Ex ante and ex post discretion over arm's length transfer prices.

I. INTRODUCTION

This paper analyzes the efficiency implications of multinationals' ability to influence, directly or indirectly, the transfer prices they use to allocate profits across divisions for tax purposes. The Internal Revenue Code [section] 482 "arm's length" regulations link

the calculation of a multinational's transfer price to a comparable transaction between uncontrolled (independent) parties. All other things equal, a multinational firm wants to set the transfer price to allocate all profit to the division in the lower tax jurisdiction. Uncontrolled parties presumably have no such tax incentive when they negotiate a price with each other. As a result, an uncontrolled transaction price represents a "fair" division of profit across divisions, at least according to the logic underlying the arm's length philosophy. Under the regulations, the firm derives a fair transfer price for tax purposes by observing (or inferring) the price of an actual (or hypothetical) uncontrolled sale of a good comparable to the good transferred between the multinational's divisions.

Despite these regulations, a firm can still shift income in two ways. First, because it is difficult to identify comparable, uncontrolled transactions, the tax auditor often can identify only a range of acceptable transfer prices. Hence, the auditor cannot always detect a deviation from the "true" arm's length price (ex post discretion). Second, because the firm's investment choices may affect the definition of comparability, the firm can manipulate the transfer price ex ante by distorting investment. (1) The common perception is that the lax enforcement and/or ambiguity of existing regulations give firms too much ex post discretion over the tax transfer price (e.g., Graham 1992; Judis 1993). A comprehensive analysis of ex post discretion, however, must also consider its effect on ex ante investment.

In the model, two divisions in different tax jurisdictions produce and sell a product. The upstream (selling) division begins the manufacturing process. The downstream (buying) division completes the manufacturing and sells the product outside the group. The firm makes capital investments in each division at time 0. The investments are specializations that are valuable only if the divisions trade with each other, and these investments determine each division's variable cost in subsequent periods. There is no agency conflict in the firm.

I analyze two different types of arm's length price: comparable profit method (CPM) price and comparable uncontrolled price (CUP). Under the CPM, the transfer price must produce a rate of return for the division that is equal to the rate of return of a similar independent firm. Under the CUP method, the transfer price must be equal to that of a similar or identical transaction between independent firms. I analyze both methods, deriving the optimal level of capital in both divisions given different tax rates and different levels of ex post discretion.

The study's main result is that the firm's opportunistic selection of the ex post transfer price may have efficiency benefits. That is, allowing the firm discretion to deviate from the "true" arm's length price in subsequent periods may mitigate its incentive to choose less efficient investments at time 0. Although in some cases ex post income shifting exacerbates investment distortions, in others, it alleviates it. A key distinction between the types of income shifting underlies the result that ex post discretion may have efficiency benefits. Ex ante discretion over the transfer price affects the absolute amount of income the firm divides between jurisdictions, but ex post discretion affects only the relative allocations. In other words, ex ante shifting determines the size of the pie; ex post shifting determines the relative sizes of the two pieces. Choosing more efficient inputs may result in a regulatory transfer price that is less favorable for tax purposes, but that also produces higher pre-tax profits. From the firm's perspective, the higher pre-tax profits may dominate the higher average tax rate. The results suggest that in some cases an increase in ex post discretion may lead to increases in the tax revenues of both jurisdictions. In any case, the interplay between the types of shifting implies that ex post shifting has an efficiency, as well as a distributional, role.

The paper builds on the research of Halperin and Srinidhi (1987). They examine the implications of different methods of calculating the CUP transfer price for manufactured goods on single-period resource allocation decisions. Halperin and Srinidhi (1996) perform a similar analysis for the arm's length regulations for intangibles, using both the CUP method and the CPM. In both papers, the authors (1) suppress intrafirm incentive problems, (2) implicitly assume that regulators can perfectly enforce arm's length rules, which eliminates ex post shifting, and (3) characterize the ex ante resource allocation distortions the regulations induce.

For both the CPM and the CUP method, I also analyze a pure ex ante shifting setting such as Halperin and Srinidhi (1987, 1996). In the CUP setting, the production distortion depends only on the tax rate differential. With the CPM, the distortion also depends on the after-tax comparable rate of return, the firm's discount rate, and its depreciation policy. This result holds because increasing investment has conflicting effects on a division's rate of return. By reducing variable cost, higher investment increases profits (the numerator in the rate of return calculation). However, it also increases the capital base (the denominator in the calculation). Therefore, tax rate differences alone do not determine the direction of the investment distortions. Because the CPM results depend on firm-specific factors, analysis of the CUP method yields the most general conclusions.

The study's main contribution is that it is the first to consider ex ante and ex post shifting simultaneously. Analysis of the feedback between the types of discretion is important because the results undermine the notion that ex post discretion benefits only the firm and the lower tax rate jurisdiction. To the extent that regulators wish to promote efficiency as well as a fair distribution of income, the improved efficiency that may accompany higher ex post discretion benefits them. The improved efficiency also mitigates, and potentially fully offsets, the reduction in tax receipts suffered by the higher tax rate jurisdiction.

Although Samuelson (1982) and Kant (1988) model similar issues, neither paper examines the interaction between the two types of shifting. Both papers consider a multinational firm that produces a product domestically and sells it both domestically and abroad. Neither model requires that the firm transfer the product at the domestic selling price, an assumption that violates current institutional practice and limits the applicability of the results.

In Samuelson (1982), the transfer price must lie between the marginal cost of production and the domestic selling price. The range of defensible prices represents ex post discretion. He shows that the firm distorts production decisions to influence the range of defensible prices, which is ex ante shifting. However, he analyzes the interaction in only one direction. He shows how changes in inputs affect the ex post range of prices, but does not show how changes in the ex post range affect the firm's ex ante incentive to distort inputs.

Kant (1988) models the effect of varying ex post discretion, but omits ex ante shifting. Because the firm's input decisions cannot affect the exogenously specified arm's length price in his model, there is no ex ante shifting. Although the firm can choose an arbitrary transfer price, the probability of regulatory intervention increases as the selected transfer price deviates from the arm's length price. The firm balances income shifting against penalty avoidance. Kant (1988) models ex post discretion by varying the parameters of the function governing the probability of regulatory intervention, but does not address the interaction between the two modes of income shifting.

The paper is also related to Harris and Sansing (1998) and Sansing (1999). These papers extend the tax transfer pricing literature by explicitly modeling the economic environment that permits the coexistence of both vertically integrated and independent firms. As Sansing (1999) observes, "only transactions for which neither organizational structure dominates will feature both the need for a transfer price for the vertically integrated group and a pair of comparable independent firms from which to derive a transfer price." Harris and Sansing (1998) model the CUP method. They find that it allocates a disproportionate (compared to independent firms) amount of income to the manufacturer. They also show that it distorts production decisions if tax rate differences exist, and that these distortions may in turn alter the choice of organizational structure. Sansing (1999) abstracts away from efficiency effects and instead examines the relative distributional properties of the CUP method and the CPM. He finds that the CUP method allocates relatively less taxable income to the domestic parent than the CPM because the CPM directly compensates the parent for its relatively larger share of the costs of the relationship-specific investments.

The economic decisions of uncontrolled firms provide the benchmark for arm's length prices. Only by explicitly modeling these decisions, then, can Harris and Sansing (1998) and Sansing (1999) compare the CUP and CPM profit allocations to each other and to the allocation between independent firms. In contrast, I make no comparisons across the CUP method and CPM, restricting my analysis within categories. For expositional clarity, I assume the existence of economic conditions supporting the coexistence of both vertically integrated and independent firms rather than deriving the conditions explicitly. In other words, I take the return of the comparable uncontrolled firm (CPM) and the divisions' external opportunities (CUP method) as given and fixed. (2) Furthermore, I suppress agency considerations that might favor one organizational form over the other.

The results of the paper are potentially of interest to accounting and tax researchers, tax auditors, regulators, and policymakers. Earlier research in accounting and economics demonstrates that literal adherence to the Treasury Regulations induces firms to distort resource allocations. Also, the business press suggests that lax enforcement of regulations and/or regulations too ambiguous to enforce strictly lead to substantial underpayment of taxes. For example, transfer pricing was briefly an issue in the 1992 presidential campaign, with Bill Clinton claiming that improved IRS enforcement of existing regulation could net $45 billion in tax revenues (see Graham 1992; Judis 1993). Although the magnitude of the alleged underpayment may be politically motivated hyperbole, the claim illustrates the common perception about the urgency of reducing firms' ex post choice over tax transfer prices.

This study shows that, contrary to popular perception, ex post discretion may have efficiency benefits. Although still acting opportunistically, firms may react to increased latitude in choosing transfer prices by producing more efficiently. These results can help policymakers understand the implications of transfer pricing regulations. They could also help tax auditors decide how to allocate scarce resources. For example, ex post shifting exacerbates ex ante distortions under the CUP method if the downstream tax rate is higher, but alleviates them if the upstream rate is higher. Tax authorities would derive more benefits from stricter enforcement in the former setting. Finally, the study is of interest to management accountants who design and implement transfer pricing policies in that it enriches the understanding of the institutional context and implications of transfer pricing policy.

