Vertical Integration and Operational Flexibility

Moretto, Michele; Rossini, Gianpaolo Working Paper Vertical Integration and Operational Flexibility Quaderni - Working Paper DSE, No. 631 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Moretto, Michele; Rossini, Gianpaolo (2008) : Vertical Integration and Operational Flexibility, Quaderni - Working Paper DSE, No. 631, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4616 This Version is available at: https://hdl.handle.net/10419/159472 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ Vertical Integration and Operational Flexibility ∗ Michele Moretto † Gianpaolo Rossini ‡ March 12, 2008 Abstract The main aim of the paper is to highlight the relation between flexibility and vertical integration. To this purpose, we go through the selection of the optimal degree of vertical disintegration of a flexible firm which operates in a dynamic uncertain environment. The enterprise we model enjoys flexibility since it can switch from a certain amount of disintegration to vertical integration and viceversa. This means that the firm never loses vertical control, i.e., the ability to produce all inputs even when it buys them in the market. This sort of flexibility makes for results which are somehow contrary to the Industrial Organization recent literature and closer to the Operations Research results. In this sense we provide a bridge between the two approaches and rescue Industrial Organization from counterintuitive conclusions. Keywords: vertical integration, outsourcing, entry, flexibility JEL Classification:.L24; G 31; C61 ∗ We wish to thank Laura Vici for the provision of vertical integration statistics. The paper will be presented at the 6th Annual IIOC at Marymount University Washington, DC, May 16-18, 2008. We acknowledge the financial support of the Univerisities of Bologna and Padova within the 2007-8 60% scheme. † University of Padova, Department of Economics, Via del Santo, 22; Padova, Italy, michele.moret[email protected] ‡ University of Bologna, Department of Economics, Strada Maggiore, 45; Bologna, Italy, [email protected] 1 1 Introduction A crucial question in both Industrial Organization and Operations Research concerns the extension of the control of a vertical production process and its flexibility. This topic has come recently to the limelight (Acemoglu, Aghion, Griffith and Zilibotti, 2005; Acemoglu, Johnson and Mitton, 2005; Rossini, 2005, 2007) following the current wave of international outsourcing (Antràs and Helpman, 2004; Grossman and Helpman, 2002, 2005) with many firms increasing vertical disintegration on a crossborder basis by buying a growing chunk of inputs from independent foreign enterprises. Even though most of current outsourcing is due to lower labour costs in the countries where portions of the vertical process of production is offshored, vertical disintegration is often chosen for reasons not directly linked to labour costs. As Grossman and Helpman (2002) pointed out, vertical disintegration may allow finer specialization in input production due to scale economies in R&D and production. This effect may be amplified by trade deepening, owing to better opportunities to concentrate on fewer stages of production. Nonetheless, vertical disintegration may also be adopted as a cushion against external shocks. By sharing the vertical control of production with other, preferably foreign, enterprises, each firm may enjoy a more flexible and less risky production organization, especially whenever different stages of the vertical production process require bearing relevant sunk costs. On the contrary, high quality firms may decide to increase their degree of vertical integration to make sure that quality standards are more likely met. All these considerations are bound to explain i) why vertical disintegration shows variable and opposite trends across time and industries, ii) why we observe outsourcing among similar countries and within the same country, iii) why there are many examples of insourcing 1 , i.e., increased vertical integration, for instance, as a result of vertical mergers 2 . 