A. The Main Assumptions

Demographics and Household Behavior—The world consists of two countries, home and foreign. In each period t, the world economy is populated by a continuum of infinitely lived households between 0 and $N_t^W$. (A superscript W denotes world variables. Foreign variables are starred.) Each household consumes home and foreign goods, supplies labor, and holds financial assets. As in Weil (1989a, b), households are born on different dates owning no assets, but they own the present discounted value of their labor income. The number of households in the home economy, N_t grows over time at the exogenous rate n, i.e., N_t+1 = (1 + n)N_t. We normalize the size of a household to 1, so that the number of households alive at each point in time is the economy’s population. Foreign population grows at the same rate as home population. We assume that the world economy has existed since the infinite past and normalize world population at time 0 so that $N_0^W=1$.

Households at home and abroad have perfect foresight, though they can be surprised by initial, unexpected shocks. Households maximize intertemporal utility functions. The period utility function in both countries is logarithmic in consumption of a constant elasticity of substitution (CES) world consumption basket and in the amount of labor effort supplied by the household. Domestic households have discount factor β, 0 < β < 1. Foreign households have discount factor αβ, 0< α ≤ 1. When α < 1, foreign households are more impatient than domestic households.⁵

Goods Market and Production—There are two goods in the world economy. Each country is fully specialized in the production of a country-specific good, performed by a continuum of atomistic, perfectly competitive, infinitely lived firms. Home firms, producing the home good, occupy the interval [0, a]; foreign firms, producing the foreign good, are in the range (a, 1]. Firms produce output using labor as the only factor of production according to a linear technology that is subject to multiplicative, country-wide productivity shocks.

Asset Market—Households in both countries trade a riskless real bond denominated in units of the world consumption basket domestically and internationally.⁶

B. Households

Consumers have identical preferences over a real consumption index (C) and leisure (1‒L, where L is labor effort supplied in a competitive labor market, and we normalize the endowment of time in each period to 1). At any time t₀, the representative home consumer j born in period ν ∈ [−∞, t₀] maximizes the intertemporal utility function:

with 0 < ρ <1.

The consumption index is $C_t^{{\nu }^j}={\left[a^{\frac{1}{\omega }}{\left(C_{\mathit{\text{Ht}}}^{{\nu }^j}\right)}^{\frac{\omega -1}{\omega }}+{\left(1-a\right)}^{\frac{1}{\omega }}{\left(C_{\mathit{\text{Ft}}}^{{\nu }^j}\right)}^{\frac{\omega -1}{\omega }}\right]}^{\frac{\omega }{\omega -1}}$, where ω > 0 is the intratemporal elasticity of substitution between consumption of domestic and foreign goods (C_H and C_F, respectively). Foreign agents consume an identical basket of goods. Trade in goods is free. There are no transportation and transaction costs.

The representative home consumer enters a period holding bonds purchased in the previous period.⁷ He or she receives interest on these bond holdings, earns labor income, consumes, and purchases new bonds with which he or she will enter the next period. Letting $B_{t+1}^{v^j}$ denote the representative home consumer’s holdings of bonds entering t + 1, the period budget constraint is:

where r_t is the risk-free world real interest rate between t − 1 and t, and w_t is the real wage, both in units of the consumption basket.⁸

The representative home consumer born in period ν maximizes the intertemporal utility function (1) subject to the constraint (2). Dropping the j superscript (because symmetric agents make identical choices in equilibrium), optimal labor supply is given by:

which equates the marginal cost of supplying labor to the marginal utility of consumption generated by the corresponding increase in labor income.

The first-order condition for optimal holdings of bonds yields the Euler equation:

for all ν ≤ t.

As usual, first-order conditions and the period budget constraint must be combined with the appropriate transversality condition (omitted) to ensure optimality.

Foreign consumers maximize a similar intertemporal utility function and are subject to an analogous budget constraint as home consumers. The only difference is that the discount factor of foreign households is αβ. Otherwise, a similar labor-leisure tradeoff, Euler equation, and transversality condition hold for foreign households.

C. Firms

Output supplied at time t by the representative home firm i is a linear function of labor demanded by the firm:⁹

where Z_t is exogenous, economy-wide productivity. Production by the representative foreign firm is a linear function of $L_t^{i*}$, with productivity $Z_t^{*}$.

