4.1. Addressing the Moral Hazard
The individual will only accept the contract if their net benefit is greater than their outside option, which we normalise to zero. This is referred to as the participation constraint in contract theory and is expressed in this case as:
From here, the individual chooses the level of maintenance expenditure
c which maximises their expected utility. This is obtained by first substituting
in Equation (
6) into the individual’s participation constraint in Equation (
9), taking the first derivative of the resulting equation with respect to
c, and equating the result to zero. The utility maximising level of
c is therefore characterised by the following:
or in words, when the first derivative of
is equal to some negative constant. In our model, this is an impossibility as the first derivative of
is assumed to always be positive. The interpretation of this result can be seen in the agent’s participation constraint in Equation (
9), namely that a rise in the asset price only increases the individual’s cost burden and so they have a very strong incentive to drive down the asset price neglecting maintenance.
Ideally, the financier has perfect monitoring ability and can compel the individual to commit as many resources as possible to asset maintenance, ensuring that the asset appreciates with 100% probability. Unfortunately this can be very difficult to do and might not be worth the cost. The go-to contract theory solution for problems such as this then, is incentive alignment, namely for the financier to offer sufficient incentives to the individual. These incentives should have a positive relationship with the interests of the financier, which in this case is capital appreciation from the asset.
One of the benefits of a mushārakah contract is that the underlying asset usually does provide the individual with some intrinsic benefit. Possibilities include a desire from the individual to also benefit from capital appreciation of the asset or to utilise the asset to generate profits. The financier can take advantage of the second in a multiperiod setting by leasing the asset to the individual and settle for a lower asset repurchase price. However, in this two period model we assume for brevity that both parties seek to benefit from capital appreciation of the asset. The contract theory framework encourages us to identify points like these, allowing us to gain a deeper understanding of the object of analysis.
Suppose then that the individual’s income depends on the quality of the underlying asset which we assume is perfectly represented by the price of the asset at a given time. Let this be represented by the individual’s income increasing with the price of the asset at a decreasing rate such that the individual’s participation constraint can be expressed as:
and therefore each party’s expected rate of return becomes:
Similar to before, we find the level of
c chosen by the individual to maximise their expected utility by firstly substituting in
from Equation (
6) and
based on Equation (
11) into the new participation constraint, Equation (
12). We then take the first derivative of the resulting equation with respect to
c and then equate it to zero. This yields:
which is positive as long as
. Interestingly, this expression also implies that if the potential asset price change
A becomes larger, the more the individual will spend on asset maintenance. This is a sensible result as even though the model indicates that the magnitude of the asset price change is exogenous, the direction of the change is solely determined by the individual’s maintenance expenditure. Therefore, if the individual recognises that there is a sufficiently large benefit from an asset price change, they will be more motivated to ensure that it becomes a benefit to them instead of a loss.
4.2. Discussion of Results
The purpose of the model is to demonstrate simply how important contract theory is to IFSs, especially considering that there will always be an emphasis on their contractual nature. For example, there are many ways to effect a debt security cash flow such as through murābaḥah (cost plus sale) and ’ijārah contracts. However, each contract has different legal and therefore financial properties which can be made more pronounced when viewed from the perspective of contract theory.
Applying contract theory to a simplified version of the DM contract allows us to go beyond the standard amortisation schedule lamented by
Asadov et al. (
2018). From Equation (
15), we come to our first result which might not immediately clear.
Proposition 1. The individual has no ready incentive to spend on asset maintenance.
This somewhat mirrors a standard contract theory result common to principal-agent settings; as presented for example in
Mas-Colell et al. (
1995), agents prone to moral hazard should not have incentives which are constant relative to their task. Put simply it makes sense for people to work harder if they are rewarded more for doing so. This leads us to our next result.
Proposition 2. An individual will only spend a non-zero amount on asset maintenance if (i) they also benefit from the asset’s appreciation and (ii) the marginal benefit of asset appreciation θ is greater than the financier’s share of the asset.
In other words the agent is only motivated to maintain the asset if they expect that their financial benefit from the DM contract offsets their financial cost. This can therefore be used as a signal of sorts by prospective DM financiers for choosing which finance seekers to partner with—we shall discuss this more in the next section.
The final main result we wish to present has already been elaborated upon above.
Proposition 3. Given an equal magnitude of asset appreciation and depreciation A, an increase in A will increase the optimal level of spending on asset maintenance .
Summarising the implications of the model so far, it is not a good idea for financiers to make DM contracts with just anyone as there are inherent moral hazards. While Islamic commercial law (ICL) has various legal devices for anticipating such problems as discussed by
Usmani (
1998),
Al-Zuhayli (
2007), and
Jobst (
2009), enforcing them can be excessively costly. From a contract theory perspective, it is therefore more prudent to understand the incentive structure of the simple DM contract and only offer it in circumstances when it robust against inherent moral hazards. An example of such a circumstance is when the prospective seekers of finance can make good enough use out of the underlying asset. We take this idea a step further by considering a two-type economic screening approach; the financier offers one DM contract for finance seekers who are highly productive with the asset they wish to finance and another for those who are not so productive. This allows seekers of finance to self-select and reveal how effectively they can use the asset they wish to finance, therefore also signalling to the financier how prone they are to the moral hazards discussed.
