Given the four capacity cases introduced in Section 5.1 now I will consider the effect of variation in firms’ capacity levels on pricing in a fixed period and also how prices change in remaining time for fixed levels of capacity. This provides interesting results on price monotonicity and additionally sets the groundwork for an assessment of the ex-ante game of capacity production (Section 6.4).
First, consider the monopoly situation. Without consumer heterogeneity, as in Dudey (1992), monopoly prices are always equal to valuationv. In my model pricing differs, as seen in Figure 6. Here monopoly prices (vertical axis) are plotted for different levels of fixed monopoly capacities and remaining time periods (horizontal axes). Immediately we can see that monopoly prices weakly increase in time and weakly decrease in capacity, ce-teris paribus. Intuitively, given fixed capacities a monopolist’s reservation value increases if she has more time to sell her goods. Contrary, in a given period her reservation value
31In Section 5.6, which is on market power, this is discussed in more detail.
is lower, the more capacity she holds. However, if capacity becomes excessive relative to remaining time (xM > t), monopoly prices are constant and equal to the final-period price. Hence, the following result.32
Result 6. Under monopoly, prices are weakly decreasing in capacity and weakly increas-ing in remainincreas-ing time.
Figure 6: Comparative Statics of Capacity and Time (Monopoly)
Parameters: v= 100,δ= 1,µ= 10, myopic consumers. Monopoly pricespM for different capacity levels ofxM and different remaining time periodst.
Second, to see how duopoly prices in a given period change in capacity level variation we have to distinguish between the different capacity cases. For this consider the example in Figure 7 fort= 10 showing firm 1’s price (vertical axis), given variation in its own and also in firm 2’s capacity levels (horizontal axes). If there are individual excess capacities, i.e. each firm can serve the entire expected demand (in the figurexi≥10 =tfori= 1,2), pricing is unaffected by the exact capacity levels because the period is essentially treated like the final one and firms post the static total-competition price (compare Proposition 3). Contrary, if there is at least one active firm that cannot serve all expected demand, i.e. min
i {xi} < t and i ∈ J(ω), today’s prices and hence selling probabilities do have implications for reservation values, i.e. future prices and selling probabilities. Specifically, if together the firms cannot serve the entire demand, i.e. capacities are scarce (in the
32This result is robust to forward-looking consumers, for this consider Figure A.3 in Appendix A.3.
5. Comparative Statics and Discussion 111 figure ifx1+x2≤10 =t), a firm’s price decreases in its own and also in the competitor’s capacity. Intuitively, lower own or lower competitor capacity and hence lower aggregate capacity means higher reservation values because firms are more confident to sell out eventually and hence can post higher prices.33 However, if we have aggregate excess capacities and rather symmetric capacities (in the figure x1 ≈x2 ∈ [6,9]), pricing can be extremely aggressive (compare Section 5.1), leading to possible non-monotonicity of prices in capacity levels, i.e. a steep decrease followed by an increase of prices in any firm’s capacity.34
Figure 7: Comparative Statics of Capacity Levels (Duopoly)
Parameters: t= 10,v= 100, δ= 1, µ= 10, myopic consumers. Pricep1 int= 10 under duopoly for different capacity levels ofx1 andx2.
These observations prove the following findings, which are robust to forward-looking consumers, as illustrated in Figure A.4 of Appendix A.3.
Proposition 7. Under duopoly in a fixed period, prices are neither monotone in own nor in competitor capacity.
Result 7. If capacities are scarce in a fixed period, prices decrease in own and in com-petitor capacity.
33Note for very low capacities that it might be that pi> v, as the firms are comfortable to “bet” on high taste realizations as long as the deadline is still relatively far.
34Similarly, ifxj> tbutxiis just lower thant, e.g. such thatx1= 9<10 =t < x2, firms’ reservation values are relatively low and they are eager to sell because if no sale occurs, in the subsequent periods there will be individual excess capacities and hence total-competition pricing until the deadline.
Figure 8: Comparative Statics of Remaining Time (Duopoly)
Parameters: v= 100, δ= 1, µ= 10, myopic consumers. Pricep1 under duopoly for different levels of capacityx1 and different remaining time periodst, whilex2= 5 is fixed.
Finally, I consider how under duopoly a firm’s price changes in remaining timet, for fixed levels of capacity. Figure 8 shows firm 1’s price (vertical axis) for various fixed levels of capacityx1in remaining timet(horizontal axes), whereby competitor capacity is fixed atx2 = 5. Similar to the analysis above we see that prices are constant intand equal to the final-period total-competition price when capacities are individually excessive (here for x1 ≥5 =x2 ≥t). Further, prices increase in remaining selling time if capacities are scarce (here if x1+ 5 ≤ t) because, intuitively, more time to sell their goods increases firms’ reservation values. Only when capacities are on aggregate excessive, prices might not be monotone in remaining time. As discussed above, specifically if capacities are sufficiently symmetric (here for example x1 = x2 = 5 < 6 = t), prices become very aggressive (even negative) while they increase again to the total-competition price as soon as capacities become individually excessive relative to remaining time (here for examplex1=x2 = 5 =t). The intuition is that firms “race" to become the smaller firm (compare Section 5.1). Finally, note that without consumer heterogeneity (Dudey, 1992) prices can also be non-monotonic in capacity and remaining time, yet remain constant and equal to the monopoly price under scarcity. The following findings summarize these observations.35
35This is robust to forward-looking consumers. See Figure A.5 in Appendix A.3 and refer to the discussion in Section 6.2 for why under individual excess capacities prices decrease in remaining time.
5. Comparative Statics and Discussion 113 Proposition 8. Under duopoly with fixed capacities, a firm’s price is not monotone in remaining time.
Result 8. If capacities are scarce and fixed, prices increase in remaining time.
The analysis of this section shows that prices will typically increase after a sale and decrease otherwise. This feature is common with the literature of monopoly and oligopoly dynamic pricing models, for example in Talluri and Van Ryzin (2006), Hörner and Samuelson (2011) or Dilme and Li (2016). However, my model dynamics explain additional price volatility, too. As Propositions 7 and 8 suggest, under competition prices can go up or down in the next period, regardless of trade actually having taken place. Technically, this non-monotonicity emerges because any firm could sell (or not) in a given period due to consumer heterogeneity, such that different capacity cases with distinct pricing can occur.