Hedging Interaction Under Cost Uncertainty:
Does a firms need to hedge increase with the extent of
hedging in the industry?
Menne C. Mennes
∗ABSTRACT
This paper demonstrates that in imperfectly competitive markets, the incentives of an individual firm to hedge its input costs risk increases as more firms hedge, and decreases as more firms choose not to hedge. The extent of hedging in an industry has real effects on the product output markets in which firms operate. Industries become more competitive as the degree of hedging increases. Consequently, a firm’s incentive to hedge cannot be considered in isolation from its industry. The results predict that industries facing input cost volatility should show homogeneity in hedging choices.
Keywords: Hedging, financial options, input price uncertainty
JEL classification: L13, G32, D43
∗Student at the Department of General Economics, University of Groningen. E-mail address:
m.c.mennes@hotmail.com.
1.
INTRODUCTION
Corporate risk management and hedging are increasingly seen as critical aspects in the business management process of financial and non-financial corporations. According to the Modigliani and Miller (1958) paradigm, corporations with risk neutral preferences do not benefit from hedging their risks. The extensive use of derivatives is since founded by the existence of several market imperfections and agency problems. Academic interest in hedging has mainly focused on demand uncertainty. Little attention has been given to cost uncertainty, as the costs of production are frequently assumed to be constant. Also models in corporate risk management usually analyze the hedging decision of a firm in isolation of other firms in its industry. However, in a recent empirical paper Nain (2004), describes that firms respond to competitors hedging policies and that hedging affects product market outcomes. Motivated by these observations, I investigate in this paper an additional rationale for firms to hedge their risk induced by the level of hedging in the firms industry.
I develop a duopoly model with two differentiated non-financial rivalling firms competing a la Cournot. The two firms compete in a non-cooperative manner using a single hedgeable commodity as an input and producing a homogenous output. The two firms are assumed to differ in the level of efficiency in transforming the input into the output. Both firms are faced with the same uncertain input prices (they face cost uncertainty). But because the inefficient firm uses more inputs to produce the same amount of output as the efficient firm it has a higher degree of risk exposure, distinguishing the model from models based on demand uncertainty. As an example one could think of two competing power plants with different degrees of efficiency using an input like coal or oil to produce electricity; the energy produced has to be sold immediately because storage is not possible. And because the product is completely homogenous the market price will fully depend on the quantity of electricity produced.
market price and the input prices. The fact that the inefficient firm has a larger input cost risk exposure along with the introduction of fixed costs, financial constraints and high costs of financial distress will lead to a risk neutral incentive for the inefficient firm to hedge its input price risk. I show that within the setting that the inefficient firm hedges its inputs and the efficient firm does not, the efficient firm facing a high cost state will not experience an offsetting change in the product market price. The inefficient firm has a competitive advantage in a high cost state because of its cheaper access to inputs. The efficient firm will, in order to be able to compete with the inefficient firm in a high cost state, find it profitable to also hedge, in spite of not having a risk neutral incentive to hedge initially. Therefore, under certain circumstances, the efficient firm will find it profitable to also hedge, in spite of not having a risk neutral incentive to hedge initially, in order to be able to compete with the inefficient firm in a high cost state.
A further contribution of my paper is showing how hedging using option contracts has real effects on the product markets. The reasons for choosing call option contracts and not forwards as hedging instruments as many preceding papers is motivated by the unique hedging possibilities that call options provide. Call options yield no future obligations to the holder but do provide the holder with the right to buy an agreed amount of assets at a predetermined price (the exercise price) on a future date. Another reason for choosing to model options explicitly as hedging instrument is that I am not aware of any comprehensive studies on the corporate use of options as instruments to hedge cost price uncertainty.
external debtholders is calculated. Section 5 outlines the expected payoffs for the duopoly model, Section 6 gives the empirical implications of the results and Section 7 concludes.
2.
REVIEW OF THE LITERATURE
Since the Modigliani and Miller (1958) paradigm, the extensive use of derivatives is conventionally explained by several capital market imperfections and agency problems. Existing theories have relied upon the existence of taxes (Mayers and Smith 1982), asymmetric information between firms and outside investors (DeMarzo and Duffie 1991), firm and manager’s risk-aversion (Stulz 1984) and a reduction in costs of expected financial distress by hedging (Smith and Stulz 1985). Froot, Scharfstein, and Stein (1993) show that hedging can reduce the probability that a firm will have insufficient internal funds to finance positive NPV projects that may arise and avert the need to issue securities at a discount to obtain financing. Another rational for hedging risk is the mitigation of the agency cost of debt, particularly because of the underinvestment problem (Myers 1977) and risk shifting (Jensen and Meckling, 1976). For a complete survey of existing hedging rationale read the papers by Michael (1998) or Oosterhof (2001).
A study by Allaz and Villa (1993) is one of the first to mention the interaction between firms while making their hedging decisions. They show that firms competing a la Cournot will partially hedge, opposed to fully hedge, using forward contracts for strategic reasons. The forward contracts serve as a commitment device for the firms to obtain a larger market share in the spot market, they try to act as a Stackelberg leader and force their rivals to cut output on the spot market. The selling of forward contracts leads to a prisoners’ dilemma type of problem. The spot price will decrease when all firms sell forward. In contrast to what Allaz and Villa assume Albaek (1990) finds that Stackelberg duopolies can be rationalized under cost uncertainty in one-shot models when direct information sharing is prohibited.
leaves no upside potential left for a firm, but can be used strategically to force rivals to cut output. Willems (2005) in his paper, investigates the efficiency effects of option contracts in a Cournot oligopoly, extending the work of Allaz and Vila (1993). In his model, firms may hedge using option contracts with a specific, exogenous, strike price. He shows that the use of call option contracts makes markets more competitive but to a smaller extent than forward contracts do.
In line with Allaz and Villa (1993), a recent empirical paper by Nain (2004) finds that hedging affects product market outcomes and that firms respond to their competitors hedging activities. The paper demonstrates that the extent of hedging in an industry has real effects on the product markets in which firms operate. Consequently, a firm’s incentive to hedge cannot be considered in isolation from its industry. Industry prices are less responsive to costs shocks in industries where hedging is more common. In these industries, profits of an unhedged firm are more sensitive to cost shocks because price changes do not offset the cost shocks. Thus, a firm’s perceived need to hedge increases with the extent of hedging in the industry. The exogenous costs shocks are faced by all firms and reduce or increase firms marginal costs. Hedging is modelled in a way similar to the paper by Froot, Scharfstein, and Stein (1993). Marginal costs are modelled as
k k cu
~
~ = + , with k~the common shock with E(k~)=0and Var(k~)=σ2. Nain also finds empirically that if a firm chooses to remain unhedged while many of its competitors do hedge it suffers a value discount.
industry choose to hedge, and increases as more firms choose not to hedge. This in contradiction with the findings of Nain (2004).
