2.1. Households

The lifetime utility of the representative dynastic household in the home country is

(1) \begin{align} U=\sum ^{\infty }_{t=0} \frac{\ln{C_{t}}}{(1+\rho )^t}, \end{align}

where $C_t$ denotes the consumption of final goods in period $t$ , and $\rho \gt 0$ is the subjective discount rate. The household maximizes lifetime utility (1) subject to the following asset accumulation equation:

(2) \begin{align} A_{t+1}=(1+r_t)A_{t}+w_{t}L_t+b_{t}\left (\bar{L}-L_{t}\right ) +\Pi _{t}-T_{t}-C_{t}, \end{align}

where $A_{t}$ is household assets, $r_t$ is the real interest rate, $w_t$ is the wage rate, $\bar{L}$ is the inelastic labor supply, $L_{t}$ is the number of employed workers, $b_t$ is the unemployment benefit provided to unemployed workers, and $T_t$ is a lump-sum tax levied by the government. The household owns all domestic final good producers and receives $\Pi _t$ in dividend income each period. In addition, all household members enjoy equal consumption levels, regardless of their employment status, eliminating consumption and employment uncertainty (Merz, Reference Merz1995; Andolfatto, Reference Andolfatto1996). We adopt the final good as the model numeraire and set its price equal to one.

Solving the household’s dynamic optimization problem, we derive the following Euler equation:

(3) \begin{align} \frac{C_{t+1}}{C_{t}}=\frac{1+r_{t+1}}{1+\rho }. \end{align}

Perfect international capital mobility ensures a common evolution for home and foreign household consumption, as the interest rate is equalized across countries: $C_{t+1}/C_{t}$ $=$ $C^{\ast }_{t+1}/C^{\ast }_{t}$ $=$ $(1+r_{t+1})/(1+\rho )$ , where variables associated with foreign are indicated with an asterisk. The foreign household faces a similar utility maximization problem and therefore has similar demand conditions.

2.2. Government

A relatively passive role is set for the government: distributing unemployment benefits and levying a lump-sum tax $T_{t}$ on the household to balance the fiscal budget. The balanced-budget condition is

(4) \begin{align} T_{t}=b_{t}\left (\bar{L}-L_{t}\right ). \end{align}

To avoid degeneration of unemployment benefits along the balanced growth path, we assume that the unemployment benefit $b_{t}$ is linked proportionately with final output $Y_{t}$ ; that is, $b_{t}=mY_{t}$ , where $m\gt 0$ is a policy parameter that sets the ratio of the unemployment benefit to final good output in the home country. Similarly, the balanced-budget condition for the foreign government is $T^{\ast }_{t}=b^{\ast }_{t} \left (\bar{L}^{\ast }-L^{\ast }_{t}\right )$ , with $b^{\ast }_{t}=m^{\ast }Y^{\ast }_{t}$ , where $m^{\ast }\gt 0$ . Note that we allow unemployment benefit policies, $m$ and $m^{\ast }$ , to differ across countries.

2.3. Final good production

Final goods are produced for a competitive international market characterized by free trade. Following Chang and Hung (Reference Chang and Hung2016), the mass of final good firms is fixed to one in each country. We adopt a Cobb-Douglas formation for the aggregate production function. For example, in the home country we have

(5) \begin{align} Y_{t}=Z_{t}^\alpha L_{t}^\beta, \end{align}

where $Z_{t}$ is an intermediate composite and $L_{t}$ is labor employment. The factor intensities associated with the intermediate composite and labor are $\alpha \in (0,1)$ and $\beta \in (0,1)$ . Following Palokangas (Reference Palokangas1996, Reference Palokangas2005) and Chang et al. (Reference Chang, Shaw and Lai2007), we assume that the production function exhibits decreasing returns to scale, with $\alpha + \beta \lt 1$ , thereby allowing final good producers to generate positive profits for the labor union and the employer federation to negotiate over.

The intermediate composite is formulated as a CES aggregator over the varieties available in period $t$ :

(6) \begin{align} Z_{t}=\left (\int ^{n_t}_{0} x_{i,t}^{\epsilon } di +\int ^{n^{\ast }_t}_{0} x_{j,t}^{\epsilon } dj \right )^{\frac{1}{\epsilon }}, \end{align}

where $x_{i,t}$ is the demand for intermediate variety $i$ of the $n_t$ mass of varieties produced in home, $x_{j,t}$ is the demand for variety $j$ of the $n^{\ast }_t$ mass of varieties produced in foreign, and the constant elasticity of substitution between any given pair of varieties is measured by $1/(1-\epsilon )$ , with $\epsilon \in (0,1)$ . Final good production exhibits “international” returns to scale, as introduced by Ethier (Reference Ethier1982). For example, an increase in the mass of varieties produced in home expands $Y_t$ by more than $n_t x_t$ . We denote the total mass of intermediate varieties by $N_t \equiv n_t + n_t^{\ast }$ .

The profit function of the representative final good producer in the home country is

(7) \begin{align} \Pi _{t}=Y_{t}-w_{t}L_{t}-\int ^{n_t}_{0} p_{i,t}x_{i,t}di-\int ^{n^{\ast }_t}_{0} p^{\ast }_{j,t} \tau x_{j,t}dj, \end{align}

where $p_{i,t}$ is the price of intermediate variety $i$ produced in home, $p^{\ast }_{j,t}$ is the price of variety $j$ imported from foreign, and $\tau \gt 1$ is an iceberg trade cost, under which $\tau$ additional units must be shipped for every unit sold in an export market.

