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Entry and exit decisions under uncertainty: The limiting deterministic case
economics letters ELSEVIER
Economics Letters 51 (1996) 77-82
Entry and exit decisions under uncertainty" The limiting deterministic case Hans Christian K o n g s t e d Institute of Economics, University of Copenhagen, Studiestraede 6, DK-1455 Copenhagen K, Denmark Received 14 August 1995; accepted 2 October 1995
Abstract
This paper establishes the general deterministic limit that corresponds to Dixit's model of entry and exit decisions under uncertainty. The interlinked nature of decisions is shown to be essential also in the deterministic limit. A numerical example illustrates the result.
Keywords: Entry-exit policies; Irreversible investment; Price volatility JEL classification: D92; E22
I. Introduction
Recent literature on capital investment decisions emphasizes the combination of irreversibility and economic uncertainty as a key determinant of investment dynamics, e.g. the comprehensive treatment in Dixit and Pindyck (1994). In the original formulation, McDonald and Siegel (1986) consider a firm deciding when to invest in a project. In that one-sided case comparative static results regarding a change in the volatility of the project value can be readily obtained, see Dixit and Pindyck (1994, ch. 5). However, in some applications, investment must be considered partly reversible rather than fully irreversible. The basic model for that case is Dixit (1989a) in which a firm makes entry and exit decisions facing a stochastic output price process. In this case of interlinked decisions, analytical comparative static expressions regarding the volatility of the price process are more difficult to interpret, see Dixit and Pindyck (1994, p. 221). Investigations, therefore, usually rely on numerical calculations (see Dixit (1989a, section V) and Dixit and Pindyck (1994, section 7.1.D) for examples). The present paper establishes the deterministic limit that corresponds to the basic model of interlinked entry and exit decisions. Previous results regarding the deterministic limit of the one-sided case, Dixit and Pindyck (1994, p. 138) and Leahy (1993, p. 1108), are extended to 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 7 8 7 - 3
78
H.C. Kongsted / Economics Letters 51 (1996) 77-82
the case of interlinked decisions. The paper is also a generalization of the analysis of Dixit (1989a, p. 630) to the case of non-zero drift in the output price process. The limiting case of no uncertainty may be of interest in itself. Perhaps more importantly, it is helpful in interpreting the numerical results appearing in the literature regarding the volatility of the price process. Thus, the deterministic case provides an approximation to low levels of uncertainty for which numerical calculations can be quite difficult. The stochastic model of interlinked entry and exit decisions of Dixit (1989a) is briefly outlined in Section 2. Section 3 provides the main analysis of entry and exit triggers for the deterministic case. Section 4 illustrates the convergence of the stochastic model solution towards the deterministic limit by a numerical example. Section 5 concludes by noting some applications where this result could prove useful.
2. The model of entry and exit
Consider a firm being in either one of two states: in the market (denoted by I) producing and selling a unit flow of output (in continuous time); or outside the market (denoted by O) and idle. The firm is risk-neutral and discounts the future at the instantaneous rate p. Costs are specified by three components: a flow production cost, C > 0; a fixed cost, k > 0, incurred when entering the market; and another fixed cost, l, incurred when leaving. The latter c o m p o n e n t could represent the resale value of assets acquired upon entry, in which case l < 0. However, assuming that k + l > 0 rules out a strategy of repeated entries and exits. For the exit alternative to be relevant at all a further condition, l < C / p , is required. Uncertainty derives from the exogenous output price process, {P,}, specified as a geometric Brownian motion: d P / P = I~ dt + or d W ,
(1)
/~ being the drift parameter and or the instantaneous standard deviation of the process. The value of an active firm is bounded if/~ < p, which will be assumed in the following, dW is the increment of a standard Wiener process so that E(dW) = 0 and E(dW 2) = dt; see Malliaris and Brock (1982, p.36). In the limiting deterministic case, o-= 0, the price is simply growing (declining) exponentially at the rate tz > 0 (~ < 0 ) . The optimal rule for switching between I and O is determined for the stochastic case in Dixit (1989a), specifying two trigger prices: the exit trigger, PL, at which a firm leaves the market; and the entry trigger, PH, at which an idle firm becomes active. There is a range of inaction as PL < PH- Moreover, PH exceeds the standard Marshallian entry level, C + p k , and PL is likewise below the standard Marshallian exit level, C - pl. The solution also determines two option values: the value to an active firm of being able to leave if the price develops adversely; and the value to an idle firm of being able to enter once market conditions improve sufficiently. Exercise of each of these options means moving into a different state a n d purchasing the other option, making the entry and exit decisions interlinked. The properties of the solution when varying the parameters of the price process, o- and ~, are examined in Dixit (1989a). First, the option values and, therefore, the range of inaction,
H.C. Kongsted I Economics Letters 51 (1996) 77-82
79
(PL' PH)' increases with or. Second, for a given value of o-, both PL and PH are lowered by an increase in /z. Numerical calculations in Dixit (1989a, Fig. 4) show a non-linear downwardsloping relation b e t w e e n / z and the trigger prices. Third, the limits as o---~ 0 for the case/z = 0 are the standard ones, PH---~ C + p k and PL---~ C - pl. The following section establishes the deterministic limit for general values of/z.
