The presence of externalities within a household implies that consumption depends on the de-cision of other household members. I consider a static discrete response model which is an extension of the probit model. In this paper, I restrict my attention to households with two members and show that, conditional on a given correlation coefficient of unobserved character-istics, the model parameters are fully identified despite the existence of multiple Nash equilibria in a noncooperative game between two household members.
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2.1 Discrete Response Model
For household i, there are two members j ∈ {1, 2}. Let the binary variable yij ∈ {0, 1} denote the subscription decision of individual j in household i. yij = 1 if and only if the individual subscribes to a telephone network. The demand is assumed to be
yij = 1 ⇔ [x′ijβ+ εij] + yi(3−j)[x′iγ] > 0, (1)
where (3 − j) is the index for the other member in the household. Let Yi = yi1+ yi2 be the total number of subscribers in the household.
The terms in the first bracket of equation (1) represents the own effect of the consumption choice. The vector xij is the set of observed characteristics, and εij represents characteristics unobserved to the econometrician. The observed characteristics xij includes variables both at the household and the individual levels. It also includes a constant term. Therefore, a nonempty common subset of xij exists for the two members in a household. Denote these household-level characteristics by xi. At least one of the covariates in xij must be at the individual level for the identification of the model (i.e. xi $ xij).
The term in the second bracket of equation (1), x′iγ, is the consumption externality due to the other household member’s choice. It is reciprocal between these two members. I normalize the externality to be zero if the other member do not subscribe. The externality is assumed to be completely captured by observed characteristics xi.3 My model reduces to the probit model if the externality vanishes (γ = 0).
The unobserved characteristics (εi1, εi2) are assumed to be jointly normally distributed,
3In theory, I can extend the model to include a unobserved part of consumption externality so that externality can expressed as x′iγ+ ηi. But the extended model is computationally intensive. When I use a simplified version of the extended model to gauge the importance of unobserved externality, the variance of unobserved externality ηi is estimated to be only 0.002 of the variance of unobserved characteristics εij. Therefore, I decide to ignore the effect of unobserved externality ηifor the rest of the paper.
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independently across households.
The variance of εij is normalized to one.
The unobserved characteristics of individuals within a household are likely to be positively correlated. The correlation coefficient ρ in (2) is identified only through functional form specifi-cation, not from observed data. It cannot be accurately estimated. As a result, I apply the idea developed in Altonji et al. (2005) and use the correlation of observed characteristics to provide information about the correlation of unobserved ones. See Section 2.4 below for more details about assumptions on the unobserved factor.
2.2 Nash Equilibria
Consider a simultaneous-move non-cooperative game.4 This is similar to the incomplete model discussed in Tamer (2003). Figure 1 shows the set of equilibria for positive externality (x′iγ >0) conditional on observed characteristics (xi1, xi2, ηi) and unobserved characteristics (εi1, εi2).
There are multiple Nash equilibria when (εi1, εi2) ∈ (−x′i1β−x′iγ,−x′i1β)×(−x′i2β−x′iγ,−x′i2β).
Both (yi1, yi2) = (0, 0) and (yi1, yi2) = (1, 1) are equilibria in this region. Nonetheless, the model predicts the exact probability for (yi1, yi2) = (0, 1) and (yi1, yi2) = (1, 0). The probability of the event (0, 0) is bounded by
Pr {εi1 <−x′i1β− x′iγ, εi2 <−x′i2β} ∪ {εi1 <−x′i1β, εi2 <−x′i2β− x′iγ}|x′iγ >0
and
Pr(εi1 <−x′i1β, εi2 <−x′i2β|x′iγ >0).
4The set of Nash equilibria under a cooperative game is included in the set of Nash equilibria under a non-cooperative game. Consequently, the results under a non-cooperative game can be viewed as imposing an equilibrium selection rule on the results under a non-cooperative game.
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(yi1, yi2) = (0, 1)
(yi1, yi2) = (1, 0) (yi1, yi2) = (0, 0)
(yi1, yi2) = (1, 1)
(yi1, yi2) = (0, 0) or (yi1, yi2) = (1, 1)
εi1 εi2
−x′i1β
−x′i1β− x′iγ
−x′i2β
−x′i2β− x′iγ
s s
s s
Figure 1: Simultaneous-move non-cooperative game for positive externality
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(yi1, yi2) = (0, 1)
(yi1, yi2) = (1, 0) (yi1, yi2) = (0, 0)
(yi1, yi2) = (1, 1)
(yi1, yi2) = (0, 1) or (yi1, yi2) = (1, 0)
εi1 εi2
−x′i1β− x′iγ
−x′i1β
−x′i2β− x′iγ
−x′i2β
s s
s s
Figure 2: Simultaneous-move noncooperative game for negative externality
On the other hand, when the externality is negative (x′iγ <0), there are multiple equilibria of (0, 1) and (1, 0) if (εi1, εi2) ∈ (−x′i1β,−x′i1β− x′iγ) × (−x′i2β,−x′i2β− x′iγ). (See Figure 2.) The model gives the exact probabilities of (yi1, yi2) = (0, 0) and (yi1, yi2) = (1, 1).
