In this section, we analyze the firm’s problem under two different pricing strategies: dynamic pricing and price commitment. We analyze the firm’s optimal decisions and compare the profits achieved under each pricing strategy in the next subsections.
Dynamic pricing
In the dynamic pricing setting, the firm announces the price of the second period at the end of the first period when the value of uncertain parameter β is known. Under this strategy, we solve the problem backward by maximizing the profit in the second period first,
$$ {\pi}_2=\left[{p}_2\left(c\left(\alpha +\beta {\lambda}_1\right)\right)\right]\left(\overline{v}\underset{\_}{v}\right). $$
(5)
Since \( \underset{\_}{v} \) is the threshold valuation of not buying in the second period, it can be solved by setting \( {U}_2=\theta \left(\ \underset{\_}{v}{p}_2\right)=0 \). In addition \( {\lambda}_1=1\overline{v} \), so we obtain:
$$ {\pi}_2=\left[{p}_2c+\alpha +\beta \left(1\overline{v}\right)\left]\right(\overline{v}{p}_2\right). $$
(6)
It is obvious that the above profit function is concave in p_{2}. So we derive the optimal p_{2} as a function of \( \overline{v} \) and β,
$$ {p}_2^{\ast}\left(\overline{v},\beta \right)=\frac{\overline{v}+c\alpha \beta \left(1\overline{v}\right)}{2}. $$
(7)
Under the framework of rational expectation, the firm decides the price on the belief of customers’ strategic behavior while the customers make their decisions by forecasting the firm’s pricing decisions and other customers’ choice. So given this \( {p}_2^{\ast } \), the customer with the threshold \( \overline{v} \) will find it indifferent to purchase in each period. In other words, \( \overline{v}{p}_1=\theta \left(\overline{v}E\left[{p}_2^{\ast}\left(\overline{v},\beta \right)\right]\right) \). Thus,
$$ \overline{v}=\frac{2{p}_1\theta \left(c\alpha \right)}{2\theta }. $$
(8)
In this case, the firm’s total expected profit is given by:
$$ \pi =\left({p}_1c\right)\left(1\overline{v}\right)+E\left[{\pi}_2^{\ast}\left({p}_1,\beta \right)\right], $$
(9)
where \( {\pi}_2^{\ast}\left({p}_1,\beta \right)=\frac{{\left(\overline{v}+\alpha c\right)}^2+2\left(\overline{v}+\alpha c\right)\left(1\overline{v}\right)\beta +{\left(1\overline{v}\right)}^2{\beta}^2}{4} \) is derived by plugging (7) into (6). The following proposition characterizes the optimal prices in each period. Note that under dynamic pricing strategy with uncertain consumption externality, if the firm sells in both periods, the optimal price in the second period is an expectation instead of a certain value due to the realization of the cost uncertainty.
Proposition 1.
Under the dynamic pricing strategy, when
\( \alpha \le \frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)
, the firm sells in both periods; when
\( \alpha >\frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)
, the firm only sells in the second period. Let
\( {p}_1^D \)
and
\( {p}_2^D \)
denote the optimal prices.

(1)
When the firm sells in both periods,
$$ {p}_1^D=\frac{\left(2\theta \right)\left(2\theta {\sigma}^2\right)+\alpha \left[2\theta \left(2\theta {\sigma}^2\right)\right]+\left[2\theta \left(\theta +{\sigma}^2\right)\right]c}{2\left(32\theta {\sigma}^2\right)},E\left[{p}_2^D\right]=\frac{2\theta {\sigma}^2+\alpha \left(3\theta +{\sigma}^22\right)+\left(43\theta {\sigma}^2\right)c}{2\left(32\theta {\sigma}^2\right)}. $$

(2)
When the firm only sells in the second period,
\( {p}_2^D=\frac{1+c\alpha }{2} \)
.
Proposition 1 implies that when the speed of technology advancement is high (e.g. \( \alpha >\frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)), strategic customers expect a low production cost and thus a low price in the second period. In this case, the firm maximizes the profit by selling the product only in the second period. Otherwise, the firm sells in both periods.
Corollary 1.
Under the dynamic pricing strategy, when the firm sells in both periods,

(1)
\( \frac{\partial {p}_1^D}{\partial_{\alpha }}>0 \), \( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\alpha }}>0 \) if and only if σ^{2} > 2 − 3θ;

(2)
\( \frac{\partial {p}_1^D}{\partial_{\sigma^2}}<0 \)
,
\( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\sigma^2}}<0 \)
;