3. Deterministic entry and exit triggers The cases to be considered in the following combine the I and O states of the firm with positive and negative values of the drift parameter,/z. First, examine a case in which 0 < / z < p and assume that the firm is currently idle. Immediate entry is profitable if P exceeds full cost, C + pk, whereas, otherwise, the firm will wait for the price to rise to that level. Entry at prices below C + pk is inoptimal because initially the flow of profits, P - C, would be less than the interest paid on entry costs, pk. This familiar argument defines a deterministic entry trigger, M H =- C + pk, f o r the region 0 < tx < p.
Second, retain the assumption that 0
(1//z) log((C + p k ) / P ) > 0
starting at time zero. A possible strategy is then to remain in the market obtaining a present value of V~L(P) - P / ( p - tx) - C/p. The alternative, which is preferable at low price levels, is exiting now and re-entering at time T L when the price is P = C + pk. This has a present value of VoL(P ) =- e-PTL((C + p k ) / ( p - tz) - (C/p) - k) - I .
The deterministic exit trigger, M E, for the region 0 < tz < p is then defined by equality of VIE(ME) and VoL(Mc) and the pair of inequalities, VoL(P ) > V1L(P) for P < M E and VoL(P ) < V~L(P) for M E < P < C - p l . A priori, M L must be smaller than C - p l , the exit trigger corresponding t o / z = 0, because a firm faced with a positive drift is willing to remain in the market at lower prices. U n d e r the maintained assumptions that k + l > 0 and l < C/p and the fact that 0 < / z < p, a unique exit trigger satisfying these conditions can be shown to exist) Substituting the definition of T c into the equation V~L(ML)= VoL(Mc) and manipulating the resulting expression yields: ME
P
p
tz ( C - p l )
+
\C+pk}
P(C +pk)
"
(2)
F r o m (2) the present case can be compared with a one-sided exit decision that disregards the possibility of re-entry. The exit trigger for that case would be a Marshallian trigger corrected 1The proof is simple but tedious. An appendix with the details can be obtained from the author.
