Unlike the current measures in the literature, where corruption is constructed as an index, this paper provides a formula for quantifying corruption. By using option pricing techniques, the paper shows that the monetary value of a corrupt activity is equivalent to a regular bond and an embedded European call option. This formula is very important because it could be used to gauge the level resources lost to corrupt activities, and also to determine the level of “tax” that could be levied at corrupt-government officials. Results in the paper show that a government committed to reducing corruption should institute measures that will reduce the level and the volatility of the price of the goods in the parallel markets. The paper also finds that a government could reduce corruption by cutting interest rates, which would spur growth and render corruption as an unprofitable exercise.
Despite the long history of corruption, modern research on economics of corruption began about 40 years ago with the work of [
The purpose of this paper is to provide a method of measuring the size of corruption. This is done by providing a formula for pricing a corrupt activity. This is a departure from the literature where the focus of research is on the aggregate determinants of corruption. Corruption is an activity that is done in secrecy and hard to detect. However, in most developing countries, there are “secret” markets for most of the government’s goods and services that are difficult to obtain through the normal channels. These markets are run by government officials or their proxies. Although the official prices of these goods are known and set by the government, the prices of the goods and services in these markets are determined by the government official based on the maximum rent that could be extracted. The price in the secret market could be higher or lower than the official set price, depending on the demand for the goods or services.
The question asked in the paper is: in a corrupt environment, what is the price a corrupt government official must pay for the right to sell a government goods or services at a future date? This question is very important because in a country where it is very difficult for a government to curb corruption, it could impose a price or a special “tax” on its corrupt officials. The special taxes collected by the government could then be used to provide public goods for the citizens, or under some circumstance returned to those who purchased the goods or services. In the spirit of Coase theorem, the significant contribution of this paper is an attempt to find a market solution to the problem of corruption. The theoretical basis of the paper is to provide a pricing formula to value a corrupt activity. It draws on the option pricing literature by considering a corrupt activity as a European call option.
In this paper, corruption is defined as “the use of public office for private gains, where an official (the agent) entrusted with carrying out a task by the public (the principal) engages in some sort of malfeasance for private enrichment which is difficult to monitor for the principal” ( [
This paper also assumes that the punishment for detecting corruption is negligible since there are a lot of corrupt officials. [
This paper is organised as follows. Section 2 presents a brief review of issues related to corruption. Section 3 lays out a model for pricing a corruption activity. Concluding remarks are made in Section 4.
Corruption, or the perception of corruption, has been observed to have a pervasive impact on a nation’s economic development. [
Just as it is impossible not to taste the honey (or the poison) that finds itself at the tip of the tongue, so it is impossible for a government servant not to eat up, at least, a bit of the king’s revenue. Just as fish moving under water cannot possibly be found out either as drinking or not drinking water, so government servants employed in the government work cannot be found out (while) taking money (for themselves) ( [
Dante and Shakespeare also give prominence to the negative effects of corruption in their works and in the Bible, Jesus Christ condemns tax collectors for their corrupt practices (I Timothy 4:5).
In the literature, corruption is observed to grow out government regulations and controls, as government officials embark on rent-seeking activities which in turn slows down economic growth ( [
Trade restrictions could also fester corruption. Some countries impose import restriction on certain goods so as to protect home industry. However, the quantity restriction on the flow of foreign goods makes import licences for the restricted goods very valuable and therefore attracts bribes from importers. On the other hand home producers of the good could also corrupt influential lawmakers to maintain such restriction even if it is through the imposition of tariffs.
Another source of corruption in some countries is the tax system. [
Price controls, differential exchange rates, exchange rate controls and lower paid civil servants are part of the contributing factors of corruption.2 Keeping the price of government goods and services below their market value allows government officials to engage in rent-seeking activities. Corruption grows in countries with differential exchange rates―one for importers, one for exporters, one for multinationals, one for tourists, one for investors, etc.―because individuals or firms would pay a bribe to corrupt official to get the “best” rate. In countries where public servants are paid significantly lower than their counterparts in the private sector and sanctions are non-existent for corrupt officials, public servants tend to take bribes in performing their duties.
