Abstract
We study a multi-unit auction model in which bidders are privately informed about the maximum number of units they are willing to trade (which we refer to as ‘capacity’). No matter how big or small, private information on capacities changes the nature of the equilibrium as compared to when private information is on costs (or valuations). Also, the uniform-price and discriminatory auctions are not revenue equivalent, in contrast to when costs are independently drawn. In particular, with independently drawn capacities (and possibly costs), the discriminatory format reduces payments to firms relative to the uniform-price format. Our analysis is motivated by the performance of future electricity markets in which renewable energies will be predominant, but the set-up also applies to a variety of contexts (from central bank liquidity auctions to emissions trading).
Ambitious environmental targets, together with decreasing investment costs, have fostered the rapid deployment of renewable energy around the world. Installed renewable capacity has more than doubled over the last ten years and it is expected to further increase during the coming decade.1 How will electricity markets perform in the future once renewables become the predominant source of energy?
Whereas competition among conventional fossil-fuel generators is by now well understood (e.g., Green and Newbery, 1992; von der Fehr and Harbord, 1993; Borenstein, 2002), much less is known about competition among wind and solar producers (which we broadly refer to as renewables). Competition-wise, there are two key differences between conventional and renewable technologies. First, the marginal cost of conventional power plants depends on their efficiency rate as well as on the volatile price at which they buy fossil fuel. In contrast, the marginal cost of renewable generation is constant (and essentially zero), as plants produce electricity out of freely available natural resources (e.g., wind or sun). Second, the availability of renewable power plants depends on weather conditions that are difficult to predict (Gowrisankaran et al., 2016),2 unlike conventional power plants that are always available in the absence of outages. Hence, the move from fossil-fuel generation toward renewable sources will imply a change in the competitive paradigm. Whereas the previous literature has analysed environments in which marginal costs are private information, but production capacities are publicly known (Holmberg and Wolak, 2018), the relevant setting will soon be one in which, during an increasing number of hours a day, marginal costs will be known (and essentially zero), but firms’ available capacities will be private information.
In this paper, we build a model in which available capacities are firms’ private information, and we apply it to electricity markets. Producers compete to serve demand by submitting price-quantity pairs, which indicate the minimum price at which they are willing to produce up to the committed quantity. Firms’ production is dispatched in increasing price order until total demand is satisfied. The price they receive for their output depends on the auction format in place: either a uniform-price auction, which pays the winning producers at the market-clearing price, or a discriminatory auction, which pays each producer at their bid. We characterise and compare bidding behaviour and market outcomes under these two auction formats.
Under the uniform-price auction, firms exercise market power by offering all their capacity at a price above marginal cost or by withholding capacity. When a firm’s realised capacity is below total demand, the firm adds a markup over its marginal costs that is decreasing in its realised capacity. This reflects the standard trade-off faced by competing firms, as decreasing the price leads to an output gain (quantity effect), but it also depresses the market price if the rival bids below (price effect). Since firms gain more from the quantity effect when their realised capacity is large, they are more willing to sacrifice part of their markup in exchange for selling at capacity. When a firm’s realised capacity is above total demand, it exercises market power by withholding output to let the rival firm set a higher market price.
These equilibrium properties imply that market prices are lower at times of high capacity availability relative to demand. Thus, in our model, price volatility is inherently linked to market power and not to capacity uncertainty per se. In the absence of market power, prices would remain unchanged at marginal cost regardless of capacity realisations. Our model also predicts that all else equal, an increase in capacity investment shifts the whole distribution to the right, which depresses expected market prices until they converge towards marginal costs.3
Bidding behaviour under the discriminatory auction is similar to under the uniform-price format, with two main differences. First, firms tend to offer higher prices relative to the uniform-price auction, given that they are always paid according to their bid. And, for the same reason, firms do not gain from withholding output as this does not affect the price they receive.
The comparison across auction formats also shows that they are not revenue equivalent. In particular, the discriminatory auction leads to lower firms’ profits and higher consumer surplus than the uniform-price format. The reason is that, under both auction formats, a firm that has a higher capacity realisation is willing to offer a lower price (quantity effect). However, having a higher capacity also implies that, conditionally on having a larger capacity than the rival, and hence a lower bid, the rival’s expected capacity goes up while its expected price offer goes down. As a result, the price that the low bidder expects to receive under the uniform-price auction is reduced, thus weakening the firm’s incentives to bid aggressively. This effect is not present in the discriminatory format, given that each firm is paid according to its bid.
In the light of the literature on multi-unit auctions (Ausubel et al., 2014), the absence of revenue equivalence might not seem surprising. However, in line with Holmberg and Wolak (2018), we find that revenue equivalence does hold with known capacities, and privately known and independently distributed costs. Why is it that the source of private information—whether on costs or on capacities—matters to the extent that revenue equivalence holds in the former, but not in the latter case? The reason is that, when private information is on costs only, the output allocated to the low and high bidders is the same regardless of their private information. This implies that the quantity and price effects depend on the firms’ costs only through their optimal bids. This is in contrast to when private information is on capacities, in which case the quantity and price effects further depend on the firms’ types through the output allocation conditionally on having the low or the high bid. This difference makes asymmetric information about capacities different from asymmetric information about costs when they are both independently drawn across firms.
While analysing the polar cases with private information on either capacities or costs, allows us to isolate the distinct channels by which they affect equilibrium bidding, it is also illustrative to combine the two sources of private information. Such an analysis reveals that the predictions of the model with independently drawn costs become similar to those of the model with independently drawn capacities, as soon as we allow both costs and capacities to be privately known. In contrast, the predictions of our baseline model with unknown capacities extend naturally beyond the polar case, including the equilibrium characterisation and the revenue comparison across auction formats.
To understand how private information changes the nature of the equilibrium, we also characterise two extreme benchmarks: competition when all the information is either publicly known or unknown. In this regard, we show that the impact of private information is similar across models with privately known costs or capacities, despite the differences in the equilibrium bidding behaviour highlighted above. Overall, we find that more information (be it on costs or capacities) strengthens firms’ market power. Since private information introduces asymmetries, firms compete less fiercely as compared when they do not observe their costs or capacities. In contrast, when they observe each other’s private information, they can condition on it to ease rivalry while sharing ex ante expected profits symmetrically. As a consequence, the highest (lowest) profits are obtained when information is publicly known (unknown), while equilibrium profits under private information are in between those polar cases.
Furthermore, we show that an increase in information precision regarding the rival’s capacity leads to less competitive outcomes. This result suggests that firms might be better off exchanging private information (be it on costs or capacities) to sustain higher equilibrium profits, at the consumers’ expense. This prediction is in line with the model of Hansen (1988), in which private information on costs enhances competition through an output expansion effect. In his model, unlike ours, demand elasticity implies that such lower prices can make both consumers and producers better off.
Regarding the performance of future electricity markets, our model shows that renewable energy will help mitigate potential market power concerns, but some degree of it might nevertheless remain. In those hours of the day when renewable sources dominate, market prices will smoothly go down as more investment is carried out, but they will not converge to marginal costs unless there exists sufficient excess capacity. The exercise of market power will exacerbate the natural volatility that would come from changes in available renewable output because markups are higher when realised capacity is low. A final insight is that, when marginal costs are fairly similar (as is the case for renewables), market transparency might exacerbate market power without delivering efficiency benefits.
Within our framework, we also analyse further developments that are likely to be important in future electricity markets, such as the increase in demand elasticity brought about by dynamic pricing policies or the deployment of storage facilities. In particular, our main equilibrium characterisation with inelastic demand extends very naturally to environments with a downward-sloping demand function. Indeed, while equilibrium properties remain the same, we show that an increase in demand elasticity reduces the maximum price that firms are willing to offer, thereby making their bid functions flatter. Ultimately, demand elasticity reduces prices, increases the pace at which prices converge towards marginal costs, and it is likely to reduce the price volatility across time.
Previous papers have analysed competition among renewable power sources under capacity uncertainty (Acemoglu et al., 2017; Kakhbod et al., 2021). These papers, unlike ours, assume Cournot competition, i.e., firms exercise market power by withholding output.4 Acemoglu et al. (2017) focused on the effects of common ownership of conventional and renewable plants. They showed that it weakens the price-depressing effect of renewables, as strategic firms respond to an increase in renewables by withholding more conventional output. This effect vanishes under the scenario considered in our paper (i.e., when renewables are enough to cover total demand), as exercising market power through conventional power plants would not affect market outcomes. Kakhbod et al. (2021) focused on the heterogeneous availability of renewable sources across locations, and showed that firms withhold more output when their plants are more closely located, i.e., when their output is highly and positively correlated.
Our paper also contributes to the literature on multi-unit auctions. Existing papers differ in whether bidders submit a finite or an infinite number of price-quantity pairs. Among the latter, Wilson (1979) was the first to characterise equilibrium bidding for share auctions, while Klemperer and Meyer (1989) were the first to characterise the equilibria in continuous supply functions.5 Although these approaches often result in a parsimonious equilibrium characterisation, they do not match the usual rules in electricity markets, where firms can submit a limited number of steps.6 This difference has relevant implications, since the possibility to submit an infinite number of price-quantity pairs implies that ties do not occur at the margin.
Our paper assumes that bid functions are constrained to be step-wise and, therefore, firms offer a finite number of price-quality pairs. Earlier papers in this literature assumed complete information on costs and capacities (von der Fehr and Harbord, 1993; Fabra et al., 2006) while the most recent ones allow for privately known marginal costs (with possibly affiliated signals), which are assumed to be constant up to a publicly known capacity (Holmberg and Wolak, 2018).7 A few papers allow for multiple steps, but either assume complete information (de Frutos and Fabra, 2012) or allow for privately known costs (or valuations), but under very specific examples (Engelbrecht-Wiggans and Kahn, 1998). Furthermore, these papers focus on the uniform-price auction. In contrast, our model proposes a general yet tractable model of discrete uniform-price and discriminatory auctions that introduces a different (albeit relevant) source of private information, i.e., instead of costs, our model allows for privately known capacities. This comes at the cost of abstracting from other model ingredients, such as signal affiliation and costs or bids with multiple steps.8
Our model also contributes to the analysis of a wide range of auction settings in which bidders are privately informed about the maximum number of units they are willing to buy or sell. Beyond electricity auctions, this is the case of Treasury Bill auctions (Hortaçsu and McAdams, 2010; Kastl, 2011), in which banks are privately informed about their hedging needs and, consequently, on the volume of bonds they aim to acquire. Other examples include emission permit auctions (Cantillon and Slechten, 2018), spectrum auctions (Milgrom, 2004), Central Banks’ liquidity auctions (Klemperer, 2010), electricity capacity markets (Fabra, 2018; Llobet and Padilla, 2018), or auctions for renewable investments (Cantillon, 2014; Fabra and Montero, 2022), to name a few.9
The remainder of the paper is structured as follows. Section 1 describes the model with privately known capacities and interprets it in the context of electricity markets. Section 2 solves the model under the uniform-price and discriminatory auctions. Section 3 studies the impact of private information and information precision on equilibrium outcomes. Section 4 revisits the model with privately known costs under both auction formats. Section 5 compares the equilibria and the market outcomes in models in which private information is on capacity, cost, or both. Section 6 extends the model to allow for demand elasticity and storage. Section 7 concludes. All proofs are relegated to the Appendix.
