Economica (2004) 71, 141–153
Firm’s Output under Uncertainty and
Asymmetric Taxation
By ITZHAK ZILCHAw and RAFAEL ELDORz
wThe Berglas School of Economics, Tel Aviv University
Hertzeliya, Israel
zInterdisciplinary Center,
Final version received 30 July 2002.
We consider competitive firms operating under price uncertainty when taxation is asymmetric
(i.e. profits are taxed at a higher rate than losses are compensated). In the absence of risk
sharing tools, the larger ‘gap’ in taxation may either lower or increase the firm’s optimal
output, depending on whether the relative measure of risk aversion is less than or larger than 1.
In the presence of risk sharing markets optimal output does not depend on the tax rates, nor
on the price distribution or the firm’s attitude towards risk. The firm will engage in hedging
and, as a result, will lower its expected taxes.
INTRODUCTION
The effect of price uncertainty on the operation of competitive firms has been
studied extensively in the literature. One strand of this literature studies the role
of financial markets, i.e. the engagement of firms operating under uncertainty
in hedging, and their effect on the production decision (see e.g. Holthausen
1979; Feder et al. 1980; Moschini and Lapan 1992; Broll et al. 1999). Another
strand of the literature studies the impact of taxation on a firm’s optimal
activities in the presence of uncertainty (see e.g. Domar and Musgrave 1944;
Stiglitz 1969; Sandmo 1971; Smith and Stulz 1985; Green and Talmor 1985; De
Marzo and Duffie 1995; Faig and Shum 1999). This work integrates these two
strands of the literature and studies the impact of taxation (when the tax is
asymmetric) on production, with and without access to futures/forward
markets.
Upon examining the corporate tax codes of many countries, we see a clear
asymmetry between the taxation of profits and the compensation for losses (see
e.g. Corporate Taxes: A Worldwide Summary 1997; Auerbach 1983, 1986;
Altshuler and Auerbach 1990). Considering a single year in isolation, we find in
most countries a clear gap between tax rates on profits and losses; i.e. losses
are ‘compensated’ at a lower rates than profits are taxed by the tax authorities.
This asymmetry in taxation is not mitigated significantly even when we
consider a multiperiod framework (see Section IV). We claim here that ‘tax
asymmetry’ has a significant effect on firm’s optimal production level and its
market value in the case where futures markets (or option markets) do not
exist. Studies regarding firms’ investments in risky assets under taxation has
begun with the seminal paper of Domar and Musgrave (1944/1994), followed
by Stiglitz’s analysis (1969), which incorporated risk aversion explicitly. Green
and Talmor (1985) describe tax liabilities as the negative position of a call
option; that is, the government owns a call option on the profits of the firm.
This emanates from asymmetry in taxation and the lack of hedging tools
available to the firm.
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When hedging tools exist, one of the main argument supporting firms’
engagement in hedging mentioned in the literature is that, by using hedging, the
firm lowers its average tax and hence raises its value (see, Smith and Stulz 1985;
Froot et al. 1993; Graham and Rogers 2000b). We demonstrate that under
asymmetric tax such risk sharing tools play an important role in determining
optimal production. This provides another argument supporting firms’ hedging
phenomena (see e.g. Mian 1996), where various explanations of hedging by
corporations are offered (see also Pirrong 1995).
Assuming that firms operate under price uncertainty, we shall concentrate
here on the effects of asymmetric taxation on optimal output and hedging
behaviour. First we examine the ‘benchmark case’, called the symmetric tax
case, where it is assumed that the tax rate imposed on profits and the subsidy
rate for losses are equal. In the case of tax asymmetry, we show the following.
(a) When risk sharing tools are not available, increasing asymmetry in taxation
lowers the optimal output if the relative measure of risk aversion is less than 1,
but raises the output if it is larger than 1. This result differs, in some cases, from
the related results discussed in the literature in various models (see e.g. Stiglitz
1969; Green and Talmor 1985). (b) The ‘separation property’ is generalized to
the asymmetric tax case: when forward/futures markets for this good exist, the
optimal production level does not depend on the tax rate, nor on the price
distribution or firm’s attitude towards risk. (c) Compared with the ‘symmetric
tax’ case, the optimal hedging, under tax asymmetry, reduces the variability of
the profits.
The paper is organized as follows. Section I considers a ‘benchmark case’:
competitive firms facing price uncertainty under symmetric taxation with
futures market. The impact of asymmetric taxation on optimal production is
analysed in Section II. Section III considers the role of forward/futures markets
when taxation is asymmetric. Section IV discusses our framework in the
multiperiod case. Section V contains some concluding remarks. Proofs are
relegated to the Appendix.