Corruption has been observed to impact negatively on economic growth ( [
Corruption has negative effects on productivity enhancing growth. For example, computers meant for the work place that are diverted for personal use by government officials, in some countries, do affect output growth negatively. Corruption also cut into profit margins of productive investments relative to rent seeking investments. As pointed out by [
Extending the corruption literature to the open economy, [
Corruption also affects economic efficiency. In the literature there are proponents who argue that corruption is the oil needed to grease the economic engine of an over- regulated economy. [
Another efficiency argument in favour of corruption in some developing countries is to consider corruption as “speed money.” A payment of bribe to a government official could hasten the movement of files in a bureaucracy. Corrupt officials take advantage of slow grinding bureaucracy by varying the size of bribes in accordance to the time preferences of the clients. The argument here is that in a country with bad and heavy regulations, the use of “speed money” to circumvent bad government control is like deregulation and therefore a good thing. Note that in some cases government regulations could require frequent contacts between government officials and citizens. Citizens may have to spend a lot of time in acquiring government licences. To cut down the processing time, citizens could pay bribes to officials and thus economic efficiency is achieved. Theoretically, [
The efficiency-enhancing view of corruption is not shared by all in the literature. As [
Empirically, the literature has also investigated the effects of corruption on economic growth. Using data from some developed and developing countries, [
Examining the effects of corruption on income and the Gini coefficient of income distribution, [
In the empirical studies cited above, the corruption variable is based on a qualitative measure rather than a quantitative one. The qualitative measure or corruption index is constructed from either published reports on corruption, such as newspapers or the internet, or questionnaire-based surveys. A popular index used by researcher and the business community is the Transparency International index.5 This index measures the perception of corruption on a scale of 0 to 10, with 0 indicating that most transactions or relations in the country are tainted by corruption and 10 indicates the country is free of corruption.
Despite the usefulness of the corruption index it has some drawbacks. First, it measures perceptions of corruption and not the size of corruption. This imposes a potential problem as the index may be fraught with perception bias, as perceptions of corruption change from country to country and from culture to culture. Second, the index is not helpful to governments or economic planner who would like to know the resource cost of corruption as the index fails to provide one. In the next section we would attempt to provide a method of measuring the size of corruption. This is done by providing a formula for pricing a corrupt activity.
Corruption is a secret activity that is very difficult to detect in some countries. Even when discovered, the monetary value of the transaction tends to be elusive. In some countries around the world, “secret” or unofficial markets exist for most of the government produced goods and services that are difficult to obtain through the normal process. These markets, which are run by government officials or their proxies, exist because government officials restrict the quantity of the goods or services sold in the “normal” markets. Furthermore, although the official price of the good is set by the government, the price in the unofficial markets fluctuates, depending on the maximum rent that could be extracted by the government official. With this observation as the back drop, we attempt to provide a model for valuing corruption. The focus of this section is to answer the question: in a corrupt environment, what is the maximum rent a corrupt government official could extract from the sale of the good or service at a future date?
In this paper, we consider a homogenous government produced goods. Following [
A corrupt official is faced with two options when he or she decides to sell the government good. The official could decide to sell the good by a multiple of the government price and then pocket proceeds without accounting for it. This first option is what [
Let K be the government set price of the good, and P be the price (government price plus bribe) of the good in the secret market. The options facing the government official can then be summarized as:
where a is a positive number. The first part of Equation (1) summarises the first option while the second part capture the second option. What Equation (1) suggests is that a rational corrupt official will choose which ever options give the maximum return. Equation (1) could then be rewritten as:
If we assume that corruption activity will take place T period from now, then we can interpret the present value of the corrupt official’s gains from the sale of a unit of government produced good or service as being equivalent to a bond with face value aK and a call option on the good, in the secret market, at an exercise price of (1 + a)K. With this interpretation, we can proceed to use the option pricing literature to provide a formula for quantifying the value of a corruption activity.