1. The Model
Consider a market in which two ex ante symmetric firms |$i=1,2$| compete to serve a perfectly price-inelastic demand |$\theta \gt 0$|. Firms can produce at a constant marginal cost |$c\ge 0$| up to their available capacities, which are assumed to be random. In particular, the available capacity of firm i, denoted as |$k_{i}$|, is distributed according to |$G(k_{i})$|, with positive density |$g(k_{i})$| in the whole interval |$[ \underline{k},\overline{k}]$|. We assume that |$2\underline{k}\ge \theta$| to make sure that there is always enough available capacity to cover total demand. Firm i can observe its capacity realisation, but not that of its rival, which is independent from its own.
Firms compete on the basis of the bids submitted to an auctioneer. Each firm simultaneously and independently submits a price-quantity pair |$(b_i,q_i)$|, where |$b_i$| is the minimum price at which it is willing to supply the corresponding quantity |$q_i$|. We assume that |$b_{i}\in [ 0,P]$|, where |$P$| denotes the ‘market reserve price’. We also assume that firms cannot offer to produce above their available capacity or below their minimum capacity, |$q_{i}\in [\underline{k},k_i ]$| for |$i=1,2$|.10
The auctioneer ranks firms according to their price offers, and calls them to produce in increasing rank order. In particular, if firms submit different prices, the low-bidding firm is ranked first. If firms submit equal prices, firm i is ranked first with probability |$\alpha ( q_{i},q_{j})$| and it is ranked second with probability |$1-\alpha ( q_{i},q_{j})$|. We assume a symmetric function |$\alpha ( q_{i},q_{j}) =\alpha ( q_{j},q_{i}) \in ( 0,1)$|.11 If firm i is ranked first, it produces |$\min \lbrace \theta ,q_{i}\rbrace ,$| while if it is ranked second, it produces |$\max \lbrace 0,\min \lbrace \theta -q_{j},q_{i}\rbrace \rbrace$|, where |$j\ne i$|.
Firms, which are assumed to be risk neutral, bid so as to maximise their individual expected profits, given their realised capacities.
1.1. Interpreting the Model
Before solving the model, and given our primary motivation, we now discuss how to interpret it in the context of electricity markets. Needless to say, electricity markets are complex institutions, which differ across jurisdictions in several aspects, including market rules or market structure. While our stylised model does not include all those ingredients, it nevertheless captures some of the main driving forces that will shape market performance once renewable energies become predominant. In Section 6 we discuss the implications of allowing for some of such ingredients.
We have assumed that firms compete by submitting step-wise supply functions to serve a known and inelastic demand. This structure resembles competition in most electricity markets in practice, where firms submit a finite number of price-quantity pairs to an auctioneer, who then allocates output and sets market prices according to such bids. Because of tractability reasons, our model restricts the number of steps that firms can use to just one.13
By the time firms submit their bids, they have very precise information about the demand, as system operators regularly publish highly accurate demand forecasts before the market opens.14 Furthermore, electricity demand is highly price inelastic in the short run because electricity retail prices rarely reflect movements in spot market prices. Even where they do, consumers typically do not have strong incentives or the necessary information to optimally respond to the high-frequency price changes.15
In current electricity markets, renewable energy sources co-exist with conventional technologies. While our model does not explicitly capture the co-existence of various energy sources, the conventional technologies are implicitly present in the model through P, which can be interpreted as the marginal costs of coal or gas plants. Under this interpretation, our model can capture the fact that dominant players could own both renewable and conventional power plants, given that we focus on those periods in which the latter are not needed to cover demand. In future stages of the energy transition, and consistently with most real electricity markets, P could be interpreted as an explicit price cap, or as an implicit one that triggers regulatory intervention.
One of the core assumptions of our model is that firms’ capacities are subject to random shocks. In the context of electricity markets, the capacity of each renewable plant is subject to common and idiosyncratic availability shocks. In our model, these shocks could be captured by decomposing the available capacity of firm i in two additive components, |$k_i\equiv \beta \kappa + \varepsilon _i$|. The parameter |$\beta \in [0,1]$| is the common shock component that affects the availability of each firm’s nameplate capacity |$\kappa$|. The idiosyncratic shock |$\varepsilon _i$| can be thought of as being distributed according to |$\Phi (\varepsilon _i|\kappa )$| in an interval |$[\underline{\varepsilon },\overline{\varepsilon }]$|, with |$E(\varepsilon _i)=0$|. As a result, firm i’s available capacity |$k_i$| is distributed according to |$G(k_{i})= \Phi (k_i-\beta \kappa |\kappa )$| in the interval |$k_{i}\in [ \underline{k},\overline{k}]$|, where |$\underline{k}=\beta \kappa +\underline{\varepsilon }$| and |$\overline{k}=\beta \kappa +\overline{\varepsilon }$|, in line with the assumptions of the model. According to this interpretation, firms’ available capacities are correlated through the common shock component, albeit imperfectly so due to the presence of idiosyncratic shocks.
While electricity system operators typically publish forecasts of the common weather shocks at the national or regional level, the idiosyncratic components remain each firm’s private information. Indeed, through the monitoring stations installed at the renewable plants’ sites, firms have access to local weather measurements that are not available to their competitors. Beyond weather conditions, the plants’ availability might be subject to random outages and maintenance schedules that only their owners are aware of. Accordingly, in the presence of private information, each firm is better informed about its own available capacity than its competitors. Our model applies even to those settings in which the amount of private information is relatively small.16
To illustrate this claim empirically, we have collected data from the Spanish electricity market to perform and compare forecasts of a plant’s production, with and without firms’ private information. In particular, we have obtained proprietary data of six renewable plants corresponding to their hourly production and their own available forecasts at the time of bidding, for two years. We have also gathered the forecasts provided by the Spanish System Operator (Red Eléctrica de España) and the one-day ahead predictions of the Spanish weather agency (Agencia Estatal de Meteorología or AEMET) at the provincial level, which is the most disaggregated data publicly available close to the plant’s location.17 We have used OLS to forecast each plant’s hourly availability, with and without the firms’ proprietary local data. Figure 1 plots the distribution of the forecast errors and Table 1 summarises the mean and SDs of the corresponding forecast errors. The evidence is consistent with firms possessing private information that allows them to significantly improve the precision of the forecasts of their own plants’ available capacities.18 Interestingly, when the private forecast is used, the national forecast, while still statistically significant, has a small economic impact on the prediction.
Kernel Distribution of the Forecast Errors Using Public (Solid) or Private Information (Dashed).
Notes: This figure depicts the densities of the forecast errors of the specifications in Table 1. Both distributions are centred around zero, but the SD is larger when only publicly available data are used.
Forecast Errors with Public versus Private Information.
| Variables | (1) | (2) |
|---|---|---|
| Public forecast | 0.582|$^{***}$| | 0.070|$^{***}$| |
| (0.035) | (0.021) | |
| Private forecast | 0.657|$^{***}$| | |
| (0.008) | ||
| Observations | 36,671 | 36,671 |
| R2 | 0.520 | 0.826 |
| SD of the error | 0.18 | 0.11 |
| Variables | (1) | (2) |
|---|---|---|
| Public forecast | 0.582|$^{***}$| | 0.070|$^{***}$| |
| (0.035) | (0.021) | |
| Private forecast | 0.657|$^{***}$| | |
| (0.008) | ||
| Observations | 36,671 | 36,671 |
| R2 | 0.520 | 0.826 |
| SD of the error | 0.18 | 0.11 |
Notes: The dependent variable is the plant’s hourly production normalised by its nameplate capacity. Both regressions include weather data (temperature, wind speed and atmospheric pressure) as well as plant, hour and date fixed effects. The robust standard errors are in parentheses. *** p|$\lt $| 0.01. One can see that using the plant’s own forecast significantly reduces the forecast error, with the |$R^2$| increasing from 0.520 to 0.826. When the private forecast is used, the public forecast is still statistically significant, but it has a small impact on the prediction.
Forecast Errors with Public versus Private Information.
| Variables | (1) | (2) |
|---|---|---|
| Public forecast | 0.582|$^{***}$| | 0.070|$^{***}$| |
| (0.035) | (0.021) | |
| Private forecast | 0.657|$^{***}$| | |
| (0.008) | ||
| Observations | 36,671 | 36,671 |
| R2 | 0.520 | 0.826 |
| SD of the error | 0.18 | 0.11 |
| Variables | (1) | (2) |
|---|---|---|
| Public forecast | 0.582|$^{***}$| | 0.070|$^{***}$| |
| (0.035) | (0.021) | |
| Private forecast | 0.657|$^{***}$| | |
| (0.008) | ||
| Observations | 36,671 | 36,671 |
| R2 | 0.520 | 0.826 |
| SD of the error | 0.18 | 0.11 |
Notes: The dependent variable is the plant’s hourly production normalised by its nameplate capacity. Both regressions include weather data (temperature, wind speed and atmospheric pressure) as well as plant, hour and date fixed effects. The robust standard errors are in parentheses. *** p|$\lt $| 0.01. One can see that using the plant’s own forecast significantly reduces the forecast error, with the |$R^2$| increasing from 0.520 to 0.826. When the private forecast is used, the public forecast is still statistically significant, but it has a small impact on the prediction.
As shown by these results, firms’ forecast errors have SDs that remain significant even when firms have private information about their capacities. Nevertheless, the day-ahead market price and output allocation are computed using firms’ committed quantities, even when these differ from their actual ones. Hence, for bidding purposes, what matters is that each firm knows exactly how much output it has offered in the day-ahead market, and not necessarily how much it will be able to produce in real time. This is consistent with our model since we have assumed that firms know their available capacity before submitting their bids.
2. Solving the Model
In this section we characterise the symmetric Bayesian Nash equilibria (BNE) of the uniform-price and discriminatory auctions when capacities are private information. When |$\underline{k} \ge \theta$|, the characterisation of the equilibrium is trivial. Since either firm can cover total demand regardless of their realised capacities, Bertrand forces drive equilibrium prices down to marginal costs. For this reason, in the rest of the paper we turn attention to the remaining cases.
