Identification and counterfactuals in dynamic models of market entry and exit

Abstract

This paper addresses a fundamental identification problem in the structural estimation of dynamic oligopoly models of market entry and exit. Using the standard datasets in existing empirical applications, three components of a firm’s profit function are not separately identified: the fixed cost of an incumbent firm, the entry cost of a new entrant, and the scrap value of an exiting firm. We study the implications of this result on the power of this class of models to identify the effects of different comparative static exercises and counterfactual public policies. First, we derive a closed-form relationship between the three unknown structural functions and the two functions that are identified from the data. We use this relationship to provide the correct interpretation of the estimated objects that are obtained under the ‘normalization assumptions’ considered in most applications. Second, we characterize a class of counterfactual experiments that are identified using the estimated model, despite the non-separate identification of the three primitives. Third, we show that there is a general class of counterfactual experiments of economic relevance that are not identified. We present a numerical example that illustrates how ignoring the non-identification of these counterfactuals (i.e., making a ‘normalization assumption’ on some of the three primitives) generates sizable biases that can modify even the sign of the estimated effects. Finally, we discuss possible solutions to address these identification problems.

Appendices

A. Extensions of the basic model

1.1 A.1 Model with no re-entry after market exit and no waiting before market entry

Some models and empirical applications of industry dynamics assume that a new entrant has only one opportunity to enter and an incumbent can not reenter after exit from the market (e.g., Ryan (2012)). This model is practically the same as the basic model presented in section 2.1, with the only difference that the value of not entering for apotential entrant is 0 and the value of exiting for an incumbent is the scrap value, i.e., v(0, 0, z _t ) = 0 and v(0, 1, z _t ) = $sv(\mathbf {z}_{t}^{c})+\varepsilon _{t}^{sv}$.

1.2 A.2 Model with time-to-build and time-to-exit

In this version of the model, it takes one period to make entry and exit decisions effective, though the entry cost is paid at the period when the entry decision is made, and similarly the scrap value is received at the period when the exit decision is taken. Now, a _t is the binary indicator of the event “the firm will be active in the market at period t+1”, and k _t = a _t−1 is the binary indicator of the event “the firm is active in the market at period t”. For this model, the one-period profit function is:

$$\Pi_{t}=\left\{ \begin{array}{lc} k_{t}[vp(\mathbf{z}_{t}^{v})-fc\left(\mathbf{z}_{t}^{c}\right) +sv\left(\mathbf{z}_{t}^{c}\right) ]+\varepsilon_{t}(0) & \text{if }a_{t}=0 \\ & \\ k_{t}[vp(\mathbf{z}_{t}^{v})-fc\left(\mathbf{z}_{t}^{c}\right) ]-\left(1-k_{t}\right) ec\left(\mathbf{z}_{t}^{c}\right) +\varepsilon_{t}(1) & \text{if }a_{t}=1 \end{array} \right.$$

(A.1)

Given this structure of the profit function, we have that the Bellman equation, optimal decision rule, and CCP function are defined exactly the same as above in Equations (6), (9), and (10), respectively.

1.3 A.3 Model with investment

The basic model and the previous extensions assume that the only dynamic decision of a firm is to be active or not in the market. However, our (non) identification results extend to more general models where incumbent firms make investments in product quality, capacity, etc. Here we present a relatively simple model with investment. Suppose that there is a quasi-fixed input, say capital, and the firm decides every period the amount of capital to use. Let a _t ∈ {0, 1, ⋯ , K} denote the firm’s decision at period t where K is the largest possible capital level. When a _t is zero, the firm is inactive in the current period. The firm’s variable profit depends on the current amount of capital a _t , e.g., the amount of capital may affect the quality of the product and therefore demand, and also variable costs. The fixed profit depends both on current capital a _t and on the amount of capital installed at previous period, k _t ≡ a _t−1. The one-period profit function of this firm is:

$$\Pi_{t}=\left\{ \begin{array}{lc} -ic\left(0, k_{t}, \mathbf{z}_{t}^{c}\right) +\varepsilon_{t}(0) & \text{if } a_{t}=0 \\ & \\ vp(a_{t}, \mathbf{z}_{t}^{v})-fc\left(a_{t}, \mathbf{z}_{t}^{c}\right) -ic\left(a_{t}, k_{t}, \mathbf{z}_{t}^{c}\right) +\varepsilon_{t}(a_{t}) & \text{if }a_{t}>0 \end{array} \right.$$

