Taxation of firms with unknown mobility

Transcription

1 Taxation of firms with unknown mobility Johannes Becker Andrea Schneider University of Münster University of Münster Institute for Public Economics Institute for Public Economics Wilmergasse 6-8 Wilmergasse Münster Münster Germany Germany August 20, 2012 Abstract This papers analyzes optimal taxation of firms when their true degree of mobility is unknown. We consider a two-period model with a revenue maximizing government facing an industrial sector with an unknown number of mobile firms. Taxes are set in both periods. Thus, the firms response to tax policy in the first period can be exploited to optimally set taxes in the second period. Firms, however, may anticipate the government learning from past migration decisions and adjust their location choice accordingly. We derive all symmetric Bayesian equilibria with a focus on the so far neglected one where government sets a tax rate that triggers partial migration but full revelation of the true number of mobile firms. We show that, in general, the higher the degree of incomplete information, the higher the expected tax rates. Furthermore, the government may profit from an informational advantage of firms vis-a-vis the government. If government can credibly tie its future tax policy to present policy outcomes, this may lead to a Pareto improvement. JEL classification: H25, H32, H87 Keywords: corporate taxation, firm mobility, international taxation We thank Jon Fiva, Espen Moen, Christian Riis and participants at workshops, research seminars and conferences in Wuppertal, Glasgow and Hamburg for helpful comments. The usual disclaimer applies. Corresponding author. Phone Fax

2 1 Introduction The question how mobile resources are (optimally) taxed has been in the centre focus of tax-related research in the last twenty-five years. Most contributions in this literature implicitly or explicitly assume that policy-makers either know or correctly anticipate the true degree of mobility of the resources under consideration. However, making mistakes in guessing the actual degree of mobility can be costly and may have persistent negative effects on tax revenues in subsequent periods. In this paper, we ask how governments actually learn about the mobility of the entities to be taxed. We consider a two-period model of a small open economy where a revenue maximizing government levies taxes on firms in both periods. Before period 1, the number of mobile firms is unknown to the government. Mobile firms may respond to tax policy in both periods by migrating abroad. Thus, tax policy in the second period can use the information revealed by migration decisions in the first period. However, firms will anticipate the government to learn from first period migration and adjust their location decision accordingly. To put in simple terms: An individual firm anticipating fellow firms to move abroad (due to high taxes) may decide to stay if it expects the government to respond to the loss in tax base by lowering tax rates. We derive all symmetric Bayesian equilibria depending on the economic environment. Our main findings are the following. There are, in general, three types of equilibrium. The first equilibrium implies that the government sets the maximum tax rate and all mobile firms leave the country with certainty. If the fraction of mobile firms exceeds a certain threshold, government implements a second period tax rate that attracts all mobile firms back. Otherwise the maximum tax rate applies in the second period as well. In sectors with high degrees of mobility, this equilibrium maximizes the efficiency loss due to migration cost. This equilibrium most likely occurs if the tax rate abroad and mobility cost are low. The second equilibrium implies that the government sets tax rates in both periods that no mobile firm has an incentive to leave. Then, the fraction of mobile firms remains unknown. This equilibrium may occur if the tax rate abroad and mobility cost are high. The third and maybe most interesting equilibrium has the government setting a tax rate which triggers partial migration. The actual number of firms moving abroad reveals information that can be used to optimally adjust tax policy in the second period. This equilibrium prevents revenue losses due to full firm migration at the cost of lower tax rates in the first period. This equilibrium is most likely for medium foreign tax rates and low mobility cost. 2

3 We compare three different scenarios. In a benchmark scenario, the true degree of mobility is common knowledge. Then, attracting firms back in the second period is never an optimal policy. The second scenario assumes symmetric incomplete information, i.e. both government and firms do not know the true degree of mobility. In a third scenario, firms do know the number of mobile firms while government does not. Comparing these scenarios allows analyzing the role of the degree of incomplete information. We show that, in general, the higher the degree of incomplete information, the higher the expected tax rates. The reason is that the revenue maximizing government loses more revenue if taxes are low and mobility is lower than expected than in the reverse case. Furthermore, the government may profit from an informational advantage of firms vis-a-vis the government, i.e. expected revenue in the third scenario where firms know the true degree of mobility may by larger than in the second scenario. Finally, if government can credibly tie its future tax policy to present policy outcomes, this may lead to a Pareto improvement. The reason is that if the government is able to (credibly) announce a threshold of moving firms above which it lowers the tax such to attract these firms back, it may reduce equilibrium migration and, thus, migration cost and tax revenue losses. All domestic agents benefit. Corporate taxation in the presence of mobile firms has been intensively studied in the literature, see Richter & Wellisch (1996), Boadway, Cuff & Marceau (2002) and Fuest (2005). A strand of literature following Keen (2001) analyzes the desirability of special tax treatment of mobile tax bases, e.g., internationally mobile firms (see also Janeba & Smart, 2003). Of course, special treatment requires knowledge on the type of firm, which is not given in our framework. Osmundsen, Hagen & Schjelderup (1998) and Becker & Fuest (2011) explore the scope for indirect discrimination when the true degree of mobility is unknown but firms also differ in some aspect other than mobility. Finally, Baldwin & Okubo (2009), Haufler & Stähler (2009) and Davies & Eckel (2010) present models in which heterogeneous mobile firms sort themselves into high-tax and low-tax locations according to their profitability and cost structure. In contrast to these contributions, we consider homogeneous firms which only differ in mobility. Although our model does not differentiate between outsourcing, FDI or other possible international production structures but only defines mobility as the ability to shift -for some cost- profits abroad we can refer to the literature on international outsourcing to get some motivation that mobility varies across industrial sectors. Following Feenstra & Hanson (1996) there are sectors like e.g. footwear, electric and electronic machinery or instruments where production processes are separable and production stages differ remarkable in 3

