Sound Taxation? On the Use of Self-declared Value Marco A. Haan∗

Pim Heijnen Lambert Schoonbeek Linda A. Toolsema December 22, 2010

Abstract In the 16th century, foreign ships passing through the Sound had to pay ad valorem taxes, known as the Sound Dues. To give skippers an incentive to declare the true value of their cargo, the Danish Crown reserved the right to purchase it at the declared value. We show that this rule does not induce truth-telling, but does allow the authorities to effectively implement a given tax rate. Keywords: Signaling; Taxation. JEL classification: C72; H21; N7. ∗

Corresponding author, [email protected]. Phone: +31 50 3637327. Fax +31 50 363 7337. All authors are affiliated to the Department of Economics, Econometrics and Finance, Faculty of Economics and Business, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands. The authors thank two anonymous referees, the editor J¨ urgen von Hagen, Martin Besfamille, Tatiana Kiseleva and seminar participants at the University of Groningen, ASSET 2010, EARIE 2010, ESEM 2009, FEMES 2008, GAMES 2008, IIPF 2008, and the NAKE Day 2007 for useful comments. Heijnen gratefully acknowledges financial support by the Netherlands Organization for Scientific Research (NWO).

1

1

Introduction

In the 16th century, the Kingdom of Denmark controlled both sides of the Sound (Øresund), an important waterway situated between present-day Denmark and Sweden. All foreign ships passing through this strait had to make a stop in Helsingør (known in English as Elsinore, the stage for Shakespeare’s Hamlet) and pay taxes to the Danish Crown, which varied between some 15% of the value of the cargo. These taxes are often referred to as the Sound Dues. Although obviously unfamiliar with the concept of incentive compatibility, the Danish Crown was fully aware that such a tax would give skippers a strong incentive to cheat and declare a value much lower than the true one. It came up with an intriguing solution. The Crown reserved the right to purchase the cargo at the value declared by the skipper. Thus, a skipper who declared a value that was too low ran the risk of losing his cargo at a price below market value. But a skipper who declared a value that was too high ran the risk of paying too much in taxes.1 This mechanism is clearly ingenious, but it does raise a number of questions. First, what is the optimal confiscation strategy for the tax authority? Obviously, it cannot be part of a Nash equilibrium to never purchase the cargo. Then, the threat of confiscation would simply be empty. Second, is this mechanism truth-telling? That is, does it give skippers the incentive to always declare the true value of their cargo? Third, does it allow the tax authority to effectively raise the tax rate that it desires? Put differently, did this mechanism really amount to sound taxation or was there something 1

See e.g. Hill (1926, p. 83): “The King expressly reserved the option of accepting the dues or buying the article at the listed price”. Zins (1972, p. 186) argues that “... as the King of Denmark had the right to purchase goods transported through the Sound, one can assume that the shipmasters in general declared the real value of their cargoes”. Menefee (1996) provides a concise history of the Sound Dues and notes that, as of 1611, the King retained “the option of accepting the dues or buying the article at the listed price” (p. 108). The mechanism is also described in some papers in the economics literature (Gerchak and Fuller, 1992; Niou and Tan, 1994). It is also mentioned in Maczak (1972) as cited by Odlyzko (2004), and in Shasha (2007).

2

rotten in the state of Denmark? We address these questions in this paper. The relevance of our analysis goes far beyond some tax scheme that was used centuries ago. There are numerous instances where a similar tax has either been proposed or implemented. In 1891, New Zealand passed a Land and Income Tax act, based on a “self-assessment with the shrewd device of making Government’s purchase at the tax value an effective check on the owner’s assessment” (Condliffe 1930, p. 182, as quoted by Niou and Tan, 1994). Dr. Sun Yat-sen, the first provisional president when the Republic of China was founded in 1912, proposed a land tax using the exact same mechanism: landowners are taxed according to their declared value of the land, and the government can also buy the land at the same price.2 Anyone buying a house in southern Europe may face property taxes based on selfdeclared value.3 Other examples include land tax in India around 1900 as well as in present-day Taiwan, taxes on works of art leaving Mussolini’s Italy, and British taxes on imported American Jerome clocks (see Gerchak and Fuller, 1992, and the references therein). In our model we assume that, using the terminology of the Sound Dues example, the skipper knows the true value of his cargo, but the tax authority does not. We analyze this situation as a signaling model with asymmetric information. Although obviously related, our paper does not fit the optimal taxation 2

