Reciprocal Import Tariffs in the Monopolistic Competition Open Economy

Abstract

We use the standard Krugman’s one-sector trade model, with unspecified variable-elasticity preferences. We study the impact of reciprocal import tariffs on welfare among symmetric countries (a free-trade agreement). We show that, without transport costs, any ad valorem tariffs or subsidies are always harmful. This is also true under “flatter” demands, satisfying the realistic assumptions of increasingly elastic demand (IED) and decreasingly-elastic utility (DEU).

Appendix

Proof of Theorem 1. i) Reaction of consumption. To find how consumptions in $K+1$ symmetric countries respond to tariff $\tau$, we denote total derivatives as $x_{\tau }^{\prime }\equiv \frac{dx}{d\tau },\,y_{\tau }^{\prime }\equiv \frac{dy}{d\tau }$ and $R_{z}^{\prime }\equiv R^{\prime }\left( z\right)$. Our symmetric-equilibrium equations are:

Free entry:

$$ \pi (x,y)\cdot \lambda \equiv R(x)+K\frac{R(y)}{\tau }-c\cdot \left( x+Ky\right) \lambda -f\lambda =0, $$

FOC:

$$ R^{\prime }(x)=c\lambda ,\,\,\,R^{\prime }(y)=c\tau \lambda . $$

Thus,

$$ \frac{R\left( x\right) }{R^{\prime }\left( x\right) }+K\frac{R\left( y\right) }{R^{\prime }\left( y\right) }=x+Ky+\frac{f}{c}. $$

Totally differentiating the latter equations in $\tau$ (and applying $\frac{d\pi }{dx}=0,\frac{d\pi }{dy}=0$ or Envelope Theorem to the third equation) we get

$$ R^{\prime \prime }(x)x_{\tau }^{\prime }=c\lambda _{\tau }^{\prime },\,\,\,R^{\prime \prime }(y)y_{\tau }^{\prime }=c\lambda +c\tau \lambda _{\tau }^{\prime } $$

and

$$ \lambda _{\tau }^{\prime }=-\frac{KR(y)\left( R^{\prime }\left( x\right) \right) ^{2}}{\left( R^{\prime }\left( y\right) \right) ^{2}\left( c\left( x+Ky\right) +f\right) }. $$

It follows that the total derivatives of consumptions are (20) and (21).

(ii) Reaction of output. Under general tariff, to find its impact on sales (output), we combine the changes in x and y:

$$\begin{aligned} q_{\tau }^{\prime }=x_{\tau }^{\prime }+Ky_{\tau }^{\prime }=\frac{K\left( \tau ^{2}R(x)R^{\prime \prime }(x)-R(y)R^{\prime \prime }(y)\right) }{\tau ^{2}R^{\prime \prime }(y)R^{\prime \prime }(x)\left( x+Ky+\frac{f}{c}\right) }. \end{aligned}$$

(25)

We would like to know the sign of this derivative. For linear demand $R^{\prime \prime }(x)=R^{\prime \prime }(y)=constant$, so, the sign is clear: output decreases in $\tau$ on the whole interval.

To find the derivative of output $q_{\tau }^{\prime }=x_{\tau }^{\prime }+Ky_{\tau }^{\prime }$ we substitute (20) and (21) into $q_{\tau }^{\prime }=x_{\tau }^{\prime }+Ky_{\tau }^{\prime }$ and

$$ q_{\tau }^{\prime }\mid _{\tau =1}=0. $$

To find the derivative of sales at autarky ($\tau _{a}:\,\,y\left( \tau _{a}\right) =0$) we just plug $y\left( \tau _{a}\right) =0$ into our formulate and obtain

$$ x_{\tau _{a}}^{\prime }=-\frac{K\cdot R(0)}{\tau ^{2}\cdot R^{\prime \prime }(x)\left( x+\frac{f}{c}\right) }=0, $$

$$ y_{\tau _{a}}^{\prime }=\frac{K\cdot R(x)}{R^{\prime \prime }(0)\left( x+\frac{f}{c}\right) }<0, $$

$$ q_{\tau }^{\prime }<0. $$

To find the derivative of sales at global point ($\tau _{x0}:\,\,x\left( \tau _{x0}\right) =0$) we just plug $x\left( \tau _{x0}\right) =0$ into our formula and obtain

$$ x_{\tau _{x0}}^{\prime }=-\frac{KR(y)}{\tau ^{2}\cdot R^{\prime \prime }(0)\left( y+\frac{f}{c}\right) }>0, $$

$$ y_{\tau _{x0}}^{\prime }=\frac{K\tau \cdot R(0)}{R^{\prime \prime }(y)\left( y+\frac{f}{c}\right) }=0, $$

$$ q_{\tau _{x0}}^{\prime }>0. $$

So it decreases when the subsidy grows.

Global impact $q_{\tau }^{\prime }$ of $\tau$. We use that $\tau =\frac{R^{\prime }\left( y\right) }{R^{\prime }\left( x\right) }$. Substitute $\tau$ into (25):

$$\begin{aligned} q_{\tau }^{\prime }=\frac{K\left( R^{\prime }\left( y\right) \right) ^{2}}{\tau ^{2}R^{\prime \prime }(y)R^{\prime \prime }(x)\left( x+Ky+\frac{f}{c}\right) }\cdot \left( \frac{R(x)R^{\prime \prime }(x)}{\left( R^{\prime }\left( x\right) \right) ^{2}}-\frac{R(y)R^{\prime \prime }(y)}{\left( R^{\prime }\left( y\right) \right) ^{2}}\right) . \end{aligned}$$

(26)

The sign of the bracket determines the sign of the derivative. Let us introduce the function $\phi \left( z\right) \equiv \frac{R(z)R^{\prime \prime }(z)}{\left( R^{\prime }\left( z\right) \right) ^{2}}$. If function $\phi \left( \cdot \right)$ is decreasing then the derivative $q_{\tau }^{\prime }$ of the total output is negative under a positive tariff $\tau >1$.

For IED:

$$\begin{aligned} \frac{R(z)R^{\prime \prime }(z)}{\left( R^{\prime }\left( z\right) \right) ^{2}}\equiv \frac{r_{u}^{\prime }\left( z\right) \cdot z+r_{u}\left( z\right) -\left( r_{u}\left( z\right) \right) ^{2}}{\left( 1-r_{u}\left( z\right) \right) ^{2}}. \end{aligned}$$

(27)

Find derivative:

$$ -\left( \frac{r_{u}^{\prime }\left( z\right) \cdot z+r_{u}\left( z\right) -\left( r_{u}\left( z\right) \right) ^{2}}{\left( 1-r_{u}\left( z\right) \right) ^{2}}\right) ^{\prime } $$

$$ =-\frac{\left( {\displaystyle \mathcal {E}_{r_{u}^{\prime }}\left( z\right) }+2\right) \cdot \left( 1-r_{u}\left( z\right) \right) +2\cdot r_{u}^{\prime }\left( z\right) \cdot z}{\left( 1-r_{u}\left( z\right) \right) ^{3}}\cdot r_{u}^{\prime }\left( z\right) . $$

Proof of Theorem 2 (about welfare)

Recall that any consumer’s welfare to be studied is expressed through consumptions as

$$ W_{\tau }\left( x,y\right) =\frac{u\left( x\right) +Ku\left( y\right) }{f+c(x+Ky)}. $$

Using notations $C_{q}\equiv C(x+Ky)\equiv f+c(x+Ky)$, we estimate the welfare total derivative $W_{\tau }^{\prime }$ w.r.t. tariff $\tau$:

$$ W_{\tau }^{\prime }\left( x,y\right) =\frac{u^{\prime }\left( x\right) x_{\tau }^{\prime }+Ku^{\prime }\left( y\right) y_{\tau }^{\prime }}{\left( f+c(x+Ky)\right) }-\frac{u\left( x\right) +Ku\left( y\right) }{\left( f+c(x+Ky)\right) }\cdot \frac{c\left( x_{\tau }^{\prime }+Ky_{\tau }^{\prime }\right) }{\left( f+c(x+Ky)\right) }. $$

Multiplying everything by $\frac{C_{q}}{U}$ we come to

$$ \frac{C_{q}}{U}\cdot W_{\tau }^{\prime }\left( x,y\right) =\frac{u^{\prime }\left( x\right) x_{\tau }^{\prime }+Ku^{\prime }\left( y\right) y_{\tau }^{\prime }}{u\left( x\right) +Ku\left( y\right) }-\frac{c\left( x_{\tau }^{\prime }+Ky_{\tau }^{\prime }\right) }{\left( f+c(x+Ky)\right) }, $$

which can be expressed in elasticities as

$$\begin{aligned} \frac{C_{q}}{U}\cdot W_{\tau }^{\prime }=\frac{x_{\tau }^{\prime }}{x}\cdot \left[ \mathcal {E}_{U|x}-\mathcal {E}_{C|x}\right] +K\frac{y_{\tau }^{\prime }}{y}\cdot \left[ \mathcal {E}_{U|y}-\mathcal {E}_{C|y}\right] , \end{aligned}$$

(28)

Unchanging Welfare at the Point of Free Trade. Using elasticities, at free trade we plug $\tau =1,\,\,x=y$ into expression (28), substitute $\mathcal {E}_{C|y}=\mathcal {E}_{C|x}$ (which is true everywhere, not only at $x=y$) and obtain

$$ \frac{C_{q}}{U}\cdot W_{\tau }^{\prime }=\left[ \mathcal {E}_{U|x}-\mathcal {E}_{C|x}\right] \left( \frac{x_{\tau }^{\prime }}{x}+K\frac{y_{\tau }^{\prime }}{x}\right) =0 $$

because of zero change $q_{\tau }^{\prime }=x_{\tau }^{\prime }+Ky_{\tau }^{\prime }=0$ at $\tau =1$, by Theorem 1.

The global welfare change under $IED-DEU$

Using (4) and let $x_{\tau }^{\prime }\cdot \left[ A\right] +Ky_{\tau }^{\prime }\cdot \left[ B\right] \equiv \frac{C_{q}}{U}\cdot W_{\tau }^{\prime }$ we would like to prove to be negative everywhere, except the point of free trade. At free trade, this $\frac{C_{q}}{U}\cdot W_{\tau }^{\prime }$ is zero, because the first bracket is equal to the second one ($A=B$), while $q_{\tau }^{\prime }=x_{\tau }^{\prime }+Ky_{\tau }^{\prime }=0$ at $\tau =1$. Both brackets at free trade ($\tau =1,\,\,x=y=z$) are positive under DEU ($\mathcal {E}_{u}^{\prime }\left( z\right) <0$) because

$$ \left[ B\right] _{\tau =1}=\frac{u^{\prime }\left( z\right) }{2u\left( z\right) }-\frac{c}{f+c(1+K)z}=-\frac{\mathcal {E}_{u}^{\prime }\left( z\right) }{2\mathcal {E}_{u}\left( z\right) }>0, $$

and identity $z\cdot \mathcal {E}_{u}^{\prime }\left( z\right) \equiv \mathcal {E}_{u}\left( z\right) \cdot \left( 1-\mathcal {E}_{u}\left( z\right) -r_{u}\left( z\right) \right)$ that can be easily derived for any function u.

Further, under positive tariff, the second bracket $\left[ B\right]$ should increase (and remain positive) when $\tau$ increases and thereby y decreases. Indeed, differentiating $\left[ B\right]$ we get

$$ \left[ B\right] _{\tau }^{\prime }=\frac{u^{\prime \prime }\left( y\right) y_{\tau }^{\prime }}{u\left( x\right) +Ku\left( y\right) }-\frac{u^{\prime }\left( y\right) \left( u^{\prime }\left( x\right) x_{\tau }^{\prime }+Ku^{\prime }\left( y\right) y_{\tau }^{\prime }\right) }{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}+\frac{c^{2}\left( x_{\tau }^{\prime }+Ky_{\tau }^{\prime }\right) }{\left( f+c(x+Ky)\right) ^{2}}>0 $$

under DEU. First summands here is positive due to $u^{\prime \prime }\left( y\right) y_{\tau }^{\prime }>0$, see part (i) of the theorem. Two of the remaining amount are

$$ y_{\tau }^{\prime }\nu _{y}+x_{\tau }^{\prime }\nu _{x}\equiv y_{\tau }^{\prime }\left[ -\frac{\left( u^{\prime }\left( y\right) \right) ^{2}}{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}+\frac{1}{\left( \frac{f}{c}+(x+Ky)\right) ^{2}}\right] $$

$$+x_{\tau }^{\prime }\left[ -\frac{u^{\prime }\left( y\right) u^{\prime }\left( x\right) }{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}+\frac{1}{\left( \frac{f}{c}+(x+Ky)\right) ^{2}}\right] . $$

Let’s compare two of the remaining amount $\nu _{y}$ and $\nu _{x}$. If $-\frac{\left( u^{\prime }\left( y\right) \right) ^{2}}{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}+\frac{1}{\left( \frac{f}{c}+(x+Ky)\right) ^{2}}<0$ then we have $y_{\tau }^{\prime }\nu _{y}>-x_{\tau }^{\prime }\nu _{x}$ (using $\frac{\left( u^{\prime }\left( y\right) \right) ^{2}}{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}>\frac{\left( u^{\prime }\left( x\right) \right) ^{2}}{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}$, i.e., $u^{\prime }\left( z\right)$is decreasing function). Then we get that positive summand $x_{\tau }^{\prime }$ weighted with small positive or negative multiplier $\nu _{x}$, whereas the negative summand $y_{\tau }^{\prime }$ is weighted with bigger negative multiplier $\nu _{y}$, while without multipliers $x_{\tau }^{\prime }+Ky_{\tau }^{\prime }<0$ under IED by Theorem 1. Then $y_{\tau }^{\prime }\nu _{y}+x_{\tau }^{\prime }\nu _{x}>0$.

We prove that $-\frac{\left( u^{\prime }\left( y\right) \right) ^{2}}{\left( u\left( x\right) +Ku\left( y\right) \right) ^{2}}+\frac{1}{\left( \frac{f}{c}+(x+Ky)\right) ^{2}}<0$. So, bracket $\left[ B\right]$ increases in $\tau$, remaining positive, whereas its multiplier $Ky_{\tau }^{\prime }$ is negative. At the same time, for all positive tariffs $\left[ A\right] <\left[ B\right]$, because $u^{\prime }\left( x\right) <u^{\prime }\left( y\right)$, other parts of these expressions being similar. Further, consider the sum $x_{\tau }^{\prime }\cdot \left[ A\right] +Ky_{\tau }^{\prime }\cdot \left[ B\right]$, where the first positive summand is weighted with a smaller multiplier $\left[ A\right]$ (positive or negative), then the negative summand $Ky_{\tau }^{\prime }$. So, the sum remains negative (provided it was negative: $x_{\tau }^{\prime }+Ky_{\tau }^{\prime }<0$ without any multipliers).

One can extend exactly the same reasoning to the case of subsidies, where $\tau =1-s<1$, here welfare decreases in subsidy.

The global welfare change under $DED-IEU$ – just exactly mirror the proof for the previous $IED-DEU$ case. Only both signs change in the derivation: $x_{\tau }^{\prime }+Ky_{\tau }^{\prime }>0$ for $\tau \in (1,\,\tau _{a})$ under IED and $\mathcal {E}_{u}^{\prime }\left( z\right) >0$ under DEU in assessing the sign of term $\left[ A\right]$, instead of term $\left[ B\right]$, both of them starting from a negative value, and $\left[ A\right]$ decreasing.