The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity2003

We start with a dynamic industry model with heterogeneous firms. There is a representative consumer with a CES utility function over continuous good \(\omega \in \Omega\) \[U = \left[ \int_{\omega \in \Omega} q(\omega)^{\rho} \, d\omega \right]^{1/\rho}\] Consumers have love of variety for every \(\omega \in \Omega\). Goods are substitutes \(\Rightarrow \rho \in (0, 1)\). Elasticity of substitution is \(\sigma = 1/(1-\rho)\). If one good is too expensive then consumers will switch. We know from Dixit and Stiglitz (1977) that consmer behavior can be modeled considering the set of varieties consumed as \(Q \equiv U\). The model creates an aggregate price, a weighted average of all prices \[P = \left[ \int_{\omega \in \Omega} p(\omega)^{1-\sigma} \, d\omega \right]^{\frac{1}{1-\sigma}}\] and we can derive optimal consumtion and expenditure decisions \[\begin{aligned}&& q(\omega) = Q \left[\frac{p(\omega)}{P}\right]^{-\sigma} && r(\omega) = R \left[\frac{p(\omega)}{P}\right]^{1-\sigma}\end{aligned}\] where \(R = P Q = \int_{\omega \in \Omega} r(\omega)\, d\omega\). There is a continuum of firms, all producing different varieties \(\omega \in \Omega\). Production only requires labor, which is inelastically supplied. \(L\) is its aggregate level which is an index of the size of the economy. Labour is \(\ell = f + \tfrac{q}{\varphi}\) where \(q\) is output, \(f \in \mathbb{R}^{+}\) is a fixed cost, and productivity levels are indexed by \(\varphi \in \mathbb{R}^{+}\). This is the formula for costs of production. \(\tfrac{q}{\varphi}\) is the variable cost. With higher productivity \(\varphi\), the firm is able to produce one unit of \(q\) with less labor. We can also think of productivity as producing higher quality variety at equal cost. Firms choose markup \[\frac{\sigma}{\sigma - 1} = \frac{1}{\rho} \Rightarrow p(\varphi) = \frac{w}{\rho \varphi} = \frac{1}{\rho \varphi}\] where \(p(\varphi)\) is the pricing rule and \(w=1\) is the wafe. Profit is then \[\pi(\varphi) = \underbrace{r(\varphi)}_{\text{revenue}} - l(\varphi) = \underbrace{\frac{r(\varphi)}{\sigma}}_{\text{variable profit}} - f\] A higher \(\varphi\) firm will have larger output \(q\) and revenue \(r\), lower price \(p\), and higher profit \(\pi\) than firms with lower productivity \(\varphi\). Melitz characterizes equilibirum of \(M\) firms and goods. The distribution is \(\mu(\varphi)\) over productivity levels over a subset of \([0, \infty)\). The pricing rule is \[\begin{split}P = \left[ \int_{0}^{\infty} p(\varphi)^{1-\sigma} M \mu(\varphi)\, d\varphi \right]^{\frac{1}{1-\sigma}} \\ = M^{\frac{1}{1-\sigma}} \left[ \int_{0}^{\infty} p(\varphi)^{1-\sigma} \mu(\varphi)\, d\varphi \right]^{\frac{1}{1-\sigma}} \\ = \left[ \int_{0}^{\infty} p(\varphi)^{1-\sigma} \mu(\varphi)\, d\varphi \right]^{\frac{1}{1-\sigma}} \\ = p(\tilde{\varphi}) \Rightarrow P = M^{\frac{1}{1-\sigma}} p(\tilde{\varphi})\end{split}\] where \(\tilde{\varphi}\) is a weighted average of firm productivity \[\tilde{\varphi} = \left[ \int_{0}^{\infty} \varphi^{\sigma-1} \mu(\varphi)\, d\varphi \right]^{\frac{1}{\sigma-1}}\] We need a weighted average because higher productivity firms need a heavier weight as they produce