1 In June 7, 2005 the president of the Confederation of Indian Industry (CII) Sunil Bharti Mittal told Democratic presidential candidate Hillary Clinton about Indian companies outsourcing to US firms, investing in the US and creating jobs ( IANS, 2005). Moreover, in 2007 Glaxo-Smith-Kline has decided to bring back to Europe labs involved in advanced R&D located in China. Moreover, some increased insourcing has been noticed recently in some Japanese high tech firms and in some high quality - high tech US firms. These are just few examples of reverse outsourcing which can be found in newspapers and in specialized publications. 2 In the food industry, in the automotive industry in emerging countries, such as Indian Tata recently buying steel factories. 2 So far the decision as to vertically disintegrate and/or integrate has been analyzed as a once and for all choice, i.e., without contemplating any opportunity for its reversal. Yet, a question arises as to whether a firm may be better off adopting a radical vertical disintegration with the loss of control of entire phases of the production process (Helpman and Grossman, 2005) or it may be preferable to retain the ability to produce the inputs bought from independent enterprises. For instance, this may be obtained by producing in-house just a quota of the input requirement. As a matter of fact, for each level of external input procurement, there are many ways to vertically disintegrate. When a firm decides to offshore an input production to a foreign independent firm, the associated risk (due to exchange rates, foreign rules, production conditions in a remote country, quality standards, delivery time, shipment costs, etc.) may be quite high and worth some prudential behavior. An industrial real (not financial) way to decrease, or diversify away, this risk may require either keeping a share of the process of production in-house as a buffer or simply preserving the ability (know how and some facilities) to make it and, perhaps, reverse the offshoring decision by bringing back sections of the vertical chain of production whenever circumstances dictate so. All these considerations confirm that the decision to outsource is neither a simple binary choice, nor a matter of sheer input costs of production. Yet, it appears to be quite related to the extent of flexibility needed by a firm so as to safely face uncertainty. Industrial Organization (Alvarez and Stenbacka, 2007) and Operations Research (Van Mieghen, 1999) offer different answers as to the optimal timing and extent of outsourcing vis à vis internal production. The Operations Research interpretation 3 maintains that firms subcontract more as market uncertainty (risk) increases, closely paralleling the behavior of financial options, whose worth increases with uncertainty. In Industrial Organization literature, once the decision as to the vertical arrangement of a firm is taken there is no possibility of reversing it: the amount of flexibility that outsourcing is meant to provide cannot change once a certain vertical organization has been decided. Moreover, in Alvarez and 3 Van Mieghem (1999) contribution belongs to Operations Research literature which highlights the flexibility that subcontracting offers to production and capacity planning. Van Mieghem (1999) paper extends and generalizes previous studies (Li, 1992 and Mc- Cardle, 1997) which dealt only with the strategic unidimensional problem of optimal production with outsourcing, without considering capacity selection, i.e. the optimal size of investment. 3 Stenbacka (2007), at higher levels of uncertainty the adoption of outsourcing is being postponed. This result is actually puzzling, contrary to both Van Mieghem (1999) and common sense, since, in an uncertain dynamic framework, outsourcing is a way to do "production and capacity smoothing" making for the presumption that higher uncertainty should accelerate the adoption of outsourcing. But in current Industrial Organization literature the firm has an option to do outsourcing, yet not to do "production (and profit) smoothing" by switching among different levels of vertical integration to minimize profit variability. Recent literature (Acemoglu, Aghion, Griffith and Zilibotti, 2005) has cast some light on the relationship between vertical integration and size finding a direct link. A further confirmation on a different data set 4 can be found in Table 1 below, where we present indices of vertical integration (VIX) computed at firm level for years 2000 through 2004. 