The paper by Adam, Dasgupta and Titman (2004) proposes a Cournot model with no explicit hedging instrument but the possibility for a firm to stabilize its cash flow similar to the model by Froot, Scharfstein, and Stein1. The paper shows that in equilibrium some firms hedge (stabilize their cashflow), while others do not, even though all firms are ex ante identical. Firm’s incentive to hedge depends partially on the hedging choices of other firms in the same industry as the incentive to hedge decreases as more firms hedge, and increases as more firms choose not to hedge.
Similar to Adam, Dasgupta and Titman (2004) the paper by Mello and Ruckes (2005) emphasizes not only the costs of risk but also the benefits of taking risk. They investigate the optimal hedging and production strategies of financially constrained firms in imperfectly competitive markets. They find that a hedging policy that minimizes the volatility of earnings reduces a firm’s financial constraints on average, but makes it impossible for the firm to gain a significant financial advantage over its competitors. Because a financial advantage allows a firm to appropriate future market share, firms do not always hedge their entire risk exposure even in the absence of transaction costs. Mello and Ruckes come to the same conclusion that the incentive to hedge of an individual firm decrease with the level of industry hedging.
Two opposing views thus exists, the first one arguing that the incentive of an individual firm to hedge decreases as more firms in an industry choose to hedge, and increases as more firms choose not to hedge. And a second view arguing that a firms need to hedge increase with the extent of hedging in the industry.
3.
MODEL
In this section I address the investment, production and hedging decisions in a three stage differentiated duopoly between an inefficient and an efficient production firm. Both firms face identical input costs if the do not hedge their input costs. The inefficient firm, firm 1, is assumed to use more inputs then the efficient firm, firm 2, to produce the same level of output. The fact that the inefficient firm has a larger input cost risk exposure along with the introduction of fixed costs, financial constraints and high costs of financial distress will lead to a risk neutral incentive for the inefficient firm to hedge its input price risk. I investigate the reaction of firm 2 to the introduction of the risk neutral incentive to hedge for firm 1.
3.1 General outline and assumptions
Consider a duopoly of two differentiated rivalling financial firms competing in a non-cooperative manner using a single hedgeable commodity as an input and producing a homogenous output. Input cost can either be low,
c
l ,or high, ch, with both states havingthe same probability of occurring. Because both firms have to pay fixed costs F at the end of the game, the real option of not producing in the high cost state is costly and will be avoided.
Firm 1 uses more inputs to produce the same quantity of output then firm 2 does. The relative amount of inputs firm 1 needs to produce the same output as firm 2 one is given by the efficiency multiplier f , with f strictly larger then 1. If the efficient firm, firm 2, is assumed to use one unit of input producing one unit of output then firm 1 ìs assumed to need f units of input. Firm 1 is assumed to have no internal funds and will not be able to produce unless it receives external finance in the form of debt.
The market demand is formed a la Cournot quantity competition; the linear demand curve has the form of X = 1−p, with p the product price and X total industry output. The industry output X =x1+x2 is the sum of individual firms i’s output xi. At date 0, the
to hedge their input costs. If internal funds are not sufficient the firm has to seek external finance. I assume firms can hedge the uncertain input costs at fair terms using financial call options with an exogenous exercise price derived below. At date 1, the production stage, the true input costs and the contracting position of the other firm are revealed and the level of production is decided upon. In the third and last period the firms engage in Cournot style quantity competition. The profits are earned, the fixed costs F are paid and the possible bondholders are repaid. Figure 1 gives a graphical representation of the timing of the model.
If firm 1 it is not able to repay its debt at date 2, it will go bankrupt. I assume that transaction costs D>>0 exist when the firm goes bankrupt (a firm will go bankrupt if the value of the firms income is lower then the value of its debt at maturity). If the firm goes bankrupt the bondholder gets the value of its debt minus the transaction costs D . I assume there to be many bondholders and money to have no time value. Consequently bondholders do not demand any compensation for lending out their money. But bondholders will not lend to the inefficient firm as long as it has positive expected costs of financial distress2.
The bondholder will thus finance the firm, conditional on the firm buying a fixed number of options reducing the excepted distress costs to zero. In investigating the reaction of firm 2 to the hedging of firm 1, the assumption that the two competing firms are not identical to each other is essential. Because this provides the initial incentive for firm 1 to hedge while firm 2 has no initial incentive to hedge.
2 This is similar to the paper by Smith and Stulz (1985)
TIME
Investment
Stage Competition Stage
0
Firms learn the input cost distribution, try to get outside funding and decide on their level of hedging.
2
Firms engage in quantity competition on the output market and earn income. Form which they repay possible external debt and pay fixed costs. 1
Firms learn the true input cost prices, each other contracting positions and make their production decisions.
Figure 1: Timing of the Model
The concave payoff structure of debt prevents the bondholder to profit from unexpected profit. Introducing cost uncertainty combined with the risk of costly financial distress will make it harder for firms to get financed. The existence of positive expected costs of financial distress forces the firm (the shareholders) to convince the potential bondholders they will hedge as to reduce the expected costs of financial distress. Because hedging transfers wealth from shareholders to bondholders, shareholders will have no incentive to actually hedge after the bondholders have agreed to invest in the firm. I assume this agency problem not to occur (think for instance of a potential negative reputational effects for the firm if it does not honour the arrangement). See appendix A for a graphical representation of the effect of hedging on expected costs of financial distress. The efficient firm, firm 2, in contrast to firm 1 does not have this rationale for hedging its risk as it does not have any debt holders.
Like the bondholders there are assumed to be many option writers. Because money is assumed to have no time value the option writers are willing to write options providing the expected profit is not negative. The exogenous exercise price of the option contracts is derived below such that it facilitates financing for firm 1 .
Figure 2. Extensive form of hedging game
The fact that firm 1 hedges provides a risk neutral incentive for firm 2 to hedge if and only if the expected payoff for firm 2 of hedging is larger then the expected payoff of not hedging, given firm 1 hedges. And also the expected payoff of not hedging is larger then the expected payoff of hedging, given firm 1 does not hedge. If both conditions hold, firm 2 will hedge its input cost risk provided firm 1 does so. The risk neutral incentive to hedge is then “passed on” from one firm to the other.