Niou and Tan (1994) quote at length: “How indeed can the price of the land be determined? I would advocate that the landowner himself should fix the price. The landowner reports the value of his land to the government and the government levies a land tax accordingly... [T]he government makes two regulations: first, that it will collect taxes according to the declared value of the land; second, that it can also buy back the land at the same price... According to my plan, if the landowner makes a low assessment, he will be afraid lest the government buy back his land at that value and make him lose his property; if he makes too high an assessment, he will be afraid of the government taxes according to this value [...], he will certainly not want to report the value of his land too high or too low; he will strike a mean and report the true market price to the government (Sun, 1924: 177-178).” 3 See Davies (2004): “In all three countries [France, Spain and Italy], these taxes are charged on the value of the property as declared in the deed of sale, which has traditionally borne little resemblance to the price actually paid. The penalties for tax dodging in this way are severe and include compulsory state purchase at the declared value, plus fines.”

3

literature. For example, a related paper in that literature is Border and Sobel (1987). They use a set-up related to ours, but assume that it is the objective of the tax authority to capture as much wealth as possible. In our analysis, the King first determines the tax rate t∗ that he wants to implement. How this desired tax rate is determined, is immaterial for our purpose. It could be the rate that maximizes expected revenue, but the King could also use some other objective function. Given t∗ , the problem then is to find a mechanism that allows the King to indeed implement this desired tax rate, i.e., to make sure that the amount of tax that the skipper pays is indeed t∗ per unit of value of cargo. It is that problem that we study in this paper. The modern-day literature on mechanism design focuses on truth-telling mechanisms (see e.g. Fudenberg and Tirole, 1991, chapter 7): according to the revelation principle, it is sufficient to restrict attention to equilibria in which agents truthfully report their type. We show that although the tax scheme investigated here is not a truth-telling mechanism, it does allow the taxation authorities to effectively (i.e., in expected value) implement any desired tax rate, provided it is not too high. This is true in any Nash equilibrium in our model. Surprisingly, there are equilibria in which the probability that the authorities will confiscate the cargo does not depend on the declared value. In other words, there are equilibria in which the authorities use what we will refer to as a signal-independent strategy. If we restrict attention to such simple strategies, the expected payoff of the skipper is also the same across Nash equilibria. Signal-dependent equilibria also do exist. We are not the first to give a game-theoretic treatment of these issues. Niou and Tan (1994) focus on Sun Yat-sen’s land tax. There are two main differences with our approach. First, in their model the tax authority is not a strategic player, but simply follows some pre-determined rule without taking the incentives of the tax payer into account. In our paper, we model the interaction as a game in which the tax authority also behaves strategically. Put differently, whereas Niou and Tan (1994) require the tax authority to

