Exchange Rates Dynamics with Long-Run Risk and Recursive Preferences

Abstract

Standard macro models fail to explain why real exchange rates are volatile and disconnected from macro aggregates. Recent research argues that models with persistent growth rate shocks and recursive preferences can solve that puzzle. I show that this result is highly sensitive to the structure of financial markets. When just a bond is traded internationally, then long-run risk generates insufficient exchange rate volatility. A long-run risk model with recursive preferences may generate realistic exchange rate volatility, if all agents efficiently share their consumption risk by trading in complete financial markets; however, this entails massive international wealth transfers, and excessive swings in net foreign asset positions. By contrast, a long-run risk, recursive-preferences model in which only a fraction of households trades in complete markets, while the remaining households lead hand-to-mouth lives, can generate realistic exchange rate and external balance volatility.

Appendices

Appendix 1: The reformulated model

The numerical solution uses a reformulated model in which consumption and welfare are scaled by domestic output. Let $\tilde{C_{i,t}}\equiv {C}_{i,t}/{Y}_{i,t}$ and $\tilde{U_{i,t}}\equiv {U}_{i,t}/{Y}_{i,t}$ be scaled consumption and welfare in country i, and let G ^Y _i,t  ≡ (Y _i,t /Y _{i,t − 1})/exp(μ) be the growth factor of country i output between periods t-1 and t, divided by the steady state growth factor. (4) implies

$$ \tilde{U_{i,t}}={\left\{\left(1-\beta \right){\left(\tilde{C_{i,t}}\right)}^{1-\sigma }+\tilde{\beta}{\left[{E}_t{\left(\tilde{U_{i,t+1}}{G}_{i,t+1}^Y\right)}^{1-\gamma}\right]}^{\left(1-\sigma \right)/\left(1-\gamma \right)}\right\}}^{1/\left(1-\sigma \right)}, $$

(A.1)

with $\tilde{\beta}\equiv \beta \kern0.1em \exp \left(\mu \cdot \left(1-\sigma \right)\right).$ Country i’s intertemporal marginal rate of substitution (IMRS) between periods t and t + 1 (see (5)) can be written as:

$$ {\rho}_{i,t+1}\equiv \beta \kern0.1em \exp \left(\mu -\sigma \right){\left({G}_{i,t+1}^Y\right)}^{-\gamma }{\left(\frac{\tilde{C_{i,t+1}}}{\tilde{C_{i,t}}}\right)}^{-\sigma }{\left(\frac{\tilde{U_{i,t+1}}}{{\left\{{E}_t{\left(\tilde{U_{i,t+1}}{G}_{i,t+1}^Y\right)}^{1-\gamma}\right\}}^{1/\left(1-\gamma \right)}}\right)}^{\sigma -\gamma }. $$

(A.2)

Country i’s demand functions for the two output goods are

$$ \tilde{y_{i,t}^i}=\alpha {\left({p}_{i,t}/{P}_{i,t}\right)}^{-\phi}\tilde{C_{i,t}},\kern0.5em \tilde{y_{i,t}^j}=\left(1-\alpha \right){\left({p}_{j,t}/{P}_{i,t}\right)}^{-\phi}\tilde{C_{i,t}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em j\ne i, $$

(A.3)

where $\tilde{y_{i,t}^j}\equiv {y}_{i,t}^j/{Y}_{i,t}$ is country i’s demand for good j, normalized by i’s output.

The market clearing conditions for goods H and F can be expressed as

$$ 1=\tilde{y_{H,t}^H}+\tilde{y_{F,t}^H}/{y}_t\kern0.5em \mathrm{and}\kern0.5em 1=\tilde{y_{H,t}^F}\cdot {y}_t+\tilde{y_{F,t}^F}, $$

(A.4)

where y _t  ≡ Y _H,t /Y _F,t is relative country H output. Home net exports/GDP are given by:

$$ \tilde{N{X}_{H,t}}=1-\left({P}_{H,t}/{p}_{h,t}\right)\tilde{C_{H,t}}. $$

(A.5)

Without loss of generality, I set

$$ \frac{1}{2}\left({p}_{H,t}+{p}_{F,t}\right)=1, $$

(A.6)

i.e. a basket consisting of half a unit of good H and of good F is used as numéraire.