(A.2)

where $ic\left (a_{t}, k_{t}, \mathbf {z}_{t}^{c}\right )$ is the investment cost function, which represents the cost the firm incurs to change its capital level from k to a taking $\mathbf {z}_{t}^{c}$ as given. We assume $ic\left (a_{t}, k_{t}, \mathbf {z}_{t}^{c}\right ) =0$ when a _t = k _t . In this specification, $ic\left (a, 0, \mathbf {z}_{t}^{c}\right )$ represents the entry cost for a firm that decides to enter in the market with an initial level of capital equal to a. Similarly, $-ic\left (0, k, \mathbf {z}_{t}^{c}\right )$ represents the scrap value of an incumbent firm with installed capital equal to k. It is clear that our baseline model is a special case of this general model when K = 1.

1.4 A.4 Identification of the model with no re-entry after market exit and no waiting before market entry

In this extension, v(0, k, z _t ) is equal to k sv(z _t ). The system of equations corresponding to (19) is:

$$\begin{array}{ccl} \tilde{v}(k, \mathbf{z}) & = & vp(\mathbf{z}^{v})-\left[ fc\left(\mathbf{z} ^{c}\right) +ec\left(\mathbf{z}^{c}\right) \right] +k\left[ ec\left(\mathbf{z}^{c}\right) -sv\left(\mathbf{z}^{c}\right) \right] +\beta \sum_{\mathbf{z}^{c\prime }\in \mathbf{Z}^{c}}f_{z}(\mathbf{z} ^{c\prime }|\mathbf{z})sv\left(\mathbf{z}^{c\prime}\right) \\ & & \\ & & +\beta \sum_{\mathbf{z}^{\prime }\in \mathbf{Z}}f_{z}(\mathbf{z} ^{\prime }|\mathbf{z})S(\tilde{v}\left(1, \mathbf{z^\prime}\right) , F_{ \widetilde{\varepsilon }|1}) \end{array}$$

(A.3)

Define the function Q(k, z) ≡ $\tilde {v}(k, \mathbf {z})-\beta \sum _{\mathbf {z}^{\prime }\in \mathbf {Z}}f_{z}(\mathbf {z}^{\prime }|\mathbf {z })$ $S(\tilde {v}\left (1, \mathbf {z}^{\prime }\right ) , F_{\widetilde { \varepsilon }|1})$. With this new definition of Q(k, z), we have exactly the same system of equations as (20). The relationship reported in Table 1 is directly applicable to this extension.

1.5 A.5 Identification of the model with time-to-build and time-to-exit

Most of the expressions for the basic model still hold for this extension, except that now the one-period payoff π(a, k, z) has a different form. In particular, now π(1, k, z) − π(0, k, z)= − k sv(z ^c)−(1 − k) ec(z ^c), and this implies that the expression for the differential value function $\tilde {v}(k, \mathbf {z})$ is:

$$\tilde{v}(k, \mathbf{z})=-ksv(\mathbf{z}^{c})-(1-k)ec\left(\mathbf{z}^{c}\right) +\beta \sum_{\mathbf{z}^{\prime }\in \mathbf{Z}}f_{z}(\mathbf{z}^{\prime }|\mathbf{z})\left[ \bar{V}\left(1, \mathbf{z}^{\prime }\right) -\bar{V}\left(0, \mathbf{z}^{\prime }\right) \right]$$

(A.4)

Also, now we have that v(0, 1, z) − v(0, 0, z)=π(0, 1, z) − π(0, 0, z)=vp(z ^v) − fc(z ^c) + sv(z ^c). Therefore, the system of identifying restrictions (20) becomes:

$$\begin{array}{ccl} Q(k, \mathbf{z}) & = & -ksv(\mathbf{z}^{c})-(1-k)ec\left(\mathbf{z}^{c}\right) +\beta \sum\limits_{\mathbf{z}^{\prime }\in \mathbf{Z} }f_{z}(\mathbf{z}^{\prime }|\mathbf{z})[vp(\mathbf{z}^{v\prime })-fc\left(\mathbf{z}^{c\prime }\right) +sv\left(\mathbf{z}^{c\prime }\right) ] \end{array}$$

