This article was downloaded by: [175.176.173.30] On: 27 December 2015, At: 23:19
Publisher: Institute for Operations Research and the Management Sciences (INFORMS)
INFORMS is located in Maryland, USA
Management Science
Publication details, including instructions for authors and subscription information:
http://pubsonline.informs.org
Demand Uncertainty and Excess Supply in Commodity
Contracting
Dana G. Popescu, Sridhar Seshadri
To cite this article:
Dana G. Popescu, Sridhar Seshadri (2013) Demand Uncertainty and Excess Supply in Commodity Contracting. Management
Science 59(9):2135-2152. http://dx.doi.org/10.1287/mnsc.1120.1679
Full terms and conditions of use: http://pubsonline.informs.org/page/terms-and-conditions
This article may be used only for the purposes of research, teaching, and/or private study. Commercial use
or systematic downloading (by robots or other automatic processes) is prohibited without explicit Publisher
approval, unless otherwise noted. For more information, contact [email protected].
The Publisher does not warrant or guarantee the article’s accuracy, completeness, merchantability, fitness
for a particular purpose, or non-infringement. Descriptions of, or references to, products or publications, or
inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or
support of claims made of that product, publication, or service.
Copyright © 2013, INFORMS
Please scroll down for article—it is on subsequent pages
INFORMS is the largest professional society in the world for professionals in the fields of operations research, management
science, and analytics.
For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org
MANAGEMENT SCIENCE
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Vol. 59, No. 9, September 2013, pp. 2135–2152
ISSN 0025-1909 (print) — ISSN 1526-5501 (online)
http://dx.doi.org/10.1287/mnsc.1120.1679
© 2013 INFORMS
Demand Uncertainty and Excess Supply in
Commodity Contracting
Dana G. Popescu
Department of Technology and Operations Management, INSEAD, Singapore 138676, [email protected]
Sridhar Seshadri
Department of Information, Risk and Operations Management, McCombs School of Business,
University of Texas at Austin, Austin, Texas 78712, [email protected]
W
e examine how different characteristics of product demand and market impact the relative sales volume
in the forward and spot markets for a commodity whose aggregate demand is uncertain. In a setting
where either the forward contracts are binding quantity commitments between buyers and suppliers or the
forward production takes place before the uncertainty in demand is resolved, we find that a combination of
factors that include market concentration, demand risk, and price elasticity of demand will determine whether
a commodity will be sold mainly through forward contracts or in the spot market. Previous findings in the
literature show that when participants are risk neutral, the ratio of forward sales to spot sales is a function of
market concentration alone; also, the lower the concentration, the higher this ratio. These findings hold under
the assumption that demand is either deterministic or, if demand is uncertain, all production takes place after
uncertainty is fully resolved and production plans can be altered instantaneously and costlessly. In our setting,
however, we find that even a low level of demand risk can reverse the nature of supply in a highly competitive
(low concentration) market, by shifting it from predominantly forward-driven to predominantly spot-driven
supply. In markets with high concentration, the price elasticity of demand will determine whether the supply
will be predominantly spot-driven or forward-driven. Our analysis suggests various new hypotheses on the
structure of supply in commodity markets.
Key words: forward and spot market; excess supply; speculators; Cournot competition
History: Received January 26, 2012; accepted September 11, 2012, by Martin Lariviere, operations management.
Published online in Articles in Advance February 15, 2013.
1.
Introduction
the result of market forces and therefore tend to fluctuate considerably (price risk). Also, such commodities
go directly into finished goods and hence the financial impact of supply shortages is great (availability
risk). Moreover, these commodities have high innovation speed and a short product life cycle—rendering
them, in effect, perishable—so stocking them for even
moderate periods of time is risky because of steep
devaluation curves (inventory risk). If the component’s
end product has high demand uncertainty (product
demand risk), then the manufacturer’s procurement
strategy must optimally balance availability, price,
and inventory risks.
Most previous research has analyzed the procurement problem from the manufacturer’s perspective.
In this paper, however, we study commodity procurement channels at the aggregate level and show how
different characteristics of product demand and of the
market affect the relative sales volume in the forward
and spot markets for a commodity whose aggregate
demand is uncertain. The objective of this analysis is
With the ongoing trends of globalization and increasing competition, procurement has become one of
the most important functions in an organization and
is now recognized as a main source of competitive advantage. In early 2000, for example, HewlettPackard (HP) was facing high uncertainty in the
future price and availability of flash memory—a standard component of HP printers—and also in HP’s
own demand for flash memory. Recognizing the
impact of procurement on the company’s bottom line,
HP launched its Procurement Risk Management Division to better deal with supply and demand risks
(Nagali et al. 2008).
Commodities such as flash memory are often the
most challenging category of products to source
(Simchi-Levi et al. 2004) for two reasons. First, there
are many procurement choices in terms not only of
suppliers but also of channels (e.g., spot versus contract market). Second, there are many sources of risk
(Billington 2002). In particular, commodity prices are
2135
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
2136
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
not to generate normative prescriptions for managers.
Rather, its aims are (i) to enhance our understanding
of commodity spot and forward markets, and of the
factors that determine their relative size, and (ii) to
produce hypotheses that could be tested in future
empirical work.
We model the problem as a two-stage game between risk-neutral producers, spot buyers, and forward buyers (speculators) in a rational expectations
framework. Although we analyze forward buyers
only in their role as speculators, this does not preclude the possibility of agents being simultaneously
consumers of the commodity. Producers sell forward
contracts to speculators in the first period; in the second period, after observing demand, producers offer
quantities for sale in the spot market. Forward contracts mature in one period and are settled by the
actual delivery of the commodity.
We assume that forward contracts are firm quantity commitments between buyers and producers—in
other words, order cancellations and refunds are not
permitted.1 This setting is appropriate for modeling
markets such as those in the semiconductor industries, where the manufacturer typically must plan production, order raw materials, stock inventory, and
configure systems to build a particular type of component. Therefore, the changing, rescheduling, or canceling of orders may result in significant nonrecoverable
costs (for a discussion of various adjustment costs, see
e.g., Thille and Slade 2000).2
Even in the absence of cost considerations, there
may be strategic incentives for producers to require
binding forward quantity commitments. Such commitments typically dampen speculative behavior in
the forward market, which leads to an increase in producers’ spot market power and also in their overall
expected profit. From a modeling standpoint, this setting is no different from one in which production of
the contract quantity occurs before the demand uncertainty is resolved. As a result, producers do not take
long positions in the second period, during which
1
Firm quantity commitments are prevalent in some industries. For
example, Nagali et al. (2008) mention that HP often enters into such
commitments with some of its suppliers to procure components
at lower prices. Suppliers are willing to give higher discounts in
such cases because committed volumes can be scheduled during
nonpeak times and there is no inventory risk for the supplier.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
they can only sell in the spot market (i.e., producers
do not buy back their forward positions).
Absent demand uncertainty, our problem reduces
to the one studied by Allaz and Vila (1993). They
show that the ratio of forward sales to spot sales is an
affine function of the number of (identical) producers in the market and that the higher the number of
producers, the higher the ratio. If there is uncertainty
in demand and if producers can buy back some of
the contractual quantity, then the problem reduces to
the one studied by Allaz (1992). Because production
starts only after the demand uncertainty is resolved
and because production plans can be changed instantaneously at no cost, Allaz (1992) concludes that
demand risk plays no role—leaving market concentration as the only factor that determines the relative
size of forward and spot markets in expectation, just
as in the Allaz and Vila (1993) model with deterministic demand. However, such a frictionless setting may
not adequately capture the dynamics in industries
where it is costly or infeasible to downsize planned
production or where opportunities for commitment
are available to producers. It is therefore worthwhile
to reexamine the effect of demand uncertainty on the
supply structure under this new setting with binding
commitments.3
Intuitively, demand uncertainty has two opposing
effects on the size of the forward market. The positive
effect is that high growth expectations drive speculators to make advance purchases of the commodity to
sell them later in the spot market at a possibly higher
price (if demand turns out to be high). The negative
effect is that the possibility of the spot market being
oversupplied (if demand turns out to be low) can
make buyers purchase less in the forward market to
take advantage of low spot prices and avoid being left
with excess inventory. In general, the positive effect is
driven by speculation, whereas the negative effect is
driven by inventory risk.
We demonstrate that, in our setting, demand uncertainty can have a substantive effect on the sizes of
the forward and spot markets. The precise nature
of this effect depends on two main factors: (i) price
elasticity of demand at marginal cost and (ii) market concentration.4 We find that the volume traded in
the forward market usually is constant up to a certain level of demand uncertainty and then starts to
2
Such costs could include what Simon (1952, p. 265) calls “sticky
costs,” which are “costs proportional to the rate of manufacture
when this is constant, but not capable of being reduced immediately as the rate of manufacture declines.” They could also include
lump-sum adjustment costs associated with a decrease in the rate
of production (e.g., costs related to the shutdown of equipment,
special maintenance costs for inactive equipment, or labor costs of
firing). If these adjustment costs are sufficiently high, then the producers will not be willing to take long positions in the spot market.
3
See, for example, Reinganum and Stokey (1985) for a discussion on the effects of the commitment period’s length on industry
behavior at equilibrium.
4
We define market concentration as 1/n, where n is the number of
producers in the market. We shall later assume all producers to be
identical and prove that, in equilibrium, they have equal market
shares.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
decrease as uncertainty increases.5 Once it reaches a
minimum level, the volume of forward sales begins
to increase as uncertainty increases. The more inelastic the demand is, the lower the negative effect of
demand uncertainty on the forward market volume.
If demand is highly inelastic at marginal cost, then the
negative effect disappears, and the volume (absolute
and relative) traded in the forward market becomes
an increasing function of demand uncertainty. The
reason is that the more inelastic the demand, the
higher the profit-maximizing margin. A higher margin translates into a higher profit potential for speculators, so there will be strong incentives to purchase
in the forward market. As demand becomes more
elastic at marginal cost, the negative effect driven by
inventory risk starts to dominate, and so the forward
market volume (absolute and relative) decreases as
demand uncertainty increases.
The level of market concentration also plays a role,
along with the price elasticity of demand, in determining how demand uncertainty affects the forward
and spot sales. As market concentration decreases, the
competition increases in both markets. This increased
competition has a generally positive effect on total
output, but it is unclear how the relative size of
the two markets is affected. Under deterministic
aggregate demand, for instance, increased competition leads to more of the total output being sold in
the forward market (Allaz and Vila 1993). Yet the
more intense the competition, the less the power of
the suppliers in the spot market; hence, if demand is
uncertain, then increased competition dampens price
fluctuations in the spot market. Even during periods of high (aggregate) demand, the equilibrium price
will be close to the perfectly competitive price if there
are enough sellers in the market. Thus, more intense
competition implies lower profit-making opportunities for speculators who buy in the forward market.