80
H.C. K o n g s t e d / E c o n o m i c s Letters 51 (1996) 7 7 - 8 2
for drift, [ ( p - t z ) / p ] ( C - p l ) . As 0 < p , < p the second term in (2) is positive so that ML > [ ( p - I~)/p]C, i.e. recognizing the possibility of future re-entry makes the firm more willing to leave the market and increases the exit trigger. Thus, the second term in (2) captures the interlinked nature of decisions, which turns out to be essential also in the deterministic limit. Third, assume that the price drift is negative, ~ < 0. Now the exit decision of an active firm is standard: the deterministic exit trigger, M L, f o r the region/z < 0 is the flow production cost less the interest paid on exit costs, C - pl. Values of P less than C - pl inflict current losses larger than pl which will never be reversed due to the negative price drift. Fourth, if/z > 0 the price will eventually fall below C - pl even if currently above full cost. However, an idle firm realizes that future exit is possible so that it needs only to consider a finite period until the price falls to C - pl. Starting at time zero this happens at time T H - (1//~) log(C - p l ) / P ) > O. The present value derived from temporarily entering the market is VIH(P ) -
(Pe"'-C)e-"dt-k-e-'r.l, I
whereas the alternative of remaining idle has a value of zero, VoH(P ) --0. Equating these terms defines the deterministic entry trigger, M H > C + ,ok, f o r the region tx < 0, which must also satisfy V m ( P ) < VoH(P ) for C + ,ok < P < M H and VIH(P ) > VoH(P ) for P > M H. Again, the entry trigger, MH, can be shown to exist and to be unique and after some manipulations of the defining equation the following expression is obtained: MH _
p_ _- tz (C + pk) + { M . ]o/" p, ( C _ , o l ) p \ C - pl,I p •
(3)
This expression can be compressed with the entry trigger derived from a one-sided problem disregarding the re-exit possibility, [ ( p - Iz)/p](C + pk). M H is unambiguously lower because the second term in (3) is negative for /z < 0 . A firm thus enters a declining market more willingly when future exit is possible. Finally, for completeness it is noted from Section 2 that M u = C + p k and M E = C - pl for the case/z = 0. This completes the characterization of deterministic entry and exit triggers for all admissible values of ~.
4. A numerical example A numerical example illustrates the relation between the entry and exit triggers and the parameters of the price process. The other parameters of the model are fixed at p = 0.006, k = 8, l = 0, and C = 0 . 9 5 2 . The standard Marshallian entry trigger for this example is C + p k = 1, whereas the corresponding exit trigger is simply C = 0.952 because there are no exit costs. The example is derived from Kongsted (1995) representing quarterly figures for the environment of a firm engaged, e.g. in U S - J a p a n trade during the post-Bretton Woods
H.C. Kongsted / Economics Letters 51 (1996) 77-82
81
1.4!J 1.3
oo,o
1.2
•--........~. ~
I.I •r- 1.0
a.
-ff : 0 . 0 0 7 ' 5
=0.025
ff : O . 0 0 7 5
M R
0.9 -
0.8
o : 0.025 . o
= 0.050
"--'"''""
"--......
.......
--
~
,u, "
." " " '~----~- .~" : . ~". ~ . ~
0.7 0.6
I -0.006
I -0.004
,
l -0.002
I
I 0.000
a
I 0.002
,
I 0.004
I
0.006
F Fig. 1. Trigger prices for the deterministic and stochastic cases.
period. The (yearly) rate of interest equals approximately 2.5% and the annualized entry cost amounts to approximately 5% of full cost, corresponding roughly to the figures assumed in Dixit (1989b). Fig. 1 shows the generalized deterministic triggers, M L and Mia, as the solid curves. Both are highly non-linear functions of/x having a kink a t / z = 0. The fact that M E is the mirror image of M H is apparent from (2) and (3), and is also brought out by the figure. The dashed curves in Fig. 1 indicate pairs of trigger prices, PL and PH, of the interlinked stochastic case in Section 2 for several values of o-. These are comparable with the trigger prices shown as Fig. 4 in Dixit (1989a). As o- becomes smaller, PL and PH appear to converge to the deterministic triggers, M L and Mrs, respectively, verifying that these are indeed the limits as o---->0 of the stochastic model for all (admissible) values of/x. The result is particularly useful because the stochastic problem is not numerically well-behaved for smaller values of tr, whereas the calculation of the deterministic limits from (2) and (3) is without any numerical problems.
5. Applications Fundamental insights of the real options approach have proved useful in other areas, e.g. durable consumption (Bertola and Caballero, 1990), labor demand (Bentolila and Bertola, 1990), and foreign trade (Dixit, 1989b). Thus, slightly modified versions of the model in Section 2 apply to the analysis of decisions to buy or scrap a durable consumption good, to hire or fire labor, and to enter or leave a foreign market. Moreover, by a result of Leahy (1993), the analysis of the entry-exit problem of a single firm is equivalent to that of competitive equilibrium, meaning that the result of this paper applies also at the industry level.
