Journal of Economics and Finance Volume 24 Number 1 Spring 2000 Pages 56-63 Revenue Equivalence and Income Taxation Veronika Grimm and Ulrich Schmidt* Abstract This paper considers the classical independent private values model of auction theory in the presence of income taxation. We show that revenue equivalence remains valid if income taxes are proportional. Progressive and regressive taxes lead, in general, to asymmetries between bidders with the well-known consequence that revenue equivalence no longer holds. However, if symmetry of the bidders is maintained, progressive (regressive) income tax implies a higher (lower) expected revenue in first-price than in second-price auctions. (JEL D44, H22, H23) Introduction Auctions become more and more popular for selling goods if demand is uncertain or varies significantly over time. Via the internet auctions are particularly easy to run, which makes them a convenient and flexible mechanism for sales to individual as well as business customers. The U.S. company General Motors, for example, recently announced that it will reorganize its procurement and plans to run auctions for all input factors needed for production. 1 Since more and more transactions are done by using auctions, an interesting question is how taxation affects the bidding behavior of firms that participate in such auctions. As far as we know, auction theory has not yet been analyzed in the presence of taxation. This is surprising since corporate income taxes are imposed in nearly every country. In this work we investigate the impact of income taxation on revenue and efficiency properties of different auctions in the independent private values model. In our framework, firms compete for an item that gives rise to an additional profit if they win it. This could be an input factor, or something they buy as an intermediary like flowers to sell it to their customers. One could also think of procurement auctions, where succeeding in the auction leads to an additional profit which amounts to the price resulting from the auction minus the firm s cost of production. * Veronika Grimm, Department of Economics, Humboldt University at Berlin, Spandauer Str. 1, 10178 Berlin, Germany, grimm@wiwi.hu-berlin.de; Ulrich Schmidt, Department of Public Economics, Christian Albrechts University, Ohlshausenstr. 40, 24098 Kiel, Germany, u1366@bwl.uni-kiel.de. Financial support by the Deutsche Forschungsgemeinschaft is gratefully acknowledged. 1 Cf. White (1999).
Revenue Equivalence and Income Taxation 57 In all these examples the additional profit made in case of winning the auction is subject to taxation. One could argue that, if there is a resale value (in case of intermediaries) or a common technology (in case of procurement auctions or auctions of input factors), a common value rather than a private values framework might be appropriate. However, since firms certainly differ in their cost parameters and their abilities, the ex post value of winning the auction is not the same for every firm. We think this is captured best by employing independent private values. We also do not allow for externalities in the sense that, if one firm wins the auction, this affects its competitors payoffs. Still, there are enough examples where markets are sufficiently competitive to justify this assumption. In the absence of taxation, a central result in auction theory is the revenue equivalence theorem: In the symmetric independent private values framework with risk neutral bidders all four standard auction mechanisms, i.e., the first-price and second-price sealed-bid auction as well as the ascending- and descending-bid auction, yield the same expected revenue for the seller. Moreover, the resulting allocation of all four mechanisms is Pareto-efficient, which means that the bidder with the highest valuation receives the item. However, revenue equivalence as well as Pareto-efficiency of all four mechanisms no longer holds if assumptions are relaxed and one allows for affiliated valuations, asymmetries, or risk aversion of the bidders. 2 We show that the revenue equivalence theorem remains valid in the presence of a proportional income tax. Progressive or regressive taxes lead to asymmetries between firms if their initial profits and thus their marginal tax rates differ. In this case, first-price auctions are, in general, not Pareto-efficient and yield a different expected revenue compared to second-price auctions. If symmetry of the firms is preserved by assumption, both first- and second-price auctions are efficient. However, we can show that a progressive (regressive) income tax leads to a higher (lower) expected revenue in first-price than in second-price auctions. The Model We consider a static model with n risk neutral firms bidding for an input factor. Without ownership of the input factor, each firm i makes profit i, which is private information. Ownership of the auctioned input by firm i gives rise to an additional profit i. We assume independent private valuations for the input factor; i.e., i does not depend on j i j. Moreover, if firm i wins the auction this has no impact on j, j i. From the viewpoint of firm i, the other firms additional profits j i j are random variables denoted by ~ j. Firms are symmetric in the sense that their additional profits ~ j are identically distributed with distribution function F: [, + ] [0,1] and density f. Therefore, from the viewpoint of firm i, the joint distribution of the other firms additional profits is F( ) n-1, which is the probability that no firm j, j i has a higher additional profit than. Profits of each firm are taxed by a corporate income tax. Note that at the time when taxes are due there is no uncertainty about the firms profits since they are observable to the tax authority ex post. If firm i gets the input factor, the other firms have to pay T j = T( j) for all j i, and firm i pays T i = T( i i ). The marginal tax rates are given by j = ( j ) and i = ( i i ), respectively. We make the following assumption. Assumption 1. The marginal tax rate does not exceed one, i.e. τ (y) < 1 y. Furthermore, resid - ual income does not grow faster than the probability of winning, i.e. d d 1n F( ) n-1 1n (1 - ( b)) (1) d d Note that assumption 1 is always satisfied if the tax rate is proportional or progressive with > 0 everywhere. 2 These standard results of auction theory are reviewed, e.g., in Grimm (1999) and Wolfstetter (1999).