Output demand comes from domestic and foreign consumers. The demand for firm i by the representative domestic household born in period ν is $C_{\mathit{\text{Ht}}}^{v^i}={({\mathit{\text{RP}}}_t)}^{-\omega }{C}_t^v$, obtained by maximizing C subject to a spending constraint. We denote with RP_t the price of the home good in units of consumption. Atomistic, competitive home firms take this price as given. Aggregating across home households alive at time t, total demand for firm i coming from domestic consumers is $C_{Ht}^i={({\mathit{\text{RP}}}_t)}^{-\omega }a{\left(1+n\right)}^t{c}_t$, where

is aggregate per capita home consumption of the composite consumption basket.

Given identity of preferences across countries, total demand for firm i by foreign consumers is $C_{\mathit{\text{Ht}}}^{i*}={({\mathit{\text{RP}}}_t)}^{-\omega }\left(1-a\right){\left(1+n\right)}^t{c}_t^{*}$, where $c_t^{*}$ is aggregate per capita foreign consumption, the definition of which is similar to that of c_t. (Absence of transportation and transaction costs implies that the price of the home good in units of consumption is the same at home and abroad.)

Total demand for home firm i is obtained by adding the demands originating in the two countries:

where ${\stackrel{⌢}{c}}_t^W$ is aggregate (as opposed to aggregate per capita) world demand of the composite good: ${\stackrel{⌢}{c}}_t^W\equiv N_t{c}_t+N_t^{*}{c}_t^{*}$.¹⁰

Perfect competition results in prices equal to marginal costs, so that:

Using the market clearing conditions $Y_t^{\mathit{\text{Si}}}=Y_t^{\mathit{\text{Di}}},{\stackrel{⌢}{c}}_t^W={\stackrel{⌢}{Y}}_t^{\mathit{\text{SW}}}={\stackrel{⌢}{Y}}_t^{\mathit{\text{DW}}}(={\stackrel{⌢}{Y}}_t^W)$, and the expressions for firm i’s supply and demand, labor demand can be written as:

Ceteris paribus, firm i’s labor demand is a decreasing function of real output price and productivity. It is an increasing function of world consumption demand. Optimal behavior by foreign firms results in similar price and labor demand equations.¹¹

D. Aggregation

A. Steady State

The details of the solution for the steady state of our model are in Ghironi, İşcan, and Rebucci (2003). Here, we summarize the main characteristics, denoting steady-state levels of variables with overbars.

It is known, at least since Becker (1980), that a standard representative agent model with identical discount factors across agents (i.e., n = 0, α = 1) results in indeterminacy of the steady-state distribution of net foreign assets. If discount factors differ across agents with no other modification to the standard model (n = 0, α < 1), the distribution of wealth across agents ends up collapsing into one in which the most patient household owns all the wealth. Buiter (1981) and Weil (1989a) demonstrated that models with overlapping generations in which households are not linked by intergenerational altruism can deliver a non-degenerate distribution of asset holdings across countries. Our model achieves precisely the same goal by assuming n > 0 and absence of intergenerational linkages in the form of altruism or government transfers. Ghironi (2000) shows that, when α = 1, this delivers a determinate steady state and stationary dynamics of prices and aggregate per capita quantities following non-permanent shocks.

To demonstrate the influence of differences in discounting across countries, we start with the special case in which all preference parameters are identical across domestic and foreign households (α = 1) but initial steady-state productivity levels $\overline{Z}\text{and}{\overline{Z}}^{*}$ may differ (possibly as a consequence of previous, asymmetric, permanent movements in productivity).¹³

When α = 1, steady-state levels of labor effort are identical across countries $(\frac{\overline{L}}{{\overline{L}}^{*}}=1)$, and net foreign assets are zero $(\overline{B}={\overline{B}}^{*}=0)$, regardless of relative productivity $(\frac{\overline{Z}}{{\overline{Z}}^{*}})$. This happens because, when consumers’ intertemporal preferences are identical at home and abroad, given a common world interest rate, households in the two countries have identical incentives to borrow or lend. (The desired slope of the consumption profile is the same for each domestic and foreign household.) In this case, the only possible steady-state equilibrium is one in which $\overline{r}=\frac{1-\beta }{\beta }$ and net foreign assets are zero even if $\frac{\overline{Z}}{{\overline{Z}}^{*}}\ne 1$. If α = 1 and $\frac{\overline{Z}}{{\overline{Z}}^{*}}\ne 1$, domestic and foreign GDPs in units of consumption differ $(\overline{y}\ne {\overline{y}}^{*})$, and so do consumption levels $(\overline{c}\ne {\overline{c}}^{*})$, but consumption equals GDP in each country, so that net foreign assets are zero. Since $\overline{y}=\overline{w}\overline{L}\text{and}{\overline{y}}^{*}={\overline{w}}^{*}{\overline{L}}^{*}$, $\overline{L}={\overline{L}}^{*}$ when α = 1 implies that the different GDP levels generated by different productivity levels translate into different real wages and labor incomes across countries.¹⁴ In particular, the more productive country has a higher steady-state real wage and consumption and a lower relative price for the same labor effort as the less productive country.¹⁵

