• 沒有找到結果。

Demand Uncertainty and the Choice of Business Model in the Semiconductor Industry

N/A
N/A
Protected

Academic year: 2021

Share "Demand Uncertainty and the Choice of Business Model in the Semiconductor Industry"

Copied!
20
0
0

加載中.... (立即查看全文)

全文

(1)

Demand Uncertainty and the Choice of Business Model in the Semiconductor Industry∗

YINGYI TSAI∗∗

National University of Kaohsiung, Taiwan. CHING-TANG WU

National Chiao Tung University, Taiwan

Abstract

In this paper, we provide another reason that may explain the wide adoption of outsourcing approach in the semiconductor industry. We show the fab-lite business model of outsourcing wafer fabrication to foundries is optimal in the presence of demand uncertainty. This is because outsourcing helps the integrated device man-ufacturer (largely the brand-producing firm) to lower its cost of capital investment in the case of low demand and to improve its capacity allocation in the case of high demand.

JEL classification: D24; D40; D81; E22; L23.

Keywords: Outsourcing; Capital Investment; Uncertainty.

We thank the Editor, two anonymous referees, Justin Yifu Lin, Preston McAfee, Abhinay

Muthoo, Roderick James McCrorie, and Hao Wang for their valuable suggestions and comments. We would also like to thank the seminar participants at the China Center for Economic Research (CCER) for providing important comments. The authors are greatly indebted to the National Science Council, Taiwan for the research grant (#NSC 92-2115-M-39-002-).

∗∗Corresponding author. Yingyi Tsai, Department of Applied Economics, National University

of Kaohsiung, Nan-Tzu Dist., Kaohsiung 811, Taiwan. E-mail: yytsai@nuk.edu.tw

(2)

1. Introduction

In the semiconductor industry, one of the most striking changes precipitated by rapid technological progress has been wide adoption of the fab-lite business model. Fab-lite refers to integrated device manufacturers or vertically integrated firms with a corporate strategy bent toward utilizing a fabless approach.1 Figure 1 illustrates the trend of growing number of the fabless semiconductor companies since 1995. And Figure 2 shows the increasing revenue growth of the fabless companies in the semiconductor industry. 0 10 20 30 40 50 60 70 80 1995 1996 1997 1998 1999 2000 2001 2002

No. of Fabless startups No. of Fabless companies

Source: Fabless Semiconductor Association, 2003

800 400 0 No . of Fabl es s st art u p s Total No. of F abless c o m p ani es

Figure 1. Growth of the Fabless Companies

Specialization and economies of scale have been identified as the major factors explaining this change. The usual argument is that a fab-lite model allows the fabless companies and integrated device manufacturers to focus on new product development by using efficiently its in-house resources (or facilities) on the one hand,

1“Fabless” refers to the business methodology of outsourcing the manufacturing of silicon wafers.

Fabless companies focus on the design, development and marketing of their products and form alliances with foundries, or silicon wafer manufacturers. And “Integrated Device Manufacturer (IDM)” refers to a class of semiconductor companies that owns an internal silicon fab or, alter-natively, the fabrication of wafers is integrated into its business. Nonetheless, even IDMs may undertake some outsourcing activities. “Foundry” is a service organization that caters to the pro-cessing and manufacturing of silicon wafers. It typically develops and owns the process technology or partners with another company for it.

(3)

0 5 10 15 20 1995 1996 1997 1998 1999 2000 2001 2002

Semi Industry Fabless

Source: Fabless Semiconductor Association, 2003

250 200 150 100 50 0 Fab less Rev enu e ($, b illio n ) Semi Ind u st ry Rev enu es ($ , billio n )

Figure 2. Revenue Growth of the Fabless Companies

and permits both front-end foundry (for wafer fabrication) and back-end foundry (for packaging and testing) to spread the costs of capital investment over different contracts on the other.

In this paper, we argue that demand uncertainty can be a reason for outsourcing in the fast changing industry like semiconductor. Setting aside as explanations for outsourcing of cost advantage (Abraham and Taylor, 1996; Feenstra and Hanson, 1996) and corporate strategy (Deavers, 1997; Shy and Stenbacka, 2003), we focus on the effect for outsourcing of demand uncertainty. A basic underlying idea is that, in the presence of demand uncertainty, outsourcing renders a firm the flexibility to balance a trade-off between having in-house facilities shortage while demand unex-pectedly surges and excess capacities otherwise. Although this assumption appears intuitively simple, it allows us to explore the rationale of outsourcing that can pro-vide important insights into the use of fab-lite approach in a fast changing sector like semiconductor. The main feature of this analysis is how the presence of outsourcing opportunity under demand uncertainty will affect a brand-producing firm’s choices of business model and in-house capital investment. We will show, in the presence of demand uncertainty, that an outsourcing (or non-integrated) business model is op-timal. By allowing for lower in-house capital investment, outsourcing renders higher profit than it otherwise would if the option to outsource is not available. Moreover,