58 JOURNAL OF ECONOMICS AND FINANCE Volume 24 Number 1 Spring 2000 In the following we analyze the impact of income taxation on equilibrium bids in second-price and first-price auctions with independent private valuations. The auctions are regarded as noncooperative games, which means that the existence of coalitions or collusion among the firms is excluded by assumption. Second-Price Auctions Since the seminal work of Vickrey (1961), it is well known that second-price auctions are demand revealing; i.e., bidding her true valuation of the object is a dominant strategy for every bidder. For the present framework this implies that every firm i would bid i in the absence of taxation. Our first result shows that the introduction of an income tax leaves the optimal bid of every firm unchanged. Proposition 1. The introduction of an income tax with τ (y) 1 y leaves the optimal bid in sec - ond-price auctions unchanged; i.e., bidding π i is a dominant strategy for every firm i. Proof. Firm i s net valuation for the input is given by i (T( i i p) T( i )), where p is the price paid in the auction. Let b * be the symmetric equilibrium strategy. Bidding as if his additional profit was x instead of i yields the following expected profit for bidder i: y i + i b*(x) U (x, i) = [ i b * (x) (y) dy] df(x)n-1. (2) The optimal bid has to satisfy the first-order condition obtained by differentiating (2) with respect to x at x = i ; i.e., i + i du(x, i) b*(x) x = i = [ i b * (x) (y) dy] df(x)n-1 = 0. (3) dx dx From (3) it is easy to see that the equilibrium bid is b * ( i ) = i as long as the marginal tax rate is less than one. Proposition 1 shows that in second-price auctions income taxation has no impact on the allocation since the same firm receives the object for the same price. Therefore, the tax does not change the seller s revenue, and the whole burden of the tax is borne by the winning firm. Second-price auctions remain demand-revealing and Pareto-efficient. Since the winning bid has no influence on the price, maximizing gross profits also yields maximum net profits. Therefore, firms reveal their gross valuations rather than their net valuations. Note that the proof of Proposition 1 does not depend on symmetry or risk neutrality of the firms. Therefore, the result is also valid for asymmetric firms with arbitrary preferences that are consistent with first-order stochastic dominance. In our framework asymmetries can occur even if bidders are symmetric in their additional profits i, since different initial profits ilead to different marginal tax rates in the presence of progressive or regressive taxation. First-Price Auctions The analysis of first-price auctions is more complicated because no equilibrium in dominant strategies exists. The validity of the revenue equivalence theorem is due to the fact that in first-price i i
Revenue Equivalence and Income Taxation 59 auctions the expected value of the highest bid in Bayesian Nash equilibrium equals, in the absence of taxation, precisely the expected value of the second highest bid in a second-price auction. The intuition is that the equilibrium strategy of every bidder in a first-price auction is to assume that her valuation of the object is the highest and then to bid the expected value of the second highest bid conditional on this assumption. In first-price auctions, however, there is a tradeoff between potential profits and probability of winning which is not present in second-price auctions. Thus, since imposing a tax lowers potential profits at a certain bid, this may affect the optimal bid in a firstprice auction. We begin our analysis by considering a proportional income tax with tax rate. It is important to observe that a proportional tax preserves symmetry of the bidders even if their initial profits differ: The distribution function of net profits, G, can be easily derived from the distribution function of gross profits as G((1 ) = F ( ). Thus, if gross profits are identically distributed, the same holds true for net profits. This observation leads to the following result: Proposition 2. The introduction of a proportional income tax with τ < 1 leaves the optimal bid in a first-price auction unchanged. Proof. In the absence of taxation a bidder s expected total profit is given by U (b, ) = + (b)[ b], (4) where b is her bid, and (b) denotes the probability of winning the auction by bidding b. The equilibrium bid can be derived by maximizing the expected total profit with respect to b. This yields the first-order condition U b (b, ) = (b)[ b] (b) = 0. (5) With a proportional income tax, expected total net profit is U b (b, ) = (1 ) ( + (b)[ b]). (6) Obviously, maximization with respect to b also yields (5) as first-order condition. Therefore the optimal bid is identical in both cases. This result is rather obvious: Imposing proportional taxes is equivalent to an affine transformation of the bidder s utility function, which does not affect preferences. Considering Propositions 1 and 2 together immediately yields the following result: Corollary 1. If bidders additional profits are identically distributed, expected revenue of all four standard auctions is identical in the presence of a proportional income tax and does not differ from the revenue in the absence of taxation. Hence, a proportional income tax preserves revenue equivalence and Pareto-efficiency of all four auctions. Let us now consider progressive and regressive taxes. In this case different values of lead to asymmetries since marginal tax rates differ across firms. Until now the analysis of auction theory with asymmetric bidders has revealed only two general results: First, the revenue equivalence theorem no longer holds, and, secondly, the outcome of first-price auctions is not necessarily Paretoefficient. Further assumptions are required in order to obtain more specific results.