2.1. Uniform-Price Auction
To analyse the uniform-price auction, it is useful to start by assuming that |$\overline{k}\le \theta$| (small installed capacities). In this case, a firm’s capacity can never exceed total demand, implying that the low bid is always payoff irrelevant. We later analyse the case in which |$\overline{k}\gt \theta$| (large installed capacities).
2.1.1. Small installed capacities
We first consider the case in which each firm’s capacity never exceeds total demand, i.e., |$\overline{k}\le \theta$|. Our first lemma identifies three key properties that any equilibrium must satisfy in this case.
If|$\overline{k}\le \theta$|,
capacity withholding is never optimal, |$q^{*}_{i}(k_{i})=k_{i}$|;
all Bayesian Nash equilibria must be in pure strategies;
the optimal price offer of firm i, |$b^{*}_{i}(k_{i})$|, must be (strictly) decreasing in|$k_{i}$|.
The first part of the lemma rules out capacity withholding in equilibrium.19 This result follows from four observations: first, conditionally on having the low bid, the firm maximises its output by offering to sell at capacity; second, conditionally on having the high bid, its profits do not depend on its quantity offer as the firm always serves the residual demand; third, the probability of having the low bid does not depend on |$q_i$| and last, the market price remains unchanged with or without capacity withholding. It follows that expected profits are strictly increasing in |$q_i$|, and are thus maximised at |$q_i=k_i$|.
The second part of the lemma rules out non-degenerate mixed-strategy equilibria. The underlying reason is simple: a firm’s profits at a mixed-strategy equilibrium depend on its realised capacity, which is non-observable by the rival. If the competitor randomises in a way that makes the firm indifferent between two bids for a given capacity realisation, the same randomisation cannot make the firm indifferent for other capacity realisations as well. It follows that all equilibria must involve pure strategies.
The last part of the above lemma rules out bids that are non-decreasing in the firm’s capacity. When a firm considers whether to reduce its bid marginally, two effects are at play (for a given bid of the rival): a profit gain due to the output increase (quantity effect) and a profit/loss due to the reduction in the market price (price effect). On the one hand, the quantity effect is increasing in the firm’s capacity, as if it bids below the rival, it sells at capacity rather than just the residual demand. On the other hand, the price effect is independent of the firm’s capacity as, contingent on bidding higher than the rival, the firm always sells the residual demand. Combining these two effects, the incentives to bid low are (weakly) increasing in the firm’s capacity, giving rise to optimal bids that are non-increasing in |$k_{i}$|. Finally, standard Bertrand arguments imply that the optimal price offer must be strictly decreasing in |$k_{i}$|: equilibrium bid functions cannot contain flat regions, as firms would otherwise have incentives to slightly undercut those prices to expand their expected quantity without affecting the price.
Maximising profits with respect to |$b_i$| and applying symmetry, we can characterise the optimal bid at a symmetric equilibrium.20
In the numerator, the output gain from marginally reducing the firm’s bid, or quantity effect, is relevant only when the two firms tie in prices, i.e., when the rival also has capacity |$k_i$|, an event that occurs with probability |$g(k_i)$|. Reducing the bid implies that the firm sells all its capacity rather than just the residual demand, i.e., its output expands in the amount |$k_i-( \theta -k_i) =2k_i-\theta .$| In the denominator, the price loss from marginally reducing the bid, or price effect, is only relevant when the firm is setting the market price, i.e., when the rival firm’s capacity is above |$k_i$|. In this case, reducing the bid implies that the firm sells the expected residual demand, |$\theta -E(k_j|k_j\le k_i)$|, at a lower market price.
The ratio of these two effects gives shape to the bid function as described on the left-hand side of (2). A bigger quantity effect increases a firm’s incentives to undercut the rival. This means that to sustain this symmetric equilibrium, the bid function must become steeper—to require a larger bid reduction for a given quantity gain—and the markup must become smaller—to make undercutting less profitable. A smaller price effect, to the extent that it makes price increases less profitable, has a similar effect.
Figure 2 illustrates the equilibrium price offer as a function of |$k_i$|. The optimal bid ranges from P for the lowest possible capacity realisation to c for the largest one. When |$k_{i}=\underline{k},$| firm j is guaranteed to have a higher capacity and thus the lower price. Since firm i serves the residual demand with probability 1, it maximises its profits by bidding at P.21 When |$k_{i}=\overline{k}$|, firm j is guaranteed to have a lower capacity, and hence a higher price. In this case, only the price effect matters. Firm i then finds it optimal to bid at c in order to maximise its chances of selling at capacity at the rival’s price. In this case, only the quantity effect matters.
Equilibrium Price Offer.
Notes: This figure depicts the equilibrium price offer as a function of |$k_{i}$| when |$k_{i}\sim U[0.5,0.9]$|, with |$\theta =1$|, |$c=0$| and |$P=0.5$|. One can see that it starts at P for |$k_i= \underline{k}$|, and that it decreases in |$k_{i}$| until it takes the value |$c=0$| at |$k_{i}=\overline{k}=0.9$|.
Since an increase in |$k_i$| pushes the quantity and the price effects in opposite directions,22 the optimal bid function is first concave and eventually becomes convex as |$k_i$| approaches |$\overline{k}$|. In the latter case, bidding incentives approach those under Bertrand competition as, for large |$k_i$| realisations, the price effect wanes. With only the quantity effect at play, the rival’s bid must become increasingly flat at marginal cost in order to offset the firm’s strong undercutting incentives.
Finally, given equilibrium bidding, each firm’s expected profits are equal to the minimax when |$k_i=\underline{k}$|, and they are strictly higher otherwise. The reason is simple: a firm can always pretend to be smaller by withholding output and replicating the smaller firm’s bid. The fact that firms prefer to offer all their capacity means that larger firms make higher equilibrium profits than the smallest one, whose profits exactly coincide with the minimax.
This equilibrium characterisation corresponds to the polar case in which marginal costs are known and constant. In Subsection 5.2 we show that these results extend naturally to the case in which the private signal affects both capacities and costs.
Changes in parameter values affect the shape of the bid functions, thereby impacting market outcomes. Equilibrium price offers shift up as demand |$\theta$| increases. This is driven by a weaker quantity effect, i.e., the quantity gain when undercutting the rival is smaller since the residual demand is larger, and a stronger price effect, i.e., the gain from increasing the price conditionally on being the high bidder goes up because the residual demand is larger. Consequently, equilibrium prices increase. Similarly, when the capacity distribution shifts to the right in the first-order stochastic sense, the equilibrium price decreases. This result is due to two opposing effects. On the one hand, the price effect becomes stronger as, for a given |$k_{i}$| realisation, the rival’s capacity is expected to be larger, making firm i more likely to set the market price. This pushes the equilibrium price offer up. On the other hand, firms’ capacities are now larger on average, which shifts the equilibrium price offer to the right. Overall, the latter effect dominates, leading to lower equilibrium market prices. The total effect is illustrated in Figure 3.
Equilibrium Price Offers and Expected Market Price as Installed Capacity Increases.
Notes: The upper panel shows that the equilibrium price offers shift outwards as |$\kappa$| and, consequently, |$\underline{k}$| increase. The lower panel shows that the expected market price smoothly goes down as a function of |$\underline{k}$| that, together with |$\overline{k}$|, shifts out as |$\kappa$| increases. The figures assume that |$\theta =1$|, |$c=0$|, |$P=0.5$| and |$k_i\sim U[\underline{k},\underline{k} +0.2]$| for |$\underline{k}\in [0.5,0.95]$|.
2.1.2. Large installed capacities
We now turn to the case in which a single firm’s capacity might exceed total demand, |$\overline{k}\gt \theta$| (large installed capacities case). In contrast with the case of small installed capacities, withholding is now optimal for firms whose capacity exceeds total demand, |$k_i\gt \theta$|, as shown in the following proposition.
If|$\overline{k}\gt \theta$|, in equilibrium, |$b^{\ast }(k_{i})=c$|and|$q_{i}^{\ast }(k_{i})=\theta$|for all|$k_{i}\gt \theta$|, |$i=1,2$|. For|$k_{i}\le \theta$|, |$q_{i}^{\ast }(k_{i})=k_{i}$|and|$b^{\ast }(k_{i})$|is defined as in Proposition1, with|$k_{i}$|and|$k_{j}$|replaced by|$q_{i}^{\ast }(k_{i})$|and|$q^{\ast }(k_{j})$|, |$i=1,2$|.
For capacity realisations |$k_i \le \theta$|, equilibrium bidding is just as in the case with small installed capacities. However, for |$k_i\gt \theta$|, offering to supply |$k_i$| is weakly dominated by offering to supply |$\theta$|; in any event, the firm will never produce more than |$\theta$| and, conditioning on having the low price, offering |$\theta$| instead of |$k_i$| increases the chances that the rival’s higher price offer will set the market price.23 For these reasons, the equilibrium characterisation is identical to that in the previous proposition, the only difference being that the relevant distribution now has a mass point at |$\theta$|. Note that withholding increases the market price, but it implies no distortion in the quantity sold, given that the withheld capacity would never have been used.
Similarly to the case of small installed capacities, an increase in demand leads to a higher price. This effect is now enhanced as the increase in |$\theta$| makes it less likely that both firms bid at c. Indeed, as |$\theta$| goes up, the expected market price goes up. If the whole capacity distribution |$[ \underline{k},\overline{k}]$| shifts to the right, as also shown in Figure 3, expected prices smoothly converge towards marginal costs.
An interesting insight from our model is that capacity realisations determine whether firms find it optimal to compete either (i) by offering all their capacity at prices above marginal costs or (ii) by withholding capacity in order to stop prices from falling when they bid at marginal cost. As we will see next, this pattern does not arise when each bidder receives its own price offer (discriminatory pricing), rather than the market clearing price. Intuitively, firms would always bid above marginal cost in order to obtain positive profits and would then have no need to withhold. This difference will have implications for the profit ranking of both auctions.
2.2. Discriminatory Auction
We now characterise equilibrium bidding under the discriminatory auction.
In the discriminatory auction, the unique Bayesian Nash equilibrium when capacities are privately known is symmetric.