(A.5)

where Q(k, z) has exactly the same definition as before in the model without time-to-build. Given this system of equations, Proposition 2 also applies to this model with the only difference that now we have the following relationship between true functions and identified objects:

$$\begin{array}{lll} ec\left(\mathbf{z}^{c}\right) -sv\left(\mathbf{z}^{c}\right) & = & Q(1, \mathbf{z})-Q(0, \mathbf{z}) \\ & & \\ ec\left(\mathbf{z}^{c}\right) +\beta \sum\limits_{\mathbf{z}^{c\prime }\in \mathbf{Z}^{c}}f_{z}(\mathbf{z}^{c\prime }|\mathbf{z})[fc\left(\mathbf{z}^{c\prime }\right) -sv\left(\mathbf{z}^{c\prime }\right) ] & = & -Q(0, \mathbf{z})+\beta \sum\limits_{\mathbf{z}^{v\prime }\in \mathbf{Z} ^{v}}f_{z}(\mathbf{z}^{v\prime }|\mathbf{z})vp(\mathbf{z}^{v\prime }) \end{array}$$

(A.6)

The first equation is exactly the same as in the model without time to build. The second equation is slightly different: instead of current value of variable profit minus the fixed cost, vp(z ^v) − fc(z ^c), now we have the discounted and expected value of this function at the next period, i.e., $\beta \sum _{\mathbf {z}^{\prime }\in \mathbf {Z}}f_{z}(\mathbf {z}^{\prime }|\mathbf {z})$ [vp(z ^v′) − fc(z ^c′)]. Table 3 reports the relationship between estimated functions and unknown structural functions.

Table 3 Interpretation of Estimated Structural Functions in the Model with Time-to-Build

1.6 A.6 Identification of the model with investment

Under mild regularity conditions the CCPs P(a|k, z) are identified, and given CCPs we can also identify the differential value function $\tilde {v}(a, k, \mathbf {z})\equiv$ v(a, k, z)− v(0, k, z). The model implies the following restrictions:

$$\begin{array}{lll} && \tilde{v}\left(a, k, \mathbf{z}\right) = vp(a, \mathbf{z}^{v})-fc\left(a, \mathbf{z}^{c}\right) -ic\left(a, k, \mathbf{z}^{c}\right) +ic\left(0, k, \mathbf{z}^{c}\right) \\ && +\beta \sum\limits_{\mathbf{z}^{\prime }\in \mathbf{Z} }f_{z}\left(\mathbf{z}^{\prime }|\mathbf{z}\right) \left[ \bar{V}\left(a, \mathbf{z}^{\prime }\right) -\bar{V}\left(0, \mathbf{z}^{\prime }\right) \right] \end{array}$$

(27)

The same logic used to derive (18 ) implies that $\bar {V}\left (k, \mathbf {z}\right ) =$ v(0, k, z)+$S(\tilde {v}\left (., k, \mathbf {z}\right ) ,$ $F_{\widetilde { \varepsilon }|k})$ where $S(\tilde {v}\left (., k, \mathbf {z}\right ) ,$ $F_{ \widetilde {\varepsilon }|k})\equiv$ v(0, k, z)+$\int \max _{a>0}\{0$ , $\tilde {v}\left (a, k, \mathbf {z}\right ) -\widetilde { \varepsilon }(a)\}$ $dF_{\widetilde {\varepsilon }|k}\left (\widetilde { \varepsilon }\right )$. Using the definition of v(a, k, z), we have that v(0, k, z) − v(0, 0, z)= π(0, k, z) − π(0, 0, z)= − ic(0, k, z ^c), that is the scrap value of a firm with k units of capital. Therefore, we have that $\bar {V}\left (a, \mathbf {z}\right ) -\bar {V}\left (0, \mathbf {z}\right ) =$ − ic(0, k, z ^c)+$S(\tilde {v}\left (., k, \mathbf {z}\right ) ,$ $F_{ \widetilde {\varepsilon }|k})-$ $S(\tilde {v}\left (., 0, \mathbf {z}\right ) ,$ $F_{\widetilde {\varepsilon }|0})$. Define the function Q(a, k, z)≡$\tilde {v}\left (a, k, \mathbf {z}\right ) -$ $\beta \sum _{ \mathbf {z}^{\prime }}f_{z}\left (\mathbf {z}^{\prime }|\mathbf {z}\right )$ $[S(\tilde {v}\left (., k, \mathbf {z}\right ) , F_{\widetilde {\varepsilon }|k})-$ $S(\tilde {v}\left (., k, \mathbf {z}\right ) ,$ $F_{\widetilde {\varepsilon }|k})]$ , that is identified from the data. Then, the restrictions shown in (27) can be written as:

$$\begin{array}{lll} && Q\left(a, k, \mathbf{z}\right) = vp(a, \mathbf{z}^{v})-fc\left(a, \mathbf{z }^{c}\right) -ic\left(a, k, \mathbf{z}^{c}\right) +ic\left(0, k, \mathbf{z} ^{c}\right)\\ && -\beta \sum\limits_{\mathbf{z}^{c\prime }\in \mathbf{Z} ^{c}}f_{z}\left(\mathbf{z}^{c\prime }|\mathbf{z}\right) ic\left(0, a, \mathbf{z}^{c\prime }\right) \end{array}$$

(A.7)

Note that Q(k, z) in Equation (20) is a special case when K = 1 and a = 1. Using this expression, we can also obtain a system of equations that correspond or extend the ones in (21). For any a>0 and any (k, z):

$$\begin{array}{lll} && -ic\left(a, k, \mathbf{z}^{c}\right) +ic\left(0, k, \mathbf{z}^{c}\right) +ic\left(a, 0, \mathbf{z}^{c}\right) = Q\left(a, k, \mathbf{z}\right) -Q\left(a, 0, \mathbf{z}\right) \\ &&fc\left(a, \mathbf{z}^{c}\right) +ic\left(a, k, \mathbf{z}^{c}\right) -ic\left(0, k, \mathbf{z}^{c}\right) +\beta \sum\limits_{\mathbf{z}^{c\prime }\in \mathbf{Z}^{c}}f_{z}\left(\mathbf{z}^{c\prime }|\mathbf{z}\right) ic\left(0, a, \mathbf{z}^{c\prime }\right) \notag\\ &&= -Q\left(a, k, \mathbf{ z}\right) +vp\left(a, \mathbf{z}^{v}\right) \end{array}$$

(A.8)

The same logic used to prove Proposition 2 implies no identification of structural cost functions fc(a, z ^c) and ic(a, k, z ^c). However, we can still identify the difference between entry cost (ic(a, 0, z ^c)) and scrap value ( − ic(0, k, z ^c)) for the same z as we have

$$\begin{array}{ccc} Q\left(k, k, \mathbf{z}\right) -Q\left(k, 0, \mathbf{z}\right) & = & ic\left(0, k, \mathbf{z}^{c}\right) +ic\left(a, 0, \mathbf{z}^{c}\right) \end{array}$$

(A.9)

The relationship between estimated functions under some normalization and structural cost functions is similar to that of the baseline model. For example, when we normalize scrap value (i.e., $\widehat {ic}\left (0, k, \mathbf {z}^{c}\right ) =0$), our estimates of investment cost function $\widehat {ic}\left (a, k, \mathbf {z}^{c}\right )$ and fixed cost function fc(k, z ^c) are written as

$$\begin{array}{ccl} \widehat{ic}\left(a, k, \mathbf{z}^{c}\right) & = & ic\left(a, k, \mathbf{z} ^{c}\right) +ic\left(0, a, \mathbf{z}^{c}\right) -ic\left(0, k, \mathbf{z} ^{c}\right) \\ & & \\ \widehat{fc}\left(k, \mathbf{z}^{c}\right) & = & fc\left(k, \mathbf{z} ^{c}\right) -ic\left(0, k, \mathbf{z}^{c}\right) +\beta \mathbb{E} \left[ ic(0, k, \mathbf{z}_{{\scriptsize t+1}}^{c})|\mathbf{z} _{{\scriptsize t}}=\mathbf{z}\right] \end{array}$$

(A.10)

B. Proofs of Lemmas and Propositions

Proof of Lemma 1.