(4)

we also explore the employment implications of outsourcing with a production func-tion incorporating both labor and capital. Our results show that low-wage need not drive an increase in the growing outsourcing activities. Hence, the general belief based upon cost advantage of low wage in explaining outsourcing is yet to be fur-ther studied. Throughout, in our discussion of outsourcing, we assume that only the manufacturing segment of end product is outsourced, that outsourcing market is competitive, namely, there is a mass of subcontractors competing for contracts, and that outsourcing is at a brand-producing firm’s disposal, but with a setup cost. The paper is organized as follows. After presenting the basic model in the next section, we characterize the optimal capital investment under uncertainty with no outsourcing as a benchmark. In Section 3, we present results on the choices of busi-ness model in equilibrium, and explore the implications for these decisions of demand uncertainty. Section 4 studies optimal capital investment under the business models of integrated and non-integrated production with demand uncertainty. In Section 5, we discuss our results and the role and applicability of the various assumptions that we make on production technology and distribution of random variable, thus providing some informal defense for interpreting our model as a qualitative charac-terization of reality. And we conclude in Section 6.

2. The Basic Model

Consider a brand-producing firm facing an inverse demand2 of

P = XY−ε, (1)

where P is the price, Y total production, ε ∈ (0, 1) an elasticity parameter, and X denotes an exogenous, absolutely continuous positive bounded random variable on a complete probability space (Ω, F , P).3

2This formulation implies the brand-producing firm alone faces overall demand shocks. For

similar characterization, see, Caballero (1991) for demand at the individual level facing a single competitive firm; and Pindyck (1993) for industry-wide demand shock facing a large number of equal-sized firms.

3The assumption of a bounded X suggests market demand should not tend to infinite. 4

(5)

An important message that emerges from Equation (1) suggests the brand price consists of two components: product design (for brand features) and basic manu-facture (for total supply). Hence, if we interpret X as brand-specificity,4 and Y the manufacturing segment for the brand, then this inverse demand function captures the impact on price of brand quality and quantity. Alternatively, Equation (1) char-acterizes a price reflecting the “qualitative” aspect of market demand owing to the indefinite outcome of “product innovation”.5

Production of Y requires the use of labor (L) and capital (K) by a linearly ho-mogeneous technology Y = K1−βLβ, where β ∈ (0, 1) denotes the share of L. To

make the point that capital investment is irreversible and not easily expandable,6 we assume production facilities must be installed before actual production can take place. Hence, the brand-producing firm first decides the amount of capital invest-ment K (K ≥ 0), it then chooses, upon the realization of actual market demand, labor employment L (L ≥ 0). The cost to capital and labor is denoted by γ and ω, respectively.

In the presence of demand uncertainty, the brand-producing firm’s problem is whether to adopt an integrated business model - in which case the firm produces Y using its own facilities to serve the market demand; or a non-integrated (or out-sourcing) model - in which case market demand is served with in-house production Y and, possibly, the purchase of y from other firms in the primary market.

In order to make our point in a manner as simple as possible, we assume the brand-producing firm withholds to itself the design of product, and decides whether to produce in-house or to purchase it from an independent specialist firm the manu-facturing segment of end product. Hence, in the present model we define outsourcing to mean that the brand-producing firm purchases the basic manufacturing compo-nent instead of carrying out the production of such compocompo-nent at its own facility, given an identical technology. We further assume, for simplicity, a unitary marginal cost for each unit of the manufacturing component. Thus, with the possibility to

4Alternatively, X consists of the variations in consumer taste, the changes in technology, and

even a changing market environment.

5See Levhari and Peles (1973) for a justification of this characterization on “product innovation”. 6Abel, Dixit, Eberly and Pindyck (1996) and Dixit and Pindyck (1998) argued that

”expand-ability” of capital investment in the future gives rise to call option while investigating the relations between optimal investment and uncertainty.

(6)

outsource basic production, the brand-producing firm faces a cost structure of (C + y)I{y>0}(y) =

(

C + y, if y > 0,

0, if y = 0,

where I{y>0} is the indicator function of {y > 0}, the price per unit outsourcing

output is normalized to one, and C represents the setup costs incurred in establishing an outsourcing partnership with the suitable subcontractor or in monitoring the contracts.7

Hence, the brand-producing firm’s total production cost is T C(K, L, y) =

(

ωL + γK, if y = 0 (in-house production)

ωL + γK + (C + y), if y > 0 (outsourcing production) (2) The choices of business model and capital investment are made in the context of uncertainty. Market demand condition is not known until the firm enters actual production, given the chosen production mode (cf. Sandmo, 1971; Pindyck, 1988). This timing reflects that outsourcing can serve as a device of mitigating the gap between unexpected demand shock and in-house production constraints underlying in capital investment, and that any ex post adjustment is not possible since it is costly to alter the decisions over business model or capital investment in the light of new market information.