In the general case α ≤ 1, we can write the solution for $\overline{r},\overline{B}$, and cross-country ratios of any pair of other endogenous variables $\overline{x}\text{and}{\overline{x}}^{*}$ as functions of the steady-state productivity ratio $\frac{\overline{Z}}{{\overline{Z}}^{*}}$. The characteristics of these functions depend on the values of structural parameters, and the steady-state levels of $\overline{r},\overline{B}$, and other endogenous variables can be obtained numerically given assumptions on $\overline{Z}\text{and}{\overline{Z}}^{*}$.¹⁶ Consider the following two examples:

1. If α ≤ 1 and $\overline{Z}={\overline{Z}}^{*}=1$, a wide range of plausible parameterizations yields $\overline{B}>0({\overline{B}}^{*}<0)$, $\overline{c}>{\overline{c}}^{*},\overline{L}<\overline{L^{*}},\overline{w}>{\overline{w}}^{*},\overline{\mathit{\text{RP}}}>{\overline{\mathit{\text{RP}}}}^{*},\overline{y}<{\overline{y}}^{*}$. If domestic agents are more patient than foreign, they accumulate steady-state assets, which make it possible to sustain relatively higher consumption with a smaller labor effort. Lower labor supply generates a higher equilibrium real wage and relative price. The labor effort differential prevails on the relative price differential in generating lower GDP at home than abroad, where higher GDP is required to pay interest on the accumulated debt.

2. If α < 1 and $\overline{Z}=1$ is sufficiently smaller than ${\overline{Z}}^{*}$, the same plausible parameter values as in Example 1 yield $\overline{B}>0({\overline{B}}^{*}<0),\overline{c}<{\overline{c}}^{*},\overline{L}<{\overline{L}}^{*},\overline{w}<{\overline{w}}^{*},\overline{\mathit{\text{RP}}}>{\overline{\mathit{\text{RP}}}}^{*},\overline{y}<{\overline{y}}^{*}$. Sufficiently higher productivity in the more impatient country causes the steady-state real wage differential to switch sign, so that the real wage is now higher in the foreign economy. This induces foreign agents to consume more, and their consumption rises above that of domestic agents, with an increase in the size of the foreign economy’s debt.

Heterogeneity in discounting across countries implies the presence of tilts in the steady-state consumption and labor effort profiles of individual households at home and abroad. To see this, observe that, when $\alpha <1,\beta \left(1+\overline{r}\right)>1\text{and}\alpha \beta \left(1+\overline{r}\right)<1$. In conjunction with the Euler equation (4) and its foreign counterpart, this implies that the steady-state consumption profiles of individual home households display an upward tilt, whereas there is a downward tilt in the steady-state consumption profiles of foreign households. The Euler equation and the labor-leisure tradeoff for an individual household make it possible to verify that the steady state is also characterized by a downward (upward) tilt in the labor effort of individual home (foreign) households.¹⁷

The tilt of individual consumption profiles determines whether a country is a steady-state creditor or debtor. The proof is in Ghironi, İşcan, and Rebucci (2003), but the intuition is simple: Given a constant real wage, the only way for home households to sustain an increasing consumption profile with decreasing labor effort is by accumulating assets. Since there is no home household with negative financial assets in steady state, home aggregate per capita net foreign assets must be positive. As we shall see, the tilt of steady-state individual consumption profiles has important consequences also for the dynamics of the economy in response to shocks.

B. Dynamics

The aggregate model of Section II.B can be safely log-linearized around the steady state. The assumptions that n >0 and newborn households enter the economy with no assets generate stationary dynamics following non-permanent shocks because the steady state is uniquely determined.¹⁸

We solve the log-linear model with the method of undetermined coefficients following Campbell (1994). In what follows, we use sans serif fonts to denote percent deviations from the steady state and focus on the model solution in terms of the minimum state vector, which at time t consists of the predetermined levels of net foreign assets and the (gross) risk-free real interest rate (the endogenous states) and the current levels of domestic and foreign productivity (the exogenous states), i.e., ${\left[{\text{B}}_t,{\text{r}}_t,{\text{Z}}_t,{\text{Z}}_t^{*}\right]}^{\prime }$.¹⁹ The solution of the model can then be written as ${\left[{\text{B}}_{\text{t}+1},{\text{r}}_{t+1},{\text{X}}_t^{\prime },{\text{X}}_t^{{*}^{\prime }}\right]}^{\prime }=\Xi {\left[{\text{B}}_t,{\text{r}}_t,{\text{Z}}_t,{\text{Z}}_t^{*}\right]}^{\prime }$, where X_t $({\mathrm{X}}_t^{*})$ is a (column) vector of endogenous, non-state, home (foreign) variables and Ξ is a matrix of elasticities of endogenous variables to the endogenous and exogenous components of the state vector.