By definition Q(k, z) is equal to $\tilde {v}(k, \mathbf {z})-$ $\beta \sum _{\mathbf {z}^{\prime }}f_{z}(\mathbf {z}^{\prime }|\mathbf {z})$ [$S(\tilde {v}\left (1, \mathbf {z}\right ) , F_{\widetilde {\varepsilon }|1})-$ $S(\tilde {v}\left (0, \mathbf {z}\right ) , F_{\widetilde {\varepsilon }|0})$ ], where $\tilde {v}\left (k, \mathbf {z} \right ) =F_{\widetilde {\varepsilon }|k}^{-1}\left (P(k, \mathbf {z})\right )$ and $S(\tilde {v}\left (k, \mathbf {z}\right ) , F_{\widetilde {\varepsilon } |k})\equiv$ $\int _{-\infty }^{\tilde {v}(k, \mathbf {z})}[\tilde {v}(k, \mathbf {z })-\widetilde {\varepsilon }]\mathbf {\ }dF_{\widetilde {\varepsilon }|k}\left (\widetilde {\varepsilon }\right )$. These expressions imply the mapping in Equation (23). We can describe mapping $\widetilde {q}(\widetilde {P};\beta , f_{z})$ as the composition of two mappings: (1) the mapping from CCPs to differential values, i.e., $\tilde {v}\left (k, \mathbf {z} \right ) =F_{\widetilde {\varepsilon }|k}^{-1}\left (P(k, \mathbf {z})\right )$; and (2) the mapping from differential values to Q ^′ s, i.e., Q(k, z)=$\tilde {v}(k, \mathbf {z})-\beta \sum _{\mathbf {z}^{\prime }}f_{z}(\mathbf {z}^{\prime }|\mathbf {z})$ $[S(\tilde {v}\left (1, \mathbf {z} \right ) , F_{\widetilde {\varepsilon }|1})-$ $S(\tilde {v}\left (0, \mathbf {z} \right ) , F_{\widetilde {\varepsilon }|0})]$. The first mapping is point-wise for every value of (k, z), and in our binary choice model it is obviously invertible. Proposition 1 in (Hotz and Miller 1993) establishes the invertibility of that mapping for multinomial choice models. Therefore, we should prove the invertibility of the second mapping, between the vector $\widetilde {v}\equiv \{\widetilde {v}(k, \mathbf {z}):$ for all (k, z)} and the vector $\widetilde {Q}\equiv \{Q(k, \mathbf {z}):$ for all (k, z)}. Define this mapping as $\widetilde {Q}=\widetilde {g}(\widetilde {v})$ where $\widetilde {g}(\widetilde {v})\equiv \{g(k, \mathbf {z}, \widetilde {v}):$ for all (k, z)} and:

$$\begin{array}{ccl} g(k, \mathbf{z}, \widetilde{v}) & \equiv & \widetilde{v}(k, \mathbf{z})-\beta \sum\limits_{\mathbf{z}^{\prime }\in \mathbf{Z}}f_{z}(\mathbf{z}^{\prime }| \mathbf{z})\left(\int\nolimits_{-\infty }^{\widetilde{v}(1, \mathbf{z} ^{\prime })}[\widetilde{v}(1, \mathbf{z}^{\prime })-\widetilde{\varepsilon }] \mathbf{\ }dF_{\widetilde{\varepsilon }|1}\left(\widetilde{\varepsilon } \right) \right) \\ & & \\ & + & \beta \sum\limits_{\mathbf{z}^{\prime }\in \mathbf{Z}}f_{z}(\mathbf{z} ^{\prime }|\mathbf{z})\left(\int\nolimits_{-\infty }^{\widetilde{v}(0, \mathbf{z}^{\prime })}[\widetilde{v}(0, \mathbf{z}^{\prime })-\widetilde{ \varepsilon }]\mathbf{\ }dF_{\widetilde{\varepsilon }|0}\left(\widetilde{ \varepsilon }\right) \right) \end{array}$$