I. THE BENCHMARK CASE: TAX SYMMETRY
WITH
FUTURES MARKETS
Consider a price-taking firm which produces a commodity under random price
P~. The firm determines its output Q at time 0, production is completed at time
1, where the spot price P~ is realized and transactions take place. The firm’s
technology of production gives rise to a cost function C(Q) which satisfies:
C(0) ¼ 0, C 0 (Q)40 and C00 (Q)X0. We assume that the firm is risk-averse with
a von Neumann–Morgenstern utility function defined on profits U(p), where
U(0) ¼ 0, U 0 40 and U00 o0. Thus, the firm maximizes expected utility of net
profits. In this section we assume that the same tax rate t applies to both profits
~ stands for the random
and losses; i.e. the firm’s net profits are ð1 tÞ~
p; where p
profits. This is the case considered by Sandmo (1971) where risk-sharing
activities such as futures markets were not available.
Assume that the firm has access at date 0 to forward/futures market (for
delivery at date 1) for the good being produced, and denote by Pf the forward/
futures price. We assume in the following that there are no transaction costs.
The firm can hedge its profits’ risk by selling or buying forward X units of this
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good at date 0 (hence X can be positive or negative). The firm chooses its
production level Q and its hedging level X in a way that maximizes its expected
utility of net profits; i.e.
pðQ; XÞ
max EU ½ð1 tÞ~
Q;X
~ðQ; XÞ ¼ P~Q þ ðPf P~ÞX CðQÞ:
s:t: p
Necessary and sufficient conditions for optimum (Qn,Xns ) are
pðQn ; Xsn Þ ¼ 0;
ð1Þ
E P~ C 0 ðQn Þ U 0 ð1 tÞ~
ð2Þ
E ðPf P~ÞU 0 ð1 tÞ~
pðQn ; Xsn Þ ¼ 0:
From (1) and (2) we can derive a known result that under symmetric tax the
‘separation property’ holds; i.e.
Theorem 1 (Separation with Symmetric Tax). When the tax rate is
symmetric the separation theorem holds; i.e. in the presence of futures market
the firm’s output is determined by: C 0 (Qn) ¼ Pf. In particular, Qn is independent of the tax (subsidy) rate.
Now let us show that the optimal hedging behaviour obtained in the no-tax
case can be generalized to the symmetric tax case as well. Denote by Xns the
optimal hedging policy in the symmetric tax case. Then from (2) we derive
pðQn ; Xsn Þ ¼ Cov½P~; U 0 ðð1 tÞ~
pÞ:
EðPf P~ÞEU 0 ð1 tÞ~
Therefore, using the same arguments as the no-tax case (see e.g. Kawai and
Zilcha 1986), one obtains
ð3Þ
Xsn xQn , Pf xE P~:
Note however that, even though Qn is independent of the attitude towards
risk and the distribution of the random price (with futures markets), the
optimal hedging policy Xns depends on both, as well as on the tax rate t. Let us
consider now the behaviour of Xns as a function of the tax rate t. According to
modern finance theory (see e.g. Black 1976) in most cases examined empirically
the futures price embodies a non-negative risk premium, i.e. Pf 4E P~:
However, under unbiased futures market , namely when Pf ¼ E P~, the hedging
level equals the total output (this can be verified easily from equation (3)),
hence it does not depend on the tax rate. Thus, let us consider now the case
where Pf oE P~. Denote by Rr(p) the relative measure of risk aversion for this
firm.
Proposition 1. Assume that Pf oE P~: The behaviour of the optimal hedging
policy Xsn ðtÞ as a function of the (symmetric) tax rate t depends on the
monotonicity of Rr(p) as follows:
ð4Þ
8
o0 if Rr ðpÞ is increasing
@Xsn ðtÞ <
¼ 0 if Rr ðpÞ ¼ constant
@t :
40 if Rr ðpÞ is decreasing
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The economic rational underlying Proposition 1 is as follows. Under
symmetric tax 1 t multiplies the profits, hence any change in 1 t varies the
riskiness in a proportional manner. Increasing the tax rate t lowers the expected
profits, but also lowers its variance; thus the resulting hedging policy depends
on the measure of relative risk aversion (owing to the multiplicative nature of
1 t). Assume that the relative risk aversion is increasing; this implies more
aversion to riskiness. Since higher t results in lower variance, in this case the
exposure to risk, given by Qn–Xns , increases. For constant relative risk aversion
the two opposing effects are balanced, hence Qn–Xns remains unchanged as t
increases.
This result, concerning the behaviour of the optimal hedging policy in
relation to the monotonicity of the relative measure of risk aversion, is
reminiscent of the result obtained by Sandmo (1971), assuming symmetric
taxation, relating the optimal output (with respect to the tax rate t) to this
measure (of course, in the absence of hedging markets). Sandmo has shown
that when no futures markets exist the optimal output Qn(t) increases in t if
Rr(p) is increasing, and decreases in t if Rr(p) is decreasing. However, in the
presence of futures markets Qn is independent of the tax rate t (and the utility
function as well!) while the firm’s optimal hedging behaviour, Xns (t), is reversed
as shown in (4).