However, there could be a substantial downside in the
form of inventory risk—that is, if demand turns out
to be low and speculators are left with excess inventory. Given the reduced profit-making opportunities,
speculators will not be willing to take on inventory
risk; therefore, as uncertainty increases, the forward
market’s volume will decline and the spot market’s
volume will increase. However, we will show that this
5
In this paper we use the following definition of “increasing uncertainty”: a random variable X is more uncertain than a random variable Y if X has more weight in the tails than does Y (i.e., if X
can be obtained from Y via a mean-preserving spread; Rothschild
and Stiglitz 1970). We shall use the terms “more uncertain” and
“riskier” synonymously. For certain types of distributions used in
our numerical examples (e.g., normal, uniform, and U-shaped distributions), increasing uncertainty is equivalent to increasing standard deviation.
2137
response requires demand to exhibit some degree of
price elasticity at marginal cost.
Based on the analysis presented here, we formulate
new hypotheses on the structure of the supply in commodity markets. These hypotheses posit a shift in the
supply structure whenever the demand risk of a product changes (e.g., because of higher innovation speed
or product maturity), the price elasticity of product
demand changes (e.g., because of the introduction
of substitutes), or the market concentration changes.
Furthermore, the direction of the shift due to changes
in market concentration need not be the direction predicted by the riskless theory.
The rest of this paper is organized as follows. Section 2 summarizes the literature. In §3 we set up
the duopoly model and characterize the equilibrium
of the spot and forward market game. In §4 we
present the comparative statics of the equilibrium and
provide an explicit representation of the equilibrium
results for a special case. In §5 we generalize these
results to the case of more than two producers, and in
§6 we formulate testable hypotheses implied by our
analysis. Our conclusions are presented in §7, and all
(nontrivial) proofs are gathered in the appendix.
2.
Literature Review
Research has established an extensive body of knowledge about the role of forward markets and the factors that affect forward trading. An important stream
of research focuses on the hedging role, whereas
another focuses on the strategic role of forward markets. By guaranteeing price and availability of products, forward contracts appeal to risk-averse buyers
and sellers. The hedging benefits of forward contracts have been extensively used to explain why forward markets exist and why sometimes the majority
of transactions between suppliers and manufacturers
take the form of bilateral forward contracts negotiated
under inferior information, even though both parties
have access to a liquid spot market (Dong and Liu
2007, Kawai 1983).
In this paper we focus on the strategic rather
than the hedging role and derive our results under
risk neutrality. On the strategic side, forward markets limit the power of suppliers in the spot market
and increase consumer surplus. Typically, forward
contracting makes the suppliers worse off because
it reduces the profitability of trades in commodities
even further. A risk-neutral monopolistic supplier, for
instance, will not engage in forward trading when
there is no demand uncertainty. However, if there
is uncertainty in demand and information asymmetry, then a monopolistic supplier might transact in
the forward market (Mendelson and Tunca 2007).
In oligopolies, because of competition, a forward
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
2138
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
market will always exist. Although suppliers overall
are better off without a forward market, a supplier
who deviates and trades in the forward market can
increase his profit. In equilibrium, therefore, all suppliers engage in forward trading (Allaz 1992, Allaz
and Vila 1993). In addition to the strategic role of
forward markets under competition, transaction costs
and nonlinearities in production costs have been proposed as alternative explanations for forward trading
under risk neutrality (Williams 1987).
Another relevant line of research concerns the role
of speculators and/or secondary markets. Lee and
Whang (2002) analyze the role of secondary markets
for a single manufacturer and many resellers. These
authors find that the manufacturer, if given a choice,
will select only low-margin items for trade in the
secondary market, whereas specialized, high-margin
products will be sold directly by the manufacturer to
end consumers. In contrast, we find that the opposite
phenomenon is observed in oligopolistic commodity markets: speculators pursue more (less) aggressively the products with high (low) profit potential.
Because of competition, producers are willing to sell
large quantities to speculators in the forward market, which often results in overproduction of the profitable but risky products; the risky but less profitable
products will be sold directly by producers to spot
buyers. Chod and Rudi (2006) model two firms that
invest in capacity or inventory under uncertain market conditions, but have the option, as more accurate
demand information becomes available, to trade the
excess/deficit inventory in a secondary market. These
authors examine how different trade mechanisms in
the secondary market (bargaining equilibrium versus
price equilibrium) affect the investment decisions of
the two firms. Su (2010) studies a monopolistic firm
selling a fixed capacity in the presence of speculators
and strategic buyers. He finds that speculative trading increases the firm’s expected profit, but might also
lead to lower capacity investment by the firm. In contrast with these papers, our model ignores the capacity investment decisions of firms and focuses instead
on the division between the spot and contract sales,
absent any capacity constraints. Milner and Kouvelis
(2007) show how contract markets are affected by
the existence of a secondary market in which participants can trade to clear their inventory positions.
They study a setting with a monopolistic supplier and
many buyers and find that buyers benefit from inventory pooling, whereas the supplier might try to counteract these benefits by restricting spot availability of
the product, thus pushing buyers into signing longterm contracts. Whereas their paper models a monopolistic setting for a storable product, we analyze a
competitive market for nonstorable commodities.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Finally, our paper contributes to the growing literature in operations management that studies procurement and distribution in commodity markets. Wu
and Kleindorfer (2005) analyze the optimal portfolio of long-term option contracts and spot market
transactions for competing heterogeneous suppliers
and a single buyer; they show that flexible contracts
mitigate some of the inefficiencies associated with
pure forward contracts. Spinler and Huchzermeier
(2006) also consider options and show how flexible
contracts help market participants better respond to
uncertain market conditions and better plan their production capabilities. Pei et al. (2011) study the optimal contract structure (fixed versus flexible volume,
linear versus nonlinear pricing) between a supplier
and a manufacturer in the presence of spot trading.
The market structure in our paper differs from the
ones in these other papers; also, we analyze pure
forward contracts, and not flexible option contracts.
Goel and Gutierrez (2007) analyze the optimal procurement policy from spot and forward markets for
a storable commodity under stochastic demand and
exogenous spot prices, showing how information on
futures prices can be used to reduce inventory cost.
Secomandi (2010) studies the optimal inventory trading strategy of a storable commodity with an exogenous spot price, subject to space and capacity limits.
Unlike these papers, we consider endogenous spot
prices that depend on the realized demand and on the
joint decisions of competing sellers.
3.
The Duopoly Model
Consider two identical producers i = 11 2 that sell a
perishable homogeneous good. The good is traded in
both a forward market (period 0) and a spot market (period 1). Demand is stochastic and is realized
in period 1. Let qfi denote the forward sales, and
let qsi denote the spot sales of firm i, i = 11 2. Then
Qf = qf1 + qf2 is the cumulative forward sales volume,
and Qs = qs1 + qs2 is the cumulative spot sales volume
(i.e., quantity sold). We denote by Pf the forward price
in period 0, and we denote by Ps the spot price in
Period 1. We assume that the two firms have identical
linear cost functions given by
c 1 4Q5 = c 2 4Q5 = bQ1
(1)
where b is the marginal cost.6
6
A more general model formulation would be to consider different
cost functions for forward and spot production (i.e., cs1 4Q5 = cs2 4Q5 =
bs Q for spot production and cf1 4Q5 = cf2 4Q5 = bf Q for forward production, with bs ≥ bf ). However, for tractability we disregard in our
analytical treatment possible differences in marginal cost of production in the two markets. We can still show numerically that
the main takeaways from our analysis are robust to differences in
marginal costs (provided such differences are not excessive).
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2139
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Similarly to Allaz and Vila (1993) and Mendelson
and Tunca (2007), we assume that the spot demand
function is affine with a random intercept and slope
normalized to unity. This may also be interpreted as
a linear approximation of the actual demand function
(Corbett and Karmarkar 2001). Thus, it holds that
D = D4P 1 …5 = a − P + …1
for P > 01
(2)
where P is the price, a is the known component of
the demand intercept, and … is the uncertain component (which is drawn from a continuous distribution
with mean 0 and standard deviation ‘). The density
and distribution functions of … are, respectively, f 4…5
and F 4…5.
We assume the price is nonnegative. In other words,
the commodity is not a waste product, and any excess
can be scrapped with no cost or value. This “free disposal” assumption is appropriate if disposal costs are
negligible and/or if there is unlimited demand for
the product when its price is zero. Also, we restrict
the support of the distribution of … to be included
in 4−a + b1 ˆ5, with a > b. This means that there is
always some demand (howsoever low) for the product when its price is equal to marginal cost.
We can rewrite the demand function as D =
D4P 5 + …, where D4P 5 = a − P is the riskless demand
function (i.e., the demand function of the riskless theory). This structural form of the demand has the property that the probability of D differing from D4P 5 by
a certain amount is independent of P ; that is, neither
firm can affect the amount of uncertainty by changing
the price (Mills 1959).
3.1. The Spot Market Game
There are three types of players in our model: spot
buyers, speculators, and producers. Spot buyers purchase in the spot market only after the uncertainty in
their demand is resolved. Speculators, who are risk
neutral, buy in the forward market and then sell to
spot buyers. Speculators do not hold stock and will
sell (or discard) the entire quantity purchased in the
forward market to the spot buyers at the marketclearing price.7 We assume that there are many speculators and that they set prices competitively (see, e.g.,
7
Kawai 1983). Producers transact in the forward market with the speculators and in the spot market with
the spot buyers.
Given that speculators purchased Qf in period 0,
the demand faced by producers in the spot market in
period 1 is given by
We analyze forward buyers only in their role as speculators, but
this does not mean agents cannot simultaneously be consumers of
the commodity. The same analysis applies if, upon the realization
of the demand in period 1, speculators consume part of the forward
purchases and sell the excess (or buy the deficit) in the spot market.
Similar assumptions are employed by Danthine (1978), among others. Moreover, we assume that there are many speculators transacting in the forward and spot markets. In other words, the amount q
that a speculator buys in the forward market is extremely small
compared with Qf . Hence, speculators do not have enough market power to individually affect the spot price later in the game,
which explains why they clear all their inventory at the marketclearing price.
Ds = D − Qf
⇐⇒ Ds = a + … − Qf − Ps 0
(3)
The two producers enter a Cournot game in period 1
(spot market). Note that if a + … − Qf ≤ 0, then Ds ≤ 0
for any Ps ≥ 0. Because we do not allow price to be
negative and do not consider negative demand, from
(3) we can rewrite, without loss of generality, the spot
demand faced by producers as follows:
Ds = 4a + … − Qf 5+ − Ps 1
(4)
where A+ = max401 A5 denotes the positive part of A.
Although this expression does not guarantee nonnegativity of price, we will show in Proposition 1 that
price is nonnegative in equilibrium.