82
H.C. Kongsted / Economics Letters 51 (1996) 77-82
The limiting deterministic case is particularly useful as a point of reference in comparing different volatility scenarios. In the labor d e m a n d context of Bentolila and Bertola (1990, p. 393), for example, the effect of increased d e m a n d uncertainty in E u r o p e in terms of average e m p l o y m e n t is investigated. In some cases, the deterministic limit itself can be considered a policy option, e.g. in Kongsted (1995) where the effect of exchange rate volatility in foreign trade is examined on the basis of a similar framework.
Acknowledgements Financial support from Carlsbergfondet is gratefully acknowledged.
References Bentolila, S. and G. Bertola, 1990, Firing costs and labour demand: How bad is Eurosclerosis, Review of Economic Studies 57, 381-402. Bertola, G. and R. Caballero, 1990, Kinked adjustment costs and aggregate dynamics, in: O. Blanchard and S. Fisher, eds., NBER macroeconomics annual (MIT Press, Cambridge, MA) 237-295. Dixit, A.K., 1989a, Entry and exit decisions under uncertainty, Journal of Political Economy 97, no. 3, 620-638. Dixit, A.K., 1989b, Hysteresis, import penetration, and exchange rate pass-through, Quarterly Journal of Economics 104, no. 2, 205-228. Dixit, A.K. and R.S. Pindyck, Investment under uncertainty (Princeton University Press, Princeton, NJ). Kongsted, H.C., 1995, Long-run effects of exchange rate volatility in foreign trade with entry costs, Working Paper, Institute of Economics~ University of Copenhagen. Leahy, J.V., 1993, Investment in competitive equilibrium: The optimality of myopic behavior, Quarterly Journal of Economics 108, no. 4, 1105-1133. Malliaris, A.G. and W.A. Brock, 1982, Stochastic methods in economics and finance (North-Holland, Amsterdam). McDonald, R.L. and D.R. Siegel, 1986, The value of waiting to invest, Quarterly Journal of Economics 101, 707-727.
Economics Letters 51 (1996) 77-82
Entry and exit decisions under uncertainty" The limiting deterministic case Hans Christian K o n g s t e d Institute of Economics, University of Copenhagen, Studiestraede 6, DK-1455 Copenhagen K, Denmark Received 14 August 1995; accepted 2 October 1995
Abstract
This paper establishes the general deterministic limit that corresponds to Dixit's model of entry and exit decisions under uncertainty. The interlinked nature of decisions is shown to be essential also in the deterministic limit. A numerical example illustrates the result.
Keywords: Entry-exit policies; Irreversible investment; Price volatility JEL classification: D92; E22
I. Introduction
Recent literature on capital investment decisions emphasizes the combination of irreversibility and economic uncertainty as a key determinant of investment dynamics, e.g. the comprehensive treatment in Dixit and Pindyck (1994). In the original formulation, McDonald and Siegel (1986) consider a firm deciding when to invest in a project. In that one-sided case comparative static results regarding a change in the volatility of the project value can be readily obtained, see Dixit and Pindyck (1994, ch. 5). However, in some applications, investment must be considered partly reversible rather than fully irreversible. The basic model for that case is Dixit (1989a) in which a firm makes entry and exit decisions facing a stochastic output price process. In this case of interlinked decisions, analytical comparative static expressions regarding the volatility of the price process are more difficult to interpret, see Dixit and Pindyck (1994, p. 221). Investigations, therefore, usually rely on numerical calculations (see Dixit (1989a, section V) and Dixit and Pindyck (1994, section 7.1.D) for examples). The present paper establishes the deterministic limit that corresponds to the basic model of interlinked entry and exit decisions. Previous results regarding the deterministic limit of the one-sided case, Dixit and Pindyck (1994, p. 138) and Leahy (1993, p. 1108), are extended to 0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0 1 6 5 - 1 7 6 5 ( 9 5 ) 0 0 7 8 7 - 3
78
H.C. Kongsted / Economics Letters 51 (1996) 77-82
the case of interlinked decisions. The paper is also a generalization of the analysis of Dixit (1989a, p. 630) to the case of non-zero drift in the output price process. The limiting case of no uncertainty may be of interest in itself. Perhaps more importantly, it is helpful in interpreting the numerical results appearing in the literature regarding the volatility of the price process. Thus, the deterministic case provides an approximation to low levels of uncertainty for which numerical calculations can be quite difficult. The stochastic model of interlinked entry and exit decisions of Dixit (1989a) is briefly outlined in Section 2. Section 3 provides the main analysis of entry and exit triggers for the deterministic case. Section 4 illustrates the convergence of the stochastic model solution towards the deterministic limit by a numerical example. Section 5 concludes by noting some applications where this result could prove useful.