In the next section we examine the production decision under the more
realistic assumption of tax asymmetry.
II. TAX ASYMMETRY: NO FUTURES MARKETS
Consider now the behaviour of this competitive firm under asymmetric
taxation. We examine first the case where risk-sharing markets are absent.
Denote by t1 the tax rate on profits and by t2 the subsidy rate on losses. Even
though we consider here a one-shot decision problem, we claim that in a
multiperiod framework of our model the asymmetry in taxation still exists in
most countries. (See our discussion in Section IV; see also Corporate Taxes:
Worldwide Summary, 1997). However, risk sharing financial tools, used to
lower expected tax liability (see Section III), exist only in the more developed
countries. To simplify our notations we shall take the tax rates to be t1 ¼ t40
and t2 ¼ 0; however, all our results hold whenever the subsidy rate is lower
than the tax rate applied to profits, i.e. for any t14t2. The firm chooses its
~n
production Q in a way that maximizes its expected utility of the net profit p
(the gross profits minus the tax), namely
ð5Þ
~n max EU ½p
Q
(
~n ðQÞ ¼
s:t: p
ð1 tÞ½P~Q CðQÞ
P~Q CðQÞ
~ðQÞX0
if p
~ðQÞo0:
if p
Denote by Qn the optimal output for this asymmetric tax case. Note that
~n ðQÞ ¼ ð1 tÞ~
under symmetric tax, i.e. when p
pðQÞ; the optimal output
depends on the tax rate t. Clearly, the same phenomenon occurs here, where t
represents the tax on profits while we keep the subsidy level at t2 ¼ 0. Let us
show first that, in the absence of risk sharing tools, moving from symmetric to
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asymmetric taxation lowers the firms optimal output. In particular, our case
differs from the symmetric tax case (see Sandmo 1971): the decline in output
holds if the relative measure of risk aversion Rr ðpÞ is less or equal to 1.
Claim 1. Assume that Rr ðpÞ41. In the absence of risk sharing markets, the
firm’s optimal output in the asymmetric tax case is lower than its output under
symmetric tax.
Thus, under tax asymmetry, in the absence of risk sharing arrangements,
the firm attempts to avoid states with losses since it is penalized by the
government in such states; that is, the government does not participate in these
loses as it does with profits. Thus, under asymmetry in taxation the firm
reduces its expected losses by choosing its marginal cost C 0 (Q) to be lower than
the expected price E P~. In this way the firm lowers its pre-tax profits as well. In
fact, it equates the after-tax profits with the net loss by decreasing its output,
thereby increasing ex ante its profit margin, creating ‘partial self-hedge’. In the
next section, when we introduce risk sharing tools, this will not be the case.
Note that our proof applies whenever the relative measure of risk aversion is
less or equal to 1.
Green and Talmor (1985) describe risk-neutral firm with multiplicative
shocks to the production function and show that, in the absence of futures
markets, tax liability (described as a negative position of a call option) creates
an incentive to underinvestment in the more risky project. In our framework,
where the firm faces random price, we also find that under certain conditions
(i.e. when the relative measure of risk aversion Rr ðpÞ is less than 1) optimal
output declines as tax asymmetry increases. In this case, such behaviour
demonstrates the incentive of the firm to generate a larger gap between
marginal cost and expected price which creates a ‘safety margin’ at the
optimum, i.e. some type of a hedging tool.
Under risk neutrality, tax asymmetry creates concavity in the profit
function (see Smith and Stulz 1985), hence the ‘risk aversion case’ result applies
and production declines (see Eldor and Zilcha 2002). By decreasing its output
the firm increases ex ante its profit margin, creating some ‘partial selfinsurance’. Higher output may increase the expected gross (i.e., before-tax)
profits, but in this case the net (i.e. after-tax) profits decline because of the
higher tax rate on positive profits.
Let us consider now the effect of the ‘gap’ in tax rates (i.e. t) on the optimal
output. Denote by Qn(t) the optimal output of the firm for a given t, hence it is
the solution to (5). In the symmetric tax case it was shown by Sandmo (1971,
pp. 68–9) that the effect of a higher tax rate (or subsidy rate) t on firm’s optimal
output depends on the monotonicity of the relative measure of risk aversion
Rr(p): if Rr(p) is increasing, Qn(t) increases in t, while decreasing Rr(p) results in
a lower Qn(t) as t increases. We shall show that under asymmetric taxation the
size of Rr(p) plays an important role in determining the way Qn(t) varies as the
tax rate t increases.