a larger share of total output. From this average firm we can derive total revenue, output, profit and overall price level of the entire industry \[\begin{aligned}P = \underbrace{M^{\frac{1}{1-\sigma}} p(\tilde{\varphi})}_{\text{price}} && R = \underbrace{M \, r(\tilde{\varphi})}_{\text{revenue}} && Q = \underbrace{M^{\frac{1}{\rho}} q(\tilde{\varphi})}_{\text{output}} && \Pi = \underbrace{M \, \pi(\tilde{\varphi})}_{\text{profit}}\end{aligned}\] Now we will understand firm entry and exit in this industry. There is a large and unbounded pool of prospects. Exit is free. To enter, firms need to invest fixed cost \(f_e \in \mathbb{R}^{+}\). Then, firms draw their initial productivty \(\varphi\) from distribution \(g(\varphi)\) with support over \((0, \infty)\) and CDF G(\(\varphi)\). After drawing a bad \(\varphi\), firms may choose to exit freely, or face probability \(\delta\) of a bad shock that would force them to exit. The model assumes that ex-ante survival probability and \(\mu(\varphi)\) are determined exogenously by \(\delta\) and \(g(\varphi)\). The modul focuses on constant productivity \(\varphi\) and aggregate variables in stationary equilibirum. Thus, \(\pi(\varphi)\) is constant. \[v(\varphi)=\max\left\{0,\sum_{t=0}^{\infty}(1-\delta)^t\pi(\varphi)\right\}=\max\left\{0,\frac{\pi(\varphi)}{\delta}\right\}\] is the value function of a firm, which represents the present value of future flows assuming no time discounting. Thus \(\varphi^{*}=\inf\left\{\varphi \mid v(\varphi)>0\right\}\) identifies the cutoff level of firms. This is the lowest productivity allowed for a firm to survive. We call \(\pi(\varphi^{*})=0\) the zero-profit condition (ZCP). \(\pi(0)=-f\) is negative and all firms that draw \(\varphi < \varphi^{*}\) exit. For remaining firms \(\mu(\varphi)\) is \(g(\varphi)\) on \([ \varphi^{*}, \infty)\) \[\mu(\varphi)=\begin{cases}\dfrac{g(\varphi)}{1-G(\varphi^{*})} & \text{if } \varphi \geqslant \varphi^{*} \\ 0 & \text{otherwise}\end{cases}\] The probability of entering the market is \(p_{in} \equiv 1-G(\varphi^{*})\). Aggregate productivity \(\tilde{\varphi}\) is then \[\tilde{\varphi}(\varphi^{*})=\left[\frac{1}{p_{in}}\int_{\varphi^{*}}^{\infty}\varphi^{\sigma-1}g(\varphi)\,d\varphi\right]^{\frac{1}{\sigma-1}}\] Therefore average profit \(\tilde{\pi}\) and average revenue \(\tilde{r}\), much like \(\tilde{\varphi}\) are determined by cutoff level \(\varphi^{*}\) so \[\begin{aligned}\tilde{r} = \underbrace{r(\tilde{\varphi}) = \left(\frac{\tilde{\varphi}(\varphi^{*})}{\varphi^{*}}\right)^{\sigma-1} r(\varphi^{*})}_{\text{average revenue}} && \tilde{\pi} = \underbrace{\pi(\tilde{\varphi}) = \left(\frac{\tilde{\varphi}(\varphi^{*})}{\varphi^{*}}\right)^{\sigma-1} \frac{r(\varphi^{*})}{\sigma} - f}_{\text{average profit level}}\end{aligned}\] ZCP implies relationship between \(\bar{\pi}\) and \(\varphi^{*}\) \[\pi(\varphi^{*}) = 0 \Longleftrightarrow r(\varphi^{*}) = \sigma f \Longleftrightarrow \bar{\pi} = f\left(\left(\frac{\tilde{\varphi}(\varphi^{*})}{\varphi^{*}}\right)^{\sigma-1} - 1\right) = f k(\varphi)\] Beside cutoff firm, all firms have \(\pi \in \mathbb{R}^{+} \Rightarrow \bar{\pi} \in \mathbb{R}^{+}\). Let \[\bar{v} = \sum_{t=0}^{\infty}(1-\delta)^t \bar{\pi} = \left(\frac{1}{\delta}\right)\bar{\pi} = \int_{\varphi^{*}}^{\infty} v(\varphi)\,\mu(\varphi)\,d\varphi\] represent the present value of average profit flows and average value of firms conditional on entry. Firms will continue to enter as long as the expected value of profits minus entry cost is non-negative. We call this the net value of entry \(v_e\) \[v_e = p_{in}\,\bar{v} - f_e = \frac{1 - G(\varphi^{*})}{\delta}\,\bar{\pi} - f_e \Rightarrow \bar{\pi} = \frac{\delta f_e}{1 - G(\varphi^{*})}\] We then have the free entry (FE) and ZCP conditions. They represent two different relations that link average profit \(\bar{\pi}\) and cutoff productivity \(\varphi^{*}\). Notice the FE curve is increasing is cut from above by ZCP at a unique point in \((\varphi, \pi)\) space. This is an equilibrium. \[\begin{aligned}\underbrace{\bar{\pi}=f\,k(\varphi^{*})}_{\text{ZCP}} && \text{and} && \underbrace{\bar{\pi}=\frac{\delta f_e}{1-G(\varphi^{*})}}_{\text{FE}}\end{aligned}\] Since it is stationary equilibrium, we need a mass \(M_e\) of new entrants such that \(p_{in} M_e\), the proportion of new entrants who are succesful replace \(\delta M\), the mass who exit due to a bad shock i.e. \(p_{in} M_e = \delta M\). Then aggregate labor is divided as follows \[L=\underbrace{L_p}_{\text{production}}+\underbrace{L_e}_{\text{entrants}}\] where \(L_p\) is labor used in production and \(L_e\) is labor used in investment by new entrants. Payment to productive labor must be \(L_p = R-\Pi\), as it is the difference between aggregate revenue and profit. Then, the labor usedfor entrants is \(L_e=M_e f_e \Rightarrow\) \[\begin{split}p_{in}M_e=\delta M \Rightarrow M_e=\frac{\delta M}{p_{in}} \\ L_e=\frac{\delta f_e M}{p_{in}}=\left(\frac{\delta f_e}{p_{in}}\right)M=\bar{\pi}M=\Pi \\ L=L_p+L_e=\Pi+L_p=\Pi+R-\Pi=R\end{split}\] Then we can also determine the mass \(M\) of producing firms as follows \[\begin{split}R=M\bar{r} \Rightarrow M=\frac{R}{\bar{r}}=\frac{L}{\sigma(\bar{\pi}+f)} \\ P=M^{\frac{1}{1-\sigma}}p(\tilde{\varphi})=\frac{M^{\frac{1}{1-\sigma}}}{\rho\tilde{\varphi}}\end{split}\] All firm-level variables \(\varphi^{*}, \tilde{\varphi}, \bar{\pi},\) and \(\bar{r}\) are independent of \(L\). Mass of firms \(M\) increases proportionally with country size \(L\). However \(\mu(\varphi)\) and \(\varphi^{*}\) are unchanged. \[W = \frac{w}{P} = \frac{1}{P} = P^{-1} = M^{\frac{1}{\sigma-1}} \rho \tilde{\varphi}\] is welfare per worker. It is higher in bigger countries because of increased variety. Welfare is defined as the purchasing power of the worker. So country size \(L\) affects the number of firms but not the selection. We now turn to analyzing trade. Theworld is comprised of \(n+1 \geqslant 2\) countries like the one we just described. Firms divide sales between domestic and foreign countries. With no costs, the open economy replicates outcomes of a larger closed economy. There are two trade costs. There are fixed costs to enterimg foreign markets (standards, distribution channels, etc). Let \(\tau > 1\) be the amount of units of a good that have to be shipped for 1 unit to arrive at the export destination (iceberg tarde costs). Suppose we study an economy that can trade with \(n \geqslant 1\) other countries. For each export destination there