3.2 Firm Profit
Firms are Cournot competitors in the output market. Each firm observes the hedging choices of the other firm and learns the true level of input costs before deciding on it’s the level of production. Nash equilibrium requires each firm’s output choice to be a best response to the output choices of the other firm.
such that is has an expected profit equal to zero: ( ) 2 1 ) ; (c s c s
o h = h− . Because the option
will only be exercised in a high input costs state the total amounts firm i spends on a number kiof options, Hi, equals the difference between the exercise price s and the high
input price price ch multiplied by the chance of the particular state occurring and the
amount of options bought. The amount a firm i spends on options, Hi, is given by:
i h i h i c k s c s k H ( ) 2 1 ) , , ( = − (1)
The amount of options firm 1 buys, k , can not be chosen freely when it is assumed to be 1 financially constrained as it has to be large enough such that the firm does not go bankrupt in any possible state. Firm 2 chooses its level of hedging, k , freely and therefore also the 2
amount it spends on option contracts.
The distribution of input costs and the knowledge of the efficiency of the rival firms are assumed to be public information and is learned at date 0 by both firms. The firms then maximize their expected income. Using backward induction the expected net profit of the two firms can be calculated.
Firm 1 is forced to finance its project with debt which it has to repay at the end of the game, and it is forced to buy a fixed amount of option contracts hedging its risk exposure. The net profit of firm 1 under high input costs is given by equation (2). The firm earns the market price times the amount of output it sells. The inputs are priced against high costs, ch, but if
the option contracts bought at date 0 are exercised the firm receives the inputs for the lower exercise price. The firms profit is deduced with the amount spend on options, H1, the fixed costs, F , the and the debt repayment, I . Firm 1’s net profit under high input costs is given by equation (2): I F H x fc k s c f x x x s f c np h = − h − h h + h − − h h − − − 1 1 1 1 2 1 1( , , ) (1 ) ( ) (2)
In the low input costs state the option contract is not exercised because the spot price is lower then the exercise price. The net profit function of firm 1 becomes:
Firm 2’s profit differs from firm 1 profit in the following ways: the level of efficiency in transforming its inputs into the output, the amount of options it buys, and in the fact that it does not have to repay its bondholders. Apart from the amount of options bought and the bondholder repayment the net profit under high input costs is otherwise similar to net profit of firm 1 under high input costs:
F H x c sk s c x x x s k c np h = − h − h h + h − − h h − − 2 2 2 2 2 1 2 2( , , ) (1 ) ( ) (4)
The net profit of firm 2 under low costs is derived in the same way as equation (3) apart form the efficiency of transforming the inputs into the output:
F H x c x x x k c np l = − l − l l − l l − − 2 2 2 2 1 2 2( , ) (1 ) (5)
The fixed costs, F , payable at the end of the game at date 2 have such a level that firm 1 will be forced to hedge because it otherwise will not be able to pay its bondholders. The fixed costs are thus assumed to be higher then firm 1 profits if it does not hedge in the case of high costs (case 1.1). But lower then firm 1 profits in case of low costs and no hedging (case 2.1)3. l l l h h h h f fc c fc c c F 1,1.1 2 2 1,2.1 9 ) 2 1 ( 9 ) 2 1 ( ) , ( ≡π = − + < − + =π (6)
defining πi,state as the profit of firm i in one of the eight states depicted in figure 2 before subtracting fixed costs and repayments to bondholders. The level of fixed costs causes firm 1 to go bankrupt in the high cost state if it decides to not hedge.
If in a certain state one of the firms has far lower input costs then its rival, the firm will be able to produce so much that it becomes unprofitable for the rival, despite of the fixed costs, to produce at all. The firm with lower input costs will then gain a monopoly position on the output market. Because these large differences in costs will yield meaningless equilibria I assume that when input prices for firms differ they still have to be sufficiently “near” to each other. The options exercise price must be sufficiently close to the high input price such that firm 1 does not get a monopoly position in the high input cost state if firm 2 does not hedge. This situation can be prevented by assuming that the exercise price is sufficiently close to the high input price4. Equation (7) gives the this “nearness” assumption.
2 1 fs
ch ≤ + (7)
The low input costs are lower then the high input costs and the exercise price by definition lies in-between the two. Firm 1 produces its output f times less efficient then its rival firm 1. Combining these features with the “nearness” assumption stated above yields two conditions concerning the relative size of marginal costs for both firms:
1 2 1 0<cl < fcl ≤s< fs<ch ≤ + fs ≤ (8) 1 0<ch < fch ≤ (9)
4.
OPTION EXERCISE PRICE AND LEVEL OF DEBT
The option exercise price has to ensure that the inefficient firm, firm 1, is able to repay its debt in all possible states. In order to derive the level of the exercise price I first calculate the level of the external debt that is needed.
4.1 Level of debt
Firm 1 needs to borrow an amount from bondholders such that it is able to buy inputs and options in the production phase at date 0. Due to the fixed costs F faced by firm 1 at the end of the game the real option of not producing is costly and will be avoided. The firm will therefore be forced to hedge its input risk in order to be able to receive debt and produce. Provided firm 1 hedges its input risk it still does not know ex ante in which of the four feasible states it produces (states 1.4, 2.4, 1.3, and 2.3). The amount of debt the firm needs depends on the state it operates in. The amount that firm 1 needs in the state with asymmetric high costs (state 1.3) is equal to the amount spend on hedging and the costs of the inputs (the fixed costs do not have to be paid until state 2) . Equation (10) shows the size of the loan needed by firm 1 in state 1.3.
h
h f k s H fsx
I1..3( , 1, )= 1+ 1,1..3 (10)
with Istate the loan needed by firm 1 in a specific state. In the symmetric high cost state (state 1.4) the firm needs debt of the size equal to Ih
4 . 1 .
h
h f k s H fsx
I1.4( , 1, )= 1+ 1,1.4 (11)
In the two low cost states (state 2.2 and 2.3) firm 1 has an identical demand for funds because marginal cost in both states are equal and thus also the amounts produced.
l l l l l l l l c f k I c H fc x H fc x I2..3( , , 1)= 2.4( )= 1+ 1,2..3 = 1+ 1,2.4 (12) The firm does not ex ante know in which state it will produce and thus will need to borrow the larger amount of the three in order to be sure that it will have enough liquidity to produce the optimal amount of products. Assuming moral hazard to be no issue, the firm gets enough debt from its bondholder to finance its operations in all states, if the firm finds itself in a state in which less liquidity is needed it is assumed to pay the surplus back directly.
