Strategic Investment under Incomplete Information

A.1 Derivation of WiF(x;xjC,xiF)

Here, I explain how to derive WiF(x;xjC,xiF). I use the standard investment model as described by Dixit and Pindyck (1994). The Bellman equation for the follower before the leader invests is

Using Ito’s lemma, the Bellman equation leads to the following differential equation:

The general solution to the above equation is

where λ is

The value function needs to satisfy two boundary conditions. The first one is

This condition is necessary to make sure that when x=0, the firm has no value. It follows from this condition that B=0. The second condition is necessary to make sure the value function for the follower is continuous at the threshold the leader invests.

Note that in eq. (4), for x=xjC,

and what remains is ViF(xjC;xiF). Hence, the function described for x<xjC is the solution to eq. (38). □

A.2 Lemmas Related to Section 4

Proof of Lemma 4.1. In order to show that supFi=supFj, suppose it does not hold and without loss of generality assume supFi<supFj. Then it is optimal for firm i with the investment cost IU to invest at supFi. As x gets close to supFi, firm j knows that firm i will invest in any moment with probability one. Since it is optimal for firm i with investment cost IU to invest at supFi, it would be optimal for firm j with investment cost arbitrarily close to IU to invest at supFi or before. This contradicts the initial assumption that supFi<supFj. □

Proof of Lemma 4.2. Suppose that in equilibrium xL(IL)<infFj. It follows from D10−D00>D11−D01 that xiL<xiF for any value of investment cost, and value of xL(IL) is less than any possible value of xjF. Also, from k∗∗=1, it follows that for firm i with the investment cost IL, investing as a first mover at xL(IL), is preferred to waiting for simultaneous investment at any threshold larger than xL(IL). Thus, there is no reason firm i with the investment cost IL to invest later than xL(IL) and it is optimal to invest at xL(IL) and it should be that infFi=xiL(IL). Since firm j knows that firm i invests at xL(IL) or before with zero probability, it is also optimal for firm j with the investment cost IL to invest at xL(IL). This contradicts with the assumption that xL(IL)<infFj. As a result infFj≤xL(IL) for j=1,2.

Note that it cannot be the case that infFi<infFj≤xL(IL), because firm i with the lowest investment cost prefers to invest at any threshold in (infFi,infFj), rather than infFi. Hence, infFi=infFj. In conclusion, if D10−D00>D11−D01 and k∗∗=1, in any equilibrium, infFi=infFj≤xL(IL). □

Proof of Lemma 4.3. The first order derivative is f′(y)=1β−1yβ−1−1. It is zero only at y=1 and the second order derivative is positive. Hence, for any y≠1, f(y)>f(1)=0. □

Proof of Lemma A.1. Since D10>D11, from eqs (10) and (9) it follows that xiL<xiS. Given that firm j has not invested yet, firm i anticipates that there are two possible outcomes after investing at xiI, firm j either invests at xiI or waits and invests after a delay. If firm i knew with certainty that firm j would not invest at xiI, then it would not be optimal to differ the investment after xiL is reached, and the optimal xiI could not be larger than xiL. Also, if firm i knew with certainty that firm j would invest immediately at xiI, then it would not be optimal to differ the investment after xiS is reached, and the optimal xiI could not be larger than xiS. The payoff keeps decreasing after the optimal threshold is reached. Since xiL<xiS, when firm i does not know which of these two possible outcomes will happen, it is never an optimal choice to delay the investment after xiS is reached. □

A.3 Proof of Proposition 4.4.

It follows from D10−D00>D11−D01 that xiL<xiF. Adding this to the assumption infFj≤xiL guarantees that infFj<xiF. For any xiI∈[infFj,xiF], the payoff function is

The derivative of the above function with respect to xiI is

I use eqs (8), (9) and (10) to replace any Dninj with a combination of xiF, xiL and xiS. This gives more tractable and intuitive equations. Also remember that x and xˆ are smaller than any investment threshold. The derivative of WiI(x,xˆ;xiI) with respect to xiI at any xiI∈[infFj,xiF], is

Setting the above derivative equal to zero, gives eq. (16).

If xiF<supFj, for any xiI in the interval [xiF,supFj], the payoff function is

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

From Lemma 4.3 it follows that f(1)=0, as a result

The value of xiL is less than xiS by definition and xiL<xiF as I showed before. Also from xjI(Ij)≤xF(Ij), by replacing Ij with Ij(xiI), it follows that xiI<xF(Ij(xiI)). Hence, value of the derivative in eq. (48) is negative for any xiI∈(xiF,supFj].