is an additional pre-period cost \(f_{ex} \in \mathbb{R}^{+}\) to their fixed cost \(f\). \[\begin{aligned}p_d(\varphi)=\underbrace{\frac{w}{p\rho}=\frac{1}{\rho\varphi}}_{\text{domestic}} && \text{and} && p_x(\varphi)=\underbrace{\frac{\tau}{\rho\varphi}=\tau p_d(\varphi)}_{\text{foreign}}\end{aligned}\] are the two pricing rules for their respective markets, while the revenue is \[\begin{aligned}r_d(\varphi)=\underbrace{R(P\rho\varphi)^{\sigma-1}}_{\text{domestic}} && \text{and} && r_x(\varphi)=\underbrace{\tau^{1-\sigma}r_d(\varphi)}_{\text{foreign}}\end{aligned}\] the, a firm's combined revenue depends on export status \[r(\varphi)=\begin{cases}r_d(\varphi), & \text{if no exports} \\ r_d(\varphi)+n\,r_x(\varphi)=\left(1+n\tau^{1-\sigma}\right)r_d(\varphi) & \text{if it exports}\end{cases}\] Remember firms earn variable profit \(\frac{r_{x}(\varphi)}{\sigma}\) in every period. Firms either export to all countries or they do not export. Exogenous factors remain unchanged by trade. Same \(g(\varphi)\) and \(\delta\) as pre-entry. To analyze profits we can split these into two types, since firms that export also produce domestically \[\begin{aligned}{\pi}_d(\varphi)=\underbrace{\frac{r_d (\varphi)}{\sigma}-f}_{\text{domestic}} && \text{and} && {\pi}_x(\varphi)=\underbrace{\frac{r_x (\varphi)}{\sigma}-f_x}_{\text{foreign}} \end{aligned}\] Then we can write profit as \[\pi(\varphi)= {\pi}_d(\varphi)+\max \{0,n\cdot {\pi}_x (\varphi) \}\] Domestic profit is variable profit from domestic sales minus fixed cost \(f\). Foreign is variable profit from sales to one foreign country minus the per-period export cost \(f_x=\delta f_{ex}\). With \(\pi_x(\varphi)\geqslant 0\) there are exports to all \(n\) countries. Firm value is \(v(\varphi) = \max\{0, \frac{\pi(\varphi)}{\delta}\}\). There are two important cutoffs, which we will name survival and export \[ \begin{align}\varphi^{*}=\underbrace{\inf\{\varphi \mid v(\varphi) \in \mathbb{R}^{+}\}}_{\pi_d(\varphi^{*})=0} && \text{and} && \varphi^{*}_{x}=\underbrace{\inf\{\varphi \mid \varphi \geqslant \varphi^{*} \: \text{ and } \: \pi_{x}(\varphi) \in \mathbb{R}^{+}\}}_{\pi_x(\varphi^{*}_x)=0}\end{align} \] If \(\varphi^{*}_{x}=\varphi^{*}\) then all firms in the industry export. The cutoff firm would have \(\pi(\varphi^{*})=\pi_d(\varphi)+n\pi_x(\varphi)=0\) and \(\pi_x(\varphi)\in\mathbb{R}^{+}\). Let us set both profits to zero and use the ZCP \[\begin{split}\frac{f_x}{f}=\frac{r_x(\varphi_x^{*})}{r_d(\varphi^{*})}=\tau^{1-\sigma}\frac{r_d(\varphi_x^{*})}{r_d(\varphi^{*})}=\tau^{1-\sigma}\left(\frac{\varphi_x^{*}}{\varphi^{*}}\right)^{\sigma-1} \\\Rightarrow \tau^{1-\sigma}\left(\frac{f_x}{f}\right)=\left(\frac{\varphi_x^{*}}{\varphi^{*}}\right)^{\sigma-1}\Rightarrow \varphi_x^{*}=\varphi^{*}\tau\left(\frac{f_x}{f}\right)^{\frac{1}{\sigma-1}}\end{split}\] Note that \(\tau\left(\tfrac{f_x}{f}\right)^{\frac{1}{\sigma-1}}>1 \iff \varphi_x^{*}>\varphi^{*}\). We call this the partitioning of firms by export status. Here, some firms with \(\varphi \in (\varphi^{*}, \varphi_x^{*})\) will produce exclusively domestic. They do not export as their export profits \(\pi_x(\varphi)\) would be negative. Firms with \(\varphi \geqslant \varphi_x^{*}\) earn positive profits from both