In the solution, we assume that productivity levels at home and abroad obey AR(1) processes in all periods after the time of an initial impulse (t = 0 in the impulse responses below): Z_t = ϕ Z_{t− 1}; ${\mathrm{Z}}_t^{*}=\varphi {\mathrm{Z}}_{t-1}^{*},0\le \varphi \le 1$. Of course, if ϕ = 1, impulses to productivity cause the economy to eventually settle at a new steady state that differs from the initial one.

Two important implications for off-steady-state dynamics emerge from our model. First, non-zero steady-state net foreign assets introduce an additional channel through which the past history of the economy matters for current dynamics relative to the model with zero steady-state assets. The predetermined, risk-free interest rate is an additional state variable in the solution. The intuition is simple. If steady-state net foreign assets are zero (if α= 1), the effect of the interest burden on previously accumulated debt is lost in the log-linearization of the laws of motion for domestic and foreign net foreign assets. This is no longer the case when steady-state assets differ from zero. This implies that the effect of net foreign asset accumulation on cross-country differences in the levels of other endogenous variables is amplified relative to a model with zero steady-state net foreign assets.

Second, worldwide productivity shocks – which have no impact on the current account in the symmetric version of the model – affect net foreign asset accumulation. This is reflected in the impossibility to write the deviation of net foreign assets from the steady-state as a function of the cross-country productivity differential. The effect of worldwide shocks on asset accumulation comes through various channels: The tilts in the consumption profiles of individual households described above trigger asset trade for given world interest rate as a consequence of heterogeneous discounting of future utility. In turn, the resulting differences in consumption dynamics across countries generate differences in relative price and GDP movements. Finally, if the shocks are not permanent, they also affect the world interest rate and the interest rate burden (or income) on previously accumulated assets, further altering the current account.

A. Parameterization

We interpret periods as quarters and choose the following benchmark parameter values: β = 0.99 (a standard choice), α = 0.9999 (so that the foreign discount factor is 0.9899), ω = 3, ρ = 0.33, a = .5 (countries have equal size), n = 0.01$\overline{Z}=1,\text{and}{\overline{Z}}^{*}=1.29$.

We choose α very close to 1 because even small differences between the foreign and home discount factors result in very large steady-state net foreign asset positions in the model of this paper. To avoid overstating the effect of non-zero steady-state assets, we choose a value of α such that the long-run ratio of debt to quarterly GDP for the foreign economy is approximately 35 percent on an annualized GDP basis. The value of ω is in (the lower portion of) the range of estimation results from the trade literature on the United States and OECD countries (Feenstra, 1994; Harrigan, 1993; Shiells, Stern, and Deardorff, 1986; Lai and Trefler, 2002).²⁰ The choice of ρ implies that households in both countries spend one third of their time working in the symmetric steady-state world. Our benchmark parameter values are plausible for G-7 countries.²¹ If we think of the United States as the relatively impatient country, consistent with the evidence in favor of a lower propensity to save for U.S. households relative to European and Asian ones, our data suggest that the labor productivity gap between the United States and the rest of the G-7 has been approximately 30 percent. We discuss the consequences of different values for α,ω, and the ratio $\frac{\overline{Z}}{{\overline{Z}}^{*}}$ below.

The benchmark parameter values above result in the steady-state configuration of Example 2 above, which is consistent with several stylized facts for the U.S. vis-à-vis the rest of the G-7: $\overline{B}>0({\overline{B}}^{*}<0),\overline{c}<{\overline{c}}^{*},\overline{L}<{\overline{L}}^{*},\overline{w}<{\overline{w}}^{*},\overline{RP}>{\overline{RP}}^{*},\overline{y}<{\overline{y}}^{*}$. Relative consumer impatience causes the model-U.S. economy to accumulate a steady-state debt against the rest of the world. Nevertheless, higher productivity results in higher real wage, GDP, consumption, and labor effort (the latter is higher than abroad for the need to pay interest on the accumulated debt). Larger U.S. GDP comes with a lower price of U.S. goods relative to the patient economy (home). Numerical values for the steady-state levels of variables are in Table 1, which also displays the values of the elasticities of endogenous variables to the state vector in the log-linear model solution. (Elasticities to productivity depend on the persistence of shocks, ϕ. We assume ϕ= 1 below.)