4 their skill-intensity in which the outsourcing propensity is in general higher than in other sectors. Inspired by the empirical literature our theoretical model captures two characteristics of mobility: First, mobile and immobile firms coexist and second, the government has some ex ante beliefs about the degree of mobility but does not know the current value. The remainder of the paper is organized as follows. The next section lays out the model. We then proceed and derive the equilibria in a model where mobility is known, as a case of reference. Then, the model is solved for unknown mobility. 2 The model Assume a two-period model in a world with a large number of countries. We focus on policy questions in one of them, called the domestic country. The domestic country s industrial sector consists of a unit mass of firms indexed by i [0, 1]. There are two types of firms, mobile ones and immobile ones. The fraction of mobile firms, µ, is a random variable chosen by nature. The corresponding distribution function is denoted by F (µ) and has a continuous corresponding density function F (µ) = f (µ). All firms have an exogenously given pre-tax profit which is independent of firm location and normalized to unity. Whereas immobile firms remain completely passive, mobile firms choose their location in both periods. Moving implies a relocation cost c, independent of the direction of migration. Firms are profit maximizers and, thus, choose the location in which after-tax profits are highest. The domestic government is assumed to maximize its tax revenue 2 by choosing tax rates τ t [0, 1] where t {1, 2} denotes the time index. Tax discrimination between immobile and mobile firms is ruled out, i.e. the government has to choose one tax rate for all firms. When firms move abroad, they choose the location with the lowest tax rate (since pre-tax profits are equal everywhere) which is denoted by τ r [0, 1] and assumed to be constant over time. There are no strategic aspects which can be justified by the small country assumption. For simplicity, we assume that the domestic 2 From the viewpoint of the literature there are essentially two alternatives for government objective functions. First, the maximization of national income which would, in this model, imply zero tax rates in both periods. Second, maximization of the representative household s utility assuming the existence of a public good. For simplicity, we opted for the maximization of tax revenue which can be interpreted either by reflecting a leviathan government or an infinitely large welfare weight on public good provision. 4

5 government does not want to (or is not able to) attract foreign firms. We discuss such an extension in section 3. The timing of decisions is as follows: Stage 0: Nature draws µ [0, 1]. Stage 1: Government sets the tax rate for the first period, τ 1 [0, 1]. Stage 2: Each mobile firm i sets σ 1i [0, 1] which is the probability of moving abroad in period 1. Whereas σ 1i remains unobservable for all agents (including the government) except for firm i itself, the fraction of moving firms in period 1, µ 1, is common knowledge. After firms have migrated, production takes place and after-tax profits as well as tax revenue of period 1 are realized. Stage 3: Government sets the tax rate for the second period, τ 2 [0, 1]. Stage 4: Mobile firms that have not moved in the first period set σ 2i [0, 1] which is the probability of moving abroad in period 2. Firms that have moved in period 1 set σ 2i [0, 1] which is the probability of moving back. Again, σ 2i and σ 2i remain unobservable, firm migration itself is common knowledge. After the firms migration decision production takes place and after-tax profits as well as tax revenue are realized. We solve the game by backward induction (a mechanism design approach is discussed in section 3). At stage 4, each firm that has not moved in the first period compares its potential after tax at home and abroad. The former is given by 1 τ 2 and the latter by (1 c) (1 τ r ). Similarly, each firm that has moved in the first period compares its profit of staying abroad, 1 τ r, with its after tax profit when it moves back, (1 c) (1 τ 2 ). 3 Lemma 1 summarizes the optimal decision at stage 4. Lemma 1. Given τ 2, each mobile firm that has not moved in the first period moves in the second period with probability σ2i where { σ2i 0, if τ 2 τ 2 = (1) 1, if τ 2 > τ 2 with τ 2 (1 τ r )c + τ r, (2) 3 Note that taxes are levied on profits net of relocation cost. 5

6 and each mobile firm that has moved in the first period moves back to the domestic country in the second period with probability σ 2i where { σ 2i 1, if τ 2 τ 2 = (3) 0, if τ 2 > τ 2 with τ 2 τ r c 1 c. (4) Proof : The proof follows directly by comparing the relevant net profits considering moving cost. This gives us some first results: First, the decision of each mobile firm in the second period does not depend on the fraction of mobile firms but is entirely determined by the government s choice of τ 2. Second, the order of the tax rates is τ 2 τ r τ 2 which is a consequence of the relocation cost. Third, a necessary condition to realize the tax rate τ 2 is that the transaction cost is not higher than the foreign tax rate, i.e. c τ r. In order to avoid a degeneration of the problem, we assume that τ 2 cannot become negative and τ 2 is always strictly lower than unity which requires Assumption 1: Assumption 1. c τ r and τ r (0, 1). We now turn to stage 3, the government s second choice of the tax rate, τ 2 [0, 1]. At this stage, τ 1 and µ 1 have been determined. Given the firms moving decisions in the second period the government s revenue function is: τ 2 (1 E(µ µ 1 )) if 1 τ 2 > τ 2 R 2 (µ 1, τ 2 ) = τ 2 (1 µ 1 ) if τ 2 τ 2 > τ 2 (5) τ 2 (1 cµ 1 ) if τ 2 τ 2 0 where E(µ µ 1 ) denotes the expected value of µ given the observed number of moving firms, µ 1. Thus, the choice of the second period tax rate is conditional on first period firm migration for two reasons. First, observing µ 1 allows for a better estimate of µ. Second, even if µ is known by the government, a sufficiently large number of moving firms may lead to a choice of τ 2, i.e. the tax rate that attracts migrated firms back to the domestic country. Government chooses τ 2 in order to maximize tax revenue. It actually chooses out of { τ 2, τ 2, 1} because increasing τ 2 within the brackets indicated in (5) 6