Section II introduces the model. Sections III and IV present the analysis. Section V discusses the results. Finally, Section VI summarizes the paper.

II. THE MODEL

Upstream and downstream affiliates of a controlled group, one affiliate incorporated in the U.S., the other incorporated in another country, combine in the manufacture and sale of a product. The upstream division begins production and transfers the intermediate good to the downstream division. The downstream division finishes production and sells the product outside the group at price p.

At time 0, the firm makes variable cost-reducing capital investments of [k.sub.u] and [k.sub.d], where the subscripts u and d designate upstream and downstream, respectively. The capital investments allow the divisions to produce a total of q units over N years at variable costs [v.sub.u]([k.sub.u])/unit and [v.sub.d]([k.sub.d])/unit, respectively, with [v'.sub.u](*) and [v'.sub.d](*) < 0, and [v".sub.d](*) and [v".sub.d](*) > 0. I assume that neither division's investment affects the other's cost: [differential][v.sub.u](*)/[differential][k.sub.d] = [differential][v.sub.d](*)/[differential][k.sub.u] = 0. Without loss of generality I set q = 1. I will subscript the time period by n [member of] {1, 2, ..., N}. Let the proportion of the total units produced and sold in period n be [[gamma].sub.n]. That is, in year n, the firm produces [[gamma].sub.n] units, where [[SIGMA].sup.N.sub.n=1] [[gamma].sub.n] = 1. The firm's depreciation factor in period n is [[delta].sub.n], with [[delta].sub.n] > 0 and [[SIGMA].sup.N.sub.n=1] [[delta].sub.n] = 1. Let [[lambda].sub.n][k.sub.j] be the book value of asset j at the beginning of period n. Therefore, [[lambda].sub.n] = 1 - [[SIGMA].sup.n-1.sub.j=1] [[delta].sub.j]. Table 1 in Appendix A provides a glossary of notation.

The investment is relationship-specific and has value only if the divisions trade internally. For example, the capital investments could involve specialization of the intermediate product that has no value unless the other manufacturing facility conforms to the specialization. The work of Williamson (1985) motivates this assumption. By organizing as a single firm, economic entities may avoid the "hold-up" problem associated with relationship-specific investment. Hence, it is a sensible setting in which to examine transfer pricing.

I assume that the firm is a price-taker and that the price p is an intertemporal constant. (3) In period n, the investments yield pre-tax accounting income of p - [v.sub.u]([k.sub.u]) - [v.sub.d]([k.sub.d]) - [[delta].sub.n]([k.sub.u] + [k.sub.d]). The divisional profits depend on the transfer price, [T.sub.n]. The upstream division's pre-tax income in period n is [[gamma].sub.n][[T.sub.n] - [v.sub.u]([k.sub.u])] - [[delta].sub.n][k.sub.u]; the downstream division's is [[gamma].sub.n][p - [T.sub.n] - [v.sub.d]([k.sub.d])] - [[delta].sub.n][k.sub.d]. The upstream division's income is taxed at [t.sub.u] and the down-stream's at [t.sub.d]. Let [[tau].sub.j] = 1 - [t.sub.j]. The firm's cash flow in period n is:

[CF.sub.n] = [[gamma].sub.n][[tau].sub.u]([T.sub.n] - [v.sub.u]([k.sub.u])) + [[tau].sub.d](p - [T.sub.n] - [v.sub.d] (k.sub.d]))] + [t.sub.u][[delta].sub.n][k.sub.u] + [t.sub.d][[delta].sub.n][k.sub.d]]

equivalent to:

[CF.sub.n] = [[gamma].sub.n][[tau].sub.d](p - [v.sub.d]([k.sub.d])) - [[tau].sub.u]([v.sub.u]([k.sub.u])] + [t.sub.u] [[delta].sub.n][k.sub.u] + [t.sub.d][[delta].sub.n][k.sub.d] + [[gamma].sub.n]([[tau].sub.u] - [[tau].sub.d])[T.sub.n].

Let the firm's discount rate be i and [D.sub.n] = (1 + i)[sup.-n]. The firm selects [k.sub.u] and [k.sub.d] to maximize the net present value of the cash flows. (4) The net present value over the life of the investment is:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The first term represents the time 0 investments, the second term is the present value of the after-tax margin, the third term is the present value of the depreciation tax benefit, and the last term is the present value of the after-tax net transfer payments. As a benchmark, consider the case in which the transfer price is independent of the level of capital investment. In this case, the last term in the objective function in Equation (1) is irrelevant.

Benchmark 1: Let [k.sup.*.sub.u] and [k.sup.*.sub.d] be the optimal levels of upstream and downstream capital investment if the transfer price is independent of the capital levels. The following first-order conditions define [k.sup.*.sub.u] and [k.sup.*.sub.d]:

(2) v'([k.sup.*.sub.u]) -1 + [[SIGMA].sup.N.sub.n=1] [D.sub.n][[delta].sub.n][t.sub.u]/[[SIGMA].sup.N.sub.n=1] [D.sub.n][[gamma].sub.n][[tau].sub.u]

(3) v'([k.sup.*.sub.u]) -1 + [[SIGMA].sup.N.sub.n=1] [D.sub.n][[delta].sub.n][t.sub.d]/[[SIGMA].sup.N.sub.n=1] [D.sub.n][[gamma].sub.n][[tau].sub.d]

The investment is not tax-neutral; both [k.sup.*.sub.u] and [k.sup.*.sub.d] are decreasing in the respective tax rates. (5) This is a variation of the well-known result in Sandmo (1974). In the next two sections I examine the effect of the specific transfer pricing regulations on the firm's investment decision.

III. COMPARABLE PROFIT METHOD

I first examine the comparable profit method (CPM). Since 1994, the IRS has allowed this method, which allocates income for tax purposes based on the rate of return of a comparable uncontrolled firm rather than on the price of a comparable uncontrolled transaction. Specifically, the CPM transfer price equates the rate of return on the division's assets to the rate of return on assets of a comparable uncontrolled firm (Treasury Reg. [section] 1.482-5(a)). That is, one works backward from the comparable rate of return to arrive at the appropriate transfer price.

I assume that a comparable firm exists. That is, I assume that economic conditions permit the simultaneous existence of vertically integrated and independently organized firms. (6) I do not model the independent firm, but rather take its rate of return as given. In particular, the rate of return is independent of the vertically controlled firm's input choices and the amount of ex post discretion. These might be related if, for example, there were strategic interactions.

Upstream Division

The Treasury Regulations define the accounting rate of return under the CPM as operating profits divided by the book value of assets. The pre-tax operating profit for the upstream division in period n is [[gamma].sub.n]([T.sub.n] - [v.sub.u]([k.sub.u])) - [[delta].sub.n][k.sub.u]. Let [R.sub.n] be the accounting rate of return of the comparable firm in period n. Equating [[[gamma].sub.n]([T.sub.n] - [v.sub.u]([k.sub.u])) - [[delta].sub.n][k.sub.u]]/([[lambda].sub.n][k.sub.u]) to [R.sub.n] yields a transfer price in period n of:

(4) [T.sub.n] = [[lambda].sub.n]/[gamma].sub.n][R.sub.n][k.sub.u] + [v.sub.u] (k.sub.u]) + [[delta].sub.n]/[[gamma].sub.n][k.sub.u].

Because the firm makes its investment decisions at time 0, it must predict, based on its knowledge of comparable firms in its industry, the comparable rates of return ([R.sub.1], [R.sub.2], ..., [R.sub.N]) in future periods and the implied transfer prices ([T.sub.1], [T.sub.2],..., [T.sub.N]). Discounting the stream of transfer payments in Equation (4) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The discounted sum of the revenue transfers is partially composed of the discounted sum of the accounting depreciation expenses; in essence, the CPM regulations monetize the depreciation expenses. Let [R.sup.*] = [[SIGMA].sup.N.sub.n=1] [D.sub.n][[lambda].sub.n][R.sub.n]. The variable [R.sup.*] aggregates the firm's predictions about the future comparable rates of return. As the following expression for the discounted stream of transfer payments indicates, one can interpret [R.sup.*] as a "cumulative" accounting rate of return on original capital:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Because it chooses the level of upstream capital, the firm has ex ante control over the present value of the transfer payments to the upstream division, as described in the following lemma. It is convenient for the following lemmas and propositions to define [DELTA] = 1 - [[SIGMA].sup.N.sub.n=1] [D.sub.n][[delta].sub.n].

Lemma 1: If the comparable profit method is applied to the upstream division, then the present value of the transfer payments to the upstream division is increasing in [k.sub.u] at [k.sub.u] = [k.sup.*.sub.u] if [[tau].sub.u][R.sup.*] > [DELTA], fixed if [[tau].sub.u][R.sup.*] = [DELTA], and decreasing if [[tau].sub.u][R.sup.*] < [DELTA].