Using Equations (1) and (2), the brand-producing firm’s profit is given by π(K, L, y) = ( πN(K, L) = X(K1−βLβ)1−ε− wL − γK, in-house πO C(K, L, y) = X(K1−βLβ + y)1−ε− wL − γK − (C + y), outsourcing (3) Equation (3) highlights the problem facing a brand-producing firm in the presence of demand uncertainty, that is, it has to balance a trade-off between an irreversible capital investment with possible idle capacity and the option of avoiding such in-vestment but having to incur a cost for outsourcing partnership and even paying the subcontractor a premium.

7Grossman and Helpman (2002) provide an intuitive justification for this formulation since

“... there are fixed costs associated with ... searching for a potential supplier”. Further, this characterization of total outsourcing cost is similarly captured by Shy and Stenbacka (2003), who modeled outsourcing in terms of a trade-off between the “make-or-buy” decision, except that we consider here a unitary marginal cost per unit outsourcing output.

(7)

Suppose a business model of integrated production (alternatively, outsourcing pro-duction is not possible) is chosen. This characterization corresponds to a standard model of optimal capital investment under uncertainty in which a monopolist facing uncertain future demand chooses the amount of capital investment. The following proposition characterizes the optimal capital investment in a model of integrated production.

Proposition 1. If it chooses an integrated production mode, then the brand-producing firm raises its capital investment, KN, with a greater expected market demand, i.e.,

KN = Hβ,ε  E h X1−β(1−ε)1 i1−β(1−ε)ε , for some positive constant Hβ,ε.

Proof. See Appendix A. 

Proposition 1 implies that higher expected market demand will lead to greater capital investment. Thus, the driving force for a high demand deserves careful investigation. Successful quality improvement or new product features provide an example. Levhari and Peles (1973) showed formally, in a deterministic setting, that quality improvement (as a form of product innovation) is able to raise market demand. This question of interpretation is important as it bears upon the issue of the nature of uncertainty. Indeed, if we interpret X as an indefinite outcome of product innovation, the brand-producing firm will increase its capital investment when it expects to successfully deliver new invention (or improve upon product quality) even it is not possible to outsource. Further, the above result also suggests that the brand-producing firm will increase its capital investment as the market demand becomes more volatile.8

8Given two positive random variables X

1and X2, each with probability distribution µ1and µ2.

Using F¨ollmer and Schied (2002), we know that if µ1 is uniformly preferred over µ2, and X1 and

X2 have the same mean, there exists a “mean preserving spread” Q such that µ2 = µ1Q. Since we have obtained, for given X1and X2, that the relative optimal capital investments K1N and K2N

exhibiting KN

1 ≤ K2N. Hence, it follows that the brand-producing firm will increase the capital

investment as X becomes “riskier”, i.e., more volatile.

(8)

3. Choices of Business Model

Proposition 2 below establishes the conditions under which the business model of non-integrated production (implying the possibility to outsource) is chosen in the presence of demand uncertainty.

Proposition 2. For any C and K, there exists a critical XC∗(K) such that (i) yO(K) > 0 if X > X∗ C(K), and (ii) yO(K) = 0 if X ≤ X ∗ C(K), where XC∗(K) =      1 1 − ε  β w 1−ββε Kε, if C = 0, sup{X : πO C(K, LO(K), yO(K)) ≤ πN(K, LN(K))}, if C > 0,

Proof. See Appendix B. 

y X X* C>0 y X X* C=0 (a) (b)

Figure 3. The Impact of C on X∗

Proposition 2 implies that outsourcing takes place only when the realized market demand is sufficiently large, and that the setup cost of outsourcing has a decisive impact on firm’s choice of business model. Figure 3 illustrates the role of the out-sourcing setup cost in affecting the outout-sourcing amount y.

Appropriately interpreted, a sufficiently small setup cost implies a negligibly low price of contracting with a compatible supplier. Thus, the outsourcing firm is able to work with subcontractor(s) without much difficulty whenever the realized market demand exceeds its in-house capacity limit. This explains the continuous curve characterizing y and X as shown in Figure 3(a). An increase in the setup cost implies it becomes more costly to engage in outsourcing activities. The brand-producing firm now faces a trade-off between gains from outsourcing with rising

(9)

setup cost and losses due to capacity shortage when market demand surges. The discontinuity of y at XC∗(K) in Figure 3(b) implies the outsourcing firm will not use outside resources unless the gains from outsourcing exceeds the setup cost at the marginal level.