7 does not change firm behavior. We assume that the government sets the lower tax rate whenever two tax rates imply the same tax revenue. Now, consider stage 2, the firm s decision on σ 1i. For simplicity, we assume for the rest of the paper that the moving decisions of the mobile firms in the first period are symmetric, i.e, the moving probabilities of the mobile firms in the first period σ 1i are equal which is stated in Assumption 2: Assumption 2. σ 1i σ 1 i. By deciding on σ 1, firms take the government s second period behavior into account although each individual firm acts as a price-taker, i.e. does not assume that its individual behavior changes any other agent s behavior. However, each firm anticipates that if the number of moving firms is large enough, government may have an incentive to set τ 2 in the second period. Let p τ2 denote the probability that the government sets τ 2 in the second period. The representative firm maximizes expected profits by choosing σ 1 such that max σ 1 (1 σ 1 ) [1 τ 1 + δ [p τ2 (1 τ 2 ) + (1 p τ2 ) (1 τ 2 )]] +σ 1 [(1 τ r ) (1 c) + δ (1 τ r )]. (6) Note that, if the firm has stayed in the first period, the second period payoff is 1 τ 2 independent of the firm s second period location decision since 1 τ 2 = (1 τ r ) (1 c). Similarly, if the firm has moved in the first period, the second period payoff does not depend on the second period location decision because (1 τ r ) = (1 τ 2 ). Further note that each individual firm takes p τ2 as given. However, p τ2 depends on the aggregate choice of σ 1. Finally, at stage 1, the government decides which tax rate τ 1 to set in the first period. It takes firm behavior in the first and second period into account and anticipates that it can adjust its policy after observing the current µ. In the following, we consider three scenarios. As a benchmark scenario, we analyze the case of complete information where both government and all firms know the mobility parameter µ. As a second step, we assume that there is symmetric incomplete information on µ, i.e. neither the government nor the firms know µ (of course, each mobile firm knows whether it is mobile or not but the aggregate number of mobile firms in unknown). Finally, we consider the case in which firms know µ but government does not. Afterwards, we intensively compare the equilibria derived under all three scenarios. 7

8 2.1 Complete information In this section, we analyze the subgame perfect equilibria for the case of complete information, i.e. the domestic government and all firms know µ. Stage 4 is summarized in Lemma 1. We can therefore directly turn to stage 3. Here, complete information implies E(µ µ 1 ) = µ. With µ 1 = σ 1 µ, it follows from (5) that the government chooses an optimal second period tax rate τ 2 according to Lemma 2. Lemma 2. (i) If no mobile firm has moved in the first period, i.e. σ 1 = 0, government sets τ2 = τ 2 iff µ 1 τ 2 and τ2 = 1 otherwise. (ii) If at least some firms have moved in the first period, i.e. σ 1 > 0, government sets τ 2 if µ 1 µ 1 τ2 = 1 otherwise τ 2 if µ 1 < µ 1 and µ 1 σ 1(1 τ 2 ) 1 σ 1 τ 2 (7) with { σ1 (1 τ 2 ) µ 1 (σ 1 ) max, τ } 2 τ 2 1 τ 2 cσ 1 τ 2 c τ 2 (8) Proof : The proof follows directly by comparing tax revenues as given in equation (5) applying µσ 1 = µ 1. For later use, it is helpful to deal with some properties of the above defined threshold µ 1 (σ 1 ). First note that µ 1 (σ 1 ) cannot fall below µ min 1 τ 2 τ 2 τ 2 c τ 2 (9) σ 1 (1 τ 2 ) as the first argument on the right hand side, 1 τ 2 cσ 1, monotonically increases in σ 1. Secondly, the largest level of σ 1 that ensures µ min 1 is given by σ1 min τ 2 τ 2 τ 2 τ 2 + τ 2 (1 τ 2 )(1 c) (the level of σ 1 equating the two arguments in (8)). If σ1 min µ < µ min 1, a firm cannot reach the threshold by a further increase in σ 1. Now, turn to stage 2 where firms decide on the optimal moving behavior in the first period, σ 1, solving equation (6). If all firms stay, i.e. σ 1 = 0, they anticipate that the government will never have an incentive to set 8

9 τ 2. Therefore expected payoffs are given by 1 τ 1 + δ (1 τ 2 ) and we can conclude, considering (6) with p τ2 = 0, σ 1 = 0 if τ 1 τ 2 δ [ τ 2 τ r ] τ 1. (10) If all firms leave, i.e. σ 1 = 1, expected payoffs are (1 τ r ) (1 c)+δ (1 τ r ). If µ is large enough firms can ensure p τ2 = 1 giving an optimal moving probability σ 1 = 1 if τ 1 > τ 2 + δ [τ r τ 2 ] τ 1 or µ < µ 1 (1) and τ 1 > τ 1. (11) If µ is sufficiently large such that the government can be forced to set τ 2 in the second period, a third alternative arises. Firms can choose σ 1 = µ min 1 /µ and, thus, trigger a second period tax rate of τ 2. Then, expected payoffs are (1 σ 1 ) [1 τ 1 + δ (1 τ 2 )] + σ 1 [(1 τ r ) (1 c) + δ (1 τ r )] (12) which requires, though, that the two terms in square brackets, i.e. the payoff from staying and leaving be equal, i.e. τ 1 = τ 2 + δ [τ r τ 2 ]. This gives an optimal firms behavior σ1 = µmin 1 µ if τ 1 = τ 1 and µ µ min 1 /σ1 min. (13) If µ < µ min 1 /σ1 min, probability p τ2 is zero. Thus, choosing a mixed strategy like in (12) is only a rational option if µ µ min 1 /σ1 min. Accordingly, firms choose out of σ1 {0, 1}. For µmin 1 /σ1 min µ < µ 1 (1), firms may actually force the government to choose τ 2. However, if they all choose σ 1 = 1, government will not choose τ 2. If τ 1 (τ 1, τ 1 ), firms choose σ 1 = 1 although choosing σ 1 = µmin 1 µ would make them better off as a group. However, due to a lack of coordination capacity the symmetric choice of µ min 1 /µ is not feasible. For τ 1 (τ 1, τ 1 ) there can t be an equilibrium. The reason is that firms prefer to stay if τ 2 = τ 2 and to leave otherwise. However, τ 2 is only chosen if the number of leaving firms is sufficiently large. This is only the case if τ 1 τ 1. At stage 2, consequently, firms choose σ1 out of { 0, µ min 1 /µ, 1 } if there are enough mobile firms to ensure τ 2 = τ 2 and out of {0, 1} otherwise. Accordingly, at stage 1, government chooses the optimal tax rate τ1 out of {τ 1, 1} if µ < µ min 1 /σ1 min and out of {τ 1, τ 1, 1} otherwise. With R (τ 1, τ 2 ) denoting the tax revenue as a function of tax rates the government s optimization problem in the first period is given by max τ 1 R (τ 1, τ 2 ) s.t. Lemma 2. We can now state the following Proposition 1. 9

