Money and Exchange Rates in the Grossman-Weiss-Rotemberg Model

Questions & answers

Q1. What puzzle motivates the paper, and what does it test?

The paper asks whether the Grossman-Weiss-Rotemberg model of segmented asset-market participation can reproduce three well-documented features of exchange rates since the breakdown of Bretton Woods: that “movements in nominal exchange rates have been much larger than the inflation, money growth, or interest rate differentials across…countries,” that “at all but very low frequencies, these movements in nominal exchange rates are associated with movements in real exchange rates of roughly the same size and persistence,” and that both series are highly persistent (Section 1, Introduction). The authors document these facts directly in Tables 1-2 using monthly G-7-versus-dollar data from January 1973 to December 1994: the mean annualized standard deviation of real exchange-rate changes across the G-7 is 37%, nominal and real exchange-rate changes have a correlation of 0.98, and real exchange rates decay with “a half-life of roughly three years” (Section 2).

Q2. What is the Grossman-Weiss-Rotemberg mechanism, and why does market segmentation matter?

In this class of model, agents “visit the asset market infrequently to obtain currency to finance extended shopping trips in the goods market,” so that at any point in time only a fraction of agents are present when the government conducts an open-market operation (Section 1). Because positive nominal interest rates make it costly to carry idle cash between asset-market visits (Proposition 1: “bonds dominate currency in the asset market”), any new money the government injects goes disproportionately to the subset of agents who happen to be trading in the asset market that period, giving monetary injections “a disproportionate impact on these agents’ currency holdings and consumption and on nominal and real asset prices as well” (Section 1). The authors adopt Lucas’s (1990) trick of organizing agents into “families” of a parent and many children to keep the model tractable in a genuinely stochastic environment, which earlier work by Grossman-Weiss and Rotemberg could not do – their distributional effects “complicate the analysis and effectively limit these studies to one-time unanticipated shocks in deterministic settings.”

Q3. How is the model structured, and what pins down who receives a monetary injection?

Shopping trips are stochastic in length: each period, a child either continues shopping with probability γ or returns home to the asset market with probability (1 − γ), and all currency a returning child has left unspent is destroyed (Section 3). Because parents hold no cash across periods when interest rates are positive (Proposition 1), the entire money stock is held by children, and under log utility a constant child spends a fixed fraction (1 − βγ) of currency each period (Proposition 2), so per-period velocity is constant at (1 − βγ) – when γ = 0 (one-period shopping trips) the model collapses exactly to the standard constant-velocity cash-in-advance model (Section 4). Proposition 3 shows that monetary injections are divided among the currently-returning children in proportion to a term z(s^t) that depends on current and past preference/wealth shares, and the elasticity of this share with respect to money growth, ∂z/∂μ · μ/z = βγμ/(1 − βγμ²), is increasing in γ – the longer average shopping trips are, the fewer agents are in the asset market at any moment, and the larger the impact of a given injection on their currency and consumption.

Q4. What does Proposition 4 say about nominal and real exchange rates, and what two channels does monetary policy operate through?

Proposition 4 derives closed-form expressions for nominal interest rates, exchange rates, real interest rates, and the real exchange rate, Z(s^t) = z(s^t)y(s^t)/[z(s^t)y(s^t)]**, as functions of the shares z(s^t), z*(s^t) of domestic and foreign output consumed by agents currently in the asset market (Eqs. 4.9-4.14). The authors state the mechanism directly: “monetary policy has two effects on exchange rates in this model. First, in this model, as in a quantity-theory model, an increase in the domestic money stock…results in…a depreciation of the domestic currency…. Second, in this model, unlike a quantity-theory model, an increase in the domestic money stock also affects the real exchange rate” because agents currently in the asset market receive a disproportionate share of the new money, raising their relative consumption of the domestic good and lowering its relative price (Section 4). Because the real exchange rate depends only on the current relative shares z and z*, “transitory movements in the money growth differential across countries will induce transitory movements in the real exchange rate while persistent movements…will induce persistent movements,” and nominal and real exchange-rate changes are correlated because both load on the same money-growth differential.

