To Launder or Not to Launder: Modelling How the Value of Dirty Income Impacts the Marginal Deterrence of AML Policy

Samuel Haak

doi:10.1515/rle-2024-0115

Article

To Launder or Not to Launder: Modelling How the Value of Dirty Income Impacts the Marginal Deterrence of AML Policy

Samuel Haak

Published/Copyright: July 11, 2025

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From the journal Review of Law & Economics Volume 21 Issue 3

Abstract

This paper develops a mathematical model of the criminal’s decision to launder money, in order to analyze the relationship between anti-money laundering (AML) policy and the incentive to commit crime. The model includes a measure of the value of dirty income for direct usage in consumption and investment, which is contrasted with the value of laundered money. The results demonstrate that increases in the strictness of AML policy cause criminals to shift away from laundering and towards spending dirty income directly. To the extent that dirty income provides a positive utility on the margin, deterrence effects from AML will be incomplete, as criminals are deterred from crime but not from laundering. Additionally, the value of dirty income is shown to present a confounding influence on the use of changes in money laundering volume to assess criminal welfare. These results suggest that efforts to improve AML regulations may be less effective at combating crime than previously supposed, and highlight the need for further research to understand how criminals use dirty income.

Keywords: money laundering; AML; microeconomics; crime; public policy

JEL Classification: K42; H10; G38

Corresponding author: Samuel Haak, Doctoral Candidate in Economics, Department of Accounting, Finance and Economics, Griffith University, Brisbane, Australia, E-mail: samuel.haak@griffithuni.edu.au

Acknowledgments

I wish to thank Andreas Chai (Griffith University), Nicholas Rohde (Griffith University), Alex Robson (Griffith University) and Kaiwen Leong (Griffith University) for supervising the writing of this paper. I would also like to thank Robert Alexander (University of the Sunshine Coast), Jackson Mejia (Massachusetts Institute of Technology), and an especially helpful anonymous peer-reviewer for their valuable comments. Needless to say, the usual disclaimer applies.

Appendix A: Proofs and Elaborations

A.1 Quadratic Utility Model

A.1.1 Assumptions

The following assumptions are utilized for the quadratic utility model in Equation (6):

Assumption 1a: 0 < M < Y
Assumption 1b: b > d
Assumption 2: β[dY + M(b − d − c − E _θ(θ)f)] < 1
Assumption 3: d > 1/βY
Assumption 4: b − c − θ(θ)f > 1/βY.

Assumptions 1a and 1b are also utilized in the case of the generalized Masciandaro model in Appendix A.2, while Assumptions 2–4 ensure the tractability of the model and the meaningful interpretation of results.

A.1.2 Simplification to Equation (1)

Proposition 1.

If:

d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β [ d Y f M + f M 2 ( b − d − c ) ] = 0 ,

then Equation (6) reduces to Equation (1).

Proof.

Starting with Equation (6), we take expectations over θ where E _θ(θ) = p to obtain:

E [ U ] = ( 1 − p ) d Y + M ( b − d − c ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p d Y + M ( b − d − c − f ) − β 2 [ d Y + M ( b − d − c ) − f M ] 2 .

Next, we expand the final quadratic term using the identity (x − y)² = x ² + y ² − 2xy, where x = dY + M(b − d − c) and y = fM:

E [ U ] = ( 1 − p ) d Y + M ( b − d − c ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p d Y + M ( b − d − c − f ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p β 2 f 2 M 2 + 2 d Y f M + 2 f M 2 ( b − d − c ) .

Grouping and rearranging terms to isolate the structure of Equation (1), we have:

E [ U ] = ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) + ( 1 − p ) d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p d ( Y − M ) + M ( b − f ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p β 2 f − 1 f M 2 + β 2 2 d Y f M + 2 f M 2 ( b − d − c ) .

We may also simplify using the identity (1 − p)z + pz = z, where:

z = d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 ,

and further rearrange to obtain:

E [ U ] = ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) + d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β [ d Y f M + f M 2 ( b − d − c ) ] .

Now finally, substituting for:

d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β [ d Y f M + f M 2 ( b − d − c ) ] = 0 ,

we are left with

E [ U ] = ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) ,

which is equivalent to Equation (1). □

A.1.3 Simplifying Parameter Restrictions

In this section, I show the parameter restrictions necessary to reduce the quadratic utility model to the original Masciandaro model. Such a limited case is unlikely to occur in reality, however, as it entails numerous, powerful assumptions–for example, that the constant term β governing the individual’s degree of risk aversion is exactly equal to the probability of laundering conviction; that the probability and severity of punishment should multiply to exactly 1/2; and that it is optimal for the individual to launder none of their criminal funds, even when the value of dirty income is zero.

Proposition 2.