The result holds for all accounting depreciation policies (the depreciation factor [[delta].sub.n]'s, which imply the book value [[lambda].sub.n]'s) and patterns of economic usage (the [[gamma].sub.n]'s). An increase in [k.sub.u] lowers the rate of return by increasing the depreciation expense and the capital base, and increases the rate of return by lowering variable costs. Which effect dominates at [k.sup.*.sub.u] depends on the relative sizes of the after-tax return to a unit of investment ([[tau].sub.u][R.sup.*]) and a term relating to the present value of the depreciation write-off ([DELTA]). If [[tau].sub.u][R.sup.*] is sufficiently high, setting [k.sub.u] > [k.sup.*.sub.u] increases the discounted present value of the transfer payments. Otherwise, setting [k.sub.u] > [k.sup.*.sub.u] decreases it. The results in the CPM hinge on the relation between [[tau].sub.u][R.sup.* and [DELTA]. The next lemma analyzes it for a special case.

Lemma 2: If the comparable rates of return are constant ([R.sup.n] = R, for all n), then the present value of the transfer payments to the upstream division is increasing in [k.sub.u] at [k.sub.u] = [k.sup.*.sub.u] if [[tau].sub.u]R > i, fixed if [[tau].sub.u]R = i, and decreasing if [[tau].sub.u]R < i.

In this special case, [[tau].sub.u][R.sup.*] = [DELTA] is equivalent to [[tau].sub.u]R = i, regardless of the depreciation policy. That is, if the after-tax comparable rate of return is equal to the discount rate, then deviations from [k.sup.*.sub.u] do not affect the amount of income shifted across divisions via the transfer payments. In practice, i, the discount rate the firm uses for capital budgeting purposes, and [[tau].sub.u]R, the after-tax rate of a comparable firm, are not necessarily equal. Which is greater, however, likely depends on firm-specific factors, limiting the generality of the CPM results. I argue informally that one would expect [[tau].sub.u]R to be less than i, implying that [[tau].sub.u][R.sup.*] < [DELTA], for two reasons. First, the accounting rate of return R is likely to understate the true rate of return because accounting depreciation is generally more accelerated than economic depreciation. (7) Second, anecdotal evidence suggests that firms set artificially high cut-off rates for capital projects (Brealey and Myers 1996, 220). If [R.sub.n] varies, then the difference between [[tau].sub.u][R.sup.*] and [DELTA] is still closely related to the difference between the average after-tax comparable rate of return and the discount rate, but the relation is not exact. Furthermore, it depends on the depreciation policy.

Lemma 1 shows that, unlike the benchmark case, the transfer price in this scenario may depend on [k.sub.u]. Substituting the expression for the present value of transfer payments (Equation [5]) into the benchmark objective function (Equation [1]) yields a net present value of:

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The benchmark investments [k.sup.*.sub.u] and [k.sup.*.sub.d] maximize the sum of the expressions in the first line of the new objective function. The term on the second line represents the distortion to the firm's incentives induced by application of the regulations. Its sign depends on the relative tax rates and on the condition in Lemma 1. The next proposition compares the CPM investments to the benchmark investments.

Proposition 1: The comparable profit method, applied to the upstream division:

(i) Induces overinvestment in upstream capital relative to the benchmark case if [t.sub.d] > [t.sub.u] and [[tau].sub.u][R.sup.*] > [DELTA], or [t.sub.u] > [t.sub.d] and [[tau].sub.u][R.sup.*] < [DELTA]

(ii) Induces underinvestment in upstream capital relative to the benchmark case if [t.sub.u] > [t.sub.d] and [[tau].sub.u][R.sup.*] > [DELTA], or [t.sub.d] > [t.sub.u] and [[tau].sub.u][R.sup.*] < [DELTA].

(iii) Induces no distortion in upstream capital if [[tau].sub.u][R.sup.*] = [DELTA].

(iv) Induces no distortion in downstream capital.

Because of the assumption that the downstream capital has no effect on the upstream variable cost, the calculation of the upstream division's return depends only on upstream capital. Therefore, the method induces no distortion in [k.sub.d]. (8) If the upstream division has the lower tax rate, then the firm has incentive to shift income upstream with a higher transfer price. Lemma 1 shows that the firm can accomplish this by increasing upstream capital if [R.sup.*] is sufficiently high, decreasing it otherwise. That is, the firm shifts income ex ante by distorting the investment choices. Based on the earlier discussion, I argue that underinvestment is the more plausible outcome. If the downstream division has the lower tax rate, then the firm shifts income downstream with lower transfer payments.

An interesting implication of Proposition 1 is that the CPM induces no productive distortions, regardless of tax rate differences, if [[tau].sub.u][R.sup.*] = [DELTA], a result that will not hold under the CUP method. As Lemma 2 shows, this is equivalent to [[tau].sub.u]R = i in the special case of constant [R.sub.n]. Ex ante shifting incentives arise from the firm's ability to affect the present value of transfer payments by distorting the investment from [k.sup.*.sub.u]. If the after-tax return, discount rate, and depreciation schedules are exactly calibrated, then deviations from [k.sup.*.sub.u] do not affect the aggregate payments. Hence, there are no distortions.

Thus far, I have implicitly assumed that regulators can enforce the "true" arm's length price. The investment distortions in Proposition I are purely the function of ex ante income shifting. I now relax this assumption and explore the implications of ex post shifting. Each set of capital investments at time 0 implies the exact transfer price in Equation (4) in period n. However, the enforcement of transfer pricing regulations is imperfect. In the comparable profit method, the need to identify comparable firms and to adjust the rate of return to reflect material differences between the controlled and uncontrolled entities are significant practical challenges (Treasury Reg. [section] 1.482-1[d](2)). In practice, tax auditors calculate an arm's length range based on multiple comparables, making no transfer pricing adjustment if the firm's operating profits fall within this range. (9)

I implement this aspect of the institutional environment by assuming that the firm can defend a rate of return in period n of [R.sup.n] + [phi], with [phi] [member of] [-[PHI], [PHI]]. (10) The variable [PHI] represents the maximum ex post discretion the firm has over the transfer price. The introduction of ex post discretion implies that the transfer price in period n is:

(7) [[lambda].sub.n]/[[lambda].sub.n] [R.sub.n][k.sub.n] + [v.sub.n] ([k.sub.u]) + [[delta].sub.n]/[[gamma].sub.n] [k.sub.u]]] + [phi] [[lambda].sub.n]/[[gamma].sub.n] [k.sub.u].

As a result, ex post discretion maps into the following range of prices in period n:

(8) [v.sub.u] ([k.sub.u]) + [k.sub.u] [[lambda].sub.n]/[[gamma].sub.n] ([R.sub.n] - [PHI]) + [[delta].sub.n]/[[gamma].sub.n]] [less than or equal to] [T.sub.n] [less than or equal to] [v.sub.u] ([k.sub.u]) + [k.sub.u] [[[lambda].sub.n]/[[gamma].sub.n] ([R.sub.n] + [PHI]) + [[delta].sub.n]/[[gamma].sub.n]].

If [PHI] = 0, then the range collapses to the expression in Equation (4). I assume that the firm knows with certainty the amount of ex post discretion tolerable (or undetectable) to tax regulators. Implicitly, the penalties associated with violating this bound are severe enough that the firm never exceeds it. The modeling captures both dimensions of discretion. First, a firm, via its input choices, determines the standard of comparability. Second, even with a given standard of comparability, the auditors' estimate of the arm's length price is imprecise. Although the certainty assumption simplifies the model, the results do not depend on the firm's knowing at time 0 the level of ex post discretion in future periods. (11)

The firm's income shifting is a sequential process. In periods 1 through n the firm, given its choice of [k.sub.u] and [k.sub.d] at time 0, chooses the ex post transfer price (i.e., chooses the [phi]) that minimizes the taxes of:

(9) [t.sub.d] (p - [T.sub.n] - [v.sub.d] ([k.sub.d])) + [t.sub.u] ([T.sub.u] - [v.sub.u] (k.sub.u])).

Substitution of the expression for [T.sub.n] in Equation (7) into Equation (9) and differentiation with respect to [phi] yields ([[lambda].sub.n]/[[gamma].sub.n]) [k.sub.u]([t.sub.u] - [t.sub.d]). If [t.sub.d] > [t.sub.u], then the tax is decreasing in [phi] and the firm sets the highest transfer price: [phi] = [PHI]. The reverse holds if [t.sub.u] > [t.sub.d].

