Electronic Companion to Tax-Effective Supply Chain Decisions under China s Export-Oriented Tax Policies Optimality Equations of EI Strategy In this part, we derive the optimality equations for the Export-Import EI) strategy. Specifically, we discuss the following cases: Case 1. ˆp g ˆp gc. In this case, selling a unit in the overseas market is more profitable than reimporting and selling it in the domestic market. We thus have three subcases as follows: Subcase 1a. ˆp g > ŝ g ˆp gc > ŝ gc. This is an extreme subcase in which, for an exported unit, salvaging it in the overseas market is more attractive than reimporting it back to the domestic market. This situation may occur when the cost of reimporting is very expensive, for instance, either b gc or t P is high.) Thus, regardless of the demand realization, all exported x units will be sold or salvaged in the overseas market. Clearly, in this subcase, P becomes two separate newsvendor problems with the following optimal solutions: = Fg 1 ˆp g ˆp g ŝ g ], y = Fc 1 ˆp c ˆp c ŝ c while the allocation of to the two markets is 1 = and 2 =. Similarly, the optimal solution to P EI1 is = Fg 1 ˆp g ˆp g ŝ g Subcase 1b. ˆp g ˆp gc > ŝ g ŝ gc. In this subcase, the exported x units are first used to satisfy the demand in the overseas market. Upon demand realization, a quantity of x 2 = x ξ g ξ c y is then reimported to satisfy the unmet demands in the domestic market. ]. ], Finally, any unsold exported products are salvaged in the overseas market. We can show that the optimal solutions, and y, to problem P satisfy ˆp g ˆp g ŝ g ) F g ˆp gc ŝ g ) ˆp c ˆp c ŝ c ) F c y ) ˆp gc ŝ g ) +y ξ g f c ξ c ) f g ξ g ) dξ c dξ g =, 1) +y ξ g y f c ξ c ) f g ξ g ) dξ c dξ g =. 11) The first term of 1), ˆp g ˆp g ŝ g ) F g ), can be viewed as the marginal profit for selling in the overseas market with respect to an increase in. In the second term, ˆp gc ŝ g ) represents the added net profit for reimporting and selling a unit in the domestic market instead of salvaging 1
that unit in the overseas market. Therefore, the second term of 1) represents the marginal profit generated by reimporting. The interpretation for 11) is similar. The allocation of to the two markets in problem P is 1 = ξ g ξc y, x 2 = ξ g ξc y. For problem P EI1, the optimal solution satisfies ˆp g ˆp g ŝ g ) F g x EI1 ) + ˆp gc ŝ g ) f c ξ c ) f g ξ g ) dξ c dξ g =. 12) ξ g The result can be viewed as setting y = in problem P.) The allocation of to the two markets is Subcase 1c. 1 = ξ g ξc, 2 = ξ g ξc. ˆp g ˆp gc > ŝ gc > ŝ g. This subcase is similar to the previous one, except that any surplus from the exported x units will be reimported and salvaged in the domestic market. Thus, the optimality equations are the same as those of the previous subcase 1)-11) and 12)), except that ŝ g is replaced by ŝ gc. The allocation of to the two markets in problem P is 1 = ξ g, 2 = ξ g. The allocation of in problem P EI1 is identical to the above. Case 2. ˆp gc > ˆp g. In this case, reimporting and selling a unit in the domestic market is more profitable than selling the unit in the overseas market. We have the following three subcases: Subcase 2a. ˆp gc > ŝ gc ˆp g > ŝ g. This represents an extreme situation in which selling and salvaging the product in the overseas market is not desirable. All of the products both x and y) will therefore be sold in the domestic market. The optimal and y for problem P can be determined by solving ˆp gc ˆp gc ŝ gc ) F c + y ) =, ˆp c ˆp c ŝ c ˆp gc + ŝ gc ) F c y ) ˆp gc ŝ gc ) F c + y ) =. The allocation of to the two markets is 1 = and 2 =. Problem P EI1 in this subcase becomes a newsvendor problem for the domestic market, and is a newsvendor solution of = Fc 1 2 ˆp gc ˆp gc ŝ gc ].