60 JOURNAL OF ECONOMICS AND FINANCE Volume 24 Number 1 Spring 2000 Due to the poor results in the case of asymmetric bidders, we first focus on the impact of progressive and regressive taxes on bidding behavior in the symmetric case. Later on we will give two examples that deal with asymmetric bidders. We obtain symmetry by assuming that i = j i, j, which is known to all firms. Without loss of generality, we set i = 0 i. The average tax rate will be denoted by t(y). By definition, a tax is progressive (regressive) if t (y) > 0 (< 0) y. Proposition 3. The introduction of a progressive (regressive) income tax with τ (y) < 1 y raises (lowers) the optimal bid in a first-price auction. Proof. Without loss of generality we characterize symmetric equilibria where all bidders play the same strategies. Let b * (. ) denote the symmetric equilibrium strategy. Assume that b * is monotone increasing (we will confirm this later on). If all other firms play the strategy b *, the expected net profit of a firm who bids as if its additional profit were x instead of is given by U (x, ) = [ b * (x) T ( b(x))] F(x) n-1. (7) If b * (x) is the equilibrium strategy, (7) must attain its maximum at x =. The necessary condition is obtained by differentiating (7) with respect to x and setting the derivative equal to zero at x = : du(x, ) x = = (n 1)F( ) n-2 f ( )( b * ( ) T( b * ( ))) F n-1 ( )(1 T ( b * ( ))) (8) dx which yields b * ( ) = (n 1)f ( ) 1 t ( b* ( )) ( b( )). (9) F ( ) 1 ( b * ( )) Obviously b * is monotone increasing in, as asserted. Let us denote the optimal bid in the absence of taxation by ^b ( ). Analogously to (9), we obtain b ^ * ( ) = (n 1)f ( ) ( b ^ * ( )). (10) F ( ) As a boundary condition we have b( ) = b( ^ ) =,and b and b ^ are increasing in. Comparing (9) and (10) implies b ( ) > (<) b ( ^ ) and thus b( ) > (<) b ^ ( ) for all > i ff (1-t (y)) / (1- (y)) > (<) 0 y [, + ]. It is easy to show that this is equivalent to t (x) > (<) 0 x [, + ]. Thus, a progressive (regressive) tax leads to higher (lower) optimal bids in a first-price auction. It remains to show that the second-order condition holds. For this purpose we show that b * is a global maximizer of U, i.e. U(b, ) is increasing in b for b < b * and decreasing for b > b *. 3 Let (b) be the expected payment of bidding b. Then, analogously to (5), we can rewrite expected utility from bidding b with valuation as U(b, ) = (b) (b) (b)t( b). (11) When we characterized the optimal bid, we implicitly used the first-order condition obtained by differentiating (11) with respect to b: U b (b, ) = (b) (b) (b)t( b) + (b)t ( b) = 0. (12) 3 This second-order condition is known as pseudoconcavity. Cf. Wolfstetter (1999).