4.3. Screening Individuals
Another layer can be added to the above analysis by considering a financier who wishes to screen prospective individuals on the basis of their ability to benefit from the underlying asset. This is particularly attractive to the financier because the previous section indicates that individuals who can better benefit from the underlying asset facilitate better capital appreciation. This approach is called screening and we provide a simple two-type example based on
Bolton and Dewatripont (
2004).
Basov (
2013) provides a treatment of screening for continuous types. The general two-type screening problem involves a proportion
of individuals with a significantly higher marginal benefit
of asset price appreciation than others such that
. The financier must therefore offer two different contracts such that the “high-type” individuals will always choose one whereas the “low-type” individuals will always choose the other. The financier’s utility function then becomes:
where
and
are the initial contributions of the high and low-type finance seekers while
and
are the repurchase values of the high and low-type finance seekers respectively. The appropriate participation constraints are:
and the following incentive compatibility constraints are:
Note that this setup can accommodate the moral hazard case discussed in the base model simply by setting
. The next step is to scrutinise the constraints for redundancy and whether or not any are binding. Constraint (
17) is redundant because it is guaranteed by constraints (18) and (
19). Constraint (20) on the other hand is redundant in the sense that it is not feasible for low-type individuals; they will never choose high-type contracts because it ends up costing them more. We then consider that the remaining constraints, (18) and (
19), bind because the financier will want to increase the required initial capital for both types as high as possible i.e. until the constraints bind:
The financier’s optimisation problem is therefore characterised by Equations (
16), (
21) and (
22). Substituting in the latter two along with
,
,
, and
results in the following modified problem for the financier:
To obtain a closed-form solution, we assume that the probability function has the following form:
such that the optimal level of asset maintenance spending
is:
and the corresponding probability of asset appreciation is:
Substituting in
and
for high and low-type finance seekers respectively:
Therefore the optimal level of initial capital that the financier will impose upon the high-type finance seekers can be obtained by differentiating the financier’s maximisation problem with respect to
and equating to zero:
This result for the high-type finance seekers is significant for reasons that will be discussed later. For now, it is substituted back into Equation (
27) and the optimal level of initial capital to be imposed upon the low-type finance seekers is obtained in a similar manner:
The results are as expected for an economic screening approach; high-type finance seekers would be more than happy to bear all of the financing if they could whereas low-type finance seekers would make a markedly lower contribution. Furthermore, we summarise the sources of the the gap between both types of finance seekers in the following proposition:
Proposition 4. The difference in initial capital that both types would be willing to contribute depends on (i) the proportion of high types to low-types (ii) the difference in marginal benefit of asset appreciation between both types and (iii) the full initial price of the asset itself.
We complete the screening analysis by comparing the expected rates of return received under both types of finance seekers and ultimately determining how much is received by the financier. Substituting in
and
into the expected rates of return for both parties from Equations (
13) and (
14) results in:
Therefore, the expected rates of return for low-type finance seekers are as follows:
whereas for high-type finance seekers, the expected rates of return are:
However, note that the expected returns associated with the high-type finance seekers in Equations (
36) and (
37) can be misleading. The model suggests that high-type finance seekers would prefer to buy out the financier’s share as soon as possible and so it is very possible for the financier to simply break even. This does approximate real-world conditions but poses a problem for the model. We remedy this by introducing a limit component associated with the high-type finance seeker’s initial capital contribution. We can therefore differentiate between two cases, namely one in which the high-type finance seekers do buy out the financier’s share and another in which they don’t. The above expected rates of return are therefore applicable only when they do not buy out the financier. If they do buy out the financer, the finance seekers do not share any capital appreciation or in flows from the asset with the financier and the financier breaks even with zero return.
Now focusing on the financier, their total expected inflow from DM contracts, based on Equation (
16), can be expressed as follows:
which becomes the following if
:
The financier’s total expected rate of return from DM contracts with both types is:
If
, it becomes:
Otherwise, it is simply equal to
:
We conclude this section with our final proposition and a discussion related to the implications of the screening model especially in relation to the implied returns from the DM contracts.
Proposition 5. Financiers can be discouraged from entering into a DM contract with finance seekers whose type levels are too high as they have an incentive and possibly the (potential) resources to limit the financier’s share in the gains from the contract.
This should be clear from our above discussion regarding
, namely that high-type finance seekers would ideally buy out the financier’s share in the underlying asset as soon as possible. This is more generally reflected in the equations related to how the type levels and type differential
affect the expected rates of return. Firstly, Equation (
34) suggests that the model has an implied constraint
as otherwise the logarithmic function in the model would take on a negative value. This implied constraint applies to all equations after, suggesting that there is an upper bound on the types and type differential which is line with the idea that sufficiently high type levels might in fact be bad for the financier. This can especially be seen in Equation (
41) representing the financier’s total return; the level of
serves only to reduce the financier’s rate of return. This result is surprisingly in line with contract theory to the extent that the solution for the standard principal-agent problem such as in
Mas-Colell et al. (
1995) is to agree on a 50-50 split between the principal and the agent. The economic screening approach suggests that a similar situation seems to be ideal for the financier, namely one in which the type levels and disparity of the prospective finance seekers is not too high. The desired result is that entering into such DM financial markets would allow the financier to secure a sufficient stake in assets but be partnered with finance seekers who can use the assets effectively enough.