Contingent trade policy and economic efficiency

Abstract

This paper models the competition for a domestic market between one domestic and one foreign firm as a pricing game under incomplete cost information. As the foreign firm incurs a trade cost to serve the domestic market, it prices more aggressively, giving rise to the possibility of an inefficient allocation. In spite of asymmetric information, we can devise a contingent trade policy to correct this potential market failure. National governments, however, make excessive use of such a policy due to rent shifting motives, thus creating another inefficiency. The expected inefficiency of national policy is found to be comparatively larger (lower) at low (high) trade costs. Hence contingent trade policy conducted by national governments is preferred only when trade costs are high.

Appendices

Appendix 1: Equilibrium pricing strategies without policy intervention

In case of entry, denote $\gamma , \gamma \in [0,1 - t]$ as the critical foreign type which is indifferent between entry and no entry. We will determine $\gamma$ below. Given that the domestic firm knows the size of $\epsilon$ and observes this investment, it will update its beliefs if it observes entry such that the foreign types which enter will be uniformly distributed between 0 and $\gamma$. Consequently, the expected profits of both firms are equal to

$$\begin{aligned} \pi _1(p_1; c_1)&= \left( 1 - \frac{\phi _2(p_1)}{\gamma }\right) (p_1 - c_1), \\ \pi _2(p_2; c_2)&= (1 - \phi _1(p_2))(p_2 - c_2-t). \end{aligned}$$

(A.1)

First, let us establish that both firms will employ a price strategy such that the optimal price functions have a common upper and lower bound for those prices by which each firm is able to win demand. Let the lower (upper) bound be denoted by ${\underline{p}} ({\overline{p}})$. If $p_i = {\underline{p}}$, firm i will win with certainty, so there is no reason to undercut this price. This confirms the common lower price bound, and hence $\phi _1(0) = \phi _2(0) = {\underline{p}}$. Suppose that the first-order conditions (2) are fulfilled for all $p_i \in [{\underline{p}}, {\overline{p}}]$. We will now establish that

$$\begin{aligned} {\overline{p}}&= {} \frac{1 + t + \gamma }{2}, \\ \phi _1({\overline{p}})&= {} \frac{1 + t + \gamma }{2}, \quad \phi _2({\overline{p}}) = \gamma \\ \phi _1(p_1)&= {} c_1, \forall p_1 \in [{\overline{p}}, 1] \end{aligned}$$

(A.2)

are part of the equilibrium pricing strategies. Note that (A.2) specifies that the domestic firm charges its cost for all prices above ${\overline{p}}$; in these cases, the domestic firm cannot win the market and will be beaten by the foreign firm with probability one. As we have assumed that the first-order conditions hold up to ${\overline{p}}$, we have to prove that no firm is better off by charging a higher price. As for the domestic firm, $\pi _1({\overline{p}}; \,{\overline{p}}) = 0$ because it will win with zero probability. A higher price leads also to zero profits as it does not change the zero win probability; hence, the domestic firm has no incentive to deviate from this strategy. The foreign firm is supposed to charge ${\overline{p}}$ for $c_2 = \gamma$. Given that the domestic firm charges its cost for all prices above ${\overline{p}}$, the foreign firm profit is equal to

$$\begin{aligned} \pi _2({\overline{p}};\, \gamma ) = (1 - {\overline{p}})\,({\overline{p}} - \gamma - t) = \frac{(1 - t - \gamma )^2}{4} \end{aligned}$$

(A.3)

if it follows the prescribed strategy and

$$\begin{aligned} \pi _2(p_2 > {\overline{p}};\, \gamma ) = (1 - p_2)\,(p_2 - \gamma - t) \end{aligned}$$

if it charges a higher price. Maximizing $\pi _2(p_2 > {\overline{p}};\, \gamma )$ over $p_2$ leads to an optimal $p_2 = {\overline{p}}$, and hence also the foreign firm has no incentive to deviate.