2. The model of entry and exit
Consider a firm being in either one of two states: in the market (denoted by I) producing and selling a unit flow of output (in continuous time); or outside the market (denoted by O) and idle. The firm is risk-neutral and discounts the future at the instantaneous rate p. Costs are specified by three components: a flow production cost, C > 0; a fixed cost, k > 0, incurred when entering the market; and another fixed cost, l, incurred when leaving. The latter c o m p o n e n t could represent the resale value of assets acquired upon entry, in which case l < 0. However, assuming that k + l > 0 rules out a strategy of repeated entries and exits. For the exit alternative to be relevant at all a further condition, l < C / p , is required. Uncertainty derives from the exogenous output price process, {P,}, specified as a geometric Brownian motion: d P / P = I~ dt + or d W ,
(1)
/~ being the drift parameter and or the instantaneous standard deviation of the process. The value of an active firm is bounded if/~ < p, which will be assumed in the following, dW is the increment of a standard Wiener process so that E(dW) = 0 and E(dW 2) = dt; see Malliaris and Brock (1982, p.36). In the limiting deterministic case, o-= 0, the price is simply growing (declining) exponentially at the rate tz > 0 (~ < 0 ) . The optimal rule for switching between I and O is determined for the stochastic case in Dixit (1989a), specifying two trigger prices: the exit trigger, PL, at which a firm leaves the market; and the entry trigger, PH, at which an idle firm becomes active. There is a range of inaction as PL < PH- Moreover, PH exceeds the standard Marshallian entry level, C + p k , and PL is likewise below the standard Marshallian exit level, C - pl. The solution also determines two option values: the value to an active firm of being able to leave if the price develops adversely; and the value to an idle firm of being able to enter once market conditions improve sufficiently. Exercise of each of these options means moving into a different state a n d purchasing the other option, making the entry and exit decisions interlinked. The properties of the solution when varying the parameters of the price process, o- and ~, are examined in Dixit (1989a). First, the option values and, therefore, the range of inaction,
H.C. Kongsted I Economics Letters 51 (1996) 77-82
79
(PL' PH)' increases with or. Second, for a given value of o-, both PL and PH are lowered by an increase in /z. Numerical calculations in Dixit (1989a, Fig. 4) show a non-linear downwardsloping relation b e t w e e n / z and the trigger prices. Third, the limits as o---~ 0 for the case/z = 0 are the standard ones, PH---~ C + p k and PL---~ C - pl. The following section establishes the deterministic limit for general values of/z.
3. Deterministic entry and exit triggers The cases to be considered in the following combine the I and O states of the firm with positive and negative values of the drift parameter,/z. First, examine a case in which 0 < / z < p and assume that the firm is currently idle. Immediate entry is profitable if P exceeds full cost, C + pk, whereas, otherwise, the firm will wait for the price to rise to that level. Entry at prices below C + pk is inoptimal because initially the flow of profits, P - C, would be less than the interest paid on entry costs, pk. This familiar argument defines a deterministic entry trigger, M H =- C + pk, f o r the region 0 < tx < p.
Second, retain the assumption that 0
(1//z) log((C + p k ) / P ) > 0
starting at time zero. A possible strategy is then to remain in the market obtaining a present value of V~L(P) - P / ( p - tx) - C/p. The alternative, which is preferable at low price levels, is exiting now and re-entering at time T L when the price is P = C + pk. This has a present value of VoL(P ) =- e-PTL((C + p k ) / ( p - tz) - (C/p) - k) - I .