(ii) If|$\overline{k}\gt \theta$|, in equilibrium,|$q_{i}^{\ast }(k_{i})=k_{i}$|, and|$b_{i}^{\ast }(k_{i})$|is given by (3), with|$k_{i}$|and|$k_{j}$|respectively replaced by|$\min \lbrace \theta ,k_{i}\rbrace$|and|$\min \lbrace \theta ,k_{j}\rbrace$|for|$i,j=1,2$|, |$i\ne j.$|
Unlike the uniform-price auction, a higher price offer under the discriminatory auction always allows the firm to obtain a higher price for its output, even when the rival firm bids higher. Hence, firms now face stronger incentives to increase their price offers. In particular, under the discriminatory auction, the optimal bid is always strictly above marginal cost, even when |$k_{i}=\overline{k}$|. Because the price that firms receive does not depend on the quantity sold, they do not need to withhold output when their capacities exceed total demand. Their profits are simply the same if they withhold capacity or not.
2.3. Comparison across Auction Formats
Since firms submit higher bids under the discriminatory auction, it follows that the market clearing price is higher than under the uniform-price auction (see Figure 4). However, this is not enough to rank payments under the two formats given that, under the discriminatory auction, firms do not receive the market clearing price, but their own bid. Indeed, as we show in our next result, the uniform-price auction yields higher payments to firms. Since costs are the same across formats, this also implies that firms make higher profits.
Firms obtain a higher expected payment under the uniform-price auction relative to the discriminatory auction.
Comparison between the Optimal Price Offers across Auctions.
Notes: The figure depicts the equilibrium price offers under the discriminatory auction (solid) and the uniform-price auction (dashed). One can see that firms always offer, for a given realised capacity, higher prices under the discriminatory auction. Parameter values: |$k_i\sim U[0.5,0.9]$|, |$c=0$|, |$P=0.5$| and |$\theta =1$|.
In order to interpret this result, it is useful to highlight the different bidding incentives triggered by the two formats. Under both formats, having a large capacity is good news as the firm is likely to have the low bid and thus sell at capacity. However, under the uniform-price auction, it also brings bad news, as it means that the rival’s capacity is also likely to be large and, hence, conditionally on having the low bid, the firm will receive a lower price for its capacity, weakening firms’ incentives to bid more aggressively. Note that this is the case despite capacity realisations being independent. The reason is that, conditionally on firm i having the low bid, the distribution of the rival’s capacity is truncated at |$k_i$|. Hence, a higher |$k_i$| also implies a higher expected |$k_j$|. In contrast, this second effect is not present under the discriminatory format, given that firms are paid according to their own bid, regardless of their rivals’ capacity.
At the lowest capacity realisation, |$\underline{k}$|, firms make minimax profits under the two auction formats. However, the previous argument indicates that, for higher capacity realisations, firms’ profits increase faster under the uniform-price auction. The result in Proposition 4 immediately follows.
The previous arguments do not rely on capacity withholding and, indeed, they completely characterise the result in the small capacity case. Interestingly, allowing for withholding strengthens the result when capacities are large. The reason is that, under the discriminatory auction, firms have no incentives to withhold capacity because they receive their own bid. The equilibrium outcome is thus independent of whether this possibility is considered. However, as shown in Proposition 2, in the uniform-price auction firms find it optimal to withhold output to |$q_i=\theta$| when their capacity exceeds that level. This decision is akin to a leftward shift in the distribution of capacities, which leads to higher equilibrium prices.
3. The Impact of Private Information
In this section we aim to understand the effect of private information on bidding behaviour and market outcomes under a uniform-price auction. We perform two types of analyses. First, we compare the equilibrium outcomes when capacities are privately known with those that arise, either when capacities are publicly known or when they are unknown to both firms prior to bidding. Second, we analyse the effects of information precision on equilibrium outcomes.
3.1. Known versus Unknown Capacities
First, suppose that firms observe realised capacities prior to submitting their price offers. The following lemma characterises the level of profits that can be sustained in symmetric pure-strategy equilibria.
Suppose that realised capacities are publicly known prior to bidding. There exist symmetric pure-strategy Nash equilibria, resulting in expected joint profits|$(P-c)\theta$|. These profits are sustained by the following bidding profiles: for|$i,j=1,2$|and|$j\ne i,$|
if|$k_{i}\gt k_{j}$|, |$b_{i}^{\ast }(k_{i})=P$|and|$q_{i}^{\ast }(k_{i})=k_{i}$|, while|$b_{j}^{\ast }(k_{j})\in [ c,\underline{b}_i]$|and|$q_{j}^{\ast }(k_{j})=\min \lbrace \theta ,k_{j}\rbrace$|, with|$\underline{b}_i$|low enough so as to make undercutting by firm i unprofitable;
if |$k_{i}=k_{j}=k$|, firms play mixed strategies, with expected joint profits|$2(P-c)(\theta -k)$| if |$k\lt \theta$|and|$0$|otherwise.
The game with known capacities allows firms to sustain equilibria in which all their output is sold at P. These equilibria are characterised by asymmetric bidding once the capacities are realised, even though the equilibrium is ex ante symmetric. Indeed, the small capacity firm bids low enough so as to make undercutting by the large firm unprofitable.24 This firm maximises profits over the residual demand by bidding at the highest possible price, P. Since both firms are equally likely to be the small or the large capacity firm, they share profits symmetrically. Observing realised capacities allows firms to overcome the coordination problem as to which firm bids low or high and this, in turn, allows them to attain maximum profits.25
Consider now the case in which firms do not observe realised capacities prior to bidding. They first choose prices before capacities are realised, and then choose their quantity offers once they have observed them.26 The following lemma shows that the unique symmetric equilibrium involves mixed-strategy pricing.
If realised capacities|$(k_{i},k_{j})$|are known after firms have made their price offers, the unique symmetric Bayesian Nash equilibrium involves mixed strategies, with firms randomising their prices in the interval|$(c,P)$|. Expected equilibrium joint profits are|$2(P-c)[ \theta -E(k|k\le \theta )]G(\theta )$|.
Since price offers cannot be conditioned on capacities, in a symmetric equilibrium both firms would either charge equal prices or use the same mixed strategy to randomise their prices. The former is ruled out by standard Bertrand arguments, implying that the only symmetric equilibrium involves mixed strategies. Since at P the rival firm is bidding below with probability 1, and since all the prices in the equilibrium support yield equal expected profits, it follows that at the unique symmetric equilibrium each firm makes expected profits equal to |$(P-c)(\theta -E[k|k\le \theta ])G(\theta )$|. Note that the high bidder only makes positive profits when the rival’s capacity turns out to be below |$\theta$|, i.e., with probability |$G(\theta )$|.
We are now ready to rank expected prices arising at the symmetric equilibria across all three information treatments.
If firms play symmetric Bayesian Nash equilibria, expected prices are the highest with publicly known capacities, and the lowest with unknown capacities. Expected equilibrium prices with privately known capacities lay in between.
When capacities are private information, the fact that bidding incentives differ across firms allows them to avoid fierce competition, but not as much as if both capacities were known: large (small) firms find it in their own interest to bid low (high), but not as low (high) as if they knew with certainty that the rival firm was bidding higher (lower). When capacities are unknown to both firms, they face fully symmetric incentives and they end up competing fiercely. As a result, private information leads to higher prices than in the case with unknown capacities, but lower than when capacities are publicly known. This suggests that firms would be better off if they could exchange their private information regarding their available capacities.27
3.2. Information Precision
Given the equilibrium characterisation in Proposition 1, one may be tempted to conclude that an improvement in information precision leads to more competitive bidding, a conclusion that would be at odds with our previous results. Indeed, based on Proposition 1, as the range |$[ \underline{k},\overline{k}]$| shrinks, equilibrium profits converge to those with symmetric and known capacities, for which the symmetric equilibrium involves mixed strategies, giving rise to very low profits.
However, this approach is misleading, as making the range |$[ \underline{k}, \overline{k}]$| narrower, not only improves information precision, but also increases the likelihood of capacities being ex post symmetric. Since increased symmetry leads to more competitive outcomes, this latter effect confounds the true impact of information precision on bidding behaviour.
A similar issue arises under private information on costs when demand is price elastic. In this case, Hansen (1988) showed that a sealed-bid auction yields lower prices than an open auction. Lagerlöf (2021) observed that the sealed-bid auction and the open auction are analogous to Bertrand competition with and without private information, respectively. Building on Hansen (1988), Lagerlöf (2021) then concluded that less precise information gives rise to a less competitive outcome.
We cannot use the approach in Hansen (1988), as he derived his result from applying the revenue equivalence theorem between sealed-bid and open auctions for a fixed quantity. In contrast, in our multi-unit setting with private information on capacities, the discriminatory and uniform-price auctions, which are the analogues of the first-price (or sealed-bid) and second-price (or open) auctions, are not revenue equivalent, as shown in Lemma 4. Furthermore, the equilibrium under the uniform-price auction differs with known or privately known capacities, as shown in Proposition 5. Hence, to disentangle the effects of information precision from those of increased symmetry, we need to extend our model to allow for ex ante asymmetric capacities.
Suppose for simplicity that firm i’s capacity is uniformly distributed in |$[\underline{k}_{i},\overline{k}_{i}]$| and that firms’ aggregate capacity is always enough to cover total demand, |$\underline{k}_{1}+\underline{k}_{2}\ge \theta$|. Our next proposition characterises equilibrium bidding for the case in which the two firms’ installed capacities are small, |$\overline{k}_2\le \overline{k}_{1}\le \theta$|.
Assume that|$k_i$|is uniformly distributed in|$[\underline{k}_i,\overline{k}_i]$|with|$\overline{k}_1-\underline{k}_1=\overline{k}_2-\underline{k}_2$|. If|$\overline{k}_1\le \theta$|, in equilibrium each firm offers all its capacity, |$q^{*}_i(k_i)=k_i$|for|$i=1,2$|. Furthermore,
- if|$\overline{k}_{2}\ge \underline{k}_{1}$|, there exists an equilibrium in which price offers are characterised bywhere$$\begin{eqnarray} b_{i}^{\ast }(k_{i})=\begin{cases} P & {\rm if}\, \underline{k}_{2}\le k_{i}\le \underline{k}_{1}, \\ b^{\ast }(k_{i}) & {\rm if} \, \underline{k}_{1}\lt k_{i}\lt \overline{k}_{2}, \\ c & {\rm if}\, \overline{k}_{2}\le k_{i}\le \overline{k}_{1}, \end{cases} \end{eqnarray}$$and(4)$$\begin{eqnarray} b^{\ast }(k_{i})=c+(P-c)\exp ( -\omega (k_{i})) \end{eqnarray}$$$$\begin{eqnarray} \omega (k_{i}) \equiv \int _{\underline{{k}}_{1}}^{k_{i}}\frac{2k-\theta }{ \int _{k}^{\overline{k}_{2}}(\theta -k_{j})dk_{j}}dk; \end{eqnarray}$$
if|$\overline{k}_2\lt \underline{k}_1$|the only pure-strategy Bayesian Nash equilibrium is asymmetric, and it coincides with that characterised in Lemma2(i).