Let:

E [ U 1 ] ≔ ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) E [ U 2 ] ≔ ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) + d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β [ d Y f M + f M 2 ( b − d − c ) ] .

If d = 0, β = p, b = f, pf = 1/2 and c ≥ (1 − p)/2p, then E[U ₁] = E[U ₂].

Proof.

To begin, we define:

A ≔ d ( Y − M ) − β 2 [ d Y + M ( b − d − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β d Y f M + f M 2 ( b − d − c ) ,

and X as the set of conditions:

X ≔ d = 0 β = p b = f p f = 1 2 c ≥ 1 − p 2 p .

Now, by Proposition 1, we know that:

A = 0 ⇒ E [ U 1 ] = E [ U 2 ] .

Therefore, if it can be shown that X ⇒ A = 0, then by transitivity of implications we have:

X ⇒ A = 0 ⇒ E [ U 1 ] = E [ U 2 ] X ⇒ E [ U 1 ] = E [ U 2 ] .

To evaluate whether X ⇒ A = 0, we first calculate the value of A under the condition that d = 0:

A = 0 ( Y − M ) − β 2 [ 0 Y + M ( b − 0 − c ) ] 2 + p M ( b − f ) + β 2 f − 1 f M 2 + p β 0 Y f M + f M 2 ( b − 0 − c ) = β p f − 1 2 ( b − c ) ( b − c ) M 2 + p M ( b − f ) + p f β f 2 − 1 M 2 .

Next, we impose β = p, b = f and pf = 1/2:

A = p 1 2 − 1 2 ( f − c ) ( f − c ) M 2 + p M ( f − f ) + 1 2 1 / 2 2 − 1 M 2 = p 1 2 − 1 2 ( f − c ) ( f − c ) M 2 − 3 8 M 2 .

Letting x ≔ f − c and rearranging to isolate M ²:

A = p x 1 2 − 1 2 x − 3 8 M 2 = p 2 x − x 2 − 3 8 M 2 .

Note that coefficient on M ² contains the quadratic function x − x ², which reaches its maximum value at the vertex:

− 1 2 ( − 1 ) = 1 2 .

Substituting this value into x − x ² gives the function’s maximum value:

1 2 − 1 2 2 = 1 4 .

Therefore, we have:

x − x 2 ≤ 1 4 ⇒ p 2 ( x − x 2 ) − 3 8 ≤ − 1 4 ,

since 0 ≤ p ≤ 1. Due to the negative coefficient on M ², it follows that for any M > 0, A < 0. However, if we allow M ≥ 0, then A = 0 if and only if M = 0, which occurs only when M* = 0 solves the individual’s utility maximization problem. To test this, we restrict the value of the expected utility function E[U ₂] according to the first four conditions in X:

E [ U 2 ] = ( 1 − p ) ( f M − c M ) + p ( − c M − f M 2 ) + p 2 x − x 2 − 3 8 M 2 ,

calculate the first and second derivatives with respect to M:^[18]

∂ E [ U 2 ] ∂ M = ( 1 − p ) f − c + 2 p 2 x − x 2 − 7 8 M ∂ 2 E [ U 2 ] ∂ M 2 = 2 p 2 x − x 2 − 7 8 ,

and evaluate these derivatives at M = 0:

∂ E [ U 2 ] ∂ M M = 0 = ( 1 − p ) f − c . ∂ 2 E [ U 2 ] ∂ M 2 M = 0 = 2 p 2 x − x 2 − 7 8 .

For M = 0 to represent a global optimum, we need:

∂ E [ U 2 ] ∂ M M = 0 ≤ 0 ∂ 2 E [ U 2 ] ∂ M 2 M = 0 < 0 .

Recalling that x = f − c and substituting f = 1/2p and c = (1 − p)/2p, the first and second derivatives become:

∂ E [ U 2 ] ∂ M M = 0 = ( 1 − p ) 1 2 p − 1 − p 2 p = 0 ∂ 2 E [ U 2 ] ∂ M 2 M = 0 = p 1 2 p − 1 − p 2 p − 1 2 p − 1 − p 2 p 2 − 7 4 = p − 7 4 < 0 ,

where the inequality follows from 0 ≤ p ≤ 1. Note that for all values of c ≥ (1 − p)/2p, we have E [ U 2 ] M ≤ 0 and E [ U 2 ] M M < 0 , meaning that X ⇒ M* = 0. Therefore, we conclude that:

X ⇒ M * = 0 ⇒ A = 0 ⇒ E [ U 1 ] = E [ U 2 ] X ⇒ E [ U 1 ] = E [ U 2 ] .

□

A.2 Generalized Masciandaro Model

A.2.1 Setup

Consider the following expected utility function for the prospective launderer:

(11) E [ U ] = ( 1 − p ) [ b M + d ( Y − M ) − c M ] + p [ b M + d ( Y − M ) − c M − f M α ] ,

or, equivalently:

(12) E [ U ] = d Y + M ( b − d − c ) − p f M α .