^{Table 1.}

The Benchmark Solution

^{Table 1.}

The Benchmark Solution

Steady-State Levels
$\overline{r}=.01015554120$	$\overline{B}=.5588806661$
$\overline{L}=.3299443011$	${\overline{L}}^{*}=.3300470075$
$\overline{w}=1.045291364$	${\overline{w}}^{*}=1.238564933$
$\overline{y}=.3448879285$	${\overline{y}}^{*}=.4087846498$
$\overline{c}=.3449748265$	${\overline{c}}^{*}=.4086977516$

View Table

^{Table 1.}

The Benchmark Solution

Steady-State Levels
$\overline{r}=.01015554120$	$\overline{B}=.5588806661$
$\overline{L}=.3299443011$	${\overline{L}}^{*}=.3300470075$
$\overline{w}=1.045291364$	${\overline{w}}^{*}=1.238564933$
$\overline{y}=.3448879285$	${\overline{y}}^{*}=.4087846498$
$\overline{c}=.3449748265$	${\overline{c}}^{*}=.4086977516$

B. A Permanent World Productivity Shock

We focus on the case of a permanent increase in world productivity because this is where the implications of our model are most striking for the parameter values above.²²

In the familiar symmetric model with zero steady-state net foreign assets, a permanent increase in world productivity results in no movement in net foreign assets. GDP, the real wage, and consumption in both countries increase immediately by the full amount of the shock. There are no changes in labor effort, relative prices, and the terms of trade $(\frac{{\mathit{\text{RP}}}_t}{{\mathit{\text{RP}}}_t^{*}})$. Anticipating the permanent consequences of the shock, agents in both countries simply find it optimal to consume the entire consumption value of the increase in productivity in all periods without adjusting their labor effort.²³

In contrast, asymmetry of the steady state results in interesting dynamics following a permanent shock to world productivity. This is illustrated in Figure 1, which shows the impulse responses to a 1 percent, permanent increase in productivity at home and abroad (i.e., ϕ = 1).

Permanent World Productivity Shock, Benchmark

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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Permanent World Productivity Shock, Benchmark

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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Permanent World Productivity Shock, Benchmark

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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The home economy accumulates assets over time in response to the shock, the foreign economy accumulates debt. Eventually, the increase in home assets (foreign debt) converges to an amount equal to the increase in world productivity. A permanent productivity shock has no effect on the risk-free interest rate.²⁴ It follows that the dynamics in Figure 1 do not originate in the effect of changes in the interest rate on the burden of (income from) the initial steady-state debt (assets).²⁵ Home and foreign households find it beneficial to engage in further asset trade without changes in the interest rate due to asymmetric income effects that stem from different discounting of future utility across countries and the implied tilt in the consumption profiles of individual households.

To see this, note that, after a permanent productivity shock that has no effect on the world interest rate, the home current account is determined by:

where ${\tilde{w}}_t$ is the annuity value of the real wage at the steady-state interest rate $\overline{r}$ $({\tilde{w}}_t\equiv \frac{\overline{r}}{1+\overline{r}}{{\displaystyle {\sum }_{s=t}^{\infty }\left(\frac{1}{1+\overline{r}}\right)}}^{s-t}{w}_s)$, and W_t is beginning-of-period aggregate per capita wealth at the interest rate $\overline{r}(W_t\equiv \left(1+\overline{r}\right)B_t+{\displaystyle {\sum }_{s=t}^{\infty }{\left(\frac{1}{1+\overline{r}}\right)}^{s-t}{w}_s})$. The foreign current account obeys a similar equation, with αβ replacing β, and it satisfies the constraint $\mathit{\text{aC}}{A}_t+\left(1-a\right)C{A}_t^{*}=0$. The term $w_t-{\tilde{w}}_t$ in (17) captures the effect of consumption smoothing. If the real wage is above its permanent level and is expected to decline, consumption smoothing pushes the current account into surplus. The term $\frac{\beta \left(1+\overline{r}\right)-1}{1+\overline{r}}{W}_t$ captures the effect of consumption tilting. Relative patience of home households implies an upward tilt in individual household consumption profiles since $\beta \left(1+\overline{r}\right)>1$. Therefore, ceteris paribus, consumption tilting contributes to home current account surplus. Conversely, the downward tilt of foreign household consumption profiles implied by $\alpha \beta \left(1+\overline{r}\right)<1$ pushes the foreign current account in the direction of deficit.