In addition to the above assumption we shall make the usual assumptions for modelling continuous time asset pricing models: 1) in the secret market, the trading of the government good is traded continuously; and 2) the price of the good follows a continuous time diffusion process of the form:
where ap is the instantaneous average return of holding one unit of the good, σp is the instantaneous standard deviation of rate of change of the price and dzp has a standard normal distribution with mean of zero and variance dt. Note that ap and σp may be functions of P and t. However, for the purpose of this exercise we shall assume they are constants. Given that the government price is fixed, we must also mention that it is the size of the bribe that is responsible for the diffusion process.
Given the secret market price of the government produced good, P, let the value of the corrupt activity at any time be C(P, t). Then applying Ito’s lemma the drift and the diffusion of the value of the activity is given as:
which upon simplifying, yields:
where
Proposition 1: The partial differential equation governing the corruption activity is:
Proof: See Appendix.
Remarks: Intuitively, the differential equation governing the valuation of the corruption activity is explained as follows. The first term on the left hand side of Equation (8) captures the Jensen’s inequality effect coming from the variance of the price of the commodity. The second term represents the risk-adjusted expected drift of the price of the good. The third term reflects the shrinking time to maturity. The last term represent the net flows to the corrupt official. It is important to note that the drift term of the price of the good plays no role in the valuation of the corruption activity. However, the variance of the price of the goods plays an important role.
Proposition 2: Under the set of assumptions given earlier, the present value of the gains from corruption activity is:
where E(P, K, τ) is an European call option to purchase the government produced good in the secret market at an exercise price of (1 + a)K. The value of the call option is
where
and N(.) is the cumulative normal distribution function.
Proof:
Based on Equation (2), it is shown that the present value of the corrupt official’s gains from the sale of a unit of government produced good or service is a equivalent to a bond with face value aK and a call option on the good, in the secret market, at an exercise price of (1 + a)K. The first term of Equation (10) corresponds to the present value of a bond and the second term is the value of the call option. The call value option follows from [
Remarks: Propositions (2) in very important on a number of reasons. First, it provides us with a method of placing a monetary value on a corruption activity. Second, for budgetary purposes, a government facing a pervasive level of corruption could use this formula to gauge the level resources lost to corruption activity. Third, the formula could be used to tax government officials for their corrupt activities. Fourth, the recognition of corruption activity as a European call option provides an analytical framework for improved understanding of the corruption literature. In the next section, we will examine how the underlying parameters of the model influence corrupt activities.
In this section, we attempt to learn more about the factors that influence corruption activity, and the implied policy implications.
Proposition 3: The value of corruption activities rises with the increase in the “secret” market price of the government produced goods.
Proof:
Differentiating Equation (10) with respect to P and using Equation (A16) in the appendix:
Remarks: The results suggest that a strong demand for the good in the secret market would enhance the value of corrupt activities. The result corroborates the view that government officials could deliberately restrict the supply of goods and services so as to profit substantially from corruption activities.
On the other hand, a government interested in curtailing corruption could embark on policies that could contribute to the fall in the price of the good in the secret market. Such policies could include more government officials supplying the government produced goods. For example, the government could decentralize the issuing of passports by having more passport offices. Because of the difficulty for several government officials to collude, such a measure would reduce the amount of bribe and therefore the price the good in the “secret” market which in turn leads to a fall in monetary value of corruption.
Proposition 4: Corruption activities increases with the rise in the volatility of “secret” market price of the goods.
