3.1. Households

In our model, there are two types of households, ones that comprise skilled workers and ones that comprise unskilled workers.Footnote ⁹ In the following, we describe the problem of a skilled worker’s household, with analogous equations holding for unskilled workers’ households. The utility of a skilled worker’s household is given by:

(4) \begin{equation} E_{t}\left \{ \sum _{k=0}^{\infty }\gamma ^{k}\left [log\left (C_{t+k}^{S}\right )S-\sum _{i=1,2}\frac{\left (H_{it+k}^{S}\right )^{1+\upsilon }}{1+\upsilon }S_{it+k}\right ]\right \}, \end{equation}

where $C_{t+k}^{S}$ is the consumption of each worker, $H_{it+k}^{S}$ is the hours supplied by each worker that is employed in sector $i$ , $S_{it+k}$ is the number of workers in sector $i$ , $S=S_{1t}+S_{2t}$ is the total number of workers in the household, $\gamma$ is the subjective discount factor, and $\upsilon$ is the inverse of the Frisch elasticity of labor supply. So every household member receives the same consumption, but labor supply might differ across sectors.

Every period the household faces the following budget constraint written in terms of final consumption:

(5) \begin{eqnarray} & & SC_{t}^{S}+A_{t}^{S}+Q_{t}Af_{t}^{S}+\frac{\eta }{2}(A_{t}^{S})^{2}+Q_{t}\frac{\eta }{2}(Af_{t}^{S})^{2}\nonumber \\ & & =\sum _{i=1,2}w_{it}^{S}H_{it}^{S}S_{it}+A_{t-1}^{S}(1+r_{t-1})+Q_{t}Af_{t-1}^{S}(1+r_{t-1}^{*})+S\Pi _{t}+TA_{t}^{S} \end{eqnarray}

The left-hand side includes household expenditure on consumption $SC_{t}^{S}$ , H bonds $A_{t}^{S}$ , and F bonds $Af_{t}^{S}$ . F bonds are in terms of the foreign consumption good and thus adjusted by the real exchange rate $Q_{t}\equiv e_{t}P_{t}^{*}/P_{t}$ , defined as the relative price of F goods versus H goods. The nominal exchange rate $e_{t}$ is normalized to $1$ , since our model does not include any nominal rigidities. Note that households have to pay a quadratic adjustment cost for H bonds $\frac{\eta }{2}(A_{t}^{S})^{2}$ and F bonds $Q_{t}\frac{\eta }{2}(Af_{t}^{S})^{2}$ . These costs are paid to financial intermediaries whose only function is to collect these transaction fees and rebate them to the households in a lump-sum fashion. The purpose of these adjustment costs is to assure stationarity of the steady state (see GM for more details).

The right hand side of equation 5 includes the sources of income such as the wage income from both sectors $\sum _{i=1,2}w_{it}^{S}H_{it}^{S}S_{it}$ , interest income on H bond holdings $A_{t-1}^{S}(1+r_{t-1})$ and F bond holdings in last period $Q_{t}Af_{t-1}^{S}(1+r_{t-1}^{*})$ , profit transfers from a mutual fund that owns all firms $\Pi _{t}$ (to be defined in more detail below), and the bond adjustment cost rebate $TA_{t}^{S}$ . $r_{t-1}$ and $r_{t-1}^{*}$ are the real interest rates on H and F bond holdings.

The household chooses how much to consume, how many hours to work in both sectors, and how much H and F bonds to buy by optimizing its utility subject to the budget constraint. The optimization problem implies the following optimality conditions:

(6) \begin{equation} \frac{\left (H_{it}^{S}\right )^{\upsilon }}{\left (C_{t}^{S}\right )^{-1}}=w_{it}^{S}\text{ for }i=1,2, \end{equation}

(7) \begin{equation} (1+\eta A_{t}^{S})=\gamma E_{t}\left [\left(\frac{C_{t+1}^{S}}{C_{t}^{S}}\right)^{-1}(1+r_{t})\right ], \end{equation}

(8) \begin{equation} (1+\eta Af_{t}^{S})=\gamma E_{t}\left [\left(\frac{C_{t+1}^{S}}{C_{t}^{S}}\right)^{-1}(1+r_{t}^{*})\frac{Q_{t+1}}{Q_{t}}\right ]. \end{equation}

The first condition determines optimal labor supply by equating the marginal rate of substitution between leisure and consumption to the real wage. The other two conditions are the Euler equations that determine the optimal demand for H and F bonds, respectively.

As mentioned above, we distinguish two specifications concerning the mobility of workers across sectors. In our baseline case, workers are fully mobile in the long run but fully immobile in the short run. In the other case, workers are fully mobile across sectors both in the short run and in the long run, without any restrictions.Footnote ¹⁰ In the case of short-run mobility, the number of workers in each sector is pinned down by a further optimality condition which assures that the value generated by a worker is the same in each sector (following from maximizing utility with respect to $S_{1t+k}$ ):

(9) \begin{equation} w_{1t}^{S}H_{1t}^{S}\left (C_{t}^{S}\right )^{-1}-\frac{\left (H_{1t}^{S}\right )^{1+\upsilon }}{1+\upsilon }=w_{2t}^{S}H{}_{2t}^{S}\left (C_{t}^{S}\right )^{-1}-\frac{\left (H_{2t}^{S}\right )^{1+\upsilon }}{1+\upsilon }. \end{equation}

Under short-run mobility, equation 9 holds at any time. Under short-run immobility, it still pins down the steady-state distribution of workers across sectors, but no longer holds in the short run.

The composition of the aggregate consumption bundle is the same for all workers, only the quantity of consumed goods differs across skilled and unskilled workers. Therefore, in the following, we will omit the indices for the worker’s skill class to avoid cumbersome notation. The aggregate consumption bundle $C_{t}$ is a Cobb–Douglas composite of the goods produced in the two sectors:

(10) \begin{equation} C_{t}=C_{1t}^{\alpha }C_{2t}^{1-\alpha }, \end{equation}

where $\alpha$ is the share of good 1 in the consumption bundle for both H and F. We obtain relative demand functions for each good from the expenditure minimization problem of the household. The implied demand functions are

(11) \begin{equation} C_{1t}=\alpha \frac{P_{t}}{P_{1t}}C_{t}\,\,\,\,and\,\,\,\,C_{2t}=(1-\alpha )\frac{P_{t}}{P_{2t}}C_{t}, \end{equation}

where $P_{t}=\left (\frac{P_{1t}}{\alpha }\right )^{\alpha }\left (\frac{P_{2t}}{1-\alpha }\right )^{1-\alpha }$ is the price index that buys one unit of the aggregate consumption bundle $C_{t}$ .

Each sector good is a consumption bundle defined over a continuum of varieties $\Omega _{i}$ , both domestic and imported,

(12) \begin{equation} C_{it}=\left [\int _{\omega \varepsilon \Omega _{i}}c_{it}(\omega )^{\frac{\theta -1}{\theta }}d\omega \right ]^{\frac{\theta }{\theta -1}}, \end{equation}

where $\theta \gt 1$ is the elasticity of substitution between goods. At any given time, only a subset of varieties $\Omega _{it}\in \Omega _{i}$ is available in each sector. The consumption-based price index for each sector is $P_{it}=\left [\int _{\omega \varepsilon \Omega _{i}}p_{it}(\omega )^{1-\theta }d\omega \right ]^{\frac{1}{1-\theta }}$ and the household demand for each variety is

(13) \begin{equation} c_{it}(\omega )=\left (\frac{p_{it}(\omega )}{P_{it}}\right )^{-\theta }C_{it}. \end{equation}

It is useful to redefine the demand functions in terms of aggregate consumption units. To this end, let us define $\rho _{it}(\omega )\equiv \frac{p_{it}(\omega )}{P_{t}}$ and $\psi _{it}\equiv \frac{P_{it}}{P_{t}}$ as the relative prices for individual varieties and for the sector bundles, respectively. Then, we can rewrite the demand functions for goods and sector bundles as $c_{it}(\omega )=\rho _{it}(\omega )^{-\theta }C_{it}$ and $C_{it}=\alpha _{i}\psi _{it}^{-1}C_{t}$ , respectively, with $\alpha _{1}=\alpha$ and $\alpha _{2}=1-\alpha$ .

3.2. Firms

3.2.1. Production

Within each sector, there is a continuum of firms, each producing a different variety. The number of firms and varieties is endogenous and determined by the entry of firms. Newly entering firms face a sunk entry cost $f_{e}$ in effective labor units. The production technology is assumed to be Cobb–Douglas in the two inputs of production:

(14) \begin{equation} y_{it}=Z_{t}z\left (S_{it}H_{it}^{S}\right )^{\beta _{i}}\left (L_{it}H_{it}^{L}\right )^{(1-\beta _{i})} \end{equation}

where $Z_{t}$ is the aggregate productivity, which is the same in both sectors, $z$ is the firm-specific productivity, and $S_{it}$ and $L_{it}$ are the numbers of skilled and unskilled workers used in a firm, whereas $H_{it}^{S}$ and $H_{it}^{L}$ are the hours worked per worker. Since firm-specific variables are now indexed by the firm-specific productivity $z$ , we can omit the variety-index $\omega$ from now on.

As explained above, we use the extensive margin of labor supply to model comparative advantage and the intensive margin of labor supply to model adjustments in labor supply in response to business cycle shocks. Aggregate productivity follows an AR(1) process with autocorrelation $\rho _{z}$ and is subject to i.i.d. shocks $\varepsilon _{z}$ , following a normal distribution with mean 0 and standard deviation $\sigma _{z}$ . $\beta _{i}$ is the share of skilled labor required by a firm to produce one unit of output $y_{it}$ in sector $i$ . Sector 1 is assumed to be skill-intensive and sector 2 unskilled-intensive which implies that $1\gt \beta _{1}\gt \beta _{2}\gt 0$ . The labor market is assumed to be perfectly competitive implying that the hourly real wage of both skilled and unskilled workers equals their marginal products. Furthermore, workers are perfectly mobile across firms implying that all firms pay the same wage. Consequently, relative labor demand can be described by the following condition:

(15) \begin{equation} \frac{w_{it}^{S}}{w_{it}^{L}}=\frac{\beta _{i}}{(1-\beta _{i})}\frac{L_{it}H_{it}^{L}}{S_{it}H_{it}^{S}}, \end{equation}

which says that the ratio of the skilled hourly real wage $w_{it}^{s}$ to the unskilled hourly real wage $w_{it}^{L}$ for sector $i$ is equal to the ratio of the marginal contribution of each factor into producing one additional unit of output. Note that this condition implies that relative demand for labor is the same across firms and is independent of firm-specific productivity.

3.2.2. Firm heterogeneity

Firms are heterogeneous in terms of their productivity $z_{i}$ . The productivity differences across firms translate into differences in the marginal cost of production. Measured in the units of the aggregate consumption bundle, the marginal cost of production is $\frac{\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{z_{i}Z_{t}}$ .

Prior to entry, firms are identical and face a sunk entry cost $f_{e}$ . Upon entry firms draw their productivity level $z_{i}$ from a common distribution $G(z_{i})$ with support on $[z_{min},\infty )$ . This firm productivity remains fixed thereafter. As in GM, there are no fixed costs of production, so that all firms produce each period until they are hit by an exit shock. This exit shock is independent of the firm’s productivity level, so $G(z_{i})$ also represents the productivity distribution of all producing firms.

Exporting goods to F is costly and involves both an iceberg trade cost $\tau \geq 1$ and a fixed cost $f_{x}$ , again measured in units of effective labor. In real terms, these costs are $\frac{f_{x}\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}}$ . The fixed cost of exporting implies that not all firms find it profitable to export.

All firms are monopolistic competitors and face the residual demand curve, equation 13. They set prices as a proportional markup $\frac{\theta }{\theta -1}$ over marginal cost. Let $p_{d,it}(z_{i})$ and $p_{x,it}(z_{i})$ denote the nominal domestic and export prices of an H firm in sector $i$ . We assume that the export prices are denominated in the currency of the export market. Prices in real terms are then given by:

(16) \begin{equation} \rho _{d,it}(z_{i})=\frac{p_{d,it}(z_{i})}{P_{t}}=\frac{\theta }{\theta -1}\frac{\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}z_{i}},\rho _{x,it}(z_{i})=\frac{p_{x,it}(z_{i})}{P_{t}^{\ast }}=\frac{1}{Q_{t}}\tau \rho _{d,it}(z_{i}). \end{equation}

Profits, expressed in units of the aggregate consumption bundle of the firm’s location, are the sum of domestic $d_{d,it}(z_{i})$ and export profits $d_{x,it}(z_{i})$ , such that $d_{it}(z)=d_{d,it}(z_{i})+d_{x,it}(z_{i})$ , and,

(17) \begin{equation} d_{d,it}(z_{i})=\frac{1}{\theta }\left (\frac{\rho _{d,it}(z_{i})}{\psi _{it}}\right )^{1-\theta }\alpha _{i}C_{t} \end{equation}

(18) \begin{equation} d_{x,it}(z_{i})=\begin{array}{cc} \begin{array}{c} \frac{Q_{t}}{\theta }\left (\frac{\rho _{x,it}(z_{i})}{\psi _{it}^{*}}\right )^{1-\theta }\alpha _{i}C_{t}^{\ast }-\frac{f_{x}\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}},\end{array} & \mbox{if firm}{z_{i}} \mbox{exports}\\ 0 & \mbox{otherwise.} \end{array} \end{equation}

A firm will export if and only if it earns non-negative profits from doing so. For H firms, this will be the case if their productivity draw $z_{i}$ is above some cutoff level $z_{x,it}=\mbox{inf}\{z\;:\;d_{x,it}\gt 0\}$ . We assume that the lower bound productivity $z_{min}$ is identical for both sectors and low enough relative to the fixed costs of exporting so that $z_{x,it}$ is above $z_{min}$ . Firms with productivity between $z_{min}$ and $z_{x,it}$ serve only their domestic market.

3.2.3. Firm averages

In every period, a mass $N_{d,it}$ of firms produces in sector $i$ of country H. The number of exporters is $N_{x,it}=\left [1-G(z_{x,it})\right ]N_{d,it}$ . It is useful to define two average productivity levels, an average $\tilde{z}_{d,it}$ for all producing firms in sector $i$ of country H and an average $\tilde{z}_{x,it}$ for all exporters in sector $i$ of country H:

\begin{equation*} \tilde {z}_{d,it}=\left [\int _{z_{min}}^{\infty }z^{\theta -1}dG(z)\right ]^{\frac {1}{(\theta -1)}},\,\,\,\tilde {z}_{x,it}=\left [\int _{z_{x,it}}^{\infty }z^{\theta -1}dG(z)\right ]^{\frac {1}{(\theta -1)}}. \end{equation*}

As in Melitz (Reference Melitz2003), these average productivity levels summarize all the necessary information about the productivity distributions of firms.

We can redefine all the prices and profits in terms of these average productivity levels. The average nominal price of H firms in the domestic market is $\tilde{p}_{d,it}=p_{d,it}(\tilde{z}_{d,it})$ and in the foreign market is $\tilde{p}_{x,it}=p_{x,it}(\tilde{z}_{x,it})$ . The price index for sector $i$ in H reflects prices for the $N_{d,it}$ home firms and F’s exporters to H. Then, the price index for sector $i$ in H can be written as $P_{it}^{1-\theta }=[N_{d,it}\left (\tilde{p}_{d,it}\right )^{1-\theta }+N_{x,it}^{*}\left (\tilde{p}_{x,it}^{*}\right )^{1-\theta }]$ . Written in real terms of aggregate consumption units, this becomes $\psi _{it}^{1-\theta }=[N_{d,it}\left (\tilde{\rho }_{d,it}\right )^{1-\theta }+N_{x,it}^{*}\left (\tilde{\rho }_{x,it}^{*}\right )^{1-\theta }]$ , where $\tilde{\rho }_{d,it}=\rho _{d,it}(\tilde{z}_{d,it})$ and $\tilde{\rho }_{x,it}^{*}=\rho _{x,it}^{*}(\tilde{z}_{x,it}^{*})$ are the average relative prices of H’s producers and F’s exporters.

Similarly, we can define $\tilde{d}_{d,it}=d_{d,it}(\tilde{z}_{d,it})$ and $\tilde{d}_{x,it}=d_{x,it}(\tilde{z}_{x,it})$ such that $\tilde{d}_{it}=\tilde{d}_{d,it}+\left [1-G(z_{x,it})\right ]\tilde{d}_{x,it}$ are average total profits of H firms in sector $i$ .

3.2.4. Productivity distribution of firms

Productivity $z$ follows a Pareto distribution with lower bound $z_{min}$ and shape parameter $k\gt \theta -1$ : $G(z)=1-\left (\frac{z_{min}}{z}\right )^{k}$ . Let $\nu =\left \{ \frac{k}{\left [k-(\theta -1)\right ]}\right \} ^{\frac{1}{\theta -1}}$ , then average productivities are

(19) \begin{equation} \tilde{z}_{d,it}=\nu z_{min}\mbox{ and }\tilde{z}_{x,it}=\nu z_{x,it}. \end{equation}

The share of exporting firms in sector i in H is

(20) \begin{equation} \frac{N_{x,it}}{N_{d,it}}=1-G(z_{x,it})=1-\left (\frac{\nu z_{min}}{\tilde{z}_{x,it}}\right )^{k}. \end{equation}

Together with the zero export profit condition for the cutoff firm, $d_{x,it}(z_{x,it})=0$ , this implies that average export profits must satisfy

(21) \begin{equation} \tilde{d}_{x,it}=\left (\theta -1\right )\left (\frac{\nu ^{\theta -1}}{k}\right )\frac{f_{x}\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}}. \end{equation}

3.2.5. Firm entry

We assume that all firms in a given country are owned by a mutual fund who invests in new firms on behalf of the entire population, collects all the profits, and distributes the surplus of profits over firm investment in a lump-sum fashion.Footnote ¹¹ To set up a new firm, the sunk entry cost $\frac{f_{e}\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}}$ has to be paid. Note that entry costs can differ between sectors due to different factor intensities and due to inter-sectoral wage differentials.

All firms are subject to exit shocks, which occur with probability $\delta \in (0,1)$ at the end of each period. We assume that entrants at time $t$ only start producing at time $t+1$ , which introduces a one-period time-to-build lag in the model. Thus, a proportion $\delta$ of new entrants will never produce. The number of existing firms is denoted by $N_{d,it}$ and the number of newly entering firms by $N_{e,it}$ . Then the law of motion for the stock of producing firms can be written as:

(22) \begin{equation} N_{d,it}=(1-\delta )(N_{d,it-1}+N_{e,it-1}). \end{equation}

The present discounted value of expected profits is

(23) \begin{equation} \widetilde{v}_{it}=E_{t}\sum _{s=t+1}^{\infty }\left [\gamma ^{s-t}(1-\delta )^{s-t}\left (\frac{C_{s}}{C_{t}}\right )^{-1}\widetilde{d}_{is}\right ]. \end{equation}

Firm profits are discounted using the aggregate stochastic discount factor adjusted for the probability of firm survival $(1-\delta )$ . Note that equation 23 can be written in recursive form as:

(24) \begin{equation} \widetilde{v}_{it}=\gamma (1-\delta )E_{t}\left [\left (\frac{C_{t+1}}{C_{t}}\right )^{-1}\left (\widetilde{v}_{it+1}+\widetilde{d}_{it+1}\right )\right ]. \end{equation}

Entry occurs until firm value is equal to the entry cost:

(25) \begin{equation} \widetilde{v}_{it}=\frac{f_{e}\left (w_{it}^{S}\right )^{\beta _{i}}\left (w_{it}^{L}\right )^{1-\beta _{i}}}{Z_{t}}. \end{equation}

Finally, the surplus of the mutual fund is given by:

(26) \begin{equation} \Pi _{t}=\widetilde{d}_{1t}N_{d,1t}+\widetilde{d}_{2t}N_{d,2t}-\widetilde{v}_{1t}N_{e,1t}-\widetilde{v}_{2t}N_{e,2t} \end{equation}

3.3. Market clearing conditions, aggregate accounting, and trade

Market clearing requires that total production in each sector must equal total income so that:

(27) \begin{equation} N_{d,it}\left (\frac{\widetilde{\rho }_{d,it}}{\psi _{it}}\right )^{1-\theta }\alpha _{i}C_{t}+Q_{t}N_{x,it}\left (\frac{\tilde{\rho }_{x,it}}{\psi _{it}^{*}}\right )^{1-\theta }\alpha _{i}C_{t}^{\ast }+\widetilde{v}_{it}N_{e,it}=w_{it}^{S}S_{it}H_{it}^{S}+w_{it}^{L}L_{it}H_{it}^{L}+\widetilde{d}_{it}N_{d,it}. \end{equation}

Total production of the sector includes the production of the aggregate consumption bundle and the production of new firms. Total income generated by the sector includes wage earnings and profits.

The trade balance is defined as total exports minus total imports:

(28) \begin{equation} tb_{t}=\sum _{i=1}^{2}\left [Q_{t}N_{x,it}\left (\frac{\widetilde{\rho }_{x,it}}{\psi _{it}^{*}}\right )^{1-\theta }\alpha _{i}C_{t}^{*}-N_{x,it}^{*}\left (\frac{\widetilde{\rho }_{x,it}^{*}}{\psi _{it}}\right )^{1-\theta }\alpha _{i}C_{t}.\right ] \end{equation}

Let us define aggregate bond holdings in H as $A_{t}\equiv \sum _{i=1}^{2}(A_{it}^{S}+A_{it}^{L}$ ) and $Af_{t}\equiv \sum _{i=1}^{2}(Af_{it}^{S}+Af_{it}^{L})$ for H and F bonds, respectively. Similarly, aggregate bond holdings in F are $A_{t}^{*}\equiv \sum _{i=1}^{2}(A_{it}^{*S}+A_{it}^{*L}$ ) and $Af_{t}^{*}\equiv \sum _{i=1}^{2}(Af_{it}^{*S}+Af_{it}^{*L})$ . In equilibrium, the international net supply of bonds is zero for H bonds such that $A_{t}+A_{t}^{*}=0$ and for F bonds such that $Af_{t}+Af_{t}^{*}=0$ . Then net foreign assets evolve according to the following law of motion:

(29) \begin{equation} A_{t}+Q_{t}Af_{t}=\left (1+r_{t-1}\right )A_{t-1}+\left (1+r_{t-1}^{*}\right )Af_{t-1}Q_{t}+tb_{t}. \end{equation}

Finally, if workers are mobile between sectors, then, at each point of time the sum of workers employed in both sectors equals the exogenous worker endowment for skilled and unskilled workers, respectively, such that $S=\sum _{i=1}^{2}S_{it}$ and $L=\sum _{i=1}^{2}L_{it}$ . Equivalent equations hold for F.

5.1. Aggregate productivity shock at Home

We start our discussion with our baseline case where workers are assumed to be immobile across sectors. This assumption is in line with recent empirical evidence suggesting that workers are immobile both geographically and across sectors. Furthermore, in section 6 we provide new empirical evidence that sectoral employment reacts only weakly to productivity shocks. Nevertheless, we will discuss the role of worker mobility in more detail further below (section 5.5) and present a version of the model with costly worker mobility in section 6.

Figure 3 shows the development of selected variables at Home in response to a drop in domestic aggregate productivity. The figure reports data-consistent measures of GDP, sector output, sector revenue, sector firm investment, and sector wages, correcting for the change in the number of varieties.Footnote ¹⁴

Figure 3.

Domestic productivity shock. Impulse responses to a decline in domestic aggregate productivity. Variables are measured in %-deviations from steady state. Solid lines either refer to aggregate variables (first and second rows) or sector 1, the exporting sector(all other rows) while dashed lines refer to sector 2, the import-competing sector.

A negative productivity shock leads to a contraction in output because production becomes less efficient and real marginal costs ( $(w_{it}^{S})^{\beta _{i}}(w_{it}^{L})^{1-\beta _{i}}/\left (z_{i}Z_{t}\right )$ ) rise. Output contracts in both sectors but the contraction is much larger in the import-competing sector than in the exporting sector, so that output and revenue become relatively more concentrated in the exporting sector. Thus, the temporary fall in Home productivity leads to a temporary increase in specialization. Note that this is very much in line with the stylized facts developed in section 2, where we show empirically that in the USA comparative disadvantage sectors are significantly more responsive to changes in aggregate productivity than comparative advantage sectors. According to the empirical analysis, the reaction of import-competing sectors to shocks in aggregate productivity is 1.5 times stronger than the reaction of exporting sectors in the year that the shock hits. Our simulation matches this result very well.

The shift in specialization is driven by movements in relative demand and relative prices. As Figure 3 shows, the fall in productivity leads to an appreciation in the welfare-consistent RER, meaning that foreign products get cheaper relative to domestic products. This is the case because the fall in productivity makes Home production less efficient. Marginal costs increase and thus the price level at Home increases relative to the price level at Foreign. Figure 3 also reports the data-consistent RER which moves in the opposite direction of the welfare-consistent RER, illustrating that the appreciation of the welfare-consistent RER is driven by the reduction in the number of Home’s varieties. Note that the development of both measures of the RER is the same as in GM (see figure III on page 892).

The appreciation of the welfare-consistent RER is associated with an increase in Home’s terms of trade. As in the standard Heckscher–Ohlin model, an increase in the terms of trade translates into an increase in specialization. Our model is more complicated since it incorporates both inter- and intra-industry trade, but the basic mechanism remains intact. In the standard Heckscher–Ohlin model, the exporting sector only exports and does not import, while the import-competing sector only imports and does not export. This is different in our model due to the co-existence of comparative advantage and love of variety, but it is still the case that exports are more concentrated in the exporting sector.Footnote ¹⁵ Since exports are concentrated in the exporting sector, this sector depends relatively more on foreign demand and relatively less on domestic demand. To the contrary, the import-competing sector depends relatively more on domestic demand.Footnote ¹⁶ Thus, in response to a negative shock to aggregate domestic productivity the import-competing sector faces a relatively larger decline in demand. Relative demand and the relative price of the two sectors move in favor of the exporting sector. Note, however, that the movement in relative prices is relatively mild, implying that concentration in terms of revenue increases only little more than concentration in terms of output. This is again in line with our empirical analysis in section 2.

Naturally, the shift in the relative demand for the products of both sectors has implications for the relative demand for production factors. While the negative productivity shock lowers the demand for labor and firms in general, the shift in relative product-demand lowers factor-demand more in the import-competing sector. If workers and firms were mobile, they would migrate from the import-competing sector to the exporting sector. In the absence of this possibility to migrate, it is only hours worked and firm investment that can react. Consequently, firm investment goes down in both sectors, but it goes down more in the import-competing sector. Analogously, hours worked, along with wages, go down in both sectors but more in the import-competing sector. Note, however, that the sectoral shifts in firm investment and hours worked are very small relative to the general decline of both variables.

Figure 3 also shows that the hours worked go down more for unskilled workers than for skilled workers, which is in line with the data.Footnote ¹⁷ This is explained by the distribution of profits from the mutual fund. In response to the negative productivity shock, the mutual fund reduces investment in new firms, and therefore can redistribute more profits to the households, even though firm profits actually go down. Thus, the transfers of the mutual fund are countercyclical and stabilize consumption. This mechanism is analogous to a decision to dissave during times of crises to smooth consumption. The cyclicality of consumption is important for the cyclicality of the labor supply because labor supply is determined by both the real wage and the marginal utility from consumption (see equation 6). Due to the stabilizing effects of the mutual fund, consumption goes down by less, the marginal utility from consumption goes up by less, and thus labor supply goes down by more.

For unskilled workers, the stabilizing effect of the mutual fund is more important, because unskilled workers have a lower wage income and thus the transfers of the mutual fund make up a bigger portion of their income. The marginal utility from consumption increases by less for unskilled than for skilled workers, and therefore they reduce their labor supply more in response to the drop in the real wage.Footnote ¹⁸ Arguably, the large volatility of hours worked for unskilled workers in the data is not all due to voluntary labor supply decisions but to a certain extent also due to insufficient labor demand. This aspect could be captured by including labor market frictions which is left for future research.

The diverging development of the labor supply of unskilled and skilled workers in turn contributes to the sectoral shift in production. Since unskilled workers are relatively more important in the import-competing (unskilled-intensive) sector, this shift in the relative labor supply decreases output in the import-competing sector relative to output in the exporting sector.

5.2. Inequality

Our model, in deviating from the standard representative household framework, allows us to discuss inequality across various margins. Figure 4 shows the development of wage and income differences across sectors, across skill classes, and in the whole economy (for a precise definition of these inequality measures, see the appendix).

Figure 4.

Inequality. Impulse responses to a decline in domestic aggregate productivity.

As discussed above, the shift in relative demand for both sectors that follows a decrease in aggregate productivity raises the relative demand for labor in the exporting sector. Because workers are immobile across sectors, this is reflected in a temporary increase in the wage of the exporting sector relative to the import-competing sector.Footnote ¹⁹

The aggregate productivity shock also has consequences for the wage inequality between skilled workers and unskilled workers. As explained above, the drop in aggregate productivity leads to a shift in relative demand from the import-competing sector to the exporting sector. Since the exporting sector is skill-intensive, this also implies relatively more demand for skilled workers than for unskilled workers. As a consequence, the share of total wage income (the wage times hours worked) that goes to skilled workers increases temporarily.

Figure 4 also reports that the skill premium goes down which seems to contradict the argument above. Note, however, that this can be explained by the stronger reduction in hours worked of unskilled workers. This has the effect of making unskilled workers relatively scarcer and that pushes down the skill premium. So the shift in relative demand raises the relative wage income of skilled workers. Nevertheless, the decrease in the supply of unskilled labor lowers the skill premium.

These developments are reflected in overall wage inequality measured by the Gini coefficient, which is based on the deviation of income from a totally equal distribution. Since the Gini coefficient measures income inequality and not just wage inequality, it depends more on the relative wage income of skilled and unskilled workers, and not so much on the skill premium, and thus goes up. Note, however, that the Gini coefficient quantitatively changes only little.

Let us stress again that this increase in wage inequality is driven to a large extent by the different responses of skilled and unskilled workers to the technology shock: Unskilled workers reduce their labor supply by more than skilled workers. This reduces their income by more and wage inequality goes up. Since unskilled workers can enjoy more leisure, wage inequality is only an imperfect measure of welfare inequality in this setting. To this end, Figure 4 also shows a Gini of “utility inequality” that is constructed in the exact same way as the income Gini, but based on the total period utility of a worker rather than just his income. This makes a big difference, the inequality of utility actually goes down in response to the aggregate productivity shock reflecting the larger increase in leisure for unskilled workers.

5.3. Productivity shocks at Foreign

Figure 5 shows the effects of a temporary productivity shock at Foreign for the economy at Home. As expected, the development of prices is the exact opposite as for the domestic productivity shock. Home’s welfare-consistent RER depreciates because Foreign produces less efficiently and thus its marginal costs increase.Footnote ²⁰ Consequently, Home’s import-competing sector faces lower competition from abroad and can expand its revenue and output, while its exporting sector faces decreased foreign demand and thus contracts. Correspondingly, the relative price of both sectors moves in favor of the import-competing sector.

Figure 5.

Foreign productivity shock. Impulse responses to a decline in foreign aggregate productivity. Variables are measured in %-deviations from steady state. Solid lines either refer to aggregate variables (first and second row) or sector 1, the exporting sector, (all other rows) while dashed lines refer to sector 2, the import-competing sector.

Although the expansion in the import-competing sector is relatively larger than the contraction in the exporting sector, the larger size of the latter implies that domestic GDP goes down. That domestic GDP is negatively affected by the foreign shock already suggests that the correlation of GDP in both countries is positive, something that is in line with empirical data but often a challenge for RBC models to replicate.

Figure 6 also reports some measures of inequality. These develop in the opposite direction compared to the domestic shock. The shift in relative demand toward the import-competing sector implies that wages in the import-competing sector go up relative to those in the exporting sector, and that the demand for unskilled workers rises. This implies that the skill premium and the relative wage income that goes to skilled workers decrease. Overall wage inequality is largely driven by the inequality between skilled and unskilled workers and therefore decreases, even though inter-sector inequality increases. Again our welfare-Gini goes in the opposite direction, driven by the development of leisure.

Figure 6.

Inequality. Impulse responses to a decline in foreign aggregate productivity.

5.4. Business cycle statistics

This section presents (partially) new evidence on business cycle statistics for the US-economy vis-a-vis its Chinese counterpart and compares them to the same statistics generated by our model for the productivity shock processes specified in section 4. In doing so, we focus specifically on the novel aspect of our analysis, the distinction between comparative advantage and comparative disadvantage sectors.

We report statistics for data on the USA and China for relevant aggregate variables as well as some sector-specific variables for the USA. All data series are log-transformed, quarterly, seasonably adjusted, hp-filtered (with smoothing parameter 1600), and span the period from 1992q1 to 2007q3. The period of analysis is restricted by the availability of data for China and exclusion of the Great Recession period. We report (real) GDP, which is nominal GDP deflated by the CPI, investment I, which is nominal gross fixed capital formation deflated by the CPI, and consumption C, which is nominal consumption expenditure deflated by the CPI.Footnote ²¹ We also report the bilateral real exchange rate of the USA vis a vis China defined as $RER_{US,CHN}=\frac{eCPI^{CHN}}{CPI^{US}}$ , where $e$ is the nominal exchange rate in dollars per yuan and an increase in $RER_{US,CHN}$ corresponds to a real depreciation of the dollar.Footnote ²²

We also report measures on the trade balance as a percent of GDP, where $tb/GDP$ is the net exports between China and the USA divided by nominal US GDP.Footnote ²³ $Y_{CD}$ and $Y_{CA}$ are averages of industrial production over comparative disadvantage and comparative advantage sectors, respectively, where comparative advantage refers to net exporting sectors and comparative disadvantage refers to net importing sectors as outlined in our calibration approach in section 4.Footnote ²⁴ The measure of the relative sector price, $P_{CD}/P_{CA}$ , is the log of the comparative disadvantage sector price minus the log of the comparative advantage sector price; due to data limitations this is an annual series, hp-filtered with smoothing parameter 100 and compared to annual real GDP in the tables.Footnote ²⁵ Finally, we also report the US skill premium, defined as the ratio of the hourly wage of skilled to unskilled workers and $Hours^{S}/Hours^{L}$ , defined as the relative hours worked of skilled to unskilled workers.Footnote ²⁶ We report these two measures because they are important indicators in our model and data on them are available.

The second and fourth columns in Table 1 report the relevant moments based on the data described above. In section 2, we have shown that in the USA import-competing sectors are more volatile conditional on shocks to aggregate productivity. Table 1 confirms that the same is true for unconditional volatilities. The standard deviation of output in comparative disadvantage sectors is about $70\%$ larger than the standard deviation of output in comparative advantage sectors. Sector output is positively correlated with GDP but far from perfectly. Sector output is more volatile than GDP. This is to be expected given that variations at the sector level might offset each other to a certain extent. The price of comparative disadvantage sectors relative to the price of comparative advantage sectors is also volatile and procyclical, i.e., during a recession the price of comparative advantage sectors goes up relative to the price of comparative disadvantage sectors.

Table 1.

Baseline model versus data

Notes: The first row reports the relative standard deviation of real GDP for the model and the standard deviation of real GDP for the data in percent. The rest of the rows report the standard deviations of relevant variables relative to the volatility of real GDP. Model-based moments are based on raw data while data moments are based on hp-filtered logged data.

The results for the aggregate variables are standard. Investment is more volatile than GDP, while consumption is less volatile. Both variables are almost perfectly correlated with GDP. The US–China RER is more volatile than GDP and recessions in the USA tend to be associated with RER depreciations, in the sense than US goods become cheaper relative to Chinese goods. GDP in China is more volatile than GDP in the USA and the correlation between both is $0.3$ . This is in line with Fidrmuc and Korhonen (Reference Fidrmuc and Korhonen2015) who summarize previous research on China’s business cycle correlation with other countries and report a mean correlation coefficient of 0.245 based on 24 surveyed papers. So the business cycle comovement between the USA and China is significant, albeit lower than the one among developed countries (see, e.g., Backus et al. (Reference Backus, Kehoe and Kydland1992) or Cunat and Maffezzoli (Reference Cunat and Maffezzoli2004)). The trade balance between the two countries is surprisingly stable. Our analysis also shows that the skill premium is less volatile than GDP and weakly procyclical, which is in line with the evidence in Keane and Prasad (Reference Keane and Prasad1993), Lindquist (Reference Lindquist2004), and Castro and Coen-Pirani (Reference Castro and Coen-Pirani2008), and that the ratio of hours worked of skilled workers to hours worked of unskilled workers is less volatile than GDP, less than $1$ and countercyclical, confirming that hours worked of unskilled workers are more volatile than hours worked of skilled workers.