markets. This industry has two interesting probabilities \[\begin{aligned} p_{in} = \underbrace{1-G(\varphi^{*})}_{\text{probability of survival}} && \text{and} && p_x = \underbrace{\frac{1-G(\varphi_x^{*})}{1-G(\varphi^{*})}}_{\text{probability of exporting}}\end{aligned}\] In this open economy there are more varieties. The total mass available to the consumers is \(M_t=M+n \cdot M_x\) where \(n\) is the number of partners in trade, \(M\) is the mass of domestic firms, and \(M_x=p_x M\) is the mass from each \(n\) country that is productive enough to export there. Now, just like before, we have to aggregate. Remember the aggregate productivity formula. We then call \(\tilde{\varphi}=\tilde{\varphi}(\varphi^{*})\) and \(\tilde{\varphi}_x=\tilde{\varphi}(\varphi^{*}_x)\) be average productivity of all firms and exporting firms.. We have to define \[ \tilde{\varphi}_t = {\left\{ \frac{1}{M_t} \left[M\tilde{\varphi}^{\sigma-1}+nM_x {(\tau^{-1}\tilde{\varphi}_x)}^{\sigma-1} \right] \right\}}^{\frac{\sigma}{\sigma-1}} \] the open economy weighted average reflecting relative mass (\(M\) and \(nM_x\)) of domestic firms and foreign exporters, as well as the lost in transit shrinkage from \(\tau\). Thus \(\tilde{\varphi}_t\) can be thought of as the weighted average productivity of all firms competing in the country. It summarizes effects of \(\mu(\varphi)\), the distribution of productivity levels on the aggregate outcome.\[ \begin{aligned} P=\underbrace{M_{t}^{\frac{1}{1-\sigma}}p(\tilde{\varphi}_t)=M_{t}^{\frac{1}{1-\sigma}} \frac{1}{p\tilde{\varphi}_t}}_{\text{aggregate price}} && R=\underbrace{M_t r_d(\tilde{\varphi}_t)}_{\text{expenditure}} && W = \underbrace{\frac{R}{L}M_{t}^{\frac{1}{\sigma-1}}p\tilde{\varphi}_t}_{\text{welfare per worker}} \end{aligned} \] We did this before for the closed economy. We can also get average profit and revenue \[ \begin{aligned} \bar{r}=\underbrace{r_d(\tilde{\varphi})+p_x n r_x (\tilde{\varphi}_x)}_{\text{revenue}} && \text{and} && \bar{\pi}=\underbrace{\pi_d(\tilde{\varphi})+p_x \pi_x (\tilde{\varphi}_x)}_{\text{profit}} \end{aligned} \] Both are the sum of domestic measure and the probability of exports multiplied by export measure. Just like in the closed economy ZCP implies relationship between \(\bar{\pi}\) and \(\varphi^{*}\) \[\begin{split}\pi_d(\varphi^{*})=0 &\iff \pi_d(\tilde{\varphi})=f k(\varphi^{*}) \\\ \pi_x(\varphi_x^{*})=0 &\iff \pi_x(\tilde{\varphi}_x)=f_xk(\varphi_x^{*})\end{split}\] we already defined \(k(\varphi)\). We also get from ZCP \(\varphi^{*}_x\) as a function of \(\varphi^{*}\) \[ \frac{r_x(\varphi^{*}_x)}{r_d(\varphi^{*})} = \tau^{1-\sigma} {\left( \frac{\varphi^{*}_x}{\varphi^{*}} \right)}^{\sigma-1} = \frac{f_x}{f} \iff \varphi^{*}_x = \varphi^{*} \tau {\left( \frac{f_x}{f} \right)}^{\frac{1}{\sigma-1}}\] and then we can express \(\bar{pi}\) as a function of surivival cutoff \(\varphi^{*}\) \[\begin{split}\bar{\pi} &= \pi_d(\tilde{\varphi})+p_x\,n\pi_x(\tilde{\varphi}_x) \\\ &= f\,k(\varphi^{*})+p_x\,n f_x\,k(\varphi_x^{*})\end{split}\] which is the new ZCP for an open economy. Present value and net value are the exact same as before so we do not need to re-write those. The equilibrium mass of firms is \[ M = \frac{R}{\bar{r}}=\frac{L}{\sigma(\bar{\pi}+f+p_x