The dynamics of output growth and of relative output depend on the assumed exogenous output process. Under the baseline output process (12) we have

$$ \begin{array}{c}\hfill \ln \left({G}_{H,t}^Y\right)={z}_{H,t-1}-\kappa \cdot \ln \left({y}_{t-1}\right)+{\varepsilon}_{H,t}^Y,\kern0.5em \ln \left({G}_{F,t}^Y\right)={z}_{F,t-1}+\kappa \cdot \ln \left({y}_{t-1}\right)+{\varepsilon}_{F,t}^Y,\hfill \\ {}\hfill \mathrm{where}\kern0.5em {z}_{i,t}={\rho}^z{z}_{i,t-1}+{\varepsilon}_{i,t}^z\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em i=H,F\kern0.5em \mathrm{and}\kern0.6em \ln \left({y}_t\right)=\left(1-2\kappa \right)\kern0.1em \ln \left({y}_{t-1}\right)+{z}_{H,t-1}-{z}_{F,t-1}+{\varepsilon}_{H,t}^Y-{\varepsilon}_{F,t}^Y.\hfill \end{array} $$

(A.7)

When the first-difference stationary output process (13) is assumed, then

$$ \begin{array}{c}\hfill \ln \left({G}_{H,t}^Y\right)={\rho}^{\varDelta Y} \ln \left({G}_{H,t-1}^Y\right)-\kappa \cdot \ln \left({y}_{t-1}\right)+{\varepsilon}_{H,t}^{\varDelta Y},\kern0.5em \ln \left({G}_{F,t}^Y\right)={\rho}^{\varDelta Y} \ln \left({G}_{F,t-1}^Y\right)+\kappa \cdot \ln \left({y}_{t-1}\right)+{\varepsilon}_{F,t}^{\varDelta Y},\hfill \\ {}\hfill \varDelta \kern0.1em \ln \left({y}_t\right)={\rho}^{\varDelta Y}\varDelta \kern0.1em \ln \left({y}_{t-1}\right)-2\kappa \kern0.1em \ln \left({y}_{t-1}\right)+{\varepsilon}_{H,t}^{\varDelta Y}-{\varepsilon}_{F,t}^{\varDelta Y}.\hfill \end{array} $$

(A.8)

Finally, under the trend-stationary endowment process (14) we have

$$ \ln \left({G}_{i,t}^Y\right)={\xi}_{i,t}-{\xi}_{i,t-1}\kern0.5em \mathrm{and}\kern0.5em \ln \left({y}_t\right)={\xi}_{H,t}-{\xi}_{F,t},\kern0.5em \mathrm{where}\kern0.5em {\xi}_{i,t}\equiv \ln \left({Y}_{i,t}\right)-\mu \cdot t\kern0.5em \mathrm{obeys}\kern0.5em {\xi}_{i,t}={\rho}^Y{\xi}_{i,t-1}+{\varepsilon}_{i,t}^{TS}. $$

(A.9)

• I. In model variants with efficient risk sharing, the net foreign assets/GDP ratio obeys

$$ \tilde{NF{A}_{H,t+2}}={E}_t{\rho}_{H,t+1}\left\{\left[\tilde{NF{A}_{H,t+2}}-1+\left({P}_{H,t+1}/{p}_{H,t+1}\right)\tilde{C_{H,t+1}}\right]{\pi}_{H,t+1}\right\},\mathrm{with}{\pi}_{H,t+1}\equiv \left({P}_t/{P}_{t+1}\right)\left({p}_{H,t+1}/{p}_{H,t}\right)\kern0.1em {G}_{H,t+1}^Y \exp \left(\mu \right). $$

(A.10)

Equations (1),(2),(6), (A.1)–(A.6) and the exogenous output process ((A.7), (A.8) or (A.9)) determine $\left\{\tilde{U_{H,t}},\kern0.1em \tilde{U_{F,t}},\kern0.1em \tilde{C_{H,t}},\kern0.1em \tilde{C_{F,t}},\kern0.1em \tilde{y_{H,t}^H},\kern0.1em \tilde{y_{F,t}^H},\kern0.1em \tilde{y_{H,t}^F},\kern0.1em \tilde{y_{F,t}^F},{G}_{H,t}^Y,\kern0.1em {G}_{F,t}^Y,\kern0.1em {y}_t,{p}_{H,t},{p}_{F,t},{P}_{H,t},{P}_{F,t},{q}_t,RE{R}_t,\kern0.1em {\tilde{NX}}_{H,t},\kern0.1em {\tilde{NFA}}_{H,t+1}\right\},$ in the model variants with efficient risk sharing. Once these variables have been solved for, it is easy to determine other variables of interest. E.g., the growth rate of consumption is $\varDelta \ln \kern0.1em {C}_{i,t+1}=\varDelta \ln \kern0.1em \tilde{C_{i,t+1}}+\varDelta \ln \kern0.1em {Y}_{i,t+1}$ etc.

