# 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 $\alpha =0$
• Do you know that the quantity responses are purely from producers to cost shocks? i.e. they look like they could be used as instruments?
– jayk
Nov 19, 2014 at 0:49
• Just to clarify, what do you mean by "switching costs?" Nov 19, 2014 at 0:52
• Is $-\alpha S$ only given to the firm from consumers who pay the switching cost? And do consumers only consume one unit per period?
– jayk
Nov 19, 2014 at 2:50
• @jmbejara Switching costs mean if I bought from firm A last period then I must incur a cost of S to buy from a different firm this period Two examples: (i) if I want to change mobile phone provider then I might have to get a new phone number, which my friends will not know; (ii) if I used to own a Mac and I switch to a Windows PC then I have to learn how to use the new operating system. Both are examples of costs involved in switching provider that serve to partially lock consumers into their original provider. Nov 19, 2014 at 10:41

## 2 Answers

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:

$$U(V^B_i - P_B^\prime -S)> U(V^A_i - P_A)$$ but: $$U(V^B_i - P_A - S)< U(V_i^A-P_A)$$ This means: $$q^\prime - q = \int_V1_{U(V^B - P_B^\prime - S)> U(V^A-P_A) >U(V^B - P_A - S)}dF(V^A,V^B)$$ 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.

Assume that agents engage in a long term contract with a firm for services. They share a common discount factor $\beta$. When they engage a firm they face a switching cost of $S$ but receive an upfront refund of $F\cdot S$ where $0\leq F\leq1$. In every period (including the first) customers pay a price $P$ for the account services. This price is locked in as long as the firms costs do not change but they do not expect costs to change. (may not be needed) The bank faces a per period cost of $C$ of providing an account and faces a demand schedule as follows: $Q_{d}=A-\frac{DP+EC}{S}$

where $D$ and $E$ are constants. (This needs to be motivated and I think I can do that). Firms are aware of their effects on demand and maximize profits: $\Pi = \max_P \{[F\cdot S + \sum_{t=0}^{\infty}\beta^{t}(P-C)](A - \frac{DP+EC}{S})\}$

Because everything is constant, we can replace the summation the closed form for the geometric series: $\Pi = \max_P \{[F\cdot S + (\frac{P-C}{1-\beta} )](A - \frac{DP+EC}{S})\}$

To optimize: $0=\frac{\partial\pi}{\partial P}=\left[F\cdot S+\left(\frac{P-C}{1-\beta}\right)\right]\left(-\frac{D}{S}\right)+\left[\frac{1}{1-\beta}\right]\left(A-\frac{DP+EC}{S}\right)$

$=\left[-\frac{D\cdot F\cdot S}{S}-\frac{D}{S}\left(\frac{P-C}{1-\beta}\right)\right]+\left[\frac{1}{1-\beta}\right]\left(A-\frac{DP+EC}{S}\right)$

$\Rightarrow0=\left[-D\cdot F\left(1-\beta\right)-\frac{PD-CD}{S}\right]+A-\frac{DP+EC}{S} =-D\cdot F\left(1-\beta\right)-\frac{PD}{S}+\frac{CD}{S}+A-\frac{DP}{S}-\frac{EC}{S}$

$\Rightarrow\frac{2DP}{S}=-D\cdot F\left(1-\beta\right)+\frac{C}{S}\left(D-E\right)+A$

$\Rightarrow P^{*}=\frac{-D\cdot F\left(1-\beta\right)+\frac{C}{S}\left(D-E\right)+A}{\frac{2D}{S}} =\frac{-D\cdot F\left(1-\beta\right)S+C\left(D-E\right)+AS}{2D}$

$\Rightarrow \frac{\partial P^{*}}{\partial C}=\frac{D-E}{2D}=a_{1}$

Which by the assumption of the question is known.

$q*=q(p^{*}) = A-\frac{DP^{*}+EC}{S}$

$\frac{\partial q^{*}}{\partial C}=\frac{\partial}{\partial C}\left[A-\frac{DP^{*}+EC}{S}\right]=\frac{\partial}{\partial C}\left[A-\frac{D}{S}P^{*}-\frac{E}{S}C\right]=-\frac{D}{S}\frac{\partial P^{*}}{\partial C}-\frac{E}{S}$

$=-\frac{D}{S}\cdot\frac{D-E}{2D}-\frac{E}{S}=-\frac{D-E}{2S}-\frac{E}{S}=-\frac{D-E}{2S}-\frac{2E}{2S}=\frac{-D-E}{2S}=a_{2}$

Which again by the assumption of the question is known.

Is it possible to use $a_{1}$ and $a_{2}$ to solve for $S$?

$a_{1}=\frac{D-E}{2D}$

$a_{2}=\frac{-D-E}{2S}=\frac{-D-E}{2D}\cdot\frac{D}{S}=\frac{D-2D-E}{2D}\cdot\frac{D}{S}=\left(\frac{D-E}{2D}-1\right)\cdot\frac{D}{S}=\left(a_{1}-1\right)\cdot\frac{D}{S}$

$S=\left(\frac{a_{1}-1}{a_{2}}\right)\cdot D$

So if I can figure out a way to find D I can identify S. But with the pieces I have ($\partial\pi/\partial C$, $\partial P^*/\partial C$, and $\partial q^*/\partial C$), I don't see how to do that.

I read j-kahn's solution as proposing in the work above that E is zero but if E is zero this does not identify S.

• Does $D$ matter except as it is relative to $S$? This is the equivalent of being limited in the estimation of utility parameters, as I was saying below.
– jayk
Nov 19, 2014 at 19:43
• I'm interested in a dollar measure of S. D/S is one of the derivatives of the demand curve so I can see why it could be interesting but that isn't an economically important quantity in my problem.
– BKay
Nov 19, 2014 at 19:52
• Whatever works for your problem. I can't think of a good solution to distinguishing demand slope from adjustment costs, unless you can track consumers or possibly observe the entire market, along with entry. I edited my response to reflect the way you're solving this, but with a more general statement. I get a slightly different answer, $\frac{d q^*}{dC} = \frac{\delta q}{\delta P}\left(2\frac{dP^*}{dC}-1\right)$. It's entirely possible there's an arithmetic error in my stuff but I don't see where right now.
– jayk
Nov 19, 2014 at 20:06
• And I've just realized why our answers differ, it's because I didn't take into account permanent demand.
– jayk
Nov 19, 2014 at 20:12
• You can still get identification without the contract, through the same methods (demand response to price change + price response using the envelope theorem) but I don't think you'll have a convenient closed form, so you may need to solve it numerically.
– jayk
Nov 19, 2014 at 20:20