The sequence of decisions is as follows: in period 0,
producers and speculators form expectations of the
demand in period 1. Each producer then decides
what quantity (qf1 and qf2 , respectively) will be sold
in the forward market. Speculators buy Qf , where
Qf = qf1 + qf2 . Production of the forward quantity (Qf )
is initiated. In period 1, production of Qf is completed.8 Uncertainty in demand is resolved (i.e., … is
realized). Each producer determines what quantity
(qs1 and qs2 , respectively) will be sold in the spot market. This quantity could potentially be zero, if the
demand realization is sufficiently low. Production of
Qs begins, where Qs = qs1 + qs2 . Producers deliver q 1 =
qf1 + qs1 and q 2 = qf2 + qs2 , respectively. Speculators sell
Qf to spot buyers. Spot buyers buy Q ≤ Qs + Qf ,
˙ + for a given realizawhere Q = Qs + Qf − 4Qf − a − …5
˙ + for which there
tion …˙ of …. Any amount 4Qf − a − …5
is no demand at a nonnegative price will be discarded
at no cost.
The producers’ optimization problem in period 1
is to decide what quantity of goods to sell given the
realization of the demand and the amount of forward
contracts Qf already purchased by the speculators.
Producer i’s period 1 profit-maximization problem
can therefore be written as
max 84Ps − b5qsi 90
qsi ≥0
8
(5)
As explained in the introduction, we assume that forward contracts are firm quantity commitments, such that buybacks and
order cancellations are not allowed. From a modeling point of view,
this is similar to assuming that production of the forward quantity
takes place before the uncertainty in demand is resolved.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2140
Table 1
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Ds , D, Ps , and qsi as Functions of Realized …˙
(Where z 2= Qf − a + b, v 2= Qf − a)
Spot market equilibrium
˙ P5
D4…1
˙ Qf 5
Ps 4…1
˙ Qf 5
qsi 4…1
−Ps
a + …˙ − P
0
0
v < …˙ ≤ z a + …˙ − Qf − Ps
a + …˙ − P
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
˙ Qf 1 Ps 5
Ds 4…1
…˙ ≤ v
…˙ > z
a + …˙ − Qf − Ps
a + …˙ − P
a + …˙ − Qf
a + …˙ − Qf + 2b
3
The expected amount of excess supply is given by (8),
and the probability of having excess supply is given
by (9). Observe that the higher the volume of forward
sales, the higher the expected excess supply and also
the higher the probability of having excess supply in
the spot market:
0
E… 4Qex 5 =
a + …˙ − Qf − b
3
Solving for the Nash equilibrium quantities and
prices in this game yields the result given in Proposition 1.
Proposition 1. The Nash equilibrium of the game in
period 1 is unique. The equilibrium quantities are as
follows.
˙ Qf 5 = qs2 4…1
˙ Qf 5 =
1. If …˙ > Qf − a + b, then qs1 4…1
˙ Qf 5 = 4a + …˙ − Qf + 2b5/3.
4a + …˙ − Qf − b5/3; also, Ps 4…1
˙ Qf 5 = qs2 4…1
˙ Qf 5 = 0;
2. If …˙ ≤ Qf − a + b, then qs1 4…1
also,
˙ Qf 5 = a + …˙ − Qf if …˙ > Qf − a, and
(a) Ps 4…1
˙ Qf 5 = 0 if …˙ ≤ Qf − a.
(b) Ps 4…1
In Table 1 we represent the demand and inverse
demand functions as well as the equilibrium val˙ From
ues under all possible scenarios of realized ….
Proposition 1 it follows that Q̄s , the producers’ total
expected spot sales volume in period 1, is a decreasing function of Qf , the producers’ forward sales volume in period 0:
Q̄s 2= E… 4Qs 5 = 2
Z
ˆ
Qf −a+b
a + … − Qf − b
3
¡ Q̄s
2
= − F c 4Qf − a + b5 < 01
¡Qf
3
dF 4…51
(7)
where F c 4x5 = 1 − F 4x5. Proposition 1 also shows that
whenever …˙ ≤ Qf − a + b, demand in the spot market will be zero at a price greater than or equal to
marginal cost. In this case, the producers will not
sell in the spot market, and the only sales will come
from speculators. The speculators will clear their
˙ Qf 5 =
inventory at the market-clearing price (i.e., Ps 4…1
4a + …˙ − Qf 5+ ). In this case we say there is “excess” supply in the spot market because the spot price is below
manufacturing cost (Ps < b). The amount of excess
supply for a realization …˙ is Qex = 4Qf − 4a + …˙ − b55+ .
Qf −a+b
−a+b
4Qf − a + b − …5 dF 4…5
= 4Qf − a + b5F 4Qf − a + b5
Z Qf −a+b
−
… dF 4…51
From (5) and (4) it is evident that in equilibrium
Ps ≥ 0, because qsi ≥ 0 for i ∈ 801 19. Given (4) and a
˙ we can rewrite the problem as follows:
realization …,
max 64a + …˙ − Qf 5+ − qsi − qsj − b7qsi 0
(6)
qsi ≥0
Z
(8)
−a+b
Pr4Qex > 05 = Pr4Ps < b5 = Pr4… < Qf − a + b5
= F 4Qf − a + b50
(9)
Thus, the spot market can be either a competitive
spot market when demand is low or an oligopolistic Cournot spot market when demand is high. Speculators make transactions in both markets, whereas
producers transact only in the latter. Note that this is
different from Allaz (1992), where producers always
transact in the spot market. When demand is high,
similar to our model predictions, producers will take
a short position in the spot market. However, when
demand is low, producers will take a long position
in the spot market by buying back their forward
positions at the cash price. Because production of the
forward orders has not started by the time the spot
transactions are completed and because production
plans can be adjusted at no cost, as assumed by Allaz
(1992), then it is “rational” for producers to buy back
some of the forward positions and not produce the
commodity, thus limiting the excess supply in the
spot market.9 However, if either production of forward orders has already been completed, or significant investment in raw materials has been made, or
other opportunities for commitment are available to
producers, as assumed in our model, then producers
will no longer take a long position in the spot market when demand is low, but rather choose not to
transact, hence the different spot and forward market
equilibria under the two models.
9
There is often a strategic reason for producers credibly committing not to buy back their forward positions in the spot market.
As explained in the introduction, such a commitment increases
the speculators’ inventory risk. Consequently, speculators will purchase less in the forward market; in turn, this increases producers’
power in the spot market. For a wide range of parameters, our
numerical experiments show that producers are overall better off
committing to forward production. That being said, commitment is
not a subgame perfect equilibrium, but rather a Nash equilibrium
in a path strategy space (for a discussion of the differences between
these, see, e.g., Reinganum and Stokey 1985).
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2141
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
3.2. The Forward Market Game
Let Pf be the price of a forward contract for one
unit of good in period 0. Speculators must decide the
optimal quantity of forward contracts to buy at this
price knowing that the demand in period 1 is stochastic. In a rational expectations framework (Muth 1961),
speculators will purchase the quantity in the forward
market that makes the forward price equal to the
expected spot market price. In other words, as long
as there is a profit-making opportunity (i.e., a forward price lower than the expected spot price), there
will be a risk-neutral speculator willing to act upon it.
If the forward price is lower than the expected spot
market price, then speculators will demand more forward contracts, which in turn will trigger an increase
in the price of a forward contract. Conversely, if the
forward price is higher than the expected spot market price, then speculators will buy fewer forward
contracts, which will decrease the price of a forward
contract. At any given moment, speculators and producers have the same information about the spot
market demand distribution; hence, they form the
same expectation of the spot market price as a function of the forward sales. If their expectations differed from the theory’s predictions, then, as shown by
Muth (1961), there would be profit-making opportunities for an insider such as selling a forecast to the
firms, operating a competing firm, or speculating in
inventory. For a given forward price Pf , we can use
Proposition 1 to obtain the forward market-clearing
condition under rational expectations:
Pf = E… 4Ps 50
(10)
Substituting the expression for the spot market price
from Proposition 1, we obtain
Pf 4Qf 5 =
Z
a + … − Qf + 2b
ˆ
Qf −a+b
+
Z
Qf −a+b
Qf −a
3
dF 4…5
4a + … − Qf 5 dF 4…50
(11)
Expression (11) can be interpreted as the inverse
demand function for forward contracts because it
expresses the forward price as a function of the
total forward sales.10 We remark that Pf is a strictly
decreasing function of Qf , for Qf > 0.
Next, we solve the producers’ profit-maximization
problem and determine the optimal forward quantity
10
Expression (11) is equivalent to the expression Pf 4Qf 5 =
Rˆ
R Qf −a+b
44a + … − Qf + 2b5/35 dF 4…5 + max8Q −a1 −a+b9 4a + … − Qf 5 dF 4…5,
Qf −a+b
f
˙ Qf 5 = 0, for … ≤
because (i) f 4…5 = 0 for … ≤ −a + b, and (ii) Ps 4…1
Qf − a (from Proposition 1, part 2(b)). It is easy to show that this
expression is differentiable for all Qf ∈ 401 ˆ5.
to sell in period 0. The total expected profit of producer i is
E… 4çi 5 = qfi 4Pf − b5 + E… 4qsi 4Ps − b551
(12)
where the profit, forward price, spot quantities, and
spot price have the respective arguments: çi 4qf1 1 qf2 5,
Pf 4Qf 5, qsi 4…1 Qf 5, and Ps 4…1 Qf 5. Recall that Qf =
qf1 + qf2 . Under the assumption of risk neutrality, the
producer maximizes his expected profit:
max E… 4çi 50
(13)
qfi
We find that the forward market equilibrium is
always symmetric (i.e., qf1 = qf2 ). More importantly,
we find that there always exists a strictly Paretodominant Nash equilibrium. Proposition 2 gives a formal statement of the equilibrium result.
Proposition 2. There always exists a forward market
Nash equilibrium. Any forward market equilibrium is symmetric and is given by qf1 = qf2 = Q/2, where Q ≥ 0 is a
fixed point of G4 · 5:
Rˆ
G4Q5 = 2
Q−a+b
F c 4…5 d… − 9
R Q−a+b
Q−a
F 4…5 d…
3 + 6F 4Q − a + b5 − 9F 4Q − a5
0
(14)
When the distribution of … has an increasing hazard (or failure) rate (see, e.g., Lariviere 2006), we can
prove the equilibrium is unique.
Corollary 1. If the distribution of demand has an
increasing hazard rate, then the equilibrium is unique.
In general, not all fixed points of G4 · 5 constitute
an equilibrium, but any equilibrium must be a fixed
point of G4 · 5. The smallest fixed point of G4 · 5 in
601 ˆ5 is a Nash equilibrium and is also the Paretodominant Nash equilibrium, as stated in the following
proposition.
Proposition 3. If there exist multiple Nash equilibria,
then the one in which forward sales are the lowest strictly
dominates the others in the Pareto sense.
Because we have assumed all producers to be identical, it follows that they will all be strictly better off
under the Pareto-dominant equilibrium. This result
is important because it shows that the existence of
multiple equilibria does not create a problem: there
will always be one equilibrium that is strictly dominant (in the Pareto sense) for all producers. Moreover,
the individual rationality of the producers will dictate
that this equilibrium will be selected in the market.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2142
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Figure 1
Forward vs. Spot Sales as a Function of Demand Uncertainty (… ∈ 8−L1 H91 L = a − b1 H ∈ 601 2007)
(a, b) = (100, 1)
(a, b) = (100, 30)
60
180
160
50
40
Quantity
120
Quantity
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
140
100
80
30
20
60
40
10
20
0
0
0
20
40
60
80
100
120
140
160
0
20
40
Std. dev.