Proposition 2. Increasing the tax rate t results in (a) a higher optimal output
Qn(t) if the relative measure of risk aversion Rr(p)41; (b) a lower output Qn(t)
if Rr(p)o1; (c) unchanged output if Rr(p) ¼ 1.
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To understand this result, let us examine first the symmetric tax case.
Sandmo (1971) has demonstrated that in this case raising t may result either in
higher output or in lower output, depending on whether Rr(p) is increasing or
decreasing. Considering the asymmetric case, using claim 1 above, if Rr(p)41
and is decreasing in p, the decline in Qn(t) is only reinforced owing to
asymmetry. But this is not the case when Rr(p)41: here larger ‘gap’ between
tax and subsidy results in higher output of this firm. To gain further insight
into this counterintuitive result (for example, when it is compared with the riskneutral case) we should look more closely at the proof itself: at each optimum
Qn(t) the mean of P~ C 0 ðQn Þ, weighted by the marginal utility, is equated to 0.
Owing to tax asymmetry, in states of nature where the price assumes higher
realizations these marginal utilities are evaluated at the after-tax profits (i.e.
U 0 ½ð1 tÞ~
p), while for the lower price realizations they are evaluated at the
untaxed losses (i.e. U 0 ½~
p). Since the marginal utility is decreasing, as we raise
the tax rate t, for higher realizations of the price, the weights U 0 ½ð1 tÞ~
p are
raised more sharply than at the lower price realizations. Since Qn(t) needs to
equate the weighted average of P~ C 0 ðQn Þ to 0, when Rr(p)41 it must
increase. Thus, for highly risk averse firms the output increases as the tax rate
increases.
For risk-neutral firms under asymmetric tax, Qn(t) declines as the tax rate t
increases (see Eldor and Zilcha 1998): by lowering output, the firm reduces the
number of states of nature in which loss occurs, as t goes up. The after-tax
profit increases as the firm lowers its optimal production level. Proposition 2
demonstrates that under risk aversion the firm’s reaction to a ‘larger
asymmetry’ in taxation will depend on the size of Rr(p). Such dependence
on the relative measure of risk aversion was also obtained in the case of
demand for risky asset under symmetric taxation (see Stiglitz 1969, pp. 269–70),
as the proportional wealth tax increases the demand for the risky assets
increases. Clearly, the asymmetric tax case differs from the proportional tax
case imposed at the same rate on profits and losses (and hence reduces the
variance of the profit function); for example, in the case of the risk-neutral
firm the effect of higher tax rate is different. In our case, increasing the tax
on (positive) profits, keeping the compensation rate when losses occur
fixed (assumed to be 0), makes the net profit function ‘less risky’.
Owing to the risk aversion assumption, the level of activity of the firm (as in
the ‘demand for risky asset’ case) depends on the relative measure of risk
aversion.
Note that, for risk averse firms satisfying Rr(p)41, in the absence of risk
sharing markets, lowering the corporate tax on profits while leaving
compensation for losses unchanged results in a lower output!
In the following section we introduce futures markets and derive the firm’s
optimal behaviour. We will show that in this case the above effects of taxation
asymmetry on optimal output will disappear.
III. TAX ASYMMETRY: THE ROLE
OF
FUTURES MARKETS
The literature dealing with firms’ behaviour in the presence of forward/futures
markets has concentrated mainly on explaining why firms hedge (see e.g. for
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example Mian 1996; Smith and Stulz 1985; Froot et al. 1993), or the effect of
hedging (under tax asymmetry) on firms’ values and expected tax liabilities.
The main argument supporting hedging is that hedging results in lower average
taxes for the firm and hence raises its value. Graham and Smith (1999)
demonstrate empirically that in the United States many firms have concave
after-tax profit functions (hence justifying hedging under tax asymmetry, since
it lowers the expected tax liabilities). Other arguments are related to the
‘informational effect’ of hedging: often it is used by the management to signal
information to shareholders about performance (see De Marzo and Duffie
1995). Hedging is also explained by managerial risk aversion (see Smith and
Stulz 1985). Taking into account tax asymmetry, it has been empirically shown
by Graham and Rogers (2000a, b) that a firm’s hedging could also result from
expected financial distress costs. Albuquerque (2000) studies the hedging
behaviour of equity firms that face a convex tax schedule. All the above papers
consider issues related to the hedging policy of firms in the face of tax
asymmetry, while the present paper examines mainly its effect on optimal
production.
Assume that the above risk-averse firm, facing price uncertainty, has access
to a forward market (at date 0, for deliveries at date 1 when production is
completed). We assume in what follows that, for given P~, Pf and the tax at rate
t, the firm produces and participates in the forward market as well. For
simplicity, let the support of the random P~ be ½0; P and the price density
function f(p) be a continuous function.