In vector form, we can express this mapping as:

$$\widetilde{g}(\widetilde{v})\equiv \left[ \begin{array}{c} \widetilde{g}(0, \widetilde{v}) \\ \widetilde{g}(1, \widetilde{v}) \end{array} \right] =\left[ \begin{array}{c} \widetilde{v}(0, .)-\beta \mathbf{F}_{z}\widetilde{e}(\widetilde{v}) \\ \widetilde{v}(1, .)-\beta \mathbf{F}_{z}\widetilde{e}(\widetilde{v}) \end{array} \right]$$

where F _z is the transition probability matrix with elements f _z (z ^′|z), and $\widetilde {e}(\widetilde {v} )\equiv \{\widetilde {e}(\mathbf {z};\widetilde {v}):$ for all z} with $\widetilde {e}(\mathbf {z};\widetilde {v})=$ $\int \nolimits _{-\infty }^{ \widetilde {v}(1, \mathbf {z})}[\widetilde {v}(1, \mathbf {z})-\widetilde { \varepsilon }]\mathbf {\ }dF_{\widetilde {\varepsilon }|1}\left (\widetilde { \varepsilon }\right ) -$ $\int \nolimits _{-\infty }^{\widetilde {v}(0, \mathbf {z })}[\widetilde {v}(0, \mathbf {z})-\widetilde {\varepsilon }]\mathbf {\ }dF_{ \widetilde {\varepsilon }|0}\left (\widetilde {\varepsilon }\right )$. The mapping $\widetilde {g}(\widetilde {v})$ is globally invertible if and only if its Jacobian matrix $\mathbf {J}(\widetilde {v})\equiv \partial \widetilde {g}(\widetilde {v})/\partial \widetilde {v}^{\prime }$ is non-singular for every value of $\widetilde {v}$. It is simple to show that this Jacobian matrix has the following form:

$$\mathbf{J}(\widetilde{v})=\mathbf{I}-\beta \left[ \left(\begin{array}{c} 1 \\ 1 \end{array} \right) \otimes \left(\mathbf{F}_{z}\frac{\partial \widetilde{e}(\widetilde{v})}{\partial \widetilde{v}^{\prime }}\right) \right]$$

where I is the identity matrix. Given the form of function $\widetilde {e}(\mathbf {z};\widetilde {v})$, it is straightforward to show that $\partial \widetilde {e}(\mathbf {z};\widetilde {v})/\partial \widetilde {v}(0, \mathbf {z})=$ $F_{\widetilde {\varepsilon }|0}(\widetilde {v}(0, \mathbf {z}))=$ − P(0, z), and $\partial \widetilde {e}(\mathbf {z};\widetilde {v} )/\partial \widetilde {v}(1, \mathbf {z})=$ $F_{\widetilde {\varepsilon }|1}(\widetilde {v}(1, \mathbf {z}))=$ P(1, z). Therefore,

$$\frac{\partial \widetilde{e}(\widetilde{v})}{\partial \widetilde{v}^{\prime } }=\left[ -diag\{P(0, .)\};diag\{P(1, .)\}\right]$$

where diag{P(k, .)} is diagonal matrix with elements {P(k, z)} for every value of z. The Jacobian matrix $\mathbf {J}(\widetilde {v })$ is invertible for every value of $\widetilde {v}$.

Proof of Proposition 1.

The proof of this Proposition 1 is a direct application of Proposition 4 in Aguirregabiria (2010) to our model of market entry and exit. Proposition 4 in Aguirregabiria (2010) applies to a general class of binary choice dynamic structural models with finite horizon, and it builds on previous results by Matzkin (1992, 1994).

Proof of Proposition 2.

(i) No identification. Let ec, fc, and sv be the true values of the functions in the population. Based on these true functions, define the functions: ec ^∗(z ^c)=ec(z ^c) + λ(z ^c); sv ^∗(z ^c)=sv(z ^c) + λ(z ^c), and $fc^{\ast }\left (\mathbf {z}^{c}\right ) =fc\left (\mathbf {z}^{c}\right ) -\lambda \left (\mathbf {z}^{c}\right ) +\beta \sum _{\mathbf {z}^{c\prime }\in \mathbf {Z}^{c}}f_{z}(\mathbf {z}^{c\prime }|\mathbf {z})$ λ(z ^c′), where λ(z ^c)≠0 is an arbitrary function. It is clear that ec ^∗, sv ^∗, and fc ^∗ also satisfy the system of Equations (20). Therefore, ec, fc, and sv cannot be uniquely identified from the restrictions in Equations (20). (ii) Identification of two combinations of the three structural functions.We can derive equations in Equations (21) after simple operations in the system (20).

