Investment, Subsidies, and Universal Service: Broadband Internet in the United States

A.1 Derivation of Continuation Values

To derive equation (12), I start with equation (7):

where p x f ( S ′ ) ≡ Pr ( κ x > β V C f ( S ′ ; θ ) ) is the probability that a fast firm exits when the state is S′. The fact that E κ x ′ | κ x ′ > V C f ( S ′ ; θ ) = V C f ( S ′ ; θ ) + μ x is a property of the exponential distribution. Rewriting equation (18) in matrix notation, we have

where M ^f is the Markov transition matrix conditional on a fast incumbent remaining in the market, and P x f is a vector of probabilities that a fast incumbent exits the market at each state. This implies that

which is equation (12). This first derivation follows closely from Pakes, Ostrovsky, and Berry (2007). The remaining derivations require some adaptations to fit my model.

To derive equation (13), I start with equation (6):

Following the derivation above, we know that

Therefore,

For ease of notation, let the continuation value associated with remaining a slow provider be represented by

and let the continuation value associated with upgrading be represented by

Therefore,

Next, we must evaluate E κ u ′ κ u ′ | κ u ′ ≤ B − A . To do so, we note that

Therefore,

and finally,

In matrix notation, this becomes

Equations (14) and (15) are derived analogously.

A.2 Pseudo-likelihood Probabilities

The actions available to potential entrants are wait, enter as a slow provider, or enter as a fast provider. The probability that a potential entrant waits in state S is

where F κ e ′ ( ⋅ ) is the CDF of an exponential distribution with mean μ _e . In matrix form, this becomes

The probability that a potential entrant enters (as either a slow or fast provider) is therefore

Then, the probability that a potential entrant enters as a slow provider in state S, conditional on entering, is

In matrix form, this becomes

where M ^ef and M ^es are Markov transition matrices conditional on a potential entrant entering as a fast and slow provider, respectively, and F κ f ( ⋅ ) is the CDF of an exponential distribution with mean μ _f .

Thus, the unconditional probability that a potential entrant enters as a slow provider is

and the probability that a potential entrant enters as a fast provider in each state is then

The actions available to slow incumbents are exit, upgrade to become a fast provider, or remain a slow provider. The probability that a slow incumbent exits in state S is

where F κ x ′ is the CDF of an exponential distribution with mean μ _x . In Matrix form,

Then, the probability that a slow incumbent upgrades in state S, conditional on not exiting, is

In matrix form,

where M ^u and M ^rs are Markov transition matrices conditional on a potential entrant upgrading and remaining a slow provider, respectively, and F κ u ( ⋅ ) is the CDF of an exponential distribution with mean μ _u . The unconditional probability that a slow incumbent upgrades is therefore

and the probability that a slow incumbent remains a slow incumbent is

The actions available to fast incumbents are exit or remain a fast provider. The probability that a fast incumbent exits in state S is

In matrix form,

and

A.3 Discretization of the State Space

Estimation of the model requires the state space to be finite. Therefore, I must first discretize any continuous variables in the state space. These variables include the market’s population and the distance from the market’s centroid to the nearest DSL central office. To do this, I assign each market’s population to a bin according to the quartile its population falls within. I create three bins, with populations in the first quartile labeled as small markets, populations in the second and third quartiles labeled as medium markets, and populations in the fourth quartile labeled as large markets. I then assign each bin a value according to the mean population within that bin, so that all small markets are assigned a population of 2525.311, medium markets are assigned a population of 4446.572, and large markets are assigned a population of 7132.233. Similarly, I assign each market’s distance to the nearest DSL central office to a bin according to whether it is above or below the median distance, thus creating two bins. I again assign each bin the value of the mean distance within that bin, so that all markets close to a central office are assigned a distance of 1.988 and all markets far from a central office are assigned a distance of 10.564.

In addition, in order to reduce the dimension of the state space to a feasible level, I further discretize the number of firms in each technology-type pair in N. I assign the number of slow DSL firms in the market to one of two bins, according to whether N^d,s = 0 or N^d,s ≥ 1. I assign the second bin a value equal to the mean of all observations that fall within the bin (1.091). I discretize the number of slow cable firms, fast DSL firms, and fast cable firms in the same manner, with bin values of 1.000, 1.193, and 1.092, respectively. Finally, I assign the number of DSL potential entrants to one of two bins with values of 5.753 and 7.107, according to whether N^d,p is less than or greater than the median number (7) of DSL potential entrants. I assign the number of cable potential entrants to one of two bins with values of 7.874 and 9.000, according to whether N^c,p is less than or greater than the median number (8) of cable potential entrants.