The allocation of to the two markets is the same as that in problem P. Subcase 2b. ˆp gc > ˆp g > ŝ gc ŝ g. In this subcase, the exported x units are first used to satisfy any unmet demand in the domestic market and then sold to meet the demand in the overseas market. Any unsold exported quantity will be salvaged in the domestic market. optimal and y to problem P satisfy We can show that the ˆp gc ˆp gc ˆp g ) F c + y ) +y ξ g ˆp g ŝ gc ) f c ξ c ) f g ξ g ) dξ c dξ g =, 13) ˆp c ˆp c ŝ c )F c y ) ˆp gc ˆp g ) F c + y ) F c y )) ˆp g ŝ gc ) +y ξ g y f c ξ c ) f g ξ g ) dξ c dξ g =. 14) The allocation of to the two markets is 1 = ξ c y ) ] + + ξg, 2 = ξ c y ) ] + + ξg. In problem P EI1, the first-order condition indicates that is the solution to ˆp gc ˆp gc ˆp g ) F c x EI1 ) ξ g ˆp g ŝ gc ) f c ξ c ) f g ξ g ) dξ c dξ g =. 15) The allocation of to the two markets is given by Subcase 2c. 1 = ξ c ξg, 2 = ξ c ξg. ˆp gc > ˆp g > ŝ g > ŝ gc. This subcase is similar to the previous one, except that any exported but unsold products will be salvaged in the overseas market instead of being reimported back to the domestic market. The optimality equations of P and P EI1 are therefore identical to those in the previous subcase 13)-14) and 15), respectively), except that ŝ gc is replaced by ŝ g. The allocations of and to the two markets are for problem P and 1 = ξ c y ) +, x 2 = ξ c y, for problem P EI1. 1 = ξ c, 2 = ξ c, 3
Justification of Assumption of Proposition 7 We remark that the assumption of Proposition 7, i.e., ˆp c ŝ c ˆp gc maxŝ gc, ŝ g ), 16) though cumbersome, is not overly restrictive. To see this, recall that ˆp c and ŝ c are the net profit and net salvage value, respectively, for selling/salvaging P in the domestic market directly; ˆp gc is the net profit for selling P in the domestic market through an export and reimport operation; and ŝ g is the net value for salvaging P in the overseas market. Since an exported unit can either be salvaged in the overseas market or be reimported and savaged in the domestic market, maxŝ gc, ŝ g ) represents the net salvage value for a unit of exported P. Thus, the left-hand side of 16) represents the net profit gained by the firm who is able to sell an extra unit of P in the domestic market from the production amount y initially planned for the domestic market. Similarly, the right-hand side of 16) represents the net profit from selling an extra unit of P in the domestic market from the initial amount x to be exported through export and reimport operation). Note that if P is reimported and eventually consumed in the domestic market, certain VAT and tariffs will be levied. Thus ˆp gc and ŝ gc are typically much smaller than ˆp c and ŝ c, respectively see 3)). Thus, it is straightforward that condition 16) would easily hold in this situation. On the other hand, if P is reimported as a bonded component and thus the selling and salvage prices for the reimported units satisfy 5), it is easy to show that 16) holds in this case as well. Proofs of Propositions We present below the proofs of the propositions in the paper. Proof of Proposition 1: i) Consider a buyer of product P in the China market who will use P to assemble another product. Let u be