60
80
100
120
140
160
Std. dev.
Expected additional spot sales
Forward sales
3.3. Benchmark Case: Riskless Demand
In a duopoly, under riskless demand (i.e., D = D4P 5),
the volume of forward sales equals the volume of spot
sales. The following proposition states the result for
the riskless theory (Allaz and Vila 1993).
Table 2
Proposition 4. If D = D4P 5, then the equilibrium is
unique, and the forward and spot equilibrium outcomes are,
respectively, qf1 = qf2 = 4a − b5/5 and qs1 = qs2 = 4a − b5/5.
Q̄s
Under risky demand, however, the total output
need no longer be sold in equal shares in the two markets. Also, depending on parameter values, the total
forward sales under demand uncertainty can be either
less than, equal to, or more than the forward sales
corresponding to the riskless theory. The following
example illustrates the difference between the risky
and riskless demand cases.
Example 1. Let D = D4P 5 + …. Assume … can take
only the values H and −L with respective probabilities L/4L + H 5 and H /4L + H 5.11 Then the equilibrium
forward and expected spot sales of the producers are
summarized in Table 2. For L = a − b we can see that,
11
We can also construct a U-shaped distribution, as a continuous
approximation of a two-point distribution, to be consistent with the
assumptions we made at the start with respect to F 4…5. Consider
the following distribution with support included in 6−L1 H 7. The
probability density function is given by

2H
x
L


− +1−
x ∈ 6−L1−L+51




 4H +L5
f 4x5 = 0
x ∈ 6−L+1H −‚71



2L2
xL
L


+1−
x ∈ 4H −‚1H 71
H 4H +L5 H

where L > 43a5/5 and ‚ = 4H 5/L. For small enough ‚, the total
forward sales under the risky demand scenario will be either Qf =
244H 6274a − b5 − 24L7 + L634a − b5 + 8‚75/481H + 15L55 if b is high or
Qf = 2443L4a + H − b5 − 27bH − ‚L5/415L55 if b is low.
Special Case: Two-Point Distribution … ∈ 8−L1 H9
Low b
Qf
a + H − b − 9bH/L
5
4a + H − b5L + 6bH
2
54H + L5
2
High b
4a − b549H + L5 − 8LH
27H + 5L
64a − b543H + L5 + H49H + 7L57L
2
427H + 5L54H + L5
2
as H increases and approaches ˆ, the limit of the forward sales is 424a − b55/27 if b is high or ˆ if b is low;
in the limit, the expected spot sales equal 424a − b55/3
if b is high or 424a + 5b55/5 if b is low. Under riskless
demand, however, Qf = Qs = 424a − b55/5.
Figure 1 plots the forward and spot sales for the
two sets of parameters 4a1 b5 = 41001 15 and 4a1 b5 =
41001 305. We infer from this simple example that
demand uncertainty does have an impact on both the
absolute and relative size of the forward and spot
markets. In the next section we analyze more generally the effect of uncertainty on the size of the forward
market and on the supply structure.
4.
Comparative Statics of the
Forward Market Equilibrium
In this section we examine in more detail the effect
of demand uncertainty on the forward and spot market equilibria and provide an explicit representation
of the equilibrium results and comparative statics for
the special case of uniformly distributed ….
4.1. Effect of Demand Uncertainty
We examine the effect of demand uncertainty on
forward trading. We are interested in analyzing the
impact on forward sales of making the demand
distribution more risky. In particular, we analyze
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2143
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
the impact on forward sales when the uncertainty
in demand changes via a mean-preserving spread
(Sandmo 1971).
Suppose now that the demand is D = a + …ˆ − P ,
where …ˆ = ƒ…. Then the distribution of the random
component of the intercept is Fˆ 4x5 = F 4x/ƒ5, and ƒ > 1
is the spread parameter. An increase in ƒ from the
point ƒ = 1 will have the following effect on the total
forward sales:
Z ˆ
Z z
¡qfi −1
…
dF
4…5
+
9
… dF 4…5
=
−K
¡ƒ ƒ=1
z
v
i
+ 3qf 42zf 4z5 − 3vf 4v55 1 (15)
where K 2= ¡ 2 E… 4çi 5/¡4qfi 52 ≤ 0. Recall that z = Qf −
a + b, and v = Qf − a. We define
Z ˆ
R1 2=
… dF 4…51
s
R2 2=
Z z
… dF 4…5 + 8
… dF 4…5 + 6qfi zf 4z51
−a+b
−a+b
Z v
i
… dF 4…5 + qf vf 4v5 1
R3 2= −9
Z
s
−a+b
where s 2= z + 34Pf − b5. Then
¡qfi ¡ƒ = −K −1 6R1 + R2 + R3 70
(16)
ƒ=1
There are three effects that together determine the
overall impact of uncertainty on the size of the forward market: the speculation effect (R1 ), the inventory
effect (R2 ), and the option effect (R3 ). In region 6s1 ˆ5,
speculators make a profit, because the spot price is
above the forward price. Hence, R1 captures the effect
of speculation, and it is always positive. Effect R2
captures the inventory risk. In region 6−a + b1 s5, the
spot price is lower than the forward price, and so
speculators incur a loss due to the devaluation of
the commodity. Effect R3 reflects the value of the
“free disposal” option: because the spot price cannot
be negative, inventory losses are bounded (i.e., the
spot price is zero for … < v; see Table 1). Absent this
free disposal option—that is, if the spot price can be
negative and unbounded—this term will disappear.12
12
To avoid burdening our model with extra parameters, we ignore
any potential salvage value or cost of disposal for the excess
supply. Yet, incorporating a small constant positive salvage value
(or unit cost of disposal) should not, in theory, affect our results;
in particular, the R3 term would still be significant. However,
if the salvage value (respectively, the cost of disposal) is high,
then demand uncertainty will always have a positive (respectively,
negative) effect on forward sales. We include an analysis of the
model with salvage value (respectively, cost of disposal) in the
electronic companion (available at http://faculty.insead.edu/dana
-popescu/documents/ecompanion.pdf).
For very low uncertainty we have
34a − b5
24a − b5
1 z≈−
< 01
5
5
Z z
34a − b5
− b < 01
… dF 4…5 < 01
v≈−
5
v
Z ˆ
… dF 4…5 > 00
Qf ≈
(17)
z
Then R1 and R3 will be positive, whereas R2 will be
negative. However, it is not straightforward to discern the sign of the cumulative effect R1 + R2 + R3 . On
the one hand, if the marginal cost b is relatively small
compared with a − Qf and if the probability mass in
any interval of size b is negligible, then the second
and third terms are negligible, and so an increase in
uncertainty will lead to an increase in forward sales;
in particular, if b/a is zero (or close to zero), then
forward sales will be an increasing function of the
uncertainty parameter. On the other hand, if the value
of b/a is significant and the probability mass in an
interval of size b is nonnegligible, then an increase in
uncertainty may lead to a decrease in forward sales.
This can occur only if the ratio b/a differs from zero
by a “nonnegligible” amount (the parameters of the
problem dictate when b/a becomes significant). This
is the same as requiring that b/4a − b5 be significant
or that the price elasticity of demand at marginal cost
be significant in absolute value.13
As remarked in the foregoing discussion, at the
present level of generality we cannot make precise statements about the marginal effect of demand
uncertainty on the size of the forward market. But
we can make precise statements about this effect for
highly inelastic or elastic goods. More than that, we
can characterize the effect of demand uncertainty on
the relative size of the forward and spot markets. The
following proposition summarizes our results.
Proposition 5. There exist „ ≤ „¯ ∈ 40115 such that
(i) if b/a < „, then ¡Qf 4a1 b1 ƒ5/¡ƒ ≥ 0 and
¡4Q̄s 4a1 b1 ƒ5/Qf 4a1 b1 ƒ55/¡ƒ ≤ 0;
¯ then ¡Qf 4a1 b1 ƒ5/¡ƒ ≤ 0 and
(ii) if b/a > „,
¡4Q̄s 4a1 b1 ƒ5/Qf 4a1 b1 ƒ55/¡ƒ ≥ 0.
Proposition 5 states that if demand is highly inelastic at a price equal to marginal cost, then the more
uncertain the demand, the higher is the quantity sold
in the forward market and the lower is the relative volume of additional spot sales. In contrast, for
products with elastic demand, the more uncertain the
demand, the lower the quantity sold in the forward
market and the higher the expected additional spot
sales.
13
The (expected) price elasticity of demand at marginal cost is
ED 4b5 = 4b/D4b554dD4P 5/dP 5 = −b/4a − b5.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
2144
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
By (8) and (9) we know that the greater the forward
sales, the greater the expected excess supply and the
probability of excess supply. Also, according to Proposition 4, riskless products will be sold in both markets
in equal quantities irrespective of the price elasticity
of demand at marginal cost. Combining these results
with the results from Proposition 5, we can conclude
that risky products with highly inelastic demand at a
price equal to marginal cost will mainly be sold in the
forward market. For such products we expect to see
frequent periods of excess supply in the spot market.
In contrast, risky products with less inelastic demand
will mainly be sold directly by producers to spot buyers, and there will rarely be excess supply.
The more inelastic demand is, the higher the profitmaximizing margin for producers. Also, the more
uncertain the inelastic demand, the greater the price
fluctuations in the spot market. Our results therefore demonstrate that forward buyers will buy more
of the profitable and risky products. On the supply side, competition makes producers willing to sell
large quantities to forward buyers, which, combined
with the preference of forward buyers, often results in
the overproduction of profitable but risky products.
4.2. An Illustration: Uniform Distribution
Here we give explicit representations of the equilibrium described in §3 and of the comparative statics described in §4.1 for … uniformly distributed
in 6−ˆ1 ˆ7. Table 3 summarizes the results. As we
already established in Corollary 1, there is a unique
equilibrium in the forward market, because the uniform distribution has an increasing hazard rate. For
the uniformly distributed demand intercept, if mean
demand is constant, then the parameter ˆ becomes
the spread parameter. Hence, there is a direct correspondence between the value of ˆ and the level
of demand uncertainty as measured by a meanpreserving spread.
Table 3 shows that the forward quantity is independent of the degree of uncertainty ˆ up to a
threshold value. Beyond that threshold, forward sales
first decrease with ˆ but then start to increase as ˆ
increases. Depending on the specific values of the
demand parameters and marginal cost, we observe
Table 3
the same behavior as described in §4.1. If b is small,
then the middle region disappears and forward sales
increase as uncertainty increases. If b is large, then
the last region disappears and forward sales decrease
as uncertainty increases. For intermediate values of b,
at lower levels of uncertainty the inventory risk will
dominate the speculative effect, hence the negative
impact of uncertainty on forward trading; at higher
levels of uncertainty, the speculative effect coupled
with the option effect (i.e., price cannot be negative)
will be dominant, explaining the positive impact of
demand uncertainty on forward trading.