4 Data used come from Osiris, a database set up by the Bureau Van Dijk. 4 Table 1: Vertical integration indices 5 small 6 medium 7 large 8 2000 VIX .28 .34 .38 obs 1075 836 1938 SD .36 .35 .37 cv 1.28 1.01 .97 2001 VIX .28 .33 .40 obs 1219 839 1908 SD .36 .35 .38 cv 1.29 1.06 .95 2002 VIX .28 .34 .41 obs 1256 815 1780 SD .37 .35 .37 cv 1.32 1.03 .90 2003 VIX .27 .34 .40 obs 1323 787 1704 SD .36 .36 .37 cv 1.33 1.06 .93 2004 VIX .27 .34 .37 obs 1321 716 1637 SD .36 .36 .38 cv 1.34 1.03 1.02 As it can be seen, vertical integration increases as firms get larger. The variability of the index of vertical integration goes down with size even though it remains quite high, somehow blurring the revelation power of data. Nonetheless, our empirical evidence is consistent with what found on a different empirical basis by Acemoglu, Aghion, Griffith and Zilibotti (2005), 5 obs stands for number of observations, SD stands for standard deviation, cv stands for the coefficient of variation which is the ratio of SD over the sample mean of the VIX index. 6 Less than 300 employees. 7 Between 300 and 1000 employees. 8 More than 1000 employees. 5 even though the result on variability does not have, to our knowledge, any comparability in the literature. On the trace of recent mentioned literature and evidence presented in Table 1, we try to explain the amount of flexibility acquired by firms and the observed relationship between size and vertical integration. We wish to explore the choice of the extent of flexibility that can be secured by the mix between outsourcing and insourcing in a dynamic uncertain framework when the scale of production changes and input price volatility varies. We shall analyze the choice of the vertical arrangement together with the entry decision when the firm is able to revise its vertical commitment if market conditions require it. This will partially bridge the gap between Operations Research and Industrial Organization and propose a unified interpretation of flexibility in terms of vertical disintegration and/or integration. In the next section we present the model. In the third part we go through the entry process. In the fourth we analyze the choice of capacity. In the fifth we go through some comparative statics. The epilogue contains a concluding summary. 2 The model We consider a flexible vertical integration arrangement (i.e. FV I)in an industry in which the market price of the final good is certain and given to the firm. 9 To perform its task the company buys a unit of a fundamental input for each unit of output (perfect vertical complementarity). The firm can either produce entirely a perfectly divisible intermediate good in-house at the marginal cost d t or buy a share α∈(0,1] of the input at the market price c t . The enterprise may costlessly switch from making the input, when ˆc t ≡αc t +(1−α)d t rises above d t ,to buying it, if ˆc t falls below d t .Therefore, once decided the portion of input to procure from an independent enterprise, the instantaneous profit is: π t = max[(p−d t ),(p−ˆc t )]X(1) ≡[p−d t + max(d t −ˆc t ,0)] X 9 The output price may be constant due to regulation. 