol{x}). \] Collect $v_i^*(k,\boldsymbol{x})$, $V_i^*(\boldsymbol{x})$, and $\boldsymbol{\pi}_{i}^{-i}(k,\boldsymbol{x})$ across all $\boldsymbol{x} \in \mathcal{X}$ into two $m_x \times 1$ vectors and one $(K^{N-1} \cdot m_x) \times 1$ vector, respectively, as \[ \boldsymbol{v}_{ik}^* \coloneqq

,\quad \boldsymbol{V}_{i}^* \coloneqq

,\quad \boldsymbol{\pi}_{ik}^{-i} \coloneqq

. \] Collect $\boldsymbol{P}_{-i}^{*}(\boldsymbol{x})^{\top}$ across all $\boldsymbol{x} \in \mathcal{X}$ into a $m_x \times (K^{N-1}\cdot m_x)$ matrix as \[ \boldsymbol{P}_{-i}^{*} \coloneqq

, \] Let $\boldsymbol{Q}_{ik}^*$ be the $m_x \times m_x$ matrix with the $(\ell,m)$-th entry $Q_i^*(\boldsymbol{x}^m|\boldsymbol{x}^\ell,k)$. With this notation, we can stack ((ref)) at $a_i=k$ across all states $\boldsymbol{x} \in \mathcal{X}$ as

This corresponds to ((ref)) in the single-agent model.

We proceed to derive the equation corresponding to ((ref)). The following lemma is a multiple-agent counterpart to Lemma 1 of am11em, and its proof is provided in the Appendix. Collect $P_i^*(a_i|\boldsymbol{x})$ into the vector $\boldsymbol{P}_i^*(\boldsymbol{x}) \coloneqq (P_i^*(1|\boldsymbol{x}),\ldots,P_i^*(K|\boldsymbol{x}))^{\top}$.

Let $\boldsymbol{\psi}_{ik}^*$ be the $m_x \times 1$ vector whose $j$th element is $\psi_{ik}(\boldsymbol{P}_i^*(\boldsymbol{x}^j))$. Collecting ((ref)) across all $\boldsymbol{x} \in \mathcal{X}$ yields the following equation, which corresponds to ((ref)) in the single-agent model:

Identifying equations

We derive the identifying equation for $\beta_i$ from ((ref)) and ((ref)). Subtracting ((ref)) from ((ref)) at $k=K$ gives $\boldsymbol{V}_i^* - \boldsymbol{\psi}_{iK}^* = \boldsymbol{P}_{-i}^{*} \boldsymbol{\pi}_{iK}^{-i} + \beta_i \boldsymbol{Q}_{iK}^*\boldsymbol{V}_i^*$, and we obtain

Eliminating $\boldsymbol{v}_{ik}^*$ from ((ref)) and ((ref)) gives $\boldsymbol{P}_{-i}^{*} \boldsymbol{\pi}_{ik}^{-i} = -\boldsymbol{\psi}_{ik}^* + (\boldsymbol{I} - \beta_i \boldsymbol{Q}_{ik}^*) \boldsymbol{V}_i^*$. Substituting ((ref)) into the right hand side gives

All terms in ((ref)) are functions of observables except for $(\boldsymbol{\pi}_{ik}^{-i}, \boldsymbol{\pi}_{iK}^{-i}, \beta_i)$. Combining ((ref)) for $k=1,\ldots,K-1$ gives a system of $(K-1) \cdot m_x$ linear equations in the unknowns $\{ \boldsymbol{\pi}_{ik}^{-i} \}_{k=1}^K$, which together have $K^N \cdot m_x$ elements. Thus, if $\beta_i$ were known, an additional $(K^{N} - K + 1) \cdot m_x$ linear restrictions on $\{ \boldsymbol{\pi}_{ik}^{-i} \}_{k=1}^K$ would be required to identify the payoff function pesendorfer08restud.

We introduce two types of restrictions commonly used in applications: (a) a known payoff for one action (typically non-entry) and (b) the irrelevance of other firms' lagged actions. These correspond to equations (17) and (16) in pesendorfer08restud, respectively. The first restriction normalizes the payoff for one action.

