Capital accumulation and the welfare gains from trade

Abstract

We measure the gains from a trade cost reduction in a model with dynamic accumulation of factors. We show that the tight link between import intensity and gains from trade that exists in static models breaks down along transition paths in dynamic models. When trade costs are reduced, the need to accumulate factors temporarily shifts spending from consumption to investment. Import intensity may rise or fall along the transition path, depending on the relative import intensity of consumption and investment. Calibrating the model to the US economy, we find that investment is more import intensive than consumption, so that import intensity is falling along the transition path even as consumption is rising. Therefore, while higher import intensity is associated with higher consumption when comparing steady states (as in static models), it is associated with lower consumption along a given transition path. We also consider the case of endogenous firm creation as another form of investment and factor accumulation, and again find a negative relationship between consumption and import intensity along the transition path.

Appendices

Appendix A: Model characterization

In this appendix, we characterize of equilibria for each of the models presented in Sect. 2 and the model from Sect. 5. We consider the symmetric country cases in all three.

1.1 A.1 Dynamic model

An equilibrium for the dynamic model with capital accumulation consists of allocations $\{c_t,c_{ht}, c_{ft},x_t, x_{ht}, x_{ft}, y_t, l_t, k_{t+1} \}_{t=0}^{\infty }$ as described in Sect. 2.1. Here we remove j subscripts that denote country because they are symmetric.

The investment choice for the household is given by the following Euler equation:

$$\begin{aligned} \begin{aligned} \left( \frac{c_{t}}{c_{t+1}}\right) ^{-\sigma } =&\frac{ A_{t+1}\alpha k_{t+1}^{\alpha -1}+ (1-\delta )\nu ^{\frac{1-\rho }{\rho }}\left( 1+\frac{\nu }{1-\nu }\tau _{t+1}^{\frac{\rho }{\rho -1}} \right) ^{\frac{\rho -1}{\rho }} }{ \nu ^{\frac{1-\rho }{\rho }}\left( 1+\frac{\nu }{1-\nu }\tau _t^{\frac{\rho }{\rho -1}} \right) ^{\frac{\rho -1}{\rho }} }\\&\times \beta \left( \frac{1+\frac{\mu }{1-\mu }\left( p_{t+1}\tau _{t+1} \right) ^{\frac{\rho }{\rho -1}}}{1+\frac{\mu }{1-\mu }\tau _{t}^{\frac{\rho }{\rho -1}}} \right) ^{\frac{1-\rho }{\rho }}, \end{aligned} \end{aligned}$$

(40)

The intra-temporal decision for consumption is:

$$\begin{aligned} c_{f,t} =c_{h,t} \tau _t^{\frac{-1}{1-\rho }} \frac{\mu }{1-\mu }, \end{aligned}$$

(41)

The intra-temporal decision for investment is:

$$\begin{aligned} x_{f,t} =x_{h,t} \tau _t^{\frac{-1}{1-\rho }} \frac{\nu }{1-\nu }, \end{aligned}$$

(42)

The consolidated budget constraint implies:

$$\begin{aligned} c_{h,t}+x_{h,t}+ \tau _t (c_{f,t}+x_{f,t})=y_t, \end{aligned}$$

(43)

The capital accumulation equation is:

$$\begin{aligned} k_{t+1}=(1-\delta )k_t+x_{h,t}\nu ^{\frac{\rho -1}{\rho }} \left( 1+\frac{\nu }{1-\nu }\tau _t^{\frac{\rho }{\rho -1}} \right) ^{\frac{1}{\rho }}, \end{aligned}$$

(44)

Lastly, labor market and the definitions of output, consumption, and investment imply:

$$\begin{aligned} \begin{aligned}&l_t = L,\\&y_t = A_t k_t^{\alpha }l_t^{1-\alpha },\\&c_t=\left( \mu ^{\rho -1} c_{ht}^{\rho }+(1-\mu )^{\rho -1} c_{ft}^{\rho }\right) ^\frac{1}{\rho },\\&x_t=\left( \nu ^{\rho -1} x_{ht}^{\rho }+(1-\nu )^{\rho -1} x_{ft}^{\rho }\right) ^\frac{1}{\rho },\\&p_t = 1\\ \end{aligned} \end{aligned}$$

(45)

The steady state of the dynamic model can be fully solved in closed form, which we characterization here explicitly.

