Pricing a new substitute good

21 Dec 2020

Suppose you’re offering a new product for a specific type of consumer. Your product is a substitute for an existing product they already purchase. Suppose also that it’s a “durable” good, in the sense that the relevant choice is made infrequently. How should you price your product?

When they choose between your product and the existing one, the maximum price you can charge is the difference between their benefits from your product and their payoff from the existing product (i.e. their surplus from choosing you). The logic is simple: if you charge any more, the consumer is better off sticking with the existing product. The closer a substitute your product is for the existing product, the closer the maximum price you can charge is to the cost of the existing product. If the consumer is uncertain about the benefits they’ll get from your product, you need to give them a discount if they’re risk averse and can charge them a premium if they’re risk seeking.

Choice with no uncertainty

The consumer’s payoff from the existing product \(j\) is \(v_j\). The payoff is composed of the benefits (\(b_j\)) minus the costs (\(c_j\)). The marginal cost of producing one unit of your product is \(c\). It provides the consumer with benefit \(b_i \leq b_j\) at price \(p\), for a payoff of \(v_i = b_i - p\). The consumer’s choice \(x \in \{0,1\}\) (i.e. {don’t buy, buy}) maximizes their value,

\[\begin{align} \max_{x \in \{0,1\} } ~~ (1-x)v_j + x(b_i - p) \end{align}\]

meaning they choose your product (\(x=1\)) if the price is lower than the difference between the benefits of your product and the payoff of the other:

\[\begin{align} x^* &= \begin{cases} 1 \text{ if } b_i - p > v_j \\ 0 \text{ if } b_i - p \leq v_j.\\ \end{cases} \end{align}\]

i.e. they’ll choose your product if \(p < b_i - v_j\): you can price up to their surplus from choosing you. If your profit from the sale is \((p-c)x\), then the best you can do is to price as close to \(b_j - v_j\) as possible and capture maximal surplus. In the special case where your product provides identical benefits as the existing product, the condition becomes \(p < c_j\), i.e. you just have to beat the cost the consumer pays for the existing product. More generally, the closer \(b_i\) is to \(b_j\) the more your maximum price is dominated by the cost of the existing product. This shortcut is especially valuable when \(b_j\) (or \(b_i\)) is hard to calculate.

The same logic carries through if the choice is over streams of payoffs from repeated single-choice purchase occasions, but the maximum price becomes the difference in discounted streams of net payoffs.

Uncertainty over benefits

What if the consumer isn’t certain about the benefits your product provides relative to the existing product? We’ll model this by saying \(b_i\) is a random variable with CDF \(F(b)\), where \(F(b)\) reflects the consumer’s subjective beliefs over the benefits from your product. If the consumer is indifferent to uncertainty in the benefits your product provides (risk neutral, as shown above) then \(b_i\) should be replaced with the consumer’s expected benefit from your product, \(E[b_i]\). This changes a little if the consumer is risk averse and would pay to reduce the variance in their payoffs, or risk seeking and would pay for greater variance in their payoffs. We’ll represent their preferences over payoffs with a monotone increasing utility function \(u\) (measured in payoff units), which is strictly concave when they’re risk averse and strictly convex when they’re risk seeking. They now solve

\[\begin{align} \max_{x \in \{0,1\} } ~~ (1-x)u(v_j) + x(E[u(b_i - p)]) \end{align}\]

which changes their choice rule to

\[\begin{align} x^* &= \begin{cases} 1 \text{ if } E[u(b_i - p)] > u(v_j) \\ 0 \text{ if } E[u(b_i - p)] \leq u(v_j).\\ \end{cases} \end{align}\]

If the consumer’s utility is quasilinear (e.g. their utility of money is linear), we can write their payoff from your product as \(E[u(b_i)] - p\). The new maximum price you can charge is \(E[u(b_i)] - u(v_j)\). Jensen’s inequality tells us this is strictly smaller than the maximum price you could charge a risk-neutral consumer, \(u(E[b_i]) - u(v_j)\).