Regardless the sign of consumption externality, the exact probability of observing one sub-scriber in a household (Yi = yi1+ yi2 = 1) for given observed characteristics (xi1, xi2) can be obtain from the model.
Pr (Yi= 1|xi1, xi2; β, γ) =
Pr εi1<−x′i1β− x′iγ, εi2>−x′i2β + Pr εi1 >−x′i1β, εi2<−x′i2β− x′iγ
− 1{x′iγ <0} Pr −x′i1β < εi1 <−x′i1β− x′iγ,−x′i2β < εi2 <−x′i2β− x′iγ , (3)
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where 1{·} denotes an indicator function.
However, the exact probabilities of no subscriber (Yi = 0) and two subscribers (Yi = 2) in a household are unknown when consumption externality is positive because we do not know how individuals choose among multiple Nash equilibria. Without loss of generality, we only need to focus on the probability Pr(Yi = 0|xi1, xi2) because Pr(Yi = 2|xi1, xi2) can be obtained from 1 − Pr(Yi = 0|xi1, xi2) − Pr(Yi = 1|xi1, xi2). The probability of no subscriber in a household is bounded in an interval. The upper bound occurs when individuals always fail to coordinate their decisions in the event of multiple Nash equilibria.
PU(xi1, xi2; β, γ) ≡ Pr(εi1 <−x′i1β, εi2 <−x′i2β). (4)
The lower bound is achieved when individuals can perfectly coordinate.
PL(xi1, xi2; β, γ) ≡ Pr(εi1<−x′i1β, εi2 <−x′i2β)
− 1{x′iγ >0} Pr(−x′i1β− x′iγ < εi1 <−x′i1β, −x′i2β− x′2γ < εi2 <−x′i2β). (5)
2.3 Identification
Although there are multiple Nash equilibrium in my econometric model, the parameters can be pointwise identified as long as the correlation coefficient of the unobserved characteristics ρ in (2) is known. My model is similar to Tamer (2003), which is identified when we have data on the individual decisions {(yi1, yi2)}. However, the data set that I use only reports the total consumption in a household (Yi = yi1+ yi2), not individual choices. The following theorem is an extension of Theorem 1 in Tamer (2003).
Theorem 1. Suppose that there exists a regressor of individual characteristics (xi1k, xi2k) with xi1k 6= xi2k, βk 6= 0 and such that the conditional distribution of xi1k|x−i1k has an everywhere positive Lebesgue density where x−i1k = (xi11, . . . , xi1,k−1, xi1,k+1, . . . , xi1K). Then the parame-ter vectors, β and γ, are identified for any given covariance matrix of unobserved characparame-teristics
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if both the matrices xi1 and xi2 have full column rank.
Proof. In equation (3), I have shown that the exact probabilities of Yi = 1 can be obtained for any given observed characteristics (xi1, xi2).
Without loss of generality, assume βk >0. Let b be different from β and r be different from γ. Suppose bk>0. As xi1k goes to minus infinity for given x−i1k, both xi1kβk and xi1kbk go to minus infinity. Because xi2 has full rank, there exists x∗i2 such that x∗′i2β 6= x∗′i2b. Consequently, as x−i1k → −∞,
Pr (Yi = 1|xi1, x∗i2; β, γ) ≃ Pr εi2>−x∗′i2β
6= Pr εi2>−x∗′i2b ≃ Pr (Yi= 1|xi1, x∗i2; b, γ) .
This implies the vector β is identified.
Now, let xi1k go to positive infinity. Both xi1kβk and xi1kbk go to positive infinity. Because xi2 has full rank, there exists x∗∗i2 such that x∗∗′i2 β + x∗∗′i γ 6= x∗∗′i2 b+ x∗∗′i r where x∗∗i is the projection of x∗∗i2 onto the space spanned by household-level characteristics. As x−i1k → +∞, I have
Pr (Yi = 1|xi1, x∗∗i2; β, γ) ≃ Pr εi2<−x∗∗′i2 β− x∗∗′i γ
6= Pr εi2<−x∗∗′i2 b− x∗∗′i r ≃ Pr (Yi = 1|xi1, x∗∗i2; b, r) .