Because xiL≥infFj, for xiI<infFj≤infXF, the derivative ddxiIWiI(x,xˆ;xiI), which is equal to ddxiIEIj|xˆ[WiL(x;xiI,xF(Ij))], has a positive value. Also, choosing xiI>supFj is not any better than choosing supFj, because the other firm invests by the time supFj is reached.

The payoff function is continuous everywhere and differentiable inside Fj, the derivative is positive for xiI<infFj and negative for xiI>xiF. Hence, the best choice of xiI for firm i exists and it is in interval [infFj,xiF], where the derivative given by eq. (46) is equal to zero. □

A.4 Proof of Proposition 4.5.

It follows directly from Proposition 4.4 that, if I1(x)=I2(x)=I(x) as characterized in eq. (22), then it is an equilibrium and I−1(.)≤xF(.). Note that all the assumptions of Proposition 4.4 hold in this equilibrium.

To show the uniqueness, first I show that there is no equilibrium in which xiI>xiF for any firm with any investment cost. There is no equilibrium in which xF(Ij)<xjI(Ij) for any Ij in G, because it contradicts Lemma 4.2. Consider the only remaining case, an equilibrium in which xF(Ij)−xjI(Ij) changes sign at some Ij∈G. Let I˜j be the smallest zero of xF(Ij)−xjI(Ij) and without loss of generality assume that I˜i≤I˜j. It follows from the assumption of this proposition that xiL<xiF, hence, Lemma 4.2 guarantees that xF(IL) is greater than infFj. As a result xF(Ij)−xjI(Ij)>0 for any Ij less than I˜i, for j=1,2. Thus eq. (46) gives the value of left derivative of payoff function. For a firm i that invests at I˜i in equilibrium, it follows that xiF=xiI(I˜i)<xF(Ij(xiI(I˜i))). Thus, the derivative at x˜i=xiI(I˜i) from left side is

and has a negative value. Hence, there is some value of xiI less than x˜i that is preferred to x˜i and this contradicts with x˜i being chosen in equilibrium. As a result, in any equilibrium, it holds that xiI≤xiF, and any equilibrium must be a solution to problem eq. (18). Since the boundary conditions for the functions I1(x) and I2(x) are the same, there is no asymmetric solution that solves eq. (18). See Lambrecht and Perraudin (2003) for more details on a similar problem.

In conclusion, there is no equilibrium other than what is characterized by eq. (22) and always it holds that I−1(.)≤xF(.). □

A.5 Proof of Proposition 4.7.

To prove Proposition 4.7, first note that it follows from D11−D00<D10−D00≤D11−D01 that xiF≤xiL<xiS.

Since infXF≤xiF≤xiL, any xiI smaller than infXF is also smaller than xiL. Hence, for any xiI<infXF, the derivative ddxiIWiI(x,xˆ;xiI), which is equal to ddxiIE[WiL(x;xiI,xF(Ij))], has a positive value and no xiI<infXF is an optimal choice for firm i’s investment threshold.

For any xiI∈[infXF,mininfFj,supXF], the payoff is

and the derivative of the payoff function with respect to xiI is

If xiI<xiF, then eq. (52) is positive, because xiI<xiF≤xiL<xiS. If infFj<xiF, for any xiI<xiF in support of xjI, the payoff function is

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

Lemma 4.3 guarantees that fxiIxiF is positive. From xF(Ij)≤xjI(Ij) it follows that G(xF−1(xiI))−G(Ij(xiI))≥0. Since xiI<xiF≤xiL<xiS, value of the derivative given by eq. (54), remains positive for values of xiI in interval [infFj,xiF). Thus for any xiI<xiF, the derivative is positive and optimal xiI cannot be smaller than xiF.

At this point I assume that Fj∩XF≠∅. This assumption is equivalent to xF(IU)>infFj. I will later find the optimal xiI for the other case.

From Lemma A.1, it follows that optimal xiI cannot be larger than xiS. Since xiF≤supFj, if supFj<xiS, firm i is indifferent between investment at any xiI∈[supFj,xiS], because this choice does not affect firm i’s investment threshold. Hence, the optimal xiI is interval [xF(IL),supFj].