Using Assumption (ref), we derive a system of linear equations from ((ref)). Define $d_i(\beta_i) \coloneqq \det(\boldsymbol{I} - \beta_i\boldsymbol{Q}_{iK}^*)$ and $\boldsymbol{A}^*_{ik}(\beta_i) \coloneqq (\boldsymbol{I} - \beta_i \boldsymbol{Q}_{ik}^*)\operatorname{adj}(\boldsymbol{I} - \beta_i\boldsymbol{Q}_{iK}^*)$. Rewrite ((ref)) as

Note that ((ref)) is a system of $m_x$ equations, with the right hand side known from Assumption (ref). Define \[ \boldsymbol{\Pi}_i \coloneqq

, \quad \boldsymbol{\Psi}_i^* \coloneqq

. \] Let $q_1 \coloneqq(K-1) m_x$ and $m_{\Pi} \coloneqq \dim(\boldsymbol{\Pi}_i) = (K-1)\cdot K^{N-1}\cdot m_x$. Define the $q_1 \times m_{\Pi}$ block-diagonal matrix $\overline{\boldsymbol{P}}_i^*$ and the $q_1 \times ( K m_x)$ matrix $\boldsymbol{A}_i^*(\beta_i)$ as \[ \overline{\boldsymbol{P}}_i^* \coloneqq \left[\smash{\underbrace{ \left.

\right.}_{K-1 times}} \left.

\right. \right] , \quad \boldsymbol{A}_i^*(\beta_i) \coloneqq

. \] Stacking ((ref)) for $k=1,\ldots,K-1$, we obtain

This system of $q_1$ linear equations in $m_{\Pi}$ unknowns summarizes the restrictions imposed by the model and Assumption (ref).

We introduce a second restriction on $\boldsymbol{\Pi}_i$. Split the state variable as $\boldsymbol{x}_t = (S_t,\boldsymbol{a}_{t-1})$, where $S_t \in \mathcal{S}$ denotes an exogenous state variable (e.g., market size) with $m_s \coloneqq |\mathcal{S}|$.

Assumption (ref) generates $(K-1) \cdot |\mathcal{A}_{-i}| \cdot m_s \cdot |\mathcal{A}_i|\cdot (|\mathcal{A}_{-i}|-1) = (K-1) \cdot (K^{N-1}-1) \cdot m_x$ linear restrictions. Let $q_2 \coloneqq (K-1) \cdot (K^{N-1}-1) \cdot m_x$ denote the number of these restrictions. If $\beta_i$ is known, Assumptions (ref) and (ref) together just identify $\boldsymbol{\Pi}_i$ because $q_1+q_2 = (K-1)\cdot m_x +(K-1) \cdot (K^{N-1}-1) \cdot m_x = (K-1) \cdot K^{N-1} \cdot m_x = m_\Pi$, provided that a suitable rank condition holds.\footnote{pesendorfer08restud define the payoff function as $\pi_i(\boldsymbol{a},\boldsymbol{x})$, where $\boldsymbol{x} = (\boldsymbol{x}_{1}, \ldots, \boldsymbol{x}_{N})$ and $\boldsymbol{x}_{i} \in \mathcal{X}_i$ with $|\mathcal{X}_i|=L$. They show that the payoff function is identified if $\beta_i$ is known, assumptions similar to Assumptions (ref) and (ref) hold, $L \geq K$, and a rank condition is satisfied, where pesendorfer08restud state this condition as “$L \geq K+1$” in because they define $\mathcal{A}_i$ as $\{0,1,\ldots,K\}$. In our setting, the condition $L \geq K$ is automatically satisfied because the state variable $\boldsymbol{x}$ includes the lagged action profile $\boldsymbol{a}_{t-1}$.}

Identification of $\beta_i$ by additional economic restrictions

Assumptions (ref) and (ref) identify $\boldsymbol{\Pi}_i$ if $\beta_i$ is known. To identify $\beta_i$, we require additional restrictions on $\boldsymbol{\Pi}_i$. To this end, we can leverage homogeneity and other assumptions introduced in Section (ref). Furthermore, models of dynamic games often exhibit structural properties that impose additional constraints not present in single-agent models, which facilitate the identification of $\beta$.

Henceforth, we impose Assumption (ref) on the payoff function and write it as $\pi_i(a_{it},\boldsymbol{a}_{-it},S_t,a_{i,t-1})$. Our first additional restriction is the exchangeability of other firms' actions.

Assumption (ref), or a variant thereof, holds when firm decisions are based on aggregate industry states rather than on the individual states of other firms, a popular specification in the literature (see, e.g., EricsonPakes1995, pakes07rand, am07em, ryan2012, collardwexler2013demand, Benkard2015, igami2017estimating, and igami2020mergers).\footnote{ Benkard2015 assumes that a firm's profit depends on its own status and an aggregate industry state, measured by the number of firms at each quality level, while igami2017estimating defines aggregate states as the counts of firms in each of the four technological categories: (1) “old only,” (2) “both,” (3) “new only,” and (4) “potential entrant.”} For example, Assumption (ref) holds if the payoff depends on $\boldsymbol{a}_{-it}$ only through the sum $\sum_{j \neq i} a_{jt}$. When $K=2$, Assumption (ref) implies $(2^{N-1}-N)\cdot m_s \cdot 2$ restrictions. It requires that the number of firms $N \geq 3$.