4

commit to some strategy in advance, we do not require such commitment. Second, Niou and Tan (1994) focus on truth-telling: they study to what extent the scheme, or some variation of it, can induce the tax payer to always tell the truth. They show that any extension of the scheme that does induce truth-telling requires the use of costly audits. We use a weaker requirement: rather than requiring the tax payer to always tell the truth, we study whether the scheme at least yields on average the revenue that the tax authority was aiming for, given the true value of the object. Arguably, if a scheme is not truth-telling but it does yield the required revenues, there is little reason for the tax authority to worry. Moreover, we show that with this weaker requirement, the tax authority does not need to perform any costly audits. The remainder of this paper is structured as follows. In the next section we present our model. Section 3 derives a number of properties that hold in any Nash equilibrium of the model. We show that if a Nash equilibrium exists, it necessarily has the King being indifferent between his two possible actions. Moreover, his expected payoff is equal across equilibria. In Section 4 we show that an equilibrium exists, even if we restrict attention to the case in which the King’s action does not depend on the declaration that the skipper makes. We also show how to construct such equilibria. Section 5 looks at signal-dependent equilibria. In Section 6 we show that the King can effectively implement any desired tax rate. Section 7 concludes. There, we also discuss other applications of the framework that we develop in this paper.

2

The Sound Dues Game

We have two risk-neutral players, a Skipper and a King (the Danish Crown). We will refer to the Skipper as being female and to the King as being male. The timing of the game is as follows. First, Nature decides on the type v of the Skipper, which is the true value of the cargo that she transports. This value is private information. It is common knowledge that v is drawn from a probability density function f (v) with support [0, v]. Second, the Skipper 5

sends some message m, which is the value of the cargo that she declares to the King. For ease of exposition but without loss of generality, we assume that m is restricted to [0, v]. Third, the King takes a decision d ∈ {a, r}: he either Accepts (a) or Rejects (r) the message m of the Skipper. The payoff to the Skipper is denoted uS (d, v, m), that of the King uK (d, v, m). If the King plays Accept, the Skipper pays a tax tm which is based on her announced value m, with the true tax rate t ∈ (0, 1). In that case we have uS (a, v, m) = −tm,

uK (a, v, m) = tm.

(1)

If the King plays Reject, he buys the cargo at price m. The Skipper then obtains m, but loses the true value v, while the King pays m but obtains the cargo: uS (r, v, m) = m − v,

uK (r, v, m) = φv − m,

(2)

with φ ∈ (0, 1] a given parameter. In the simplest case φ = 1 and we have a zero-sum game: the value of the cargo for the King equals that for the Skipper. We do however also allow for the case that φ < 1 so the value for the King is lower, for example since he faces some transaction costs from appropriating, storing, and selling a cargo. In the remainder of this paper, we will often distinguish between the zero-sum and the non-zero-sum version of our game. A strategy for the Skipper assigns to each type v a probability density function zv (m) on the set of messages [0, v]. Beliefs of the King assign to each message m a probability density function of possible Skipper types βm on [0, v]. A strategy for the King is a function which, given beliefs βm , maps the interval of messages into the unit interval, i.e., p : [0, v] → [0, 1], where p(m) denotes the probability with which the King plays Reject if he observes signal m. Both players want to maximize their ex ante expected payoff. Notice that these payoffs depend on a joint distribution of v and m which, in general, may be degenerate. We will investigate perfect Bayesian Nash equilibria of the game (see e.g. Fudenberg and Tirole, 1991).

6

3

Equilibrium Properties

3.1

Introduction

Before deriving full equilibria of our game, we can already establish some properties that any equilibrium necessarily satisfies. In particular we show that in any equilibrium, the King is indifferent between playing Accept and Reject (Section 3.2), and has the same expected revenue (Section 3.3).