Proof:
Differentiating Equation (10) with respect to the variance of P,
Remarks: The result demonstrates that corrupt government officials profit a great deal from increased volatility of the price of the good in the secret market. The increased volatility of the price increases the value of the call option implied in the value of the corrupt activity. An explanation for this is that a call option has no downside risk as the value of the call is zero irrespective of how far it finishes out of the money. Hence, an increase in the volatility of the price of the good goes to increase the chances that the call option will expire in the money. The rise in the volatility of the price of the good also increases the general uncertainty in the secret market which in turn enhances the gains from corruption.
To curb corruption, the government could embark on measures that will reduce the volatility of the price of the good in the secret market. One measure would be for the government allowing competition in the supply of the good. Such an action would not only reduce the level of bribe but also the volatility of the price in the secret market, and eventually decrease the incentive to engage in corrupt activities.
Proposition 5: A rise in the government set price, K, may or may not reduce the gains from corruption activity.
Proof:
Differentiating Equation (10) with respect to K and using Equation (A28) of the appendix:
which is clearly indeterminate.
Remarks: As mentioned earlier a corrupt official has two options: corruption with theft and corruption without theft. Corruption with theft gives the implied “bond” feature to the gains from corruption and corruption without theft gives the embedded call value in the gains. Proposition (5) is explained by the fact that an increase in government set price leads to an increase in the value of the “bond” implied in the gains from corruption and a decrease in the value of the embedded call option. The value of the call falls because the rise in the government set price increases the probability that the call finishes out of the money. The net impact of the rise in the government set price on corruption activity depends on which effects dominate. In other words, if official’s accountability of the goods and services is weak then we may see corruption with theft rising. In which case the first term in Equation (13) will dominate the second term and therefore corruption activity will rise with the increase in K.
Corollary: A rise in the government set price, K, would reduce the monetary value of corruption if it is backed by tough accountability for all government-produced goods sold by officials.
Remarks: If the government mechanism of ensuring that all goods supplied to government officials are properly accounted for then officials would find it very difficult to carry out corruption with theft activities. Thus in Equation (13) a would be close to 0, and the impact of the rise in K on corruption activity will be negative. The policy implication of this result is that, in an environment of unstoppable levels of corruption, a government could reduce corruption by raising its price of goods while maintaining strong accounting standards.
Proposition 6: The rise in the factor (a) by which the government official, engaged in corruption with theft, sells the good or service for leads to an increase in the gains form corruption.
Proof:
Differentiating Equation (10) with respect to a and using Equation (A34):
since
Remarks: Following the explanation given for the remarks made for Proposition 5, an intuitive explanation for this result is that a rise in the parameter a, enhances the value of the implied bond and decreases the embedded call value. However, in this case, the net impact is positive because the bond dominates the call. Although corruption without theft falls, the implication of the results is that the rise in the parameter a raises the rewards to corruption because of the increase in corruption with theft.
Proposition 7: Interest rates have indeterminate effects on the gains from corrupt activities.
Proof:
Differentiating Equation (10) with respect to the interest rate, r, and making use of Equation (A40) yields:
Remarks: The result shows that rising interest rates have an ambiguous impact on corruption. Again, following the remarks for Proposition 5, the rise in interest rates reduces the value of the implied bond in the gains of corruption activity, discouraging corruption with theft. At the same time the rewards from corruption without theft rises since the exercise price on the embedded option reduces. The impact on corruption activity depends on which effect dominates. Intuitively, interest rate increases tends to slowdown the economy and consequently raise the level of unemployment. Using the efficiency wage argument, increased unemployment pool would serve as a deterrent to officials who would want to keep their jobs from engaging in corruption with theft for fear of being fired. If that is the case then a would be close to zero and therefore the impact of the rise in interest rate on corruption activity would be positive.
Corollary: In a country with strict accounting standards for all government-produced goods sold by officials, a rise in the interest rate would result in higher corruption.