Interestingly, this proposition shows that ex ante capacity asymmetries move equilibrium bidding behaviour from the symmetric equilibria provided in Proposition 1 to the asymmetric equilibria provided in Lemma 2. When the capacity intervals do not overlap, as in part (ii), one firm sets the market price at P and the other one chooses a sufficiently low bid that avoids undercutting. In equilibrium, firms cannot offer the same bid, as it occurs in Proposition 1, because that rests on firms being uncertain about the identity of the large firm and, therefore, about the identity of the low bidder.
In contrast, when the capacity intervals overlap, this uncertainty re-emerges for capacities in the range |$[\underline{k}_1,\overline{k}_2]$|. Over this interval, the equilibrium price offers resemble those in Proposition 1, with firms pricing at P for |$k_i=\underline{k} _1$| and at c for |$k_i=\overline{k}_2$|. For smaller capacity realisations, firm 2 bids at P. For higher capacity realisations, firm 1 bids at c. As a result, both price offers are continuous in the realised capacities. Figure 5 illustrates these bids.28
Equilibrium Price Offers When Firms are Ex Ante Asymmetric.
Notes: This figure depicts the equilibrium price offers as a function of |$k_{i}$| when |$k_{1}\sim U[0.6,0.9]$| and |$k_{2}\sim U[0.5,0.8]$|, |$\theta =1$|, |$c=0$| and |$P=0.5$|. One can see that the equilibrium is symmetric only in the area of capacity overlap, [0.6, 0.8]. For larger capacities [0.8, 0.9], the large firm bids at c (upper panel), whereas for smaller capacities [0.5, 0.6], the small firm bids at P (lower panel).
These equilibria survive in the large installed capacities case when capacity withholding becomes optimal, as stated next.
If|$\overline{k}_1\gt \theta$|, in equilibrium each firm offers|$q_i^{*}(k_i)=\min \lbrace \theta ,k_i\rbrace$|and prices according to Proposition6, where the relevant threshold in part (ii) of the proposition, |$\overline{k}_2$|, is replaced by|$\min \lbrace \theta ,\overline{k}_2\rbrace$|.
For the same reasons explained in the ex ante symmetric capacities case, firms always find it optimal to withhold capacity whenever their realised capacity exceeds |$\theta$|. As a result, firms behave in equilibrium as if their capacities were capped with a mass point at |$\theta$|.
This equilibrium characterisation allows us to conclude that, keeping aggregate capacity as given, an increase in firms’ ex ante asymmetries results in higher expected prices. As firm 2 becomes smaller in expected terms, it bids at P with a higher probability, raising the expected equilibrium price. In the limit, when asymmetries are such that there is no capacity overlap, |$\underline{k}_{1}\gt \overline{k}_{2}$|, the market price is P with probability 1.29
What do these results tell us about the effects of information precision? To answer this question, suppose first that firms’ capacities are so asymmetric that their intervals never overlap, |$\underline{k}_{1}\gt \overline{k}_{2}$|. By Proposition 6, the equilibrium price in this case is P with probability 1. Introducing a small amount of uncertainty around asymmetric capacities would have no impact on bidding behaviour or market outcomes as long as the intervals do not overlap. Otherwise, adding more uncertainty would eventually imply that |$\overline{k}_{2}\gt \underline{k}_{1}$|, giving rise to equilibria with bids below P. As in Proposition 5, the expected market price would start falling below P, the more so the more noisy the forecasts about the rival’s capacity become.
We can thus conclude that the less precise the signal about the rival’s capacity, the weaker the market power, in line with our previous conclusions regarding the impact of private information.
4. Auctions with Privately Known Costs
We now turn to the case where marginal costs are private information, but capacities are known. This set-up allows us to highlight how the different sources of private information affect equilibrium outcomes.
In particular, suppose that both firms have the same capacity, denoted by k, while their costs are the realisation of two independent random variables. Firm |$i=1,2$| has a cost |$c_i$| drawn from a distribution |$F(c_{i})$| in the interval |$c_{i}\in [ \underline{c},\overline{c}]$|, with a strictly positive density |$f(c_{i})$| in the whole support and |$\underline{c}\ge 0$|. Firm i observes its own idiosyncratic cost, but not that of its rival, i.e., costs are private information. We assume that |$2k \ge \theta$| to ensure that firms always have enough combined capacity to cover total demand and competition is meaningful. As in the benchmark case, demand |$\theta$| is assumed to be price inelastic and bids cannot be raised above a price cap P. This set-up is equivalent to that of Holmberg and Wolak (2018), with minor variations.30
It is immediate, using arguments similar to those in the baseline model, that firms always find it optimal to withhold capacity if |$k\gt \theta$|. Hence, in equilibrium the quantity offered by firms is always |$q=\min \lbrace \theta ,k\rbrace$|. For this reason, in what follows we assume without loss of generality that |$k \le \theta$|.
Our analysis follows the same structure as in previous sections. We first characterise equilibrium bidding in uniform-price and discriminatory auctions, and then compare the equilibria across auction formats. We end by analysing the effects of private information on equilibrium bidding and market outcomes.
4.1. Uniform-Price Auction
Our first result is analogous to Lemma 1 for the case in which costs are privately known.
When costs are private information,
all Bayesian Nash equilibria must be in pure strategies;
theoptimal price offer of firm i, |$b^{*}_{i}(c_{i})$|, must be (strictly) increasing in|$c_{i}$|.
We can rule out non-degenerate mixed-strategy equilibria for reasons similar as when capacities are privately known. Quite intuitively, the optimal price offer must be increasing in the firm’s cost. A firm with a lower cost bids more aggressively since its profit margin is higher and, therefore, it benefits more from an increase in the quantity sold.
The next result characterises the bid function in the symmetric equilibrium of the game.
It is easy to verify that |$b(\overline{c})=P$| and |$b(\underline{c})=\underline{c}$|. When the firm has the highest possible cost, it sells the residual demand and sets the equilibrium price with probability 1. As a result, the firm finds it optimal to choose the highest possible price. At the other extreme, when the firm has the lowest possible cost, it always sells at capacity and it never sets the equilibrium price. As a result, it is a dominant strategy for the firm to offer the lowest possible price. Note that the quantity effect and the price effect show up in the numerator and denominator of the exponent term, respectively. However, unlike in the model with privately known capacities, these effects are invariant to the firm’s private information.
4.2. Discriminatory Auction
Note that |$H(c_{i})$| represents firm i’s expected output when its cost is |$c_i$|. It thus captures the price effect. Comparison of the equilibrium bids under the uniform-price and the discriminatory auctions shows that the former are always lower, as depicted in Figure 6. Intuitively, as we already discussed in the case with privately known capacities, a firm in a uniform-price auction offers lower prices knowing that, conditionally on being the low bidder, the price will be set by the rival. However, unlike that case, the two effects now exactly compensate each other, giving rise to revenue equivalence between the two auction formats.
Comparison between the Optimal Price Offers across Auctions When Costs are Privately Known.
Notes: The figure depicts the equilibrium price offers under the uniform-price auction (solid) and the discriminatory auction (dashed) when costs are privately known. One can see that firms always offer, for a given realised cost, higher prices under the discriminatory auction. Parameter values: |$c_i\sim U[0.2,0.4]$|, |$k=0.7$|, |$P=0.5$| and |$\theta =1$|.
When costs are independent, expected payments to firms are the same under the uniform-price and discriminatory auction formats.
The reasoning goes as follows. At a symmetric equilibrium, small changes in costs affect prices but, due to the envelope theorem, this does not directly affect profits. Furthermore, contingent on having either the low or the high cost, the quantity produced by each firm is independent of its private information. Thus, since the probability that the two firms have the same cost is zero, the bid ranking and quantities allocated to the two firms are not affected by small cost shocks. Hence, revenue stays unchanged. The only effect of private information on profits is through changes in the cost of production, but this effect is the same across auction formats.
4.3. Known versus Unknown Costs
In this section we briefly show that a counterpart of the results provided in Subsection 3.1 regarding the effects of information on bidding behaviour also go through in this case.
We start by characterising equilibrium profits when cost realisations are publicly known.
Suppose that realised costs are publicly known prior to bidding. There exist symmetric pure-strategy Nash equilibria, resulting in expected joint profits|$[P-E(c)]\theta$|. These profits are sustained by the following bid profiles: for|$i,j=1,2$|and|$j\ne i,$|
if|$c_i\gt c_j$|, |$b_{i}^{\ast }(c_i)=P$|and|$b_{j}^{\ast }(c_j)\in [ c,\underline{b}_i]$|, with|$\underline{b}_i$|low enough so as to make undercutting by firm iunprofitable;
if|$c_{i}=c_{j}=c$|, firms play mixed strategies, with expected joint profits|$2(P-c)(\theta -k)$|.
The equilibrium when costs are unknown to both firms before they make their offers is characterised as follows.
If realised costs|$(c_{i},c_{j})$|are known only after firms have made their offers, the unique symmetric Bayesian Nash equilibrium involves mixed strategies, with firms randomising their prices in the interval|$(\underline{p},P)$|, where|$\underline{p}\gt \underline{c}$|. Expected equilibrium joint profits are|$2[P-E(c)](\theta -k)$|.
Combining the previous results, we can rank expected equilibrium prices across information treatments.
If firms play symmetric Bayesian Nash equilibria, expected prices are the highest with publicly known costs and the lowest with unknown costs. Expected equilibrium prices with privately known costs lay in between.
Several comments are in order. As in the case with heterogeneous capacities, more information allows firms to coordinate their behaviour, giving rise to higher prices. However, it is important to note that both when costs are private information and when they are publicly known, the equilibrium bid of the firm with the highest cost is always higher than that of the rival. As a result, the most efficient firm sells at capacity, leading to productive efficiency. When costs are unknown, however, since firms can condition neither on their own nor on the rival’s cost, the identity of the firm that produces at capacity is independent of the cost realisations. This inefficiency reduces total welfare.
5. The Source of Private Information
In this section, we show that the source of private information matters. To do so, we first compare equilibrium outcomes across the two polar cases characterised above—either with private information on capacities or costs—and we then merge both sources of private information to shed light on the robustness of our results once we move away from the polar cases.
5.1. Private Information on Capacities or Costs
Let us start by highlighting the insights that are common to both polar cases; namely, the impact of private information and information precision on the intensity of competition (Propositions 5 and 10). As we have shown, in a uniform-price auction, firms can attain more collusive outcomes when they can condition their bids on the realisation of a random variable (be it on capacities or costs) than when they cannot. Hence, moving from the case in which capacities or costs are privately known to one in which they are publicly known weakens competition in both cases. A similar impact arises when moving from a setting in which both capacities and costs are unknown to one in which they are privately known. Furthermore, as we show in Subsection 2.2 for the case of privately known capacities, the more precise the signal about the rival’s capacity or cost, the weaker the competition.