Final good firms set their inputs of $x_{i,t}$ and $x_{j,t}$ to maximize profit (7), generating the following home country demand conditions for varieties produced in home and foreign:

(8) \begin{align} x_{i,t}& = p_{i,t}^{\frac{-1}{1-\epsilon }}P_{Z,t}^{\frac{\epsilon }{1-\epsilon }}\alpha Y_t,& x_{j,t}& = (\tau p^{\ast }_{j,t})^{\frac{-1}{1-\epsilon }}P_{Z,t}^{\frac{\epsilon }{1-\epsilon }}\alpha Y_t, \end{align}

where the home-country price index for intermediate varieties is

(9) \begin{align} P_{Z,t} = \left ( \int _0^{n_t} p_{i,t}^{-\frac{\epsilon }{1-\epsilon }} d i + \varphi \int _0^{n^{\ast }_t}{p_{j,t}^{\ast }}^{-\frac{\epsilon }{1-\epsilon }} d j \right )^{-\frac{1-\epsilon }{\epsilon }}, \end{align}

and $P_{Z,t} Z_t = \alpha Y_t$ . We index the level of trade costs using the freeness of trade $\varphi \equiv \tau ^{-\frac{\epsilon }{1-\epsilon }} \in (0,1)$ ; that is, $d \varphi/ d \tau \lt 0$ , with $\varphi = 0$ describing prohibitively high trade costs and $\varphi = 1$ indicating free trade between countries. Final good producers in foreign solve a similar optimization problem generating analogous demand conditions for intermediate varieties.

2.4. Labor unions and collective bargaining

Central to the labor market in our framework are the negotiations between the labor union and the employer federation that determine national employment and the wage rate. As in Pemberton (Reference Pemberton1988), Chang et al. (Reference Chang, Shaw and Lai2007), and Chu et al. (Reference Chu, Cozzi and Furukawa2016), we consider a managerial labor union, in which union members desire a high wage rate while union leaders aim for a large membership. Formally, the union’s objective function in the home country is set as

(10) \begin{align} O_{t}=\left (w_{t}-b_{t} \right )^{\chi } L_{t}, \end{align}

where $\chi \gt 0$ measures the weight the union places on incremental wage income from employment (i.e., the wage rate minus the unemployment benefit).

The bargaining problem involves choosing $w_{t}$ and $L_{t}$ to maximize the following Nash product: $O_{t}^{\psi }{\Pi _{t}}^{1-\psi }$ , where $\psi \in (0,1)$ denotes the relative bargaining power of the labor union. Solving this bargaining problem, we obtain the following optimal conditions for the wage rate and employment:

(11) \begin{align} w_{t}-b_{t}&=\chi \left (w_{t}-\beta \frac{Y_{t}}{L_{t}} \right ), \end{align}

(12) \begin{align} \frac{w_{t}L_{t}}{Y_{t}}&=\beta +\psi (1-\alpha -\beta ), \end{align}

where we have used (7) and (10). The contract curve (11) describes the locus of points for which the union’s indifference curve and the firm’s isoprofit curve are tangent in $(w_{t}, L_{t})$ space. The rent division curve (12) indicates the labor union’s negotiated share of income.

The equilibrium level of employment is obtained through substitution of the rent curve (12) and $b_{t}=mY_{t}$ into the contract curve (11):

(13) \begin{align} L=\frac{1}{m}\left [\beta +(1-\chi )\psi (1-\alpha -\beta ) \right ]. \end{align}

Hence, we find that employment is decreasing in union wage orientation ( $\chi$ ). The relationship between union bargaining power and national employment is summarized as follows.

Following Chu et al. (Reference Chu, Cozzi and Furukawa2016), we use $\chi$ to describe the relative importance that the union places on wages over membership. When $\chi \gt 1$ , the contract curve has a negative slope, and we refer to the union as wage-oriented. In this first case, a rise in union bargaining power shifts the rent curve, causing a contraction in national employment. Alternatively, when $\chi \lt 1$ , the contract curve has a positive slope, and we refer to the union as employment-oriented. In this second case, an increase in bargaining power leads to an expansion in national employment.

Analogous conditions are obtained for the foreign labor market: $w_t^{\ast } - b_t^{\ast } = \chi ^{\ast } (w_t^{\ast } - \beta Y_t^{\ast }/L_t^{\ast })$ , $w^{\ast }_{t}L^{\ast }_{t}/Y^{\ast }_{t}=\beta +\psi ^{\ast }(1-\alpha -\beta )$ , and $L^{\ast }=(1/m^{\ast })\left [\beta +(1-\chi ^{\ast })\psi ^{\ast }(1-\alpha -\beta ) \right ]$ , where we allow union wage orientation ( $\chi$ , $\chi ^{\ast }$ ) and bargaining power ( $\psi$ , $\psi ^{\ast }$ ) to differ between countries.Footnote ¹

2.5. Intermediate good production

Intermediate firms employ final goods in the production of horizontally differentiated varieties for supply to the home and foreign markets under monopolistic competition (Dixit and Stiglitz, Reference Dixit and Stiglitz1977). Each firm survives for two periods. In the first, firms invest in market entry and process innovation, and in the second they produce an intermediate product variety.

The production technologies of intermediate firms located in home and foreign are

(14) \begin{align} X_{i,t}&=\theta ^{\gamma }_{i,t} I_{x,i,t}, & X^{\ast }_{j,t}&={\theta ^{\ast }_{j,t}}^{\gamma } I^{\ast }_{x,j,t}, \end{align}

where $X_{i,t}$ and $X^{\ast }_{j,t}$ are firm-level outputs, $I_{x,i,t}$ and $I^{\ast }_{x,j,t}$ are firm-level inputs of final goods, $\theta _{i,t}$ and $\theta ^{\ast }_{j,t}$ are firm-specific productivity coefficients, and $\gamma \in (0,1)$ is the output elasticity of productivity. Each intermediate firm produces to meet the combined demands from final good firms in home and foreign. For example, a home-based intermediate firm sets supply equal to $X_{i,t}=x_{i,t}+\tau x^{\ast }_{i,t}$ .