This model represents a more generalized form of Equation (1), with the addition of the parameter d, the application of benefits in both scenario (conviction and non-conviction), and alterations made to the punishment technology. For the latter case, it is assumed that social harm is an increasing and convex function of the amount of money laundered, and that the authorities aim to capture this convexity to some degree in the size of the punishment.^[19] This leads to a more generalized punishment of the form fM ^α, with α > 1.

A.2.2 Relations

First, we differentiate Equation (12) with respect to the parameters in the model:

(13) ∂ E [ U ] ∂ b = M ≥ 0 ∂ E [ U ] ∂ d = Y − M ≥ 0 ∂ E [ U ] ∂ c = − M ≤ 0 ∂ E [ U ] ∂ p = − f M α ≤ 0 ∂ E [ U ] ∂ f = − p M α ≤ 0 ∂ E [ U ] ∂ f = − p M α ≤ 0 .

If we restrict attention to amounts of laundering for which 0 < M < Y (Assumption 1a), then all of these inequalities become strict. In words, we can see that increases in both the value of clean income (b) and the value of dirty income (d) are positively related to overall expected utility, while the effect of an increase in the laundering cost parameters (p, f, c and α) is negative. Next, the expected utility function for money laundering is differentiated twice with respect to M, giving the necessary and sufficient conditions for a global maximum:

(14) ∂ E [ U ] ∂ M = b − d − c − α p f M α − 1

and

(15) ∂ 2 E [ U ] ∂ M = − ( α − 1 ) α p f M α − 2 < 0 ,

since α > 1. As in the original Masciandaro model, the function reaches its maximum where the individual chooses the utility maximizing amount of money laundering M*, which here is equal to:

(16) M * = b − d − c α p f 1 a − 1 .

In contrast to Equation (2), Equation (16) clearly shows that the relationship between the values of clean and dirty income to the individual and the transaction costs of laundering (b − d − c) is crucial in determining the utility maximizing amount of laundering. One interpretation of this is that because laundering carries an additional transaction cost, the amount that the individual will launder can be reduced to zero, even when dirty income is less valuable than clean income. Specifically, the utility maximizing amount of laundering is zero when:

(17) d = b − c .

Of course, if dirty income is assumed to provide no value, then the individual will always launder some positive amount, provided that the value of clean income outweighs the transaction costs of laundering (b > c).

Lastly, by taking first-order derivatives of (16) with respect to each parameter, we can see that the utility maximizing amount of laundering is increasing with respect to the value of clean income, and decreasing with respect to the value of dirty income, the transaction costs of laundering, and the AML policy parameters:

(18) ∂ M * ∂ b = ( b − d − c ) 1 α − 1 − 1 ( α − 1 ) ( α p f ) 1 α − 1 > 0 ∂ M * ∂ d = − ( b − d − c ) 1 α − 1 − 1 ( α − 1 ) ( α p f ) 1 α − 1 < 0 ∂ M * ∂ c = − ( b − d − c ) 1 α − 1 − 1 ( α − 1 ) ( α p f ) 1 α − 1 < 0 ∂ M * ∂ p = − ( b − d − c ) 1 α − 1 p ( α − 1 ) ( α p f ) 1 α − 1 < 0 ∂ M * ∂ f = − ( b − d − c ) 1 α − 1 f ( α − 1 ) ( α p f ) 1 α − 1 < 0 ∂ M * ∂ α = − b − d − c α p f 1 a − 1 α − 1 + α ⁡ ln b − d − c α p f α ( α − 1 ) 2 < 0 ,

for amounts of laundering where 0 < M < Y.

A.2.3 Simplification to Equation (1)

Proposition 3.

If d(Y − M) + pbM = 0 and α = 2, then Equation (11) reduces to Equation (1).

Proof.

We begin with Equation (11), which is expanded to a more easily manipulable form:

E [ U ] = ( 1 − p ) ( b M − c M ) + ( 1 − p ) d ( Y − M ) + p ( − c M − f M α ) + p d ( Y − M ) + p b M .

Using the identity (1 − p)x + px = x, with x = d(Y − M), we may simplify this expression to:

E [ U ] = ( 1 − p ) ( b M − c M ) + p ( − c M − f M α ) + d ( Y − M ) + p b M .

Substituting d(Y − M) + pbM = 0 and α = 2, we have:

E [ U ] = ( 1 − p ) ( b M − c M ) + p ( − c M − f M 2 ) ,

which is equivalent to Equation (1). □

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Received: 2024-09-12

Accepted: 2025-06-02

Published Online: 2025-07-11

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Articles in the same Issue

https://doi.org/10.1515/rle-2024-0115

Keywords for this article

money laundering; AML; microeconomics; crime; public policy