~ðQ; XÞ ¼ P~Q þ ðPf P~ÞX CðQÞ. The firm chooses its
Denote by p
production level Q and its hedging level X in a way that maximizes
ð6Þ
~n ;
maxQ;X EU ½p
where
~n ¼
p
ð1 tÞ½P~Q þ ðPf P~ÞX CðQÞ
P~Q þ ðPf P~ÞX CðQÞ
~ðQ; XÞX0
if p
~ðQ; XÞo0:
if p
Denote by (Qn, Xn) the optimum for problem (6). Surprisingly, the Separation
theorem remains valid in the asymmetric tax case as well.
Proposition 3 (Separation result for the asymmetric tax case). Consider the
above competitive firm operating under asymmetric tax. If futures markets are
available, then its optimal output Qn is given by: C 0 (Qn) ¼ Pf ; Qn that is
independent of its attitude towards risk, its belief about the price distribution
and the tax rates.
The fact that tax is asymmetric will effect the optimal hedging policy of the
firm, but not its optimal production level. This generalization of the separation
result is significant, since in almost all countries firms operate under tax
asymmetry.
Let us consider now the hedging behaviour of the firm in our case. We shall
extend to our case the known result regarding optimal hedging behaviour,
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which relates negative or positive risk premia in the futures price to
‘overhedging’ or ‘underhedging’ (see e.g. Kawai and Zilcha 1986). In our case
the role of hedging tools is more important than the symmetric tax case, since it
is utilized to reduce the number of states of nature in which the firm suffers
losses; in this way the firm reduces the impact of tax asymmetry. Now we can
state the following proposition.
Proposition 4. Given the firm’s optimal decisions (Qn, Xn), the hedging
policy Xn satisfies the following conditions.
(a) If Pf oE P~, then XnoQn.
(b) If Pf oE P~, then Xn ¼ Qn, i.e. full hedging.
(c) If Pf 4E P~, then Xn4Qn
Comparing the hedging policy under symmetric tax with the asymmetric
tax case, i.e. Xsn ðtÞ X n ðtÞ, we come to the following conclusions. Whenever
X n ðtÞoCðQn Þ=Pf , the sign of @X n ðtÞ=@t differs from that of @Xsn ðtÞ=@t for the
case where Rr(p) is constant or decreasing. On the other hand, if XnXC(Qn)/
Pf, the sign of @X n =@t is determined by the monotonicity of Rr(p), in the same
way as in the symmetric tax case. For example, consider the case of constant
relative risk aversion. In this case Xsn ðtÞ does not vary with the tax rate, while
for certain levels of Pf, Xn(t) will decrease with t.
IV. DISCUSSION
OF THE
MULTI-PERIOD CASE
We shall now consider the extension of our model to the case where the firm
operates in a multi-period framework rather than the one-period analysis we
have considered so far. In this case, the level of taxes payed by the firm on each
date may be affected by any losses occured in the previous periods. Although in
most countries firms can carry forward losses (in a limited way), only in few
cases are firms allowed by the tax authorities to carry back losses (up to three
years in the United States, the United Kingdom, the Netherlands and
France), hence we are far from removing the taxation asymmetry. To
substantiate our claim, let us indicate the following points. (a) In most
countries (as can be seen from Corporate Tax: Worldwide Summary (1997)
there is no tax loss carry-back. Thus, a firm that suffers some losses during a
certain period cannot deduct part of the losses from taxes paid in earlier years.
(b) When only tax loss carry-forward applies, there are interest and
opportunities forgone owing to the delay in the (uncertain) future compensation. (c) When tax loss carry-back is applicable it is limited in time, and it is not
relevant for young firms.
We can add the following comments regarding the multi-period case. (1) If
tax loss carry-forward and carry-back are allowed without limitation (a case
that does not exist in reality), the asymmetry in taxation diminishes
significantly. (2) When hedging is feasible, then whenever the firm suffers
significant losses in date t its output in date t þ 1 will be higher, compared with
the case where no losses occured. In this case the tax loss carry-forward
provides some reduction in taxation of future profits. (3) Some of the results we
obtained for the one-shot optimization case may fail to hold in the many-period
case, particularly if some correlation between prices in subsequent dates exists.
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V. CONCLUDING REMARKS
The effects of taxation on a firm’s production (or investment) and hedging
behaviour has been discussed extensively in the literature. However, the effect
of tax asymmetry on production in the presence of hedging markets has not
received sufficient attention. This paper concentrates on the optimal output
with and without hedging tools, and demonstrates that asymmetry in taxation
differs from the symmetric tax case, emphasizing the role played by the risk
sharing markets. We do not consider here the firm’s financing decision, a topic
that is suitable for a more extended model. However, such an extension of our
framework would not vary the results, since it would not affect significantly the
(random) profits distribution function.