Private information on capacities also affects bidding incentives through the failure rate, which is increasing in |$k_i$| when |$G(k_i)$| is log concave. However, (7) further depends on |$k_i$| as it affects the quantities produced by firm i, conditionally on being the low or high bidder. This additional effect is not present when private information is on costs, given that the low and high bidders produce the same regardless of their private information.
To see this in more detail, note first that |$k_i$| impacts the quantity effect in (7) through the output loss from being undercut, i.e., the |$(2k_i-\theta )$| term. This means that, abstracting from the density term that is also present in (8), the quantity effect in (7) is stronger the higher |$k_i$|, while in (8) it equals |$(2k-\theta )$| independently of |$c_i$|. Turning to the price effect, an increase in |$k_i$| reduces the firm’s expected residual demand contingent on the rival bidding below, i.e., |$(\theta -E(k_j|k_j\le k_i))$| is decreasing in |$k_i$|. In contrast, the residual demand faced by the high bidder in the model with privately known costs remains constant at |$(\theta -k)$| regardless of the firm’s own cost.
Such differences in bidding incentives explain why the shape of the optimal bid functions differs across models. When private information is on costs, the optimal bid function in Figure 6 is concave for all cost realisations. In contrast, when private information is on capacities, the optimal bid function in Figure 2 turns from being concave for low capacity realisations to being convex for high capacity realisations.
The above differences also underline an important dimension of the comparison between models, namely, the revenue ranking between the uniform-price and the discriminatory auction formats. The standard revenue equivalence result applies when costs are private information as long as they are independent across firms (Proposition 9). In contrast, when capacities are private information, the uniform-price auction yields higher firm payoffs as compared to the discriminatory auction, despite capacity draws being independent (Proposition 4). The reason is that the rival’s capacity is payoff relevant for each firm beyond the effect on its bids. In the natural case when costs are positively affiliated, the opposite ranking holds, with the discriminatory auction resulting in higher firm payoffs (Holmberg and Wolak, 2018). This is in line with the result of Milgrom and Weber (1982) for the single-unit case, under which the first-price auction delivers a higher price than the second-price auction when costs are positively affiliated.
5.2. Private Information on Capacities and Costs
To further understand the fundamental differences between the effects of the two sources of private information and the robustness of the model predictions, it is illustrative to move away from the polar cases discussed above. For this purpose, we now allow both costs and capacities to be privately known, as they both depend on the realisation of a private signal z distributed according to a function |$M(z)$| with density |$m(z)$|, which is strictly positive if and only if |$z\in [\underline{z},\overline{z}]$|. We denote the cost and capacity of a firm with type z as |$c(z)$| and |$k(z)$|, respectively, with |$c^{\prime }(z)\le 0$| and |$k^{\prime }(z)\ge 0$|. This model embeds the polar cases with unknown capacities when |$c^{\prime }(z)=0$| for all z, or with unknown costs when |$k^{\prime }(z)=0$| for all z. To simplify the exposition, we focus on the case of small installed capacities, so that |$k(\underline{z})\gt {\theta }/{2}$| and |$k(\overline{z})\lt \theta$|.
Our first result characterises the equilibrium bid functions in the uniform-price auction.
The equilibrium takes a similar form as in the baseline model with unknown capacities and constant marginal costs (Proposition 1). Indeed, for |$c( z_{i}) =c$| for all |$z_{i}$|, then |$\Gamma ( z_{i}) =c$| and |$b_{i}^{\ast }(z_{i})$| becomes (1). However, for strictly decreasing costs, the markup term falls faster in |$k_{i}$| than in the baseline model. Intuitively, as |$k_{i}$| goes up, the firm has an additional incentive to bid low in order to sell at capacity as it can now do so at a lower cost. Note that the resulting equilibrium is fully efficient as the low bidder is the high capacity and low cost firm. Similarly, it can be shown that, when |$k(z_i)=k$| for all |$z_i$|, the expression becomes (6), i.e., the equilibrium bid function in the case of unknown costs (Proposition 7).
In line with our previous discussion, the equilibrium characterisation in Proposition 11 also uncovers the different ways in which private information on capacities or costs affects the bid function. The trade-off between the quantity and the price effect is captured by the ratio |$a(z)$|, which depends only on the realised capacities. In contrast, the effect of cost heterogeneity is captured by the expression |$\Gamma (z)$| in the markup term, which depends on the sensitivity of the cost function to changes in z. Of course, the markup term also depends on realised capacities through |$a(z)$|, given that the effect of cost changes depends on the quantity produced.
We can now move to characterising the equilibrium bid function under the discriminatory auction.
Again, the equilibrium takes the same form as under the uniform-price auction, with a key difference: the denominator in expression |$a(z)$| now reflects a stronger price effect. As already noted in previous sections, this difference implies that firms submit lower bids under the uniform-price auction. In the polar case with unknown costs, this effect is exactly compensated by the fact that the uniform-price auction pays all production at the highest bid, leading to revenue equivalence (Proposition 9). Instead, in the polar case with unknown capacities, the latter effect dominates, allowing the discriminatory format to reduce firms’ payments (Proposition 4). Our next result shows that this conclusion is robust to adding private information on costs. In contrast, the revenue equivalence result under the polar case with unknown costs breaks down as soon as private information on capacities is introduced, no matter how small.
Firms obtain a (weakly) higher expected payment under the uniform-price auction relative to the discriminatory auction if and only if|$k^{\prime }(z) \ge 0$|for some values of z. The comparison is strict if and only if|$k^{\prime }(z)=0$|for all values of z.
Holmberg and Wolak (2018) found that expected prices were lower in the uniform-price auction, as compared to the discriminatory format when privately known costs are positively affiliated and capacity is stochastic, but independent of the cost realisations. The previous proposition suggests that their result might not be robust to situations in which the positive cost shocks also increase firms’ available capacities.
6. Extensions
Our stylised model can be used to shed light on the performance of future renewable-dominated electricity markets. However, it omits several ingredients that are likely to be relevant in this context: the possibility that demand becomes more price elastic as dynamic pricing and automation devices get more broadly deployed in the future and the likely expansion of storage facilities. We consider these two extensions in turn.
6.1. Price-Elastic Demand
Consider first the impact of allowing for price-elastic demand. As it is standard in oligopoly models, this will partially mitigate market power. However, beyond reducing expected prices, demand elasticity will also affect price volatility through its effect on the shape of the optimal biding function. In particular, the optimal bid function will tilt downwards, starting at the profit-maximising price of the residual monopolist (taking into account the expected capacity of the rival)31 and ending at marginal cost. Our next proposition characterises optimal bidding behaviour in the uniform-price auction.
Suppose that market demand|$D( p)$|is downward sloping, continuously differentiable, log concave and such that|$D( c) \lt 2\underline{k}$|. Firms compete in a uniform-price auction.
- If |$\overline{k}\lt D(c)$|, at the unique symmetric Bayesian Nash equilibrium when capacities are privately known, firm i offers all its capacity at a price that is strictly decreasing in |$k_{i}$| and which is decreasing in the price elasticity of demand. The optimal bid function starts at |$b_{i}( \underline{k})$|, implicitly defined bywhere|$\epsilon ( b_{i}(\underline{k}))$|is the price elasticity of the residual demand|$D(b_{i})-E[ k]$|at a price|$b_{i}(\underline{k}),$|and it ends at|$b_{i}(\overline{k})=c.$|For each|$k_i$|, the optimal bid function is below expression (2).$$\begin{eqnarray} \frac{b_{i}(\underline{k})-c}{b_{i}(\underline{k})}=\frac{1}{\epsilon ( b_{i}(\underline{k})) }, \end{eqnarray}$$
If|$\overline{k} \ge D(c)$|, the bid function is as defined in part (i) for all|$k_i\lt D(c)$|. For|$k_i \ge D(c)$|, firm i bids at c and withholds capacity to|$q_i=D(c)$|.
The previous result shows that the main features of the model extend to environments with a downward-sloping demand.32 Firms find it optimal to withhold capacity when they can individually cover the whole market at the competitive price. Instead, for lower capacity values, they offer all their capacity at prices that decrease in their capacity all the way down to marginal cost, c. Interestingly, the highest price offer they ever submit, |$b(\bar{k})$|, is lower for higher |$E(k)$|, as this makes the residual demand more elastic. Therefore, when the distribution of capacities moves to the right—in the first-order stochastic sense—the bid function starts at a lower price, in contrast to the inelastic demand case in which the highest bid stays constant at P (Figure 3).
As the bid function becomes flatter, price dispersion diminishes as compared to the inelastic demand case. However, demand elasticity also enlarges price differences across periods (with more or less abundant renewable energy available and higher or lower demand), as these shocks do not only shift the bid functions outwards and inwards (as in Figure 3), but also change their slopes. Whether demand elasticity results in higher or lower price volatility will depend on the interplay between these two effects.
6.2. Energy Storage
The deployment of storage facilities will allow for supply management. Firms owning storage capacity will engage in price arbitrage by moving production from periods when renewables’ capacity is high (and/or demand is low) to periods when it is low (and/or demand is high). The extent of this shift will depend on the storage capacity and the market power of the firms that operate it. Our model can capture the effects of storage through a dampening in the variation of the |$\theta$| across time, which could be interpreted as demand net of storage.
Following the comparative statics we derived from Propositions 1 and 2, storage will reduce price differences across periods, both through the direct effects on net demand as well as the indirect effects on the markups. Whether average prices go up or down crucially depends on whether the price-depressing effect in the high priced periods is stronger than the price-increasing effect in the low priced ones, as well as on the degree of market power that storage firms can exert. Andrés-Cerezo and Fabra (2022) analysed this issue, albeit in a different set-up. They concluded that storage depresses average prices since its effects are stronger when markups are higher. This result would suggest that the combination of storage and uncertain renewable power sources would also depress average prices, as prices and markups are higher when renewable power availability is low relative to demand.
7. Concluding Remarks
In this paper we have analysed equilibrium bidding in multi-unit auctions when firms’ production capacities are private information. We have allowed changes in capacity availability to shape the bid functions, both through changes in the prices and the quantities offered by firms.