the unit production cost and q be the unit revenue at the buyer side. Besides, let h P c and h P g be the unit inventory holding costs of un-bonded P and bonded P, respectively. Assume the buyer sells z units in total. With Option 1, the buyer purchases P from the domestic market at p c, and thus his profit is q + v q u h P c )z p c + v p c )z v qz + v p c z = q u h P c )z p c z, where the first term on the left-hand side is the gross sales, the second term is the purchasing cost of P, and the last two terms are the output and input VAT. With Option 2, the buyer imports P 4
at price p gc and thus has a profit of q + v q u h P c )z 1 + t P + v 1 + t P ))p gc z v qz + v 1 + t P )p gc z =q u h P c )z 1 + t P )p gc z, where the second term on the left-hand side is the total cost for importing P including tariffs and taxes) and the last two terms again represent the output and input VAT. Comparing the profits under two procurement options, we conclude that the buyer will be indifferent at 1 + t P )p gc = p c. Then, 3) follows. ii) Now consider the case in which the buyer of P uses it to assemble an exported product. With Option 1, P is domestically purchased at price p c. Note that the buyer pays no output VAT since the final product is exported. The profit under this option is thus q u h P c )z p c + v p c )z T 1, where T 1 = v p c z + qz )v v 1 ) is the tax payable. On the other hand, with Option 2, the buyer pays neither taxes nor tariffs on the imported P nor output VAT. Thus, the buyer s profit is q u h P c δ p )z p gc z T 2, where the tax payable T 2 of this option is T 2 = qz p gc z)v v 1 ) and as defined, δ p represents the added logistics and trading cost to the buyer in this option as opposed to Option 1. Comparing the two profits under Options 1 and 2, we see that the buyer would be indifferent if p c = 1 v + v 1 )p gc + δ p. Then, 4) follows. Proof of Proposition 2: By comparing 2) with 1), we know that when 1 v + v 1 )p gc p c + t C2 + v v 1 )w 2 b gc b gc and 1 v + v 1 )s gc s c + t C2 + v v 1 )w 2 b gc b gc, 17) the BI strategy is more attractive than the EI strategy. With the assumptions of t P t C2 and s c 1 + t C2 )w 2 and the fact that s c < p c, the left-hand sides of both inequalities in 17) are smaller than or equal to v + v 1 t P 1 + t P s c + t C2 + v v 1 )w 2 v + v 1 t C2 1 + t C2 s c + t C2 + v v 1 )w 2 = t C2 + v v 1 1 + t C2 1 + t C2 )w 2 s c ) b gc b gc. Thus, we conclude that the BI strategy is more attractive than the EI strategy. 5
Proof of Proposition 3: Suppose δ. In this case, we have 1 v + v 1 )p gc p c + t C2 + v v 1 )w 2 b gc b gc) = δ p + δ. Similarly, we can show 1 v + v 1 )s gc s c + t C2 + v v 1 )w 2 b gc b gc). By discussions in the proof of Proposition 2 i.e., 17) holds), we conclude that the BI strategy is more attractive than the EI strategy. Suppose now δ >. To show i), note that when δ p = δ and δ s = δ, we have ˆp gc = ˆp gc and ŝ gc = ŝ gc. Thus, problems defined by 2) and 1) are identical. Clearly, π EI δ p, δ s ) = π BI. To prove ii), first note that the expected profit π EI δ p, δ s ) is a non-increasing function of the costs δ p and δ s respectively. Thus, suppose π EI δp, 1 δs) 1 = π EI δp, 2 δs) 2 = π BI and δp 1 δp. 2 We have