3.2

The King’s Indifference

We have the following: Proposition 1. In any equilibrium the King is indifferent between Accept and Reject, for every message m. Proof. First note that the result trivially holds for those m for which the King strictly mixes between the two actions. Next, let p(m) = 1 for some m and let v be a type of Skipper that selects this signal m with positive probability in equilibrium. Note that, given the signal m, the King plays Reject with certainty, implying a payoff uS (r, v, m) for the Skipper. We will argue now that for the given v and m we must have g(v, m) ≡ uK (r, v, m) − uK (a, v, m) = φv − (1 + t)m ≤ 0. Suppose otherwise, i.e., suppose that φv − (1 + t)m > 0. We then can find an m0 > m and m0 sufficiently close to m such that both uS (a, m0 , v) = −tm0 > m − φv > m − v = uS (r, m, v) and uS (r, m0 , v) = m0 − v > m − v = uS (r, m, v). Hence, regardless of the value of p (m0 ), it is always optimal for a Skipper of type v to deviate to m0 , which cannot be possible in equilibrium. Therefore, p(m) = 1 implies that g(v, m) ≤ 0 for all types v of Skipper that play action m in equilibrium. Furthermore, in the case that p(m) = 1, given m, the expected payoff of the King from playing Reject should be at least as large as the expected payoff from playing Accept: E [uK (r, v, m) |m] ≥ E [uK (a, v, m) |m], or equivalently, E [g (v, m) |m] ≥ 0. This contradicts with g(v, m) ≤ 0 unless E [g (v, m) |m] = 0. The proof for the case p(m) = 0 for some m goes along the same lines. 7

3.3

The King’s Payoff

Using Proposition 1, we immediately have: Proposition 2. In any equilibrium the ex ante expected payoff of the King equals

φt E [v]. 1+t

Proof. We know from Proposition 1 that in any equilibrium the King is indifferent between Reject and Accept for every message m: E [uK (r, v, m) |m] = E [uK (a, v, m) |m]. Using (1) and (2), we can rewrite this as E [uK (r, v, m) |m] = E [φtv − t × uK (r, v, m)|m]

(3)

or

φt E [v|m] . (4) 1+t The ex ante expected payoff of the King in equilibrium is the expected value E [uK (r, v, m) |m] =

over m. Using Theorem 6.5.4 (a) of Ash (1972), we obtain φt E [E [uK (r, v, m) |m]] = E [v] , 1+t which completes the proof.

(5)

Hence, in any equilibrium, the ex ante expected payoff of the King can be calculated without knowing the precise equilibrium strategies of both players; it is a function of only φ, t and the expected value of v. As an aside, Proposition 2 immediately implies that in any equilibrium of the zero-sum game (with φ = 1), the ex ante expected equilibrium payoff t of the Skipper equals − 1+t E [v]. It is well-known that any equilibrium of

zero-sum games with complete information has the same ex ante equilibrium payoff (see e.g. Luce and Raiffa, 1957, Appendix 2). Here, it turns out that this property also holds in our zero-sum game with asymmetric information.

4 4.1

Signal-independent Equilibria Definition

In the previous section we showed that if equilibria exist, they necessarily have the King being indifferent between Accept and Reject, and the expected 8

payoff of the King being equal to

φt E [v]. 1+t

But of course these observations

are of little use if an equilibrium fails to exist. In this section, we therefore show the existence of equilibrium. For simplicity, for now we restrict attention to equilibria in which the action of the King is independent of the message m that he receives. We will refer to this as a signal-independent strategy: Definition 1. The King uses a signal-independent strategy if his action does not depend on the message of the Skipper: p(mi ) = p(mj ) for all mi , mj ∈ [0, v]. We call an equilibrium signal independent if it has the King using a signal-independent strategy. There are a number of reasons to focus on such strategies. First, since these strategies are simple, they require relatively little sophistication for the players in this game, and are relatively easy to analyze.4 Second, there are obvious practical advantages to using a signal-independent strategy. The rule is easy to communicate to the agent in charge of the actual collection of taxes, and it is easy to monitor. The rule does not depend on the actual distribution of values, which implies it is stable through time and across circumstances. Third, we will show that in the zero-sum version of our game, the unique separating equilibrium is signal independent, and hence it is also a natural candidate for our non-zero-sum analysis. Fourth, as the main purpose of this section is to show existence of equilibrium, it is sufficient if we can already do so while restricting attention to signal-independent equilibria.

4.2

Analysis

We now have: 4

Wilson (1987) criticizes most of the game theory literature as relying too much on common-knowledge assumptions, and argues for the use of simple mechanisms that do not require such assumptions. This is known as the Wilson Doctrine (see also e.g. Chung and Ely, 2007). Although our signal-independent strategies do not strictly satisfy Wilson’s requirements, they are at least a step in that direction.