The evidence regarding the hedging behaviour of firms is very limited in
most countries. In particular, such information is hard to gather in countries
where tax asymmetry is strong (e.g. in most developing economies). It is
claimed by Nance et al. (1993) in the United States, where tax asymmetry is
very weak (owing to the availability of loss carry-back and carry-forward in the
United States), using a sample of 169 firms, that 104 firms use some hedging
tools. Another study by Mian (1996) uses data from the financial reports of
firms to the stock exchange in the United States. It demonstrates that firms
classified as ‘hedgers’, according to their 1992 annual reports, constitute a
significant portion of the sample considered in Mian’s study, and that hedging
activities exhibit economies of scale. Moreover, the conjecture that tax
consideration plays an important role in this hedging activity cannot be
rejected. This sample includes mainly large firms that are already well
diversified in their activities and are profitable; thus, the asymmetry in taxation
is a weak motive in this case. Geczy et al. (1997) show that firms might use
currency derivatives to reduce cash flow variations. Moreover, firms with
extensive foreign exchange rate exposure and economies of scale in hedging
also use currency derivatives.
We consider our argument regarding the role of hedging when taxes on
profits and ‘compensation’ for losses are asymmetric as another important
reason for this observed evidence about hedging. The literature dealing with
this topic (see e.g. Smith and Stulz 1985; Green and Talmor 1985; Auerbach
1986; Faig and Shum 1999) does not mention explicitly the effects of lower
expected taxes on production and the role played by futures/forward markets
on economic efficiency. Our framework is suitable to studying the behaviour of
exporting firms under random exchange rate (i.e. when the price volatility
results from the random exchange rate). In this case one can derive from our
analysis the impact of asymmetric taxation on international trade. For
example, if in the home country exporting firms face asymmetric tax, then
introducing foreign exchange futures markets results in higher output, i.e. a
higher volume of international trade.
It was shown by Green and Talmor (1985) that, in the absence of hedging
markets, asymmetry in taxation enhances mergers between firms. This follows
from benefits in tax liabilities of the two firms before and after the merger.
However, our analysis indicates that the existence of hedging tools will make
merger considerations motivated by tax benefits less attractive, owing to the
smoothing in profits attained from using these hedging markets.
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APPENDIX: PROOFS
Proof of Proposition 1
Differentiating equation (2) with respect to t, we obtain
ðA1Þ
~U 00 ½ð1 tÞ~
EfðPf P~Þ p
pg
@Xsn
:
¼
2 00
~
@t
ð1 tÞE fðPf PÞ U ½ð1 tÞ~
pg
Let us find the sign of the numerator since the denominator is negative. Assume first
that Rr(p) is increasing. Then
pÞ for P4Pf
Rr ½ð1 tÞp4Rr ð
~ ¼ ð1 tÞ½ðPf Qn CðQn Þ: Therefore
where p
pU 00 ½ð1 tÞp4Rr ð
pÞU 0 ½ð1 tÞp for p4
p ðor P4Pf Þ:
Thus, we have
ðA2Þ
pÞðPf PÞU 0 ½ð1 tÞp
pðPf PÞU 00 ½ð1 tÞp4Rr ð
for all PoPf :
However, (A2) holds also for all P4Pf as well. Therefore
EðPf P~ÞpU 00 ½ð1 tÞp4Rr ð
pÞEfðPf P~ÞU 0 ½ð1 tÞ~
pg ¼ 0
which implies that @Xsn =@to0:
&
Proof of the Claim 1:
^ the optimal output for the symmetric tax case, i.e. when t ¼ t1 ¼ t2
Denote by Q
Define
Z
VðQÞ ¼ ½P~ C 0 ðQÞU 0 fð1 tÞ½P~Q CðQÞg:
^ is the solution to V(Q) ¼ 0; i.e. VðQ
^Þ ¼ 0. Note that
It is easy to see that Q
V(Q) is a strictly decreasing function of Q since it is the derivative of
EUfð1 tÞ½P~Q CðQÞg ð1 tÞ1 , which is a strictly concave function of Q.