4. Optimal Capital Investment under Uncertainty

In the presence of uncertainty, the brand-producing firm chooses the capital in-vestment to maximize its expected profit conditional on the available information. Thus, the brand-producing firm faces, depending upon whether it is possible to outsource or not, an optimal capital investment problem of

max K≥0 ( EπN(K, LN(K)) , in-house EπO C(K, LO(K), yO(K))I{X>X∗ C(K)}+ π N(K, LN(K))I {X≤X∗ C(K)} , outsourcing (4) Proposition 3 characterizes the firm’s optimal capital investment under demand uncertainty with two models of integrated and non-integrated production.

Proposition 3. In equilibrium, the capital investment levels with outsourcing and without, denoted by KO

C and KN respectively, satisfy the following properties: (1)

KO

C increases in C. (2) KCO tends to KN as C → ∞.

Proof. See Appendix C. 

Proposition 3 suggests a brand-producing firm will reduce its capital investment when the option of outsourcing is available. Nevertheless, it raises such investment for a higher setup cost of outsourcing. In the extreme, as the cost of outsourcing becomes prohibitively high, the amount of capital investment under outsourcing approximates that in the absence of outsourcing. This result could therefore be interpreted as searching for the “compatible” partner in the outsourcing relationship. We have, thus, identified the conditions under which capital investment are chosen in both business models of integrated and non-integrated production.

5. Discussion

Sections 5.1 and 5.2 state, and comment upon, some of the main properties of the equilibrium results as described in Propositions 2 and 3. In Section 5.4 we argue

(10)

that our results are robust to modification in the technology of the basic production and in the distribution of the random variable.

5.1. Implications for Outsourcing Choices of Uncertainty. To explore the implications for choices of business models of demand uncertainty, we further inves-tigate the properties of XC∗(K).

Lemma 1. (1) XC∗(K) is strictly increasing both in C and in K. (2) For any C ≥ 0, XC∗(K) decreases in w and is independent of γ.

Proof. See Appendix D. 

Lemma 1 provides an important insight into debates over the nature of outsourc-ing. If we interpret XC∗(K) as the firm boundary, then Part (1) of Lemma 1 implies that idiosyncratic investment and industry-specific characteristics, such as capital-and/or labor-intensity for production and knowledge content in the product, and their interactions play a critical role in determining the outsourcing choices. Indeed, a low setup cost implies the brand-producing firm need not devoting much effort while monitoring the existing (or establishing for) outsourcing relationship. Hence, the boundary beyond which it chooses for outsourcing is low as the setup cost drops. Further, the critical value of this boundary is also affected by factor price. Part (2) of Lemma 1 suggests the higher the wage rate, the lower the boundary beyond which outsourcing occurs. Appropriately interpreted, the results characterize the growing off-shoring of basic production abroad from the developed economies subsequent to an increase in their domestic wages, in particular, under demand uncertainty. No-tice, nevertheless, that outsourcing boundary is independent of capital price since it was incurred (a sunk cost) prior to actual production.

5.2. Implications for Job Losses of Outsourcing. The following Lemma char-acterizes the conditions for factor employment, given the chosen business model and optimal capital investment.

Lemma 2. (1) If  γ 1 − β 1−β w β β ≥ 1, then KO 0 = 0, LO0 = 0 and yO= (X(1 − ε))1ε. Moreover, KO

C = KN for all C if and only if X = 0 a.s.; and if P(X > 0) > 0, 10

(11)

KO 0 < KN. (2) If  γ 1 − β 1−β w β β < 1 and X1−β(1−ε)1 ≤ 1 − β γ  β w 1−ββ E[X1−β(1−ε)1 ] a.s., (5)

then KCO = KN for all C. Conversely, if (5) does not hold almost everywhere, K0O < KN.

Proof. See Appendix E. 

The above results can provide an important insight into the issue of job losses in the presence of outsourcing. Indeed, Lemma 2 highlights that the extent of outsourcing is determined crucially by the marginal cost of in-house production relative to the price of purchasing from the specialized subcontractor. Part (1) reflects that the brand-producing firm may simply purchase the basic manufacturing component and not enters the primary market at all if using own facilities is relatively costly. And Part (2) characterizes a firm’s choice under situations when there is substantial divergence between the expected market demand and the realized one. This result offers a justification as to why brand-producing firm does not opt for complete outsourcing and retains some in-house production capacities (or capital investment) even though the outsourcing opportunity is available.

5.3. Total Output under Integrated and Non-integrated Production. Theorem 1. In equilibrium, there exists YC∗(KO

C) ≥ X ∗ C(KCO) such that (1) if X > YC∗(KO C), then YCO+ yO > YN. (2) if X ≤ YC∗(KO C), then YCO+ yO≤ YN.

Proof. Using the results in Appendices A-E, it is straightforward to verify this result.  Intuitively, greater uncertainty implies a higher required return on the use of outside resources if outsourcing is adopted. Hence, when the realized market demand is sufficiently high, the possibility to outsource provides the firm an avenue for profit increases by raising total output.