9

Proposition 3. In any signal-independent equilibrium the King uses the strategy p = p∗ ≡ t/(1 + t), and the ex ante expected payoff of the Skipper t equals − 1+t E [v].

Proof. Suppose the King uses a signal-independent strategy. First, suppose he sets p = p∗ ≡ t/(1 + t). In that case, the Skipper is indifferent between all messages m ∈ [0, v], since p∗ × uS (r, v, m) + (1 − p∗ ) × uS (a, v, m) = −

v 1+t

is independent of m. Next, suppose p > p∗ . Then p×

∂uS (r, v, m) ∂uS (a, v, m) + (1 − p) × = p(1 + t) − t > 0, ∂m ∂m

which implies that all types v of the Skipper will choose m = v. However, we see that uK (r, v, v) < uK (a, v, v) for all v, which implies that the King then wants to switch to Accept, i.e., set p = 0 < p∗ . We thus obtain a contradiction. A similar argument shows that the King cannot play some p < p∗ . The ex ante expected payoff for the Skipper in any signal-independent equilibrium is given by E[p∗ × uS (r, v, m) + (1 − p∗ ) × uS (a, v, m)] = −

t E[v], 1+t

where the expectation on the LHS is taken with regard to both v and m, the expectation on the RHS is taken with respect to v only, and the equality follows from Theorem 6.5.4 (a) of Ash (1972). Hence, while we already know that the payoff for the King is equal across equilibria, the restriction to signal-independent equilibria also pins down the expected payoff for the Skipper. It is now easy to show that, even if we restrict attention to signalindependent strategies, there is a multiplicity of equilibria (see below). Two types of equilibrium are of particular interest. In a separating equilibrium, each type of Skipper chooses a different signal. Upon observing her signal, 10

the King can thus infer the true type of the Skipper. In a pooling equilibrium each type of Skipper selects the same signal m, which makes m completely uninformative. We can show the following: Proposition 4. If the King focuses on signal-independent strategies, the game has a unique separating equilibrium and a unique pooling equilibrium. Proof. Using Proposition 1 we know that in any equilibrium the King is indifferent between Accept and Reject for every message m. This implies the following restriction: E [uK (a, v, m) |m] − E [uK (r, v, m) |m] = φE [v|m] − (1 + t)m = 0,

(6)

for every m. The conditional distribution of v given m follows from the strategy of the Skipper. Using Proposition 3, we see that the King will use the strategy p(m) = p∗ . As a result, the Skipper is indifferent between any message m. In a separating equilibrium we have E [v|m] = v, i.e., in such an equilibrium, the conditional distribution of v given m is very simple, since v is known with certainty given m. It follows that there exists a unique separating equilibrium in which the Skipper plays the strategy m∗ (v) ≡ φv/(1 + t). In a pooling equilibrium each type of Skipper selects the same signal, and the conditional distribution of v given m is simply equal to the unconditional distribution of v. We easily see that there exists a unique pooling equilibrium in which all types of Skipper send the message m∗∗ ≡ φE[v]/(1 + t). To see that many more equilibria do exist, even if we restrict attention to signal-independent equilibria, note the following. First, from the proof of Proposition 3 we have that the Skipper is necessarily indifferent between all possible messages she can send. For any v, there always exists a unique m∗ (v) that would make the King indifferent between Accept and Reject, should he be able to observe v. The unique separating equilibrium has a Skipper of type v exactly sending that message m∗ (v). Upon observing m,