5.
Oligopoly and Perfect Competition
Here we extend the equilibrium results of §3 to the
case of n > 2 producers with identical linear cost functions. The spot market equilibrium is given by our
next proposition.
Proposition 6. The Nash equilibrium of the Cournot
game in period 1 is unique. The equilibrium quantities and
price are as follows.
1. If …˙ ≤ Qf − a + b, then qsi = 0 for i = 11 0 0 0 1 n; also,
˙ Qf 5 = 4a + …˙ − Qf 5+ .
Ps 4…1
˙ Qf 5 = 4a + …˙ − Qf − b5/
2. If …˙ > Qf − a + b, then qsi 4…1
˙ Qf 5 = 4a + …˙ − Qf +
4n + 15 for i = 11 0 0 0 1 n; also, Ps 4…1
nb5/4n + 15.
The forward market equilibrium always exists. Our
next proposition characterizes the equilibrium.
Proposition 7. There always exists a forward market Nash equilibrium. Any forward market equilibrium is
symmetric and is given by qfi = Q/n for i = 11 0 0 0 1 n,
where Q is a fixed point of G4 · 5:
Rˆ
R Q−a+b
4n−15 Q−a+b F c 4…5d… −4n+152 Q−a F 4…5d…
G4Q5 = n
0
4n+15+4n2 +n5F 4Q−a+b5−4n+152 F 4Q−a5
An immediate corollary of this proposition is the
following:
Corollary 2. (a) If b/a > 0 (i.e., if demand is not perfectly inelastic at marginal cost), then, as the number of
producers becomes very large (i.e., as market concentration approaches zero), the ratio of forward sales to residual
spot sales approaches zero; that is, as n → ˆ, we have
Qf /Q̄s → 0.
Special Case: Uniform Distribution (… ∼ U6−ˆ1 ˆ7)
ˆ ≤ 434a − b55/5
434a − b55/5 < ˆ ≤ 43a + 4b5/5
Qf
24a − b5
5
242a − 2b − ˆ5
7
Q̄s
24a − b5
5
34a − b + 3ˆ52
98ˆ
¡Qf /¡ˆ
=0
<0
ˆ > 43a + 4b5/5
p
5a + 22b + 5ˆ − 3 4a + ˆ − 2b52 + 64b4b + ˆ5
8
p
348b4a − 2b5 + 4a − 10b + ˆ54a − 10b + ˆ + 4a + ˆ − 2b52 + 64b4b + ˆ555
64ˆ
p
< 0 for ˆ ≤ −a − 30b + 10pb4a + 13b5;
> 0 for ˆ > −a − 30b + 10 b4a + 13b5
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2145
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
The result in Corollary 2 should be contrasted with
the corresponding result in the riskless theory. Absent
demand uncertainty, the ratio of forward sales to
(residual) spot sales, Qf /Q̄s , is equal to n − 1. So when
there are many producers competing in the market,
an overwhelming proportion of the producers’ output will be sold in the forward market. However,
this need not be the case when demand is uncertain.
If demand is less inelastic at marginal cost and there
are many producers in the market, then nearly all of
the producers’ output will be sold in the spot market. Thus, demand uncertainty can reverse the nature
of supply in a highly competitive market, by shifting
it from predominantly forward-driven to spot-driven
supply. The intuition behind this result is not difficult to grasp. When there are many producers competing in the spot market, the spot price will likely
be very close to the competitive price even if demand
turns out to be high. Hence, there will be relatively
small fluctuations of the spot price above marginal
cost when demand is high. However, the spot price
can still drop to zero if demand is low enough. Facing
an appreciable downside against a relatively inconsequential upside, speculators will purchase less in the
forward market under demand uncertainty.
This result can also be observed in Figures 2 and 3.
In Figure 2 we plot the percentage difference (i.e.,
64Qf − Qfd 5/Qfd 7 × 100%) between forward sales under
uncertain demand (Qf ) and forward sales under
deterministic demand (Qfd ) for different values of
the demand intercept’s standard deviation as well
as various levels of elasticity and market concentration. By Proposition 5 we know that, in a duopoly,
if demand is highly inelastic at marginal cost then
the volume of forward sales is increasing in the level
of uncertainty; however, this relation is reversed for
higher levels of elasticity. From Figure 2 we make
two observations. First, for middle-range values of
price elasticity, the forward sales volume is U-shaped
in the level of uncertainty (see, e.g., ED 4b5 = −00053
for n = 2 and n = 6). Second, the two thresholds of
price elasticity—below and beyond which forward
sales are respectively increasing and decreasing with
uncertainty—depend on the level of market concentration. Specifically, the lower the concentration (i.e.,
the higher the n), the more inelastic the demand must
be for uncertainty to have a positive effect on forward sales; when the price elasticity at marginal cost
is ED 4b5 = −00013, for example, demand is inelastic
enough for uncertainty to have a positive effect on
forward sales in a duopoly, but not enough in less
concentrated markets (see the bar graphs for n = 6
and n = 30). The converse statement also holds: the
lower the concentration, the lower the threshold of
Percentage Difference Between Forward Sales Under Uncertain Demand and Forward Sales Under Deterministic Demand for n = 2 (Top),
n = 6 (Bottom Left), and n = 30 (Bottom Right)
Figure 2
20
b = 10
ED(b) = – 0.111
15
b=5
ED(b) = – 0.053
b = 1.3
ED(b) = –0.013
b=1
ED(b) = –0.01
b=5
ED(b) = –0.005
%
10
5
0
–5
–10
15
20
25
30
35
40
45
50
40
40
20
20
0
%
60
%
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
(b) If b = 0 (i.e., if demand is perfectly inelastic at
marginal cost), then, as the number of producers becomes
very large, the ratio of forward sales to residual spot sales
also becomes very large; that is, as n → ˆ, we have
Qf /Q̄s → ˆ.
0
– 20
– 20
– 40
– 40
5
10
15
20
25
30
35
40
45
50
– 60
0
5
10
15
20
25
30
35
40
45
50
Notes. We assume the distribution of … is (one sided) truncated normal, TN4Œ1 1 ‘1 1 −a + b1 ˆ5, where Œ1 and ‘1 are chosen such that E4…5 = 0 and E4…2 5 = ‘ 2 .
We use a = 100.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2146
Figure 3
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Relative Size of Forward Market Under High Demand Uncertainty and Different Levels of Elasticity and Market Concentration
1.0
1.0
b = 20, ED (b) = –0.25
b = 10, ED (b) = –0.111
0.8
0.8
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
0.6
n=6
n = 12
n = 30
0.4
0.2
0.6
0.4
0.2
0
0
30
40
50
30
40
50
1.0
1.0
b = 1, ED (b) = –0.01
b = 2, ED (b) = –0.02
0.8
0.8
0.6
0.4
n = 2 0.6
n=6
n = 12 0.4
0.2
0.2
0
0
30
40
50
30
40
50
Note. See note to Figure 2.
elasticity required for uncertainty to have a negative
effect on forward sales. For example, when the price
elasticity at marginal cost is ED 4b5 = −00053, price elasticity of demand is high enough (in absolute value)
for uncertainty to have a negative effect on forward
sales in a competitive market (n = 30 producers) but
not high enough in more concentrated markets (n = 2
and n = 6 producers).
In Figure 3 we plot the relative size of the forward market (i.e., Qf /4Qf + Q̄s 5) when there is high
uncertainty of demand under various levels of market concentration and price elasticity. The upper two
panels present bar graphs for goods with less inelastic
demand. With these types of goods at medium or low
levels of market concentration, we note that the lower
the concentration, the smaller the relative size of the
forward market. Yet this positive relation is not likely
to hold for high levels of concentration (i.e., for low n)
unless demand is highly elastic. From the riskless theory we know that, under deterministic demand, the
ratio Qf /Qs is increasing in the number of producers.
Even though demand uncertainty (when combined
with significant levels of price elasticity of demand
at marginal cost) will reduce this ratio considerably,
for low n it will likely not be decreasing in n unless
elasticity is extremely high.
The lower two panels present bar graphs for
goods with more inelastic demand. Here, the trend is
reversed: the lower the concentration, the higher the
relative size of the forward market (at medium and
high levels of concentration). That inverse relation is
not likely to hold for markets with extremely low
concentration (i.e., for high n) unless demand is perfectly inelastic. This claim follows from Corollary 2,
by which—for markets with perfect competition—the
size of the forward market approaches zero unless
demand is perfectly inelastic.
As a further illustration of our results, consider
a simplified example involving two commodities,
A and B. Demand for commodity A is less inelastic than the demand for commodity B. We shall analyze the relative and absolute sizes of forward and
spot markets for these two commodities. The values
in Table 4 are chosen for illustrative purposes only
and do not match any real values; note further that
all information required for the analysis is summarized in this table. Results are plotted in Figure 4 for
the three scenarios described in Table 4: deterministic
demand (S1), medium demand uncertainty (S2), and
high demand uncertainty (S3).
Under deterministic demand, almost all of commodity A will be sold in the forward market. The
reason is that competition will drive sellers to sell in
this market to secure their market share. As demand
uncertainty increases, however, the fraction of forward purchases will decline. Because demand for
commodity A is less inelastic, and because there are
many sellers in the market, it follows that buyers are
reluctant to buy much in the forward market before
realizing their actual demand. The opposite behavior
is seen in the market for commodity B. As uncertainty increases, the size of the forward market
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2147
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Relative and Absolute Sizes of Forward and Spot Markets
Absolute size of forward and spot markets
24,000,000
22,000,000
20,000,000
18,000,000
16,000,000
14,000,000
12,000,000
10,000,000
8,000,000
6,000,000
S1
S2
S3
S1
Commodity A
S2
S1
S3
Commodity B
Forward volume
Table 4
Demand and Market Characteristics for Commodities A and B
Commodity A
Market concentration
Average quantity sold (price)
Standardized demand function
Marginal cost
Demand uncertainty (‘ /a × 100%)
Scenario 1 (%)
Scenario 2 (%)
Scenario 3 (%)
0
30
50
0
30
50
Notes. We consider a linear approximation of demand with slope normalized
to unity and additive uncertainty. We assume the distribution of … is (one
sided) truncated normal, TN4Œ1 1 ‘1 1 −a + b1 ˆ5, where Œ1 and ‘1 are chosen
such that E4…5 = 0 and E4…2 5 = ‘ 2 .
will increase—in both relative and absolute values—
because demand for this commodity is inelastic and
market concentration is high; hence, both price risk
and the incentive for speculation are also high.
6.