Proof of Proposition 3.

Under the conditions of Proposition 3, we have that Equation (24) becomes:

$$\begin{array}{lll} && Q(k, \mathbf{z};\mathbf{\theta }^{0}+\Delta_{\theta }) = Q^{0}(k, \mathbf{ z})+\Delta_{vp}(\mathbf{z}^{v})-\\ &&\left[ \Delta_{fc}\left(\mathbf{z} ^{c}\right) +\Delta_{ec}\left(\mathbf{z}^{c}\right) \right] +k \left[ \Delta_{ec}\left(\mathbf{z}^{c}\right) -\Delta_{sv}\left(\mathbf{z }^{c}\right) \right] \\ && + \beta^{0}\sum\limits_{\mathbf{z}^{sv\prime }\in \mathbf{Z} }f_{z, sv}^{0}(\mathbf{z}^{sv\prime }|\mathbf{z})\Delta_{sv}(\mathbf{ z}^{sv\prime }) \end{array} $$

(28)

The researcher knows all the elements in the right-hand-side of this equation, and therefore Q(k, z; 𝜃 ⁰ + Δ _𝜃 ) is identified. Given $\widetilde {\mathbf {Q}}(\mathbf {\theta }^{0}+\Delta _{\theta })$, we can use the inverse mapping $\widetilde {q}^{-1}$ to get $\widetilde {\mathbf {P}}(\mathbf {\theta }^{0}+\Delta _{\theta })=$ $\widetilde { q}^{-1}(\widetilde {\mathbf {Q}}(\mathbf {\theta }^{0}+\Delta _{\theta });\beta ^{0}, f_{z}^{\ast })$.

Proof of Proposition 4.

Under the conditions of Proposition 4 (and making Δ _vp = Δ _fc = Δ _ec = Δ _sv = $\Delta _{sv}=\Delta _{f_{z_{,nosv}}}=0$, for simplicity but without loss of generality), Equation (24) becomes:

$$\begin{array}{ccl} Q(k, \mathbf{z};\mathbf{\theta }^{0}+\Delta_{\theta }) & = & Q^{0}(k, \mathbf{ z})+\beta^{0}\sum\limits_{\mathbf{z}^{sv\prime }\in \mathbf{Z}}\Delta _{f_{z, sv}}(\mathbf{z}^{sv\prime }|\mathbf{z})sv^{0}(\mathbf{z} ^{sv\prime }) \\ & & \\ & + & \Delta_{\beta }\sum\limits_{\mathbf{z}^{sv\prime }\in \mathbf{Z} }[f_{z, sv}^{0}(\mathbf{z}^{sv\prime }|\mathbf{z})+\Delta_{f_{z, sv}}(\mathbf{ z}^{sv\prime }|\mathbf{z})]sv^{0}(\mathbf{z}^{sv\prime }) \end{array} $$

(29)

Since the scrap value sv ⁰ is not identified, none of the terms that form Q(k, z; 𝜃 ⁰ + Δ _𝜃 ) − Q ⁰(k, z) are identified.

Proof of Proposition 5.

It follows simply from equation (25).

Proof of Proposition 6.

Suppose $\mathbf {z}_{2}^{c}$ enters only in the entry cost ec(z ^c). Under the conditions of Proposition 6, the system of Equations (21) becomes:

$$\begin{array}{rcl} fc\left(\mathbf{z}_{1}^{c}\right) +sv\left(\mathbf{z}_{1}^{c}\right) -\beta \sum\limits_{\mathbf{z}_{1}^{c\prime }\in \mathbf{Z}^{c}}f_{z}(\mathbf{z}_{1}^{c\prime }|\mathbf{z}^{v}, \mathbf{z}_{1, }^{c}\mathbf{z} _{2}^{c})sv\left(\mathbf{z}_{1}^{c\prime }\right) & = & -Q(1, \mathbf{z})+vp(\mathbf{z}^{v}). \end{array}$$