11

the King then exactly knows which type of Skipper he is facing, which is v(m) = m∗−1 (m) = (1 + t)m/φ. For the King to be willing to mix between Accept and Reject, however, it is not necessary that he can exactly infer the type of Skipper he is facing from observing the message m that the Skipper sends. It is sufficient that he faces the required type v on average, given m. Consider for example the situation in Figure 1. Suppose that Skipper types are uniformly distributed. Suppose that, for this particular situation, the function m∗ (v) is given by the straight line. In the separating equilibrium the Skipper exactly plays m∗ (v). But she may just as well add some noise to her signal. Suppose that a type v plays a message m that is drawn from a uniform distribution between the two dotted curves. From the figure, it is then easy to see that, for the King, the expected value of v given m is not affected. Hence, this also constitutes an equilibrium, in particular, a semi-pooling equilibrium. mu(v) m

v

Figure 1: An illustration of a semi-pooling equilibrium Our unique pooling equilibrium takes this argument to the extreme. It can even be an equilibrium for the Skipper to play the same message m∗∗ , regardless of her type – as long as, given the distribution of types, m∗∗ is 12

constructed in such way that the King is exactly indifferent between Accept and Reject.

4.3

The Zero-Sum Game

Consider the zero-sum game, with φ = 1. Assume that the King focuses on differentiable strategies p(·), which are not necessarily signal-independent. Using Proposition 1, we know that in any equilibrium the King is indifferent between Accept and Reject for every message m. It then easily follows that in a separating equilibrium we must have m∗ (v) = v/(1 + t). Second, for a Skipper of type v, message m∗ (v) must be optimal, i.e., m∗ (v) ∈ argmaxm H(m, v), where H(m, v) ≡ p(m)(m − v) + (1 − p(m))(−tm).

(7)

Differentiating H(m, v) with respect to m, we find the corresponding firstorder condition p0 (m)(m − v) + p(m) + p0 (m)tm − t(1 − p(m)) = 0.

(8)

We thus have: Proposition 5. Consider the zero-sum game and suppose that the King focuses on differentiable strategies. Then the game has a unique separating equilibrium. In this equilibrium the King’s strategy is independent of the signal m that the Skipper sends. Proof. Substituting v = (1 + t)m into (8) yields p(m) = p∗ = t/(1 + t). The result follows directly. Hence, the separating equilibrium in the zero-sum game has the King playing a signal-independent strategy.

5

Signal-dependent Equilibria

For completeness, we also briefly consider equilibria in which the King does not restrict himself to using signal-independent strategies. For simplicity, we focus on separating equilibria. Doing so, we have the following: 13

Proposition 6. Suppose that φ < 1 and that the King uses differentiable strategies. We then have a separating equilibrium in which the King sets φ

p(m) = km 1−φ +

t , 1+t

with k negative and sufficiently close to zero. Proof. Using Proposition 1, it is easy to see that in a separating equilibrium we must have m∗ (v) = φv/(1 + t). Next, m∗ (v) must maximize the expected payoff H(v, m) of a Skipper of type v, defined in (7). Substituting v = (1 + t)m/φ into the first-order condition (8), we find µ ¶ 1 1 1− mp0 (m) + p(m) = . φ 1+t

(9)

This differential equation has a solution φ

p(m) = km 1−φ +

t , 1+t

(10)

where k is a constant. We have a signal-dependent strategy if k 6= 0. We have to make sure that p(m) is a proper probability. Note that 0 < p(0) < 1. Further, p(m) ∈ (0, 1) for m ∈ (0, φv/(1 + t)) if k is sufficiently close to zero. Next, verifying the second-order condition for the Skipper’s maximization problem, we see that ∂ 2 H(m∗ (v), v) kφv = 2 ∂m 1−φ

µ

φv 1+t

¶ 3φ−2 1−φ

.