By the definition of comparability, the discounted sum of the portion of the transfer price in brackets in Equation (7) is the expression in Equation (5). The discounted sum of the portion of each period's transfer price attributable to ex post discretion is [[SIGMA].sup.N.sub.n=1] [D.sub.n][[lambda].sub.n][phi][k.sub.u]. Let [mu] = [[SIGMA].sup.N.sub.n=1] [D.sub.n][[lambda].sub.n]. The firm's ability to deviate ex post from the true arm's length price generates extra aggregate return on capital of [mu] [phi]. The last term of the objective function (Equation [6]) is now:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the sign on [mu][PHI] depending on the relative tax rates. Anticipating the optimal selection of [phi] in subsequent periods, the firm selects at time 0 the [k.sub.u] and [k.sub.d] that maximize the net present value of the cash flows. The next proposition shows the effect of ex post discretion on the distortions to investment.

Proposition 2: If the comparable profit method, applied to the upstream division, induces overinvestment, then increasing ex post discretion ([PHI]) exacerbates the distortion. If it induces underinvestment, then increasing the ex post discretion mitigates the distortion.

Note that the introduction of ex post discretion alters the conditions in Proposition 1 because it affects the after-tax return on an incremental unit of capital. If [t.sub.d] > [t.sub.u], then the condition on the comparable rate of return is [[tau].sub.u]([R.sup.*] + [mu][PHI]) > [DELTA]; if [t.sub.u] > [t.sub.d], then it is [[tau].sub.u]([R.sup.*] - [mu][PHI]) > [DELTA]. To understand the proposition, consider a scenario in which the upstream division has the lower tax rate and [[tau].sub.u]([R.sup.*] + [mu][PHI]) > [DELTA]. Because [t.sub.d] > [t.sub.u], shifting income via a high transfer price is desirable. Lemma 1 implies that selecting [k.sub.u] > [k.sup.*.sub.u] shifts the furture defensible range of transfer prices higher. Furthermore, ex post discretion reinforces ex ante incentives, exacerbating the production distortion. In this case, the transfer price is equal to the upper bound of the range in Equation (8). The amount by which increases in upstream investment raise the transfer price is increasing in ex post discretion. Hence, the firm distorts [k.sub.u] even further if [PHI] increases. Conversely, if [t.sub.u] > [t.sub.d] and [[tau].sub.u]([R.sup.*] - [mu][PHI]) < [DELTA], shifting income downstream via a low transfer price is optimal. Selecting [k.sub.u] > [k.sup.*.sub.u] shifts the future defensible range of transfer prices lower. The firm chooses the lower bound of the range as the transfer price. The amount by which decreases in upstream investment lower this transfer price is decreasing in discretion. Hence, the ability to shift income ex post mitigates the ex ante incentives. The firm reduces the distortion in [k.sub.u] as ex post discretion increases.

Previous transfer pricing papers demonstrate that arm's length regulations distort the firm's decisions. Proposition 2 shows that if one considers both forms of income shifting simultaneously, then ex post shifting, perceived to be a practice that serves only to undermine regulations, may have the benefit of alleviating production distortions. In fact, the proposition suggests that the tax revenues of both jurisdictions increase in the amount of ex post discretion if the more efficient input selections increase the size of pre-tax profits enough to offset the higher tax-rate division's lower share of those profits.

The firm deviates within, rather than from, the transfer pricing rule. If discretion implied that the firm could choose an arbitrary transfer price, then the regulation-induced investment distortion would vanish in the limit. As discretion increased, the firm's objective function would eventually converge to the benchmark case (Equation [1]) in which the transfer price is independent of the capital levels. The required adherence to the method of calculating the transfer price produces the asymmetry in the comparative statics.

Downstream Division

The analysis proceeds in a symmetric way if one applies the comparable price method to the downstream jurisdiction. Pre-tax operating profits for the downstream division in period n are [[gamma].sub.n](p - [v.sub.d]([k.sub.d]) - [T.sub.n]) - [[delta].sub.n][k.sub.d]. Equating the rate of return to the predicted [R.sub.n]'s and aggregating yields the following expression for the present value of the transfer payments to the upstream division as a function of downstream investment [k.sub.d]:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [R.sup.*] is defined as in the previous section. In this scenario, the firm controls the present value of the transfer payments to the upstream division by changing downstream investment.

Lemma 3: If the comparable profit method is applied to the downstream division, then the present value of the transfer payments to the upstream division is decreasing in [k.sub.d] at [k.sub.d] = [k.sup.*.sub.d] if [[tau].sub.d][R.sup.*] > [DELTA], fixed if [[tau].sub.d][R.sup.*] = [DELTA], and increasing if [[tau].sub.d][R.sup.*] < [DELTA]. In the special case of constant [R.sub.n] = R, the discounted transfer payments are decreasing in [k.sub.d] at [k.sub.d] = [k.sup.*.sub.d] if [[tau].sub.d]R > i, fixed if [[tau].sub.d]R = i, and increasing if [[tau].sub.d]R < i.

This lemma combines the results of Lemmas 1 and 2 for the upstream division. The sign on the condition reverses because a higher downstream return corresponds to a lower transfer price. Substituting the expression for the present value of transfer payments (Equation [10]) into the benchmark objective function (Equation [1]) yields a net present value of:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The benchmark investments [k.sup.*.sub.u] and [k.sup.*.sub.d] maximize the sum of the expressions in the first line of the new objective function. The last term represents the distortion to the firm's incentives induced by application of the regulations. Its sign depends on the relative tax rates and on the condition in Lemma 3. The next proposition compares the CPM investments to the benchmark investments in the pure ex ante setting.

Proposition 3: The comparable profit method, applied to the downstream division:

(i) Induces underinvestment in downstream capital relative to the benchmark case if [t.sub.d] > [t.sub.u] and [[tau].sub.d][R.sup.*] > [DELTA], or [t.sub.u] > [t.sub.d] and [[tau].sub.d][R.sup.*] < [DELTA].

(ii) Induces overinvestment in downstream capital relative to the benchmark case if [t.sub.u] > [t.sub.d] and [[tau].sub.d][R.sup.*] > [DELTA], or [t.sub.d] > [t.sub.u] and [[tau].sub.d][R.sup.*] < [DELTA].

(iii) Induces no distortion in downstream capital if [[tau].sub.d][R.sup.*] = [DELTA].

(iv) Induces no distortion in upstream capital.

As in the upstream case, tax rate differentials need not distort investments if [[tau].sub.d][R.sup.*] and [DELTA] calibrate exactly. As I discussed in the previous section, [[tau].sub.d][R.sup.*] < [DELTA] is the most plausible scenario, but the opposite could hold for firm-specific reasons. The modeling of ex post discretion is also similar. In period n, the CPM transfer price is:

(12) [T.sub.n] = p - [v.sub.d] ([k.sub.d]) - [[delta].sub.n]/[[gamma].sub.n] [k.sub.d] - [[lambda].sub.n]/[[gamma].sub.n] [k.sub.d][R.sub.n].

The introduction of ex post discretion implies that the transfer price in period n is:

(13) [p - [v.sub.d] ([k.sub.d]) - [[delta].sub.n]/[[gamma].sub.n] [k.sub.d] - [[lambda].sub.n]/[[gamma].sub.n] [k.sub.d][R.sub.n]] - [phi] [[lambda].sub.n]/[[gamma].sub.n] [k.sub.d].

Maximum discretion of [PHI] implies a range of prices of:

(14) p - [v.sub.d] ([k.sub.d]) - [[delta].sub.n]/[[gamma].sub.n] - [[delta].sub.n]/[[gamma].sub.n] [k.sub.d] - [[lambda].sub.n]/[[gamma].sub.n] [k.sub.d] ([R.sub.d] + [PHI]) [less than or equal to] [T.sub.n] [less than or equal to] p - [v.sub.d] ([k.sub.d]) - [[delta].sub.n]/[[gamma].sub.n] [k.sub.d] - [[lambda].sub.n]/[[gamma].sub.n] [k.sub.d] ([R.sub.d] - [PHI]).

Higher ex post discretion implies that the firm can choose from a wider range of defensible prices. Also, the firm's time 0 capital investment choices shift the interval higher or lower. By the definition of comparability, the discounted sum of the portion of the transfer price in brackets in Equation (13) is the expression in Equation (10). The last term of the objective function is now:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with [mu] defined as before and the sign on [mu] [PHI] depending on the relative tax rates. The next proposition shows the effect of ex post discretion on the distortions to investment.

Proposition 4: If the comparable profit method, applied to the upstream division, induces overinvestment, then increasing ex post discretion ([PHI]) exacerbates the distortion. If it induces underinvestment, then increasing the ex post discretion mitigates the distortion.

Again, the introduction of ex post discretion alters the conditions in Proposition 3. If [t.sub.d] > [t.sub.u], the condition on the comparable rate of return is [[tau].sub.d]([R.sup.*] -- [mu] [PHI]) > [DELTA]; otherwise the condition is [[tau].sub.d]([R.sup.*] + [mu] [PHI]) > [DELTA]. Proposition 4 is symmetric to Proposition 2. Ex post discretion may relieve ex ante distortions, again suggesting the possibility of higher tax revenues in both jurisdictions resulting from higher ex post discretion.