(3)
\( {p}_1^D \)
can be increasing or decreasing in θ,
\( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\theta }}>0 \)
.
First, as technology advances (α increases), the firm produces the product at a lower unit cost and thus obtains a higher profit in the second period. So the firm increases the price in the first period to shift sales from the first period to the second period to earn more profits (\( \frac{\partial {p}_1^D}{\partial_{\alpha }}>0 \)). Regarding the secondperiod price, there are two effects. On one hand, as α increases, since the firstperiod price increases, strategic buyers with relatively high valuations change their purchasing time from the first period to the second period. Then the firm is allowed to increase the secondperiod price to reap more profits from these customers, which is called as the superior effect. On the other hand, some potential customers with low valuations who decided not to purchase originally may decide to buy in the second period. Then the firm may decrease the secondperiod price to attract such customers, which is called as the inferior effect. When the cost reduction is uncertain or customers are patient (σ^{2} > 2 − 3θ), the superior effect dominates, the firm mainly targets at customers changing their purchasing time and thus \( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\alpha }}>0 \).When cost reduction is certain and customers are impatient (σ^{2} ≤ 2 − 3θ), the inferior effect dominates, the firm mainly targets at customers who just enter the market in the second period, and thus \( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\alpha }}\le 0 \).
Second, as uncertainty increases (σ^{2} increases), the uncertain component βλ_{1} plays a more important role in the secondperiod production cost. Then the firm decreases the firstperiod price (i.e., \( \frac{\partial {p}_1^D}{\partial_{\sigma^2}}<0 \)). The reason is explained as follows: On one hand, if the uncertain cost is realized to be cost reduction (i.e. positive β), the firm should increase initial sales by decreasing the firstperiod price in order to generate a greater cost reduction in the second period. On the other hand, if the uncertain cost is realized to be cost increase (i.e. negative β), the firm can always use the secondperiod price as a lever to alleviate this effect. Thus the gain of increasing the firstperiod price in the former case is larger than the loss in the latter case. Resulting from a lower firstperiod price and a smaller remaining market (\( \overline{v} \) is smaller), the firm should decrease the secondperiod price in expectation. This is different from the findings in Shum et al. (2016). They find that price level is independent on cost uncertainty because they assume the uncertainty does not depend on initial sales.
Third, when customers are more strategic, or equivalently, more likely to wait (θ increases), the firm will decrease the firstperiod price to alleviate customers’ waiting behaviors in most cases (i.e., \( \frac{\partial {p}_1^D}{\partial_{\theta }}<0 \)). However, when α is so large that the second period is very attractive to the customers, the firm will increase the firstperiod price to encourage customers with high valuation to buy in the second period (i.e., \( \frac{\partial {p}_1^D}{\partial_{\theta }}\ge 0 \)). It is intuitive when the customers are more strategic, the firm is able to increase the price in the second period (i.e., \( \frac{\partial E\left[{p}_2^D\right]}{\partial_{\theta }}>0 \)).
The next proposition characterizes the firm’s optimal expected profit under dynamic pricing.
Proposition 2.
Let π
^{D}
be the optimal expected profit of the firm under dynamic pricing.

(1)
When the firm sells in both periods,
$$ {\pi}^D=\frac{{\left(2\theta \right)}^2{\sigma}^2+2\alpha \left[2\left(2\theta \right)\theta {\sigma}^2\right]+{\alpha}^2\left(4+{\theta}^2{\sigma}^2\right)c\left\{2{\left(2\theta \right)}^2+2\alpha \left[2\left(2\theta \right)\theta \right]2\left(1+\alpha \right){\sigma}^2\left[{\left(2\theta \right)}^2{\sigma}^2\right]c\right\}}{4\left(32\theta {\sigma}^2\right)}. $$

(2)
When the firm only sells in the second period,
\( {\pi}^D=\frac{{\left(1+\alpha c\right)}^2}{4} \)
.
Corollary 2.
Under the dynamic pricing strategy, when the firm sells in both periods,

(1)
\( \frac{\partial {\pi}^D}{\partial_{\alpha }}>0 \)
.

(2)
\( \frac{\partial {\pi}^D}{\partial_{\sigma^2}}>0 \)
.