10 Τ r c Figure 1: Equilbria with complete information; µ R(0, 1)δ = 0.5; - All mobile firms stay, + All mobile firms move Propostion 1. Given the second period strategy of the firms as described in Lemma 1, for every set up of tax rate τ r in the rest of the world, relocation costs c and time preference δ there is a unique subgame perfect equilibrium ( (σ 1, σ 2, σ 2 ), (τ 1, τ 2 )) : ( ) All stay: If µ (1 τ r ) 1 c 1+δ : σ1 = 0 and (τ 1, τ 2 ) = (τ 1, τ 2 ). 4 All move: Otherwise: σ1 = 1 and (τ 1, τ 2 ) = (1, 1). Firms optimal behavior in period 2 is given in Lemma 1. Proof : Under complete information there is no possibility to learn from the firm s first period moving behavior. Thus, costly relocation cannot be an equilibrium strategy. For this reason there are only two possible equilibria: A first one, in which all firms stay in both periods and a second one, in ( ) 4 Note that, since (1 τ r ) 1 c > 1 τ 1+δ 2, the government s strategy to set τ 2 in the second period is subgame perfect. 10

11 which all mobile firms leave and then stay in the rest of the world. Which of these two equilibria is optimal depends on the current number of mobile firms. The critical value (1 τ r )(1 c/(1+δ)) results by comparing R(τ 1, τ 2 ) and R(1, 1). For reasons of completeness the appendix gives a more detailed proof of the proposition formally comparing relevant tax revenues. Figure 1 illustrates the proposition for the case of a uniformly distributed µ. For the case of complete information we can summarize: The higher the current µ, i.e. the more mobile the industrial sector, the higher the probability that all mobile firms stay. For a very mobile sector, i.e. µ 1, a situation where no firm moves becomes equilibrium independent of the other environmental parameters δ, c and τ r. A situation without relocation is also more probable for a high foreign tax rate and high migration cost. Thus, quite intuitively, we observe more relocation of firms when foreign tax rates are low and relocation is cheap. Finally, the higher the preference for the future, δ, the lower the probability that no firm moves. 2.2 Incomplete information: The symmetric case In this section we modify the above presented model by assuming that the actual number of mobile firms, µ, is unknown to all agents. Thus, both the government and the firms can only infer an estimation of µ from the actual number of moving firms in the first period, µ 1. This change in the information structure of the game makes the concept of subgame perfect equilibria used in the case of complete information insufficient for the remaining analysis. Therefore, we apply the concept of Bayesian Equilibrium in both scenarios with incomplete information. Strictly speaking we have to define government s and firms expectations about the current number of mobile firms for the first and second period decisions. However, as expectations of the government and the firms in the first period are always given by the unconditioned expected value, E(µ), and firms second period decisions do not depend on the expected number of mobile firms, we only describe the government s expectations after the first period when denoting a Bayesian Equilibrium. At stage 4, the second period tax rate is known and Lemma 1 applies. We may therefore directly turn to stage 3. Assuming symmetric incomplete information about µ, the government can take the perspective of an individual firm and induce the level of σ 1. As a consequence, observing µ 1 reveals all information with, however, one exception. Observing µ 1 = 0 may mean that 11

12 σ 1 = 0 or µ = 0 (or both). Then, in (5), E(µ µ 1 ) = µ 1 /σ 1 if σ 1 > 0 and E(µ µ 1 ) = E(µ) if σ 1 = 0. Lemma 3 summarizes the government s tax setting behavior in the second period. Lemma 3. (i) If no mobile firm has moved in the first period, i.e. σ 1 = 0, the government sets a second period tax rate τ 2 iff E(µ) 1 τ 2 and τ 2 = 1 otherwise. (ii) If at least some mobile firms have moved in the first period, part (ii) of Lemma 2 applies. Proof : See proof of Lemma 2. Now turn to stage 2 where firms decide on their relocation probability σ 1. As in the case of complete information, the payoff is 1 τ 1 + δ (1 τ 2 ) if all firms stay, and (1 τ r ) (1 c) + δ (1 τ r ) if all firms leave, see (6). Again, firms may opt for a mixed strategy trying to trigger a second period tax rate of τ 2, however this time without knowing the actual number of mobile firms, µ. Then, expected payoffs are (1 σ 1 ) [ 1 τ 1 + δ [ p s τ 2 (1 τ 2 ) + ( 1 p s τ 2 ) (1 τ2 ) ]] +σ 1 [(1 τ r ) (1 c) + δ (1 τ r )] (14) where p s τ 2 p s τ 2 (σ 1 ) denotes the probability that under symmetric incomplete information the government implements τ 2 in the second period. Considering that the payoffs of staying and leaving (the two terms in square brackets) are equal the moving probability that implies indifference of the firms, ˆσ 1 s, is defined as solution of τ 1 = τ 2 δ ( τ 2 τ r ) + p s τ 2 δ ( τ 2 τ 2 ) (15) with p s τ 2 = 1 F ( µ 1 /σ 1 ). The probability reaches a minimum at σ 1 µ min implying p s τ 2 ( µ min ) = 0 and a maximum, denoted by p s τ 2 at σ 1 = σ1 min yielding p s τ 2 = 1 F ( µ min /σ1 min ) (recall that µ1 rises in σ 1 ). Note that, if all firms leave, σ 1 = 1, the probability is given by p s τ 2 (1) < p s τ 2. The equilibrium moving probability in the first period, σ1 s, depends on the first period tax rate choice. Thus, firms choose 0 if τ 1 τ 1 σ1 s = 1 if τ 1 > τ 1 δ [ τ 2 τ 2 ] (1 p τ2 ) τ 1 s (16) ˆσ 1 s (τ 1) if τ 1 (τ 1, τ 1 ] 12