n
The optimal output Q under asymmetry satisfies the following necessary and
sufficient first order condition:
Z
½P~ C0 ðQn Þ U 0 fð1 tÞ½P~Qn CðQn Þg
ð1 tÞ
A
Z
þ
½P~ C 0 ðQn ÞU 0 ½P~Qn CðQn Þ ¼ 0;
A
where A is the set of all states of nature in which P~ C 0 ðQÞ40: Now consider the
function
f ðyÞ ¼ yU 0 ðyxÞ U 0 ðxÞ:
It is clear that f(1) ¼ 0. Also, it can be verified directly that f 0 (y)40 if xo0. Moreover,
it is easy to see that, for all x40, f 0 (y)40 if Rr(yx)41. Hence under our assumptions,
for any value of x, f(y)o0 for all yo1. This implies that
ð1 tÞU 0 fð1 tÞ½P~Qn CðQn ÞgU 0 ½P~Qn CðQn Þ:
Using this result together with the previous equation, we obtain
Z
½P~ C0 ðQn ÞU 0 fð1 tÞ½P~Qn CðQn Þg
A
Z
½P~ C 0 ðQn ÞU 0 fð1 tÞ½P~Qn CðQn Þg40;
þ
A
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FIRMS’ OUTPUTS UNDER UNCERTAINTY
151
n
^. This
that is, V(Q )40. But, owing to the monotonicity of V(Q), we derive that Qn oQ
proves our assertion. &
Proof of Proposition 2
Let us differentiate the first-order condition corresponding to problem (5) with respect
to t. Then Z
Z dQn
½P~ C0 ðQn ÞU 0 fð1 tÞ½P~Qn CðQn Þg þ ð1 tÞ
C00 ðQn ÞU 0
dt
A
A
Z n
dQ
½P~Qn þ CðQn Þ þ ð1 tÞ½P~ C0 ðQn Þ
þð1 tÞÞ
dt
A
Z
Z
n
dQ
dQn
þ
¼0
C00 ðQn ÞU 0 ½P~Qn CðQn Þ
½P~ C0 ðQn Þ2 U 00
þ
dt
dt
A
A
But this impies that
Z
Z
dQn 1
¼
f½P~ C 0 ðQn Þ½U 0 ð1 tÞ~
pþð1 tÞg
D A
dt
n
f½P~ C0 ðQn Þ~
pU 00 ½ð1 tÞ~
pg;
A
^¼P~Qn CðQn Þ and
where p
Z
Z
~ U 00
ðC00 ÞU 0 þ ð1 tÞ2
½P~ C 0 ðQn Þ2 p
D ¼ ð1 tÞ
A
A
Z
þ
fC00 U 0 þ ½P~ C0 ðQn Þ2 U 00 g:
A
Since we assume that C00 X0 and U00 o0, it is easy to verify that the denominator is
negative, i.e. that Do0. Now since on the set A we have P~ C 00 ðQn ÞX0, the sign of the
numerator will depend upon the sign of U 0 ½ð1 tÞ~
p þ ð1 tÞ~
pU 00 ½ð1 tÞ~
p: In fact, we
can rewrite the above expression as
Z
dQn 1
ðA3Þ
¼
½P~ C 0 ðQn Þf1 Rr ½ð1 tÞ~
pg:
D
dt
A
Thus, it is negative if Rro1, positive if Rr41 and equal 0 if Rr ¼ 1. This proves our
assertion. &
Proof of Proposition 3
To verify that first-order conditions are sufficient for the optimum, let us note that the
maximand in problem (6) is strictly concave in Q and X. To see this, let us define
ð1 tÞX if XX0
tðXÞ ¼
X
if Xo0;
where r(X) is a concave function and the maximand in (6) can be rewritten as
ðA4Þ
pðQ; XÞÞ:
max EU ½rð~
Q;X
~ is concave in Q and X, r is concave while U is strictly concave, we see that the
Since p
maximand is strictly concave in (Q, X). By the strict concavity of the maximand in (A4),
there is a unique optimum (Qn, Xn) to this optimization. Now let us define
A ¼ PjPðQn X n Þ þ Pf X n CðQn Þ 6¼ 0 :
By our assumptions, the probability of the event A, given Qn, Xn, is 1. Although r(X) is
differentiable everywhere except at X ¼ 0, since Pr{A} ¼ 1, we obtain
~ ðQn ; X n Þ P~ C0 ðQn Þ U 0 r p
~ðQn ; X n Þ
ðA5Þ
E r0 p
¼ 0;
0
0 n
n
n
n
~ ðQ ; X Þ Pf P~ U r p
~ðQ ; X Þ
¼ 0:
ðA6Þ
E r p
r The London School of Economics and Political Science 2004
152
ECONOMICA
[FEBRUARY
Now, from
(A6),
we
(A5) and
obtain
~ðQn ; X n Þ U 0 r p
~ðQn ; X n Þ ¼ 0;
ðA7Þ
Pf C0 ðQn Þ EA r0 p
which is possible only if C 0 (Qn) ¼ Pf .