If xiF<infFj, the derivative for any xiI∈[xiF,infFj] is given by eq. (52). For any xiI in [max{xiF,infFj},xF(IU)], the payoff function is

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

For any xiI in [xF(IU),supFj], the payoff function is

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

Given firm j’s behavior, let Lj(xiI) be a function that equals to G(xF−1(xiI)) for any xiI∈XF and is equal to one for xiI>xF(IU). For any xiI≥xF(IL), define

Remember that for xiI<infXF, I defined Ij(xiI)=IL. Combining eqs (52), (56) and (58) together, for any xiI∈[xF(IL),supFj] that xiI>xiF, it follows that

The function gj(xiI) is continuous. From eq. (59), since 1−G(Ij(xiI))−[1−Lj(xiI)]−[Lj(xiI)−G(Ij(xiI))]=0, it follows that gj(xiI) is positive for any xiI∈[xF(IL),xiL]. From Assumption 4.6 and eq. (58) it follows that gj(.) is strictly decreasing in the interval [xiL,supFj], hence, if gj(xiI) is zero at any xiI in this interval, then that value of xiI is the optimal choice for firm i. Otherwise, gj(.) is positive in [xiL,supFj] and since the other firm definitely invests by the time supFj is reached, any xiI≥supFj is optimal.

From Lemma A.1 it follows that the zero of gj(.) cannot be larger than xiS. Hence, if xiS<xF(IU), the optimal xiI solves eq. (24). From eq. (59) it follows that if xF(IU)≤xiS, then gj(.) is zero at xiI=xiS and is positive for values of xiI smaller than xiS.

In conclusion, if xiS<xF(IU), then the optimal xiI solves eq. (24), if xF(IU)≤xiS<supFj, then the optimal xiI is xiS, and if supFj≤xiS, then any xiI≥supFj is optimal.

Consider the case that Fj∩XF=∅. The derivative for any xiI∈[xiF,xF(IU)] is given by eq. (52). For any xiI∈[xF(IU),infFj], the expected payoff function is equal to WiS(x;xiI) and its derivative is

For any xiI in Fj, the payoff function is

and the derivative of WiI(x,xˆ;xiI) with respect to xiI is

If xiS<xF(IU), Equations (61) and (63) are negative and the optimal xiI is the zero of eq. (53) and solves

From eqs (61) and (63) it follows that if xF(IU)≤xiS, then the derivative is zero at xiI=xiS and is positive for values of xiI smaller than xiS. If xiS<xF(IU), since for xiI<infXF, I defined Ij(xiI)=IL, then the zero of eq. (52) is the same as solution to eq. (24). If xF(IU)≤xiS<supFj, then the optimal xiI is xiS, and if supFj≤xiS, then any xiI≥supFj is optimal. □

A.6 Proof of Proposition 4.9.

First I show that for any xI(I,xˉ) defined by Definition 4.8, Assumption 4.6 holds. For any xI, from eq. (25) it follows that

The derivative in Assumption 4.6 equals to the left side of the above equation. The right side of the above equation is negative for any xI, hence any solution to eq. (25) satisfies the condition in Assumption 4.6. Thus, it follows directly from Proposition 4.7 that for any xˉ∈{xF(IU),xS(IU)}, if firm j’s investment threshold is given by xI(.,xˉ), then it is optimal for firm i with investment cost I to invest at xI(I,xˉ). Hence, investment at xI(I,xˉ) is an equilibrium. Also it follows that xI(I,xˉ) is not smaller than xF(I). □

A.7 Proof of Proposition 4.10.

To find the limit for the case that D10−D00≤D11−D01, note that as IU gets unboundedly large, IU−α converges to zero, hence, from eq. (28) it follows that in the limit xI(I) solves

The solution to above equation is xI(I)=ββ−1r−μM2I.

For the case that D10−D00>D11−D01 and k∗∗=1, from eq. (27) it follows that the function xI(I) in the limit must solves

The function xI(I)=ββ−1r−μM2I is a solution to the above. Note that xI′(I)=ββ−1r−μM2.

Equation (27) is derived from the results of Proposition 4.5 which guarantees that xI(I)≤xF(I). Also eq. (27) is derived from the results of Proposition 4.9 which guarantees that xI(I)>xF(I). This inequalities will continue to exist in the limit as IU becomes unboundedly large. Hence, it holds that

or equivalently M2<D11−D01≤M1. □

A.8 Proof of Lemma 5.1.

First I show that the sign of EIi,Ij[UiC(Ii,Ij)]−EIi,Ij[UiI(Ii,Ij)] is not an explicit function of r and μ. From eqs (8), (9), (10) and (29), it follows that xiF, xiS, xiL and xiI are constant proportions of (r−μ). These thresholds have no other explicit relationship with r and μ.