Testable Hypotheses
Given the limiting properties of the forward and spot
equilibria derived in §4 (Proposition 5) and §5 (Corollary 2) or borrowed from the riskless theory (Allaz
and Vila 1993), we can map out the dominant supply channel in terms of three factors—demand risk (left
vertical axis in Figure 5), market concentration (horizontal dimension), and price elasticity of demand at
marginal cost (right vertical axis)—under the extreme
cases of perfectly elastic (b → a) and perfectly inelastic
(b = 0) demand. It is clear from the figure that the
main supply channel in markets with perfect competition is the spot channel unless demand is perfectly
inelastic or riskless. In duopolies, the main supply
channel is the forward channel only when elasticity is
low. These limiting properties of the spot and forward
market equilibria, when combined with the additional
S3
S1
S2
S3
Commodity B
Spot volume
Figure 5
Commodity B
0.05 (20 suppliers) 0.33 (3 suppliers)
20 mil. ($2)
20 mil. ($4)
12 mil. ($4)
19 mil. ($12)
7−P +…
164 − P + …
(a = 7)
(a = 164)
b = $105
b = $105
S2
Commodity A
Main Supply Channel Depending on Market
Concentration, Demand Risk, and Price Elasticity of
Demand at Marginal Cost
Market concentration
Demand risk
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Relative size of forward and spot markets
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
0.50
0.45
0.40
0.35
0.30
High
(duopoly)
Low
(Perfect competition)
Spot
Spot
Forward
Forward
Forward = Spot
Forward
High
None
Perfectly
elastic
Perfectly
inelastic
Perfectly
elastic
Perfectly
inelastic
Elasticity
Figure 4
insights gained from numerical analysis, allow us to
formulate several hypotheses regarding the supply
structure in commodity markets.
In particular, we use Proposition 5 and the numerical experiments summarized in Figure 2 to formulate
the following hypotheses regarding the relative size
of the forward and spot markets.
Hypothesis 1. For commodities with highly inelastic
demand, the greater the uncertainty in demand, the larger
the relative size of the forward market compared with the
residual spot market.
Hypothesis 2. For commodities with elastic demand,
the greater the uncertainty in demand, the smaller the relative size of the forward market compared with the residual
spot market.
We also use Corollary 2 and the numerical experiments summarized in Figure 3 to formulate Hypotheses 4 and 5, which address the effects of market
concentration on forward and spot trading under
demand uncertainty. These contrast with Hypothesis 3, which addresses those effects under deterministic demand (as implied by the results of Allaz and
Vila 1993).
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2148
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Hypothesis 3. For commodities with low demand
uncertainty, the more intense the competition, the larger
the relative size of the forward market compared with the
residual spot market.
Hypothesis 4. For commodities with high demand
uncertainty and with elastic demand, the more intense
the competition, the smaller the relative size of the forward market compared with the residual spot market ( for
medium and low levels of market concentration).
Hypothesis 5. For commodities with high demand
uncertainty and with highly inelastic demand, the more
intense the competition, the larger the relative size of the
forward market compared with the residual spot market
( for medium and high levels of market concentration).
Finally, Equation (9) and Proposition 5 allow us to
formulate the following hypothesis regarding the frequency of excess supply in the spot market.
Hypothesis 6. For commodities with high demand
uncertainty, the more inelastic the demand, the higher the
frequency of excess supply in the spot market.
7.
Conclusions
In this paper we build a theoretical model to identify the main factors that determine the relative volume of transactions in the forward and spot markets
for a commodity whose aggregate demand is uncertain. We assume a setting where forward contracts
are firm quantity commitments between buyers and
suppliers and/or production of the forward quantity
takes place before uncertainty in demand is resolved,
such that producers do not buy back their forward
positions.
First, we show that there always exists a symmetric
equilibrium in the forward and spot markets. In the
spot market, the equilibrium is unique. In the forward
market, we demonstrate that if the distribution of
demand has an increasing hazard rate, then the equilibrium is unique. For all other cases we show that
the equilibrium in which forward sales are the lowest
is the strictly Pareto dominant Nash equilibrium.
Second, we show that demand uncertainty can have
a substantive impact on the sizes of the forward and
spot markets. This is in contrast to previous findings
in the literature that show that demand uncertainty
does not affect forward trading if production takes
place after uncertainty in demand is resolved and if
production plans can be altered instantaneously and
costlessly.
The precise nature of demand uncertainty’s effect
depends on two main factors: market concentration
and the price elasticity of demand at marginal cost.
We find that in highly concentrated markets, commodities with high demand uncertainty and highly
inelastic demand will mainly be sold through forward contracts, often leading to excess supply in
the spot market. For these products, either (i) the
manufacturing cost is too high, so it is not worth
betting against uncertainty, or (ii) there are many
suppliers in the market, so that fluctuations in the
spot price above marginal cost are not significant.
However, commodities with more elastic demand will
mainly be sold directly by producers to spot buyers and will seldom be in excess supply. These are
commodities featuring significant price fluctuations
capable of consuming most of the end-product margins. In a market with low concentration we find
that uncertainty in demand can reverse the nature
of supply—shifting it from (predominantly) forwardto spot-driven supply—provided demand is not perfectly inelastic.
Third, we use our analytical results and numerical experiments to formulate new hypotheses on
the supply structure in commodity markets and on
the propensity of excess supply in the spot market. These hypotheses indicate that there will be a
shift in the supply structure whenever (i) a product becomes more or less risky, (ii) the price elasticity of demand for a product changes, and/or
(iii) the market becomes more or less competitive.
Further research is needed to test these hypotheses
empirically.
Acknowledgments
The authors thank the department editor and review team
for their effort and valuable input. In particular, the authors
thank the associate editor for the numerous suggestions
and insights that improved this paper significantly. They
also acknowledge the helpful comments from colleagues at
INSEAD and from participants at the Stern School of Business seminars and the 2012 M&SOM Conference.
Appendix
Proof of Proposition 1. Producer i’s profit-maximization problem can be written as
j
max 864a + …˙ − Qf 5+ − qsi − qs − b7qsi 90
qsi ≥0
(18)
This profit function is concave in qsi . The first-order conditions (FOCs) for producers 1 and 2 are as follows:
• If a + …˙ − Qf > b (i.e., if …˙ > z) then,
qs1 =
a+ …˙ −Qf −qs2 −b
qs2 =
a+ …˙ −Qf −qs1 −b
0 (19)
2
2
Solving this system of two equations with two unknowns
yields
and
qs1 = qs2 =
a + …˙ − Qf − b
0
(20)
3
Also, we know from the inverse demand function that
Ps = a + …˙ − 4Qf + Qs 5 = 4a + …˙ − Qf + 2b5/3.
Thus, if …˙ > z, then qs1 = qs2 = 4a + …˙ − Qf − b5/3 and Ps =
4a + …˙ − Qf + 2b5/3.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2149
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
• If 4a + …˙ − Qf 5+ ≤ b, then also a + …˙ − Qf ≤ b (i.e., …˙ ≤ z).
j
In this case we see that 4a + …˙ − Qf 5+ − qsi − qs − b < 0 for any
qsi > 0. This implies qs1 = qs2 = 0.
Thus, the producers will not sell in the spot market. The
speculators will clear their inventory (Qf ) at the marketclearing price, which is Ps = 4a + …˙ − Qf 5+ . This follows from
two assumptions: (1) a competitive market for speculators
and (2) the free disposal option. We have assumed that there
are many speculators, each holding a very small portion
of Qf ; hence, a speculator cannot individually influence the
price. As a result, each speculator will sell his entire inventory. Second, because we assume free disposal, the price
cannot become negative. This means that if a + …˙ − Qf < 0
(i.e., …˙ ≤ v), then Ps = 0.
To summarize, if v < …˙ ≤ z, then Ps = a + …˙ − Qf . If …˙ ≤ v,
then Ps = 0. In both these cases, qs1 = qs2 = 0. ƒ
Proof of Proposition 2. Producer i’s profit-maximization problem can be written as
max qfi 6Pf − b7 + E… qsi 4Ps − b5 — Ps > b Pr4Ps > b50 (21)
qfi
But Pf = E… 4Ps 5 = E… 4Ps — Ps > b5 Pr4Ps > b5 + E… 4Ps — Ps ≤ b5 ·
Pr4Ps ≤ b5. Then (21) can be rewritten as
max qfi E… 4Ps — Ps > b5Pr4Ps > b5+E… 4Ps — Ps ≤ b5Pr4Ps ≤ b5−b
For qf1 and qf2 to be optimal quantities, it is necessary that Qf
also satisfy the SOC given in (25). Summing for i = 11 2 and
rearranging the terms, we find that Qf satisfies
3Qf 2
+ 18F 4Qf − a5 ≤ 00
+ E… qsi 4Ps − b5 — Ps > b Pr4Ps > b51
Lemma 1. G4x5 has at least one fixed point in 601 ˆ5 that
satisfies (27), and therefore there exists a symmetric Nash equilibrium for the forward market game.
Proof. We first examine the sign of the continuous function G4x5 − x on 601 ˆ5. Note that
G405−0 =
+
ˆ
a+… −Qf −b
a+… −Qf +2b
3
Qf +b−a
3
(28)
x→ˆ x−a+b
=⇒ lim
x→ˆ
x→ˆ x−a
Z
ˆ
F c 4…5 d… − 9
x−a+b
Z
x−a+b
F 4…5 d…
x−a
x→ˆ
−b dF 4…5
(23)
for i = 11 2 and Qf = qf1 + qf2 . The FOCs and second-order
conditions (SOCs) for maximizing seller i0 s profit (22) are
Rˆ c
Rz
F 4…5 d… − 9 v F 4…5 d…
i
z
qf =
1 i = 11 21
(24)
3 + 6F 4z5 − 9F 4v5
3qfi 43f 4v5−2f 4z55−4−14F 4z5+18F 4v5 ≤ 01
i = 11 20 (25)
Equation (24) indicates that the equilibrium is always
symmetric (qf1 = qf2 ). After adding the reaction Equation (24)
for qf1 and qf2 we get that Qf is a fixed point of G4x5 as
given by
Rˆ
R x−a+b
F c 4…5 d… − 9 x−a F 4…5 d…
G4x5 = 2 x−a+b
0
3 + 6F 4x − a + b5 − 9F 4x − a5
An equivalent representation of G4x5 that we sometimes
use is
Z ˆ
Z x−a+b
G4x5 = 2
… dF 4…5 + 9
… dF 4…5
x−a+b
lim 6G4x5−x7 = −ˆ < 00
x→ˆ
= −9b lim 3 + 6F 4x − a + b5 − 9F 4x − a5 & 00 (29)
Qf −a
Z
24a−b5
> 01
3
The limit in (28) follows from the fact that limx→ˆ G4x5
= −ˆ, which can be shown as follows:
Z ˆ
Z x−a+b
lim
F c 4…5 d… = 0 ∧ lim
F 4…5 d… = b
(22)
or, equivalently, as
Z ˆ
a + … − Qf + 2b
dF 4…5
max qfi
i
3
Qf +b−a
qf
Z Qf +b−a
+
4a + … − Qf 5 dF 4…5 − b
(27)
If G4x5 had at least one fixed point in 601 ˆ5 that satisfied (27), then this would immediately imply the existence
of an equilibrium. We demonstrate this fact in the following
lemma.
qfi
3f 4Qf − a5 − 2f 4Qf − a + b5 − 4 − 14F 4Qf − a + b5
x−a
− 4x − a + b5F c 4x − a + b5
− 9 4x − a + b5F 4x − a + b5 − 4x − a5F 4x − a5
−1
· 3 + 6F 4x − a + b5 − 9F 4x − a5 0
(26)
Since G4x5 − x is a continuous function on 601 ˆ5 and takes
different signs at the two ends of this interval, it follows
from the intermediate value theorem that G4x5 has at least
one fixed point in 601 ˆ5.14 Let x̌ be the smallest fixed point
of G4x5 in 601 ˆ5. From (28) it follows that the first derivative
of G4x5 − x at that point must be negative. However, the
sign of the first derivative of G4x5 − x at x̌ is given by the
sign of the expression
3x̌ 3f 4x̌ − a5 − 2f 4x̌ − a + b5 − 5 − 22F 4x̌ − a + b5
+ 27F 4x̌ − a51
(30)
where we have used the fact that G4x̌5 = x̌.
