2.1. The prototype economy

Let us work with discrete time $t$ , in which the probability of a given state of nature ( $s_t$ ) is given by $\pi _t\left(s^t\right)$ , where $s^t = (s_0,\ldots,s_t)$ represents the history of events up to and including period $t$ . We take initial state ( $s_0$ ) as given.

2.1.1. Households

Consumers maximize expected lifetime utility over per capita consumption ( $c_t$ ) and labor ( $l_t$ ) for each $t$ and $s^t$ ,

(1) \begin{equation} \sum _{t=0}^{\infty } \sum _{s^t} \pi _t\left(s^t\right) \beta ^t U\left(c_t\left(s^t\right), l_t\left(s^t\right)\right) N_t, \end{equation}

subject to the budget constraint:

\begin{equation*} c_t\left(s^t\right) + \left(1+\tau _{xt}\left(s^t\right)\right) x_t\left(s^t\right) = \left(1- \tau _{lt}\left(s^t\right)\right) w_t\left(s^t\right) l_t\left(s^t\right) + r_t\left(s^t\right) k_t\left(s^t\right) + T_t\left(s^t\right). \end{equation*}

where $\beta$ is the discount rate, $U(.)$ represents the utility function, $N_t$ is the population (which has a growth rate of $\gamma _n$ ); $1/(1+\tau _{x,t})$ is the investment wedge, $x_t$ is the per capita investment, ( $1-\tau _{l,t}$ ) is the labor wedge, $w_t$ is the real wage rate, $r_t$ is the return on capital, $T_t$ is per capita lump-sum transfers from the government to households, and $k_t$ is the capital stock, which bears adjustment costs $\phi\!\left(\frac{x_t\left(s^t\right)}{k_t\left(s^{t-1}\right)}\right) = \frac{a}{2}\left(\frac{x_t\left(s^t\right)}{k_t\left(s^{t-1}\right)}-b\right)^2$ , with $b = \delta + \gamma _A + \gamma _n$ , and has the following law of accumulation:

(2) \begin{equation} (1+ \gamma _n) k_{t+1}\left(s^t\right) = (1-\delta ) k_t\left(s^{t-1}\right) + x_t\left(s^t\right) - \phi\!\left(\frac{x_t\left(s^t\right)}{k_t\left(s^{t-1}\right)}\right). \end{equation}

where $\delta$ is the depreciation rate.

2.1.2. Firms

The final good is produced under perfect competition and firms combine capital and labor in order to maximize profits $\Pi _t$ , given the production technology $F(k_t\left(s^{t-1}\right), (1+\gamma )^t l_t\left(s^t\right))$ :

(3) \begin{equation} \max _{k_t, l_t} \Pi _t\left(s^t\right) = y_t\left(s^t\right) - r_t\left(s^t\right) k_t\left(s^{t-1}\right) - w_t\left(s^t\right) l_t\left(s^t\right) \end{equation}

Combining the first-order conditions for the household’s and firm’s maximization problems, the production function, and the resource constraint, we have the four equilibrium conditions of the model:

(4) \begin{equation} y_t\left(s^t\right) = A_t\left(s^t\right) F(k_t\left(s^{t-1}\right), (1+\gamma )^t l_t\left(s^t\right)) \end{equation}

(5) \begin{equation} -\frac{U_{l,t}\left(s^t\right)}{U_{c,t}\left(s^t\right)} = \left(1-\tau _{l,t}\left(s^t\right)\right)A_t\left(s^t\right) (1+\gamma ) F_{l,t} \end{equation}

(6) \begin{equation} \begin{aligned} & U_{c,t}\left(s^t\right) (1+\tau _{x, t}\left(s^t\right)) = \\[5pt] &\beta \sum _{s^{t+1}} \pi _t(s^{t+1}|s^t) [U_{c,t+1}(s^{t+1}) (A_{t+1}(s^{t+1}) F_{k,t} + (1-\delta )(1+\tau _{x, t+1}(s^{t+1})) + \phi _{k_{t+1}}] \end{aligned} \end{equation}

(7) \begin{equation} c_t\left(s^t\right) + x_t\left(s^t\right) + g_t\left(s^t\right) = y_t\left(s^t\right) \end{equation}

where $g_t$ is the government consumption wedge, which is assumed to fluctuate around a deterministic trend, $(1 + \gamma _A)^t$ ; $\gamma _A$ is the rate of labor-augmenting technical progress and $A_t\left(s^t\right)$ is the efficiency wedge; furthermore, $U_{c,t}$ , $U_{l,t}$ , $F_{l,t}$ , $F_{k,t},$ and $\phi _{k_{t+1}}$ represent derivatives of the utility function with respect to $c$ and $l$ , the derivatives of the production function with respect to $l$ and $k$ , and the derivative of the adjustment costs function with to $k$ , respectively.