The first and third columns of Table 1 provide the analogous moments for our benchmark model economy, subject to domestic and foreign productivity shocks as specified in section 4. It can be seen that the model does a reasonably good job in replicating many important stylized facts of the business cycle.

Importantly, all variables, except the trade balance and the RERFootnote ²⁷ , exhibit the right cyclicality, i.e., the right sign of correlation with GDP. Given that the trade balance is extremely stable over time both in the data and in the model, its wrong cyclicality is not really meaningful. In the model, the correlation between RER and GDP is close to zero while it is slightly negative in the data. Note, however, that the zero-correlation in the model is the effect of two counter-acting effects. In response to domestic shocks, the RER is countercyclical and $-0.18$ , very close to the $-0.2$ found in the data. However, in response to foreign shocks the response is procyclical and the correlation very strong ( $0.95$ ). Thus, the model probably over-predicts the importance of RER-responses to foreign shocks. Further below, we will demonstrate that labor mobility also matters for the cyclicality of the RER.

The volatility of GDP is reasonably close to the data, although the model under-predicts US GDP volatility and over-predicts Chinese GDP volatility, so that in the model China’s GDP appears much more volatile than the USA’s, whereas this is not the case in the data. The correlation of US and Chinese GDP in the model is smaller than in the data ( $0.12$ vs. $0.3$ ), but larger than in typical RBC models (without correlated shocks).Footnote ²⁸ As in GM, the model generates consumption volatility that is too low. Investment, as in GM measured as firm investment, is more volatile than GDP both in the data and in the model, but the model clearly over-predicts the volatility of investment. Note, however, that investment in model and data is defined differently. The volatility of the RER in the model is too low relative to the data. This is analogous to GM, who argue that this reflects fluctuations in the nominal exchange rate that have no impact on real variables in models with flexible prices.

Let us now turn to some of the variables that are specific to our model, the variables that pertain to the sectors. The GDP-correlation of both sectoral output measures is close to their empirical counterparts. However, the model does predict that the comparative advantage sector is more procyclical than the comparative disadvantage one, contrary to the data. Note, however, that the difference between the correlation coefficients in the data is small (0.09) and not statistically significant. In addition, the volatility of both sectoral output measures is substantially lower than in the data. In a way this is to be expected, given that we only consider aggregate productivity shocks that affect both sectors equally.Footnote ²⁹ Importantly, however, the model does a very good job in replicating the relative volatility of import-competing and exporting sectors, which is $1.73$ in the data and $2$ in the model. Thus, import-competing sectors are substantially more volatile than exporting sectors both in the model and in the data. This is reassuring since this is a central aspect of our analysis.

The model also does a fairly good job in replicating the cyclicality of the relative price of both sectors, even if the ratio is less volatile relative to the data. Concerning the skill premium and the relative hours worked of skilled and unskilled workers, the model succeeds in replicating their volatility relative to the volatility of GDP. However, both variables exhibit a GDP-correlation that is too high. In our model, the business cycle is the only factor driving both variables, while in the data obviously other factors also play an important role.

5.5. The role of worker mobility and international trade

Having established the reasonable performance of our preferred model with respect to business cycle statistics, we now compare the model with other versions of the model to highlight the role of international trade, of workers’ mobility, and of the structure of international trade. To this end, we simulate an autarky version of the model, a version in which workers are freely mobile across sectors and a one-sector version of the model (basically the model in GM).Footnote ³⁰

Tables 2 and 3 compare the relative standard deviations of selected variables and their correlations with GDP for the empirical data and the four versions of the model. On a general note, the results for GDP, investment, and consumption are very similar across the different versions of the model, but important differences arise with respect to the volatility of sectoral output, relative prices, and international business cycle comovement. We now discuss each version of the model in more detail.

Table 2.

Models versus data volatility

Notes: The first row reports standard deviations of real GDP in percent. The rest of the rows report the standard deviations of relevant variables relative to the volatility of real GDP.

Table 3.

Models versus data correlation with GDP

5.5.4. Trade openness

Naturally, the openness to trade plays a crucial role in determining business cycle comovement. To further highlight this, Figure 7 shows the volatility of GDP at Home and the correlation between the GDP of both countries for different levels of iceberg trade costs, which are the most prominent measure of openness to trade in theoretical trade models. The figure compares our benchmark model with worker immobility, the two-sector model with worker mobility, and the one-sector model.

Figure 7.

Effect of trade costs on volatility and correlation: models comparison.

The right panel of Figure 7 reveals that for all models, higher openness to trade leads to larger business cycle comovement across the two countries. Note, however, that this effect is much stronger for our benchmark model. For high trade costs, all three versions of the model yield very similar and very low cross-country correlations of GDP. As trade costs become smaller, correlations increase substantially for the benchmark model but only little for the other two models. Thus, Figure 7 confirms that it is primarily the immobility of workers across sectors that enhances business cycle comovement.

Regarding the volatility of GDP, we observe a surprising pattern (in the left panel of Figure 7). Whereas in the one-sector model, openness to trade hardly affects the volatility of GDP, it increases GDP volatility in the two-sector model with mobile workers (unless trade costs become very small) and decreases GDP volatility in the two-sector model with immobile workers. The immobility of workers pushes more of the adjustment in response to business cycle shocks toward relative prices, which dampens the volatility of the import-competing sector, stabilizing GDP to a certain extent. The freer the international trade is, the more important this channel becomes. To the contrary, when workers are mobile they move quickly to the expanding sector which enhances especially the volatility of the import-competing sector.