(12)

5.4. Comments on Production Technology and the Distribution of X. How would the results obtained in this paper change if we consider a general function of the basic production? Would the results change if a different specification for demand uncertainty is employed? We now sketch an argument that establishes the outcome is, in fact, unaffected: that is, the equilibrium choices of business model and capital investment (in an alternative setting) are identical to the equilibrium ones (in the present setting).

Clearly, the brand-producing firm decides the business model and, simultaneously, the capital investment. In the presence of demand uncertainty, the adjustment mechanism made available to the firm consists of the variable input of labor and the opportunity to outsource, if chosen. This suggests that capital investment, once installed, is independent from the variations in market demand. It is, therefore, evident that our results are robust to any modification of production technology that involves the use of capital and labor so long as the assumption of irreversibility and in-expandability for capital are retained.

We have investigated the issue of uncertainty by characterizing X as random variable. It may, however, be reasonable (in some contexts) to explore different distribution of this variable. In that case one can still define the critical value of XC∗(K) and the optimal capital investment. With such a change in the setting that defines the firm profit, our results are unaffected. Propositions 1-3 still describe the equilibrium choices.9

6. Conclusion

In this paper, we show that outsourcing provides brand-producing firms with increased flexibility in adjusting their resources as new information about demand conditions become available. This argument can easily be extended to encompass even anticipated demand shifts, such as seasonal factors. Thus, outsourcing may well reflect an effort to deal with non-perfectly positively correlated anticipated demand variations.

Assessing the importance of demand uncertainty in explaining the wide adoption of the outsourcing business model would require the following: first, evaluation of the importance of demand uncertainty in various industries; second, examination of

9However, with this change, the arguments and proofs are lengthier and restrictive. 12

(13)

the extent to which random demand components are correlated across industries; and third, investigation for the importance of production networks and adjustment costs. Although some industry studies seem to confirm both the presence of a market idiosyncratic uncertainty as well as the presence of inflexibility in capital investment, the issue at hand still begs for a more rigorous empirical analysis.

Finally, it is important to note that the interaction between demand uncertainty and such factors as cost considerations, consumer preferences and strategic interac-tions - none of which are dealt with in the present paper - may yield important new insights. In particular, the welfare implications of outsourcing can be formally ad-dressed once consumer utility is incorporated. Thus, analyses of these interactions feature high in our research agenda.

Appendix A

Proof of Proposition 1. We derive the optimal capital investment using the back-ward induction. Upon the realization of market demand, for given K ≥ 0, the firm chooses an optimal labor employment to maximize

πN(K, L) = X(K1−βLβ)1−ε− wL − γK.

It is east to verify that the optimal labor employment LN(K) exists using the first and second order conditions. Hence, the optimal choice of labor implies a firm profit of πN(K, LN(K)) = Gβ,εX 1 1−β(1−ε)K (1−β)(1−ε) 1−β(1−ε) − γK, (6)

where Gβ,ε is a positive constant depending on β and ε.

The firm then decides on its capital investment to maximize the expected profit, i.e., max K≥0E[π N(K, LN(K))] = G β,εE[X 1 1−β(1−ε)]K (1−β)(1−ε) 1−β(1−ε) − γK. (7)

Using standard techniques to analyze the maximization problem of Equation (7), it is easy to verify that the first- and second-order conditions of (7) are satisfied, and, thus, gives us the required results. Notice that the direction of change of optimal capital investment under uncertainty depends only on the convexity effect, i.e., for any 0 < β < 1, KN is convex in X (cf. Hartman, 1972).

(14)

Appendix B

Proof of Proposition 2. If outsourcing is possible (y ≥ 0), then the firm chooses, for given K, an optimal labor employment to solve

max

L,y≥0π

O

C(K, L, y), (8)

To investigate (8), we first consider an optimization problem of “definite outsourc-ing” (i.e., y > 0), that is,

max

L,y≥0π˜

O

C(K, L, y) = X(K

1−βLβ+ y)1−ε− wL − γK − (C + y). (9)

Clearly, the relation between πO

C(K, L, y) and ˜πCO(K, L, y) is given by πOC(K, L, y) =      ˜ πCO(K, L, 0) + C ≥ ˜πCO(K, L, 0), for y = 0, ˜ πO C(K, L, y), for y > 0. (10)

The optimal solution to (9) exists if ∂ ˜πOC ∂L = X(1 − ε) K 1−βLβ + y−ε βK1−βLβ−1− w = 0, ∂ ˜πOC ∂y = X(1 − ε) K 1−βLβ + y−ε − 1 = 0.