At time 0, when the shock happens, home wealth W unambiguously increases, because the real wage w increases in all periods. Given $\beta \left(1+\overline{r}\right)>1$, this tends to increase the home current account through the consumption tilting channel. However, the path of the home real wage in Figure 1 is increasing over time. Therefore, the consumption smoothing channel in (17) would dictate that the home current account should worsen. What we observe is an improvement in home’s net foreign asset position, i.e., an increase in home’s current account above the steady state. Based on equation (17), this is driven by the fact that the consumption tilting channel prevails on the pure smoothing one. The steady-state incentive of home households to postpone consumption implicit in the upward tilt of individual home consumption profiles results in home households lending more than in the initial steady state. Conversely, the relative impatience of foreign consumers induces them to anticipate consumption and borrow more against their permanently higher human wealth.

Consistent with different incentives to postpone or anticipate consuming, consumption increases in both countries at time 0, but it does so by less than the full amount of the productivity shock at home and by more in the foreign economy. Consumption then increases over time in the home economy and decreases abroad, as foreign households must pay interest on an increasing debt. In the long run, the consumption increase in both the home and foreign countries reflects the full amount of the world shock.

Different consumption responses to the shock generated by relative patience versus impatience and their general equilibrium consequences for labor effort and real wages motivate different short-run relative price movements across countries. The relative price of home goods falls, because the home real wage does not increase as much as productivity on impact, while the relative price of foreign goods rises, yielding a deterioration of home’s terms of trade. Relative prices return to the initial steady state over time. There is no long-run effect of the permanent change in asset positions on relative prices because the permanent, worldwide productivity shock eventually results in GDP and consumption increases of the same size as the shock both at home and abroad.

Equilibrium labor effort increases at home (decreases abroad), reflecting the expansionary (contractionary) effect of a lower (higher) relative price on labor demand. After the initial jump, labor effort slowly returns to the original steady state in both countries. The real wage increases on impact at home and abroad. It increases by more than productivity in the foreign economy, which explains the increase in the foreign relative price, and the decrease in equilibrium labor effort in the foreign country. Both the domestic and the foreign real wages converge over time to a higher steady-state level that reflects the full amount of the world productivity shock. Like consumption, the domestic real wage increases over time, while the foreign real wage decreases.

GDP also increases in both countries. As for consumption, in the long run, the increase reflects the full amount of the world shock, since both relative prices and labor effort return to their original levels. In the short run, GDP increases by more in the patient, less productive economy (home) and then decreases toward the new steady-state level. Foreign GDP increases over time. Therefore, changes in labor effort prevail on relative price movements in determining the direction of GDP changes.

The key for the dynamics in Figure 1 is the difference in consumption responses implied by patience versus impatience, the lending and borrowing that this generates, and the adjustment of relative prices and the terms of trade that takes place as a consequence. When households in different countries capitalize wealth effects differently due to heterogeneity in subjective discount factors, long-run consumption differs from long-run labor income in each country, and even symmetric, permanent productivity shocks end up redistributing demand across countries in a way that induces agents to adjust their labor effort over time rather than keeping it unchanged. Consumption tilting then results in accumulation of assets (or debt) during the transition dynamics. In the long run, the foreign economy has a permanently larger debt – and its new long-run consumption and GDP levels remain higher than those at home, as in the initial steady state.

C. Sensitivity Analysis

We consider four alternative parameterizations to verify the robustness of the results in Figure 1. In Scenario 1, we remove the steady-state productivity differential and let $\overline{Z}={\overline{Z}}^{*}=1$; in Scenario 2, we return to $\overline{Z}=1<{\overline{Z}}^{*}=1.29$, but consider a lower value of α, equal to 0.999; in Scenario 3, we return to α = 0.9999 and keep $\overline{Z}=1<{\overline{Z}}^{*}=1.29$, but assume ω = 1.5, in line with the international real business cycle literature (for instance, Backus, Kehoe, and Kydland, 1994); finally, in Scenario 4, we assume α = 0.9999, $\overline{Z}={\overline{Z}}^{*}=1$ and ω = 1.5.

The impulse responses after a 1 percent permanent increase in world productivity for these scenarios are in Figure 2, which reproduces also those for the benchmark parameterization (Scenario B) to facilitate comparison. The qualitative pattern of the responses is the same in all scenarios, and it is driven by the same intuitions. The impulse responses are also very similar on quantitative grounds. Only Scenario 2 deviates significantly from the benchmark, because the lower value of α accentuates the tilts in consumption profiles that are responsible for the dynamics in the figure. Even in Scenario 4, very small heterogeneity in discounting and a value of ω that is not much different from 1 still deliver dynamics that are quantitatively similar to those of the benchmark case. We conclude from this analysis that the consequences of heterogeneous discounting are robust to our assumptions on initial steady-state productivity and substitutability across home and foreign goods.