6 where pis the output market price. The profit function (1) draws on a simplified linear technology with only one input. The quantity of output is Xunits per year. When α≤1,with technology FV I the firm manufactures the output by using a linear combination of produced and procured input, while keeping the flexibility of going back to total vertical integration every time cbecomes too high. 10 The market price, c t ,of the input needed for the production of the final good is uncertain. On the contrary, the marginal cost of internal production is constant, i.e., d t =d. Finally, for the sake of simplicity, we assume that p−d > 0.Even though outsourcing may induce cost advantages, as cgoes down, the firm retains its know-how to manufacture the input in a viable and profitable way. Dynamic uncertainty in the market price of the input, c t ,boils down to a geometric Brownian motion: dc t =γc t dt +σcdz t (2) with dz t as the increment of a Wiener process (or Brownian motion), uncorrelated over time. The drift parameter is lower than the riskless interest rate, i.e., γ≤r. 11 The process dz t satisfies the conditions that E(dz t ) = 0 and E(dz 2 t ) = dt. Therefore, E(dc t )/c t =γdt and E(dc t /c t ) 2 =σ 2 dt, i.e., starting from the initial value c 0 ,the random position of the cost c t at time t > 0has a normal distribution with mean c 0 e γt and variance c 2 0 (e σ 2 t −1), which increases as we look further and further into the future. Notice that the process “has no memory” (i.e., it is Markovian), and hence i) at any point in time t, the observed c t is the best predictor of future profits, ii)c t may next move upwards or downwards with equal probability. 2.1 The value of the FV I technology With a FV I technology we have to distinguish between two opposite cases. If ˆc t > d the firm is Effectively Vertically Integrated (EV I). It does possess the facilities and produces its own input, while keeping the option of buying 10 We note that if α= 1,with a technology FV I the firm can switch between two extremes: total vertical integration and total vertical disintegration. 11 Alternatively, we could use an interest rate that includes an appropriate adjustment for risk and take the expectation with respect to a distribution of cadjusted for risk neutrality (see Cox and Ross, 1976; Harrison and Kreps, 1979; Harrison, 1985). 7 it. On the contrary, if ˆc t < d the firm is only Virtually Vertically Integrated (V V I). It buys a share αof the input while producing a portion 1−αand keeps the ability (option) to manufacture the whole input requirement if ˆc t goes up. Since for α≥0,the condition ˆc t > d implies c t > d 12 , the value of the firm is given by the solution of the following free boundary dynamic programming problems (Dixit, 1989; Dixit and Pindyck 1994; Moretto, 1996): ΓV EV I (c t ;α) = −(p−d)X, for c t > d (3) and ΓV V V I (c t ;α) = −(p−αc t −(1 −α)d)X, for c t < d, (4) where Γindicates the differential operator: Γ = −r+γc ∂ ∂c + 1 2 σ 2 c 2∂ 2 ∂c 2 . The solution of the differential equations (3) and (4) requires the following boundary conditions: lim c→∞ V EV I (c t ;α)−p−d rX= 0 (5) and lim c→0 V V V I (c t ;α)−p−(1 −α)d r−αc t r−γX= 0,(6) where p−d r Xindicates the present value of operating the firm forever while “making” the input and  p−(1−α)d r − αc t r−γ Xis the present value of operating the firm forever while “buying” a share αof the input in the market. Then, from the assumptions and the linearity of the differential equations (3) and (4), using (6) and (5), we get: 12 It is easy to show that αc t + (1 −α)d > d, α(c t −d)>0, c t > d. 8 and the firm keeps on manufacturing in-house a small fraction of the input as a sort of prudential behavior. Condition (14) can also be written as: Ac ∗β 2 =2p−d rXK, (17) which says that the trigger, letting the firm enter, can be obtained by equating the value of the option to switch from EV I to V V I to the constant value 2 p−d r XK, which depends on, among other things, the cost of the in-house input dand the sunk cost K. A larger sunk cost Kgenerates, ceteris paribus, a reduction of the optimal c ∗ ,making the firm delay entry. An increase of the cost of producing in-house the input boosts the optimal c ∗ letting the firm anticipate entry. 