IV. CUP METHOD

I next examine the comparable uncontrolled price (CUP) method of determining an arm's length transfer price. Unlike the CPM, which infers the appropriate transfer price from the rate of return on assets of unrelated parties, the CUP method derives the transfer price from similar or identical transactions involving unrelated parties (Treasury Reg. [section] 1.482-3(b)). That is, the divisions must transact at the same price that obtains in a similar transaction between independent firms. Prior to 1994, regulations gave priority to the CUP method over Other methods. The 1994 regulations diminished its dominance.

I implement the CUP method analogously to Sansing (1999). Suppose that, after making the time 0 investments, the firm sold one of the divisions to an uncontrolled party. The CUP method asks, "At what price would these (now) unrelated parties transact?" Because of the relationship-specific assets, trading with each other is more beneficial than trading with external parties because each division enjoys a lower variable cost. The transaction price would depend on the surplus created by internal trade. Even though it is optimal to trade internally, the price would also depend on the external opportunities for trade, which define the negotiation "threat" points for the divisions.

The investments [k.sub.u] and [k.sub.d] are relationship-specific and have value only if the divisions trade internally. Because the initial investments [k.sub.u] and [k.sub.d] are sunk costs after time 0, the depreciation expenses [[delta].sub.n] [k.sub.u] and [[delta].sub.n] [k.sub.d] are not relevant for the calculation of the CUP price. The variable costs of production are [k.sub.u](0) [equivalent to] [k.sub.u] and [k.sub.d] (0) [equivalent to] [k.sub.d], respectively, if the divisions trade externally. The upstream

division can sell the intermediate good outside the firm at a price of [x.sub.u]/unit. The upstream division, then, would realize pre-tax profits of [x.sub.u] - [k.sub.u] if it traded externally. The downstream firm can buy the intermediate good externally for [x.sub.d]/unit, where [x.sub.d] [greater than or equal to] [x.sub.u], and would realize pre-tax profits of p - [x.sub.d] - [k.sub.d] if it traded externally. Total pre-tax profits if both divisions traded externally, then, would be:

(15) p + [x.sub.u] - [x.sub.d] - [k.sub.d] - [k.sub.u].

Subtracting pre-tax profits from external trade (Equation [15]) from pre-tax profits under internal trade (p - [v.sub.u] ([k.sub.u]) - [v.sub.d]([k.sub.d])) yields the surplus generated by internal trade:

[k.sub.u] + [k.sub.d] - [v.sub.d] ([k.sub.d] - [v.sub.u]([k.sub.u]) - ([x.sub.u] + ([x.sub.d]) [x.sub.d] > 0.

I assume that the divisions would split the surplus equally in the hypothetical negotiation. Hence, the transfer price T must satisfy:

T - [v.sub.u] ([k.sub.u]) - ([x.sub.u] - [k.sub.u]) = 1/2 [[k.sub.u] + [k.sub.d] - [v.sub.d] (k[sub.d] - [v.sub.d] (k[sub.d]) - [v.sub.u] ([k.sub.u]) - [x.sub.u] + [x.sub.d]]

and

p - T - [v.sub.d]([k.sub.d]) - (p - [x.sub.d] - [k.sub.d]) = 1/2 [[k.sub.u] + [k.sub.d] - [v.sub.d]([k.sub.d]) - [v.sub.u]([k.sub.u]) - [x.sub.u] + [c.sub.d]],

implying that:

(16) T = 1/2 [X - [v.sub.d] ([k.sub.d]) + [v.sub.u] ([k.sub.u])],

where X = [k.sub.d] - [k.sub.u] + [x.sub.u] + [x.sub.d]

I assume that [x.sub.d] and [x.sub.u], and therefore X, are constant across time. (12) Unlike the CPM, in which the transfer price depends on the accounting depreciation expense and asset book value each period, the CUP transfer price is constant over time.

The CUP transfer price is decreasing in upstream capital and increasing in downstream capital. An increase in [k.sub.u], holding the transfer price constant, increases the surplus from internal trade by the amount of the decrease in upstream variable cost. To allocate half of the incremental surplus to the downstream division, the transfer price must decrease by half of the amount of the variable cost reduction. A symmetric argument holds for the effect of changes in the downstream division's capital investment on the transfer price.

The present value of the transfer payment is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting this expression into the benchmark objective function (Equation [1]) yields a net present value of:

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The benchmark investments [k.sup.*.sub.u] and [k.sup.*.sub.d] maximize the sum of the terms of the first line of the objective function. The term on the second line represents the distortion to the firm's incentives induced by application of the regulations. Its sign depends on the relative tax rates. The next proposition describes the underinvestment and overinvestment CUP method induces.

Proposition 5: The comparable uncontrolled price method:

(i) Induces underinvestment in upstream capital and overinvestment in downstream capital relative to the benchmark if [t.sub.d] > [t.sub.u].

(ii) Induces overinvestment in upstream capital and underinvestment in downstream capital relative to the benchmark if [t.sub.u] > [t.sub.d].

If [t.sub.d] > [t.sub.u], then the firm has incentive to shift income upstream with high future transfer prices. The firm increases the transfer price by overinvesting in downstream capital and underinvesting in upstream capital. The direction of the investment distortions reverses if [t.sub.u] > [t.sub.d]. Unlike with the CPM, changes in [k.sub.u] or [k.sub.d] do not have conflicting effects on the transfer price. Therefore, the under- and overinvestment in Proposition 5 depends only on the relative tax rates, unlike in Propositions 1 and 3.

In practice, regulations allow firms to group transactions when applying the CUP method (Treasury Reg. [section] 1.482-1[f][2](iv)), which, as in the CPM, produces a range of acceptable transfer prices. Hence, I model ex post discretion over the comparable uncontrolled price transfer price with the assumption that T = 1/2 (1 + [phi])[X - [v.sub.d]([k.sub.d]) + [v.sub.u]([k.sub.u])], where [phi] [member of] [- [PHI], [PHI]]. This translates into a range of defensible transfer prices of:

(18) 1/2 (1 - [PHI]) [X - [v.sub.d]([k.sub.u]) + [v.sub.u] ([k.sub.u])] [less than or equal to] [T.sub.n] [less than or equal to] 1/2 (1 + [PHI]) [X - [v.sub.d]([k.sub.d]) + [v.sub.u] ([k.sub.u])].

An increase in ex post shifting expands the range. The firm can shift the interval higher (lower) ex ante by increasing [k.sub.d] ([k.sub.u]).

Substitution of T into the expression for taxes in Equation (9) and differentiation with respect to [phi] yields 1/2([t.sub.u] - [t.sub.d])(X - [v.sub.d]([k.sub.d]) + [v.sub.u]([k.sub.u])). Tax minimization requires that the firm set the highest transfer price ([phi] = [PHI]) if [t.sub.d] > [t.sub.u] and the lowest transfer price ([phi] = [PHI]) otherwise. The last term in the objective function is now:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [PHI] depends on the relative tax rates. The next proposition discusses the effect of ex post discretion on investment decisions under the CUP method.

Proposition 6: Under the comparable uncontrolled price method, the distortion in both investments is increasing in the amount of ex post discretion [PHI] if [t.sub.d] > [t.sub.u] and decreasing if [t.sub.u] > [t.sub.d].

If [t.sub.d] > [t.sub.u], then the transfer price is the upper end of the range in Equation (18). Clearly, the amount by which distortions in investments increase in transfer price is increasing in ex post discretion. At the other of the range (if [t.sub.u] > [t.sub.d]), increasing [PHI] reduces the ability to shift the transfer price by changing the investments. Therefore, as with the CPM, it is possible that increases in the firm's ex post discretion could increase the tax revenues of both jurisdictions if [t.sub.u] > [t.sub.d].

V. DISCUSSION

Samuelson (1982) is the first to model distortionary ex ante income shifting, but his model of transfer pricing is inconsistent with existing regulations. Specifically, the firm need not transfer at the existing comparable uncontrolled price. Subsequent research, such as Halperin and Srinidhi (1987, 1996) and Harris and Sansing (1998), model both the CUP method and the CPM specified by [section] 482 Treasury Regulations. The settings in these papers differ, but they all yield the same basic result: Firms can distort the inputs from the no-tax optimal levels to obtain a more favorable regulatory transfer price. Propositions 1, 3, and 5 essentially replicate this earlier work, with the incremental contribution that the direction of the distortion under the CPM depends not only on the relative tax rates, but also on the relative magnitudes of the expected cumulative accounting rate of return and the present value of the depreciation tax benefit. This dependence arises because of the link between the CPM transfer price and the firm's accounting rate of return (and therefore the accounting depreciation policy). In the CUP analyses, the direction of the distortion depends only on the relative tax rates.

The Halperin and Srinidhi (1987, 1996) and Harris and Sansing (1998) papers suppress ex post discretion, implicitly assuming that the tax authority can enforce the regulatory transfer price. The Samuelson (1982) model does incorporate ex post shifting by allowing the firm to choose any price between its marginal cost of production and the price at which it sells the intermediate product domestically. The firm optimally chooses an extreme transfer price, as in my paper. Inferences from Samuelson's (1982) model are limited, however, by the fact that his model is inconsistent with current regulations. Moreover, his study does not consider the simultaneous interaction of ex ante and ex post shifting. That is, he does not analyze the effect of varying the width of the ex post range on the ex ante input distortions.