Therefore, I can identify the sum (βl+ γl) for each element γl in the vector γ. This implies γ is identified.
For bk < 0, x′i1β increases in xi1k, but x′i1b decreases. Consequently, there exists x∗i1k such that x∗′i1β = x∗′i1b. Similar to the above arguments, I can show that β and γ are both identified.
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2.4 Unobserved Characteristics
As I mentioned at the beginning of this section, the observed data cannot truly identify the covariance matrix of the unobserved characteristics (2). In fact, the correlation coefficient ρ can not be separately identified from the mean of consumption externality. To demonstrate this, consider the relationship between ρ and γ0, where γ0 is the constant term in the linear expression of consumption externality x′iγ. For positive externality,5
Pr(Yi = 1|xi1, xi2)
Consider the partial derivatives with respect to ρ and γ0, respectively.6
∂Pr(Yi = 1|xi1, xi2)
According to the implicit function theorem, the above two inequations imply
dv
5The result for negative externality is similar.
6The computational details are provided in an appendix available from the author.
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for any given value of Pr(Yi = 1|xi1, xi2) when ρ ≥ 0. Consequently, the correlation coefficient of the unobserved characteristics ρ cannot be separately identified from the externality v.
As Altonji et al. (2005) suggest, if the selection of observed characteristics is completely random from the set of all relevant factors, the correlation coefficient of observed characteristics is identical to that of unobserved characteristics,
ρ= Cov(x′i1β, x′i2β) (7)
On the contrary, if all relevant factors are included in the set of observed characteristics, the unobserved characteristics are purely random noises. In the latter case, the correlation among two household members is zero. The reality is likely to lie between the above two extreme cases. Since I have tried to include the most important variables in the set of regressors xij, the correlation of the unobserved characteristics is likely to be positive but less than that of observed characteristics.
0 ≤ ρ ≤ Cov(x′i1β, x′i2β). (8)
Equation (6) shows the derivative of externality on the correlation coefficient is negative.
Therefore, the consumption externality estimated under the assumption Cov(x′i1β, x′i2β) = ρ is an lower bound of the true value while the externalities estimated under zero correlation Cov(xi1, xi2) = 0 is an upper bound.
2.5 Semiparametric Maximum Likelihood Estimator
If consumption externality is negative, I know the exact probability of the events {Yi = 0}, {Yi = 1}, and {Yi = 2} conditional on the observed characteristics. Consequently, the usual likelihood can be computed. On the contrary, the exact probabilities of {Yi = 0} and {Yi = 2}
are unknown when externality is positive. I use a semiparametric maximum likelihood estimator, extended from Tamer (2003), to obtain the parameters in the demand model. Define the
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conditional probability of the event {Yi= 0} for observed characteristics (xi1, xi2) as
H(xi1, xi2) = Pr (Yi= 0|xi1, xi2) .
When H is known, I can write down the likelihood and the parameters (β, γ) are estimated by maximizing the logarithm of the likelihood function. For a random sample with size N ,7 the logarithm of the likelihood function is
L(β, γ; H) = 1
The unknown function H(xi1, xi2) represents the probability of observing no subscriber in a household in the event of multiple Nash equilibria. From equations (4) and (5), we know H(xi1, xi2) is bounded by the closed interval [PL(xi1, xi2), PU(xi1, xi2)], but the model cannot predict the exact probability. To overcome this difficulty, Tamer (2003) suggest to approxi-mate the unknown function by a kernel regression of the event {Yi = 0} on (xi1, xi2).8 Since H(xi1, xi2) is bounded by [PL(xi1, xi2), PU(xi1, xi2)], I truncate the result of the kernel regres-sion by the upper and lower bounds and denote the value by ˆH(xi1, xi2). Replace H in the
7The survey data I use to perform estimation is not a random sample. Therefore, I need to adjust for the sampling weights in my calculation. To ease the exposition, however, I present the estimator for a random sample in this section.
where φ is the density function of a standard normal distribution, and the metric ρ is defined as
ρ[(xi1, xi2), (xi′1, xi′2)] ≡
A bandwidth B = 0.3 is used for the following results. The parameter estimates are robust to changes in the bandwidth B.
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likelihood (9) by ˆH. I can obtain a consistent estimate of (β, γ). The asymptotic variance of the estimate can be computed from the score and Hessian of the log likelihood (9).9
Although the model is described under a simultaneous non-cooperative game, it actually includes cooperative game as a special case. If all households can coordinate their consumption decisions, the kernel estimation of H(xi1, xi2) will converge to PL(xi1, xi2) in probability. Sim-ilarly, if individuals make decisions sequentially, then the subsame-prefect equilibrium is also included in the set of Nash equilibria under a simultaneous non-cooperative game.