This implies that optimal L and y, in equilibrium, must be given by ˜ LO(K) =  β w 1−β1 K, (11) ˜ yO(K) = (X(1 − ε))1ε − β w 1−ββ K. (12)

Further, it is easy to verify the second-order condition is satisfied. Using (12), we see that ˜yO(K) ≥ 0 if and only if X ≥ ˜X(K) := 1

1 − ε  β

w 1−ββε

Kε. This implies the value of X plays a crucial role in determining the optimal solution of (9). Thus, we separate our discussion into two cases: {w ∈ Ω : X(w) ≤ ˜X(K)} and {w ∈ Ω : X(w) > ˜X(K)}.

First, on the set {w ∈ Ω : X(w) ≤ ˜X(K)}, we have ˜yO(K) ≤ 0. Hence,

the optimal solution to equation (9) occurs on the boundary y = 0. This coin-cides with the case of no outsourcing, and, thus, the optimal solution of (8) is (LO(K), yO(K)) = (LN(K), 0).

(15)

Second, on the set {w ∈ Ω : X(w) > ˜X(K)}, an interior solution holds and ( ˜LO(K), ˜yO(K)) is the optimal solution to (9). Note, however, from (10) that

πOC(K, L, 0) = ˜πCO(K, L, 0) + C. We, therefore, compare if

πOC(K, ˜LO(K), ˜yO(K)) > πn(K, LN(K)).

Since the monopolist involves outsourcing if and only if the payoff generated from doing so is greater than it otherwise would have been. For fixed K, define by FC(X, K) the payoff differences between the two outsourcing regimes, i.e.,

FC(X, K) = πOC(K, ˜L

O(K), ˜yO(K)) − πN(K, LN(K)) > 0. (13)

A straightforward calculation shows, for any X > ˜X(K), that ∂FC

∂X > 0. This implies FC(·, K) is strictly increasing to infinite on ˜X(K), ∞

 . Furthermore, notice that FC ˜X(K), K  = −C.

It follows that if C = 0, F0(X, K) > F0( ˜X(K), K) = 0 for all X > ˜X(K), and

if C > 0, FC ˜X(K), K



< 0. Since FC(·, K) is strictly increasing in X to ∞ on

 ˜X(K), ∞

, we know that FC(X, K) = 0 has a unique solution for X > ˜X(K).

And, the solution is given by

XC∗(K) = sup{X : FC(X, K) ≤ 0}. (14)

Thus, we have established that FC(X, K) > FC(XC∗(K), K) = 0 for all X > X ∗

C(K).

This result suggests that, given K ≥ 0, the firm will engage in outsourcing (yO(K) >

0) when X is sufficiently large (i.e., X > XC∗(K)).

Appendix C

Proof of Proposition 3. Due to Proposition 2, we write the monopolist’s expected profit as GC(K) := EπOC(K, L O(K), yO (K))I{X>X∗ C(K)}+ π N(K, LN (K))I{X≤X∗ C(K)}  (15) To prove Proposition 2, we proceed in the following four steps.

(16)

Step 1. The existence of KO

0 . Given C = 0, differentiating (15) with respect to K,

we have G00(K) = (1 − β) β w 1−β(1−ε)β(1−ε) (1 − ε)1−β(1−ε)1 K −ε 1−β(1−ε)U 0(K), (16) where U0(K) = E h X1−β(1−ε)1 − X∗ 0(K) 1 1−β(1−ε)  I{X≤X∗ 0(K)} i +X0∗(K)1−β(1−ε)1  1 − "  γ 1 − β 1−β w β β# 1 1−β . (17) Thus, we separate our discussion of (17) into two cases:

(i)  γ 1 − β 1−β w β β

≥ 1. Since the first term in (17) is strictly negative for K > 0, G00(K) < 0 for any K > 0. This suggests G0(K) has a global

maximum at K = 0, i.e., KO 0 = 0. (ii)  γ 1 − β 1−β w β β < 1. Due to U00(K) = M1K −(1−β)(1−ε)1−β(1−ε)  P[X ≥ X ∗ 0(K)] − "  γ 1 − β 1−β w β β# 1 1−β , with a positive constant M1. Using the fact that U0(0) = 0, we see that

U0(K) > 0 if K is small enough. Moreover, we have established that

U0(K) −→ −∞ if K → ∞. Therefore, there exists a unique K0O such

that U0(K0O) = 0, which suggests G00(K0O) = 0. Hence, G0(K) has a global

maximum at KO

0 ∈ (0, ∞).

Step 2. The existence of KO

C for general C. Differentiating (15) with respect to K,

we have

G0C(K) = M2K

− ε

1−β(1−ε)U

C(K), (18)

where M2 is a positive constant and

UC(K) = U0(K) + E h X1−β(1−ε)1 − X∗ 0(K) 1 1−β(1−ε)  I{X∗ 0(K)<X<X ∗ C(K)} i . (19) Recall FC(XC∗(K), K) = 0, and the results obtained in Step 1, we have U0(K) −→

−∞ as K → ∞. Moreover, notice E[X1ε] < ∞, the second term in (19) is strictly

positive and bounded. Thus, UC(K) < 0 as K large enough. Together with 16

(17)

UC(0) ≥ 0, we have established that there exists a zero of UC(K), which is also

global maximum of GC(K). This proves the existence of KCO for general C.