Given a constant productivity A and trade costs $\tau$, the steady state is characterized by:

$$\begin{aligned} \begin{aligned}&k^{SS} = \left( \frac{\beta \alpha A }{\left( 1-\beta (1-\delta )\nu ^{\frac{1-\rho }{\rho }}\right) \left( 1+ \frac{\nu }{1-\nu }\left( p\tau \right) ^{\frac{\rho }{\rho -1}}\right) ^{\frac{\rho -1}{\rho }}} \right) ^{\frac{1}{1-\alpha }},\\&l^{SS} = 1,\\&y^{SS} = A \left( k^{SS} \right) ^\alpha ,\\&x^{SS} = \delta k^{SS}, \\&x_h^{SS} = x^{SS} \nu ^{\frac{1-\rho }{\rho }}\left( 1+\frac{\nu }{1-\nu }\left( p\tau \right) ^{\frac{-\rho }{1-\rho }} \right) ^{-\frac{1}{\rho }},\\&x_f^{SS} = x_h^{SS} \left( p \tau \right) ^{\frac{-1}{1-\rho }} \frac{\nu }{1-\nu },\\&c_h^{SS} = \frac{y^{SS} - x_h^{SS} - p\tau x_f^{SS}}{1+(p \tau )^{\frac{\rho }{\rho -1}} \frac{\mu }{1-\mu }},\\&c_f^{SS} = c_h^{SS} \left( p \tau \right) ^{\frac{-1}{1-\rho }} \frac{\mu }{1-\mu },\\&c^{SS} = c_h^{SS} \mu ^{\frac{\rho -1}{\rho }}\left( 1+\frac{\mu }{1-\mu }(p\tau )^{\frac{\rho }{\rho -1}} \right) ^{\frac{1}{\rho }}. \end{aligned} \end{aligned}$$

(46)

1.2 A.2 Static model

An equilibrium for the model presented in Sect. 2.2 consists of allocations:

$$\begin{aligned} \begin{aligned}&y=A k^\alpha , \\&l=1, \\&c_h=y\left( 1+(p \tau )^{\frac{\rho }{\rho -1}} \frac{\mu }{1-\mu }\right) ^{-1},\\&c_f=y\left( \frac{1-\mu }{\mu }(p \tau )^{\frac{-1}{\rho -1}}+p \tau \right) ^{-1}. \end{aligned} \end{aligned}$$

(47)

Appendix B: Intermediate goods case

1.1 B.1 Static model with intermediate goods

In this appendix we develop a static version of the model from Sect. 2.1 in which the production process involves the production of an intermediate good combining domestic and foreign intermediate goods.¹⁸

The problem that households in country j solve is:

(48)

Firms

The firm in country j solves:

(49)

Finally, prices are determined by market clearing conditions that are similar to those of the dynamic model, but without time subscripts. Namely,

$$\begin{aligned} \text {Labor Market: }&l_j = L_j, \end{aligned}$$

(50)

$$\begin{aligned} \text {Trade Balance: }&y_{j} - c_{hj} - m_{hj} = p \tau (c_{fj}+m_{fj}). \end{aligned}$$

(51)

Competitive equilibrium in the static economy with intermediates

Next, we define what an equilibrium is in this economy.

Definition 3

An equilibrium in this economy are prices $\{w_j, r_j, p\}_{j\in \{h,f\}}$ and allocations $\{c_j, c_{hj}, c_{fj}, x_{hj}, x_{fj}, k_j, y_j, l_j \}_{j\in \{h,f\}}$ such that the household solves the problem (48), the firm solves problem (49), labor markets clear, Eq. (50), and trade balances, Eq. (51).