Kant (1988), on the other hand, focuses exclusively on ex post shifting. The firm can deviate from the arm's length price, but at some risk of detection and punishment by tax authorities. The firm does not always choose the most extreme transfer price, a result absent in my model. (13) By varying the probability of detection for a given deviation from the arm's length price, Kant (1988) varies ex post discretion. Changes in ex post discretion affect the firm's input choices. The Kant model does not allow ex ante income shifting, however, because the arm's length price is exogenous (independent of the firm's inputs). Because the firm cannot affect the arm's length transfer price, it produces efficiently even in the presence of tax rate differentials if ex post shifting is not available. This is in contrast to my Propositions 1, 3, and 5, which show that tax rate differentials induce input distortions under current regulations even in the absence of ex post discretion.

Propositions 2, 4, and 6, which simultaneously consider the effects of ex ante and ex post discretion in an institutionally valid context, represent the principal contribution of my study. The main insight from considering the feedback between the two is that permitting firms ex post discretion in selecting transfer prices may result in more efficient investment. This result is potentially important to the frequent debates about transfer pricing policy. Given my focus on the efficiency implications of transfer pricing policy, I discuss the welfare effects only informally. In the most plausible setting, increasing ex post discretion would have an ambiguous effect on welfare, but there is at least the theoretical possibility of unambiguous effects. This possibility exists because ex post shifting may relieve ex ante inefficiencies. (14)

VI. CONCLUSION

This paper examines the interaction between two types of income shifting in multinationals. Transfer pricing regulations link the firm's allocation of income across jurisdictions to the firm's input decisions. By distorting the inputs, the firm can generate a more favorable "true" transfer price for taxation purposes. I label this ex ante income shifting. Imperfect enforcement of transfer pricing regulations means the firm can choose a transfer price that deviates from the true arm's length price. I label this ex post income shifting. In other words, ex post discretion implies a range of future defensible transfer prices; ex ante inputs determine the central tendency of the range. Previous research considers either one of the two modes of income shifting in isolation, but not the interaction between the two.

I examine both the comparable profit method (CPM) and the comparable uncontrolled price (CUP) method of calculating an arm's length transfer price. The study's main result is that ex post shifting may mitigate ex ante investment distortions. If the firm can deviate more from the true transfer price (higher ex post discretion), then it may distort the original inputs less. Ex post income shifting replaces ex ante shifting, and the reduction in ex ante shifting has efficiency implications. Depending on the transfer pricing method used and the relative tax rates, ex post shifting either exacerbates or alleviates ex ante investment distortions. A complete analysis of ex post income shifting must consider its effect on the ex ante input distortions. Although the improved efficiency can occur under either the CUP method or the CPM, the CUP method yields the sharpest results. In the CPM, the direction of the investment distortion depends on the discount rate and depreciation policy, both of which are idiosyncratic to the firm. In the CUP method, the direction of distortion depends only on tax-rate differentials.

Although I focus on the efficiency effects of the regulations, the results suggest some distributional implications. In particular, Propositions 2, 4, and 6 imply that tax revenues in both jurisdictions could increase in the level of ex post discretion if the increased pretax profits resulting from more efficient investment decisions more than offset the higher tax-rate jurisdiction's lower share of those profits. That is, a smaller piece of a bigger pie might lead to higher tax collections. A full analysis of the distributional effects of ex post discretion would require modeling of the objective function and actions of tax authorities, which is beyond the scope of this study. Efficiency is not the only goal of regulators. While higher ex post discretion may improve efficiency, the improved efficiency may be less important to regulators than maintaining the income redistribution that the transfer pricing regulations were presumably intended to create. Only a full equilibrium analysis can derive the optimal amount of ex post discretion.

The study is also a partial equilibrium analysis in that instead of explicitly modeling the economic conditions under which both independent and vertically integrated firms exist, I impose the coexistence exogenously. This coexistence is important. If vertically integrated firms dominate independent firms, then there are no uncontrolled transactions from which to infer arm's length prices. If independent firms dominate vertically integrated firms, then transfer prices do not exist. I avoid explicitly modeling coexistence by taking as given the comparable rate of return in the CPM and the external opportunities for trade in the CUP method, and by suppressing agency considerations.

Although only a partial equilibrium analysis, this study helps to rationalize the perceived lax enforcement, or unenforceability, of transfer pricing regulations. When determining the specificity of regulations and the level of resources to allocate to enforcement, tax authorities likely trade off tax revenues, productive efficiency, and auditing and political costs. Although ex post shifting may conflict with the other goals, it may improve ex ante productive efficiency. This result could explain a relatively lax regulatory regime.

APPENDIX A

NOTATION

TABLE 1
Notation

Term                    Definition

p                       price of the finished good
[k.sub.u]               upstream capital investment
[k.sub.d]               downstream capital investment
q                       quantity produced
N                       economic life of asset
[v.sub.j]([k.sub.j])    variable cost function for division j
[[gamma].sub.n]         economic depreciation factor in period n
[[delta].sub.n]         accounting depreciation factor in period n
[[lambda].sub.n]        book value of asset at beginning of period
                          n as a percentage of historical cost
[T.sub.n]               transfer price in period n
[t.sub.u]               selling division (upstream) tax rate
[t.sub.d]               buying division (downstream) tax rate
[[tau].sub.j]           1 - [t.sub.j], where j = u, d (upstream or
                          downstream division)
i                       the firm's discount rate
[D.sub.n]               discount factor in period n
[R.sub.n]               accounting rate of return for comparable
                          firm in period n
[R.sup.*]               cumulative accounting rate of return
                          ([[SIGMA].sub.n=1.sup.N] [D.sub.n]
                          [[lambda].sub.n][R.sub.n]
[DELTA]                 factor related to depreciation expense
                          (1 - [[SIGMA].sup.N.sub.n=1] [D.sub.n]
                          [[delta].sub.n]
[phi]                   ex post discretion in setting transfer
                          prices
[PHI]                   maximum absolute value of ex post
                          discretion
[mu]                    return multiplier on ex post discretion
                          (CPM)
[X.sub.d]               external purchase price for downstream
                          division (CUP)
[X.sub.u]               external sale price for upstream division
                          (CUP)

APPENDIX B

PROOFS

Proof of Benchmark Case

Let [NPV.sub.uu] be the second derivative of the objective function with respect to [k.sub.u] and let [NPV.sub.dd] be the second derivative of the objective function with respect to [k.sub.d]. Because the transfer price is independent of [k.sub.u] and [k.sub.d], [NPV.sub.uu] = -[[tau].sub.u] [v".sub.u] ([k.sub.u]) [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] < 0 and [NPV.sub.dd] = - [[tau].sub.d] [v".sub.d] ([k.sub.d]) [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] < 0. [NPV.sub.ud] = [NPV.sub.du] = 0. Therefore, the objective function is concave and the first-order conditions characterize the maximizing values of [k.sub.u] and [k.sub.d]. The expressions in the paper are implied by the first-order conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Lemma 1

The derivative of the present value of the transfer payments with respect to [k.sub.u] evaluated at [k.sub.u] = [k.sup.*.sub.u] (the substitution in line 2) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the derivative is positive (negative) if [[tau].sub.u] [R.sup.*] > [DELTA] ([[tau].sub.u] [R.sup.*] < [DELTA]), where [DELTA] = 1 - [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[delta].sub.n].

Proof of Lemma 2

I will prove the claim that [[tau].sub.u] [R.sup.*] = [DELTA] if [[tau].sub.u] R = i by mathematical induction. The following table recalls the definitions of the variables in the paper.

n         [D.sub.n]      [[delta].sub.n]    [[lambda].sub.n]

1     [(1 + i).sup.-1]   [[delta].sub.1]    1

2     [(1 + i).sup.-2]   [[delta].sub.2]    1 - [[delta].sub.1]

3     [(1 + i).sup.-3]   [[delta].sub.3]    1 - [[delta].sub.1]
                                            -[[delta].sub.2]

n     [(1 + i).sup.-n]   [[delta].sub.n]    [[delta].sub.n]

n      [D.sub.n][[delta].sub.n]R   [D.sub.n][[delta].sub.n]

1      R[(1 + i).sup.-1]          [(1 + i).sup.-1][[delta].sub.1]

2      R[(1 + i).sup.-2]          [(1 + i).sup.-2][[delta].sub.2]
       (1 - [[delta].sub.1]

3      R[(1 + i).sup.-3]          [(1 + i).sup.-3][[delta].sub.3]
       (1 - [[delta].sub.1]
                  - [[delta].sub.2]

n      R[(1 + i).sup.-n]          [(1 + i).sup.-n][[delta].sub.n]
       [[delta].sub.n]

Because the [D.sub.n]'s must sum to 1, [[delta].sub.n] = 1 - [[delta].sub.1] - [[delta].sub.2] - ... - [[delta].sub.n-1]. I will use this substitution to obtain Equations (B.5) and (B.6) below. For arbitrary n, then, [R.sup.*] = [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[lambda.sub.n] R is [R/(1 + i)] + [R(1 - [[delta].sub.1])/(1 + i)[sup.2]] + ... + [R(1 - [[delta].sub.1] -[[delta].sub.2] - ... - [[delta].sub.n-2])/(1 + i)[sup.n-1]] + [(R [[delta].sub.n]/(1 + i)[sup.n])], which is equivalent to:

(B.1) R (1 + i)[sup.n-1] + (1 + i)[sup.n-2] (1 - [[delta].sub.1]) + ... + (1 + i) (1 - [[delta].sub.1] - [[delta].sub.2] - ... - [[delta].sub.n-2]) + [[delta].sub.n]/ (1 + i)[sup.n].