Step 3. The monotonicity of KCo. From (18) we have, for fixed K, C1 > C2 ≥ 0,

G0C1(K) − G0C2(K) = M2K

− ε

1−β(1−ε)(U

C1(K) − UC2(K)).

Recall, from Lemma 1, that X0∗(K) < XC2(K) < XC1(K). This implies UC1(K) − UC2(K) = E h X1−β(1−ε)1 − X∗ 0(K) 1 1−β(1−ε)  I{X∗ C2(K)<X<X ∗ C1(K)} i > 0. Hence G0C 1(K) > G 0 C2(K), (20)

for all K. Following the results obtained in Step 2 we know GC2(K) has at least one

local maximum (which occurs at the points such that G0C

2(K) = 0). For simplicity,

we assume that GC2(K) has two local maxima: at K1 and at K2 with K1 < K2 (For

the case with one and n local maxima, we may use the similar argument.). (i) KO C2 = K1 = 0. Clearly K O C1 ≥ 0 = K O C2.

(ii) KCO2 = K1 > 0, i.e., GC2(K1) ≥ GC2(K2). Due to (20) and since G

0

C2(K) > 0

for all K < K1, G0C1(K) > G

0

C2(K) ≥ 0 for all K ≤ K1. Following Step 2, we

know that G0C1(K) = 0 has at least one solution. Together with G0C1(K) > 0 for all K < K1, we see that all the zeros of G0C1(K) is larger than K1, i.e.,

KO C1 > K1 = K O C2. (iii) KO C2 = K2, i.e., GC2(K1) < GC2(K2). Due to GC2(K2) = GC2(K1) + Z K2 K1 G0C2(K)dK, we have Z K2 K1 G0C 2(K)dK > 0. (21)

Suppose that the global maximum of GC1(K), K

O

C1, is less than K

O

C2 = K2.

Using (20) we know G0C1(K) > G0C2(K) > 0 for all K < K1. Thus, there

exists a solution to G0C 1(K) = 0, denoted by K O C1, between K1 and K2. Furthermore, since G0C 1(K2) > G 0 C2(K2) = 0 and G 0 C1(K) < 0 as K large

enough (see Step 2), GC1(K) has at least one local maximum larger than

(18)

K2, say ¯K. Because of (20) and (21), we get GC1( ¯K) = GC1(K O C1) + Z K¯ KC1O G0C 1(K)dK > GC1(K O C1) + Z K2 KC1O G0C 1(K)dK > GC1(K O C1) + Z K2 KO C1 G0C2(K)dK > GC1(K O C1) + Z K2 K1 G0C2(K)dK > GC1(K O C1),

which clearly contradicts the result that KCO1 is the global maximum of GC1(K). This implies the global maximum for GC1(K) must occur at the

place larger than K2, i.e., KCO1 > K2 = K

O

C2.

Hence, KO C1 > K

O

C2 for C1 > C2. In other words, K

O

C is increasing in C.

Step 4. Using Equation

πCO(K, LO(K), yO(K)) = πCO(K, LO(K), yO(K))I{X>X∗

C(K)}+π

N

(K, LN(K))I{X≤X∗ C(K)},

and note that XC∗(K) −→ ∞ as C → ∞, it is easy to verify that GC(K) −→

E[πN(K, LN(K))] as C → ∞. Further, using Step 3, we know that KO

C is increasing

in C, thus KO

C ≤ Kn for all C.

Appendix D Proof of Lemma 1. (1) If C = 0, then ∂X

∗ 0(K) ∂K = ε 1 − ε  β w 1−ββε Kε−1 > 0; and if C > 0, using (14) and note that FC(XC∗(K), K) = 0, and that FC(X, K) is

continuous in K, we know XC∗(K) must satisfy X1εε(1 − ε) 1−ε ε + β w 1−ββ (1 − β)K − C = β(1 − ε) w 1−β(1−ε)β(1−ε) (1 − β(1 − ε))X1−β(1−ε)1 K (1−β)(1−ε) 1−β(1−ε) . (22)

Differentiating (22) with respect to K and C, respectively, we have both ∂X

∗ C(K) ∂K and ∂X ∗ C(K)

∂C are strictly positive for any K > 0, implying that X

C(K) is strictly

increasing both in K and C.

(2) Using (22), it is easy to verify that both of ∂X

∗ 0(K) ∂w and ∂XC∗(K) ∂w are strictly negative since A < 1. 18

(19)

Appendix E Proof of Lemma 2. (1)  γ 1 − β 1−β w β β

≥ 1. Using the results contained in Proposition 1 and Step 1 in the proof of Proposition 3, we know if

 γ 1 − β 1−β w β β ≥ 1, KN = KO

0 = 0 if and only if X ≡ 0 a.s.