13 with τ 1 and τ 1 defined in (10) and (11). If the government chooses a first period tax rate out of the interval (τ 1, τ 1 s ] firms adjust their relocation probability such that they are indifferent between staying and moving. 5 Now, consider stage 1 where the government decides on the optimal first period tax rate τ1. If all firms leave, it is always optimal to levy the highest possible tax rate, τ1 = 1. Then, according to Lemma 3, the second period tax rate will equal τ2 = τ 2 if µ 1 τ 2 1 c τ 2 and τ2 = 1 otherwise. [Note that firm migration at stage 2 fully reveals the unknown level of µ.] Expected tax revenue in this regime is given by R s (1, τ 2 ) = 1 E(µ) + δ(1 p s τ 2 ) (1 E s 1(µ 1)) (17) +δp s τ 2 τ 2 ( 1 ce s τ2 (µ 1) ). where E s 1(µ 1) E(µ σ1 s = 1, τ2 = 1) and Es τ 2 (µ 1) E(µ σ1 s = 1, τ2 = τ 2 ), respectively, denote the expected number of mobile firms when firms choose σ1 s = 1 and the second period tax rates τ 2 = 1 and τ 2, respectively, are realized. If no firm leaves, tax revenue is given by R s (τ 1, τ 2 ) = τ 1 + δ τ 2 = τ 2 + δτ r. (18) Note that such a policy is only feasible if E(µ) > 1 τ 2. Otherwise, government cannot commit to τ 2 in the second period. In an equilibrium where some firms move, government chooses an optimal tax rate τ1 out of (τ 1, τ 1 s]. Using ˆσs 1 (τ 1 ) σmin 1, tax revenue is then given by R s (τ 1, τ 2 ) = τ 1 (1 E(µ)ˆσ s 1) + δ(1 p s τ 2 p s τ 2 ) (1 E s 1(µ ˆσ s 1)) (19) +δp s τ 2 τ 2 ( 1 ˆσ s 1 E s τ 2 (µ ˆσ s 1) ) + δp s τ 2 τ 2 ( 1 cˆσ s 1 E s τ 2 (µ ˆσ s 1) ) where E s 1(µ ˆσ 1 s) E(µ σs 1 = ˆσ 1 s, τ 2 = 1), Es τ 2 (µ ˆσ 1 s) E(µ σs 1 = ˆσ 1 s, τ 2 = τ 2 ), and E s τ 2 (µ ˆσ 1 s) E(µ σs 1 = ˆσ 1 s, τ 2 = τ 2). The probability that the second period tax rate is τ 2, p s τ 2, is defined above. Analogously, p s τ 2 denotes the probability that the government realizes under symmetric incomplete information τ 2 in period 2. Of course, both probabilities depend on the equilibrium moving decsion ˆσ 1. In the case of ˆσ 1 s ( τ 1 s ), the probability that τ2 = τ 2 is zero. 5 Precisely, a small increase of τ 1 with τ 1 (τ 1, τ s 1 ] leads to a change in ˆσ s 1 of dˆσs 1 dτ 1 = [ δ ( τ 2 τ 2) dp τ 2 dσ 1 ] 1 with dp τ2 dσ 1 = f d µ dσ s 1 = f 1 τ 2 τ 2 (σ1 s)2 τ 2 c τ 2. 13

14 At stage 1, government solves the following optimization problem: max τ 1 R s (τ 1, τ 2 ) s.t. Lemma 3 where the relevant tax revenues are given in (17)-(19). Proposition 2 summarizes the three different types of Bayesian Equilibria that can appear. Propostion 2. Assume that µ is unknown to both firms and the government. For each environment (δ, c, τ r, F (µ)) there exists a unique symmetric Bayesian Equilibrium which refers to one of the three following types: 1. All firms stay, i.e. σ s 1 = 0 with tax rate choices τ 1 = τ 1 and τ 2 = τ 2. Government s expectations are E(µ µ 1 ) = E(µ). 2. All firms move, i.e. σ s 1 = 1 with tax rate choices τ 1 = 1 and τ 2 { τ 2, 1}, depending on the first period migration µ 1. Government s expectations are E(µ µ 1 ) = µ Some firms move, i.e. σ1 s (τ 1 ) [ µ min, σ1 min ] with tax rate choices τ1 (τ 1 τ 1 s] and τ 2 { τ 2, 1}, depending on the first period migration µ 1. Government s expectations are E(µ µ 1 ) = µ 1. The optimal actions in period 2 are described in Lemma 1 and Lemma 3. Proof : Existence and uniqueness of the symmetric Bayesian Equilibrium directly results by the fact that for every of the four stages we have described a unique best response of the government and the firms, respectively. Furthermore, there are no other government strategies than those indicated in the proposition by the following argument: As in the case of full information, (τ 1, τ 2 ) and (1, τ 2 ) cannot be optimal for reasons explained in the proof of Proposition 1 (see appendix). Strategy (τ 1, 1) can never be optimal by the following argument: R s (τ 1, 1) = τ 1 + δ(1 E (µ)) > R s (τ 1, τ 2 ) iff E (µ) < (1 τ 2 ) and R s (τ 1, 1) > R s (1, 1) iff E(µ) > 1 τ 1 + δp s τ 2 (1 c τ 2 ) [ E s τ 2 (µ 1) µ 1 (1) ]. As τ 1 δp s τ 2 (1 c τ 2 ) [ E s τ 2 (µ 1) µ 1 (1) ] < τ 2 the strategy (τ 1, 1) is revenue dominated by (τ 1, τ 2 ) or (1, 1). The uniqueness of the symmetric Bayesian Equilibrium, of course, relies on the assumptions that the government sets the lowest tax rate whenever two or more 14 σ1 s

15 Τ r o o o o o c Figure 2: Equilbria with symmetric incomplete information; µ R(0, 1)δ = 0.5; - All mobile firms stay, + All mobile firms move, o Some mobile firms move tax rates would imply the same tax revenue and that the firms locate per assumption in the domestic country whenever net profits are equal in the domestic country and the rest of the world. A removing of the symmetry assumption would also destroy uniqueness. Figure 2 illustrates the results of Proposition 2 when µ is uniformly distributed. Similar to the case of complete information a situation where all mobile firms move is an equilibrium for low foreign tax rates and low relocation cost. However, for low cost and a medium foreign tax rate the government finds it optimal to trigger partial migration. This third type of equilibrium can only arise under incomplete information. The higher the preference for the future the higher the set of environmental situations for which there is an equilibrium where some mobile firms move. This also holds for equilibria where all mobile firms relocate which goes in line with the results under complete information. A common feature of the equilibrium where all mobile firms move and the equilibrium where some mobile firms 15