• II. In the bonds-only model variant, the country H budget constraint can be written as

$$ {\tilde{NFA}}_{H,t+1}+\left({P}_{H,t}/{p}_{H,t}\right)\tilde{C_{H,t}}=1+\tilde{NF{A}_{H,t}}\left(1+{r}_t^A\right)/\left\{\left({p}_{H,t}/{p}_{H,t-1}\right){G}_{H,t}^Y \exp \left(\mu \right)\right\}, $$

(A.11)

given the choice of numéraire (A.6). The Euler Eq. (15) then gives

$$ \left(1+{r}_{t+1}^A\right){E}_t\left({P}_{i,t}/{P}_{i,t+1}\right){\rho}_{i,t+1}=1. $$

(A.12)

Equations (1),(2), (A.1)–(A.6), (A.11), (A.12) and the law of motion of output determine

$\left\{\tilde{U_{H,t}},\kern0.1em \tilde{U_{F,t}},\kern0.1em \tilde{C_{H,t}},\kern0.1em \tilde{C_{F,t}},\kern0.1em \tilde{y_{H,t}^H},\kern0.1em \tilde{y_{F,t}^H},\kern0.1em \tilde{y_{H,t}^F},\kern0.1em \tilde{y_{F,t}^F},{G}_{H,t}^Y,\kern0.1em {G}_{F,t}^Y,\kern0.1em {y}_t,\kern0.1em {p}_{H,t},\kern0.1em {p}_{F,t},\kern0.1em {P}_{H,t},\kern0.1em {P}_{F,t},{q}_t,\kern0.1em RE{R}_t,\tilde{N{X}_{H,t}},\kern0.1em \tilde{NF{A}_{H,t+1}},\kern0.1em {r}_{t+1}^A\right\},$ in the bonds-only economy.

III. In the model variants with hand-to-mouth (HTM) households, the scaled consumption of the country i HTM household is given by

$$ \tilde{C_{i,t}^{HTM}}\equiv {C}_{i,t}^{HTM}/{Y}_{i,t}={\lambda}_{i,t}{p}_{i,t}/{P}_{i,t}. $$

(A.13)

Note that

$$ \tilde{C_{i,t}}=\tilde{C_{i,t}^{HTM}}+\tilde{C_{i,t}^{RS}},\mathrm{where}\kern0.6em \tilde{C_{i,t}^{RS}}\equiv {C}_{i,t}^{RS}/{Y}_{i,t} $$

(A.14)

is the scaled consumption of the country’s ‘risk-sharer’ household. The scaled welfare of the ‘risk-sharer’ household $\tilde{U_{i,t}^{RS}}\equiv {U}_{i,t}^{RS}/{Y}_{i,t}$ obeys

$$ \tilde{U_{i,t}^{RS}}={\left\{\left(1-\beta \right){\left(\tilde{C_{i,t}^{RS}}\right)}^{1-\sigma }+\tilde{\beta}{\left[{E}_t{\left(\tilde{U_{i,t+1}^{RS}}{G}_{i,t+1}^Y\right)}^{1-\gamma}\right]}^{\left(1-\sigma \right)/\left(1-\gamma \right)}\right\}}^{1/\left(1-\sigma \right)}, $$

(A.15)

and her IMRS is:

$$ {\rho}_{i,t+1}^{RS}\equiv \beta \exp \left(-\mu \sigma \right){\left({G}_{i,t+1}^Y\right)}^{-\gamma }{\left(\frac{\tilde{C_{i,t+1}^{RS}}}{\tilde{C_{i,t}^{RS}}}\right)}^{-\sigma }{\left(\frac{\tilde{U_{i,t+1}^{RS}}}{{\left\{{E}_t{\left(\tilde{U_{i,t+1}^{RS}}{G}_{i,t+1}^Y\right)}^{1-\gamma}\right\}}^{1/\left(1-\gamma \right)}}\right)}^{\sigma -\gamma }. $$