The deterministic exit trigger, M E, for the region 0 < tz < p is then defined by equality of VIE(ME) and VoL(Mc) and the pair of inequalities, VoL(P ) > V1L(P) for P < M E and VoL(P ) < V~L(P) for M E < P < C - p l . A priori, M L must be smaller than C - p l , the exit trigger corresponding t o / z = 0, because a firm faced with a positive drift is willing to remain in the market at lower prices. U n d e r the maintained assumptions that k + l > 0 and l < C/p and the fact that 0 < / z < p, a unique exit trigger satisfying these conditions can be shown to exist) Substituting the definition of T c into the equation V~L(ML)= VoL(Mc) and manipulating the resulting expression yields: ME
P
p
tz ( C - p l )
+
\C+pk}
P(C +pk)
"
(2)
F r o m (2) the present case can be compared with a one-sided exit decision that disregards the possibility of re-entry. The exit trigger for that case would be a Marshallian trigger corrected 1The proof is simple but tedious. An appendix with the details can be obtained from the author.
80
H.C. K o n g s t e d / E c o n o m i c s Letters 51 (1996) 7 7 - 8 2
for drift, [ ( p - t z ) / p ] ( C - p l ) . As 0 < p , < p the second term in (2) is positive so that ML > [ ( p - I~)/p]C, i.e. recognizing the possibility of future re-entry makes the firm more willing to leave the market and increases the exit trigger. Thus, the second term in (2) captures the interlinked nature of decisions, which turns out to be essential also in the deterministic limit. Third, assume that the price drift is negative, ~ < 0. Now the exit decision of an active firm is standard: the deterministic exit trigger, M L, f o r the region/z < 0 is the flow production cost less the interest paid on exit costs, C - pl. Values of P less than C - pl inflict current losses larger than pl which will never be reversed due to the negative price drift. Fourth, if/z > 0 the price will eventually fall below C - pl even if currently above full cost. However, an idle firm realizes that future exit is possible so that it needs only to consider a finite period until the price falls to C - pl. Starting at time zero this happens at time T H - (1//~) log(C - p l ) / P ) > O. The present value derived from temporarily entering the market is VIH(P ) -
(Pe"'-C)e-"dt-k-e-'r.l, I
whereas the alternative of remaining idle has a value of zero, VoH(P ) --0. Equating these terms defines the deterministic entry trigger, M H > C + ,ok, f o r the region tx < 0, which must also satisfy V m ( P ) < VoH(P ) for C + ,ok < P < M H and VIH(P ) > VoH(P ) for P > M H. Again, the entry trigger, MH, can be shown to exist and to be unique and after some manipulations of the defining equation the following expression is obtained: MH _
p_ _- tz (C + pk) + { M . ]o/" p, ( C _ , o l ) p \ C - pl,I p •
(3)
This expression can be compressed with the entry trigger derived from a one-sided problem disregarding the re-exit possibility, [ ( p - Iz)/p](C + pk). M H is unambiguously lower because the second term in (3) is negative for /z < 0 . A firm thus enters a declining market more willingly when future exit is possible. Finally, for completeness it is noted from Section 2 that M u = C + p k and M E = C - pl for the case/z = 0. This completes the characterization of deterministic entry and exit triggers for all admissible values of ~.
4. A numerical example A numerical example illustrates the relation between the entry and exit triggers and the parameters of the price process. The other parameters of the model are fixed at p = 0.006, k = 8, l = 0, and C = 0 . 9 5 2 . The standard Marshallian entry trigger for this example is C + p k = 1, whereas the corresponding exit trigger is simply C = 0.952 because there are no exit costs. The example is derived from Kongsted (1995) representing quarterly figures for the environment of a firm engaged, e.g. in U S - J a p a n trade during the post-Bretton Woods
H.C. Kongsted / Economics Letters 51 (1996) 77-82
81
1.4!J 1.3
oo,o
1.2
•--........~. ~
I.I •r- 1.0
a.