2.2. Accounting results

What are the dynamics of the four wedges in each crisis? Using the neoclassical growth model presented in Section 2.1, we estimated the path for each wedge during the three different crises. As can be seen in Figure 2, the efficiency wedge ( $A_t$ ) fell in all three recessions. However, during the 1995 and 2008 crises, the fall was much larger relative to the one during the COVID crisis. For the $\tau _{l,t}$ , the distortion that resembles a labor income tax, the result is the opposite, however. During the COVID crisis, it fell more than during the previous episodes. This seems to be directly related to the dynamics of hours of work in Figure 1.

Figure 1. Macroeconomic variables during crises in Mexico. Notes: The blue curves are for the 1995 crisis, where 1994Q3 = 100; the red curves stand for the 2008 crisis with 2008Q3 = 100, and the black curves for the COVID recession, with 2019Q4 = 100. The horizontal axes present quarters relative to the beginning of the episode. All variables are divided per working-age population and with the adjustments described in the text.

Figure 2. The behavior of estimated distortions in the model during the crises. Notes: The blue curves are for the 1995 crisis, where 1994Q3 = 100; the red curves stand for the 2008 crisis with 2008Q3 = 100, and the black curves for the COVID recession, with 2019Q4. The horizontal axes present quarters relative to the beginning of the episode.

It is worth noting that the distortion on intertemporal decisions, $\tau _{x,t}$ , also had a larger change during the COVID crisis, followed by the 2008 crisis, relative to the 1995 crisis, as presented in the left-handed panel side in the bottom of Figure 2. This helps to explain why investment behaved so differently during these crises, as illustrated in Figure 1. Finally, the government consumption wedge ( $g_t$ ) also presents very different behaviors among crises. In the next section, we decompose the wedge to understand better what is behind those differences.

What is the relative importance of each wedge in order to explain the dynamics of output in those episodes? After measuring/estimating the distortions, it is possible to simulate the paths of the variables in the model to see which wedge helps to account for data movements. Following Chari et al. (Reference Chari, Kehoe and McGrattan2007), the marginal effects of each wedge are obtained by keeping all other wedges fixed at their steady-state values, but the one we are interested in, for example, if we want to see the contribution of the efficiency wedge, we let it to fluctuate, while the others (labor, investment, and government consumption) are held fixed. Then, we can see how much of the data behavior the model with only one distortion can account for. The procedure also works by letting two or three wedges vary simultaneously throughout time.

The simulations with only one wedge and with only one wedge off (with only one wedge remaining constant) and the observed output paths for the 1995 and 2008 crises are presented in Figures 3 and 4.Footnote ² Table 1 presents the statistics used to evaluate the BCA simulations: success ratio, root-mean-square error (RMSE), and the $\phi$ -statistic presented in Brinca et al. (Reference Brinca, Chari, Kehoe and McGrattan2016):

\begin{equation*} \phi ^y_i = \frac {1/\sum _t (y^D_t-y_{i,t})^2}{\sum _{j} (1/\sum _t (y^D_t-y_{i,t})^2)}, \end{equation*}

where $y^D_t$ is the observed data for output, $y_{i,t}$ is the simulated path for output prescribed by model $i \in [1, j]$ . The statistics lie between $0$ and $1$ , and the closer the value is to $1$ , more it accounts for observed movements in the data.

Figure 3. Simulated output trajectory during the 1995 crisis. Notes: The red curves are for the model with (without) the efficiency wedge in one-wedge (off) economies; analogously, the green curves are related to the labor wedge, the blue curves to the investment wedge, and the orange curves to the government consumption wedge. The black line is the observed data. 1994Q4 = 100.

Figure 4. Simulated output trajectory during the 2008 crisis. Notes: The red curves are for the model with (without) the efficiency wedge in one-wedge (off) economies; analogously, the green curves are related to the labor wedge, the blue curves to the investment wedge, and the orange curves to the government consumption wedge. The black line is the observed data. 1994Q4 = 100.

Table 1. BCA decomposition statistics

Success ratio: the relative frequency when simulated and observed data had the same sign; linear correlations between simulated and observed data; RMSE: root of the mean-square error; $\phi$ -statistic following Brinca et al. (Reference Brinca, Chari, Kehoe and McGrattan2016). For the two crises, we consider a 12-quarter window; we define the first quarters of the BCA exercise as 1994Q4 and 2008Q2.