(3)
\( \frac{\partial {\pi}^D}{\partial_{\theta }}<0 \)
.
It is intuitive that technology advancement reduces the unit production cost in the second period and thus increases the profit (i.e., \( \frac{\partial {\pi}^D}{\partial_{\alpha }}>0 \)). As shown in Shum et al. (2016), uncertainty is always beneficial for the firm under dynamic pricing (i.e., \( \frac{\partial {\pi}^D}{\partial_{\sigma^2}}>0 \)). The reason is similar to our discussions for Proposition 1: If the uncertain cost is realized as a reduction, it makes the firm earn more profits in the later sales due to a lower unit cost. If the uncertain cost is realized as an increase, the firm can use the price in the second period as a lever to alleviate this effect. Consistent with previous literature, the firm’s profit decreases when customers become more patient (i.e., \( \frac{\partial {\pi}^D}{\partial_{\theta }}<0 \)), because the customers would wait in anticipation of the price decrease, which hurts the firm’s profit. In other words, when the customers are more patient, the firm is not able to utilize price discrimination over time to get more profits.
Price commitment (PC)
In the price commitment setting, the firm announces the prices in both periods before customers decide when to purchase and the realization of β. The utility of a customer with valuation \( \overline{v} \) satisfies \( \overline{v}{p}_1=\theta \left(\overline{v}{p}_2\right) \), or equivalently,
$$ \overline{v}=\frac{p_1\theta {p}_2}{1\theta }. $$
(10)
As in dynamic pricing, the valuation threshold of not buying in the second period is given by setting \( {U}_2=\theta \left(\ \underset{\_}{v}{p}_2\right)=0 \). However, the firm determines both prices based on the expected uncertain cost in the second period. So the total expected profit of the firm is given by
$$ \pi =\left({p}_1c\right)\left(1\overline{v}\right)+\left({p}_2E\left[{c}_2\right]\right)\left(\overline{v}\underset{\_}{v}\right) $$
(11)
where E[c_{2}] = c − α according to the assumptions that c_{2} = c − (α + βλ_{1}) and β has a mean of zero. The following proposition characterizes the optimal prices under price commitment.
Proposition 3.
Under price commitment strategy, when
\( \alpha \le \frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)
, the firm sells in both periods; when
\( \alpha >\frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)
, the firm only sells in the second period. Let
\( {p}_1^C \)
and
\( {p}_2^C \)
denote the optimal prices.

(1)
When the firm sells in both periods,
\( {p}_1^C=\frac{2+\alpha +\left(1+\theta \right)c}{3+\theta } \)
,
\( {p}_2^C=\frac{1+\theta \alpha +2c}{3+\theta } \)
.

(2)
When the firm only sells in the second period,
\( {p}_2^C= \)
\( \frac{1+c\alpha }{2} \)
.
Different from dynamic pricing, the firm prices the product in advance in anticipation of what would happen in the future and thus only the expectation of the uncertain cost would affect the firm’s decision. Then the prices are independent on the uncertainty level σ^{2}. But with the same condition (\( \alpha >\frac{\left(1\theta \right)\left(1c\right)}{1+\theta } \)) as in dynamic pricing, the firm only sells the product in the second period and otherwise the firm sells the product in both periods.
Corollary 3.
Under the price commitment strategy, when the firm sells in both periods,

(1)
\( \frac{\partial {p}_1^C}{\partial_{\alpha }}>0 \), \( \frac{\partial {p}_2^C}{\partial_{\alpha }}<0 \).

(2)
\( \frac{\partial {p}_1^C}{\partial_{\theta }}<0 \),\( \frac{\partial {p}_2^C}{\partial_{\theta }}>0 \).
First, as technology advances (α increases), similar to that under dynamic pricing, \( \frac{\partial {p}_1^C}{\partial_{\alpha }}>0 \). However, under the price commitment, the secondperiod price is decreasing in α. This is because the firm has no ability to respond to uncertainty or in other words, the firm can only anticipate a cost reduction due to technology advance. Therefore in this case, the inferior effect always dominates and thus the price decreases when technology advances faster (\( \frac{\partial {p}_2^C}{\partial_{\alpha }}<0 \)).
Second, when the customers become more patient (θ increases), consistent with findings in Besanko and Winston (1990) and Shum et al. (2016), the advantage of price commitment is constraining the waiting behavior of the strategic customer and when the firm sells in both periods, the optimal pricing path is always markdown.^{Footnote 2} Thus when the customers become more patient, the firm commits to not decreasing the secondperiod price to constrain the waiting behavior of strategic customers and decrease the firstperiod price to encourage them to purchase early (i.e.,\( \frac{\partial {p}_1^C}{\partial_{\theta }}<0 \) and \( \frac{\partial {p}_2^C}{\partial_{\theta }}>0 \)).
The next proposition characterizes the firm’s optimal expected profit under the price commitment strategy.
Proposition 4.
Let π
^{C}
be the optimal expected profit of the firm under the price commitment.

(1)
When the firm sells in both periods,
\( {\pi}^C=\frac{1\theta +\alpha \left(1+\alpha \theta \right)c\left(1\theta \right)\left(2+\alpha c\right)}{\left(1\theta \right)\left(3+\theta \right)} \)
.

(2)
When the firm only sells in the second period,
\( {\pi}^C=\frac{{\left(1+\alpha c\right)}^2}{4} \)
.
Corollary 4.
Under the price commitment strategy, when the firm sells in both periods,

(1)
\( \frac{\partial {\pi}^C}{\partial_{\alpha }}>0 \).

(2)
\( \frac{\partial {\pi}^C}{\partial_{\theta }}<0 \).
Under price commitment the firm is not able to respond to the cost uncertainty dynamically and so the optimal expected profit does not depend on σ^{2}. It is intuitive that the firm earns more profit as technology advances and less profit when customers are more patient.