From a broad economic perspective, we have shown that the nature of private information and the strategies available to firms have a key impact on equilibrium behaviour. We have shown that, due to competition, firms find it optimal to offer more output at prices rapidly approaching marginal cost when they receive a positive capacity shock. In contrast, when costs are private information, a low cost realisation implies a slower convergence towards marginal cost bidding. Arguably more importantly, we have also shown that competition with privately known capacities does not give rise to revenue equivalence between the uniform-price and the discriminatory auctions, in contrast to models with privately known—and independently distributed—costs (Holmberg and Wolak, 2018). In particular, the uniform-price auction results in higher prices as compared to the discriminatory auction. This result extends naturally when each firm’s private signal determines both its capacity as well as its cost, as long as low costs are associated with large capacities.
Although our model applies to a variety of set-ups, it is particularly well suited to the future performance of electricity markets. We have provided suggestive evidence on the existence of private information regarding renewable plants’ available capacity, whose marginal costs are broadly known to be close to zero. Understanding competition among renewables is of first-order importance to guide policy making in this area, notwithstanding the importance of other issues such as the incentives for demand response and the deployment of storage capacity. Our model predicts that electricity prices will go down as more renewables get deployed, although some market power will remain. Promoting demand elasticity as well as investments in energy storage—ingredients that can well be incorporated into our model—will mitigate market power and improve market performance along the way.
Appendix. Proofs
Note that the previous result implies that each bid can be used by at most one capacity realisation. That is, the bid supports used for different capacity realisations do not overlap. Suppose now that a firm with capacity |$k_i$| randomises between two different bids |$b_i$| and |$\hat{b}_i$| with |$b_i\lt \hat{b}_i$|. By Bertrand arguments, it has to be that case that all bids in between are also in the randomisation support. However, since each capacity arises with probability 0, the firm will always prefer to choose the highest point in the support, |$\hat{b}_i$|, as the revenues increase, but the probability of being outbid is essentially unchanged.
Part (iii) follows directly from the first part of the previous argument. Without randomisation, a firm with capacity |$k_i$| chooses |$b_i=b_i^{min}$| and, for any |$k^{\prime }_i\gt k_i$|, it has to be the case that |$b^{\prime }_i\le b_i$|.
It follows that deviations to P are not profitable, since equilibrium profits are always strictly greater than the minimax, except for |$k_{i}= \underline{k}$| when equilibrium profits are exactly equal to the minimax.
Because there is a unique solution to the first-order condition, this implies that the profit function is quasiconcave, which guarantees the existence of the equilibrium. In particular, this rules out deviations where firms choose any lower bid, including c.
For the case with |$\overline{k}\ge \theta ,$| similar arguments as those in the proof of Proposition 2 show that, for |$k_{i}\ge \theta ,$| offering a quantity |$q_{i}\gt \theta$| is equivalent to offering |$q_{i}=\theta$|. Hence, the problem is the same as that solved above, with |$G(k_{i})$| and |$G(k_{j})$| now adjusted to |$G(q_{i}^{\ast }(k_{i}))$| and |$G(q_{j}^{\ast }(k_{j}))$|, |$i,j=1,2,i\ne j$|.
We show that expected profits under the uniform-price auction are weakly higher than under the discriminatory auction for all values of |$k_i$|, with a strict inequality for some capacities. Since the total cost and quantity are always the same, this implies that firms’ payments are higher under that format. Denote the bid and profits under the uniform-price and discriminatory auctions with the subscripts u and d, respectively. Define |$\Pi _s(k_i)=\pi _i(b^{*},b^{*}|k_i)$| for |$s=u,d$|.
Consider now the possibility of withholding. Firms’ profits under the discriminatory auction remain unchanged. In the uniform-price format, however, as withholding is equivalent to a leftward shift of |$G(k)$| in the first-order stochastic sense, this implies even higher profits, reinforcing the previous result.
The proof follows similar steps as Fabra et al. (2006) for the case in which demand and capacities are known.
Consider first the second stage of the game when realised capacities are known to be |$( k_{i},k_{j}) .$| First, suppose that |$k_{i}\gt k_{j}$| with |$k_{j}\lt \theta$|. Following Lemma 1, in all candidate equilibria we must have |$q_{j}( k_{i},k_{j}) =k_{j}\lt q_{i}( k_{i},k_{j}) =k_{i}.$| There cannot exist a pure-strategy equilibrium with |$b_{i}( k_{i},k_{j}) =b_{j}( k_{i},k_{j})$|, given that either firm would be better off slightly undercutting the other in order to increase its production with no effect on the price. Consider equilibria with |$b_{i}( k_{i},k_{j}) \gt b_{j}( k_{i},k_{j}) .$| Since, conditionally on being the higher bidder, firm i’s profits are strictly increasing in its bid, it follows that |$b_{i}( k_{i},k_{j}) =P.$| In order to discourage firm i from undercutting firm j’s bid, it must be the case that |$( P-c) ( \theta -k_{j}) \ge ( b_{j}( k_{i},k_{j}) -c) k_{i}.$| Solving for |$b_{j},$| it follows that |$b_{j}( k_{i},k_{j}) \le \underline{b}_{i}\equiv c+( P-c) {(\theta -k_{j})}/{k_{i}}.$| Since the low bid is pay-off irrelevant, and firm j is selling all its capacity at |$P,$| it does not have incentives to deviate either. In equilibrium, firm i makes profits |$( P-c) ( \theta -k_{j})$| and firm j makes profits |$( P-c) k_{j}.$|
Second, if |$\theta \le k_{j}\lt k_{i},$| one can apply the same argument as above by setting |$q_{j}( k_{i},k_{j}) =\min \lbrace \theta ,k_{j}\rbrace $|.
Last, if |$k_{i}=k_{j}=k,$| Bertrand arguments rule out any pure-strategy symmetric equilibrium. The only equilibrium is therefore in mixed strategies. Since P must be part of the equilibrium support, it follows that expected equilibrium profits for firm |$i=1,2$| are |$( P-c) ( \theta -k_{j}) .$|
To conclude the proof, consider the first stage of the game. Since both firms are equally likely to be the small or the large firm, and since the event |$k_{i}=k_{j}=k$| occurs with almost zero probability, it follows that in equilibrium firms make symmetric expected profits |$( P-c) \theta /2.$|
The proof follows similar steps as in Fabra et al. (2006) for the case in which demand is unknown and capacities are symmetric and known.
Consider the second stage of the game in which capacities are known to be |$( k_{i},k_{j}) $| and firms have to choose |$q_{i}$|. If |$k_{i}\lt \theta$|, firm i’s profits are weakly increasing in |$k_{i}$|. Therefore, firm i does not find it optimal to withhold. If, instead, |$k_{i}\ge \theta$|, for the same reason as in the benchmark model, it finds it optimal to offer |$q_{i}=\theta$|. In this case, in the first stage when firms choose prices, they behave as if their capacity had a mass point at |$\theta$|. See Proposition 2.
Consider now the first stage of the game in which firms have to choose their price offer without knowing their realised capacities. Since firms are symmetric in expected terms, a symmetric equilibrium would call them to offer the same price. However, this is ruled out by Bertrand arguments. The only equilibrium is therefore in mixed strategies. Since P must be part of the equilibrium support, it follows that expected equilibrium profits for firm |$i=1,2$| are |$( P-c) [ \theta -E(k|k\le \theta )] G(\theta ),$| given that a firm bidding at P only faces a positive residual demand in the event that its rival’s capacity is below |$\theta ,$| which occurs with probability |$G(\theta ).$|
It follows from combining the results of Propositions 1 and 2 and Lemmas 2 and 3.
We show that there is no profitable deviation from the candidate equilibrium stated in the text of the proposition.
Under the assumption that |$g_i(k_i)$| is uniformly distributed in an interval of the same length, we have |$g_i(k_i)=g_j(k_j)$| for |$k_i\in [\underline{k}_i, \overline{k}_i]$| and |$i=1,2$|. As a result, the profit functions of the two firms are identical because the bid function in this range is the same. Hence, the first condition is also the same and it coincides with (A4) in the proof of Proposition 1, leading to (4).
It remains to show that (1) |$b_2(k_2)=P$| for |$k_2\lt \underline{k}_1$| and (2) |$b_1(k_1)=c$| when |$k_1\gt \overline{k}_2$|. Regarding (1), by definition of equilibrium we have |$\pi _2(\underline{k}_1,P)\ge \pi _2(\underline{k} _1,b)$| for |$b\lt P$|. Since firm 2 can always satisfy the residual demand, it also follows that, for |$k_2\lt \underline{k_1}$|, the firm makes the same level of profits, |$\pi _2(\underline{k}_1,P)=\pi _2(k_2,P)$|. In turn, since profits are always increasing in capacity, it also follows that, for any |$b\lt P$|, |$\pi _2( \underline{k}_1,b)\gt \pi _2(k_2,b)$|. This shows that (1) is optimal.
With respect to (2), by definition of equilibrium we have |$\pi _{1}( \overline{k}_{2},c)\ge \pi _{1}(\overline{k}_{2},b)$| for any |$b\gt c$|. Furthermore, for all |$k_{1}\ge \overline{k}_{2}$|, profits increase faster with capacity when the firm bids at c than when it bids at any |$b\gt c$|, |${\partial \pi _{1}}(k_{1},c)/{\partial k_{1}}\gt {\partial \pi _{1}}(k_{1},b)/{ \partial k_{1}}$|. This shows that (2) is optimal.
Note that the previous result implies that each bid can be used by at most one cost realisation. That is, bid supports used under different cost realisations do not overlap. Suppose now that a firm with cost |$c_i$| randomises between two different bids |$b_i$| and |$\hat{b}_i$| with |$b_i\lt \hat{b}_i$|. Using again Bertrand arguments, it has to be the case that all bids in between are also in the randomisation support. However, since each capacity arises with probability 0, the firm will always prefer to choose the highest point in the support, |$\hat{b}_i$|, as the revenues increase, but the probability of being outbid is essentially unchanged.
Part (ii) follows directly from the first part of the previous argument. Without randomisation, a firm with cost |$c_i$| chooses |$b_i=b_i^{max}$| and, for any |$c^{\prime }_i\gt c_i$|, it has to be the case that |$b^{\prime }_i\ge b_i$|. |$\square$|
Denote with the subindices d and u the profits under the discriminatory and the uniform-price auctions, respectively. Define |$\Pi _s(c_i)=\pi _i(b^{*},b^{*}|c_i)$| for |$s=u,d$|.
This proof follows the structure used in the proof of Proposition 4. Define |$\Pi _s(z_i)=\pi _i(b^{*},b^{*}|z_i)$| for |$s=u,d$|.