Revenue Equivalence and Income Taxation 61 Differentiating U b with respect to yields U b (b, ) = (b)(1 T ( b)) + (b)t ( b). (13) Now suppose b < b * and ^ is the valuation that would lead to bid b if the strategy b * is played, i.e. b * ( ^) = b. By strict monotonicity of b *, it holds that > ^. Thus, we get U b (b, ) U b (b, ^ ) U b (b * ( ^ ) ^ ) 0 if it holds that U b (b, ) 0, which is the case if and only if (b) (1 T ( b)) - (b) T ( b), or, equivalently, assumption 1 holds so that (n 1)f ( ) ( b) F ( ) 1 ( b) (14) Thus we have shown that U(b, ) is increasing in b for all b < b *. In a similar way one can show that U(b, ) is decreasing in b for all b > b *. Thus, b * is a global maximizer of U(b, ) if assumption 1 holds. Note that assumption 1 always holds if the tax is either proportional or progressive in the sense that marginal tax is increasing everywhere, i.e. 0. In case of a decreasing marginal tax rate, a sufficient condition for b * to be a global maximum is that a higher valuation does not cause residual income to grow faster than probability of winning given the bidder plays the equilibrium strategy. Thus, assumption 1 rules out both distributions with locally very low density and marginal tax rates that are locally extremely decreasing. The interpretation of this result is straightforward. Determining the optimal bid in a first-price auction is the solution to a tradeoff between the probability of winning and the potential profit: A slightly higher bid raises the probability of winning but lowers the potential profit. However, the presence of a progressive tax weakens the effect of an increased bid on the profit since a lower profit also implies a lower tax rate. Therefore, the tradeoff is solved at a higher bid. 4 Nevertheless, progressive income taxes retain Pareto-efficiency of first-price auctions because, according to (9), the optimal bid is a strictly increasing function of if (y) < 1. Proposition 3 immediately yields the following result: Corollary 2. In the presence of a progressive (regressive) income tax, first-price auctions lead to a strictly higher (strictly lower) expected revenue than second-price auctions. Surprisingly, in first-price auctions on the one hand, a part of the burden of a regressive tax can be shifted to the seller, who, on the other hand, would profit from a progressive tax. The intuition is as follows. As we have shown in Proposition 2, a proportional tax does not change the optimal bid in the first-price auction. This is because the average tax to be paid does not depend on the bidder s decision on how much to bid. Thus, the tradeoff between potential profit and probability of winning the auction is still solved at the same bid as before. It is obvious that, by maximizing expected gross profits, the bidder also maximizes expected net profits. If the tax is progressive or regressive, a bidder s decision affects his average tax rate, and he will take this fact into consider- 4 There is an analogy to bidding behavior of risk-averse bidders who buy a higher probability of winning by giving up potential profit in this case. Cf. Maskin and Riley (1984).
62 JOURNAL OF ECONOMICS AND FINANCE Volume 24 Number 1 Spring 2000 ation when deriving his optimal bid. Bidding one dollar more decreases net profit in case of winning by 1-. Thus, if the tax is progressive (regressive), profits decrease at a decreasing (increasing) rate. Bidding higher becomes cheaper (more expensive) compared to a proportional tax schedule, and the tradeoff between potential profit and probability of winning is solved at a higher (lower) bid. In case of a progressive tax, the auctioneer benefits from the fact that the tax schedule makes a higher gross profit less attractive at the margin, while a regressive tax is beneficial to the bidders: A part of the tax burden is shifted to the seller because a lower bid not only increases potential profits but also decreases the average tax rate. In other words, lower bids are subsidized by regressive taxes while a progressive tax subsidizes higher bids. Bidders account for this by bidding lower (higher) in equilibrium. Note that the maximum total surplus generated by the auction is constant and equal to the highest bidder s additional profit in case of winning as long as the outcome is Pareto-efficient. This implies that, with a regressive tax schedule, surplus is shifted from the seller and the buyer to the government while a progressive tax shifts surplus from the buyer to both the government and the seller. Finally we give two examples that deal with asymmetries between bidders: Example 1 (Tax rates are step functions). Suppose the progressive tax schedule is not continu - ously increasing in profits but rather a step function as it is in many countries. Then revenue equiv - alence remains valid if firms do not change their tax bracket when winning the input factor. This follows immediately from (6) together with (5), since the first-order condition does not change in case the average tax rate is constant. Example 2. Suppose there are two bidding firms, i and j; one is subject to a proportional, and the other is subject to a progressive, tax. Then, the bidders behavior in a second-price auction does not change compared to the cases without or with proportional taxes. In a first-price auction, how - ever, for the one firm that faces the progressive tax rate, it is cheaper to place a higher bid since bidding higher decreases the average tax rate. 5 Thus, this firm will bid higher than in the case of a proportional tax. This in turn induces the proportionally taxed firm to bid higher as well. Thus, as soon as one firm is taxed progressively, expected revenue from the first-price auction increases and exceeds expected revenue in the second-price auction. Furthermore, a first-price auction dis - criminates against the proportionally taxed firm since it faces higher opportunity costs for higher bids. Conclusion In this paper we have shown that proportional income taxes leave the revenue equivalence theorem intact, while progressive and regressive taxes yield different expected revenue in first-price and second-price auctions. Moreover, non-linear taxes may distort efficiency in first-price auctions if firms differ either in their initial profits or in their corporate structure, or both. Note that, for instance, in Germany only joint-stock companies have to pay a proportional corporation tax, while all other firms are taxed by a progressive income tax. Thus, first-price auctions discriminate against joint stock companies since the worse tradeoff between potential profits and probability of winning increases the equilibrium bids of their competitors. Apart from risk aversion of the bidders and fear of collusion, this paper points out another environment where first-price auctions dominate second-price auctions in terms of revenue. While one should not overstress this superiority there are situations where the English auction performs much better theoretically one should keep in mind the effects taxes may have on bidding behavior. 5 This follows from an argument in Maskin and Riley (1985).
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