(2)  γ 1 − β 1−β w β β

< 1. As shown in the Step 1 in the proof of Proposition 3, it is easy to show that if

 γ 1 − β 1−β w β β

< 1, the equation G00(K) = 0 has two different solutions 0 and KO

0 , and

G00(K) (

> 0, if K ∈ (0, K0O),

< 0, if K > K0O. (23)

Thus, KN = K0O if and only if G00(KN) = γ EhX1−β(1−ε)1 i E h (X0∗(KN))1−β(1−ε)1 − X 1 1−β(1−ε)  I{X≥X∗ 0(KN)} i = 0.

This implies that KN = KO

0 if and only if  X0∗(KN)1−β(1−ε)1 − X 1 1−β(1−ε)  I{X≥X∗ 0(KN)} = 0, a.s., which is equivalent to X1−β(1−ε)1 ≤ X∗ 0(KN) 1 1−β(1−ε) = 1 − β γ  β w 1−ββ E[X1−β(1−ε)1 ] a.s. References

[1] Abel, A. B., Dixit, A. K., Eberly, J. C., and Pindyck, R. S. “Options, the Value of Capital, and Investment.” Quarterly Journal of Economics 111 (1996): 753-777.

[2] Abraham, K. G., and Taylor, S. K. “Firm’s Use of Outside Contractors: Theory and Evi-dence.” Journal of Labor Economics 14 (1996): 394-424.

[3] Caballero, R. “On the Sign of the Investment-Uncertainty Relationship.” American Economic Review 81 ( 1991): 279-288.

[4] Deavers, K. L. “Outsourcing: a Corporate Competitiveness Strategy, Not a Search for Low Wages.” Journal of Labor Research 18 (1997): 503-519.

[5] Dixit, A. K., and Pindyck, R., S. “Expandability, Reversibility, and Optimal Capacity Choice.” National Bureau of Economic Research, Working Paper, No. 6373, 1998.

[6] Domberger, S. The Contracting Organization: A Strategic Guide to Outsourcing. Oxford: Oxford University Press, 1998.

(20)

[7] Fabless Semiconductor Association (FSA). Publications - Outsourcing Trends, available at http://www.fsa.org/pubs/outsourcingTrends/default.asp, 2003.

[8] Feenstra, R. C., and Hanson, G. H. “Globalization, Outsourcing and Wage Inequality.” Amer-ican Economic Review 86 (1996): 240-245.

[9] F¨ollmer, H., and Schied, A. Stochastic Finance. An Introduction in Discrete Time. Berlin: Walter de Gruyter, 2002.

[10] Grossman, G., and Helpman, E. “Integration Versus Outsourcing in Industry Equilibrium.” Quarterly Journal of Economics 117 (2002): 85-120.

[11] Hartman, R. “The Effects of Price and Cost Uncertainty on Investment.” Journal of Economic Theory 5 (1972): 258-266.

[12] Lee, J., and Shin, K. “The Role of a Variable Input in the Relationship Between Investment and Uncertainty.” American Economic Review 90 (2000): 667-680.

[13] Levhari, D., and Peles, Y. “Market Structure, Quality and Durability.” The Bell Journal of Economics and Management Science 4 (1973): 235-248.

[14] Pindyck, R. S. “A Note on Competitive Investment under Uncertainty.” American Economic Review 83 (1993): 273-277.

[15] Sandmo, A. “On the Theory of the Competitive Firm under Price Uncertainty.” American Economic Review 61 (1971): 65-73.

[16] Shy, O., and Stenbacka, R. “Strategic Outsourcing.” Journal of Economic Behavior and Or-ganization 50 (2003): 203-224.

數據

Figure 1. Growth of the Fabless Companies
Figure 2. Revenue Growth of the Fabless Companies
Figure 3. The Impact of C on X ∗

參考文獻

相關文件

fostering independent application of reading strategies Strategy 7: Provide opportunities for students to track, reflect on, and share their learning progress (destination). •

Strategy 3: Offer descriptive feedback during the learning process (enabling strategy). Where the

Now, nearly all of the current flows through wire S since it has a much lower resistance than the light bulb. The light bulb does not glow because the current flowing through it

O.K., let’s study chiral phase transition. Quark

H., Liu, S.J., and Chang, P.L., “Knowledge Value Adding Model for Quantitative Performance Evaluation of the Community of Practice in a Consulting Firm,” Proceedings of

In terms of “Business Model Canvas,” the Value Proposition of Humanistic Buddhism is “to establish the Buddha’s vocation in the world.” Given that a specific target audience

There are existing learning resources that cater for different learning abilities, styles and interests. Teachers can easily create differentiated learning resources/tasks for CLD and

• Thresholded image gradients are sampled over 16x16 array of locations in scale space. • Create array of