4 The choice of capacity So far we have considered the optimal entry-timing and the optimal mix of outsourcing with capacity fixed at X. Empirical evidence presented in the introduction and coming from recent literature (Acemoglu, Aghion, Griffith and Zilibotti, 2005) invites some investigation on the trade-off between entry costs, outsourcing and capacity. To this purpose, we generalize the model allowing the firm to adjust size continuously. We suppose that capacity can be indexed by a continuum X∈X, ¯ X and that the investment costs to set up an outsourcing network go up with αand the size of the firm X. In particular, extending the cost function (9), we assume that the unit organization cost Kdepends on size. That is: I(α, X;w) = 1 2K(X;w)α 2 ,(18) where wrepresents the price of the (fixed) factors needed to set up the subcontractors network, to write contracts, to monitor input quality. The organization cost function K=K(X;w)shows the usual properties: K X (X;w)> 0,K XX (X;w)>0, K w (X;w)>0and K Xw (X;w) = 0. 20 By the definition 20 By the Shephard’s Lemma, if the firm’s conditional factor demand is positively sloped with respect ot X, we get K Xw (X;w)≥0.However, we may alternatively assume that demand for the factors required to build up the subcontractors network is independent of capacity. 15 of I, if α= 0,the organizational cost drops to zero regardless of the size of the firm. 21 Here, we consider a firm with a perfectly divisible plant of maximum size ¯ X. The firm has to decide which plant to build, taking into account entry costs I. By (16) the optimal dimension requires choosing Xfor which the constant C(X;w)≡A2 p−d rK(X;w) Xis the largest. This is equivalent to maximizing the ratio X K subject to K=K(X;w).Then, we get the following first order condition (FOC) 22 : K(X ∗ , w)−XK X (X ∗ , w) = 0.(19) From (19), a necessary condition for an optimal solution is a cost elasticity ε KX ≡ X ∗ K X (X ∗ ;w) K(X ∗ ;w) = 1,i.e., the average cost AC(X;w)≡ K(X;w) X is constant around the optimum (constant returns to scale). Then, the optimal size is always, conditionally on w, efficient, i.e., X ∗ (w) = arg min AC(X;w). If there are economies or diseconomies of scale, the optimum comes from a binary comparison between the smallest and the largest size. In the first case, the optimal policy requires waiting to invest in the largest plant in the spectrum of output capacity, i.e., X ∗ =¯ X. In the second case, investment occurs soon in the smallest plant in the spectrum, i.e., X ∗ =X. Finally, within the range where the SOC holds, an increase of wimplies an increase of the firm’s optimal size. That is, by (19) we get: ∂X ∗ (w) ∂w =−K w (X ∗ ;w)−X ∗ K Xw (X ∗ ;w) SOC >0.(20) 5 Comparative Statics The analysis performed so far could have been carried out using traditional economic analysis tools. However, our model allows for a deeper study of the effect of both the uncertainty and the project size on the entry policy as well as on the optimal outsourcing mix. 21 The results would not change if we introduced a technology cost such that I(0, X;w) = k(X)>0. 22 The second order condition (SOC) is always satisfied because of the convexity of K(X;w)with respect to X. 16 5.1 The effect of uncertainty It is possible to show that an increase in the risk concerning the input market price (σ)always entails an increase in the optimal entry trigger c ∗ ,i.e., ∂c ∗ ∂σ >0 and the firm outsources earlier. On the contrary, by (15), we easily see that α ∗ is not going to change as uncertainty soars. If we analyze the condition K∈2 p−d r X, (r−γβ 1 )d (β 1 −β 2 )r(r−γ) ,we see that on the left side it is simply determined by parameters. On the right side it depends on the degree of uncertainty. 23 The interval becomes wider as uncertainty grows making the adoption of EV I more likely. Finally, once the firm has resolved to invest in the EV I technology, we may investigate how uncertainty affects the probability of outsourcing the input manufacturing. According to (7), the probability of investing in the technology EV I is represented by the likelihood that the input price touches the critical dfrom above starting from an initial c t > d. This may be written as (Dixit, 1993, p.54): Pr(c t ) =      1if 2γ σ 2 ≤1  d c t  2γ σ2 −1 if 2γ σ 2 >1 Starting at c t in the interior of the range [d, ∞), after a “sufficient” long interval