&
Proof of Proposition 4
Let us use equation (A6) to obtain
EðPf P~ÞEA fr0 ð~
pðQn ; X n ÞÞU 0 ½rð~
pðQn ; X n ÞÞgþ
cov fPf P~; r0 ð~
pðQn ; X n ÞÞU 0 ½rð~
pðQn ; X n ÞÞgg ¼ 0:
But r 0 (X) ¼ 1 for Xo0 while r 0 (X) ¼ 1 t for X40 thus, as we vary the price P we shall
not change the monotonicity of r[p(Qn, Xn)] in P: it is increasing in P if Qn–Xn40, it is
decreasing in P if Qn–Xno0, and it is constant if Qn ¼ Xn. Since event A has probability
1, we derive that if Pf E P~o0 then the cova(. , .)40. This implies that U 0 ½rð~
pðQn ; X n ÞÞ
~ðQn ; X n Þ is increasing in P~. Thus, Qn–Xn40. Similarly,
is decreasing in P~ i.e. that p
~ðQn ; X n Þ ¼ constant, hence Qn ¼ Xn. &
Pf E P~¼ 0 implies that p
REFERENCES
ALBUQUERQUE, R. (2000). Optimal currency hedging under loss aversion. Working Paper (March),
University of Rochester.
ALTSHULER, R. and AUERBACH, A. (1990). The significance of tax law asymmetries: an empirical
investigation. Quarterly Journal of Economics, 105, 61–86.
AUERBACH, A. J. (1983). Taxation, corporate financial policy, and the cost of capital. Journal of
Economic Literature, 21, 905–40.
FFF (1986). The dynamic effects of tax law asymmetries. Review of Economic Studies, 53,
205–25.
BLACK, F. (1976). The pricing of commodity contracts. Journal of Financial Economics, 3, 167–79.
BROLL, U., WONG, K. P. and ZILCHA, I. (1999). Multiple currencies and hedging. Economica, 66,
421–32.
Corporate Taxes: A Worldwide Summary (1997). London: Price Waterhouse.
DE MARZO, P. and DUFFIE, D. (1995). Corporate incentives for hedging and hedge accounting.
Review of Financial Studies, 8, 743–71.
DOMAR, E. D. and MUSGRAVE, R (1944). Proportional income taxation and risk-taking. Quarterly
Journal of Economics, 388–422; first published 1944.
ELDOR, R. and ZILCHA, I. (2002). Tax asymmetry, production and hedging. Journal of Economics
and Business, 54, 345–356.
FAIG, M. and SHUM, P. (1999). Irreversible investment and endogenous financing: an evaluation of
corporate tax effects. Journal of Monetary Economics, 43, 143–71.
FEDER, G., JUST, R. E. and SCHMITZ, A. (1980). Futures markets and the theory of the firm under
price uncertainty. Quarterly Journal of Economics, 94, 317–28.
FROOT, K. A., SCHARFSTIEN, D. S. and STEIN, J. (1993). Risk management: coordinating
corporate investment and financing policies. Journal of Finance, 48, 33–41.
GECZY, C., MINTON, B. A. and SCHRAND, C. (1997). Why firms use currency derivatives. Journal
of Finance, 52, 1323–54.
GRAHAM, J. R. and ROGERS, D. A. (2000a). Does corporate hedging increase firm value? An
empirical analysis. Unpublished paper.
FFF and FFF (2000b). Do firms hedge in response to tax incentives? Unpublished paper.
FFF and SMITH, C. W. Jr (1999). Tax incentives to hedge. Journal of Finance, 54, 2241–62.
GREEN, R. C. and TALMOR, E. (1985). The structure and incentive effects of corporate tax
liabilities. Journal of Finance. 60, 1095–14.
HOLTHAUSEN, D. M. (1979). Hedging and the competitive firm under price uncertainty. American
Economic Review, 69, 989–995.
KAWAI, M. and ZILCHA, I. (1986). International trade with forward-futures markets under
exchange rate price uncertainty. Journal of International Economics, 20, 83–98.
r The London School of Economics and Political Science 2004
2004]
FIRMS’ OUTPUTS UNDER UNCERTAINTY
153
MIAN, S. L. (1996). Evidence on corporate hedging policy. Journal of Financial and Quantitative
Analysis, 31, 419–39.
MOSCHINI, G. and LAPAN, H. (1992). Hedging price risk with options and futures for the
competitive firm with production flexibility. International Economic Review, 33, 607–18.
NANCE, D., SMITH, C. W. and SMITHSON, C. W. (1993). On the determinants of corporate hedging.
Journal of Finance, 48, 267–84.
PIRRONG, C. (1995). The welfare costs of arkansas best: the inefficiency of asymmetric taxation of
hedging gains and losses. Journal of Furures Markets, 15, 111–29.
SANDMO, A. (1971). On the theory of the competitive firm under price uncertainty. American
Economic Review, 61, 65–73.
SMITH, C. W. Jr. and STULZ, R. (1985). The determinants of firm’s hedging policies. Journal of
Financial and Quantitative Analysis, 20, 391–405.
STIGLITZ, J. E. (1969). The effects of income, wealth, and capital gains taxation on risk-taking.
Quarterly Journal of Economics, 83, 263–83.
r The London School of Economics and Political Science 2004