4.2
Option exercise price and feasibility
I first derive the exercise price for the option contract, making sure firm 1 profit in the state with the fiercest competition, state 1.4, is non-negative. I continue checking if the exercise price is feasible. I first derive the set on which the option exercise price has a positive value. After that I derive the set on which the option exercise value provides firm 1 with a positive profit in state 1.3 as well. I finish with confirming that the set of exercise prices satisfy the relative cost assumptions of equations (7) and (9).
4.2.1 The option exercise price
to satisfy the condition that that in both states firm 1 is able to repay its bondholders. Using the level of the loan from equation (11) I derive equation (13):
1 4 . 1 , 1 4 . 1 , 1 4 . 1 x F I fsx H p h − ≥ = h + (13)
with p1,1.4the price of the homogenous output. Substituting π1,1.4in equation (13) gives the level of profit needed to repay bondholders.
0 1 4 . 1 , 1 −F−H ≥ π (14)
Doing the same for firm 1 profit in state 1.3 yields equation (15). 0 1 13 , 1 −F−H ≥ π (15)
The profit of firm 1 in state 1.3 is larger then the profit of firm 1 in state 1.45. If the
exercise price is such that firm 1 does not make a loss in state 1.4, it will also do so in state
1.3. The option price ( )
2 1 ) ;
(c s c s
o h = h− depends on the difference between exercise price
s and spot price ch as does the total amount spend on buying option contracts,
1 H . h h h h s c s x ox c H1 ( ) 1,1.4 1,1.4 2 1 ) , ( = − = (16)
The income after subtracting fixed costs should at least be equal to the repayment to the bondholders in order for the firm to not get into financial distress. The firm will, once it has received the loan and bought the options, reach the production stage. It then maximizes its income6 by producing an amount equal to
3 2 1 ) , ( 14 ., 1 s fs s f xh = − + (17)
Using equation (6), (16) and (17) one can calculate the firm 1 state 1.4 net profit after the subtraction of fixed costs and the repayment of debt holders
3 2 1 2 ) ( 9 ) 2 1 ( 9 ) 2 1 ( 2 2 1 4 . 1 , 1 s fs s c c fc s fs H F− = − + − − h+ h − h − − + − π (18)
This net profit has to be at least equal to zero in order to repay debt holders. Solving equation (18) for s , yields two solutions:
h h c c s1( )= (19.1) 5 9 ) 2 1 ( 9 ) 2 1 ( 2 4 . 1 , 1 2 13 , 1 s fs c fs h − + = > + − = π π because ch >s
5 14 8 ) 7 8 2 8 8 ( ) , ( 2 2 2 − + + + + − − − = f f f c c fc f c f s h h h h (19.2)
The first solution does not provide any hedging opportunity as the exercise price is equal to the spot price. So I focus on the second solution dependent on the relative efficiency of firm 1 and the level of high input prices. Buying an amount of options equal to (17) with an exercise price equal to equation (19.2) ensures a non-negative profit in state 1.4 for firm 1. If the firm hedges its risk exposure it will thus be able to repay its bondholders.
4.2.2 A positive exercise price
The exercise price derived in equation (19.2) is not positive on the whole set on which f and ch are defined. Because a negative exercise price has no meaningful interpretation I
restrict the analysis to the set where the exercise price is positive. Figure 3 gives the set of combinations of chand f for which the exercise price is positive. By solving equation
(19.2) for f I find the boundary for which meaningful exercise prices are found. One binding constraint within the feasible set of chand f can be found:
2 ) 1 2 ( 2 7 4 − − = f f ch (20)
Equation (20) is depicted in figure 3, it provides a boundary between negative and non negative exercise prices.
Figure 3. Set of positive exercise prices of the
financial call option. The boundary between negative and non-negative exercise prices is depicted by equation (20).
Figure 3 shows the set of combinations of chand f on which the exercise price is
non-negative. The figure shows that the higher the difference in input efficiency (the level of f ) the lower the high input prices have to be to yield a positive exercise price. This can be explained from the spending on fixed costs and option contracts which are relatively low for a lower level of the high input costs, ch.
Another thing that stands out in the graph is the nonpositive value of the exercise price for levels of both chand f near respectively zero and one. The reason is that firm 1 pays
a price equal to: ( ) 2
1 c s
o= h − for the option contracts. With a chance equal to a half it
pays f(ch − less per option contract for its inputs then it would normally have to pay. s) Momentarily setting aside from possible effects from hedging on the industries output market, firm 1’s expected gain from buying an option contract is equal to equation (21):
(
)
( 1)( ) 2 1 ) ( ) ( 2 1 f ch −s − ch −s = f − ch −s (21)For levels of the multiplier f near 1 the exercise price has to become small or even negative in order for equation (21) to yield a high enough gain from buying options for firm 1 to be able to pay its fixed costs and bondholders. Providing firm 1 knows the amount of inputs it will need, it have an expected gain from buying that amount of options because f is defined to be equal or larger then one.
4.2.3
Positive firm 1 profit in state 1.3
The option exercise price found by equation (19.2) is such that the profit in state 1.4 is non-negative, I continue with checking whether the firm has positive profits if it fully hedges in state 1.3. Using the best reply functions of both firms one finds that if option contracts were in abundance firm 1 would produce more in state 1.3 then in state 1.47. The amount of
output maximizing profits for firm 1 is larger then the amount of options bought. If firm 1 has exercised all of its options and want to buys additional inputs it has to pay the high
input price, ch. Firm 1 will only buy additional options against the high input price if this
increases its profits increase by doing so. If firm 1 thus has a non-negative profit producing the amount of output allowed by the fixed number of options it will also have non-negative profit when it decides to produce more using inputs bought against the high input price. Two best reply functions of firm 2 in state 1.3 are found by maximizing firm 2 net profits and substituting the level of firm 1 production in its profit function. Assuming firm 1 does not buy additional high priced inputs above the amount that may be bought against the exercise price the net profit function is equal to equation (22).
F x c x s fs F x c x x s f c npv h = − h − h − h h − = − − + − h − h h − 13 , 2 13 , 2 13 , 2 13 , 2 4 . 1 , 1 3 . 1 , 2 ) 3 2 1 1 ( ) 1 ( ) , , ( (22) The amount produced by firm 2 is equal to:
6 3 2 2 ) , , ( 3 . 1 , 2 h h h c f s fs s c x = + − − . Using the
amount produced by firm 2, the net profit of firm 1 can now be derived8:
9 ) 2 1 ( 3 2 1 2 ) ( 3 2 1 6 3 2 4 4 ) , , ( 2 1 3 . 1 , 1 13 , 1 h h h h h h c fc s fs s c s fs fs c s fs F H s f c npv + − − + − − − + − − + − + = − − =π (23)
Solving firm 1’s net profit in state 1.3 for ch yields the conditions for which the net profit
is non-negative. Four solutions may be found of which equation (24) is found to be binding. ) 2 1 ( 1 f ch + − = (24)
Combining equations (20) and (24) yields a feasible set of chand f depicted in figure 4.