nf_x)} \] which determines the mass of variety in every country \(M_t=(1+np_x)M\). Notice how consumers have access to more than just domestic goods. ZCP and FE curves still intersect in (\(\varphi,\pi\)) space. Equilibrium in the steady state is the point where firms entering balances firms failing, and labor supply is fully utilized. This of course because ZCP (survival) intersects FE (entry). ZCP curve shifts upwards because of the second exporting term. Thus survival cutoff \(\varphi^{*}\) is higher in equilibrium in the open economy, and so is \(\bar{\pi}\). The bar for survival is higher. Let us say this again. FE condition is identical inan open and closed economy. ZCP curve shifts up: exposure to trade induces \(\varphi^{*}>\varphi^{*}_a\) where \(a\) signifies autarky, or a closed economy. The relation, as we have said, is preserved for average profit. Firms with \(\varphi \in (\varphi^{*}_a,\varphi^{*})\) are forced to exit because they can't earn positive profits. Another selection process is that only firms with productivity above \(\varphi^{*}_x\) enter the export market. Both output and domestic market selection effects reallocate market shares towards these firms which are more productive. We also have that \(M<M_a\) despite \(M_t>M_a\). So we have less domestic firms, but more variety. Even when the exit of too many domestic firms leads to lower \(M_t\), the positive gain in productivity dominates loss of variety.Trade always generates welfare gains.

Now we examine the effects of trade on firms with different \(\varphi\) levels. Suppose a firm has \(\varphi \geqslant \varphi^{*}_a\), with \(r_a(\varphi)\in\mathbb{R}^{+}\) and \(\pi_a(\varphi)\geqslant 0\) in autarky. The firm's market shares are \(\tfrac{r_a(\varphi)}{R=L}\) and \(\tfrac{r(\varphi)}{R}\) before and after trade. The domestic share is obviously \(\tfrac{r_d(\varphi)}{R}\). Thus \[\forall \: \varphi \geqslant \varphi^{*} \Rightarrow r_d(\varphi)<r_a(\varphi)<r_d (\varphi)+nr_x(\varphi)\] So we have firstly domestic loss, since trade forces all firms to lose a portion of their domestic sales. However, for those firms who export, the foreign market revenue makes up for the loss in domestic firms. Exporter firms increase their share of industry revenues. The direction of profit change \(\Delta \pi (\varphi)\) is clear when \(\varphi<\varphi^{*}_x\), revenue and variable profit are lower. For a firm with \(\varphi \geqslant \varphi^{*}_x\) we can write \[\begin{split}\Delta\pi(\varphi)=\pi(\varphi)-\pi_a(\varphi)&=\frac{1}{\sigma}\left[(r_d(\varphi)+n r_x(\varphi))-r_a(\varphi)\right]-n f_x \\\ &=\varphi^{\sigma-1}f\left[\frac{1+n\tau^{1-\sigma}}{(\varphi^{*})^{\sigma-1}}-\frac{1}{(\varphi_a^{*})^{\sigma-1}}\right]-n f_x\end{split}\] and from the last inequality we have that the term in the bracket is positive for all \(\varphi\geqslant\varphi^{*}_x\). Thus \(\Delta\pi(\varphi)\) is an increasing function of productivity. Since \(\varphi^{*}>\varphi^{*}_a\) then \(\varphi^{\sigma-1}\) is higher in trade for exporters. Firms are partitioned into groups that gain and lose profits. There are exitors \((\varphi<\varphi^{*})\), domestic losers who lose market share and profits (\(\varphi^{*}\leqslant \varphi < \varphi^{*}_x\)), exporting