-ff : 0 . 0 0 7 ' 5
=0.025
ff : O . 0 0 7 5
M R
0.9 -
0.8
o : 0.025 . o
= 0.050
"--'"''""
"--......
.......
--
~
,u, "
." " " '~----~- .~" : . ~". ~ . ~
0.7 0.6
I -0.006
I -0.004
,
l -0.002
I
I 0.000
a
I 0.002
,
I 0.004
I
0.006
F Fig. 1. Trigger prices for the deterministic and stochastic cases.
period. The (yearly) rate of interest equals approximately 2.5% and the annualized entry cost amounts to approximately 5% of full cost, corresponding roughly to the figures assumed in Dixit (1989b). Fig. 1 shows the generalized deterministic triggers, M L and Mia, as the solid curves. Both are highly non-linear functions of/x having a kink a t / z = 0. The fact that M E is the mirror image of M H is apparent from (2) and (3), and is also brought out by the figure. The dashed curves in Fig. 1 indicate pairs of trigger prices, PL and PH, of the interlinked stochastic case in Section 2 for several values of o-. These are comparable with the trigger prices shown as Fig. 4 in Dixit (1989a). As o- becomes smaller, PL and PH appear to converge to the deterministic triggers, M L and Mrs, respectively, verifying that these are indeed the limits as o---->0 of the stochastic model for all (admissible) values of/x. The result is particularly useful because the stochastic problem is not numerically well-behaved for smaller values of tr, whereas the calculation of the deterministic limits from (2) and (3) is without any numerical problems.
5. Applications Fundamental insights of the real options approach have proved useful in other areas, e.g. durable consumption (Bertola and Caballero, 1990), labor demand (Bentolila and Bertola, 1990), and foreign trade (Dixit, 1989b). Thus, slightly modified versions of the model in Section 2 apply to the analysis of decisions to buy or scrap a durable consumption good, to hire or fire labor, and to enter or leave a foreign market. Moreover, by a result of Leahy (1993), the analysis of the entry-exit problem of a single firm is equivalent to that of competitive equilibrium, meaning that the result of this paper applies also at the industry level.
82
H.C. Kongsted / Economics Letters 51 (1996) 77-82
The limiting deterministic case is particularly useful as a point of reference in comparing different volatility scenarios. In the labor d e m a n d context of Bentolila and Bertola (1990, p. 393), for example, the effect of increased d e m a n d uncertainty in E u r o p e in terms of average e m p l o y m e n t is investigated. In some cases, the deterministic limit itself can be considered a policy option, e.g. in Kongsted (1995) where the effect of exchange rate volatility in foreign trade is examined on the basis of a similar framework.
Acknowledgements Financial support from Carlsbergfondet is gratefully acknowledged.
References Bentolila, S. and G. Bertola, 1990, Firing costs and labour demand: How bad is Eurosclerosis, Review of Economic Studies 57, 381-402. Bertola, G. and R. Caballero, 1990, Kinked adjustment costs and aggregate dynamics, in: O. Blanchard and S. Fisher, eds., NBER macroeconomics annual (MIT Press, Cambridge, MA) 237-295. Dixit, A.K., 1989a, Entry and exit decisions under uncertainty, Journal of Political Economy 97, no. 3, 620-638. Dixit, A.K., 1989b, Hysteresis, import penetration, and exchange rate pass-through, Quarterly Journal of Economics 104, no. 2, 205-228. Dixit, A.K. and R.S. Pindyck, Investment under uncertainty (Princeton University Press, Princeton, NJ). Kongsted, H.C., 1995, Long-run effects of exchange rate volatility in foreign trade with entry costs, Working Paper, Institute of Economics~ University of Copenhagen. Leahy, J.V., 1993, Investment in competitive equilibrium: The optimality of myopic behavior, Quarterly Journal of Economics 108, no. 4, 1105-1133. Malliaris, A.G. and W.A. Brock, 1982, Stochastic methods in economics and finance (North-Holland, Amsterdam). McDonald, R.L. and D.R. Siegel, 1986, The value of waiting to invest, Quarterly Journal of Economics 101, 707-727.