Identification of switching costs from price shocks What, if anything, can we learn about customer switching costs by looking at price, revenue, profit, and quantity responses of producers to cost shocks?

For example, we can define the profit equation as:

$\Pi = \sum^\infty_{t=0} [ -\alpha S + \beta^t(P - C(q))] \cdot  q(P,C,S)$

Where $q(P,C,S)$ is demand as as function of prices, costs, and switching costs respectively, $C(q)$ is total cost as a function of quantity produced, and $\beta$ is the discount rate of the firm, and $\alpha$ is the fraction of the switching cost refunded to the purchaser. Are there published or just worked out examples of choices of $q(P,C,S)$ and $C(q)$  that allow identification of S from $\partial\pi/\partial C$, $\partial P^*/\partial C$, and $\partial q^*/\partial C$? 


Consider the cost shock as an exogenous shock, but I have in mind a setting that allows a structural identification.
By switching costs, I mean that if a customer wants to first buy from firm A and later switch to firm B he pays $P_B+S$, in the period of the switch then $P_B,P_B\ldots$ in subsequent periods instead of $P_A, P_A, \ldots$.
The $-\alpha S$ term is a refund to only the purchasers in the period of initial purchase while $S$ is paid in the final period of doing business with a firm. An example of $-\alpha S$ could be getting a special deal on your Verizon cellphone because you can't use that phone with any other network or a free toaster when you open a bank account. I fixed it in the profit equation so that $S$ was multiplied by $q$ to indicate that $S$ is is the switching costs per unit of $q$. I had in mind that customers purchase either $0$ or $1$ unit of the service and $q$ aggregates their individual decisions. But I'm not wedded that refund, if it gets me somewhere I'm happy to assume that $\alpha =0$
This is not a complete argument but I'm going to give a brief sketch. Assume that producers set prices instead of quantity (which seems to be the model above). Assume that cost shocks do not affect demand, in other words $q(P,C,S) = q(P,S)$, and that they are not correlated with some kind of variation in switching costs. For simplicity, imagine also that this is a constant cost model. Then:

$$\frac{d q^\star}{d C}= \frac{\delta q}{\delta P}\frac{d P^\star}{d C}$$

Assuming switching costs are such that $\frac{\delta q}{\delta P}$ is low for high switching costs, this could give you an idea. If you enough form for $q$ you should be able to pin down $S$ using just-identified GMM (1 moment, 1 parameter). 

Let's take a more concrete example, using a more concrete problem. Assume a two period model. In the first period, prices are announced, in the second period there is a surprise shock to the costs of $B$ (WLOG they get cheaper) which $A$ does not experience. This causes a switch from $P_B$ to $P_B^\prime$. Suppose a continuum of consumers, $i$, each with a time invariant valuation of each good, $j$, $V_i^j$, jointly distributed according to $F$. Then people who switch to $B$ from $A$ this period are those individuals for whom:

$$U(V^B_i - P_B^\prime -S)&gt; U(V^A_i - P_A)$$
but:
$$U(V^B_i - P_A - S)&lt; U(V_i^A-P_A)$$
This means:
$$q^\prime - q = \int_V1_{U(V^B - P_B^\prime - S)&gt; U(V^A-P_A) &gt;U(V^B - P_A - S)}dF(V^A,V^B)$$
Note that this is basically analogous to the argument above using derivatives. With form on $V$ and $U$ estimation is straightforward. 

More complicated setups, (for instance, consumers with infinite horizons) will require more complicated arguments, but the basic identification will be the same. In this setting what you really need are forms for demand and not costs. As far as demand goes, with only this data you are restricted in what you can estimate. With linear utility, and one of the goods valuations normalized to 0, you could probably pin down the mean of $V_i^j$, and maybe a bit of its cross-sectional distribution. More detail would be possible with consumer level data.