Now we consider a continuous-time model with a demand shifter Xt at time t.
The stochastic process (Xt) follows a geometric Brownian motion with drift σ, i.e.,
dXt Xt
= σ dZt and X0 = x , (5.1)
where (Zt) is a standard Brownian motion. The firm takes It to expand factory facilities and invests Rt in developing new technology at time t. The equipment of the factory and accomplishments in research of the firm depreciate at constant rates δ and λ, respectively. The laws of motion for capital and research are
dKt= (It− δKt) dt, (5.2)
dAt = (Rt− λAt) dt. (5.3)
The fundamental value of the firm at time t is given by maximizing the expected present value of its cash flow with the discount rate r; explicitly, V is given by
V (At, Kt, Xt) = max
It,Lt,Rt
E
Z ∞ t
[PsYs− c1(Ls) − c2(Is) − c3(Rs)]e−r(s−t)ds
Ft
,(5.4)
which is subject to the constraints in (5.2) and (5.3). Here the cost functions of capital and research are considered convex such that c2(It) = ωtItuand c3(Rt) = θtRvt where u ≥ 1 and v ≥ 1 are constants. In this chapter, we will separate it into two cases: the cost function is strictly convex and is linear.
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5.1. Convex cost functions
In this section, we consider c2(It) and c3(Rt) to be convex. In other words, we take u, v > 1. V (At, Kt, Xt) satisfies the following Bellman equation:
rV (At, Kt, Xt) = max
It,Lt,Rt
PtYt− c1(Lt) − c2(It) − c3(Rt) + E[dV |Ft] dt
. (5.5)
The value of the firm is a function concerned with variables At, Kt and Xt, then apply Itˆo formula and Equations (5.1), (5.2) and (5.3) to obtain that
dV (At, Kt, Xt) = σXtdZt+ VA(Rt− λAt)dt + VK(It− δKt)dt +1
2Xt2σ2VXXdt.
Then Equation (5.5) can be rewritten as
rV (At, Kt, Xt) = (5.6)
Itmax,Lt,Rt
PtYt− ρtLt− ωtItu− θtRvt + VA(Rt− λAt) + VK(It− δKt) + 1
2Xt2σ2VXX
.
The exhaustive deriving process from (5.4) to (5.6) is shown in Appendix C. Differ-entiating the right-hand side of the above equation with respect to It, Lt and Rt, respectively, yields
VK = ωtuItu−1, (5.7)
VA = θtvRtv−1, (5.8)
L∗t =
"
β(1 − ε)XtKtα(1−ε)Aγt ρt
#1−β(1−ε)1 .
Substitute these three equations back into (5.6), then we have
rV (At, Kt, Xt) = hX
1 1−β(1−ε)
t K
α(1−ε) 1−β(1−ε)
t A
γ 1−β(1−ε)
t + (u − 1)ωtItu+ (v − 1)θtRtv
−δKtVK− λAtVA+ 1
2Xt2σ2VXX, (5.9)
where h = β(1 − ε) From Equation (5.7), we have
Kt= E1X
And from Equation (5.2), we know that Kt =
Z t 0
eδ(s−t)Isds + e−δtK0. (5.11)
Combine (5.10) and (5.11) and then use Itˆo formula. It∗ satisfies the following equation
1From mathematical point of view, we should write (5.9) as
rV (At, Kt, Xt) = hX
However, we have no much information about this nonlinear partial differential equation (existence, uniqueness, ...). Hence, we follow here the method introduced by Abel (1983).
Similarly, we can find R∗t satisfying the following equation by Equation (5.3) and also make the demand shifter Xt increase. However, for a given level of the current demand shock Xt, increased σ2 will lead to an increase in the optimal investment;
uncertainty has a positive effect on investment for capital It and research Rt. More-over, we can observe that if raising the value of σ2, the present value of the firm V (Xt, At, Kt) will also increase for a given level of Xt. This result is the same as the performance of calculation on Table (4.1) in Section 4.4.
5.2. Linear cost functions
In the last section we have got the optimal investment when cost functions of capital and research are convex such as c2(It) = ωtItu and c3(Rt) = θtRtv, that is u, v > 1. From the structures of Itand Rt, we discover that it could be an interesting situation when u = v = 1, i.e., the cost functions of Itand Rtare linear. We wonder what will happen if the cost functions are linear to It and Rt. Now we consider a
special case for u = v = 1. The value of the firm at time t would be V (Xt, At, Kt) = max
Lt,It,Rt∈R+E
Z ∞ t
(PsYs− ρsLs− ωsIs− θsRs)e−r(s−t)ds
Ft
. The maximization is also subject to the constraints in (5.2) and (5.3). With the similar procedure, the value function will be derived to be the following optimality condition
rV (At, Kt, Xt)
= max
Lt,It,Rt∈R+
PtYt− ρtLt− ωtIt− θtRt+ VA(Rt− λAt) + VK(It− δKt) + 1
2Xt2σ2VXX
= max
Lt∈R+{PtYt− ρtLt} + max
It∈R+{(VK− ωt)It} + max
Rt∈R+{(VA− θt)Rt}
−λAtVA− δKtVK+ 1
2Xt2σ2VXX. (5.12)
It is usual to assume that the investments It and Rt are bounded. However, it is interesting to consider the case where It and Rt are unbounded.
5.2.1. Investment without constraints. It is easy to get that
L∗t =
"
β(1 − ε)XtKtα(1−ε)Atγ ρt
#1−β(1−ε)1 .