The efficiency wedge alone has the best performance among one-wedge economies for both episodes. The model with only that distortion accounts for 71.3% of whole output movements during the 1995 crisis and 54.3% during the Great Recession. The investment and government consumption wedges explain around 10% of GDP fluctuations during the 1995 crisis, while their relative contributions rise to around 17% during the Great Recession.

The poor performance of the models with only government consumption raises a question. Given that the 1995 episode was a currency crisis (followed by a large increase in net exports) and that the 2008 episode was an international crisis, shouldn’t they both be transmitted through the balance of payments and/or assigning a greater role to the government consumption wedge due to an increase in net exports following exchange rate depreciation? This calls for a further investigation of the dynamics of the government consumption wedge. In Figure 5, we have decomposed the dynamics of the government consumption wedge into the changes in government consumption, exports, and imports for the 1995 and 2008 crises.

Figure 5. Decomposition of the government consumption wedge for the 1995 and 2008 crises. Notes: The blue curves are for the 1995 crisis, where 1994Q3 = 100; the red curves stand for the 2008 crisis with 2008Q3 = 100, and the black curves for the COVID recession, with 2019Q4. The horizontal axes present quarters relative to the beginning of the episode.

We can see in the top panel of Figure 5 that the fiscal response was very different between crises. In 1995, the decrease in government spending offset the increase in net exports (see that exports in the middle panel rose more than imports in the bottom panel). In 2008, however, there was an increase in both government spending and net exports. This probably helps to explain why the relative role of the government consumption wedge rises from around 10% to more than 17%.

Given the prominent role of the efficiency wedge in explaining a large share of output fluctuations in both crises, we explore in the next section what is behind the efficiency wedge and how it is related to output falls and the international transmission of crises.

3.1. Detailed economy

We introduce imported intermediate inputs in the final goods production in Schmitt-Grohé and Uribe (Reference Schmitt-Grohé and Uribe2003) small open-economy debt-elastic interest rate model. Households behave in a rational way and derive utility from consumption ( $c_t$ ) and disutility from hours worked ( $l_t$ ). As is usual in this kind of model, households have a positive but decreasing marginal utility of consumption and an increasing marginal disutility of labor. The domestic agents can use their own resources as well as foreign capital. The dynamics of foreign debt ( $d_t$ ) are given by:

(8) \begin{equation} d_t = (1 + r_{t-1}) d_{t-1} - q_t + c_t + x_t + e_t m_t. \end{equation}

where $d_t$ is the foreign debt, $r_t$ is the one-period real interest rate, $q_t$ stands for gross output, $x_t$ represents investment, $m_t$ stands for imported intermediate goods, and $e_t$ is the foreign (imported intermediate goods) prices over domestic prices. The interest rate is a function of the equilibrium interest rate and the country’s foreign indebtedness level. $r_t = r + \psi (\! \exp(d_t-d) -1)$ , where $r$ represents the steady-state real interest rate, $d$ represents the equilibrium value of foreign debt, and $\psi$ is a parameter.

Gross output is produced by combining imported intermediate goods, capital, and labor as follows:

(9) \begin{equation} q_t = m^{\mu }_t (k^{\alpha }_t l^{1-\alpha }_t)^{1-\mu }. \end{equation}

where $k$ is the stock of productive capital, which has a share of $\alpha$ in total output, and $\mu$ is the share of imported intermediate goods.

We assume $e_t$ to be exogenous, following an AR(1) process:

(10) \begin{equation} \ln e_t = ( 1 - \rho _e) \ln e + \rho _e \ln e_{t-1} + \epsilon ^{e}_t. \end{equation}

where $\rho _e$ is the persistence parameter, $e$ without the subscript $t$ is the steady-state value of $e_t$ , and $\epsilon ^{e}_t$ is a shock. The law for capital accumulation is given by:

(11) \begin{equation} k_{t+1} = (1-\delta ) k_t + x_t - \Lambda (k_{t+1}). \end{equation}

where $\Lambda (k_{t+1}) = \frac{\eta }{2}(k_{t+1} - k_t)^2$ represents capital adjustment costs.

Under the presented assumptions, the optimization problem is given by:

(12) \begin{equation} \max _{c_t, l_t, d_t, k_{t+1}} E_t \sum _{t=0}^{\infty } \beta ^t U(c_t, l_t) \end{equation}

where $U(c_t, l_t)$ is the instantaneous utility, subject to the debt dynamics from equation 8, the production technology expressed in equation 9, the law for capital accumulation (equation 11), taking the exogenous movements of the real exchange rate (equation 10) and the real interest rate as given.