(30)

The difference between this equation evaluated at $\mathbf {z}_{2}^{c1}$ and at $\mathbf {z}_{2}^{c0}$ is:

$$\begin{array}{lll} \sum\limits_{\mathbf{z}_{1}^{\prime }\in \mathbf{Z}}[f_{z}(\mathbf{z} _{1}^{c\prime }|\mathbf{z}^{v}\mathbf{, z}_{1}^{c}, \mathbf{z}_{2}^{c1})-f_{z}(\mathbf{z}_{1}^{c\prime }|\mathbf{z}^{v}\mathbf{, z}_{1}^{c}, \mathbf{z} _{2}^{c0})]sv\left(\mathbf{z}_{1}^{c\prime }\right) \notag \\ = \frac{1}{ \beta }\left[ Q(1, \mathbf{z}^{v}, \mathbf{z}_{1}^{c}, \mathbf{z}_{2}^{c1})-Q(1, \mathbf{z}^{v}, \mathbf{z}_{1}^{c}, \mathbf{z}_{2}^{c0})\right] . \end{array}$$

In matrix form, we can express this system of equations as

$$\begin{array}{ccc} \left[ \mathbf{F}_{{z_{1}^{c}}}(\mathbf{z}_{2}^{c1})-\mathbf{F}_{{z_{1}^{c}}}(\mathbf{z}_{2}^{c0})\right] \mathbf{sv} & = & \frac{1}{\beta }\left[ \mathbf{Q}(1, \mathbf{z}_{2}^{c1})-\mathbf{Q}(1, \mathbf{z}_{2}^{c0})\right] . \end{array}$$

(31)

Note that matrix $\mathbf {F}_{{z_{1}^{c}}}(\mathbf {z}_{2}^{c1})-\mathbf {F} _{{z_{1}^{c}}}(\mathbf {z}_{2}^{c0})$ does not have full column rank because any matrix that is a difference of transition matrices is singular. However, the rank of $\mathbf {F}_{{z_{1}^{c}}}(\mathbf {z}_{2}^{c1})-\mathbf {F} _{{z_{1}^{c}}}(\mathbf {z}_{2}^{c0})$ can be $|\mathcal {Z}_{1}^{c}|-1$. If that is the case, we can combine the system of Equations (31) with a normalization assumption on one single element of the vector s v (e.g., $sv(\mathbf {z} _{1}^{c0})=0$ for some value $\mathbf {z}_{1}^{c0}$) to uniquely identify the vector s v.

Proof of Proposition 7.

The identified function $\widehat { sv}\left (\mathbf {z}^{c}\right )$ is such that $\widehat {sv}\left (\mathbf {z} ^{c}\right ) =$ sv ⁰(z ^c) + κ where sv ⁰(z ^c) is the true scrap value function and κ is an unknown constant. Given Δ _β = 0, we can write Equation (29) as

$$\begin{array}{ccl} Q(k, \mathbf{z};\mathbf{\theta }^{0}+\Delta_{\theta }) & = & Q^{0}(k, \mathbf{ z})+\beta^{0}\sum\limits_{\mathbf{z}^{sv\prime }\in \mathbf{Z}}\Delta _{f_{z, sv}}(\mathbf{z}^{sv\prime }|\mathbf{z})\left[ \widehat{sv} \left(\mathbf{z}^{sv^{\prime }}\right) -\kappa \right] \\ & & \\ & = & \beta^{0}\sum\limits_{\mathbf{z}^{sv\prime }\in \mathbf{Z}}\Delta _{f_{z, sv}}(\mathbf{z}^{sv\prime }|\mathbf{z})\widehat{sv}\left(\mathbf{z}^{sv\prime }\right) \end{array}$$

where the last equality holds as we always have $\sum \nolimits _{\mathbf {z} ^{sv\prime }\in \mathbf {Z}}\Delta _{f_{z, sv}}(\mathbf {z}^{sv\prime }|\mathbf { z})=0$. Therefore, we can calculate Q(k, z; 𝜃 ⁰ + Δ _𝜃 ), and the effect of the counterfactual experiment on CCPs is identified.