Note that if (30) is negative then the SOC given in (27)
is also negative. Then q̌1f = q̌2f = x̌/2 satisfy the FOCs and
SOCs and is a Nash equilibrium. Using Qf = x̌ yields that
q̌1f = q̌2f = Qf /2 is a Nash equilibrium, where Qf is the
smallest fixed point of G4x5. This concludes the proof of
both Lemma 1 and Proposition 2. ƒ
Proof of Corollary 1. If F 4…5 is a distribution function
with increasing hazard rate (i.e., if f 4…5/F c 4…5 is increasing),
14
Here we have used the previously stated assumption that the
support of the distribution of … is included in 4−a + b1 ˆ5. That
assumption can be relaxed to accommodate either a larger or a
smaller support. If the assumption were relaxed to accommodate a
larger support, to avoid unnecessary complications in the derivation of the equilibrium results,
R ˆ one would further
R −a+b require the following condition to hold: −a+b F c 4…5 d… − 9 −a F 4…5 d… ≥ 0, such
that G405 ≥ 0.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
2150
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
then we can show that there exists a unique Nash equilibrium. We show this result by proving that G4x5 has a unique
fixed point. From Proposition 2 we know that an equilibrium must be a fixed point of G4x5. We also know from
Lemma 1 that there exists at least one fixed point of G4x5
which is an equilibrium. Hence, by showing that G4x5 has
a unique fixed point, we can conclude that the equilibrium
must also be unique.
The FOC for maximizing seller i’s profit (23) can be
expressed as
24a − b + E4… — … ≥ z55 − 5Qf F c 4z5
+ 9 24a − b + E4… — v ≤ … < z55 − 3Qf
· 4F 4z5 − F 4v55 − 18bF 4v5 = 00
(31)
Let
A4Qf 5 = 24a − b + E4… — … ≥ z55 − 5Qf F c 4z5 and
(32)
B4Qf 5 = 9 24a − b + E4… — … ∈ I55 − 3Qf F 4I5 − 18bF 4v51 (33)
where I = 6v1 z7 and F 4I5 = F 4z5 − F 4v5. Then the FOC in (31)
is equivalent to
A4Qf 5 + B4Qf 5 = 00
If f 4v5 > f 4z5 then by (38) and (39) we have
56f 4v5 − f 4z57 − 5f 4v5F 4z5 ≤ 4f 4z5F 4z5 − 9f 4z5F 4v5
=⇒ 56f 4v5 − f 4z57F c 4z5 ≤ 9f 4z56F 4z5 − F 4v57
5F c 4z5 34F 4z5 − F 4v55
≤
0
3f 4z5
f 4v5 − f 4z5
=⇒
Thus, in the interval 601 w7 on which A4x5 is decreasing,
B4x5 is also decreasing. Therefore, if A4x5 + B4x5 = 0 has
a solution in 601 w7, then that solution is unique. We will
show that A4x5 + B4x5 = 0 has no solution on 6w1 ˆ5. This,
when combined with the existence of at least one solution
to A4x5 + B4x5 = 0, proves that the solution is unique. To see
that A4x5 + B4x5 < 0 for all x > w, note that B4x5 < 0 for all
x > 0 (by (35)) and that limx→ˆ A4x5 = 0. ƒ
Proof of Proposition 3. We can write the expected
profit of seller i in period 0 as a function of q = qf1 = qf2 alone:
E… 4çi 4q55 = q
Z
ˆ
z
(35)
a + … − 2q − b
3
2
dF 4…50
(41)
The derivative of E… 4çi 4q55 with respect to q is given by
¡A4Qf 5
¡Qf
¡Qf
a + … − 2q + 2b
dF 4…5
3
z
Z z
+ 4a + … − 2q5 dF 4…5 − b
ˆ
v
+
We also have
¡B4Qf 5
Z
(34)
First note that B4Qf 5 ≤ 0 for Qf ≥ 0 because
a − b + E4… — … ∈ I5 ≤ a − b + z = Qf 0
(40)
= 3Qf f 4z5 − 5F c 4z5 and
= −9 Qf 4f 4z5 − f 4v55 − 27 F 4z5 − F 4v5 0
(36)
(37)
Under the assumption of increasing hazard rate, we know
that F c 4x5/f 4x5 is a decreasing, nonnegative function.
Hence, there is a unique w > 0 that satisfies 45F c 4w − a + b55/
43f 4w − a + b55 = w. (Trivally, h1 4w5 = w is an increasing function on 601 ˆ7, whereas h2 4w5 = 45F c 4w − a + b55/
43f 4w − a + b55 is a decreasing nonnegative function; therefore the two functions must cross in at least one point.)
For Qf ≤ w, A4Qf 5 is decreasing in Qf (and Qf ≤
45F c 4z55/43f 4z555; for Qf > w, A4Qf 5 is increasing in Qf .
Note that (37) is negative if either f 4z5 ≥ f 4v5 or Qf ≤
434F 4z5 − F 4v555/4f 4v5 − f 4z55. We now prove that for an
increasing hazard rate, Qf ≤ 45F c 4z55/43f 4z55 implies that
(37) is negative. Since z = Qf − a + b > v = Qf − a, it follows that
4f 4z56F 4z5 − F 4v57 ≥ 0
=⇒ −5f 4z5F 4v5 ≤ −5f 4z5F 4v5 + 4f 4z56F 4z5 − F 4v57
=⇒ −5f 4z5F 4v5 ≤ 4f 4z5F 4z5 − 9f 4z5F 4v50
Proof of Proposition 5. (i) We show that there exists a „¯
¯ then the forward sales are a decreasing
such that if b/a ≥ „,
function of the spread parameter ƒ. The proof proceeds in
three steps. First, we show that if Qf < b, then Qf < a − b.
Second, we show that if Qf < min4b1 a − b5, then an increase
in the spread parameter leads to a decrease in forward sales.
Third, we show that, for large enough b/a, forward sales
are close to zero; hence, Qf < b, and so any increase in the
spread parameter will decrease Qf further.
Step 1. If Qf < b, then F 4Qf − a5 = 0 and from (26) we
know that Qf is a fixed point of G4Qf 5 as given by
(38)
Our assumption of an increasing hazard rate means that
G4Qf 5 = 2
8
R Qf −a+b
−a+b
… dF 4…5−4Qf −a+b541+8F 4Qf −a+b55
3+6F 4Qf −a+b5
0
(43)
f 4v5
f 4z5
≤
F c 4v5 F c 4z5
R Qf −a+b
=⇒ f 4v5 − f 4z5 ≤ f 4v5F 4z5 − f 4z5F 4v5
=⇒ 56f 4v5 − f 4z57 − 5f 4v5F 4z5 ≤ −5f 4z5F 4v50
¡E… 4çi 4q55
z − 6q c
= −bF 4z5 +
F 4z5 − 2q4F 4z5 − F 4v55
¡q
9
Z z
1Z ˆ
… dF 4…5 + 4−v + …5 dF 4…50
(42)
−
9 z
v
Rˆ
Rˆ
Note that q ≥ 0, and z …f 4…5 d… > −a+b …f 4…5 d… = 0. Also
note
4a − b5 < 0. Finally, observe that
R z that z − 6q = −4qR−
z
4−v + …5f 4…5 d… ≤ b v f 4…5 d… = b4F 4z5 − F 4v55 ≤ bF 4z5.
v
Thus, expression (42) is negative and so E… 4çi 4q55 is decreasing in q for q > 0. ƒ
(39)
Rˆ
Since −a+b … dF 4…5 < −a+b … dF 4…5 = 0, it is easy to see that
G4Qf 5 < 0 for Qf ≥ a − b; hence, Qf < a − b. Thus, Qf < b
implies Qf < a − b.
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2151
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Step 2. If Qf < min4b1 a − b5, then v = Qf − a < −a + b and
z = Qf − a + b < 0. Now, by (15) the sign of ¡qfi /¡ƒ is given
by the sign of
ˆ
Z
… dF 4…5 + 9
z
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
z
Z
v
=8
Z
z
−a+b
Thus, Q̄s /Qf = F c 4Qf − a5. Now
¡41 − F 44Qf − a5/ƒ55 =
¡ƒ
ƒ=1
¡4Q̄s /Qf 5 ¡ƒ
… dF 4…5 + 3qfi 42zf 4z5 − 3vf 4v55
… dF 4…5 + 6qfi zf 4z50
(44)
Rz
Since z < 0, it follows that 6qfi zf 4z5 < 0. Also, 8 −a+b …dF 4…5
Rˆ
< 8 −a+b … dF 4…5 = 0. Therefore, forward sales are decreasing
in the spread parameter ƒ.
Step 3. Observe that as b → a we have G405 =
424a − b55/3 & 0. Moreover, limb→a G4Qf 5 < 0 for all Qf > 0.
Since Qf is the smallest fixed point of G4x5 in 601 ˆ5,
we have Qf → 0. Hence, Qf < b, which by Steps 1 and 2
implies that Qf is decreasing in the spread parameter ƒ.
To show that ¡4Q̄s /Qf 5/¡ƒ > 0, we show that ¡ Q̄s /¡ƒ > 0
whenever ¡Qf /¡ƒ < 0. From (7) we obtain
¡Qf 2Z ˆ
¡ Q̄s =
dF 4…5 > 00
…
−
¡ƒ ƒ=1 3 Qf −a+b
¡ƒ ƒ=1
(45)
(50)
We investigate the sign of (50).
¡Qf 4Qf − a5 −
¡ƒ ƒ=1
Rˆ
2 Q −a … dF 4…5 − 3Qf 4Qf − a5f 4Qf − a5
f
= 4Qf − a5 −
5F c 4Qf − a5 − 3Qf f 4Qf − a5
Rˆ
54Qf − a5F c 4Qf − a5 − 2 Q −a … dF 4…5
f
=
5F c 4Qf − a5 − 3Qf f 4Qf − a5
(ii) We first prove that if b = 0, then forward sales are
an increasing function of the spread parameter ƒ. Then, by
the continuity of G4x5, there must exist a „ such that the
relation holds.