As seen as in figure 3 the option contract exercise prices are positive to the left of the equation (20). Figure 4 depicts equation (24), to the right of the equation the combination of chand f yield positive profit for firm 1 in state 1.3. A higher efficiency difference
between the efficient en inefficient firms (level of f ) can be neutralized by a higher level of high input costs, because firm 1 benefits form higher marginal costs for its rival
The combinations of chand f yielding a feasible set of exercise prices are thus enclosed
by equations (20) and (24), as depicted in figure 4.
Figure 4. The feasible set of combinations of chand f is bounded to the left by equation (20) and to the right by equation (24).
4.2.4 Relative cost assumptions
I have derived an equation for the level of the exercise price and have indicated for which combinations of ch and f the set yield meaningful results. The feasible set of
combinations of chand f yields only positive exercise prices that ensure non-negative
profits for firm 1 in states 1.3 and 1.4 if it buys a fixed number of options equal to the amount given by equation (17).
Equations (7) and (9) state conditions for the relative sizes of the input costs for the two firms which have to be satisfied for the model to be effective. I correct the feasible set of combinations of chand f enclosed by equations (20) and (24) so as to satisfying
Figure 5. Correcting the feasible set of ch and f for equation (7). The left graph depicts
the graph of equation (7). The solid coloured area in the right graph is the area of the feasible set for which the conditions holds.
Second, correcting the feasible set of combinations of ch and f for condition (9)
reduces the feasible set to the one depicted in the right graph of figure 6.
Figure 6. Correcting the feasible set of ch and f after the correction for equation (7) for
equation (9). The left graph depicts the graph of condition (9). The solid coloured area in the right graph is the area of the feasible set for which both conditions (7) and (9) hold.
5.
EXPECTED PAYOFFS
In the previous sections I have derived the fixed exercise and option contract prices as well as the amount of option contracts firm 1 is forced to buy by its bondholders. I also derived the feasible set of ch and f on which this all holds. Using these derivatives I
The game played by the firms is a static game in which players simultaneously decide on their level of output, no future exists to enforce threats or promises. One can thus assume that the equilibria found in this section are not influenced by any strategic behaviour but are fully the result of the interaction in hedging decisions which both firms observe before making their output decisions. The financial situation, and thus whether a firm needs external financing in order to produce, is assumed to be public information.
This section is consists of two parts, I first derive the payoffs for firm in the states 1.2, 2.2, 1.1 and 2.1 pretending that firm 1 has no incentive to hedge because it does not need external financing. By comparing the expected payoffs for firm 2 of hedging and not hedging I show that firm 2 has no incentive to hedge. In the second part I find the payoffs for firm 2 in states 1.4, 2.4, 1.3 and 2.3 for which firm 1 is financially constrained and does hedge its risk exposure. I show that the hedging incentive is passed on from firm 1 to firm 2. Note that I derive all equilibria on the feasible set of combination of chand f and disregard all solutions that fall outside this set as these do not concur with rationality restrictions.
5.1 Firm 1 does not hedge
If firm 1 has no need for external debt and does not hedge its input costs risk, four different states are possible. I here derive only firm 2 payoffs as I am analysing its hedging incentives.
Lemma 1: If firm 1 does not hedge its input costs, firm 2 has
no risk neutral incentives to hedge its input risk.
5.1.1 No hedging by firm 2
State 1.1 High input costs
Under high input costs and no hedging by both firms the quantities produced by both firms can be found by setting equal both firms best reply functions. The quantities produced by firm 1 and 2 are respectively given by equation (25) and (26).
3
2
1
1 h h hfc
c
x
=
−
+
(25)3
2
1
2 h h hc
fc
x
=
−
+
(26)Using the net profit function of firm 2 under high input costs given by equation (4) and with the help of equations (25) and (26) the net profit of firm 2 can be calculated:
9 ) 2 1 ( 9 ) 2 1 ( ) 1 ( ) , , ( 1 2 2 2 2 1 . 1 , 2 h h h h h h h h h f s x x c x F c fc fc c c np = − − − − = − + − − + (27)
State 2.1 Low input costs
For low input costs and no hedging by both firms the net profit of firm 2 can be derived. The best quantities produced can be found in the same way as mentioned above. The net profit of firm 2 in state 2.1 is equal to:
9
)
2
1
(
9
)
2
1
(
)
1
(
)
,
(
2 2 2 2 2 1 1 . 2 , 2 h h l l l l l l l hc
fc
c
fc
F
x
c
x
x
x
f
c
np
+
−
−
−
+
=
−
−
−
−
=
(28)Expected firm 2 profit when both firms do not hedge
+
−
−
−
+
⋅
+
+
−
−
+
−
⋅
=
⋅
=
9
)
2
1
(
9
)
2
1
(
2
1
9
)
2
1
(
9
)
2
1
(
2
1
))
,
,
(
(
2 2 2 2 2 h h l l h h h h h lc
fc
c
fc
c
fc
fc
c
payoff
prob
s
c
c
np
E
(29)5.1.2
Firm 2 hedges
The expected payoff for firm 2 if it does not hedge, found by equation (29), now will be compared with the expected payoff if firm 2 does hedge.
State 1.2 High input costs
I check whether firm 2 has any incentive to buy options and thus if it prefers hedging to not hedging if firm 1 does not hedge. I first check if firm 2 has incentives to buy an amount of options less then the amount of inputs it would “normally” (when it has bought no options) use to produce the output. Buying less options then the amount the firm would “normally” use will just give the firm a cost of ( )
2 1
s
ch − in both states and a profit of (ch − in the high state s)
with a probability equal to a half. Assuming infinite small transaction costs keeps firm 2 from buying less options then it would normally produce. Note that when firm 2 buys less options then it would normally produce there will be no change in the marginal costs and thus in the amounts produced.