(11)

It follows that k must be negative. We thus have an infinite number of separating signal-dependent equilibria. To see this, note that if there is an equilibrium for k = κ, then there is also an equilibrium where k = κ0 for every κ < κ0 < 0. Also note that the signalindependent equilibrium of Proposition 4 corresponds to the case k = 0. Finally note that, from the proof, we immediately have that in the case of a zero-sum game (where φ = 1), a separating signal-dependent equilibrium does not exist, as we already observed in the previous section. 14

The expected payoff of a Skipper of type v in the equilibrium of Proposition 6 equals tv H(m (v), v) = − − kv(1 − φ) 1+t ∗

µ

φv 1+t

φ ¶ 1−φ

.

(12)

Since k is negative, it follows that the Skipper has a higher ex ante expected payoff in the signal-dependent separating equilibrium than in the signal independent equilibrium of Proposition 4. As we showed earlier, the King has the same ex ante expected payoff in both equilibria.

6

Revenue Implications

Above, we showed that equilibria in our Sound Dues game do exist, and that in any equilibrium, the ex ante expected revenue for the King equals

φt E[v] 1+t

per Skipper. With complete information the tax revenue would be tE[v]. Hence, in this sense, in any equilibrium the Skipper on average underreports her valuation by a factor

φ . 1+t

At this point the expected cost of receiving a

lower price for the cargo in case of purchase is equal to the expected benefit of paying a lower tax. The King can use this knowledge to adjust the true tax rate t and effectively implement a desired rate t∗ . Proposition 7. In the Sound Dues game, the King can effectively (i.e., in expected value) implement any desired tax rate t∗ ∈ [0, φ/2) by imposing a true tax rate t = t∗ /(φ − t∗ ). In particular, this can be done by confiscating cargo with fixed probability p∗ = t/(1 + t), regardless of its declared value. The first claim in this proposition states that the King can implement any tax rate. This holds for any Nash equilibrium. The second claim shows one particular way of doing so: by using a signal-independent strategy. The proposition shows that the taxation scheme we study is efficient in the following sense. Suppose that the King has decided that he wants to tax all cargo at a rate of t∗ . As explained in the introduction, it is immaterial and beyond the scope of this paper how this tax rate has been determined. Then 15

the proposition implies that the King can implement the desired tax revenue t∗ v per Skipper in expected value by imposing a true tax rate t = t∗ /(φ − t∗ ). This is the case even though the Skipper does not declare the true value in equilibrium. It is easy to see that in the separating equilibrium discussed in Section 4.2 each Skipper precisely pays the required tax, rather than only in expected value. Another implication is that there is no necessity to second-guess the Skipper. If the equilibrium effectively implements the desired tax rate anyhow, then e.g. hiring an appraiser is completely redundant and would only add to the costs of tax collection. Hence, the Sound toll scheme is a sound taxation scheme. Note however that the mechanism is not truth-telling, and in most cases not even truthrevealing. But for the expected tax revenue, this is immaterial.

7

Conclusion

In this paper, we studied an ingenious method used by Danish authorities in Medieval times to collect taxes. The authorities raised taxes on the basis of the declared value of the cargo of a ship, but reserved the right to purchase the cargo at that declared value. We showed that, although this is never a truth-telling and often not even a truth-revealing mechanism, it does allow the authorities to raise the desired tax revenue in expected value. Apart from raising the required tax revenue, the mechanism has other obvious advantages. It is simple, and therefore easy to communicate to any civil servant that is involved in the collection of taxes. It is relatively cheap, as it does not require costly appraisals. Of course, there are disadvantages to the mechanism as well. The tax authority does need to acquire the cargo in some cases, which implies costs of storage and resale, for example. Still, such costs are taken into account in the model we propose. The required tax revenue that the mechanism allows the government to raise, are net of such costs. Insights from this paper not only apply to the case of Medieval tax duties. 16