16 move is that the government perfectly learns the sector s mobility during the first period. However, in the case where some firms move this comes for a lower price as less relocation cost arise. In contrast to the case of complete information the results now do not depend on the current degree of mobility as this is unknown to the government (and the firms) and can therefore not determine the government s first period decision. Government s behavior in the case of incomplete information is driven by the ex ante belief on the current mobility and the expectations about information after the first period. At least for a uniformly distributed µ incomplete information seem to increase relocation in the sector as in some situations where under complete information no firm moves now at least some firms migrate while we do not find an example for the reverse case. 2.3 Incomplete information: Information advantage of the firms In this section we modify the above presented model by assuming that the actual fraction of mobile firms, µ, is unknown to the government but not to the firms. The plausibility of this scenario may be seen as limited as the government may always hire a single firm and pay them to reveal the information. However, it seems straightforward to assume that firms have some informational advantage. This section provides the benchmark for this setting by assuming that the informational advantage is perfect. Again, at stage 4, the second period tax rate is known and Lemma 1 applies. At stage 3, government has observed first period migration and may use this information in order to optimally set second period tax rates. In contrast to the case of symmetric uncertainty, government knows that firms have an informational advantage and that they will not make mistakes in setting the relocation probability σ 1. To be precise, they will only play the mixed strategy if µ is sufficiently large. Lemma 4 summarizes the government s tax setting behavior in the second period. Lemma 4. Let µ as 1 denote the fraction of leaving firms in the first period necessary for the government to set τ 2 in the second period with { E µ as as (σ 1 µ 1 ) (1 τ 2 ) 1 = max 1 c τ 2 E as (σ 1 µ 1 ), τ } 2 τ 2 (20) τ 2 c τ 2 where we used E(µ µ 1 ) = µ 1 E as (σ 1 µ 1 ). 16

17 (i) If no mobile firm has moved in the first period, i.e. σ 1 = 0, the government sets a second period tax rate τ 2 iff E(µ) 1 τ 2 and τ 2 = 1 otherwise. (ii) If 0 < µ 1 < µ as 1, government concludes that σ 1 = 1. Then, government levies τ 2 = 1. (iii) If µ 1 µ as 1, governments sets τ 2 = τ 2. Proof: Part (i) and (iii) of Lemma 4 follow directly by comparing second period tax revenues as given in equation (5). Part (ii) follows from the observation that firms will only play the mixed strategy if µ is large enough. If it is, µ 1 µ as 1. If not, firms have a strict incentive to leave, government observes 0 < µ 1 < µ as 1 and concludes that µ < µ as 1. the current fraction of mobile firms is not high enough to implement τ 2 in the second period. Thus, if a mobile firm leaves in the first period it has a strict incentive to do this as the probability that government sets τ 2 in the second period is zero. Now, consider stage 2. As before, if all firms stay the payoff is 1 τ 1 + δ (1 τ 2 ). If all firms leave, it equals (1 τ r ) (1 c) + δ (1 τ r ), see (6). If the number of mobile firms is sufficiently large, firms may choose a mixed strategy that triggers a second period tax rate of τ 2. This time, firms know the actual number of mobile firms, µ, and thus will not make mistakes (implying that the probability of the government setting τ 2 in the second period equals 1 provided that µ is large enough). Thus, the mixed strategy payoff is (1 ˆσ as 1 ) [1 τ 1 + δ (1 τ 2 )] +ˆσ as 1 [(1 τ r ) (1 c) + δ (1 τ r )] (21) with ˆσ 1 as µas 1 µ denoting the first period moving probability in a mixed strategy equilibrium under asymmetric incomplete information. Again, this requires that the payoffs from staying and leaving (the two terms in square brackets) be equal which is the case if τ 1 = τ 1 where τ 1 is the maximum first period tax rate that implies indifference of the firms in the case of complete information. As this critical tax rate could be higher than unity, an equilibrium with mixed strategies additionally requires τ 1 1. Concluding, firms equilibrium first period behavior under asymmetric incomplete information, σ1 as, is σ1 as = µas 1 µ if τ 1 = τ 1 1 and µ µ as 1. (22) 17

18 In contrast, if τ 1 τ 1 or τ 1 > 1, all firms have either a strict incentive to move or a strict incentive to stay in the first case and will stay with strict preference in the second case. Tax revenue where all firms stay, R as (τ 1, τ 2 ), and tax revenue where all mobile firms move, R as (1, τ2 ), are the same as under symmetric incomplete information. Thus, equations (11) and (10) still hold. In the latter case, E as (σ 1 µ 1 ) = 1 and, consequently, µ as 1 = µ 1 (1). If τ 1 (τ 1, τ 1 ) and µ µ as 1 there cannot exist an equilibrium by the same argumentation as under complete information. Thus, the only difference to the case of symmetric incomplete information may arise when τ 1 = τ 1 1. Then, µ as 1 µmin 1 if E as (σ 1 µ 1 ) > σ min 1. Accordingly, at stage 1, government chooses out of {τ 1, τ 1, 1}. If τ 1 τ 1, it anticipates that firms choose either σ1 as = 0 or σ1 as = 1. Expected tax revenue for τ 1 = 1 (and an optimal second period tax rate τ2 ) are the same as under symmetric incomplete information and given by (17). Expected tax revenue for τ 1 = τ 1 (and a second period tax rate τ 2 ) are also the same as under symmetric incomplete information and, thus, given by (18). Choosing τ 1 = τ 1 yields revenues of R as ( τ 1, τ2 ) = p as τ 2 [ τ 1 (1 µ as 1 ) + δ τ 2 (1 c µ as 1 )] (23) + ( 1 p as ) τ 2 [ τ1 (1 E as 1 (µ ˆσ 1 as )) + δ (1 E as 1 (µ ˆσ 1 as ))] where E as 1 (µ ˆσas 1 ) E(µ σas 1 = ˆσ 1 as, τ 2 = 1) gives the expected number of mobile firms when the mobile firms choose ˆσ 1 as and τ2 = 1 is realized, and p as τ 2 = 1 F ( µ as 1 ) denotes the probability that the government sets under asymmetric incomplete information τ 2 in the second period. This probability again depends on the choice of ˆσ 1 as. The government solves the following optimization problem: max τ 1 R as (τ 1, τ 2 ) s.t. Lemma 4 where relevant tax revenues are given in (17), (18), (23). Proposition 3 summarizes the three different types of Bayesian Equilibria that can appear. Propostion 3. Assume that µ is unknown to the government but known to firms. For each environment (δ, c, τ r, F (µ)) there exists a unique symmetric Bayesian Equilibrium which refers to one of the three following types: 1. All firms stay, i.e. σ 1 = 0 with tax rate choices τ 1 = τ 1 and τ 2 = τ 2. Government s expectations are E(µ µ 1 ) = E(µ). 18