Maximizing firm 2’s net profit assuming its marginal cost to be equal to the option exercise price yields firm 2’s best reply functions:
2 2 1 2 1 1 2 1 2 1 2 1 0 2 1 2 1 2 1 1 2 2 1 2 1 x c s x s c x x s c x x h h h h h h h h h − − − = − − − = → = − − − − (31)
Setting the appropriate best reply functions found in equations (30) and (31) equal to each others yields the level of production of both firms:
3 2 1 2 1 2 1 1 s c fc x h h h = − + + ,
3
1
2 h h hc
s
fc
x
=
−
−
+
(32)Using the quantities both firms produce and the net profit function equal to equation (4) the net profit function of firm can be derived9.
9 ) 2 1 ( 6 ) )( 1 ( 3 1 3 2 1 2 2 1 1 ) , , ( 2 2 . 1 , 2 h h h h h h h h h h f s fc c s c s fc c s fc c s fc c c np = + + − − − + − − − + − − − + (33) In state 1.2 firm 2 will thus not buy an amount of option contracts less then the amount of inputs it would “normally” buy because the assumed infinitely small transaction costs. If firm 2 would buy more option contracts then it the amount of inputs it would “normally” buy, it has a payoff equal to (33). To find the expected firm 2 profit is firm 2 hedges and firm 1 does not, I continue with deriving the profit of firm 2 of hedging when it finds itself in the low costs state (state 2.2).
State 2.2 Low input costs
Under low input costs and if firm 1 does not hedge but firm 2 does (state 2.2), the marginal costs and benefits do not differ from low costs without hedging (state 2.1, derived above). So the amounts produced will be the same. The difference between the profit of
firm 2 in the states will be that the profit will be decreased with the amount spend on options when it has decided to hedge.
Expected firm 2 profit when only firm 2 hedges
By multiplying the probability of both the high and the low input states by the expected payoff of firm 2 in these states the expected profit of firm is derived. Using (28) and (33) the expected profit of firm 2 is found. If it is the only firm to hedge its profit is positive on the whole feasible set, as can be seen in figure A2 in the appendix G. Equation (34) gives the level of firm 2 expected profit.
+ − − − + − − − + − − − + + ⋅ + − + − − − + − − + − ⋅ = ⋅ = 9 ) 2 1 ( 6 ) )( 1 ( 3 1 3 2 1 2 2 1 1 2 1 6 ) )( 1 ( 9 ) 2 1 ( 9 ) 2 1 ( 2 1 )) , ( ( 2 2 2 2 h h h h h h h h h h h h h h h h h c fc s c fc s c fc s c s c fc s c fc s c c fc fc c payoff prob s c np E (34)
I compare firm 2 expected profits from hedging and not hedging(equation (29) and (34)) in order to find if firm 2 prefers hedging its input cost risk to not hedging if firm 1 does not hedge. Figure 7 shows both the feasible set of combinations of chand f as the area
for which firm 2 prefers hedging to not hedging. The area for which firm 2 prefers hedging to not hedging lies completely outside of the feasible set I conclude that firm 2 has no incentive to hedge if firm 1 also not hedges. This
proves lemma 1.
Figure7. Area where firm 2 prefers hedging to not hedging
5.2 Firm 1 hedges
In this part of the analysis firm 1 hedges its input costs and firm 2 chooses whether to hedge or not, there are four different states in which firm 1 hedges its input costs.
Lemma 2: If firm 1 hedges its input costs firm 2 has a risk
neutral incentive to hedge its input risk.
Below I show that lemma 2 holds and that firm 2 prefers hedging its input risk to not hedging if firm 1 hedges because its expected profit of hedging is higher if it hedges. Firm 2 thus prefers the expected payoff of states 1.4 and 2.4 to the payoffs of states 1.3 and 2.3.
5.2.1 No hedging by firm 2
State 1.3 High input costs
In order to find the net profit for firm 2 under high input costs when firm 1 hedges and firm 2 does not, I need to know how many options firm 1 exercises and if it buys any additional high priced inputs to produce more output.
Lemma 3: If only firm 1 has hedged its input risk and bought options at date 0 it will
exercise all its option at date 1 in order to transform inputs to produce output. Firm 1 will not buy additional high priced inputs to produce more output.
Proof. See appendix G
2 1 2 1 0 2 1 1 2 2 1 2 1 h h h h h h h h h c x x c x x c x x − − = − − = → = − − − (35)
Setting the best reply function equal to the amount firm 1 will produce yields the amount firm 2 produces in state 1.3.
6 3 2 2 3 2 1 2 1 2 2 1 1 h h h h h c s fs x s fs c x k x − − + = + − = − − ↔ = (36)
Substituting the amount produced by firm 2 yield its profit function.
9 ) 2 1 ( 6 3 2 2 ) 6 3 2 2 3 2 1 1 ( ) , ( 2 3 . 1 , 2 h h h h h h c fc c s fs c c s fs s fs s c np + − − − − + − − − + − + − − = (37)
State 2.3 Low input costs
Under low input costs and if firm 1 hedges but firm 2 does not, the marginal costs and benefits do not differ from the other low costs states. Firm 2 profit is easily derived10
9 ) 2 1 ( 9 ) 2 1 ( ) 1 ( ) , ( 2 2 2 2 2 1 3 . 2 , 2 h h l l l l l l l h l c fc c fc F x c x x x c c np + − − − + = − − − − = (38)
Expected profit across states if only firm 1 hedging,
Using the expected firm 2 net profit in states 1.3 and 2.3 (equations (37) and (38)) yields the expected profit if firm 2 decides not to hedge
5.2.2 Firm 2 hedges
State 1.4 High input costs
In order to see if firm 2 prefers to hedge to not hedging, I first check if firm 2 has incentives to buy options by deriving the optimal level of hedging. I substitute the amount of options firm 1 buys in its profit function
Lemma 4: If both firms hedged their input risk and bought options at date 0; firm 1 will
exercise all its option at date 1 in order to transform buy inputs to produce output. Firm 1 will not buy additional high priced inputs to produce more output.
Proof. See appendix J
Using the result from lemma 4 that firm 1 buys exactly the amount of input as it owns options I can derive the firm 2 net profit function.
9 ) 2 1 ( ) 2 1 2 1 3 2 1 1 ( ) ( 2 1 ) 1 ( ) , ( 2 2 2 2 2 2 1 4 . 1 , 2 h h h h h h h h h h h c fc x s c x s fs F s c x x s x x s c np + − − − − − + − − = − − − − − − = (40)
The best reply function is easily derived.
s c s fs xh h 4 1 4 1 6 2 2 2 − − − + = (41)
Substituting the best reply function in the net profit function yields the net profit of firm 2.