For arbitrary n, [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[delta].sub.n] is [[delta].sub.1]/(1 + i)] + [[delta].sub.2]/(1 + i)[sup.2] + ... + [[delta].sub.n]/(1 + i)[sup.n]] and (1/[[tau].sub.u]) [DELTA] = (1/[[tau].sub.u]) (1 - [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[delta].sub.n]) is:

(B.2) 1/[[tau].sub.u] (1 + i)[sup.n] - (1 + i)[sup.n-1] [[delta].sub.1] - (1 + i)[sup.n-2] [[delta].sub.2] -...- (1 + i) [[delta].sub.n-1] - [[delta][sub.n]/(1 + i)[sup.n]

I want to show that Equation (B.1) equals (B.2) if R = i/[[tau].sub.u]. After substituting R = i/[[tau].sub.u] and factoring out 1/((1 + i)[sup.n] [[tau].sub.u]), one obtains the following expressions for Equations (B.1) and (B.2):

(B.3) i[(1 + i)[sup.n-1] + (1 + i)[sup.n-2] (1 - [[delta].sub.1]) + ... + (1 + i) (1 - [[delta].sub.1] - [[delta].sub.2] - ... - [[delta].sub.n-2]) + [[delta].sub.n]],

and

(B.4) (1 + i)[sup.n] - (1 + i)[sup.n-1] ([[delta].sub.1] - ... - (1 + i) [[delta].sub.n-1] - [[delta].sub.n].

As the basis for the mathematical induction, I assume that the difference between Equations (B.3) and (B.4) is 0 for the n case. The difference is a first-order polynomial in [[delta].sub.1], [[delta].sub.2], ..., [[delta].sub.n-1]. For the difference to be 0 for any possible set of [[delta].sub.n]'s the coefficient on each [[delta].sub.n] and on the intercept must be 0. After making the proper substitutions for [[delta].sub.n], the coefficients on the intercept and [[delta].sub.1], respectively, are:

(B.5) i(1 + i)[sup n-1] + i(1 + i)[sup.n-2] + ... + i(1 + i) + i - [(1 + i)[sup.n] - 1] = 0

(B.6) -i(1 + i)[sup.n-2] - i(1 + i)[sup.n-3] - ... - i(1 + i) - i + [(1 + i)[sup.n-1] - 1] = 0

Now consider the case of n + 1. The coefficients on [[delta].sub.1] and [[delta].sub.2] are, respectively:

(B.7) -i(1 + i)[sup.n-1] - i(1 + i)[sup.n-2] - ... - i(1 + i) - i + [(1 + i)[sup.n] - 1] = 0

(B.8) -i(1 + i)[sup.n-2] - i(1 + i)[sup.n-3] - ... - i(1 + i) - i + [(1 + i)[sup.n-1] - 1] = 0

Note that Equation (B.8) is equivalent to Equation (B.6), which by assumption is equal to 0. Similarly, the coefficients on [[delta].sub.3] through [[delta].sub.n] in the n + 1 case are equivalent to the coefficients on [[delta].sub.2] through [[delta].sub.n-1] in the n case, and are therefore also 0. Therefore, I need only show that the coefficients on [[delta].sub.1] and the intercept are 0 for the n + 1 case. First, I show this for [[delta].sub.1]. Rearrange Equation (B.6), the coefficient on [[delta].sub.1] in the n case as:

(1 + i) [(1 + i)[sup.n-2] - i(1 + i)[sup.n-3] - i(1 + i)[sup.n-4] - ... - i] - 1 - i.

By the induction assumption, this equals 0. Therefore, the term in square brackets must be 1. Now rearrange the coefficient on [[delta].sub.1] for the n + 1 case (Equation (B.7)) as:

(1 + i) {(1 + i)[(1 + i)[sup.n-2] - i(1 + i)[sup.n-3] - i(1 + i)[sup.n-4] - ... - i] - i} - 1 - i.

The previous argument establishes that the term in square brackets is 1. Therefore, the term in braces is also 1, and the entire term is 0. I omit the proof, which is similar, that the coefficient on the intercept is also 0 in the n + 1 case. Therefore, if the claims holds for n, it also holds for n + 1.

Now, for n = 1. For n = 1, the derivative is equal to (1/(1 + i))R - (1/[[tau].sub.u])[1 - 1/(1 + i)]. This clearly equals 0 if R = i/[[tau].sub.u]. Therefore, by mathematical induction, the claim is true.

Proof of Proposition I

Let the objective in Equation (6) be II. [II.sub.uu] = - [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [[tau].sub.n] [v".sub.u] ([k.sub.u]) < 0, [II.sub.dd] = - [[SIGMA[sup.N.sub.n-1] [D.sub.n] [[gamma].sub.n] [v".sub.d] ([k.sub.d]) < 0, and [II.sub.du] = [II.sub.ud] = 0. Therefore, the function is concave. The first derivative with respect to [k.sub.u] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At [k.sup.*.sub.u], the first term in large brackets is 0. The sign of the derivative therefore depends on both the relative tax rates and the sign of the second term in large brackets (the distortion term). As Lemma 1 establishes, the last term in large brackets is positive at [k.sup.*.sub.u] if [[tau].sub.u] [R.sup.*] > [DELTA], negative if [[tau].sub.u] [R.sup.*] < [DELTA]. Therefore, the distortion term is positive if [t.sub.d] > [t.sub.u] and[[tau].sub.u] [R.sup.*] > [DELTA] or [t.sub.d] < [t.sub.u] and [[tau].sub.u][R.sup.*] < [DELTA]. If the first-order condition of a function concave in [k.sub.u] is greater than 0 at [k.sup.*.sub.u], then the optimal value of [k.sub.u] must be greater than [k.sup.*.sub.u]. Therefore, the firm overinvests in [k.sub.u] relative to the benchmark.

If [t.sub.d] > [t.sub.u] and [[tau].sub.u][R.sup.*] < [DELTA] or [t.sub.d] < [t.sub.u] and [[tau].sub.u][R.sup.u] > [DELTA], then the second term is negative.

Hence, the optimal [k.sub.u] is less than [k.sup.*.sub.u]

The first-order condition for [k.sub.d] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfied at [k.sub.d] = [k.sup.*.sub.d]. Therefore, the regulation induces no distortion in [k.sub.d].

Proof of Proposition 2

The introduction of ex post discretion has no effect on the second-order conditions. Therefore, the function is still concave in [k.sub.u] and [k.sub.d]. If [t.sub.d] > [t.sub.u], then the distortion term in the first derivative is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [t.sub.u] > [t.sub.d], then it is instead:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The derivative becomes either more positive or less negative as [PHI] increases. Therefore, if the regulation induces overinvestment in [k.sub.u], then the overinvestment is increasing in [PHI], and if the regulation induces underinvestment, then the underinvestment is decreasing in [PHI].

Proof of Lemma 3

The derivative of the present value of the transfer payments with respect to [k.sub.d] evaluated at [k.sub.d] = [k.sup.*.sub.d] (the substitution in line 2) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the derivative is positive (negative) if [[tau].sub.d] [R.sup.*] < [DELTA] ([[tau].sub.d] [R.sup.*] > [DELTA]).

The proof of the special case is symmetric to the proof of Lemma 2 and is omitted.

Proof of Proposition 3

Now the objective is Equation (11). [II.sub.uu] = - [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [[tau].sub.d] [v".sub.u] ([k.sub.u]) < 0, [II.sub.ud] = - [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [[tau].sub.u] [v".sub.d] ([k.sub.d]) < 0, and [II.sub.ud] = [II.sub.du] = 0. Therefore, the function is concave. The first derivative with respect to [k.sub.d] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At [k.sup.*.sub.d], the first term in the large brackets is 0. The sign of the distortion term (the second term) depends on both the relative tax rates and the sign of the last term in large brackets. As Lemma 3 establishes, the last term is negative at [k.sup.*.sub.d] if [[tau.sub.d] [R.sup.*] > [DELTA], positive if [[tau].sub.d] [R.sup.*] < [DELTA]. Therefore, the distortion term is positive if [t.sub.d] > [t.sub.u] and [[tau].sub.d] [R.sup.*] < [DELTA] or [t.sub.d] < [t.sub.u] and [[tau].sub.d][R.sup.*] > [DELTA]. If the first-order condition of a function concave in [k.sub.d] is greater than 0 at [k.sup.*.sub.d], then the optimal value of [k.sub.d] must be greater than [k.sup.*.sub.d]. Therefore, the firm overinvests in [k.sub.d] relative to the benchmark.