For all $p_1, p_2 \in [{\underline{p}},\, {\overline{p}}]$, the first-order conditions for (A.1) are

$$\begin{aligned} \gamma - \phi _2(p_1) - \phi _2^{\prime }(p_1) (p_1 - c_1) = 0, \\ 1 - \phi _1(p_2) - \phi _1^{\prime }(p_2) (p_2 - c_2 - t) = 0. \end{aligned}$$

Note that each first-order condition depends on both inverse price functions. We now follow a solution concept similar to Krishna (2002) as to determine the boundary conditions and to simplify the differential equations. In equilibrium, $c_i = \phi _i(p_i)$, and using p as the argument in the inverse price functions allows us to rewrite the first-order condition as

$$\begin{aligned} (\phi _2^{\prime }(p) - 1)(p - \phi _1(p))&= \gamma - \phi _2(p) - p + \phi _1(p),\\ (\phi _1^{\prime }(p) - 1)(p - \phi _2(p) - t)&= 1 - \phi _1(p) - p + \phi _2(p) + t. \end{aligned}$$

Adding up yields

$$\begin{aligned} \frac{-d}{dp}(p-\phi _1(p))(p-\phi _2(p)-t) = 1 + t + \gamma - 2p, \end{aligned}$$

(A.4)

and integration implies

$$\begin{aligned} (p - \phi _1(p))(p - \phi _2(p) - t) = p^2 -(1 + t + \gamma )p + K, \end{aligned}$$

(A.5)

where K denotes the integration constant. We can determine K by using the upper boundary condition. For $p = {\overline{p}}$, the LHS of (A.5) is zero and we find that

$$\begin{aligned} K = \frac{(1 + t + \gamma )^2}{4}, \end{aligned}$$

so that (A.5) reads

$$\begin{aligned} (p - \phi _1(p))(p - \phi _2(p) - t) = p^2 -(1 + t + \gamma )p + \frac{(1 + t + \gamma )^2}{4} \end{aligned}$$

(A.6)

in equilibrium. Furthermore, $\phi _1(0) = \phi _2(0) = {\underline{p}}$ so that

$$\begin{aligned} {\underline{p}}({\underline{p}} - t) = {\underline{p}}^2 -(1 + t + \gamma ){\underline{p}} + \frac{(1 + t + \gamma )^2}{4} \end{aligned}$$

which leads to

$$\begin{aligned} {\underline{p}} = \frac{(1 + t + \gamma )^2}{4(1 + \gamma )}. \end{aligned}$$

(A.7)

We can use (A.6) as to rewrite the first-order conditions such that each depends on a single inverse price function only:

$$\begin{aligned} \gamma - \phi _2(p)&= \phi _2^{\prime }(p)\dfrac{p^2 -(1 + t + \gamma )p + \frac{(1 + t + \gamma )^2}{4}}{p - \phi _2(p) - t} = 0, \\ 1 - \phi _1(p)&= \phi _1^{\prime }(p) \dfrac{p^2 -(1 + t + \gamma )p + \frac{(1 + t + \gamma )^2}{4}}{p - \phi _1(p)} = 0. \end{aligned}$$

(A.8)

Equations (A.2), (A.7) and (A.8) completely describe the equilibrium behavior of both firms in terms of their inverse price functions.²¹ Hence, they represent the solution to stage II of our game, given that no intervention will occur. As for stage I, Eq. (A.3) allows us to determine the critical type $\gamma$ which will be indifferent between entry and no entry. This type’s expected profit must be equal to the investment $\epsilon$ such that

$$\begin{aligned} \gamma = 1 - t - 2\sqrt{\epsilon }. \end{aligned}$$

An interior solution requires that $2\sqrt{\epsilon } < 1 - t$. More importantly, as we deal with markets to which entry is easy, $\gamma \simeq 1 - t$ for a $\epsilon$ sufficiently close to zero. For $\gamma \simeq 1 - t$, (A.8) simplifies to

$$\begin{aligned} 1 - t - \phi _2(p)&= \phi _2^{\prime }(p)\dfrac{(1 - p)^2}{p - \phi _2(p) - t}, \\ 1 - \phi _1(p)&= \phi _1^{\prime }(p) \dfrac{(1 - p)^2}{p - \phi _1(p)}. \end{aligned}$$

(A.9)

Because prices must not fall short of overall costs, $\phi _1^{\prime }, \phi _2^{\prime } > 0$, and hence the solutions to (A.9) satisfy that the (inverse) price functions increase with the costs (prices). Solving these equations gives the inverse price functions

$$\begin{aligned} \phi _1(p)= 1 - \frac{2(1-p)}{1-(1-p)^2 K_1} \end{aligned}$$

(A.10)

$$\begin{aligned} \phi _2(p)= 1 - \frac{2(1-p)}{1-(1-p)^2 K_2} - t , \end{aligned}$$

(A.11)

where the $K_i$’s are the constants of integration. Note that the domestic firm’s price policy will no longer include a range of prices in which it will charge its cost (and win with zero probability) because

$$\begin{aligned} {\overline{p}} = 1 \text { and } {\underline{p}} = \frac{1}{2 - t} \end{aligned}$$

for $\gamma \simeq 1 - t$. Using the last condition, that is $\phi _1(0) = \phi _2(0) = 1/(2 - t)$, we find that

$$\begin{aligned} K_1 = \frac{t(2-t)}{(1-t)^2} \ge 0 \text { and } K_2 = - K_1 \le 0. \end{aligned}$$

Plugging $K_1$ and $K_2$ back into (A.10) and (A.11) and solving for p yields (3).

Appendix 2: Proof of Lemma 2

To determine the probability that an inefficient outcome occurs, contingent upon entry of the foreign firm, we define the borderline ${\tilde{c}}_2(c_1)$ between the inefficient and the efficient set of cost draws at which the resulting prices are equal. Setting $p_1$ and $p_2$ in (3) equal to each other gives

$$\begin{aligned} {\tilde{c}}_2(c_1)= 1 - {\frac{1 - {c_1}}{{\sqrt{{\dfrac{1 - \left( 2 - t \right) \,t\, \left( 2 - {c_1} \right) \,{c_1}}{{{\left( 1 - t \right) }^2}}}}}}} - t. \end{aligned}$$

(A.12)

The foreign firm prices more aggressively if ${\tilde{c}}_2(c_1) + t \le c_1$ which is equivalent to

$$\begin{aligned}&(1 - c_1)\left( 1 - {\frac{1 - {c_1}}{{\sqrt{{\dfrac{1 - \left( 2 - t \right) \,t\, \left( 2 - {c_1} \right) \,{c_1}}{{{\left( 1 - t \right) }^2}}}}}}}\right) \ge 0 \\&\qquad \Leftrightarrow \sqrt{{\dfrac{1 - \left( 2 - t \right) \,t\, \left( 2 - {c_1} \right) \,{c_1}}{{{\left( 1 - t \right) }^2}}}} \ge 1 \\&\qquad \Leftrightarrow 1- (2 - t)t(2 - c_1)c_1 \ge (1 - t)^2. \end{aligned}$$

(A.13)

Note that the LHS decreases with $c_1$ and is thus at least equal to $1 - 2t +t^2 = (1 - t)^2$ or larger which completes the proof for Lemma 2.

Appendix 3: General distribution and demand functions

We now show—adapting the proof of Proposition 4.4 in Krishna (2002)—that this result is robust when relaxing the uniform distributional assumption and allowing demand to be price elastic. Let x(p) be any downward sloping differentiable demand function with $x(1)=0$ (Fig. 4).

Fig. 4

Pricing functions

The expected profit functions of the domestic and foreign firm then take the following form:

$$\begin{aligned} \pi _1(p_1)&= (1-F_2(\phi _2(p_1))) (p_1-c_1) x(p_1), \\ \pi _2(p_2)&= (1-F_1(\phi _1(p_2))) (p_2-(c_2+t)) x(p_2); \end{aligned}$$

and the corresponding first-order conditions of profit maximization are:

$$\begin{aligned} \phi _2'(p_1)&= \frac{1-F_2(\phi _2(p_1))}{f_2(\phi _2(p_1))} \frac{x(p_1)+(p_1-c_1)x'(p_1)}{(p_1-c_1)x(p_1)}, \\ \phi _1'(p_2)&= \frac{1-F_1(\phi _1(p_2))}{f_1(\phi _1(p_2))} \frac{x(p_2)+(p_2-(c_2+t))x'(p_2)}{(p_2-(c_2+t))x(p_2)}. \end{aligned}$$

We want to establish that the foreign firm sets a lower price if it has the same (total) cost, i.e., $p_2(c)<p_1(c) \forall c \in [t,1]$. The proof proceeds by contradiction. Suppose there exists a common point; that is, for some ${\tilde{p}}\in ({\underline{p}},1)$$\phi _1({\tilde{p}})=\phi _2({\tilde{p}})+t=z$. Then the first-order conditions above imply:

$$\begin{aligned} \phi _2'({\tilde{p}})&= \frac{1-F_2(z-t)}{f_2(z-t)} \frac{x({\tilde{p}})+({\tilde{p}}-z)x'({\tilde{p}})}{({\tilde{p}}-z)x({\tilde{p}})}, \\ \phi _1'({\tilde{p}})&= \frac{1-F_1(z)}{f_1(z)} \frac{x({\tilde{p}})+({\tilde{p}}-z)x'({\tilde{p}})}{({\tilde{p}}-z)x({\tilde{p}})}, \\ \end{aligned}$$

where $F_2^{incl}$ is the foreign firm’s cost distribution defined in terms of total cost, i.e. $F_2^{incl}(c) \equiv F_2(c-t)$. Assume that $F_2^{incl}$ stochastically dominates $F_1=F$ in terms of hazard rate (not reverse hazard rate) dominance. A linearly decreasing density as implied by $F=2c-c^2$ is one example that gives rise to such dominance. Stochastic dominance together with the above derivatives of the inverse pricing functions implies that $p_1'(z)>p_2'(c)$ at any common point. This implies that there is at most one intersection. Therefore if $p_1(c)$ were less than $p_2(c)$ for some $c \in (t,1)$, then—no matter whether there is an intersection or not—this would imply that $p_2(c)>p_1(c)$ at $c=t+\epsilon$. However, we know that $p_1(0)=p_2(t)$ and hence $p_1(t)>p_2(t)$ which is a contradiction.

Appendix 4: Probability of an inefficiency

The probability of inefficiency can be best derived from two graphs in the $c_2-c_1-$space. Figure 5 shows Eq. (A.12) for $t = 0.2$ as the solid line. The broken line is the efficiency border $c_2 = c_1 - t$ where both firms are equally efficient. For $c_1 < t$, the domestic firm is the efficient one in any case. In the laissez-faire equilibrium, the foreign firm wins (loses) if ${\tilde{c}}_2 < (>) c_1$, and the domestic firm should win from a global perspective if $c_2 > c_1 - t$. The area between the two lines represents the inefficiency. Note that the size of the rectangle is $1 - t$ due to the upper bound for $c_2$. The probability of inefficiency can thus be computed as the area below the solid line minus the area below the broken line, corrected by the factor $1/(1 - t)$:

$$\begin{aligned} \frac{1}{1-t} \left( \int _0^1 {\tilde{c}}_2(c_1) dc_1 - \int _t^1 (c_1 - t)dc_1 \right) = \frac{t}{2}\left( \frac{1-t}{2-t}\right) \end{aligned}$$

(A.14)

Fig. 5

Inefficiency in the laissez-faire equilibrium

Appendix 5: Inefficiency under national policy

We want to calculate the probability of inefficiency in the case of national policy. National policy intervenes if $p_2<p_1$ and $p_2>c_1$. The intervention awards the market to the domestic firm, which is inefficient (from a global perspective) if $c_1>c_2+t$. The cost combinations that satisfy these three conditions are depicted by the color-shaded area in Fig. 6. Under the assumption of independent uniform distributions, the probability of inefficiency (unconditional on entry) amounts to the size of the color-shaded area, that is:

$$\begin{aligned} \frac{(1-t)^2}{2} - \frac{(1-(1+t)/2)(1-t)}{2} = \frac{(1-t)^2}{4} \end{aligned}$$

The probability conditional on entry by the form firm is thus $(1-t)/4$.

Fig. 6

Inefficiency under national policy

In order to calculate the expected inefficiency (conditional on entry), we integrate the distance from the diagonal over the shaded area (where the inner integral is horizontal along the line depicted in Fig. 6) and divide by the probability of entry:

$$\begin{aligned} \frac{1}{1-t} \int _{c_2=0}^{1-t} \int _{c_1=c_2+t}^{(c_2+t)/2+1/2} (c_1 - (c_2+t)) dc_1 dc_2 = \frac{1/24 - t/8 + t^2/8 - t^3/24}{1-t}. \end{aligned}$$