Permanent World Productivity Shock, Alternative Parameterizations

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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Permanent World Productivity Shock, Alternative Parameterizations

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A. Identification

To identify a permanent worldwide productivity shock, we follow King et al. (1991) and Mellander et al. (1992), and we use a common trend representation for the VAR in levels for the six variables described above, interpreting the innovation to the only common trend in the model as the worldwide productivity shock.²⁸ To proceed in this manner, we specify and test on the data (i) a long-run, cointegration relation between home and foreign productivity, and (ii) a set of four cointegration relations between productivity and the remaining variables in the system.

First, we assume that home and foreign labor productivity levels are cointegrated, and hence share a common stochastic trend, with cointegration vector given by the long-run relation observed in the data over the sample period. Thus, in log-levels, we have:

where ${\mathrm{\gamma }}_1^{\overline{Z}}>0\text{and}{\mathrm{\gamma }}_2^{\overline{Z}}\ge 0$. Such a cointegrating relation is consistent with stochastic growth models in which worldwide technological progress determines long-run growth and with models of technological diffusion (e.g., Nelson and Phelps, 1966, and Barro and Sala-i-Martin, 1995, chapter 8).

Second, we derive the remaining four cointegration relations from a linear approximation to the steady state of our theoretical model. We show in Ghironi, İşcan, and Rebucci (2003) that the model delivers three non linear steady-state relations for domestic consumption relative to foreign $(\overline{c}/{\overline{c}}^{*})$, home net foreign assets $(\overline{B})$, and the risk-free real interest rate $(\overline{r})$ as functions of the steady-state productivity ratio $(\overline{Z}/{\overline{Z}}^{*})$. Since there are no closed form expressions for these functions, we assume that $\overline{r}\text{and}\overline{B}$ are such that:²⁹

Steady-state consumption levels can then be written as:

Finally, combining (19)–(22) with (18) gives:

where the ${\gamma }^{\prime }$ coefficients are functions of the coefficients in equations (18)–(22).

The system (23) contains five linear relations in six variables. If the six variables are I(1) and the five relations are (0), these represent a set of long-run, cointegration relations, and the six variables must share a single common stochastic trend by definition of cointegration. Consistent with a range of long-run growth models, we interpret an innovation to this unique common stochastic trend as a permanent, worldwide productivity shock. With this interpretation in mind, we now turn to testing and estimation of our empirical model.

B. Cointegration Results

We conducted the cointegration analysis in a VAR in (log) levels with two lags as suggested by standard lag-selection criteria. To ensure residual normality, no autocorrelation, and homoscedasticity, we also included a set of seasonal dummy variables in this VAR. To avoid introducing too many dummy variables, we estimated the system on the sub-sample 1980:Q1–1994:Q4. Thus, our sample period stops right before the beginning of the recent period of productivity growth acceleration in the United States.³⁰

Overall, the evidence for the United States and the rest of the G-7 is supportive of our assumptions (i) and (ii) above. Table 2 reports the results of the Johansen cointegration procedure applied to the specified VAR, the five estimated cointegration relations in (23) and a test on the (overidentifying) restrictions implicit in these relations, and mispecification test statistics at the system level.³¹ Table 2, panel A shows that there is only one eigenvalue clearly close to zero in our six-variable VAR, suggesting the presence of five stationary components consistent with our hypotheses (i) and (ii) above. Further, if we impose the hypothesis of five stationary components on this six-variable system, consistent with (23), the implied over-identifying restrictions cannot be rejected by the data (and by a wide margin – see panel B). However, in a bivariate system, the null hypothesis of cointegration between home and foreign productivity levels is rejected by our relatively short time series data (results not reported). Also, in the six-variable system described above, the Johansen test on the cointegration rank of the VAR suggests the rank is three (panel A). But, if home and foreign productivity levels are entered exogenously to form a four-variable VAR system (Table 2, panel C), the Johansen test on the cointegration rank suggests that the system is full rank. We conclude from this body of evidence that cointegration of productivity levels is the one long-run relation that, among the five relations in (23), does not statistically fit our relatively short data series perfectly. Nonetheless, we view this long-run relation as an economically plausible “identifying assumption” in our empirical model.³²

^{Table 2.}

Cointegration Analysis (1980:Q1-1994:Q4)

^{Table 2.}

Cointegration Analysis (1980:Q1-1994:Q4)