If [t.sub.d] > [t.sub.u] and [[tau].sub.d][R.sup.*] > [DELTA] or [t.sub.d] < [t.sub.u] and [[tau].sub.d] [R.sup.*] < [DELTA], then the second term is negative. Hence, the optimal [k.sub.d] is less than [k.sup.*.sub.d].

The first-order condition for [k.sub.u] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

satisfied at [k.sub.u] = [k.sup.*.sub.u]. Therefore, the regulation induces no distortion in [k.sub.u].

Proof of Proposition 4

The introduction of ex post discretion has no effect on the second-order conditions. Therefore, the function is still concave in [k.sub.u] and [k.sub.d]. If [t.sub.d] > [t.sub.u], then the distortion term in the first derivative is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [t.sub.u] > [t.sub.d], then it is instead:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The derivative becomes either more positive or less negative as [PHI] increases. Therefore, if the regulation induces overinvestment in [k.sub.d], then the overinvestment is increasing in [PHI], and if the regulation induces underinvestment, then the underinvestment is decreasing in [PHI].

Proof of Proposition 5

The objective function is Equation (17). [II.sub.uu] = - 1/2 [[SIGMA].sub.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [v".sub.n] ([k.sub.u]) ([tau].sub.d] + [[tau].sub.u]) < 0, [II.sub.dd] = - 1/2 [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [v".sub.d] ([k.sub.d] ([[tau].sub.d] + [[tau].sub.u]) < 0, and [II.sub.ud] = 0. Therefore the function is concave. The derivative with respect to [k.sub.u] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At [k.sup.*.sub.u], the term in brackets is 0. If [t.sub.u] > [t.sub.d], then the second term is positive. If the first-order condition of a function concave in [k.sub.u] is greater than 0 at [k.sup.*.sub.u], then the optimal value of [k.sub.u] must be greater than [k.sup.*.sub.u]. Therefore, the firm overinvests in [k.sub.u] relative to the benchmark. If [t.sub.d] > [t.sub.u], then the second term is negative. Hence, the optimal [k.sub.u] is less then [k.sup.*.sub.u].

The derivative of the objective with respect to [k.sub.d] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

At [k.sup.*.sub.d], the term in brackets is 0. If [t.sub.d] > [t.sub.u], then the second term is positive. If the first-order condition of a function concave in [k.sub.d] is greater than 0 at [k.sup.*.sub.d], then the optimal value of [k.sub.d] must be greater than [k.sup.*.sub.d]. Therefore, the firm overinvests in [k.sub.d] relative to the benchmark. If [t.sub.u] > [t.sub.d], then the second term is negative. Hence, the optimal [k.sub.d] is less than [k.sup.*.sub.d].

Proof of Proposition 6

With the addition of ex post income shifting, [II.sub.uu] = - 1/2 [[SIGMA].sub.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [v".sub.n] ([k.sub.u]) [1 + [phi]) [[tau].sub.d] + (1 - [phi]) [[tau].sub.u]] and [II.sub.dd] = - 1/2 [[SIGMA].sup.N.sub.n=1] [D.sub.n] [[gamma].sub.n] [v".sub.d] ([k.sub.d]) [(1 + [phi]) [[tau].sub.u] + (1 - [phi] [[tau].sub.d]]. These are both negative as long as [phi] < 1. Therefore the function is still concave. Revisiting the derivatives with respect to [k.sub.u] and [k.sub.d]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If [t.sub.d] > [t.sub.u], [phi] = [PHI], then the distortion term for 1% becomes more negative as [PHI] increases, and the distortion term for [k.sub.d] becomes more positive as [PHI] increases. Therefore, the underinvestment in [k.sub.u] and overinvestment in [k.sub.d] are both more severe. If [t.sub.d] < [t.sub.u], [phi] = [PHI], the distortion term for [k.sub.u] becomes less positive as [PHI] increases, and the distortion term for [k.sub.d] becomes less negative as [PHI] increases. Therefore, both the overinvestment in [k.sub.u] and the underinvestment in [k.sub.d] are less severe.

This paper is based on an essay from my dissertation at Stanford University. I thank the members of my committee, David Baron, George Foster, and Richard Lambert (chair). I also appreciate the comments and suggestions of Jack Hughes, Richard Sansing (associate editor), and two anonymous referees. I gratefully acknowledge the financial support of the Deloitte & Touche Foundation.

Submitted April 2000

Accepted July 2001

(1) I define ex ante and ex post relative to the realization of cost and revenue.

(2) Harris and Sansing (1998) argue that if the decisions of the vertically integrated firm affect the independent transactions, the transactions are not appropriate arm's length benchmarks anyway.

(3) Since [k.sub.u] and [k.sub.d] do not affect p, all of the results hold with stochastic p.

(4) Survey evidence by Graham and Harvey (2001) indicates that firms are as likely to use NPV methods as rate of return methods such as IRR in their capital budgeting decisions. Absent compelling evidence to select one objective function over the other, I focus on NPV.

(5) The derivative of the first-order conditions with respect to [t.sub.j] is:

-([[SIGMA].sup.N.sub.n=1] [D.sub.n][[gamma].sub.n])(1 - [[SIGMA].sup.N.sub.n=1] [D.sub.n][[delta].sub.n])/ [([[SIGMA].sup.N.sub.n=1] [D.sub.n][[gamma].sub.n][[tau].sub.j]).sup.2] < 0.

(6) Independently organized firms are more susceptible to the hold-up problem in relationship-specific investment, but vertically integrated entities may have higher agency costs because of the potential separation of ownership from control. If the respective costs offset each other, then both types of organization can exist.

(7) Consider a firm that buys the same asset every period. In the steady state in which the firm holds assets in every stage of useful life, the aggregate depreciation charge is the same regardless of the speed of depreciation. However, the aggregate net book value is higher if the depreciation is slower.

(8) If the variable cost depended on both investments, then the extension would be straightforward. Suppose ([differential] [v.sub.u](*)/ [differential] [k.sub.d]) < 0. Then, decreasing [k.sub.d] would increase the upstream variable cost and hence the transfer price. Therefore, the firm would underinvest (overinvest) in [k.sub.d] if [t.sub.u] < [t.sub.d] ([t.sub.u] > [t.sub.d]).

(9) Treasury Reg. [section] 1.482-1[e][2][iii](A). More precisely, the entire range is acceptable if the comparable data are reliable. If not, the acceptable range is the interquartile range. (Treasury Reg. [section] 1.482-1 [e][2] [iii](B)).

(10) Clearly, [PHI] must be small relative to [R.sub.n]. I also assume that the maximum discretion is not so large that the firm can choose a transfer price producing losses in either division.

(11) Suppose that instead of knowing with certainty the limit of ex post discretion, the firm knows only the expected penalty associated with deviating from the true arm's length price. Let this penalty function be P([phi], [eta]) with [P.sub.[phi]], [P.sub.[phi][phi]], [P.sub.[phi][eta]] all positive. The parameter [eta] captures the severity of the regulatory regime. Increasing [eta] shifts [P.sub.[phi]] higher for all [phi], corresponding to a regime with lower ex post discretion. The reduced-form modeling is consistent with Kant (1988). In this case the firm would minimize the sum of taxes and expected penalties of:

[t.sub.d] (p - [T.sub.n] - [v.sub.d] ([k.sub.d])) + [t.sub.u] ([T.sub.n] - [v.sub.u] ([k.sub.u])) + P([phi], [eta])

and the optimal amount of discretion would be determined implicitly by the first-order condition:

[k.sub.u] ([t.sub.u] - [t.sub.u]) = [P.sub.[phi]].

Increasing [eta] is equivalent to decreasing [PHI] in the model. As [P.sub.[phi]] increases, the firm sets an ex post transfer price closer to the true price. The optimal [phi] would feed back into the firm's time 0 decision in the same manner. The richer model of ex post discretion, therefore, does not lead to different insights into the basic interaction between ex ante and ex post discretion, the primary focus of this paper.

(12) Because [x.sub.d] and [x.sub.u] are independent of [k.sub.u] and [k.sub.d], the results would hold even if they were stochastic.

(13) As I discuss in footnote 11, I could obtain this result by modeling enforcement less starkly.

(14) The absence of feedback between ex ante and ex post shifting explains the ambiguous welfare results in Kant (1988). In his paper, the improvement in welfare of one jurisdiction always comes at the expense of the other.

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Michael J. Smith
Duke University