One more restriction should be imposed to assure non-Ponzi dynamics in the system. The transversality condition is given by:

(13) \begin{equation} \lim \limits _{j \rightarrow \infty } E_t \frac{d_{t+j}}{\prod ^{j}_{s} (1+r_{s})} \leq 0, \end{equation}

which implies that the present value of the debt should be less than or equal to zero, that is, there should be no remaining debt at the limit.

The first-order conditions with respect to $d_{t+1}, c_t, l_t, k_{t+1}$ and $m_t$ are as follows:

(14) \begin{equation} \lambda _t = \beta E_t \lambda _{t+1} (1 + r_{t}), \end{equation}

(15) \begin{equation} \lambda _t = U_{c_{t}}, \end{equation}

(16) \begin{equation} -\frac{U_{l_{t}}}{U_{c_{t}}} = (1-\alpha ) (1-\mu ) \frac{q_t}{l_t}, \end{equation}

(17) \begin{equation} \lambda _t (1+ \phi (k_{t+1} - k_t)) = \beta E_t \lambda _{t+1} \big(\eta (k_{t+2} - k_{t+1}) + 1-\delta + (1-\mu ) \alpha \frac{q_{t+1}}{k_{t+1}} \big), \end{equation}

(18) \begin{equation} \frac{\mu q_t}{m_t} = e_t, \end{equation}

where $\partial U(c_t, l_t)/\partial c_t = U_{c_{t}}$ and $\partial U(c_t, l_t)/\partial l_t = U_{l_{t}}$ and $\lambda _t$ represent the Lagrange multiplier. The definition of trade balance over GDP ( $tby_t$ ) closes the model:

(19) \begin{equation} 1 - \frac{c_t}{y_t} - \frac{x_t}{y_t} = tby_t. \end{equation}

where $y_t = q_t - e_t m_t$ is the value-added output. We assume that all intermediate goods produced domestically are exported.

3.2. Shocks to relative prices as shocks to productivity: an equivalence

How does the detailed economy relate to the prototype model in Section 2.1? In Proposition 1, we can map the equilibrium allocations of the small open-economy model with imported intermediate inputs into the prototype economy used in the BCA exercises and provide a structural interpretation for the wedges (we thank the referee for his suggestions to improve the exposition of the proposition and the discussion of the results).

Proof. The efficiency wedge mapping is obtained by substituting the first-order condition with respect to $m_t$ (equation 18) for the gross production function (equation 9): $q_{t}=\left (\mu \frac{q_{t}}{e_{t}}\right )^{\mu }\left (k_{t}^{\alpha } l_{t}^{1-\alpha }\right )^{1-\mu }=\left (\frac{\mu }{e_{t}}\right )^{\frac{\mu }{1-\mu }} k_{t}^{\alpha } l_{t}^{1-\alpha }$ . According to the definition of the value-added output, we have that $y_{t}=q_{t}-e_{t} m_{t}=(1-\mu ) q_{t}=(1-\mu )\left (\frac{\mu }{e_{t}}\right )^{\frac{\mu }{1-\mu }} k_{t}^{\alpha } l_{t}^{1-\alpha }$ , or $y_{t}=A_t k_{t}^{\alpha } l_{t}^{1-\alpha }$ , where $A_t =(1-\mu )\left (\frac{\mu }{e_{t}}\right )^{\frac{\mu }{1-\mu }}$ . The labor wedge distorts the intratemporal decision. In the detailed economy, there is no such distortion. It is easy to see that if we compare equation 16 with equation 5 using the definition of GDP, $y_{t} = (1-\mu ) q_{t};$ thus, $(1- \tau _{l,t}) = 0$ . From equation 15, we have the marginal utility of consumption. The right-hand side of equation 6 presents the marginal utility of consumption, which is equal to the Lagrange multiplier, times the inverse of the investment wedge. By equating the left-hand side of equation 17 to the left-hand side of equation 6, we have that $(1+\tau _{x,t}) = (1+ \eta (k_{t+1} - k_t))$ . Moreover, making the right-hand sides of the two equations equal yields $\beta E_t[u_{c,t+1} (A_{t+1} F_{kt} + (1-\delta )(1+\tau _{x, t+1}))] = \beta E_t \lambda _{t+1} (\eta (k_{t+2} - k_{t+1}) + 1-\delta + (1-\mu ) \alpha \frac{y_{t+1}}{k_{t+1}}) \iff (1+\tau _{x,t+1}) = 1+ (1+ \eta (k_{t+2} - k_{t+1}))/(1-\delta )$ . Finally, the government consumption wedge can be calculated directly from the trade balance (equation 8 and equation 19): $y_t - x_t - c_t = g_t = (1 + r_{t-1}) d_t - d_t$ .