The difference between \(u(E[b_i])\) and \(E[u(b_i)]\) for a risk-averse consumer is the “risk discount” they need from you to compensate them for the variance they perceive in benefits from your product. If the consumer is risk-seeking, then \(u(E[b_i]) - E[u(b_i)] < 0\) and you can charge the consumer a “risk premium” for the perceived variance in benefits. All of this logic still goes through if we view \(b_i\) and \(b_j\) as suitably-discounted streams of benefits accruing over time.

Pricing satellite life extension

There’s a space angle here, of course. How much would a satellite operator pay for satellite life extension? Life extension (temporarily) substitutes for replacement. You avoid launching a whole new satellite, forgoing the benefits of potential upgrades in exchange for a lower price. Life extension changes the refresh rate (and upgrade cycle) for a constellation operator, so calculating \(b_j\) when pricing the service is not trivial. For operators who expect to eventually need a replacement, \(b_j - b_i\) reflects both the difference in benefits from the replacement vs servicing current hardware (probably favors replacement) as well as the difference in the streams of benefits from the refresh rates induced by replacing vs servicing (potentially favors life extension). On net, I’d expect \(b_j \geq b_i\).

Matt Desch, CEO of Iridium, said he’d be willing to pay just $10,000 for a deorbit. Even though the technology stacks for deorbit and life extension are similar, they’re different things so there isn’t quite an apples-to-apples comparison to be made. But let’s suppose for argument’s sake he’d be willing to pay something on that order for life extension. I interpret this price point as implying two possibilities in terms of the model:

1. Iridium’s upgrade cycle is expected to deliver a lot of benefits.
2. Iridium’s marginal cost of a replacement satellite is very low.

I have a hard time believing the marginal cost of building and launching any kind of replacement satellite for a telecom constellation is on the order of $10,000. Maybe it is for Starlink since they’re vertically integrated, but for Iridium? Even if they could pack 60-70 satellites into a single launch, a $10m launch (pretty cheap as these things go) puts \(c_j\) about an order of magnitude higher than $10k. So to the extent that $10,000 price point comes from the marginal cost of a replacement, it’s probably about spare satellites that are already on orbit.

The upgrade cycle might be expected to deliver a lot of benefits. This could be especially true if the satellites weren’t built to last very long, so that life extension would come at a time when the per-period benefits from that hardware is at its lowest. Satellite deterioration is at least partly a choice though, so some of that is really reflecting earlier decisions to commit to a certain refresh rate. Fair enough, these decisions have long lead times. A would-be servicer ought to make the case to operators early.

These possibilities aren’t exhaustive. First, the analysis above tells us the operator’s maximum willingness-to-pay. Why would any sane operator reveal that value? And if you’re trying to build a recurring relationship with these operators, it’s probably not good business to try to ruthlessly extract all the surplus from the relationship. Nobody likes that partner. So maybe Desch was trying to move early, anchor expectations (the SpaceNews article hints at this), and capture more of the surplus. It’d be very interesting to see if debris removal and life extension could auctioned efficiently, but I don’t know if operators would fully participate without some kind of regulation.

It could also reflect market power. Maybe Desch expects he’d have a lot of leverage over new servicers since they won’t have many customers. How many big constellation customers might there be in LEO? Starlink probably won’t be buying any servicing. Right now that leaves Iridium as the big player that’s already on orbit. Eventually Kuiper and OneWeb will have their assets up and their act together, but 3 potential buyers isn’t a whole lot more than 1. Orbital-use fees or some other type of operator-focused environmental regulation could help increase operators’ willingness to pay for servicing and removal in the same way Pigouvian taxes increase emissions abatement. I don’t see that kind of regulation happening any time soon. I don’t know how big a deal monopsony in orbit will be, but any environmental regulation (e.g. orbital-use fees, performance bonds, or Price-Anderson-like liability pools) which reduces entry could increase operators’ monopsony power.

I guess the bottom line is I’m skeptical that the value of servicing is as low as $10,000 per satellite. I can see why someone like Desch wouldn’t want to announce or commit to a higher number, but I find it easy to imagine there are/will be operators willing to pay more than $10,000 per satellite.