19 2. All firms move, i.e. σ 1 = 1 with tax rate choices τ 1 = 1 and τ 2 { τ 2, 1}, depending on the first period migration µ 1. Government s expectations are E(µ µ 1 ) = µ 1. { } 3. Some or all firms move, i.e. σ1 µ1 µ, 1 with tax rate choices τ1 = τ 1 and τ2 { τ 2, 1}, depending on the first period migration µ 1. Government s expectations are E(µ µ 1 ) = 1 µ µf(µ)dµ. as 1 The optimal actions in period 2 are described in Lemma 1 and Lemma 4. Proof : Existence and uniqueness are ensured by an anlogous argumentation as given in Proposition 2. Furthermore, there are no other government strategies than those indicated in the proposition by the following argument: As in the case of full information, (τ 1, τ 2 ), (1, τ 2 ) and (τ 1, 1) cannot be optimal for reasons explained in the proof of Proposition 1. Figure 3 illustrates the results of Proposition 3 for a uniformly distributed fraction of mobile firms. Similar to the previous analysis for complete and symmetric incomplete information an equilibrium where all mobile firms relocate is more probable for a low foreign tax rate and low relocation cost. Again, as under symmetric incomplete information, there is the possibility that the government triggers partial migration. However, now this does not arise for low relocation cost and medium foreign tax rates but if migration cost and the foreign tax rate are rather high. As a result if relocation cost is low and foreign tax rate is medium a some move equilibrium under symmetric incomplete information is under asymmetric incompleteness replaced by an equilibrium where no one moves. If the sector s current degree of mobility is low, so that there is no reattraction of firms in the second period, i.e. τ2 τ 2, firms benefit in this situation from their information advantage. In contrast, if relocation is more costly and the foreign tax rate is high an equilibrium where under symmetric incomplete information no firm moves is replaced by an equilibrium where some firms move making firms loose from their information advantage if the current degree of mobility is low. In contrast to the scenario with symmetric incomplete information under asymmetric incomplete information the existence of an equilibrium where the government triggers partial migration requires a low preference for the future, δ, as otherwise indifference of the firms could only be forced by a first period tax rate which is higher than unity. Again equilibria do not depend on the sector s current mobility as the government which triggers the equilibrium decision by moving first is uniformed. 19

20 Τ r o o o o o c Figure 3: Equilbria with symmetric incomplete information; µ R(0, 1)δ = 0.1; - All mobile firms stay, + All mobile firms move, o Some mobile firms move 2.4 Comparison In this subsection, we compare different aspects of the three equilibria derived above and establish a couple of corollaries. To start with, we analyze the role of information on tax setting. As discussed in the introduction, most studies on optimal taxation (implicitly) assume that it suffices that the government has correct expectations of tax base mobility. Our model, however, shows that this assumption has to be qualified. Under complete information, ( ) government chooses the strategy (τ 1, τ 2 ) whenever µ (1 τ r ) 1 c 1+δ. If government does not know µ it prefers under symmetric incomplete information (τ 1, τ 2 ) over (1, τ2 ) if R s (τ 1, τ 2 ) R s (1, τ2 ). This gives ( E(µ) > (1 τ r ) 1 c ) + δ 1 + δ 1 + δ ps τ 2 (1 c τ 2 ) ( E s τ 2 (µ 1) µ 1 (1) ) (24) As E s τ 2 (µ 1) µ 1 (1) 0. We can thus state Corollary 1. 20

21 Corollary 1. (first period tax rates) The expected tax rate in the first period under incomplete information (both symmetric and asymmetric) is larger than or equal to the expected first period tax rate under complete information implying a higher degree of first period migration under incomplete information. Proof : With E s τ 2 (µ 1) µ 1 (1) 0 and p s τ 2 0, it follows that the threshold ( ) implied by (24) cannot be fall short of (1 τ r ) 1 c 1+δ. If choosing ( τ 1 s, τ 2 ) dominates some of the two other strategies, Corollary 1 still holds since τ 1 s > τ 1. The proof analogously holds for asymmetric incomplete information. Now, consider the government s objective, maximizing tax revenue, under all three scenarios. It is straightforward and intuitive that expected tax revenue is largest under complete information. Interestingly, however, government may profit from an informational disadvantage relative to firms. To see this, note that firms do not profit from their knowledge of µ. Expected profits under government strategies (τ 1, τ 2 ) and (1, τ2 ) are equal under both scenarios with incomplete information. If government triggers a mixed strategy, firms are indifferent between moving and staying. Under both scenarios expected tax payments of those firms that stay in the country are equal. Thus, it depends on the expected number of firms that actually stay. The number of firms staying under the symmetric scenario is given by (1 ˆσ 1 s) 1 0 µf (µ) dµ whereas the equivalent number under the asymmetric scenario is (1 µ as 1 ) 1 µ f (µ) dµ. We can thus state Corollary 2. as 1 Corollary 2. (expected tax revenue) Expected tax revenue under the symmetric scenario is larger than under the asymmetric scenario if (1 ˆσ s 1) E(µ) > (1 µ as 1 ) p τ as 2 Proof : The proof follows directly from the considerations above with E(µ) = 1 0 µf (µ) dµ and p τ as = 1 2 µ f (µ) dµ as 1. Under the symmetric uncertainty scenario, firms make mistakes in the sense that, first, they choose the mixed strategy even if the true level of µ is not large enough to trigger τ 2 in the second period and, second, if µ is large, too many firms leave in the first period and return in the second. The resulting losses however are compensated by the government setting a lower 21