Remarks: A country with strict accounting standards suggests that a would be close to zero and therefore the sign for Equation (15) would be clearly positive. The monetary value of corruption activity rises because the embedded option rises in value as the rise in interest rate reduces the exercise price of the option. The implication of this result is that a government committed to a higher accounting standard could reduce corruption by cutting interest rates, which in turn spurs economic growth. The rise in economic growth would in turn curtail acts of corruption, since the acts would be less lucrative.
Corruption is a costly activity to a nation as it leads to the diversion of public resources for private use. It has the potential of shifting resources away from the productive sector of an economy. This paper has attempted to address the issue by providing a market based solution by using a financial model to quantify the value of corrupt activities. The approach taken in the paper is very innovative as it provides a solution for a country where it is very difficult for a government to curb corruption. The paper proposes that the government could impose a price or a special “tax” on its corrupt officials. The special taxes collected by the government could then be used to provide public goods for the citizens, or under some circumstance returned to those who purchased the goods or services.
Using option pricing techniques, the paper shows that the monetary value of a corrupt activity is equivalent to a regular bond and an embedded European call option. Unlike the current measures in the literature, where corruption is constructed as an index, this paper provides a formula for quantifying corruption. This formula is very important because it could be used by governments, private sectors and interested parties to gauge the level resources lost to corrupt activities. The formula could also be used to determine the level of tax that corrupt-government officials could be levied. Furthermore, the recognition of corruption as a European call option provides an analytical framework for improved understanding of the corruption literature.
Results in the paper show that a government committed to reducing corruption should institute measures that will reduce the level and the volatility of the price of the goods or services it provides in the parallel markets. The measures could include the government allowing for competition in the supply of the goods or services. Such an action would not only reduce the level of bribe but also the volatility of the price in the secret market and eventually a decrease in the incentive to engage in corrupt activities. The paper also finds that a government committed to a higher accounting standard could reduce corruption by cutting interest rates, which would spur economic growth and render corruption as an unprofitable exercise.
The author would like to acknowledge the useful comments of colleagues at the Economic Commission for Africa. However, any errors or omissions must be attributed to the author. Of course, the views expressed in this paper are those of the author and should not be attributed to the United Nations.
Atta-Mensah, J. (2016) The Valuation of Corruption. Journal of Mathematical Finance, 6, 728-746. http://dx.doi.org/10.4236/jmf.2016.65051
To prove Proposition 1, we use standard arbitrage arguments common in the options pricing literature. Construct a hypothetical portfolio by investing ω in the government produced goods and (1 − ω) in corruption activity. If L is the value of the portfolio then the instantaneous return on portfolio, dL, is:
or:
The portfolio, L, could be made riskless if ω is chosen to satisfy:
implying:
Holding
Equating Equation (A5) to the drift term of Equation (A2) with the substitution of ω we have:
Substituting Equations (6) and (7) into Equation (A6) and re-arranging yields the differential equation governing the loan contract (Equation (8)).
A.2. Properties of the Call Option1) The value of the call option
In the main text, an expression is derived for valuing a European call option as:
where
and N(.) is the cumulative normal distribution function.
The paper now turns to the sensitivity of the value of the call to the parameters.
2) The effect of the change of the “secret” market price on the call value
Differentiate the call with respect to the secret market price, P:
but
thus:
Substitute Equations (A2) and (A3) in the last part of Equation (A7):
which simplifies into:
hence:
3) The change of the variance of the “secret” market price on the call value
Differentiate the call price with respect to
or
Using Equation (A11),
substituting Equations (A8) and A(9),
which simplifies to:
hence,
4) The change of the government set price, K, on the call value
Differentiate the call value with respect to K:
which could be expressed as:
manipulating further:
which reduces to:
or
hence
5) The change of the parameter a on the call option value
Differentiate with respect to a:
simplifying,
or,
which reduces to:
or
hence
6) The change of interest rate on the call option value
Differentiate with respect to r:
rearranging:
or:
which reduces to:
or:
hence: