Trade Costs and Endogenous Nontradability in a Model with Sectoral and Firm-Level Heterogeneity

Abstract

The paper takes a first step in the direction of simultaneously incorporating sectoral and firm-level heterogeneity in the models of international trade and macroeconomics in a tractable manner: without increasing the complexity of numerical computations compared to the existing models with heterogeneity in one dimension. In a model with sectoral heterogeneity in trade costs and firm-level heterogeneity in productivity, introducing one source of heterogeneity at a time and piecing together the results implies that, on reduction in trade costs, more goods and more varieties of every tradable good become traded. In contrast, in the correctly specified model with simultaneous heterogeneity in both dimensions, while more goods do indeed become tradable, but for more than 50% of the previously traded goods, the number of traded varieties falls. The model also reconciles apparently contrasting predictions for the differences in the deviation of domestic price from the world price for the traded and nontraded goods when heterogeneity is introduced, one dimension at a time.

Appendices

Appendix A: A Small Open Economy with Production

As shown by Bergin and Glick (2009) for heterogeneity in one dimension, introducing production does not involve any additional issues. Following them, the production function for good $\left( i,j\right)$ is

$$\begin{aligned} y_{i,j}=A_{i,j}\left( l_{i,j}\right) ^{a},\quad 0\le a\le 1. \end{aligned}$$

(A.1)

The economy has a higher productivity of the variety with higher index j

$$\begin{aligned} A_{i,j}=Aj^{\beta _{a}}. \end{aligned}$$

(A.2)

With perfect competition, price equals marginal cost for each good $\left( i,j\right)$ and wage (W) is equalized across goods and varieties, which gives

$$\begin{aligned} p_{i,j}&= \frac{W}{ay_{i,j}/l_{i,j}}=\frac{W}{a\left( A_{i,j}\right) ^{1/a}} \left( y_{i,j}\right) ^{1/e}, \end{aligned}$$

(A.3)

$$\begin{aligned} y_{i,j}&= \left( \frac{a\left( A_{i,j}\right) ^{1/a}}{W}p_{i,j}\right) ^{e}. \end{aligned}$$

(A.4)

where $e=a/\left( 1-a\right)$ is the elasticity of output with respect to price. Thus, for any two varieties j and k of a good i, we have

$$\begin{aligned} \frac{y_{i,j}}{y_{i,k}}=\left( \frac{p_{i,j}}{p_{i,k}}\right) ^{e}\left( \frac{A_{i,j}}{A_{i,k}}\right) ^{1+e}, \end{aligned}$$

(A.5)

On the other hand, relating outputs and prices by consumer optimization gives

$$\begin{aligned} \frac{p_{i,j}}{p_{i,k}}=\left[ \frac{y_{i,j}}{y_{i,k}}\right] ^{-\frac{1}{ \phi }}. \end{aligned}$$

(A.6)

Equations (A.5) and (A.6) together yield

$$\begin{aligned} \frac{p_{i,j}}{p_{i,k}}&= \left( \frac{A_{i,j}}{A_{i,k}}\right) ^{-\frac{e+1 }{e+\phi }}, \end{aligned}$$

(A.7)

$$\begin{aligned} \frac{y_{i,j}}{y_{i,k}}&= \left( \frac{A_{i,j}}{A_{i,k}}\right) ^{\phi \frac{e+1}{e+\phi }}. \end{aligned}$$

(A.8)

For goods with $i\ge \bar{\imath }$, this implies

$$\begin{aligned} p_{i,N}&=\frac{p^{*}i^{\beta _{c}}}{\alpha }\left[ 1+\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }\right] ^{\frac{1}{\phi -1}}, \end{aligned}$$

(A.9)

$$\begin{aligned} p_{i,T}&= \frac{p^{*}}{\alpha }i^{\beta _{c}}, \end{aligned}$$

(A.10)

$$\begin{aligned} p_{i}&= \frac{p^{*}}{\alpha }\frac{i^{\beta _{c}}}{\left[ 1-w^{\prime }n_{i}\right] ^{\frac{1}{\phi -1}}}, \end{aligned}$$

(A.11)

where

$$\begin{aligned} w^{\prime }=\frac{\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }}{ 1+\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }}<1. \end{aligned}$$

(A.12)

Proceeding as in the case with endowment, we get

$$\begin{aligned} c_{i,N}&= n_{i}\left[ \frac{1}{n_{i}}\int _{0}^{n_{i}}c_{i,j}^{1-\frac{1}{ \phi }}dj\right] ^{\frac{1}{1-\frac{1}{\phi }}}=\frac{y_{i,n_{i}}n_{i}}{ \left[ 1+\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }\right] ^{\frac{1 }{1-\frac{1}{\phi }}}}, \end{aligned}$$

(A.13)

$$\begin{aligned} c_{i,T}&= \left( 1-n_{i}\right) \left[ \frac{1}{1-n_{i}} \int _{n_{i}}^{1}c_{i,j}^{1-\frac{1}{\phi }}dj\right] ^{\frac{1}{1-\frac{1}{ \phi }}}=\left( 1-n_{i}\right) y_{i,n_{i}} \end{aligned}$$

(A.14)

$$\begin{aligned} c_{i}&= \left[ n_{i}\left( \frac{c_{i,N}}{n_{i}}\right) ^{\frac{\phi -1}{ \phi }}+\left( 1-n_{i}\right) \left( \frac{c_{i,T}}{1-n_{i}}\right) ^{\frac{ \phi -1}{\phi }}\right] ^{\frac{\phi }{\phi -1}}=y_{i,n_{i}}\left[ 1-w^{\prime }n_{i}\right] ^{\frac{\phi }{\phi -1}.} \end{aligned}$$

(A.15)

The employment in production of traded and nontraded varieties of various goods is given by

$$\begin{aligned}&\displaystyle l_{i,N} = \left( \frac{aA}{W}\frac{p^{*}i^{\beta _{c}}}{\alpha }\right) ^{1+e}\frac{n_{i}^{1+\beta _{a}\left( 1+e\right) }}{1+\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }},\end{aligned}$$

(A.16)

$$\begin{aligned}&\displaystyle l_{i,T} = \left( \frac{aA}{W}\frac{p^{*}i^{\beta _{c}}}{\alpha }\right) ^{1+e}\frac{1}{1+\beta _{a}\left( e+1\right) }\left[ 1-n_{i}^{1+\beta _{a}\left( 1+e\right) }\right] ,\end{aligned}$$

(A.17)

$$\begin{aligned}&\displaystyle l_{i} = l_{i,N}+l_{i,T}. \end{aligned}$$

(A.18)

Note that for $i\le \bar{\imath }$, $l_{i,T}=0$ as $n_{i}=1.$ The labor market clearing condition is

$$\begin{aligned} \int _{0}^{1}l_{i}di=L\equiv 1, \end{aligned}$$

(A.19)

which can be explicitly solved to give

$$\begin{aligned} W^{1+e}=\int _{0}^{1}\left( aA\frac{p^{*}i^{\beta _{c}}}{\alpha }\right) ^{1+e}\frac{1}{1+\beta _{a}\left( e+1\right) }\left[ 1+\frac{1+e}{\phi -1} w^{\prime }n_{i}^{1+\beta _{a}\left( 1+e\right) }\right] di. \end{aligned}$$

(A.20)

Consumer’s optimization over different goods i and m,  as before, gives

$$\begin{aligned} \frac{c_{i}}{c_{m}}=\left[ \frac{p_{i}}{p_{m}}\right] ^{-\gamma }, \end{aligned}$$

(A.21)

from which, for $i\ge \bar{\imath }$, one obtains

$$\begin{aligned} n_{i}=i^{-\frac{\gamma +e}{1+e}\frac{\beta _{c}}{\beta _{a}}}\left[ \frac{ 1-w^{\prime }n_{i}}{1-w^{\prime }n_{1}}\right] ^{\frac{1}{\beta _{a}(1+e)} \frac{\gamma -\phi }{\phi -1}}n_{1}. \end{aligned}$$

(A.22)

which also defines $\bar{\imath }$, when $n_{i}$ is set to 1. The price of the marginal fully nontraded good is

$$\begin{aligned} p_{\bar{\imath }}=p_{\bar{\imath },N}=\frac{p^{*}\bar{\imath }^{\beta _{c}} }{\alpha }\left[ 1+\beta _{a}\left( \phi -1\right) \frac{e+1}{e+\phi }\right] ^{\frac{1}{\phi -1}}, \end{aligned}$$

(A.23)

and, for any other fully nontraded good $i<\bar{\imath }$, consumer optimization, as before, implies

$$\begin{aligned} p_{i}=\left[ \frac{c_{i}}{c_{\bar{\imath }}}\right] ^{-\frac{1}{\gamma }}p_{ \bar{\imath }}=p_{\bar{\imath }},\quad i\le \bar{\imath }. \end{aligned}$$

(A.24)

Solving for equilibrium again involves searching over one unknown. Given $n_{1}$, (A.22) can be solved for $n_{i}$ as function of i, and for $\bar{ \imath }$. Given $n_{i}$, (A.20) gives the corresponding value of wage rate W. Finally, the value of $n_{1}$, the share of non-traded varieties for good 1, is determined by the external balance condition or the budget constraint.

For example, in the steady state case with current account in balance, we again have

$$\begin{aligned} p_{H}c_{H}\equiv \int _{0}^{1}p_{i}c_{i}di=\theta \int _{0}^{1}p_{i}y_{i}di\equiv \theta p_{H}y_{H}, \end{aligned}$$

(A.25)

where

$$\begin{aligned} p_{i}y_{i}=\left( \frac{aA}{W}\right) ^{e}\left( \frac{p^{*}i^{\beta _{c}}}{\alpha }\right) ^{1+e}\frac{1}{1+\beta _{a}\left( e+1\right) }\left[ 1+\frac{1+e}{\phi -1}w^{\prime }n_{i}^{1+\beta _{a}\left( 1+e\right) }\right] , \end{aligned}$$

(A.26)

and

$$\begin{aligned} p_{i}c_{i}=\left( \frac{aA}{W}\right) ^{e}\left( \frac{p^{*}i^{\beta _{c}}}{\alpha }\right) ^{1+e}\left[ 1-wn_{i}\right] . \end{aligned}$$

(A.27)

Substituting for $p_{i}c_{i}$ and $p_{i}y_{i}$ from (A.26–A.27) into (A.25) and noting (i) that for $i\ge \bar{\imath }$ (A.22) gives $n_{i}$ as function of i and $n_{1}$ and $\bar{\imath }$ as a function of $n_{1}$, and (ii) that (A.20) gives W as function of $n_{i}$ (and hence $n_{1}$), one obtains one equation in one unknown, $n_{1}$. In fact, for this steady state case, W drops out of the budget constraint. However, this will not be the case out of the steady state.

Appendix B: A Two Country Model

The two countries, Home and Foreign, are identical in all respects except that their endowment of different varieties of a good is anti-symmetric as mentioned in the paper

$$\begin{aligned}&\displaystyle y_{i,j} = y\left( \frac{1+j}{\delta }\right) ^{\beta _{a}},\quad y>0,\quad \beta _{a}>0,\quad \delta >0, \end{aligned}$$

(B.1)

$$\begin{aligned}&\displaystyle y_{i,j}^{*} = y\left( \frac{2-j}{\delta }\right) ^{\beta _{a}}. \end{aligned}$$

(B.2)

As usual, variables with asterisk denote quantities for the Foreign. Transport costs are as in the small economy case. The preferences for Home’s representative consumer are

$$\begin{aligned} c^{1-\frac{1}{\gamma }}=\int _{0}^{1}c_{i}^{1-\frac{1}{\gamma }}di, \end{aligned}$$

(B.3)

where $\gamma >1$ is the elasticity of substitution across goods produced by different industries. For every sector there exists a variety $n_{i}$ such that Home imports varieties $j\in [0,n_{i})$, exports varieties $j\in (1-n_{i},1]$, and varieties $[n_{i},1-n_{i}]$ are nontraded. Thus, $c_{i},$ the index of consumption of good i can be broken up as follows

$$\begin{aligned} c_{i}^{1-\frac{1}{\phi }}&= \int _{0}^{n_{i}}c_{i,j}^{1-\frac{1}{\phi } }dj+\int _{n_{i}}^{1-n_{i}}c_{i,j}^{1-\frac{1}{\phi }}dj+ \int _{1-n_{i}}^{1}c_{i,j}^{1-\frac{1}{\phi }}dj, \nonumber \\&= n_{i}\left( \frac{c_{i,M}}{n_{i}}\right) ^{1-\frac{1}{\phi }}+\left( 1-2n_{i}\right) \left( \frac{c_{i,N}}{1-2n_{i}}\right) ^{1-\frac{1}{\phi } }+n_{i}\left( \frac{c_{i,X}}{n_{i}}\right) ^{1-\frac{1}{\phi }}, \end{aligned}$$

(B.4)

where $\phi >1$ is the elasticity of substitution among varieties of a good and

$$\begin{aligned}&\displaystyle c_{i,M} = n_{i}\left[ \frac{1}{n_{i}}\int _{0}^{n_{i}}c_{i,j}^{1-\frac{1}{ \phi }}dj\right] ^{\frac{1}{1-\frac{1}{\phi }}}, \end{aligned}$$

(B.5)

$$\begin{aligned}&\displaystyle c_{i,N} = \left( 1-2n_{i}\right) \left[ \frac{1}{1-2n_{i}} \int _{n_{i}}^{1-n_{i}}c_{i,j}^{1-\frac{1}{\phi }}dj\right] ^{\frac{1}{1- \frac{1}{\phi }}}, \end{aligned}$$

(B.6)

$$\begin{aligned}&\displaystyle c_{i,X} = n_{i}\left[ \frac{1}{n_{i}}\int _{1-n_{i}}^{1}c_{i,j}^{1-\frac{1}{ \phi }}dj\right] ^{\frac{1}{1-\frac{1}{\phi }}}. \end{aligned}$$

(B.7)

Once again, $c_{i,M},$$c_{i,N}$ and $c_{i,X}$ have the interpretation of ‘aggregate’ consumption of imported, nontraded and exported varieties of good i and, therefore, $c_{i,M}/n_{i},$$c_{i,N}/(1-2n_{i})$ and $c_{i,X}/n_{i}$ have the interpretation of ‘average’ consumption.

The consumption-based-price indices for the above defined consumption aggregates follow immediately

$$\begin{aligned}&\displaystyle p^{1-\gamma } = \int _{0}^{1}p_{i}^{1-\gamma }di, \end{aligned}$$

(B.8)

$$\begin{aligned}&\displaystyle p_{i}^{1-\phi } = \int _{0}^{n_{i}}p_{i,j}^{1-\phi }dj+\int _{n_{i}}^{1}p_{i,j}^{1-\phi }dj, \nonumber \\&\displaystyle = n_{i}p_{i,M}^{1-\phi }+(1-2n_{i})p_{i,N}^{1-\phi }+n_{i}p_{i,X}^{1-\phi }. \end{aligned}$$

(B.9)

Here p is the aggregate price level in Home and $p_{i}$ is the price index of good i. The price indices of imported, nontraded and traded varieties of good i, $p_{i,M}$, $p_{i,N}$ and $p_{i,X}$ are

$$\begin{aligned} p_{i,M}&= \left[ \frac{1}{n_{i}}\int _{0}^{n_{i}}p_{i,j}^{1-\phi }dj\right] ^{\frac{1}{1-\phi }}, \end{aligned}$$

(B.10)

$$\begin{aligned} p_{i,N}&= \left[ \frac{1}{1-2n_{i}}\int _{n_{i}}^{1-n_{i}}p_{i,j}^{1-\phi }dj \right] ^{\frac{1}{1-\phi }}, \end{aligned}$$

(B.11)

$$\begin{aligned} p_{i,X}&= \left[ \frac{1}{n_{i}}\int _{1-n_{i}}^{1}p_{i,j}^{1-\phi }dj\right] ^{\frac{1}{1-\phi }}, \end{aligned}$$

(B.12)

and

$$\begin{aligned} p_{i,T}^{1-\phi }=n_{i}p_{i,M}^{1-\phi }+n_{i}p_{i,X}^{1-\phi }. \end{aligned}$$

(B.13)

Using conditions from consumer optimization, as in the small open economy case, one can derive the prices of all varieties of good i in terms of the price $p_{i,n_{i}}$ of the marginal imported variety, $n_{i},$

$$\begin{aligned} p_{i,j}&= \left[ \frac{\left( 1+\tau _{i}\right) y_{i,j}+y_{i,j}^{*}}{ \left( 1+\tau _{i}\right) y_{i,ni}+y_{i,n_{i}}^{*}}\right] ^{-\frac{1}{ \phi }}p_{i,n_{i}},\quad j\in [0,n_{i}), \end{aligned}$$

(B.14)

$$\begin{aligned} p_{i,j}&= \left[ \frac{1+j}{1+k}\right] ^{-\frac{\beta _{a}}{\phi } }p_{i,n_{i}},\quad j\in (n_{i},1-n_{i}], \end{aligned}$$

(B.15)

$$\begin{aligned} p_{i,j}&= \left[ \frac{y_{i,j}+\left( 1+\tau _{i}\right) y_{i,j}^{*}}{ \left( 1+\tau _{i}\right) y_{i,ni}+y_{i,n_{i}}^{*}}\right] ^{-\frac{1}{ \phi }}\left[ \frac{2-n_{i}}{1+n_{i}}\right] ^{-\frac{\beta _{a}}{\phi } }p_{i,n_{i}},\quad j\in (1-n_{i},1]. \end{aligned}$$

(B.16)

Recall, due to assumed symmetry marginal exported variety is $1-n_{i}$.

Similarly, consumption of varieties of good i is given by

$$\begin{aligned} c_{i,j}&= s_{1,i}\left[ \left( 1+\tau _{i}\right) y_{i,j}+y_{i,j}^{*} \right] ,\quad j\in [0,n_{i}), \end{aligned}$$

(B.17)

$$\begin{aligned} c_{i,j}&= y_{i,j}=y\left[ \frac{1+j}{\delta }\right] ^{\beta _{a}},\quad j\in [n_{i},1-n_{i}], \end{aligned}$$

(B.18)

$$\begin{aligned} c_{i,j}&= s_{2,i}\left[ y_{i,j}+\left( 1+\tau _{i}\right) y_{i,j}^{*} \right] ,\quad j\in (1-n_{i},1], \end{aligned}$$

(B.19)

where

$$\begin{aligned} s_{1,i}&\equiv \frac{c_{i,j}}{\left( 1+\tau _{i}\right) y_{i,j}+y_{i,j}^{*}}=\frac{\left( 1+n_{i}\right) ^{\beta _{a}}}{\left( 1+\tau _{i}\right) \left( 1+n_{i}\right) ^{\beta _{a}}+\left( 2-n_{i}\right) ^{\beta _{a}}},\quad j\in [0,n_{i}), \end{aligned}$$

(B.20)

$$\begin{aligned} s_{2,i}&\equiv \frac{c_{i,j}}{y_{i,j}+\left( 1+\tau _{i}\right) y_{i,j}^{*}}=\frac{\left( 2-n_{i}\right) ^{\beta _{a}}}{\left( 2-n_{i}\right) ^{\beta _{a}}+\left( 1+\tau _{i}\right) \left( 1+n_{i}\right) ^{\beta _{a}}},\quad j\in (1-n_{i},1]. \end{aligned}$$

(B.21)

From (B.14–B.21), it is easy to see that aggregate consumption of good i and its price are functions of $n_{i}$ and $p_{i,n_{i}}$ alone. Explicitly denoting this dependence, consumer optimization across goods with some varieties traded implies

$$\begin{aligned} \frac{c_{i}\left( n_{i}\right) }{c_{1}\left( n_{1}\right) }=\left[ \frac{ p_{i}\left( n_{i},p_{i,n_{i}}\right) }{p_{1}\left( n_{1},p_{1,n_{1}}\right) } \right] ^{-\gamma }. \end{aligned}$$

(B.22)

A similar condition holds for Foreign¹⁴

$$\begin{aligned} \frac{c_{i}^{*}\left( n_{i}\right) }{c_{1}^{*}\left( n_{1}\right) }= \left[ \frac{p_{i}^{*}\left( n_{i},p_{i,n_{i}}^{*}\right) }{ p_{1}^{*}\left( n_{1},p_{1,n_{1}}^{*}\right) }\right] ^{-\gamma }. \end{aligned}$$

(B.23)

As in case of small open economy, there is a marginal fully nontraded good, $\bar{\imath }$, with $n_{\bar{\imath }}=0$, which solves

$$\begin{aligned} \frac{c_{\bar{\imath }}\left( 1\right) }{c_{1}\left( n_{1}\right) }=\left[ \frac{p_{i}\left( 0,p_{\bar{\imath },0}\right) }{p_{1}\left( n_{1},p_{1,n_{1}}\right) }\right] ^{-\gamma }, \end{aligned}$$

(B.24)

and for goods with $i<\bar{\imath }$,

$$\begin{aligned} p_{i,j}=p_{\bar{\imath },j}\quad j\in [0,1]. \end{aligned}$$

(B.25)

To begin solving for the equilibrium, first normalize $p_{1,n_{1}}=1$. Also note that for the marginal imported variety for Home, $n_{i},$

$$\begin{aligned} p_{i,n_{i}}=\left( 1+\tau _{i}\right) p_{i,n_{i}}^{*}. \end{aligned}$$

(B.26)

In light of (B.26), given $n_{1},$ (B.22–B.23) can be solved for $n_{i}$ and $p_{i,n_{i}}$ for every $i>\bar{\imath }$ where value of $\bar{\imath }$ follows from (B.24). Also, recall for $i\le \bar{\imath }$, $n_{i}=0$. The appropriate value of $n_{1}$ is found by imposing the budget constraint which on simplification gives¹⁵

$$\begin{aligned} \int _{\bar{\imath }(n_{1})}^{1}p_{i}\left( n_{i},p_{i,n_{i}}\right) c_{i}\left( n_{i}\right) di=\int _{\bar{\imath }(n_{1})}^{1}p_{i}\left( n_{i},p_{i,n_{i}}\right) y_{i}di. \end{aligned}$$

(B.27)

Note that, again, to solve for the entire equilibrium, one needs to solve only one Eq. (B.27) in one unknown $n_{1}$. However, if the two countries were dissimilar besides having anti-symmetric endowments, one will need to solve two equations in two unknowns, the marginal imported and exported variety of good 1 for Home, $n_{1}^{m}$ and $n_{1}^{x}$. In the symmetric case discussed here, $n_{1}=n_{1}^{m}=1-n_{1}^{x}$.