22 first period tax rate, i.e. τ s 1 < τ 1. Thus, in expected terms, profits are equal under the symmetric and the asymmetric setting. Uncertainty about µ implies that decisions have to be made on the basis of expected values given the distribution of µ, summarized in F (µ). From a policy perspective, it may be important how large the potential losses are if actual levels of µ deviate from expected values. The following Corollary 3 summarizes the size and the sign of maximum losses in tax revenue due to the uncertainty about µ. Corollary 3. (maximum revenue losses) Depending on the type of equilibrium, the maximum loss in tax revenue is given by: 1. All firms stay equilibrium: 1 τ 1 + δ (1 τ 2 ). 2. All firms move equilibrium: τ 1 + δ ( τ 2 τ 2 (1 c)). 3. Mixed strategy equilibrium: Both scenarios of incomplete information if µ = 0: 1 τ 1 Symmetric incomplete information if µ = 1: τ 1 τ 1 (1 ˆσ 1 s) + δ ( τ 2 τ 2 (1 ˆσ 1 sc)) Asymmetric incomplete information if µ = 1: τ 1 τ 1 (1 µ as 1 ) + δ ( τ 2 τ 2 (1 µ as 1 c)) Proof : The proof follows directly from equations (17), (18), (19) and (23). The first case arises if the government chooses (τ 1, τ 2 ) although actual mobility is zero, i.e. µ = 0. In the second case the government chooses (1, τ2 ) although µ = 1. In the mixed strategy equilibrium there are two relevant scenarios may arise: Either if there is actually no mobile firm or if all firms are mobile. In these cases the government chooses ( τ 1, τ2 ) although it should have chosen (1, τ2 ) if µ = 0 and (τ 1, τ 2 ) if µ = 1. Corollary 3 demonstrates that the mixed strategy equilibrium has lower maximum losses than the other two. This finding may be discussed in the context of loss-averse policy-makers. Under the all firm stay equilibrium, losses may be huge but voters may never find out. Under the all firm move equilibrium, losses are instantaneously revealed. The mixed strategy equilibrium has lower observable and unobservable losses. Corollary 4. (winners and losers) By a change from symmetric to asymmetric incomplete information the government never loses and it is sometimes better off. In contrast, the firms are in most cases equally well of. They benefit from their information advantage if the equilibrium type changes from 22

23 an equilibrium where all mobile firms leave to an equilibrium where all firms stay. If the change is in the reverse direction they lose. Proof: If information changes from symmetric to asymmetric incomplete information the equilibrium type chosen by the government can stay the same or change. Thus, nine different situations can arise. First, focus on the government s expected revenues. Whenever the equilibrium under symmetric information is an equilibrium where firms have strict incentives the government will not lose by a change to asymmetric information as it still could choose the initial equilibrium type implying the initial profit. If the government changes the equilibrium type it is strictly better off. Whenever under symmetric information the government implements an equilibrium where some firms move it only benefits if this type of equilibrium would imply a higher profit under asymmetric information, i.e. if R s (ˆτ 1 s, τ 2 ) > Ras ( τ 1, τ2 ) where ˆτ 1 s (τ 1, τ 1 s ] denotes the first period tax rate optimally chosen by the government to trigger partial migration. This inequality always holds by the following argument: In both scenarios firms are indifferent. However, as τ 1 as > τ 1 ˆτ 1 s it follows that p τ as > 2 p τ s 2. This is equivalent to 1 F ( µ as 1 ) > 1 F ( µ 1) and therefore F ( µ as 1 ) < F ( µ 1). This gives µ as 1 < µ 1. Of course this does not hold for a fixed moving probability of the firms but considers that moving probabilities in an equilibrium where some firms move may differ in the two scenarios. These considerations directly give R s (ˆτ 1 s, τ 2 ) < Ras ( τ 1, τ2 ) < Ras (1, τ2 ) and R s (ˆτ 1 s, τ 2 ) < Ras ( τ 1, τ2 ) Ras (τ 1, τ 2 ) where the second inequality in both cases must hold as the government optimally chooses the first period tax rate. Summarizing, the government never loses by a change form symmetric to asymmetric incomplete information. Now, consider the firms expected profits. Whenever there is no change in the equilibrium type expected profits do not change. Analogously, if the equilibrium type changes from or to an equilibrium where some firms move there is also no change in expected profits as in such an equilibrium firms are indifferent and a firm s profit of leaving is equal in both scenarios. Two cases remain to be analyzed: A change where under symmetric (asymmetric) incomplete information all firms stay while under asymmetric (symmetric) incomplete information all mobile firms move. Comparing profits of both types of equilibria directly gives that firms are better of if no one moves. 23

24 3 Discussion In this section we critically discuss aspects of the above analysis and explore the implications and the boundaries of the findings derived above. Time horizon Our model ends after two periods. Real world tax policy, however, has an infinite time horizon. In a longer time horizon, the value of learning becomes larger. Thus, the equilibrium where all firms stay and the government foregoes the opportunity of learning becomes less probably. In the other two equilibria, all information is revealed after period 2 and a stationary state is reached as long as the level of mobility is constant over time. If the mobility parameter changes over time, the game described in section 2 is repeated. If current mobility contains information on future mobility, this may render the analysis more complex. However, as a rough tendency, this assumption would again increase the value of learning and, thus, decrease the attractiveness of the all firms stay equilibrium. Foreign firms In the above model, the domestic government only deals with domestic firms. In principle, it may try to attract foreign firms as well. We have abstracted from this possibility although integrating the existence of foreign firms is straightforward. Assuming that foreign firms and domestic firms are equal with regard to pre-tax profits and mobility cost, the domestic government will consider setting a first period tax rate τ 1 = τ 2 in order to attract an expected number of foreign firms. Then, it does not learn anything about the mobility of domestic firms and has to base the decision on τ 2 on uncondition expected values of domestic mobility µ. Thus, learning takes place either with domestic firms or with foreign firms. Since learning about true mobility levels is in the centre focus of this paper, adding foreign firms simply extends the model without adding new aspects to the problem of learning. Of course, if the first period tax rate exceeds τ 1 and at least partial migration of domestic firms is triggered, setting τ 2 in the second period becomes more attractive in the presence of foreign firms. Coordination among firms The assumption of an infinitely large number of firms implies that each firm acts as a price-taker and firms cannot coordinate. Allowing for the coordination of firms among themselves fundamentally changes the game, and an intensive analysis is beyond the scope 24