Panel A: Johansen procedure (with productivity entered endogenously)
Eigenvalue	loglik.	rank
	1298.406	0
0.51379	1320.040	1
0.42826	1336.812	2
0.34351	1349.438	3
0.26254	1358.574	4
0.11075	1362.095	5
4.2240e-005	1362.096	6
Rank<=	Trace test	[Prob.]
0	127.38	[0.000] **
1	84.114	[0.002] **
2	50.569	[0.026] *
3	25.318	[0.155]
4	7.0449	[0.579]
5	0.0025	[0.960]

View Table

^{Table 2.}

Cointegration Analysis (1980:Q1-1994:Q4)

Panel A: Johansen procedure (with productivity entered endogenously)
Eigenvalue	loglik.	rank
	1298.406	0
0.51379	1320.040	1
0.42826	1336.812	2
0.34351	1349.438	3
0.26254	1358.574	4
0.11075	1362.095	5
4.2240e-005	1362.096	6
Rank<=	Trace test	[Prob.]
0	127.38	[0.000] **
1	84.114	[0.002] **
2	50.569	[0.026] *
3	25.318	[0.155]
4	7.0449	[0.579]
5	0.0025	[0.960]

Further, the view that cointegration of productivity levels is an economically plausible assumption is supported by the economic and statistical significance of the estimated cointegration relations. The coefficients of the estimated cointegration relations have all the right signs and are statistically significant (standard errors in parentheses; see Table 2, panel B).³³ These estimates suggest that productivity was growing faster in the rest of the G-7 than in the United States during our sample period. This is consistent with productivity convergence in the G-7, where the United States is the more productive economy and the remaining G-7 productivity levels are catching up (Baumol, 1986).³⁴ The results also show that the response of net foreign assets to movements in worldwide productivity is economically significant. By contrast, productivity does not affect the real interest rate in the long-run, matching the theoretical model, which predicts that permanent changes in global productivity have no effect on the real interest rate.

We conclude that the cointegration results are broadly consistent with the linear cointegration relations in (23) and allow us to identify global productivity shocks in a model- and data-consistent manner.

C. Impulse Responses

We now turn to the estimated impulse responses of the six variables in our VAR.³⁵ Figure 3 reports the point estimates and two standard error bands for all impulse responses to a one-standard-deviation innovation to the common stochastic productivity trend. The results are strikingly consistent with the main predictions of our theoretical model. Theoretical impulse responses predict that, on impact, the less patient economy reduces its foreign assets (or increases its foreign debt), while the more patient economy accumulates assets, in response to a worldwide, permanent productivity shock. Thus, if we interpret the United States as the relatively impatient economy, the strong, negative, and statistically significant empirical response of U.S. net foreign assets (LBT1S_US in Figure 3) to such a shock is consistent with the impulse responses shown in Figure 1.

VAR Response to a One-Standard Deviation Permanent Global Productivity Shock

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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VAR Response to a One-Standard Deviation Permanent Global Productivity Shock

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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VAR Response to a One-Standard Deviation Permanent Global Productivity Shock

Citation: IMF Working Papers 2005, 082; 10.5089/9781451861013.001.A001

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Consumption responses for the United States and the rest of the G-7 (LC_US and LCEX_US respectively) are also noteworthy for their consistency with the predictions of our theoretical model. On impact, consumption increases in both the United States and the rest of the G-7. Then, U.S. consumption reaches its new steady state from above, whereas foreign consumption reaches the new steady-state from below. These dynamic patterns are consistent with the predictions of our theory, if again we interpret the United States as the less patient economy, and the foreign economy as the more patient one.³⁶

Finally, our theoretical model also predicts that net foreign asset and consumption dynamics are largely driven by consumption tilting and variations in relative prices, with no dynamic response from the risk-free real interest rate (LR1_US1 in Figure 3). Although the point estimates indicate a positive initial interest rate response, the standard errors suggest that this is not statistically different from zero.

Our finding that G-7 net foreign assets respond to permanent, worldwide productivity shocks asymmetrically is consistent with other studies, although they use different empirical models. For instance, Nason and Rogers (2002) find that the Canadian current account tends to respond to worldwide shocks. Gregory and Head (1999) find that a highly persistent worldwide productivity shock has an economically significant and asymmetric impact on individual G-7 current accounts, including a negative and statistically significant impact on the U.S. current account. Based on this evidence they conclude that “asymmetries across countries that appear in the empirical analysis cannot be replicated in a calibrated version of the artificial economy in which countries are asymmetric only with regard to the parameterization of the technology shock process” (p. 427).

In sum, we find that net foreign assets and consumption do respond to permanent worldwide productivity shocks differently across countries, in a manner that is predicted by our theoretical model of consumption and foreign asset dynamics with different discount factors, but inconsistent with standard, symmetric models with equal discount factors.