π EI δp, 2 δs) 2 = π EI δp, 1 δs) 1 π EI δp, 2 δs), 1 which implies δs 2 δs. 1 Therefore, ii) follows. Now, for any fixed δ p, δs), 1 say, in the LR sub-region. Suppose π EI δp, 1 δs) 1 = π BI. We have δ p < δp 1 and therefore π EI δ p, δs) 1 π EI δp, 1 δs) 1 = π BI. So iii) follows. Proof of Proposition 4: For ease of exposition, we rewrite the optimization problem P EI as where max Π x, y) 1x)J g 1y)J c h 2g x h 2c y], x,y Π x, y) = E max ˆpc + h 2c ) y ξ c + ŝ c + h 2c ) y ξ c + ˆp g + h 2g ) x 1 ξ g x 1 +x 2 =x + ŝ g + h 2g ) x 1 ξ g + ˆp gc + h 2g ) x 2 ξ c y ŝ gc + h 2g ) x 2 ξ c y ]. That is, Π represents the total expected profit that excludes fixed and variable costs of storing the imported component C2. As discussed, the optimal solution to P EI can be obtained by comparing the solutions to the following two optimization problems which correspond to the EI1 and strategies): P EI1 : P : max Π x, ) h 2gx] J g, x max Π x, y) h 2gx h 2c y] J g J c. x,y 6
Note that it is possible that the constraint of y becomes binding in the optimal solution to P. In that case, the optimal solution to P degenerates to an EI1 structure i.e., = and y = ). Below we present two claims. Claim 1: Let h 2c = Πx, y) y x=,y=. Then, if h 2c h 2c, the optimal solution to P is = and y = i.e., the same as that to P EI1 ). Otherwise, we have y > in the optimal solution to P. To prove the claim, we note for P the optimality condition with respect to y is given by Πx, y) y h 2c + µ y =, where µ y is the Lagrange multiplier associated with y. If µ y >, y = and thus the optimal solution to P is = and y =. Substituting the solution into the optimality condition stated above, we can obtain the result of Claim 1. Claim 2: When h 2c < h 2c, Π, y ) h 2g h 2c y is decreasing in h 2c. In the case of h 2c < h 2c, as shown in the previous claim, we have y >. Thus, the result of Claim 2 holds because d Π, y ) h 2g h 2c y ] dh 2c = Π, y ) h 2g h 2c y ] = y <, h 2c where the first equality comes from the Envelope Theorem. In short, Claims 1 and 2 indicate that, when excluding the fixed costs, the optimal profit obtained from the strategy i.e., Π, y ) h 2g h 2c y ) is decreasing in h 2c when h 2c < h 2c but becomes that of EI1 i.e., Π, ) h 2g ) and thus constant when h 2c h 2c. Now we take into account the fixed costs, that is, J g in the EI1 strategy and J g +J c in. It is straightforward that π = Π, y ) h 2g h 2c y J g J c and π EI1 = Π, ) h 2g J g intersect exactly once at h 2c = h 2c which is less than h 2c ) and that at the intersection, y >. Therefore, the proposition follows. Prior to proving Propositions 5 to 7, we present a useful lemma below. Lemma 1 + y. 7
Proof: We prove the lemma by discussing all the cases presented previously. For subcases 1a and 2a, the proof is rather straightforward. Now, consider subcase 1b. We first show. Note that equation 1) implies that ˆp g ˆp g ŝ g ) F g ˆp gc ŝ g ) f g ξ g ) f c ξ c ) dξ c dξ g. ξ g On the other hand, 12) can be rewritten as ˆp g ˆp g ŝ g ) F g ˆp gc ŝ g ) f g ξ g ) f c ξ c ) dξ c dξ g =. ξ g Due to the concavity of the expected profit of EI1, we obtain. We then show + y. We have ˆp g ˆp g ŝ g ) F g + y +y ˆp gc ŝ g ) +y ξ g f g ξ g ) f c ξ c ) dξ c dξ g +y = ˆp