9 ) 2 1 ( 4 1 4 1 6 2 2 4 1 1 4 3 6 2 2 3 2 1 1 9 ) 2 1 ( 4 1 4 1 6 2 2 2 1 2 1 1 4 1 4 1 6 2 2 3 2 1 1 ) , ( 2 2 2 h h h h h h h h h h c fc s c s fs c s s fs gs fs c fc s c s fs s c s c s fs gs fs s c np + − − − − − + − + − + − + − − = + − − − − − + + − − − − + − + − − = (42)
The amount spend on hedging is derived straightforwardly.
State 2.4 Low input costs
If both firms hedge the marginal costs in the low costs state will be the same as in all other low input states. The quantities produced will thus be the same as well; the profits differ with the spending on options done a date 0.
Expected profit across states if both firms hedge
Firm 2 expected net profits if it hedges can be calculated using the firm 2 net profit functions for states 1.4 and 2.4. (using equations (38),(42) and (43)).
− − − − + − + − − − + ⋅ + + − − − − − + − + − + − + − − ⋅ = ⋅ = 2 ) ( 4 1 4 1 6 2 2 9 ) 2 1 ( 9 ) 2 1 ( 2 1 9 ) 2 1 ( 4 1 4 1 6 2 2 4 1 1 4 3 6 2 2 3 2 1 1 2 1 )) , , ( ( 2 2 2 2 s c s c s fs c fc c fc c fc s c s fs c s s fs s fs payoff prob s c c np E h h h h l l h h h h h l (44)
Figure 8 depicts the area where the difference in expected profit for firm 2 between not hedging (equation (39)) and hedging (equation (44)) is positive, given that firm 1 hedges. Because on the entire feasible set of combination of chand f the difference in expected
profit for firm 2 between not hedging is positive the expected profit of hedging is larger then not hedging. Firm 2 thus has an incentive to hedge its input cost risk because firm 1 hedges its input risk. A firm’s need to hedge thus increases with the extent of hedging in the industry, proving lemma 2.
Figure 8. Positive differences in firm 2 expected
Proposition 1: Using lemma’s 1 and 2 it can be proven that the risk neutral incentive of
firm 1 to hedge will be passed on to firm 2, inducing it to hedge as well.
Proof. Proposition 1
Lemma 1: If firm 1 does not hedge its input costs firm 2 has no risk neutral incentive to
hedge its input risk.
Lemma 2: If firm 1 hedges its input costs firm 2 has a risk neutral incentive to hedge its
input risk.
From lemma’s 1 and 2 it clear that a firm that at first does not have a risk neutral incentive to hedge its risk gets a risk neutral incentive by its rival firm engaging in hedging. A firms need to hedge thus increase with the extent of hedging in the industry.
6. EMPIRICAL IMPLICATIONS
My model is motivated by a couple of observations regarding academic interest for firm’s hedging decisions:
- Academic focus regarding hedging has mainly on demand (price) uncertainty, and little attention has been given to cost uncertainty.
- Models in corporate risk management usually analyze the hedging decision of a firm in isolation of other firms and ignore possible interaction11.
- If hedging decisions are modelled explicitly they are generally modelled as a firm having the option to buy derivatives with state independent payoffs like forwards. State dependent derivatives like options are disregarded.
Motivated by these observations, I have analysed how hedging incentives of individual firms can change the output market of a complete industry, such that the whole industry is forced to hedge. The model’s results depends on several assumptions, first the assumptions of imperfect competition. If firms are small and numerous, no production efficiency differential will be able to exist and thus the risk neutral incentive identified
11 A recent empirical paper by Nain (2004) underlines the relevance of this observation as he finds that
for the first firm to engage in hedging will be nonexistent, leading to no incentive at all to hedge risk. Therefore, my results will be primarily of use in sectors where a few significant players competing head to head. My model predicts that a large degree of homogeneity should be observed in hedging decisions in these industries.
Another assumption essential to the model is the assumption regarding the inefficient firm using more inputs to produce the same amount of output as the efficient firm. This assumption leads to an asymmetric risk exposure, distinguishing the model from models based on demand uncertainty
The degree of hedging in an industry has a real impact on the output market equations (45) and (46) give the industry output of in respectively states 1.4 and 1.3 and states 1.2 and 1.1. 12 3 4 8 ) , ( 4 . 1 h h s fs s c c Q = − − − > 12 6 2 4 8 ) , ( 3 .. 1 h h s fs s c c Q = − + − (45) 12 2 2 4 8 ) , ( 2 . 1 s c fc s c Q h = − h − h − > 12 4 4 8 ) , ( 1 . 1 h h h s fc c c Q = − − (46)
Because equation (47) holds one can conclude that industry output rises with the degree of hedging and because prices are formed in a linear fashion prices too decrease with the degree of hedging. Note that the industry output in the low input costs states does stays the same as marginal prices do not change.
0 12 ) )( 1 4 ( 2 . 1 3 . 1 > − − = −Q f c s Q h (47) The result that hedging using financial call options increases industry competition by raising quantities produced and thus lowering prices in consistent with the findings of Willems (2005) and in line with results of Allaz and Vila (1993) .
In trying to relate the results of the above-mentioned studies to the implications of my model on should keep in mind the fundamental differences in assumptions between the studies. The papers mentioned do not explicitly model a hedging instrument all papers use forward contracts as hedging instrument. The question remains if hedging using forward contracts gives significantly different results as hedging using call option contracts does. Both types of instruments can in theory eliminate the financial distress costs for the lender, leaving the upside surplus in case of advantages price movements the only difference for the firm.
7. CONCLUSION
The degree of hedging affects the extent of industry rivalry, making an industry more competitive as more firms hedge their input risk. Because the degree of hedging in an industry has a real impact on the output market firms hedging decisions can not be considered in isolation from its industry. In line with recent research my results shows that a firm’s incentive to hedge depends partially on the hedging choices of other firms in the same industry. My results show that in imperfectly competitive markets, the
incentives of an individual firm to hedge increases as more firms hedge, and decrease as more firms choose not to hedge. This rationale for firms to hedge their risk is not
consistent with a risk-neutral setting where firms use hedging as tool with which to distinguish themselves from their competitor as some recent studies have argued.
APPENDIX A
Figure A1 depicts the effect of hedging on expected costs of financial distress. If a firm does not hedge, the bondholder has positive expected costs of financial distress E[C1] (the level of expected cost of financial distress with an expected level of income of
] 1 [Y
E ). Income equal or higher then a cut-off point brings along a zero expected cost of financial distress If the firm decides to hedge and it make sure income is higher then a cut-off point, depicted as point Ys1, in the graph. Hedging brings down the expected costs of financial distress to a level of zero.