Under monopolistic competition, the large mass of intermediate varieties sold in each market eliminates strategic interaction between firms as they choose their optimal production levels. Thus, each firm maximizes operating profit on sales, $\pi _{i,t} \equiv p_{i,t}X_{i,t}-I_{x,i,t}$ , by setting price equal to a constant markup over unit cost; that is,

(15) \begin{align} p_{i,t}&=\frac{1}{\epsilon \theta ^{\gamma }_{i,t}}, & p^{\ast }_{j,t}&=\frac{1}{\epsilon{\theta _{j,t}^{\ast }}^{\gamma }}, \end{align}

for home and foreign firms. Accordingly, optimal operating profit on sales can be obtained as $\pi _{i,t}=(1-\epsilon )p_{i,t}X_{i,t}=[(1-\epsilon )/\epsilon ] I_{x,i,t}$ . Substituting the demand conditions (8), the pricing rules (15), and $X_{i,t}=x_{i,t}+\tau x^{\ast }_{i,t}$ into $\pi _{i,t}=(1-\epsilon )p_{i,t}X_{i,t}$ yields the following expressions for the optimal operating profits

(16) \begin{align} \begin{split} \pi _{i,t}&=\alpha (1-\epsilon ) p_{i,t}^{-\frac{\epsilon }{1-\epsilon }} \left ( P_{Z,t}^{\frac{\epsilon }{1-\epsilon }} Y_t + \varphi{P_{Z,t}^{\ast }}^{\frac{\epsilon }{1-\epsilon }} Y_t^{\ast } \right ), \\[5pt] \pi ^{\ast }_{j,t}&=\alpha (1-\epsilon ){p^{\ast }}_{\hspace{-1mm}j,t}^{-\frac{\epsilon }{1-\epsilon }} \left ( \varphi P_{Z,t}^{\frac{\epsilon }{1-\epsilon }} Y_t +{P_{Z,t}^{\ast }}^{\frac{\epsilon }{1-\epsilon }} Y_t^{\ast } \right ), \end{split} \end{align}

of home and foreign intermediate firms.

2.6. Process innovation

Intermediate firms employ final goods both in preparation for market entry and in process innovation. Specifically, to produce intermediate variety $i$ with productivity $\theta _{i,t}$ in home in period $t$ , a firm must invest a fixed $f$ units of final goods in market entry and a variable $I_{R,i,t-1}$ units of final goods in process innovation in period $t-1$ .

The productivities associated with the production technologies of home and foreign intermediate firms are generated by investment in process innovation as follows:

(17) \begin{align} \theta _{i,t}&=\xi K_{t-1} I_{R,i,t-1},& \theta ^{\ast }_{j,t}&=\xi K^{\ast }_{t-1} I^{\ast }_{R,j,t-1}, \end{align}

where $K_{t-1}$ and $K^{\ast }_{t-1}$ are the productivities of home and foreign firms in R&D, and $\xi \gt 0$ . Following the in-house process innovation literature (Smulders and van de Klundert, Reference Smulders and van De Klundert1995; Peretto, Reference Peretto1996, Reference Peretto2018), we assume that technical knowledge accumulates within the production technology of each firm. Specifically, national stocks of knowledge are captured by the average productivity of the intermediate production technologies employed in each country: $\theta _t \equiv (1/n_{t}) \int ^{n_t}_{0} \theta _{i,t} di$ and $\theta ^{\ast }_t \equiv (1/n^{\ast }_{t}) \int ^{n^{\ast }_t}_{0} \theta ^{\ast }_{j,t} dj$ .

We then model the productivities of home and foreign firms in process innovation as the weighted average of the productivities of observable technical knowledge in period $t-1$ , generating an intertemporal knowledge spillover from production to innovation. Knowledge spillovers in home and foreign are therefore

(18) \begin{align} K_{t-1}&=s_{t-1} \theta _{t-1}+\delta s^{\ast }_{t-1} \theta ^{\ast }_{t-1},& K^{\ast }_{t-1}&=s^{\ast }_{t-1} \theta ^{\ast }_{t-1}+\delta s_{t-1} \theta _{t-1}, \end{align}

where $s_{t-1} \equiv n_{t-1}/N_{t-1}$ and $s^{\ast }_{t-1} =1-s_{t-1} \equiv n^{\ast }_{t-1}/N_{t-1}$ . The degree of international knowledge diffusion is regulated by $\delta \in (0,1)$ , with knowledge spillovers that are completely national in scope for $\delta = 0$ , and perfect international knowledge diffusion for $\delta = 1$ .

In period $t-1$ , firms borrow from households to finance the costs of market entry and process innovation. As in Young (Reference Young1998), the net present values of home and foreign firms in period $t-1$ are

(19) \begin{align} V_{i, t-1}&=\frac{\pi _{i,t}}{1+r_{t}}-\left ( I_{R,i,t-1} +f \right ), & V^{\ast }_{j, t-1}&=\frac{\pi ^{\ast }_{j,t}}{1+r_{t}}-\left (I^{\ast }_{R,j,t-1} +f \right ), \end{align}

where we assume that fixed entry costs are symmetric across countries ( $f=f^{\ast }$ ).

Firms choose their optimal levels of investment in innovation, $ I_{R,i,t-1}$ and $I^{\ast }_{R,j,t-1}$ to maximize firm value (19) subject to the technology constraint (17). From the first-order conditions (i.e., $\partial V_{t-1}/\partial I_{R,j,t-1}=\partial V^{\ast }_{t-1}/\partial I^{\ast }_{R,j,t-1}=0$ ), we obtain

(20) \begin{align} I_{R,i,t-1} &= \frac{\eta \pi _{i,t}}{1+r_{t}},& I^{\ast }_{R,j,t-1} &= \frac{\eta \pi ^{\ast }_{j,t}}{1+r_{t}}, \end{align}

with $\eta \equiv \gamma \epsilon/(1-\epsilon )$ . These expressions imply that firms located in each country have the same productivity level, and therefore set the same prices and employment levels for production and innovation. Henceforth, we omit the indices $i$ and $j$ , with $\theta _{i,t}=\theta _{t}$ and $\theta ^{\ast }_{j,t}=\theta ^{\ast }_{t}$ . Moreover, substituting (20) into (19), we rewrite the net present values of home and foreign firms in period $t-1$ as

(21) \begin{align} V_{t-1}&=\frac{(1 - \eta )\pi _{t}}{1+r_{t}}-f, & V^{\ast }_{t-1}&=\frac{(1 - \eta ) \pi ^{\ast }_{t}}{1+r_{t}}-f. \end{align}

We assume that $\eta \equiv \gamma \epsilon/ (1-\epsilon )\lt 1$ in order to satisfy the second-order condition for the maximization of firm value.

2.7. Market entry

Firm value drives market entry and exit in the intermediate sector in each country. Specifically, with no costs incurred in the introduction of new product designs, firms enter the intermediate sector when firm value is positive ( $V_{t-1} \gt 0$ ) and exit when firm value is negative ( $V_{t-1} \lt 0$ ). Referencing (9), (16), and (21), firm value responds correctly to market entry and exit ( $\partial V_{t-1}/\partial n_t \lt 0$ and $\partial V^{\ast }_{t-1}/\partial n^{\ast }_t \lt 0$ ), and thus, the level of market entry in each country adjusts immediately, forcing firm value to zero ( $V_{t-1} = V^{\ast }_{t-1} = 0$ ) at all moments in time, in the spirit of Novshek and Sonnenschein (Reference Novshek and Sonnenschein1987).

A key implication of free market entry and exit, in combination with the integrated financial market, is common scales of production and process innovation across firms in both countries.Footnote ² Substituting (20) and (21) into $V_{t-1}=V^{\ast }_{t-1}=0$ yields equilibrium firm-level investment in process innovation as $I_R \equiv I_{R,t-1} =I^{\ast }_{R,t-1}= \eta f/(1-\eta )$ . As such, using (18) with (17), the productivity growth rates of intermediate firms in home and foreign are

(22) \begin{align} \begin{split} g_{\theta }(\tilde{\theta }_{t-1}) \equiv \frac{\theta _{t}}{\theta _{t-1}}&= \frac{K_{t-1}}{\theta _{t-1}}\xi I_{R}= \left [s_{t-1}+\delta (1-s_{t-1})\tilde{\theta }_{t-1}^{-1} \right ] \xi I_{R}, \\[5pt] g_{\theta }^{\ast }(\tilde{\theta }_{t-1}) \equiv \frac{\theta ^{\ast }_{t}}{\theta ^{\ast }_{t-1}}&= \frac{K^{\ast }_{t-1}}{\theta ^{\ast }_{t-1}} \xi I_{R}= \left [1-s_{t-1}+\delta s_{t-1} \tilde{\theta }_{t-1} \right ] \xi I_{R}, \end{split} \end{align}

where $\tilde{\theta }_{t} \equiv \theta _{t}/\theta ^{\ast }_{t}$ denotes the international productivity differential. These expressions show that firm-level productivity growth depends solely on knowledge spillovers ( $K_{t}/\theta _{t}$ and $K^{\ast }_{t}/\theta ^{\ast }_{t}$ ) and is therefore closely linked with national shares of intermediate production ( $s_t$ ) and the international productivity differential ( $\tilde{\theta }_t$ ).

2.8. Market equilibrium

We now characterize the market clearing conditions for the intermediate and final good sectors. First, with perfect capital mobility between countries, the total asset holdings of households equals total lending to home and foreign firms: $A_{t+1}+A^{\ast }_{t+1}$ $=$ $(I_{R}+f )n_{t+1}+ (I^{\ast }_{R}+f )n^{\ast }_{t+1}$ . Combining this expression with (19) and the free entry conditions $V_{t-1} = V^{\ast }_{t-1} = 0$ implies that $(1+r_{t+1}) (A_{t+1}+A^{\ast }_{t+1})$ $=$ $\pi _{t+1}n_{t+1}+\pi ^{\ast }_{t+1}n^{\ast }_{t+1}$ .

Next, the market clearing condition for intermediate goods is given by

(23) \begin{equation} \alpha Y^{w}_{t} = n_{t}p_{t}X_{t}+n^{\ast }_{t}p^{\ast }_{t}X^{\ast }_{t}, \end{equation}

where $Y^{w}_{t} \equiv Y_{t}+Y^{\ast }_{t}$ . The left-hand side is the sum of home and foreign final good firms’ expenditure on intermediate goods. The right-hand side is the total revenue of home and foreign intermediate firms.

Lastly, the market clearing condition for final goods is given by

(24) \begin{equation} Y^{w}_{t}=C^{w}_{t}+N_{t}I_{x,t}+ N_{t+1}(I_{R}+f ), \end{equation}

where $C^{w}_{t} \equiv C_{t}+C^{\ast }_{t}$ . Thus, the world supply of final goods is set to meet household consumption, total employment in intermediate production, and total employment in process innovation and market entry.

3.1. Steady-state industry location patterns

Adapting the equilibrium concept of Davis and Hashimoto (Reference Davis and Hashimoto2014), the first condition for the determination of $s_t$ and $\tilde{\theta }_t$ is obtained by recalling that free market entry and exit reduces the value of intermediate firms to zero ( $V_{t-1}=V^{\ast }_{t-1}=0$ ). Then, the integrated financial market implies that all firms have the same level of operating profit; that is, $\pi _t = \pi ^{\ast }_t$ at all moments in time. In Appendix A, we combine $\pi _t = \pi ^{\ast }_t$ with $Y_t/Y_t^{\ast } = (Z_t/Z^{\ast }_t)^{\alpha }(L_t/L^{\ast }_t)^{\beta }$ , $Y_t/Y^{\ast }_t = (P_{Z,t} Z_t)/(P_{Z,t}^{\ast } Z^{\ast }_t)$ , (9), (15), and (16) to solve for the equilibrium share of intermediate firms based on home at time $t$ as

(25) \begin{align} s_{R,t} &= \frac{(\tilde{\theta }^{\eta }_t - \varphi ) y_t - \varphi (1 - \varphi \tilde{\theta }^{\eta }_t)}{(\tilde{\theta }^{\eta }_t - \varphi )(1 - \varphi \tilde{\theta }^{\eta }_t)(1+y_t)}, & y_{t} & \equiv \frac{Y_t}{Y^{\ast }_t} = \left ( \frac{L}{L^{\ast }}\right )^{\frac{\beta \epsilon }{\epsilon - \alpha }} \left ( \frac{\tilde{\theta }^{\eta }_{t} - \varphi }{1 - \varphi \tilde{\theta }^{\eta }_{t}} \right )^{\frac{\alpha (1- \epsilon )}{\epsilon - \alpha }}, \end{align}

where $\epsilon \gt \alpha$ is required to ensure that productivity growth generates a positive rate of growth in final good output (see Section 3.2). This expression is illustrated by the $s_R$ curve in Figure 1 and has a strictly positive slope: $\partial s_{R,t}/ \partial \tilde{\theta }_t \gt 0$ .Footnote ³ An increase in the international productivity differential ( $\tilde{\theta }_t$ ) generates an expansion in the home share of intermediate firms through two channels. The first is a direct effect, whereby an improvement in the relative productivity of home-based firms expands their market share ( $\partial s_{R,t}/\partial \tilde{\theta }_t \gt 0$ ). The second channel is an indirect effect, wherein international trade costs ensure that an increase in $\tilde{\theta }_t$ raises the home country’s share of final good production ( $\partial y_t/ \partial \tilde{\theta }_t \gt 0$ ) as the relative cost of sourcing home produced intermediate goods falls. The expansion of the home market relative to foreign leads to a greater share of intermediate good producers for home ( $\partial s_{R,t}/\partial y_t \gt 0$ ). Because the home share of intermediate firms is bounded ( $s_t \in (0,1)$ ), however, there are limits on the range of values for the international productivity differential, outside of which an equilibrium with active intermediate sectors in both countries is not feasible; that is, $\tilde{\theta }_{t} \in (\tilde{\theta }_{l}, \tilde{\theta }_{u})$ .

Figure 1. Determination of the international productivity differential ( $\tilde{\theta }$ ).

Next, we derive a differential equation to describe the dynamics of the international productivity differential. Referencing (22), we have

(26) \begin{align} \frac{\tilde{\theta }_{t+1}}{\tilde{\theta }_t} = \frac{g_{\theta } (\tilde{\theta }_t)}{g^{\ast }_{\theta } (\tilde{\theta }_t)} = \frac{K_t/\theta _t}{K^{\ast }_t/\theta ^{\ast }_t} = \frac{s_t + \delta (1 - s_t) \tilde{\theta }_t^{-1}}{1 - s_t + \delta s_t \tilde{\theta }_t}. \end{align}

Thus, we naturally find that a constant productivity differential ( $\tilde{\theta }_{t+1}/\tilde{\theta }_{t} = 1$ ) is synonymous with equal productivity growth rates for home and foreign intermediate firms ( $g_{\theta } = g^{\ast }_{\theta }$ ).

With all firms employing the same level of final goods in process innovation ( $I_R = I_R^{\ast }$ ), the constant productivity differential and equalized productivity growth rates require that firms have access to the same level of knowledge spillovers ( $K_t/\theta _t = K^{\ast }_t/\theta ^{\ast }_t$ ), allowing us to derive the following steady-state condition for the home country’s share of intermediate firms using (18):

(27) \begin{align} s_K(\tilde{\theta }) = \frac{1 - \delta \tilde{\theta }^{-1}}{2 - \delta \tilde{\theta } - \delta \tilde{\theta }^{-1}}, \end{align}

where we now drop the time script to indicate that steady-state values are constant. As shown by the $s_K$ curve in Figure 1, when the international productivity differential ( $\tilde{\theta }$ ) increases, a greater share of firms is required for home to offset the improved access of foreign-based firms to technical knowledge ( $\partial s_K/\partial \tilde{\theta } \gt 0$ ). Note that $\tilde{\theta } \in (\delta, 1/\delta )$ is required for $s_K \in (0,1)$ . In addition, the slope of the $s_K$ curve is closely linked to the degree of knowledge diffusion. When there is no international knowledge diffusion ( $\delta = 0$ ), the $s_K$ curve becomes a horizontal line. Alternatively, with perfect knowledge diffusion ( $\delta =1$ ), the $s_K$ curve becomes a vertical line at $\tilde{\theta } = 1$ , and the international productivity differential vanishes.

In Appendix B, we study the local dynamics around the steady state characterized by the intersection of the $s_K$ and $s_R$ curves in Figure 1 and obtain the following proposition.

As a state variable, the international productivity differential is rising (falling) for values of $\tilde{\theta }$ to the left (right) of the $s_K$ curve. Therefore, a stable steady state requires $\partial s_K/ \partial \tilde{\theta } \gt \partial s_R/ \partial \tilde{\theta }$ to ensure that the economy moves along the $s_R$ curve towards the long-run equilibrium, as depicted in the phase diagram of Figure 1.Footnote ⁴ Examining (25), we find that $\lim _{\varphi \to 0} \tilde{\theta }_{l}=0$ and $\lim _{\varphi \to 0} \tilde{\theta }_{u} = \infty$ . Accordingly, we assume that the freeness of trade ( $\varphi$ ) is small enough to satisfy $\tilde{\theta }_{l}\lt \delta \lt 1/\delta \lt \tilde{\theta }_{u}$ , ensuring the existence of at least one steady state with the slope ranking outlined in Proposition 1.Footnote ⁵ Hereafter, we focus on equilibria that satisfy these conditions for the existence of a stable long-run equilibrium.

The long-run international productivity differential ( $\tilde{\theta }$ ) is implicitly linked with the relative employment level of the home country ( $L/L^{\ast }$ ). Referencing (25), an increase in $L/L^{\ast }$ expands the relative market size of the home country for intermediate producers ( $\partial y/ \partial (L/L^{\ast }) \gt 0$ ), causing an upward shift in the $s_R$ curve in Figure 1 ( $\partial s_{R}/\partial y \gt 0$ ). As a result, the home country’s shares of intermediate firms ( $s$ ) and final good production ( $y$ ) both increase. The rise in $s$ coincides with an increase in the international productivity differential that returns knowledge spillovers back to equality across countries ( $K/\theta = K^{\ast }/\theta ^{\ast }$ ), as the economy converges to a new long-run equilibrium. These results are summarized in the following lemma.

From the results of Lemma 2 and the mechanics of Figure 1, referencing (25) and (27), it becomes clear that equality of national employment levels generates a symmetric equilibrium with $y = 1$ , $s = 1/2$ and $\tilde{\theta } =1$ . Accordingly, when home employment rises above foreign employment ( $L/L^{\ast } \gt 1$ ), the home country has greater shares of intermediate firms ( $s \gt 1/2$ ) and final good production ( $y\gt 1$ ), and relatively productive intermediate firms ( $\tilde{\theta }\gt 1$ ). When the foreign country has a larger relative employment level ( $L/L^{\ast } \lt 1$ ), however, it has greater shares of intermediate firms ( $s \lt 1/2$ ) and final good production ( $y\lt 1$ ), and foreign-based intermediate firms are relatively productive ( $\tilde{\theta }\lt 1$ ).

As we have seen, knowledge spillovers equalize across countries in the long run. Substituting (27) back into (18), and reorganizing the result, yields the steady-state level of knowledge spillovers as follows:

(28) \begin{align} \frac{K}{\theta } = \frac{K^{\ast }}{\theta ^{\ast }} = \frac{1 - \delta ^2}{2 - \delta \tilde{\theta } - \delta \tilde{\theta }^{-1}}. \end{align}

Therefore, we determine that $K/\theta$ is convex in the international productivity differential ( $\tilde{\theta }$ ) with a minimum at $\tilde{\theta } = 1$ ; that is, $\partial (K/\theta )/ \partial \tilde{\theta } = - [\delta (1 - \tilde{\theta }^2)/(1-\delta ^{-2}) \tilde{\theta }^2 ] (K/\theta )^2$ . For $\tilde{\theta } \lt 1$ , an increase in $\tilde{\theta }$ lowers knowledge spillovers, and for $\tilde{\theta } \gt 1$ , the increase in $\tilde{\theta }$ raises knowledge spillovers. Consequently, we find that firm-level productivity in R&D is directly linked with national shares of intermediate production, with an increase in the geographic concentration of industry in one country raising the level of knowledge spillovers.

Lastly, before leaving this section, we derive the equilibrium operating profit associated with the intermediate firms based in home and foreign as

(29) \begin{align} \pi _t = \frac{\alpha (1-\epsilon ) Y^w_t}{N_t}, \end{align}

where we have used $\pi _t = (1-\epsilon )p_t X_t$ and $\pi _t = \pi _t^{\ast }$ in the market clearing condition for intermediate goods (23).

3.2. Economic growth

We now turn to the long-run rate of output growth. First, combining (3), (21), and (29) with $V_{t-1} = V^{\ast }_{t-1} = 0$ , we rewrite the dynamics of household consumption as

(30) \begin{align} \frac{C_{t+1}}{C_t} = \frac{\alpha (1 -\epsilon )(1-\eta )}{(1+\rho ) f} \frac{Y^w_{t+1}}{N_{t+1}}. \end{align}

Because we are interested in balanced growth paths that feature a constant rate of growth in consumption for home and foreign households, this expression implies that final good output and the mass of intermediate firms grow at the same rate. In addition, given that $y \equiv Y_t/Y^{\ast }_t$ is constant in a steady state with a constant international productivity differential ( $\tilde{\theta }_{t+1}/\tilde{\theta }_t = 1$ ), we find that $Y^w_{t+1}/Y^w_t = Y_{t+1}/Y_t =Y^{\ast }_{t+1}/Y^{\ast }_t = N_{t+1}/N_t$ .

Then, substituting (5), (9), (15), (17), and (28) with $\alpha Y_t = P_{Z,t} Z_t$ into $Y_{t+1}/Y_t$ delivers the long-run rate of output growth as follows:

(31) \begin{align} g \equiv \frac{Y^{w}_{t+1}}{Y^{w}_{t}}={g_{\theta }}^{\frac{\alpha \epsilon \gamma }{\epsilon -\alpha }} = \left [ \frac{(1-\delta ^2)\xi I_R}{2 - \delta \tilde{\theta } - \delta \tilde{\theta }^{-1}} \right ]^{\frac{\alpha \epsilon \gamma }{\epsilon -\alpha }}, \end{align}

where $I_R = \eta f/(1-\eta )$ . A casual examination of this expression makes it clear that knowledge spillovers are the key channel through which intermediate and final good production patterns affect the steady-state rate of output growth. As such, referencing (28), we find that the output growth rate is convex in the international productivity differential ( $\tilde{\theta }$ ) with a minimum at $\tilde{\theta } = 1$ , as depicted in Figure 2. In addition, (31) indicates that the long-run growth rate is scale-invariant as proportionate changes in the labor forces of home and foreign leave the international productivity differential and the rate of output growth unchanged.

Figure 2. Long-run output growth.

The mechanics of Figure 2, combined with Lemma 2, yield the following result.

Lemma 3 describes the key mechanism in our framework linking national labor supplies with long-run output growth through the relationship that arises between industry location patterns ( $s$ ) and the strength of knowledge spillovers ( $K/\theta$ ) from production into innovation. Returning to Lemma 2, an increase in the relative employment of home ( $L/L^{\ast }$ ) expands the home-country shares of final production ( $y$ ) and intermediate firms ( $s$ ), while raising the relative productivity of home-based firms ( $\tilde{\theta }$ ). Thus, starting from $L/L^{\ast } \lt 1$ , where initially the foreign country has a greater share of industry ( $s\lt 1/2$ ), an increase in $L/L^{\ast }$ reduces the geographic concentration of industry, as the foreign share of intermediate firms falls, weakening knowledge spillovers ( $K/\theta$ ) and slowing output growth ( $g$ ). The minimum growth rate is reached when home and foreign have equal shares of industry at $L/L^{\ast } =1$ , $s=1/2$ , and $\tilde{\theta }=1$ , as illustrated in Figure 2. Further increases in $L/L^{\ast }$ then lead to the concentration of industry in the home country, strengthening knowledge spillovers and accelerating output growth. Thus, the rate of output growth is faster when industry is concentrated in a single country than when industry is dispersed equally across countries.

The level of market entry in the intermediate sector ( $N_t$ ) depends on the international productivity differential ( $\tilde{\theta }$ ) and the rate of output growth ( $g$ ). To emphasize this point, we first use (5), (9), (15), and (25) to derive final good output in home and foreign as

(32) \begin{align} Y_t &= \frac{(\alpha \epsilon \theta _t^{\gamma })^{\frac{\alpha }{1-\alpha }} (1-\varphi ^2)^{\nu } N_t^{\nu } L^{\frac{\beta }{1-\alpha }}}{[(1-\varphi \tilde{\theta }^{\eta })(1+1/y)]^{\nu }}, & Y^{\ast }_t &= \frac{(\alpha \epsilon \theta _t^{\gamma })^{\frac{\alpha }{1-\alpha }} (1-\varphi ^2)^{\nu } N_t^{\nu }{L^{\ast }}^{\frac{\beta }{1-\alpha }}}{[(\tilde{\theta }^{\eta }-\varphi )(1+y)]^{\nu }}, \end{align}

where $\nu \equiv \alpha (1 - \epsilon )/((1-\alpha )\epsilon )\lt 1$ . An examination of these expressions shows that $N_{t+1}/N_t = Y_{t+1}/Y_t = Y^{\ast }_{t+1}/Y^{\ast }_t = g$ in the long-run equilibrium. A corollary of this result is that $C_{t+1}/C_t = g$ , and referencing (3) the steady-state interest rate is therefore constant: $1 + r = (1+\rho )g$ .

Substituting national output levels into $\pi = \alpha (1-\epsilon )Y_t^w/N_t = (1+\rho ) f g/(1-\eta )$ , we then obtain the total mass of intermediate firms along the balance growth path as follows:

(33) \begin{align} N_t &= \theta _t^{\frac{\alpha \epsilon \gamma }{\epsilon - \alpha }} \left [ \frac{\Gamma }{(1 + \rho ) g f} \right ]^{\frac{1}{1 - \nu }} \left \{ \frac{L^{\frac{\beta }{1-\alpha }}}{ [(1-\varphi \tilde{\theta }^\eta )(1+1/y) ]^{\nu }} + \frac{{L^{\ast }}^{\frac{\beta }{1-\alpha }}}{ [(\tilde{\theta }^\eta -\varphi )(1+y) ]^{\nu }} \right \}^{\frac{1}{1 - \nu }}, \end{align}

with $\Gamma \equiv \alpha ^{1/(1-\alpha )}(1 - \eta )(1 - \epsilon )\epsilon ^{\alpha/(1-\alpha )}(1-\varphi ^2)^{\nu }$ . Hence, we find that for a given level of home productivity ( $\theta$ ), our model features the standard tradeoff that arises between market entry and economic growth in endogenous growth and endogenous market structure frameworks: given the productivity differential ( $\tilde{\theta }$ ) and relative final good output ( $y$ ), a rise in $g$ necessitates a fall in $N$ . In addition, the mass of intermediate firms adjusts to absorb proportionate changes in the labor forces of home and foreign, ensuring a scale-invariant rate of output growth.

Although productivity growth drives market entry and the expansion of aggregate output along the balance growth path, adjustments in the international productivity differential ( $\tilde{\theta }$ ) generally have an ambiguous effect on market entry, as an increase in $\tilde{\theta }$ expands the final good output of home, but contracts the final good output of foreign, while generating an ambiguous effect on the rate of output growth ( $g$ ), as seen in Lemma 3.

3.3. Social welfare

Lastly, we solve for the steady-state welfare levels of households in each country. Using (1) and (3), we express the lifetime welfare of home and foreign households along the balanced growth path as

(34) \begin{align} U_0 &=\frac{1+\rho }{\rho }\left (\ln C_0 +\frac{1}{\rho }\ln g \right ), & U_0^{\ast } &= \frac{1+\rho }{\rho }\left (\ln C^{\ast }_0 +\frac{1}{\rho }\ln g \right ). \end{align}

Household welfare derives from the current level of consumption ( $C_0$ ) and the growth of consumption ( $g$ ). In Appendix C, we derive steady-state consumption as follows:

(35) \begin{align} C_0 &=(1-\alpha )Y_0 + \frac{\rho f N_0 g}{2(1-\eta )},& C_0^{\ast }&=(1-\alpha )Y^{\ast }_0+ \frac{\rho f N_0 g}{2(1-\eta )}, \end{align}

where we have set the initial value of home-firm productivity to unity ( $\theta _0 = 1$ ), and we have assumed that initial asset wealth is the same for home and foreign ( $A_0 = A_0^{\ast }$ ). With $Y_{0}$ , $Y^{\ast }_{0}$ , and $N_0$ determined by (32) and (33), from the above expressions, we find that labor union policy influences welfare through its effects on domestic output levels ( $Y_0$ and $Y_0^{\ast }$ ), market entry ( $N_0$ ), and the rate of output growth ( $g$ ).

4.1. Wage orientation of labor unions

We begin with an examination of the effects of an increase in the wage orientation ( $\chi$ ) of the home-country labor union.Footnote ⁶ The results are summarized in the following proposition.

The intuition behind Proposition 2 lies in the link between national labor supplies, the location of industry, and the strength of knowledge spillovers from production to innovation. Referring back to (13), a rise in the wage orientation of the home-country union ( $\chi$ ) leads to lower employment in home ( $d L/ d \chi \lt 0$ ), with the union’s push for higher wages depressing labor demand. Suppose that the home country initially has a larger labor supply ( $L/L^{\ast }\gt 1$ ) and therefore has greater shares of final good production ( $y\gt 1$ ) and intermediate firms ( $s\gt 1/2$ ), and higher productivity ( $\tilde{\theta }\gt 1$ ). An increase in $\chi$ then reduces the home country’s share of industry ( $s$ ) and lowers the international productivity differential ( $\tilde{\theta }$ ) as the relative labor supply ( $L/L^{\ast }$ ) falls. Consequently, referencing Lemma 3, the rise in $\chi$ weakens knowledge spillovers ( $K/\theta$ ) and slows output growth ( $g$ ) for $\tilde{\theta }\gt 1$ , and strengthens knowledge spillovers and accelerates output growth for $\tilde{\theta }\lt 1$ . The rate of output growth is minimized when $\chi = 1 + (\beta - m L^{\ast })/(\psi (1 - \alpha - \beta ))$ , where $L/L^{\ast }=1$ , $s=1/2$ , and $\tilde{\theta } = 1$ .

4.2. Bargaining power of labor unions

Next, we consider how adjustments in the bargaining power ( $\psi$ ) of the home-country labor union influence output growth ( $g$ ). Returning to (13), we find that the relationship between bargaining power and national employment depends critically on the slope of the contract curve, as described by the wage orientation ( $\chi$ ) of the union, with $d L/ d \psi \lt 0$ for $\chi \gt 1$ and $d L/ d \psi \gt 0$ for $\chi \lt 1$ . In the former case, the wage-oriented union demands higher wages as its bargaining power increases, thereby depressing labor demand. In the latter case, the employment-oriented union negotiates greater employment by allowing lower wages and expanding labor demand.

The relationship between union bargaining power and national employment has important implications for the rate of output growth, as summarized in the following proposition.

As we have seen, the rate of output growth is linked with national labor supplies through the role of industry location patterns in determining the strength of knowledge spillovers from production to innovation. When the home-country is union is wage-orientated ( $\chi \gt 1$ ), an increase in union bargaining power ( $\psi$ ) reduces the home labor supply, causing a fall in relative employment ( $L/L^{\ast }$ ) that decreases the home country share of intermediate firms. Then, knowledge spillovers ( $K/\theta$ ) and the rate of output growth ( $g$ ) fall for $L/L^{\ast } \gt 1$ , but rise for $L/L^{\ast } \lt 1$ , with the growth rate minimized at $\psi = (\beta - m L^{\ast })/((\chi - 1)(1-\alpha -\beta ))$ where $L/L^{\ast } = 1$ , $s = 1/2$ , and $\tilde{\theta }=1$ . In contrast, when the home-country union is employment-oriented ( $\chi \lt 1$ ), an increase in bargaining power ( $\psi$ ) expands the home-country labor supply, raising relative employment ( $L/L^{\ast }$ ) and the home-country share of firms ( $s$ ). In this case, knowledge spillovers ( $K/\theta$ ) and the output growth rate ( $g$ ) fall for $L/L^{\ast } \lt 1$ , but rise for $L/L^{\ast } \gt 1$ . The growth rate is maximized at $\psi = (\beta - m L^{\ast })/((\chi - 1)(1-\alpha -\beta ))$ .

Table 1. Parameter values