losers who increase market share but lose profit, and the ,pst efficient firms who increase both market share and profits. These effects are driven by labor. This is because CES keeps price elasticities constant and because it is the only factor of production. The reason why labor drives them is because trade creates new profit opportunities, and thus firms increase demand for labor, which bids the wage up, which increases the costs for everyone, and only the most productive firms can afford it.

We must also compare less to more trade, as very few countries are in autarky. The effects will be similar. Melitz investigates three mechanisms: increase in trading partners \(n\), decrease in fixed export costs \(f_x\), and decrease in variable trade costs \(\tau\). First, let us go from \(n^{\prime}>n\) countries. ZCP curve shifts up \(\varphi^{\prime} > \varphi^{*}\)and \(\varphi^{* \prime}_x>\varphi^{*}_x\). Increase in profits forces least productive firms to exit. All firms lose domestic market share \(\forall \: \varphi \geqslant \varphi^{*} \Rightarrow r_d^{\prime}(\varphi)<r_d(\varphi)\). Firms with \(\varphi < \varphi^{* \prime}_x\) lose revenue and profit. Firms that export with \(\varphi \geqslant \varphi^{* \prime}_x\) make up for their losses as \(r_d^{\prime}(\varphi)+n^{\prime}r_x^{\prime}(\varphi)>r_d(\varphi)+nr_x(\varphi)\). Some lose profits given new export costs, but most productive firms increase their profits. Market share and profits reallocate to these efficient firms. Aggregate productivity and welfare increase. A decrease in variable trade cost \(\tau\) leads to quasi-identical effects. Decrease \(\tau^{\prime}<\tau\) shifts up the ZCP curve and leads to \(\varphi^{* \prime}>\varphi^{*}\). The only difference from before will be \(\varphi^{* \prime}_x<\varphi^{*}_x\). Increased liberalization forces (again) least productive firms to exit, but from the difference in productivity level new firms will enter export market. Benefits are the same.Finally, a decrease in fixed export costs \(f_x\) induces similar changes as \(\tau\). We see \(\varphi^{*}\) rise and least productive firms exit, we see \(\varphi^{*}_x\) decrease, which adds new firms to export market. Aggregate productivity increases if these new entrants are more productive than the average. Less productive firms who don't export lose profits and market share, but this time market share and profit reallocation is not the same. When \(f_x\) falls it does not increase market share or profit of any previous exporter firms, and only new entrants increase their revenue. Welfare is higher.

This model has some unrealistic feature, such as the constant mark-ups \(\tfrac{\sigma}{\sigma-1}\) in the price elasticity \(p_d = \tfrac{\sigma}{\sigma-1}\tfrac{1}{\varphi}\), firms decisions that are independent of scale of the economy, only \(L\) as a factor of production, constant exit probability \(\delta\), and no additional uncertainty after entering the export market. Ottaviano and Melitz (2008) is next! This model was very hard to understand for me. The prose is also very difficult. My favorite part was the trade liberalization section, which I maybe found the easiest because I had digested the material a little. Classic paper.