Now we first discuss the case where the maximum value V in (5.12) is finite, then the both optimization values
max
It∈R+{(VK− ωt)It} and max
Rt∈R+{(VA− θt)Rt} are finite, which implies that
VK ≤ ωt and VA≤ θt. (5.13)
This means that the unit cost of capital and research investments is larger than their marginal profit, respectively. It is obviously that if VK < ωt, the optimal It∗ = 0, and if VK = ωt, the optimal It∗ is not unique, but in both case,
max
It∈R+{(VK − ωt)It} = 0.
Similarly, we can get
max
Rt∈R+{(VA− θt)Rt} = 0.
The optimality condition (5.12) then becomes
rV (At, Kt, Xt) = hX
1 1−β(1−ε)
t K
α(1−ε) 1−β(1−ε)
t A
γ 1−β(1−ε)
t − δKtVK− λAtVA+1
2Xt2σ2VXX. (5.14) This differential equation is much simpler than that with convex cost functions and we can find that the solution to (5.14) is given by
V (At, Kt, Xt) = E3Xt
1 1−β(1−ε)Kt
α(1−ε) 1−β(1−ε)At
γ
1−β(1−ε), (5.15)
where
E3 = 2[1 − β(1 − ε)]3
2r[1 − β(1 − ε)]2+ 2[αδ(1 − ε) + λγ][1 − β(1 − ε)] − β(1 − ε)σ2
β(1 − ε) ρt
1−β(1−ε)β(1−ε) .
Of course, we still have to check if the conditions (5.13) hold or not. Observing (5.13) and V given by (5.15), we know that if the unit cost of the capital, research and labor investment is too expensive, or alternatively, the initial investments of capital and research, K0 and A0, are too small, the firm will not invest much in capital and research. The only opportunity for the firm to increase its investment in A and K is when the unit price of capital, research or labor investment sinks such that VK = ωt or VA= θt.
As for the general case where V is not necessary to be finite, we have to consider the situation VK > ωt or VA> θt. It is easy to see that under these situations, the firm should invest as much as possible in this item, and then it results in V = ∞.
However, this case does not make much sense, since in practice all of the investment are limited. Thus, it is reasonable to consider the investment with constraints.
5.2.2. Investment with constraints. An interesting and more practical case is to consider the investment with constraints. More explicitly, we consider the case, where 0 ≤ It ≤ ¯I and 0 ≤ Rt ≤ ¯R for some positive constants ¯I and ¯R. Similarly
to the above argument, we get the firm’s optimal strategies are given by
It∗ =
0, if VK < ωt, I,¯ if VK ≥ ωt,
and R∗t =
0, if VA< θt, R,¯ if VA≥ θt.
This implies that the firm either invests nothing or invests so much as he can. Thus, the cumulative investments are given by
Kt = e−δt
K0 + ¯I Z t
0
eδsI{VK≥ωs}ds
,
At = e−λt
A0+ ¯R Z t
0
eλsI{VA≥θs}ds
, and the optimal condition (5.12) can be written as
rV (At, Kt, Xt) = hX
1 1−β(1−ε)
t K
α(1−ε) 1−β(1−ε)
t A
γ 1−β(1−ε)
t + ¯I(VK − ωt)I{VK≥ωt}
+ ¯R(VA− θt)I{VA≥θt}− λAtVA− δKtVK+ 1
2Xt2σ2VXX. Figure 5.1 shows one possible sample path of It which behaves like a step func-tion. Figure 5.2 shows the corresponding capital investment of the sample path given in Figure 5.1. We can see that in the interval where It= 0, the corresponding Kt is convex decreasing; in the interval where It = ¯I, the capital investment Kt is either increasing or decreasing with a smoother decay (which depends on the value of ¯I, K0, and δ).
It is clear to see the positive effect of uncertainty on V (At, Kt, Xt). We can also find the optimal investment and analyze the solution with respect to the parameters, just like the analysis for the general case. So the tautology is omitted here.
t1 t2 t3 It
_
Figure 5.1. One sample path of It
t1 t2 t3
K0
Figure 5.2. The corresponding Kt of Figure 5.1
Conclusion
In our analysis for discrete time, we systemized total possible conditions to six cases. Even a slight change of a single parameter will drive the case to switch to another one, then the form of the optimal investment and the present value of the firm will be completely different. The jump is fierce.
The model we use does not differ from Abel’s model in 1983 too much. The solution of a firm’s fundamental value and the effect caused by uncertainty are similar as well. The added factor research complicates the solution, but the basic result does not change. The effect of uncertainty on investment for capital and research is positive. The firm will invest more on capital and research if demand uncertainty increases, then the firm value will be raised. This is possible in reality, for the firm might invest more capital and make more effort on research to deal with the unexpected fluctuation of output demand in the market. Any sudden incident could induce a violent attack on the operation of each firm. Besides, we conclude that given the current demand shifter, increased uncertainty will also make the present value of a firm raise. This result can be observed not only from the numerical appearance of examples but also from the feature of the general formula. However, the optimal investment has a high degree of complexity so that the explicit form of the solution is not easy to find out. And this situation makes the analysis for effects of some parameters on the optimal investment policy and the present value of a firm become much more difficult. It is a obstacle remaining to overcome in the future. In addition, recent studies on this topic have highlighted the importance of irreversibility for investment. It would be a feasible direction for our study.
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