of time the process will for sure hit the barrier dif the trend is positive, but low with respect to uncertainty, or if it is negative. However, if γ is positive and sufficiently high with respect to volatility, the process may drift away and never hit d. Furthermore, higher volatility increases the probability of hitting the barrier dmaking more attractive, ceteris paribus, outsourcing. Indeed, the derivative of Pr(c t )with respect to σis unambiguously positive dPr dσ =−4σγ (σ 2 ) 2 ln d c t d c t  2γ σ2 −1 >0 All these results can be summarized in the following proposition: 23 Defining (r−γβ 1 )d (β 1 −β 2 )r(r−γ) ≡R, it can easily compute: ∂R ∂σ =8dσ 3 [8rσ 2 + (σ 2 + (σ 2 −2γ) 2 ] 3 2 ≥0. 17 Proposition 3 1) The optimal threshold to enter with the technology that allows to switch, in the future, to a partial outsourcing production is a strictly increasing function of input price volatility, i.e. ∂c ∗ ∂σ >0. This means that higher uncertainty makes for earlier entry with the EV I technology. 2) The optimal share of outsourcing at entry does not depend on input price volatility, i.e., ∂α ∗ ∂σ = 0. 3) However, once the enterprise has adopted the EV I technology, an increase of volatility boosts the probability of switching to the V V I, i.e., to outsourcing. Proof. See Appendix C The above results run counter the Industrial Organization findings (Alvarez and Stenbacka, 2007). Entry is anticipated due to the opportunity to switch to outsourcing. Once a firm has entered with an EVI technology the likelihood that it will resort to outsourcing increases with uncertainty. However, the extent of outsourcing established at the time of entry does not depend on input price volatility. This is due to the fact that the firm we consider is flexible and can change the decision to vertically integrate or disintegrate. In this sense our results provide a generalization of previous ones since our firm is flexible also in the choice of "being flexible". 5.2 The effect of the firm’s size Here we wish to explore the effects of size on entry and outsourcing decisions. By (20), an increase of the organization cost K(X;w)translates into an increase of the firm’s optimal size X ∗ . Then, we may write the following: Proposition 4 1) An increase in the entry cost which translates into a larger size X ∗ produces an entry delay, i.e., ∂c ∗ ∂w <0. 2) An increase in the entry cost making for a larger size of the firm produces a decrease in the degree of vertical disintegration [as evidence in the introduction suggests], i.e., ∂α ∗ ∂w <0. Proof. See Appendix 4 18 6 Conclusions We have analyzed the decision to outsource input production totally or partially in a dynamic uncertain environment. The enterprise analyzed must decide entry, vertical mode and capacity. The firm is flexible and it can revise the vertical organization decision if market requires to do so. This flexibility makes for results which are at odds with received results of Industrial Organization, stating, among other things, that uncertainty is going to postpone the adoption of outsourcing. In our framework outsourcing provides a sort of cushion against risk and becomes more appealing in risky conditions. Therefore, outsourcing is anticipated as uncertainty soars, as common sense suggests. Nonetheless, flexibility makes for the level of vertical integration independent of uncertainty at entry, since the firm possesses an option to vary the level of vertical integration once it is in. Even this second result is at odds with received literature and is due to the extent of flexibility the firm is thought to possess. Finally, as evidence from the introduction and recent literature suggests, when size increases, the complexity of the outsourcing network may overcome that of internal organization leading to higher vertical integration for large firms. 