Figure A1: The effect of hedging on expected costs of financial distress
With
Y1,h = Firm 1’s income in the high cost state Ys = Firm 1’s income with both firms fully hedging
Y1,l = Firm 1’s income in the low cost state E[Y1] = The expected income
I = Face value of debt at maturity E[C1] = Firm 1’s expected costs of financial
distress without hedging
APPENDIX B
firms maximize their net profit function to yield to best response functions. Firm 1 maximizes its net profit function to x1 yielding its best response function.
(
)
h h h h h h h h h h h h h fc x x fc x x fc x x I F x fc x x np − − = − − = → = − − − − − − − − = 1 2 2 1 2 1 1 2 1 1 . 1 , 1 2 1 2 1 0 2 1 ) 1 ( max ) max( (A1)The best response functions of firm 2 are derived in the same way.
(
)
2 1 2 1 0 2 1 ) 1 ( max ) max( 1 2 2 1 2 1 2 2 1 1 . 1 , 2 h h h h h h h h h h h h h c x x c x x c x x F x c x x np − − = − − = → = − − − − − − − = (A2)The intersection of the best reply functions yield the quantities both firms will produce:
3 2 1 , 3 2 1 2 1 h h h h h h fc c x c fc x = − + = − + (A3)
The profit functions for both firms are easily derived:
9 ) 2 1 ( , 9 ) 2 1 ( 2 1 . 1 , 2 2 1 . 1 , 1 h h h h h h = − fc +c
π
= − c + fcπ
(A4)The fixed costs are assumed to be smaller the firm profit in state 2.1. The derivation of firm 1 profit in state 2.1 is done in an identical way as above and yields:
3 2 1 , 3 2 1 2 1 l l l l l l fc c x c fc x = − + = − + (A5) 9 ) 2 1 ( , 9 ) 2 1 ( 2 1 . 2 , 2 2 1 . 2 , 1 l l l l l l = − fc +c π = − fc +c π (A6)
The fixed cost are assumed to be equal to firm 1 state 1.1 profit which is by definition smaller then firm 1 state 2.1 profit because of the size of the input costs.
APPENDIX C
Proof “Nearness” assumption
nonsustainable in order for the assumption to hold which only holds if the quantity
produced by firm 2 is positive. Maximizing the net profit functions of both firms, yields the best reply functions (A7) and (A8):
2 1 2 1 fs x xh = − h − (A7) h h h x c x1 =1−2 2 − (A8)
Setting the reaction functions equal to each other gives the amount produced by firm 2
0 3 2 1 2 > + − = c fs xh h (A9)
Function (A9) has to be a positive value for a monopoly not to occur. The numerator thus can not be smaller then zero which provides the exercise price with a condition that it has to be sufficiently close to the high input price.
2 1 fs
c
fs≤ h ≤ + (7)
If equation (7) holds, firm 1 does not get the chance to become a monopolist.
APPENDIX D
I assume the competition for firm 1 to be as heavy as can be. For convenience firm 2 is assumed to have as much options at its disposal as it needs. The marginal cost of producing for both firms then is equal to the exercise price s . Maximize both profit functions yields the best reply functions, intersecting those gives the quantities produced. The profit of firm 1 in state 1.4 can be calculated.
9 ) 2 1 ( and 3 2 1 , 3 2 1 2 4 . 1 , 1 4 . 1 , 2 4 . 1 , 1 s fs fs s x s fs xh = − + h = − +
π
h = − +APPENDIX E
I show here that that if option contracts are in abundance firm 1 will produce more in state 1.3 then in state 1.4. I calculate the amount firm 1 produces in state 1.3 if it has as many options at its disposal as it needs. Its marginal cost then are equal to the exercise price s , firm 2 does not hedge and thus has marginal costs equal to high input costsch.
I F H x fs x x np = − h − h − h +− − − 1 1 2 1 3 . 1 , 1 (1 ) F x c x x np = − h − h − h h − 2 2 1 3 . 1 , 2 (1 )
Quantities produced are easily derived
3 2 1 , 3 2 1 3 . 1 , 2 3 . 1 , 1 fs c x c fs xh = − + h h = − h+ (A11)
The quantity produced by firm 1 in state 1.4 is stated by equation (17):
3 2 1 ) , ( 14 ., 1 s fs s f xh = − + (17)
Because ch > the quantity produced by firms 1 in state 1.3 is clearly higher then in state s
1.4: ) , ( 3 2 1 3 2 1 14 ., 1 3 . 1 , 1 x f s s fs c fs xh = − + h > − + = h (A12)
APPENDIX F
I derive the profit of firm 1 in state 1.3. Maximizing firm 2 state 1.3 profit yielding the quantity produced
F
x
c
x
s
fs
F
x
c
x
x
npv
=
−
h−
h−
h h−
=
−
−
+
−
h−
h h−
3 . 1 , 2 3 . 1 , 2 3 . 1 , 2 3 . 1 , 2 3 . 1 , 1 3 . 1 , 2)
3
2
1
1
(
)
1
(
(A13) 6 3 2 2 3 . 1 , 2 h h fs s c x = + − −3 2 1 6 3 2 4 4 ) 6 3 2 2 3 2 1 1 ( ) 1 ( ) ( 1.3 1,1.3 1,1.3 2,1.3 1,1.3 1,1.3 3 . 1 , 1 s fs fs c s fs x fs c s fs s fs x fs x x x fs p h h h h h h h h h + − − + − + = − − − + − + − − = − − − = − = π (A14)
APPENDIX G
Lemma 3: If only firm 1 has hedged its input risk and bought options at date 0 it will
exercise all its option at date 1 in order to transform buy inputs to produce output. Firm 1 will not buy additional high priced inputs to produce more output.
Proof Lemma 3
I begin with analysing if firm 1 exercises all of its options bought at date 0 in state 1.3, the number of options bought being equal tok1.
3 2 1 1 s fs k = − +
By deriving the best reply functions of both firm and next the quantities produced I will be able to see whether firm 1 will want to exercise all of its options.
fs x x fs x x fs x x h h h h h h − − = − − = → = − − − 1 2 2 1 2 1 2 1 2 1 0 2 1 9 ) 2 1 ( ) ( 2 1 3 2 1 ) 1 ( 2 1 2 1 3 . 1 , 1 h h h h h h x fs x fs s c s fc c x np = − − − − − + − − − + (A15) F x c x x np = − h − h − h h − 2 2 1 3 . 1 , 2 (1 ) 2 1 2 1 0 2 1 1 2 2 1 2 1 h h h h h h h h h c x x c x x c x x − − = − − = → = − − − (A16)
