Static Pricing Game

Time is assumed to be discrete and periods are indexed by t = 0,1,2, . . . ,∞. There are M independent markets indexed by m = 1,2, . . . , M. Each market m is comprised of a continuum of consumers and Jm firms, where a representative consumer is indexed by i and firms are indexed byjm= 1,2, . . . , Jm. I assume that each firm produces a single differentiated good so that the notions of “firms” and “products” are synonymous. For instance, in my empirical application purchasing product j can be thought of as “choosing to have an MRI scan performed at hospital j.” In each period consumers choose to purchase one of the Jm products available or the outside option, denoted by j = 0.15 In what follows I employ the differentiated product demand system developed by Berry (1994) to specify firms’ per-period flow profits.

Omitting market subscripts, consumer i’s flow utility from purchasing productj in period tis given by:

uijt=xjtγ+θjtτ −αpjt+ξj+ijt, (3.1)

14_{For now I simply fix the value of this parameter before solving for the equilibrium of the model.}

Estimating this probability would be relatively straightforward as long as it is time-invariant and constant across firms. However, solving for a perfect Bayesian equilibrium of the dynamic game would be incredibly complex.

15_{For now I assume that the outside option is exogenous and time-invariant. In other words, the}

price ofj= 0 is determined outside of the model prior to the start of the game. While this assumption is common, it is potentially restrictive in the context of my empirical application. In each market only hospitals are considered “players” in the dynamic game. Therefore, all independent imagining facilities are not explicitly modeled. The extent of the competition between hospitals and these imaging facilities in the market for MRI scans is somewhat unclear. I plan to further investigate this issue and potentially employ a time-varying outside option in future work.

where xjt is a vector of product and market-level variables, θjt is firm j’s technology, and pjt is the price of product j. Assuming τ is positive, on average a consumer values a firm’s technology. Bothξj, a product characteristic, andijt, an idiosyncratic consumer taste variable, are unobserved to the econometrician. Thus, ξj can be interpreted as permanent unobserved heterogeneity for productj. Following Berry (1994), I denote the mean utility of productj at timet as

δjt≡xjtγ+θjtτ −αpjt+ξj (3.2)

and normalizeδ0t= 0,∀t. Assuming that eachijt is an IID draw from a Type 1 EV distribu- tion, firm j’s market share in period t is given by the standard logit equation:

sjt(δ) =

eδjt

PJ

k=0eδkt

. (3.3)

I assume that a firm’s marginal cost is constant in output and is specified as

cjt(θjt, ωjt, φ) =eφ0−φ1θjt+φ2wjt+ωjt, (3.4)

where wjt and ωjt are observable and unobservable firm-specific cost shifters, respectively. φ0, φ1, and φ2 are parameters. Adopting a new technology decreases a firm’s marginal cost,

where the extent of the cost reduction is governed by φ1. It is important to note that in

the current specification the benefit from adopting a new technology is twofold–through an increase in consumer utility and therefore demand (τ) along with a decrease in marginal cost (φ1). However, the equilibrium predictions of the model do not change qualitatively if one of

the two effects is removed.16

I assume that conditional on the vector of chosen technologies, firms subsequently engage

16_{While I assume that both effects are beneficial to an adopting firm, this might not always be the}

case. For instance, a new technology might increase a firm’s marginal cost if it requires more electricity or labor to operate compared with previous versions. In the case of MRI it is assumed that that any increase in variable costs required to operate a more advanced scanner are far outweighed by the decrease in time needed to perform a scan. Thus, it is assumed that a more advanced scanner causes a hospital’s marginal cost per scan to decrease. In addition to a decrease in the time required for a scan, consumers are also assumed to benefit from the improved comfort and detail of images afforded by a newer scanner.

in a static Bertrand pricing game to determine prices and flow profits. LettingIm denote the size of marketm, flow profit for firmj is defined as

πjt =Imsjt(pjt−cjt). (3.5)

A pure strategy Bertrand equilibrium requires that each firm’s price satisfies the following first-order condition:

pjt =cjt+

sjt |∂sjt/∂pjt|

. (3.6)

Using the chain rule it is straightforward to show that ∂sjt ∂pjt =−α∂sjt ∂δjt =−αsjt(1−sjt), (3.7) so that (3.6) simplifies to pjt =cjt+ 1 α(1−sjt) . (3.8)

The set of equilibrium prices (and corresponding market shares) is then defined as the solution to both (3.3) and (3.8) for allj∈J. To solve for the equilibrium price vector I set each firm’s price above its marginal cost and gradually decrease the prices until a fixed point is reached. Equilibrium flow profits are then calculated using (3.5).17