We can see from the definition of the efficiency wedge in the equivalence above, $A_t =$ $(1-\mu )\left (\frac{\mu }{e_{t}}\right )^{\frac{\mu }{1-\mu }}$ , that an increase in relative prices manifests itself as a deterioration in the efficiency wedge. This is a key result of this paper. By extending a simple small open-economy model with the introduction of imported intermediate goods into the gross production function, we have revealed how international shocks such as terms of trade shock and/or a nominal exchange rate shock have real effects on GDP. For instance, an increase in foreign prices relative to domestic prices produces a negative impact on production that is not matched by a proportional decrease in imports, since capital and labor are not perfect substitutes and, at least in the short run, domestic firms cannot replace the imported intermediate goods. Proposition 1 offers a solution to the puzzle presented by Kehoe and Ruhl (Reference Kehoe and Ruhl2008). Furthermore, it reveals a “missing” international link as a mechanism for the transmission of crises in the efficiency wedge and why the government consumption wedge played a smaller role in explaining output fluctuations during the 1995 and 2008 crises.

3.3. Quantitative exercise

In Section 3.1, we presented the structural model for understanding the main drivers of output during the 1995 and 2008 crises. In Section 3.2, we showed that relative price shocks generate changes in the efficiency wedge in the model, and we know from the results of Section 2.2 this is the main distortion that we need in the neoclassical growth model to account for observed GDP fluctuations. How does the model capture the dynamics of output during both crises, namely the largest fall and faster recovery in the 1995 crisis relative to the 2008 episode?

In order to answer that question, we perform a stochastic simulation of the log-linearized version of the model with a sequence of shocks on the relative prices as the deviations from the steady state and compare the prescribed path for GDP growth with the observed deviations of output from its estimated trend during two windows: 1994Q1–1997Q2 and 2008Q3–2015Q1.

Figure 8. The dynamics of output during the 1995 and the 2008 crises: data vs simulations. Notes: In both graphs, the solid line stands for log-detrended data on output, and the dashed line refers to the simulation of the log-linearized version of the detailed economy with the exogenous shocks on the real exchange rate. On the top panel, the simulation is for the 1994Q1–1997Q2 period. On the bottom panel, the simulation is for the 2008Q3–2015Q1 period.

Figure 9. Short-run interest rate during the three crises in Mexico. Notes: The blue curves are for the 1995 crisis, where 1994Q3 = 100; the red curves stand for the 2008 crisis, with 2008Q3 = 100, and the black curves are for the COVID recession, with 2019Q4. The horizontal axes present quarters relative to the beginning of the episode.

3.4. Investment and labor frictions

What are the candidates for the investment and labor wedges? For the former, we believe that shocks to the capital accumulation process, as in Cooper and Ejarque (Reference Cooper and Ejarque2000) could explain the observed path of the distortion during the aforesaid crises. The nature of the shock might differ, though. We can see in Figure 9 that the short-run interest rate rose rapidly during the 1995 balance-of-payments crisis, whereas it had the opposite movement amid the 2008 and the COVID crises. We consider this as an indication that in 1995, a shock on the country’s risk premium might have affected capital accumulation, whereas during the other two crises, the nature of the shock was more on entrepreneurs’ (and banks’) confidence and/or difficulties within a financial accelerator with nominal rigidity set up as in Gertler and Kiyotaki (Reference Gertler and Kiyotaki2010), since monetary policy was expansionary, not only via interest rates but also by alleviating credit market constraints. See Brinca et al. (Reference Brinca, Costa-Filho and Loria2020) for other sorts of detailed economies that can generate investment wedges.

Finally, for the labor wedge, we hypothesize that it can be the result of one (or a combination) of three types of frictions: financial frictions may have contributed to the labor wedge as in Mendoza (Reference Mendoza2010), as well as markups in products and labor markets as in Gali et al. (Reference Gali, Gertler and Lopez-Salido2007) and labor market policies operating in an environment with search and matching frictions with an endogenous selection mechanism that can not only influence the labor wedge but also the dynamics of aggregate productivity (Lama et al. Reference Lama, Leyva and Urrutia2022).