Q5. How does the model generate a real effect of money on interest rates, and how does this interact with the persistence of money growth?

Because a monetary injection redistributes consumption between agents currently in the asset market and those who will be there next period, changes in money growth from t to t+1 move the real interest rate away from its Fisherian value, with the real-rate elasticity to money growth given by (ρ − 1)βγz/(1 − βγ²ρ) at steady-state z (Section 4, following Eq. 5.5). The magnitude depends on both γ (how much of output the asset-market share can be moved) and ρ, the persistence of money growth: “If increases in money growth…are very persistent…then these increases…have only a small negative effect on the real interest rate…On the other hand, if increases in money growth…are transitory…then they have a large negative impact on real interest rates.” This is the mechanism the numerical calibration exploits: choosing money growth to be highly persistent keeps nominal interest rates close to their Fisherian (inflation-tracking) values even while real and nominal exchange rates move a great deal.

Q6. What does the calibrated numerical example show, and what parameter values are required to match the data?

The authors simulate the model for 10,000 periods, choosing six parameters (γ=.989, β=.997, σ_ε=.03, ρ=.98, σ_η=.03, ρ_{ε,η}=.50) to match six target moments from the G-7 data: the standard deviation of real exchange-rate changes (37% annualized), the half-life of real exchange rates (36 months), mean inflation (6.5%), mean interest rates (9.5%), and the standard deviations of inflation and of inflation differentials (6% each) (Section 5). The calibrated model reproduces nominal exchange rates “that are highly volatile relative to inflation, money growth, or interest rate differentials,” real exchange-rate movements that are “highly correlated with movements in nominal exchange rates and also very volatile,” and high persistence in both (Section 6, Conclusion). Two parameter choices are doing the work: a high persistence of money growth (ρ close to 1, so “nominal interest rates in this model are close to their Fisherian fundamentals” and correlate near 1 with inflation) and a very small average asset-market share z, “on the order of 0.03” – the share of currency and consumption going to agents currently in the asset market must be small because “it is only when this average share is small that small changes in money growth rates have large effects on the portfolios of children in the asset market,” and given observed average inflation this requires the fraction of agents in the asset market, (1−γ), to itself be small.

Q7. How does this model’s performance compare with related liquidity-effect models, and what does the comparison imply about the market-segmentation channel specifically?

The authors contrast their results directly with the open-economy extensions of the Lucas (1990) and Fuerst (1992) liquidity-effect models studied by Grilli and Roubini (1992) and by Schlagenhauf and Wrase (1995): “Grilli and Roubini (1992) show analytically that the impact of a monetary injection on the real exchange rate and the short-term interest rate is the same. Thus, monetary shocks in this alternative model cannot produce exchange rates that are substantially more volatile than interest rate differentials” (Section 4). Because the Grossman-Weiss-Rotemberg structure lets the share of agents absorbing a shock be a separate free parameter from the shock’s persistence, it can decouple real-exchange-rate volatility from interest-rate volatility in a way the simpler Lucas/Fuerst segmented-markets models cannot – which is the paper’s answer to why this particular friction, rather than other liquidity-effect models, is worth exploring for exchange rates.

Q8. What does the paper NOT establish, and what limitations do the authors flag themselves?

The authors are explicit that the exercise is exploratory rather than a test of an independently disciplined model: “This paper is an exploration of the range of outcomes that this model can produce as this parameter is varied,” explicitly likening their exercise to Lucas’s (1990) own closing observation that monetary theory needs “formulations that place a smaller burden on the idea of a fixed period” without yet explaining the underlying trading frictions (Section 1, quoting Lucas 1990). They flag directly that the key friction has not been validated against microeconomic data: “Whether or not the frictions required in this model to match exchange rate data are consistent with micro data on trading patterns is a question we leave for future research” (Section 1). The required calibration – roughly 3% of agents transacting in the asset market in any given period – is also asserted as a quantitative requirement of matching the model to aggregate moments, not independently measured.