In the working paper version (Haan et al., 2008), we build a more general framework in which one party is fully informed about the true value of some item, whereas the other party is not informed at all. One application of that framework concerns the dissolution of a public-private partnership. Suppose that a public party wants to end some joint-venture with a private party. Also assume that the private party is fully informed about the true value of the joint-venture, but the public party is not. The public party can then ask the private party to name a price m, with the understanding that the public party could either decide to sell its own share to the private party at price m, or to buy the private party’s share at price m. Also in this case, the expected revenue for the public party is the same across equilibria, and the private party does not obtain an advantage from its private information. Another application is income taxation, and in particular the application of costly tax audits. Rather than the choice of whether or not to acquire some property, the tax authority then has the choice whether or not to audit some tax return. Qualitatively, the results are the same to those in the Sound Dues case: by randomly auditing a fixed fraction of tax returns, the tax authorities can make sure that citizens pay a desired tax rate in expected value. Further applications include the use of shotgun clauses and the allocation of an indivisible item. In all these applications, a mechanism similar to that of the Sound Dues allows the uninformed party to obtain its desired revenue in expected value. The beauty of this type of mechanism is that being completely uninformed does not hurt a party at all in acquiring its share in expected value.

References Ash, R. B. (1972): Real Analysis and Probability. Academic Press, New York. Border, K. C., and J. Sobel (1987): “Samurai Accountant: A Theory of Auditing and Plunder,” Review of Economic Studies, 54(4), 525–540. 17

Chung, K.-S., and J. Ely (2007): “Foundations of Dominant-Strategy Mechanisms,” Review of Economic Studies, 74(2), 447–476. Condliffe, J. (1930): New Zealand in the Making. Allen & Unwin, London. Davies,

J.

The

Guardian,

(2004):

“Sun,

August

2,

Sea

and

Available

online

Tax,” at

http://www.guardian.co.uk/money/2004/aug/02/buyingpropertyabroad. Fudenberg, D., and J. Tirole (1991): Game Theory. mit-Press, Cambridge, Mass. Gerchak, Y., and J. D. Fuller (1992): “Optimal Value Declaration in ”Buy-Sell” Situations,” Management Science, 38(1), 48–56. Haan, M. A., P. Heijnen, L. Schoonbeek, and L. A. Toolsema (2008): “Sound Taxation? On the Use of Self-declared Value,” CeNDEF Working Paper 08-02, University of Amsterdam. Hill, C. E. (1926): The Danish Sound Dues and the Command of the Baltic: A Study of International Relations. Duke University Press, Durham, North Carolina. Luce, R., and H. Raiffa (1957): Games and Decisions. Wiley, New York. Maczak, A. (1972): Miedzy Gdanskiem a Sundem: Studia nad Handlem Baltyckim od Polowy XVI to Polowy XVII Wieku. Panstwowe Wydawnictwo Naukowe, Warsaw. Menefee, S. (1996): “The Sound Dues and Access to the Baltic Sea,” in The Baltic Sea: New Developments in National Policies and International Cooperation, ed. by R. Platz¨oder, and P. Verlaan, pp. 101–132. Kluwer Law International, The Hague. Niou, E. M., and G. Tan (1994): “An Analysis of Dr. Sun Yat-sen’s Self-assessment Scheme for Land Taxation,” Public Choice, 78, 103–114. 18

Odlyzko, A. (2004): “The Evolution of Price Discrimination in Transportation and its Implications for the Internet,” Review of Network Economics, 3, 323–346. Shasha, D. E. (2007): “The Tolls of Elsinore,” Scientific American, August, Available online at http://www.sciam.com/article.cfm?id=the–tolls– of–elsinore. Sun, Y. (1924): The Three Principles of the People. Translated into English by Frank W. Price. China Publishing Co., Taipei, Taiwan, ROC. Wilson, R. (1987): “Game-Theoretic Approaches to Trading Processes,” in Advances in Economic Theory, ed. by T. Bewley, Fifth World Congress, Econometric Society Monograph Series, No. 12, pp. 33–77. Cambridge University Press, Cambridge. Zins, H. (1972): England and the Baltic in the Elizabethan Era. Manchester University Press, Rowman & Littlefield, Totowa, N.J.

19