prob u u e= 1 H =
ψα
where0
≤ψ
≤1
. If ψ = 1, high effort has no impact on the risk level of the project.When ψ = 0, high effort can eliminate the risk from the project.
Conventionally, effort may be put into different categories. Rather than having the tedious distinctions, effort in this analysis simply reflects general managerial competence and attentiveness. In this role, effort is not necessarily directed towards a specific managerial task. The low level of effort characterizes the one that can be contracted for and verified by external investors. Showing up in the office 8 hours a day from Monday to Friday is a good illustration. The high level one includes the low level of effort plus an extra effort that can only be internally monitored. For example, a high level of effort involves costs in taking senior skilled managers from other areas and using up valuable management time. Unfortunately, such an additional amount cannot be contracted for and verified by external investors. Interestingly, when the company goes for project financing working as an agent for external investors, the cash flow received will be independent of the outcome in the market. There is no direct incentive of providing high effort unless the external investors contract for it. However, external investors do not contract for high effort because they cannot determine whether it is being supplied or not.
The company’s utility function is assumed to exhibit Arrow-Pratt constant relative risk aversion with R. (4)
U W ( ) = W
1−RBeing risk averse, 0 < R < 1, the company maximizes the expected utility of cash flow, W from the investment project.
Financing Projects
Two forms of financing and compensations are considered. The first option is self-financing and corporate-funded by the company. The cash flow varies depending on the market outcome. The other alternative is to use project financing with outside investors. The company then gets a fixed reward that is independent of market outcome.
Since the high level of effort cannot be contracted for, the company will only supply low effort. However, company may benefit even more from the risk reduction than the monetary compensation generated by the project.
Case 1: Corporate Self-Financing
For a low level of effort, the expected utility of cash flow is given by:
(5)
( ) [ ]
Correspondingly, a high level of effort yields
(6)
( ) [ ]
The company must choose both effort and output before the state of the world is known. To determine the optimal level of effort, the company calculates the maximum expected utility under the high and the low effort level comparing the two.
Similarly, high effort will have an optimal output and expected utility as below.
q a c c e
Under some specific conditions, it can be shown unambiguously that the firm will choose either a high level of effort or a low level of effort. These are demonstrated in theorem 1 and 2.
Theorem 1: A company will always choose a high level of effort if e A Bc
L≥(2 c− 3)
32 .
Proof: Only two levels of effort are actually available but we can still consider the hypothetical impact of other
effort levels on expected utility. In general,
[ ]
That is, e A Bc it follows that high level of effort increases the expected utility above that of low level of effort.
Corollary: If (2A – Bc3 ) < 0, then high effort will always be preferred.
Proof: As long as eL > 0, conditions of theorem 1 are satisfied automatically. The circumstance is favored by A being small or Bc3 being large.
Interpretation: The term A = b + c1 is small if the market size is huge (i.e., small b) or diseconomies of scale is moderate (i.e., small c1). The term Bc3 = (a – c2)c3 is large if the maximum willingness to pay (a) is high or the unit variable cost (c2) is small or the impact of effort on reducing unit variable cost (c3) is significant.
Theorem 2: If high effort has no influence on the cash flow occurrence (i.e., ψ =1), then low effort will yield higher expected utility whenever
e A Bc
H
≤ ( 2 c −
3)
otherwise, higher effort increases the likelihood raising the probability of favorable outcomes may compensate for the negative impact that high effort has on cash flow in a given state of the world.The relationship between effort and expected utility can be represented by an U-shaped curve. Sufficiently high effort if available, ultimately leads to greater cash flows than lower effort because there are no diminishing returns to effort in the model of this paper. Each additional amount of effort contributes a constant reduction in marginal and average cost. The reduction in cost encourages greater output, which magnifies the rewards of greater effort. For all possible effort levels, constant improvement in average cost from effort is unrealistic. However, by assumption, the analysis will focus on considering only two discrete effort levels whose relevant range is characterized by effort making a constant improvement in average cost due to additional effort.
Case 2: Project Financing with Fixed Management Fees paid by Outside Investors
Instead of accepting variable return through corporate financing, the company can contract out the project to a risk-neutral external investor and acts as a managing agent. The fixed management fee may include economic rent that the company would expect to earn when corporate financing is used. Since the capital market of project financing is assumed to be competitive, the expected return for external investors need only be K, the cost of capital. While the investors commit a fixed payment to the company, they assume all the risks of the project. For the poor market outcome, investors will earn less than K and earn more than K as a premium when market is favorable.
As argued earlier, the outside investors cannot monitor whether or not high effort has been provided. Therefore, the payment that investors offer will be based on the low effort expectation. Expected cash flow, E(W) from the project is the fixed management fee received by the company regardless of the state of the world, netting out the cost of
effort. That is, E W
[
u u]
B c e4 . The corresponding utility becomes
[ ]
Lemma 1: Project financing is preferred if
Note: the above expression is normalized by setting u1 = 1 and eL = 1 for simplifying purpose from now on.
Proof: By definition, company prefers seeking project financing to providing its own funds if
U E W e ( ( )
L) > EU W e (
H) > 0
. Substituting the parameterized functions for these two relationships yields theLemma 2: The company prefers project financing if low effort is already the optimal action for self-financing.
Proof: The net value created is the same whether project financing or corporate self-financing are undertaken at a low effort level. The difference is the external investors absorbing all the market risk. Comparing the resultant expected utility of project financing and corporate self-financing yields the following claim.
( ) [ ] ( )
Proof: Obviously the application of low effort must earn positive cash flow if it is worth considering. Given this is the case, from Theorem 1 since eH > eL, then
∂
For the second claim,
[ α α ]
and ψ = 1. Furthermore, since the numerator cannot be at a maximum while the denominator is at a minimum, therefore[ α α ]
Lemma 1 and 3 together imply certain restrictions where a company will choose project financing.
CONCLUSION
The model highlights the conditions under which a particular ownership structure is best suited for carrying out a capital investment. The choice to use project financing is driven by two major factors, namely the extent and nature of the uncertainty and the required amount of investment.
The results derived for the company arranging for outside financing are similar to the standard principal-agent relationship. There are, however, two main differences. First, the assumption of a large number of potential investors implies that all the negotiating power goes to the company in question and the outside investors only earn a competitive rate of return. Second, the on-going relationship suggests company must have strategic considerations in its involvement. This is why company prefers not to sell its idea or knowledge to outside investors and to have management in the project.
A range of other structural factors may influence the decision of a firm to choose internal financing versus project financing. Company will tend to prefer corporate financing of investment when effort has a significant impact on the magnitude and likelihood of favorable outcomes. Regarding investment size, the larger the capital investment required, the more likely outside project financing is preferred. The impact of the market size will depend on how large fixed capital investment is relative to the maximum surplus (i.e., the difference between maximum willingness to pay and average variable cost) available from production. If fixed capital costs are large compared with the maximum surplus, then project financing is favored in smaller markets. Conversely, if fixed capital costs are smaller relative to the maximum surplus available, project financing will be favored by larger markets. Lastly, for factors such as maximum consumer surplus, B and contribution of effort to reducing average cost, c3, project financing will be favored in an intermediate range of potential circumstances for business operations.
The analysis in this paper has focused on two extreme cases: all internal corporate financing versus complete project financing by outside investors. An interesting area of future research would be to consider an intermediate case where the company shares the financing with an outside investor each putting up part of the investment funds. This arrangement would attempt to balance benefits in risk reductions with incentives for managerial effort by the company.
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