g ŝ g ) F g ) F g + y ) ] + ˆp gc ŝ g ) f g ξ g ) f c ξ c ) dξ c dξ g +y ξ g ˆp g ŝ g ) F g ) F g + y ) ] + ˆp gc ŝ g ) F g + y ) F g ) ] = ˆp g ˆp gc ) F g ) F g + y ) ], where the first equality comes from applying 1). Comparing the above with 12) and noting the concavity of the expected profit, we conclude that + y. We can prove the remaining subcases by following the same logic as used above. For conciseness, we omit the details. Proof of Proposition 5: We note that to analyze the sign of d h 2c dh 2g, it is sufficient to analyze that of π πei1 π due to 7). From the Envelope Theorem, we obtain = E 1) 1 ξ g + 1) 1 ξ g + 1) x 2 ξ c y ) + 1) 2 ξ c y ) ] + = E x1 + ] 2 =. Similarly, we can show Therefore, we conclude π π EI1 =. πei1 =, 8
where the inequality comes from Lemma 1. Therefore, we conclude d h 2c dh 2g Proof of Proposition 6: We can show. π EI1 π = E p g w 2 ) 1 ξ g ) 1 ξ g x ) EI1 + ) + s g w 2 ) 1 ξ g x + ) 1 ξ g + p gc w 2 ) 2 ξ c 2 ξ c y ) x ) EI1 + + s gc w 2 ) 2 ξ c x 2 ξ c y ) )] +. Therefore, in determining the sign of πei1 π, we can analyze the four terms inside the expectation in the above equation. The analysis can be based on the different subcases presented previously. For subcase 1a, we see that it is a trivial case in which πei1 π = due to = = 1 = 1 and 2 = 2 =. Next, we focus on subcase 1b. In this subcase, the optimal allocation of in the EI1 structure yields while the optimal allocation in gives 1 ξ g = ξ g, ) x EI1 + 1 ξ g = ξ g ξ g ξc, 2 ξ c = ξ g ξc, ) x EI1 + 2 ξ c =, Thus, we have 1 ξ g = ξ g, ) x + 1 ξ g = ξ g ξ g ξc y, 2 ξ c y = ξ g ξc y, x 2 ξ c y =. p g w 2 ) 1 ξ g ) 1 ξ g, x ) EI1 + s gc w 2 ) 2 ξ c x 2 ξ c y ) =, due to, as shown in the above lemma. On the other hand, note that ˆp gc > ŝ g, as 9
stated in the condition for this subcase, implies that p gc w 2 > s g w 2. Thus, we get x ) EI1 + ) s g w 2 ) 1 ξ g x + ) 1 ξ g + p gc w 2 ) 2 ξ c 2 ξ c y ) x = s g w 2 ) EI1 ξ g ξ g ξc ξ g + ξ g ξc y ) x + p gc w 2 ) EI1 ξ g ξc ξ g ξc y ) x s g w 2 ) EI1 ξ g ξ g ξc ξ g + ξ g ξc y ) x + s g w 2 ) EI1 ξ g ξc ξ g ξc y ) x = s g w 2 ) EI1 ξ g ) ξ g, where the inequalities come from p gc w 2 > s g w 2, s g w 2 and. Therefore, we conclude that πei1 π holds for this subcase. The proof of the remaining subcases follows the same approach as shown above. In summary, we conclude d h 2c dv 1. Proof of Proposition 7: We can show that the desired result holds for all the subcases presented previously. For conciseness, we focus on subcase 1b, in which the optimal objective of is y π = J g J c + ˆp c µ c + ŝ c y ξ c )f c ξ c ) dξ c ˆp c + ˆp g µ g + ŝ g ξ g )f g ξ g ) dξ g ˆp g + ˆp gc ŝ g ) + ˆp gc ŝ g ) Then, we can prove +y ξ g y y ξc y ) f c ξ c ) dξ c ξg ) f g ξ g ) dξ g ξc y ) f c ξ c ) f g ξ g ) dξ c dξ g +y ξ g ξ g ) fc ξ c ) f g ξ g ) dξ c dξ g. π = ˆp c ŝ c ) F c y +y ξ g ˆp gc ŝ g ) f c ξ c ) f g ξ g ) dξ c dξ g. µ c y Similarly, we can show π EI1 ξ g = ˆp gc ŝ g ) f c ξ c ) f g ξ g ) dξ c dξ g. µ c Thus, we obtain π µ c πei1 µ c with + y, as shown in the above lemma, and the assumption of ˆp c ŝ c ˆp gc max ŝ g, ŝ gc ) = ˆp gc ŝ g. Such approach for subcase 1b can be carried over to the proof of other subcases. We omit the details for conciseness. 1