Since |$\Pi _u(\underline{z})=\Pi _d(\underline{z})$|, it follows that if |$k^{\prime }(z_i)\gt 0$| for some |$z_i$| then firms’ expected profits are strictly lower under the discriminatory format. If |$k^{\prime }(z_i)=0$| for all |$z_i$| then |${d\Pi _u(z_i)}/{dz_i}= {d\Pi _d(z_i)}/{dz_i}$|, which together with |$\Pi _u(\underline{z})=\Pi _d(\underline{z})$| implies revenue equivalence across auction formats.
Since the optimal bid is decreasing in capacity, we first note that in a symmetric equilibrium it has to be the case that |$k_{i}\gt D( b_{i}) -b_{j}^{-1}( b_{i}) .$| Towards a contradiction, suppose not. Then, |$k_{i}\lt D( b_{i}) -b_{j}^{-1}( b_{i}) =D( b_{i}) -k_{i}$|. Since firms never choose prices below marginal cost, it follows that |$2 \underline{k}\le 2k_{i}\lt D( b_{i}) \le D( c) ,$| which contradicts our initial assumption that |$D( c) \lt 2\underline{k}.$| Hence, we only need to consider cases where |$k_{i}\gt D( b_{i}) -b_{j}^{-1}( b_{i}) ,$| implying that a firm always has enough capacity to satisfy the residual demand if it turns out to be the high bidder.
If |$q_{i}=D( b_{i})$|, profits do not depend on |$k_{i}$|. Hence, the optimal bid must be independent of |$k_{i}$|. Bertrand arguments rule out that such a bid is greater than c as firms would have incentives to undercut it. It follows that the optimal bid is |$b_i(k_i)=c$| for all |$k_{i}\gt D(c)$|.
Altogether, if |$\overline{k}\lt D(c)$|, there is never withholding, |$q_i=k_i$|, and the optimal price offer is equal to the solution to (A12). The optimal price offer at |$\overline{k}$| must equal marginal cost. Since the second term in (A12) cancels out, the first term is also zero when |$b_{i}(\overline{k})=c$|.
If |$\overline{k}\gt D(c)$|, there is no withholding for capacity realisations up to |$D(c)$|, with |$q_i=k_i$| and the optimal price offer given by the solution to (A12). Instead, for all capacity realisations |$k_i\gt D(c)$|, there is withholding to |$q_i=D(c)$|, and the optimal price offer is equal to marginal cost.
Additional Supporting Information may be found in the online version of this article:
Replication Package
Notes
The data and codes for this paper are available on the Journal repository. They were checked for their ability to reproduce the results presented in the paper. The replication package for this paper is available at the following address: https://doi.org/10.5281/zenodo.7194480.
The paper has benefited from comments by Micael Castanheira, Nils May, Nicolas Schutz, Juan-Pablo Rincón, Ian Schneider and Hannes Weigt, as well as seminar participants at the University of Toulouse, ETH (Zurich), Energy Economics (Madrid), Isaac Newton Institute (Cambridge), Universidad Autónoma de Madrid, PUC (Santiago de Chile), Ecares (Brussels), the University of Mannheim, Universidad Carlos III (Madrid), the Toulouse Conference on Energy Economics, IESE (Barcelona), the 2019 EARIE Conference (Barcelona), the Hamburg Research Workshop on The Economics of the Energy System Transition, the 3rd Workshop on Advances in Industrial Organization (Bergamo), the University of Vienna and the European University Institute. We are grateful to Ángeles Mora and Nekane Martínez from Gamesa for providing the private wind forecast data. Imelda has provided excellent research assistance. All errors are our own.
This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No. 772331). Support of the Regional Government of Madrid through Grants S2015/HUM-3491 and H2019/HUM-5859 is also acknowledged.
Footnotes
In several jurisdictions, the goal to achieve a carbon-free power sector by 2050 will require an almost complete switch towards renewable energy sources. The International Renewable Energy Agency estimates that compliance with the 2017 Paris Climate Agreement will require overall investments in renewables to increase by 76% in 2030, relative to 2014 levels. Europe expects that over two-thirds of its electricity generation will come from renewable resources by 2030, to achieve a carbon-free power sector by 2050 (European Commission, 2012). Likewise, California has recently mandated that 100% of its electricity will come from clean energy sources by 2045.
In this respect, our analysis applies mainly to wind and solar power, which are the most relevant renewable technologies. However, strictly speaking, not all renewable power sources share their characteristics. Other renewable technologies are storable, such as hydroelectricity, or have a production that can be managed, very much like thermal plants (e.g., biomass plants).
Bushnell and Novan (2021) reached a similar conclusion in a counterfactual exercise that used data from the Californian electricity market.
In a context without uncertain renewables, Genc and Reynolds (2019) and Bahn et al. (2021) also assumed Cournot competition to analyse the effects of the ownership structure of renewable plants on market outcomes. The trade-offs that arise when relying on a simple and tractable set-up, like the Cournot model, versus one that more closely mimics the institutional details of electricity markets, like an auction model, have been extensively discussed in the previous literature. See, among others, von der Fehr and Harbord (1993) and Wolfram (1998).
Since then, many other works have followed. Examples following the approach of Wilson (1979) include Back and Zender (1993) in a pure common value setting and Ausubel et al. (2014) in which valuations are bidders’ private information, with possibly affiliated signals. Examples following the approach of Klemperer and Meyer (1989) include the analysis of Green and Newbery (1992) of the British electricity market and the model of Vives (2011) with private cost signals.
For instance, similarly to many other countries, firms in the Spanish electricity market cannot submit more than 25 steps per unit, though in practice they rarely use more than three. See Hortaçsu and Puller (2008) for similar evidence regarding the electricity market in Texas.
This assumption is also common in papers with infinitely many price-quantity pairs, e.g., in Ausubel et al. (2014), where bidders’ capacities are fixed and known.
Even though we allow for a single step, the fact that firms can withhold capacity is equivalent to allowing for more than one step as long as the second one is at the price cap.
Likewise, firms have private information on capacities in a wide range of markets that can be analysed through the lens of auction theory (Klemperer, 2003), e.g., the markets for hotel bookings or ride-hailing services, in which firms are privately informed about the number of empty rooms or available cars.
The implicit assumption is that withholding below |$\underline{\it k}$| would make it clear that the firm has strategically reduced output in order to raise prices, which could trigger regulatory intervention.
Hence, when firms’ quantity offers are equal, |$\alpha ( q,q)=1/2$|. We do not need to specify |$\alpha ( q_{i},q_{j})$| outside of the diagonal as this is inconsequential for equilibrium bidding.
Assuming that the market price is set at the lowest non-accepted bid when the quantity offered by the winning bid(s) equals total demand is made for analytical convenience, with no impact on equilibrium outcomes. It avoids situations where firms want to offer a quantity slightly below total demand, in order to push the market price up to the higher bid offered by the rival.
While demand is known, demand net of renewable energy is uncertain, an issue which is captured in our model.
The empirical evidence shows that this is the case in the Spanish electricity market, the only country so far where real-time pricing has been implemented as the default option for all households (Fabra et al., 2021). However, this might change once automation devices become more broadly deployed. The incentives for demand to adopt price response technologies will be enhanced with price volatility, which will likely increase, partly driven by the channels we highlight in this paper. See Section 6 for more on this.
One caveat of our model, however, is that it assumes that all firms are symmetrically informed. In reality, however, it is reasonable to expect that larger firms have more precise information about their rivals’ capacities. Intuitively, having access to more accurate forecasts could reinforce their market power, but this issue is out of the scope of the current paper.
We have also used more general specifications, such as a LASSO, with almost identical results.
In case of indifference between withholding or not, we assume without loss of generality that the firm chooses not to withhold.
The model also displays asymmetric equilibria, with one firm always setting the market price at P, while the rival chooses a price sufficiently close to c. This equilibrium may lead to coordination issues, and makes the comparison with the discriminatory auction difficult (as the unique equilibrium under that format is symmetric, as we show below).
If P were stochastic (either because it is interpreted as the marginal cost of the conventional producers or because it is the implicit price cap that triggers regulatory intervention), the equilibrium market price would maximise the high bidder’s profits, taking into account the distribution of |$P$|. Since the high bidder sells the expected residual demand, such a price would be independent of the firm’s realised capacity.
This is always the case whenever the distribution function is log concave, which holds for a large family of distribution functions.
If instead of setting the market price at the lowest non-accepted bid, we set it equal to the highest accepted bid, firm i would optimally offer to produce a quantity slightly below total demand, |$\theta$|, giving rise to the same market price and (almost) the same quantity allocation.
This holds even if |$k_{i}\gt \theta$| as in this case firms can escape Bertrand pricing by withholding output and choosing |$q_{i}(k_{i})=\theta$|. This is in contrast to Fabra et al. (2006), who predicted Bertrand competition when |$k_{i}\gt \theta .$| The difference is that they did not allow firms to choose both prices and quantities.
The only exception is when firms' realised capacities are equal. In this case, since observing realised capacities does not allow them to overcome the coordination problem, the unique symmetric equilibrium involves mixed-strategy pricing, with firms making lower expected profits. However, this case arises with a zero probability.
The same results would arise if, instead, firms commit to sell all their capacity at the chosen price once capacities are realised.
In fact, in our model firms would have unilateral incentives to share the realisation of their own capacity with the rival, as this would allow them to better coordinate and sustain higher equilibrium profits. The debate on the incentives for information transmission between firms dates back to classical papers like Vives (1984) and Gal-or (1986).
Note that this comparative statics exercise is based on the characterisation of the equilibrium with symmetric bidding. Trivially, differences in the support would not matter if we focus only on the equilibrium with asymmetric bidding described in Lemma 2(i), which exists for all support combinations.
It is important to note, however, that the characterisation of this equilibrium hinges on the density of each firm being identical in the range of capacity overlap, thanks to the assumption of uniformly and identically distributed idiosyncratic shocks. This guarantees that the two first-order conditions that characterise optimal bidding are identical, allowing us to conclude that the equilibrium price offers are symmetric. While we do not provide a characterisation for generic distribution functions, we conjecture that the nature of the equilibrium would remain similar, but explicit solutions for the optimal bids would be unlikely to come by.
In their paper, they assumed that |$P=\overline{c}$|. Also, they allowed the capacity to be stochastic, but it is unrelated to the cost shocks. These differences have no impact on the results. They also analysed the more general case where costs are affiliated, which provides interesting insights for the comparison with our model, as discussed later in the paper.
We are implicitly assuming that the price cap P is so high that it is never binding. Otherwise, the optimal bid function would still start at P.
Somogy et al. (2021) analysed the discriminatory auction with elastic demand, albeit in a simplified version of our set-up. In their model, firms can only be either capacity constrained or unconstrained, and capacity withholding is not allowed. In line with our result of decreasing bid functions, they also found that smaller (capacity-constrained) firms set higher prices than larger (unconstrained) firms.