The following assumption imposes that firm $i$'s adjustment cost of changing states is independent of other firms' actions.

Assumption (ref) is commonly imposed in empirical studies of dynamic games with adjustment costs and holds when entry or capacity adjustment costs are independent of other firms' actions. Examples include Aguirregabiria2020 and Hao2023, as well as the studies cited after Assumption (ref). This assumption provides at least $\kappa \cdot (K^{N-1} - 1)$ restrictions, where $\kappa$ is the number of $(a_{it}, S_t)$ pairs for which the assumption holds. In an entry model with $\mathcal{A}_i = \{1, 2\}$, where action $2$ denotes non-entry, Assumption (ref) holds if the entry cost $\pi_i(1,\boldsymbol{a}_{-it},S_t,1) - \pi_i(1,\boldsymbol{a}_{-it},S_t,2)$ is independent of $\boldsymbol{a}_{-it}$.

The following assumption provides a sufficient condition for identifying $\beta_i$. Under Assumptions (ref) and (ref), the additional restrictions required for this condition are supplied by Assumption (ref), Assumption (ref), or the equality restrictions discussed in Section (ref), (ref), and (ref).

Assumption (ref)(b) ensures ((ref)) does not contain redundant restrictions. We proceed to derive the identifying polynomial in $\beta_i$ under these restrictions. Because $\boldsymbol{X}_i$ has full column rank by Assumption (ref)(a), we can, after rearranging rows if necessary, write \[ \boldsymbol{X}_i =

, \] where $\boldsymbol{X}_{i1}$ is an $m_{\Pi}\times m_{\Pi}$ invertible matrix. Split $\boldsymbol{Y}_i(\beta_i)$ conformably as $\boldsymbol{Y}_{i1}(\beta_i)$ and $\boldsymbol{Y}_{i2}(\beta_i)$, so that ((ref)) becomes $\boldsymbol{X}_{i1}\boldsymbol{\Pi}_i = \boldsymbol{Y}_{i1}(\beta_i)/d_i(\beta_i)$ and $\boldsymbol{X}_{i2}\boldsymbol{\Pi}_i = \boldsymbol{Y}_{i2}(\beta_i)/d_i(\beta_i)$. The first equation implies $\boldsymbol{\Pi}_i = \boldsymbol{X}_{i1}^{-1}\boldsymbol{Y}_{i1}(\beta_i)/d_i(\beta_i)$. Substituting this into the second equation gives the following proposition.

The degree of each polynomial in this system is $m_x$ because the elements of $\boldsymbol{Y}_i(\beta_i)$ are linear function of $d_i(\beta_i)$ and $\{(\boldsymbol{I} - \beta_i \boldsymbol{Q}_{ik}^*)\operatorname{adj}(\boldsymbol{I} - \beta_i\boldsymbol{Q}_{iK}^*)\}_{k=1}^{K-1}$.

We can use inequality restrictions to refine the identified set of $\beta_i$ obtained from equality constraints. Such inequality restrictions arise from assumptions such as monotonicity, concavity, and complementarity, as discussed in Section (ref). We present two examples that are applicable to models of dynamic games. Arrange the values of $a_{it}$ so that smaller values correspond to “stronger” actions by firm $i$. The first example concerns monotonicity with respect to the firm's own lagged action.

It is often natural to assume that the payoff function is monotonic with respect to other firms' actions. Our second example incorporates this restriction. Define the partial order \(\preceq\) on the set of $N$-dimensional vectors as follows: $\boldsymbol{a} \preceq \boldsymbol{b}$ if and only if $a_i \leq b_i$ for all $i \in \{1, 2, \dots, N\}$.

The following assumption summarizes these inequality restrictions.

We derive the identified set of $\beta_i$ incorporating the inequality restrictions. The model, together with Assumption (ref), imposes the restriction $d_i(\beta_i) \overline{\boldsymbol{P}}_i^* \boldsymbol{\Pi}_i = \boldsymbol{A}_i^*(\beta_i)\boldsymbol{\Psi}_i^*$ as stated in ((ref)). Note that the expected payoff $\overline{\boldsymbol{P}}_i^* \boldsymbol{\Pi}_i$, rather than the payoff $\boldsymbol{\Pi}_i$, appears on the left hand side. To incorporate inequality constraints on $\boldsymbol{\Pi}_i$, we first express $\boldsymbol{\Pi}_i$ in the form $\boldsymbol{\Pi}_i= \boldsymbol{M}(\beta_i)$ for some matrix $\boldsymbol{M}(\beta_i)$. This transformation is feasible if Assumption (ref) and additional equality restrictions---such as those implied by Assumptions (ref) or (ref), or discussed in Sections (ref), (ref), and (ref)---provide sufficient identifying information.