(A.16)

Efficient risk sharing among the Home and Foreign ‘RS’ households implies:

$$ RE{R}_{t+1}/RE{R}_t={\rho}_{H,t+1}^{RS}/{\rho}_{F,t+1}^{RS}, $$

(A.17)

Equations (1), (2), (A.3)–(A.6), (A.13)–(A.17), and the law of motion of output determine $\left\{\tilde{U_{H,t}^{RS}},\kern0.1em \tilde{U_{F,t}^{RS}},\kern0.1em \tilde{C_{H,t}^{RS}},\kern0.1em \tilde{C_{F,t}^{RS}},\kern0.1em \tilde{C_{H,t}^{HTM}},\kern0.1em \tilde{C_{F,t}^{HTM}},\kern0.1em \tilde{C_{H,t}},\kern0.1em \tilde{C_{F,t}},\tilde{y_{H,t}^H},\kern0.1em \tilde{y_{F,t}^H},\kern0.1em \tilde{y_{H,t}^F},\kern0.1em \tilde{y_{F,t}^F},\kern0.1em {G}_{H,t}^Y,\kern0.1em {G}_{F,t}^Y,\kern0.1em {y}_t,\kern0.1em {p}_{H,t},\kern0.1em {p}_{F,t},\kern0.1em {P}_{H,t},\kern0.1em {P}_{F,t},\kern0.1em {q}_t,\kern0.1em RE{R}_t,\tilde{N{X}_{H,t}},\kern0.1em {\tilde{NFA}}_{H,t+1}\right\},$in the HTM model variants.

Appendix 2: Net foreign assets/GDP ratios of US and UK

The Table below provides annual data on net foreign assets/GDP ratios for the US and the UK (NFA measured at end of year).

Col. 1: year; Cols. 2 and 3: US and UK NFA/GDP ratios as reported in the updated and extend version of the Lane and Milesi-Ferretti (2007) dataset [LMF] (http://www.philiplane.org/EWN.html).

Col. 4: US NFA/GDP series computed by dividing the Bureau of Economic Analysis [BEA] series ‘U.S. net international investment position’ (IIP Table 1.1, in current dollars) by the BEA GDP series (current dollars, NIPA Table 1.1.5).

Year	US (LMF)	UK (LMF)	US (BEA)
1970	0.085	0.048
1971	0.071	0.072
1972	0.067	0.073
1973	0.075	0.047
1974	0.076	0.005
1975	0.077	0.002
1976	0.069	0.014	0.042895
1977	0.061	0.024	0.047260
1978	0.062	0.049	0.054431
1979	0.070	0.023	0.088238
1980	0.078	0.059	0.103707
1981	0.080	0.105	0.070692
1982	0.065	0.125	0.071260
1983	0.057	0.157	0.071877
1984	0.022	0.192	0.034682
1985	−0.006	0.205	0.023991
1986	−0.029	0.245	0.023797
1987	−0.042	0.121	0.012241
1988	−0.054	0.095	0.004089
1989	−0.064	0.088	−0.00596
1990	−0.057	−0.031	−0.02501
1991	−0.064	−0.011	−0.03941
1992	−0.079	0.012	−0.06608
1993	−0.058	0.039	−0.01770
1994	−0.056	0.029	−0.01509
1995	−0.072	−0.023	−0.03622
1996	−0.071	−0.082	−0.04053
1997	−0.103	−0.070	−0.09156
1998	−0.106	−0.137	−0.11374
1999	−0.086	−0.204	−0.10375
2000	−0.142	−0.100	−0.14943
2001	−0.189	−0.137	−0.21607
2002	−0.201	−0.123	−0.21963
2003	−0.198	−0.116	−0.19921
2004	−0.200	−0.196	−0.19254
2005	−0.164	−0.191	−0.14189
2006	−0.176	−0.310	−0.13052
2007	−0.144	−0.231	−0.08838
2008	−0.244	−0.059	−0.27145
2009	−0.183	−0.229	−0.18224
2010	−0.181	−0.252	−0.16785
2011	−0.274	−0.174	−0.28709
2012			−0.28325
2013			−0.29788