Lemma 2. If the marginal cost is zero, then Qf is an increasing function of the spread parameter ƒ, and Q̄s /Qf is a decreasing
function of the spread parameter ƒ.
ƒ=1
¡44Qf − a5/ƒ5 = −f 4Qf − a5
¡ƒ
ƒ=1
¡Qf − 4Qf − a5
= −f 4Qf − a5
¡ƒ ƒ=1
¡Qf = f 4Qf − a5 4Qf − a5 −
0
¡ƒ ƒ=1
=
34Qf − a5F c 4Qf − a5 − 3Qf F c 4Qf − a5
5F c 4Qf − a5 − 3Qf f 4Qf − a5
¡4Q̄s /Qf 5 =⇒
< 00
¡ƒ
ƒ=1
<0
(51)
(46)
Hence, if b = 0, then forward sales are an increasing function
of the spread parameter, and, moreover, Q̄s /Qf is a decreasing function of the spread parameter. Then, by the continuity of Qf 4a1 b1 ƒ5, there exists a „ such that if 4b/a5 < „, then
the forward sales are an increasing function of the spread
parameter and the ratio of spot to forward sales is a decreasing function of the spread parameter. This completes the
proof of Lemma 2 and Proposition 5. ƒ
Rˆ
We show that v …f 4…5 d… − 3qfi vf 4v5 > 0. If v ≤ 0, then
Rˆ
3qfi vf 4v5 ≤ 0, and, since v …f 4…5 d… ≥ 0, the inequality
holds. Assume now v > 0. We know that
Rˆ c
Rˆ
F 4…5 d…
…f 4…5 d…
v
qfi = v c
=− + v c
0
(47)
3F 4v5
3
3F 4v5
Proof of Proposition 7. The proof for the case of n producers follows the same steps as that for two producers.
We show how the results change as a function of n. Seller
i’s maximization program in Period 0 is now
Z ˆ
Z z
a+… −Qf +nb
f 4…5 d… + 4a+… −Qf 5f 4…5 d… −b
max qfi
n+1
v
z
qfi
• We show that Qf is an increasing function of the
spread parameter ƒ. For b = 0 we have
¡qfi ¡ƒ =−
ƒ=1
¡ 2 E… 4çi 5
¡4qfi 52
−1 Z
ˆ
v
…f 4…5 d… −3qfi vf 4v5 0
Rˆ
Therefore, v …f 4…5 d… = 43qfi + v5F c 4v5. But by (30) we have
6qfi f 4v5 ≤ 5F c 4v5, so 3qfi f 4v5 < 205F c 4v5, and then
Z
v
ˆ
…f 4…5 d… − 3qfi vf 4v5 > 43qfi
c
c
+ v5F 4v5 − 205vF 4v5
= 3qfi F c 4v5 − 105vF c 4v5 = 105aF c 4v5 > 00
(48)
• We now show that if b = 0, then ¡4Q̄s /Qf 5/¡ƒ < 0. Note
that when b = 0 we have
Q̄s =
2Z
3
ˆ
Qf −a
Z
ˆ
z
a + … − Qf − b
2
n+1
f 4…5 d…
for i = 11 0 0 0 1 n. The FOC and SOC become
Rz
Rˆ
4n − 15 z F c 4…5 d… − 4n + 152 v F 4…5 d…
i
qf =
and
(52)
4n + 15 + 4n2 + n5F 4z5 − 4n + 152 F 4v5
n
4n + 152 qfi f 4v5 −
f 4z5 − 24n + 4n2 + n + 15F 4z5
n+1
− 4n + 152 F 4v55 ≤ 01
(53)
respectively, and G4Qf 5 now has the following form:
4a + … − Qf 5 dF 4…51
Z ˆ
2
Qf = c
4a + … − Qf 5 dF 4…50
3F 4Qf − a5 Qf −a
+
(49)
4n−15
G4Qf 5 = n
Rˆ
Qf −a+b
F c 4…5 d… −4n+152
4n+15+4n2 +n5F 4Q
f
R Qf −a+b
Qf −a
F 4…5 d…
−a+b5−4n+152 F 4Q
f
−a5
0
Popescu and Seshadri: Demand Uncertainty and Excess Supply in Commodity Contracting
2152
Management Science 59(9), pp. 2135–2152, © 2013 INFORMS
Downloaded from informs.org by [175.176.173.30] on 27 December 2015, at 23:19 . For personal use only, all rights reserved.
Now we can apply the same arguments as for the case of
two producers. ƒ
Proof of Corollary 2. (a) For this proof we will use
the fact that the support of … is 4−a + b1 ˆ5 and 0 <
R x−a+b
dF 4…5 < 1 for all x ∈ 401 ˆ5.
−a+b
Note
that we have F 4−a + b5 = F 4−a5 = 0 and
R −a+b
F
4…5
d… = 0. For Qf = 0 we have
−a
Rˆ
n4n − 15 −a+b F c 4…5 d… 4n2 − n54a − b5
G405 =
=
n+1
n+1
and limn→ˆ G405 = ˆ.
R Qf −a+b
For Qf > 0 and Qf −a F 4…5 d… > 0 we have
lim G4Qf 5
n→ˆ
4n − 15
= lim n
n→ˆ
Qf −a+b
4n + 15 + 4n2
Rˆ
=
Rˆ
Qf −a+b
F c 4…5 d… − 4n + 152
+ n5F 4Qf
R Qf −a+b
Qf −a
F 4…5 d…
− a + b5 − 4n + 152 F 4Q
f
− a5
c
F 4…5 d…
F 4Qf − a + b5 − F 4Qf − a5
R Qf −a+b
4n + 25 Qf −a F 4…5 d…
= −ˆ0
− lim
n→ˆ F 4Q − a + b5 − F 4Q − a5
f
f
We know that G4Qf 5 is continuous, G405 → ˆ, and
G4Qf 5 → −ˆ for Qf > 0. We also know from Proposition 7 that for a given n, we can find Qf ∈ 401 dn 5 such that
G4Qf 5 = Qf . Then it must be that as n grows large, dn → 0,
and hence Qf → 0 and Q̄s → a − b. It is easy to see that
Qf /Q̄s → 0.
(b) For b = 0 we have F 4Qf − a + b5 = F 4Qf − a5 and
R Qf −a+b
Rˆ
F 4…5 d… = 0 and G4Qf 5 = 4n4n − 15 Q −a F c 4…5 d…5/
Qf −a
f
44n + 15F c 4Qf − a55 and limn→ˆ G4Qf 5 = ˆ for all Qf ≥ 0
such that F c 4Qf − a5 < 1. If the support of … is infinite (i.e.,
F c 4Qf 5 < 1 for Qf ∈ 601 ˆ5), then for a large n, we have
Qf → ˆ (because we know that there exists a fixed point
of G4Q5, there must exist Qf ≥ 0 such that G4Qf 5 = Qf ; but
since limn→ˆ G4Q5 = ˆ for all Q ≥ 0, then as n grows large,
Qf → ˆ) and Qf /Q̄s → ˆ. This result (i.e., Qf /Q̄s → ˆ as
n → ˆ) would still hold if the support of … had a finite
upper bound. ƒ
References
Allaz B (1992) Oligopoly, uncertainty and strategic forward transactions. Internat. J. Indust. Organ. 10:297–308.
Allaz B, Vila J-L (1993) Cournot competition, forward markets and
efficiency. J. Econom. Theory 59:1–16.
Billington C (2002) HP cuts risk with portfolio approach. Purchasing
131:43–45.
Chod J, Rudi N (2006) Strategic investments, trading, and pricing
under forecast updating. Management Sci. 52:1913–1929.
Corbett CJ, Karmarkar US (2001) Competition and structure in
serial supply chains with deterministic demand. Management
Sci. 47:966–978.
Danthine JP (1978) Information, futures prices, and stabilizing speculation. J. Econom. Theory 17:79–98.
Dong L, Liu H (2007) Equilibrium forward contracts on nonstorable commodities in the presence of market power. Oper.
Res. 55:128–145.
Goel A, Gutierrez G (2007) Integrating commodity markets in the
optimal procurement policies of a stochastic inventory system.
Working paper, University of Texas at Austin, Austin.
Kawai M (1983) Spot and futures prices of nonstorable commodities
under rational expectations. Quart. J. Econom. 98:235–254.
Lariviere MA (2006) A note on probability distributions with
increasing generalized failure rates. Oper. Res. 54:602–604.
Lee H, Whang S (2002) The impact of the secondary market on the
supply chain. Management Sci. 48:719–731.
Mendelson H, Tunca TI (2007) Strategic spot trading in supply
chains. Management Sci. 53:742–759.
Mills ES (1959) Uncertainty and price theory. Quart. J. Econom.
73:116–130.
Milner J, Kouvelis P (2007) Inventory, speculation, and sourcing
strategies in the presence of online exchanges. Manufacturing
Service Oper. Management 9:312–331.
Muth JF (1961) Rational expectations and the theory of price movements. Econometrica 29:315–335.
Nagali V, Hwang J, Sanghera D, Gaskins M, Pridgen M, Thurston T,
Mackenroth P, Branvold D, Scholler P, Shoemaker G (2008) Procurement risk management (PRM) at Hewlett-Packard Company. Interfaces 38:51–60.
Pei PP, Simchi-Levi D, Tunca TI (2011) Sourcing flexibility, spot trading, and procurement contract structure. Oper. Res. 59:578–601.
Reinganum J, Stokey N (1985) Oligopoly extraction of a common
property natural resource: The importance of period of commitment in dynamic games. Internat. Econom. Rev. 26:161–173.
Rothschild M, Stiglitz J (1970) Increasing risk: I. A definition.
J. Econom. Theory 2:225–243.
Sandmo A (1971) On the theory of the competitive firm under price
uncertainty. Amer. Econom. Rev. 61:65–73.
Secomandi N (2010) Optimal commodity trading with a capacitated
storage asset. Management Sci. 56:449–467.
Simchi-Levi D, Kaminsky P, Simchi-Levi E (2004) Managing the
Supply-Chain (McGraw-Hill, New York).
Simon HA (1952) On the application of Servomechanism theory in
the study of production control. Econometrica 20:247–268.
Spinler S, Huchzermeier A (2006) The valuation of options on
capacity with cost and demand uncertainty. Eur. J. Oper. Res.
171:915–934.
Su X (2010) Optimal pricing with speculators and strategic consumers. Management Sci. 56:25–40.
Thille H, Slade ME (2000) Forward trading and adjustment costs in
Cournot markets. Cahiers d’Economie Politique 37:177–195.
Williams J (1987) Futures markets: A consequence of risk aversion
of transactions costs. J. Political Econom. 95:1000–1023.
Wu DJ, Kleindorfer PR (2005) Competitive options, supply contracting, and electronic markets. Management Sci. 51:452–466.