We first derive the identified set of $\beta_i$ when only Assumptions (ref) and (ref) provide equality restrictions.

Under Assumption (ref), we can express $\boldsymbol{\Pi}_i$ as $\boldsymbol{\Pi}_i = \boldsymbol{X}_{ai}^{-1}\boldsymbol{Y}_{ai}(\beta_i)/d_i(\beta_i)$. Applying Assumption (ref) to both sides and noting that $d_i(\beta_i)>0$ lead to the following proposition.

The following proposition characterizes the identified set when additional equality restrictions satisfying Assumptions (ref) are available and Assumption (ref) holds.

Numerical Example 2: Dynamic Game Model

This section illustrates the application of the identifying restrictions to the dynamic game model discussed in Section (ref). We consider a market with $N = 3$ firms. In each period, firm $i$ decides whether to operate ($a_{it} = 1$) or not ($a_{it} = 2$). The current-period payoff for firm $i$, $\widetilde\pi_i(\boldsymbol{a}_{t}, \boldsymbol{x}_t,\boldsymbol{\varepsilon}_{it})$, is given by \[ \theta_{RS} \log S_{t} - \theta_{RN} \log \left(1+\sum_{j\neq i}(2-a_{jt})\right) -\theta_{FC,i} -\theta_{EC}(1-a_{i,t-1})+\varepsilon_{it}(1), \quad \text{if } a_{it} = 1, \] and $\varepsilon_{it}(2)$ if $a_{it} = 2$. The pair $(\varepsilon_{it}(1), \varepsilon_{it}(2))$ is drawn from an i.i.d.\ type-I extreme value distribution. The market size $S_{t}$ follows an exogenous first-order Markov process.\footnote{The state space of $S_{t}$ is $\{2,6,10\}$. The transition probability matrix of $S_{t}$ is $

. $} We set the firm-specific discount factors to $(\beta_1,\beta_2,\beta_3) = (0.8, 0.9, 0.95)$. The parameter values are set to $\theta_{RS}=1.0$, $\theta_{EC}=1.0$, $\theta_{FC,1}=1.0$, $\theta_{FC,2}=0.9$, and $\theta_{FC,3}=0.8$.

Figures (ref) and (ref) plot the identifying polynomials in $\beta$ implied by the assumptions in Section (ref) over the range $\beta \in [0.75, 1.05]$ for firm 1 and 2, respectively. The result for firm 3, not presented here, is qualitatively similar. The panels titled “Irrelevance of Other Firms' Lagged Actions and Exchangeability” and “Irrelevance of Other Firms' Lagged Actions and Independence of Entry” plot 6 and 8 identifying polynomials in $\beta$ formed by applying Assumptions (ref), (ref), (ref) and Assumptions (ref), (ref), (ref), respectively. In both figures, all curves intersect the horizontal axis at the true value of $\beta$, $0.8$ for firm 1 and $0.9$ for firm 2, indicating the strong identifying power of these assumptions. These are the only values in $[0,1)$ at which all curves intersect the horizontal axis.

The panel titled “Monotonicity in Other Firms' Actions” depicts the polynomials in $\beta$ for which equation ((ref)) holds with equality under Assumptions (ref), (ref), and (ref). The yellow region indicates the values of $\beta$ for which the corresponding inequality is satisfied. In this model, the inequality constraints are not informative because they imply only the bound $\beta \leq 1$. The inequality constraints based on Assumptions (ref), (ref), and (ref) also imply the bound $\beta \leq 1$.

The panel titled “Linearity in Parameters” applies the linear-in-parameter assumption, which yields 20 identifying restrictions for $\beta$. In both figures, all curves intersect the horizontal axis at the true value of $\beta$, $0.8$ for firm 1 and $0.9$ for firm 2. These are the only values in $[0,1)$ at which all curves intersect the horizontal axis.

Conclusion

This paper studies the identification of discount factors and payoff functions in standard stationary infinite-horizon dynamic discrete choice models. We show that commonly used nonparametric assumptions on per-period payoffs---such as homogeneity, monotonicity, concavity, and zero cross-differences---provide substantial identification power for the discount factor. Leveraging these nonparametric shape restrictions, we derive equality and inequality constraints in the form of finite-order polynomial equations that characterize the identified set. We further extend the analysis to dynamic games and highlight the identifying power of assumptions such as the irrelevance of other firms' lagged actions, exchangeability, and independence of adjustment costs from other firms' actions. An important direction for future research is to develop practical estimation procedures to implement the identification strategies presented herein. Such methods would facilitate empirical analysis and enhance the applicability of dynamic discrete choice models to real-world economic questions and counterfactual policy interventions.