19 A Proof of Proposition 1 Since ˆ A=αA, the optimal vertical arrangement is given by: α ∗ = arg max NPV EV I (c t , α)(21) = arg max p−d rX+αAc β 2 t −K 2α 2 . Then, the FOC is: Ac β 2 t −Kα = 0 (22) while the SOC is always satisfied. From (22) it is immediate to show that: α ∗ =1if c t ≤˜c A K c β 2 t if c t >˜c(23) where ˜c≡ K A  1/β 2 .Finally, by substituting α ∗ in (7) and (9), we get (11) in the text. B Proof of Proposition 2 The operating constraints to find Cand c ∗ are: F(c ∗ ) = NPV EV I (c ∗ , α ∗ (c ∗ )) (24) and F ′ (c ∗ ) = NPV EV I c (c ∗ , α ∗ (c ∗ )).(25) We distinguish two cases according to the value taken by the optimal trigger c ∗ : •If c ∗ ≤˜c(and d < c ∗ ), if the firm invests, it will choose α= 1 and the two conditions (24) and (25) become: Cc ∗β 2 =p−d rX+Ac ∗β 2 −K 2 and β 2 Cc ∗β 2 −1 =Aβ 2 c ∗β 2 −1 . 20 However, Cc ∗β 2 cannot simultaneously satisfy value matching and smooth pasting conditions with p−d r X+Ac ∗β 2 − K 2 .In other words, to make c ∗ ≤˜c, we need p−d r X≥ K 2 ,which implies α= 1.Then, it is not possible to obtain c ∗ . •If c ∗ >˜c(and d < ˜c), by (23) the firm will choose α < 1. Then the equations (24) and (25) become: Cc ∗β 2 =p−d rX+1 2 A 2 Kc ∗2β 2 and β 2 Cc ∗β 2 −1 =A 2 Kβ 2 c ∗2β 2 −1 . By some substitutions we get: A 2 Kc ∗2β 2 =p−d rX+A 2 Kc ∗2β 2 −K 2A K 2 c ∗2β 2 = 2p−d rX. Then, we have that: Ac ∗β 2 =2p−d rXK (26) C=A2p−d rK X > 0.(27) Finally, substituting (26) into (23) we get the value of α ∗ in the text. Let us now consider the conditions which guarantee the existence of the optimal trigger c ∗ . Recalling that A˜c β 2 =K, by (26), the first condition c ∗ >˜cis satisfied iff: 2p−d rXK < K. Therefore, since K > 0,we may write: p−d rX < K 2. 21 From (8) we get Ad β 2 = 1 β 1 −β 2 (r−γβ 1 )d 1 r(r−γ) ,and the second condition d < ˜cis satisfied iff: K < 1 β 1 −β 2 (r−γβ 1 )d1 r(r−γ). Putting all results together, we get: 2p−d rX < K < 1 β 1 −β 2 (r−γβ 1 )d1 r(r−γ). C Proof of Proposition 3 Let us consider first the effect of uncertainty on c ∗ .From (26) and the implicit function theorem we get: ∂c ∗ ∂σ =− ∂A ∂σ c ∗β 2 +A ∂β 2 ∂σ (ln c ∗ )c ∗β 2 Aβ 2 c ∗β 2 −1 where A= 1 β 1 −β 2 (r−γβ 1 )d 1−β 2 1 r(r−γ) .Taking the derivative of Awith respect to σwe obtain: ∂A ∂σ =− ∂β 1 ∂σ − ∂β 2 ∂σ (β 1 −β 2 ) 2 (r−γβ 1 )d 1−β 2 1 r(r−γ)− −1 β 1 −β 2 γ∂β 1 ∂σ d 1−β 2 + (r−γβ 1 )∂β 2 ∂σ (d 1−β 2 ) ln d1 r(r−γ) =1 r(r−γ) 1 (β 1 −β 2 )(r−γβ 1 )d 1−β 2 − ∂β 1 ∂σ − ∂β 2 ∂σ (β 1 −β 2 )−γ (r−γβ 1 ) ∂β 1 ∂σ −∂β 2 ∂σ ln d =A−1 (β 1 −β 2 ) ∂β 1 ∂σ + ( 1 (β 1 −β 2 )−ln d)∂β 2 ∂σ −γ (r−γβ 1 ) ∂β 1 ∂σ . Since ∂β 1 ∂σ <0and ∂β 2 ∂σ >0, d < 1is a sufficient condition to get ∂A ∂σ >0,even though we know that (as shown in Dixit and Pindyck, 1994, pp. 189-190 Figure 6.1) the result holds also for much higher values of d. Therefore, ∂A ∂σ >0, ∂β 2 ∂σ >0and c ∗ > d let us conclude that ∂c ∗ ∂σ >0. 22 Consider now the effect of uncertainty on α ∗ .Since the firm enters when c t =c ∗ ,from (23) and (26) it is immediate to show that ∂α ∗ ∂σ = 0.To verify this, we take the derivative of α ∗ = 1 K Ac ∗β 2 with respect to σ: ∂α ∗ ∂σ =1 K∂A ∂σ c ∗β 2 +A∂β 2 ∂σ (ln c ∗ ) + β 2 1 c ∗ ∂c ∗ ∂σ c ∗β 2 . This expression can be simplified by substituting 1 c ∗ ∂c ∗ ∂σ , i.e.: 1 c ∗ ∂c ∗ ∂σ =− ∂A ∂σ c ∗β 2 +A ∂β 2 ∂σ (ln c ∗ )c ∗β 2 Aβ 2 c ∗β 2 =− ∂A ∂σ Aβ 2 − ∂β 2 ∂σ (ln c ∗ ) β 2 . Then: ∂α ∗ ∂σ =1 K∂A ∂σ c ∗β 2 +A∂β 2 ∂σ (ln c ∗ )− ∂A ∂σ A−∂β 2 ∂σ (ln c ∗ )c ∗β 2  =1 K∂A ∂σ c ∗β 2 +A− ∂A ∂σ Ac ∗β 2  =1 K∂A ∂σ c ∗β 2 −∂A ∂σ c ∗β 2 = 0. D Proof of Proposition 4 Let us consider first the effect of won c ∗ . From (14) we get: ∂c ∗ ∂w =1 β 2  2 p−d r X ∗ (w)K(X ∗ (w); w) A  1/β 2 −1 × 1 A2p−d rX ∗ (w)K(X ∗ (w); w) −1/2 p−d r ∂X ∗ (w)K(X ∗ (w); w) ∂w Therefore the sign of ∂c ∗ ∂w is driven by the sign of − ∂X ∗ (b)K(X ∗ (b);b) ∂b ,i.e., ∂c ∗ ∂w ∝−∂X ∗ (w)K(X ∗ (w); w) ∂w =− ∂X ∗ (w) ∂w K(X ∗ (w); w)+ +X ∗ (w)K X (X ∗ (w); w) ∂X ∗ (w) ∂w +K w (X ∗ (w); w)<0. 23 Let us consider now the effect of won α ∗ . From (15) we get: ∂α ∗ ∂w =2 p−d r X ∗ (w) K(X ∗ (w), w) −1/2 p−d r ∂ X ∗ (w) K(X ∗ (w),w) ∂w . Again the sign of ∂α ∗ ∂w is driven by the sign of ∂ X∗(w) K(X∗(w),w) ∂w ,i.e., ∂α ∗ ∂w ∝∂ X ∗ (w) K(X ∗ (w),w) ∂w ≡∂ 1 K X (X ∗ (w),w) ∂w =−(K XX (X ∗ (w), w)) ∂X ∗ (w) ∂w −K Xw (X ∗ (w), w) K X (X ∗ (w), w) 2 <0 where the first equality follows from the FOC. 24