1.2 B.2 Isolating the transition channel

We denote the total expenditure on intermediate goods:

$$\begin{aligned} E_{mj} = q_j m_j. \end{aligned}$$

(52)

We can compute real income in the static model with intermediates directly from the equilibrium characterization (see Appendix 1). Real income is equal to:

$$\begin{aligned} c_j = (y_j-E_{mj}) \mu ^{\frac{\rho -1}{\rho }} (\lambda _j^C)^{\frac{1-\rho }{\rho }}. \end{aligned}$$

(53)

The change in real income in this case, $\Delta ^I_j$ (the superscript I stands for intermediates), that a consumer gets from moving from a regime of high trade costs H to a regime of low trade costs L is given by:

$$\begin{aligned} \Delta ^I_j = \left( \frac{\lambda _{jL}^{C}}{\lambda _{jH}^{C}} \right) ^{\frac{\rho -1}{\rho } } \left( \frac{y_{jL}-E_{mjL}}{y_{jH}-E_{mjH}} \right) . \end{aligned}$$

(54)

We see that this equation is substantially different than that from the baseline static model given by Eq. (19). As in the dynamic model, we see that the welfare-relevant measure of imports is the domestic share of expenditure restricted to consumption goods, rather than overall. This captures the composition channel discussed in the decomposition before. Likewise, the second term in this equation is the same as the capital channel in that it reflects the expenditure on consumption rather than total output.

Finally, the elasticity channel is captured in this model in exactly the same way as in the dynamic model if all of the following are true: the trade elasticity and domestic expenditure shares for consumption are the same in both models, consumption as a share of gross output is the same in both models, and the domestic expenditure share of investment in the dynamic model is the same as the domestic expenditure share of intermediates in this static model.¹⁹ This is the version of the model in which intermediate inputs play exactly the same role that the endogenous accumulation of capital plays in the dynamic model.

In that case, the analogue of Eq. (21) is:

$$\begin{aligned} \frac{\Delta ^D_j}{\Delta ^I_j} = \left( \sum ^\infty _{t=0} \beta ^t \left( \frac{c_{jt}}{c_{jL}} \right) ^{1-\eta }\right) ^{\frac{1}{1-\eta }}. \end{aligned}$$

(55)

This is exactly the same formula for the transition channel discussed before. Moreover, under the set of assumptions made above, we can use exactly the same set of computed results from before to quantify the difference between the dynamic model of capital accumulation and the static model with intermediate goods. On Table 2, if one subtracts the terms in the row labeled “Decomposition” from the row labeled “Dynamic Gains”, that gives the natural logarithm of the change in real income in the static model with intermediate goods. Given that the only remaining channel in this case is the transition, then the fact that transitions from high to low trade costs are always costly in this environment implies that the model with intermediate goods will always have higher gains from trade than the model with capital accumulation.

1.3 B.3 Static model with intermediates

An equilibrium for the model presented in Sect. 1 consists of allocations:

$$\begin{aligned} \begin{aligned}&l=1, \\&x_h=\left( \alpha A \nu ^{\alpha \frac{\rho -1}{\rho }}\right) ^{\frac{1}{1-\alpha }} \left( 1+\frac{\nu }{1-\nu }(p\tau )^{\frac{\rho }{\rho -1}} \right) ^{\frac{\alpha -\rho }{\rho (1-\alpha )}},\\&x_f=x_h\frac{\nu }{1-\nu }(p\tau )^{-\frac{1}{1-\rho }},\\&q=x_h\nu ^{\frac{\rho -1}{\rho }}\left( 1+\frac{\nu }{1-\nu }(p\tau )^{\frac{\rho }{\rho -1}} \right) ^{\frac{1}{\rho }},\\&c_h=\alpha ^{\frac{\alpha }{1-\alpha }}A^{\frac{1}{1-\alpha }}(1-\alpha )\nu ^{\frac{\alpha }{1-\alpha }\frac{1-\rho }{\rho }} \frac{ \left( 1+\frac{\nu }{1-\nu }(p\tau )^{\frac{\rho }{\rho -1}} \right) ^{\frac{\alpha }{1-\alpha }\frac{1-\rho }{\rho }} }{\left( 1+\frac{\mu }{1-\mu }(p\tau )^{\frac{\rho }{\rho -1}} \right) },\\&c_f=c_h\frac{\mu }{1-\mu }(p\tau )^{-\frac{1}{1-\rho }},\\&y=A m^\alpha L^{1-\alpha }. \end{aligned} \end{aligned}$$

(56)