Durable Goods and the Forward-Looking Theory of Consumption:
Estimates Implied by the Dynamic Effects of Money
William D. Lastrapes
Department of Economics
Terry College of Business
University of Georgia
Athens, Georgia 30602 last@terry.uga.edu
706 542-3569
Todd B. Potts
Department of Economics
Indiana University of Pennsylvania
Indiana, Pennsylvania 15705 potts@iup.edu
May 10, 2005
Abstract: In this paper, we analyze the effects of money on the market for durable goods. Using quarterly US data, we estimate the dynamic responses of the price and quantity of durable goods and housing to money supply shocks, assuming only that money is neutral in the long-run. We then match these responses to those predicted by an equilibrium model of the markets for durable goods; the model relies on the standard theory of intertemporal consumer choice and the trade off between durable and non-durable goods. We find that money has important dynamic effects on the markets for durable goods, and that the theory provides a reasonable framework for interpreting these responses for plausible parameter values.
JEL Classifications : E40, E21, D91
Key words : Intertemporal choice, impulse response functions, housing, user cost

Durable Goods and the Forward-Looking Theory of Consumption:
Estimates Implied by the Dynamic Effects of Money
1. Introduction
This paper examines the link between money and the markets for durable consumer goods.
Because a durable good is an asset, as well as a direct source of utility, the demand for such a good is likely to be more sensitive to interest rates than the demand for non-durable goods. And because fluctuations in market interest rates are largely due to monetary policy, at least in the short-run, we would expect monetary policy to have potentially important effects on the markets for durable goods. Our interest in this paper is quantifying such effects, and interpreting them through a model of durable goods market equilibrium, the main feature of which is the standard theory of forward-looking consumption behavior. Not only will this objective help us better understand the monetary transmission mechanism, it can shed light on the validity and usefulness of this theoretical framework.
Our approach to studying this issue is relatively new, although there are precedents in the literature. We first estimate a standard VAR model of the economy, including variables measuring aggregate prices and expenditures of durable goods (including housing) as well as other macro variables, and use weak identifying restrictions – based on long-run monetary neutrality – to estimate the dynamic effects of money supply shocks on the system. We then choose parameters of the theoretical model to match as closely as possible the dynamic responses of durable goods prices and expenditures as predicted by the theory to those estimated from the weakly-identified VAR. We are thus able to determine whether, for reasonable parameter values, the predictions of the model are consistent with observed reality (at least as measured by the VAR model). From the alternative point of view, our empirical strategy goes beyond the typical VAR innovation accounting exercise to interpret the estimated impulse responses in terms of a specific theoretical model.
1
Our method is based on the work of Campbell (1994) in that we analytically solve for equilibrium values of the model using log-linear approximations of the Euler equations and budget constraints. This allows us to focus on the dynamic effects of shocks to the system, and directly
1 Our approach is different from, but similar in spirit to, other approaches for combining the restrictions of theory with the statistical power of VAR models, such as Ireland (2004).

compare the model’s predictions to the estimates from the linear time series model. Our work also follows Rotemberg and Woodford (1997), Christiano, Eichenbaum and Evans (2005), and Smets and Wouters (2002) in estimating theoretical parameters from the estimated responses to a limited number of shocks, specifically, money supply shocks. As in Smets and Wouters (2002), we also condition the theoretical response functions for the durable goods variables on the estimated responses of driving variables; in our case, these external variables are expenditures on non-durable goods and the real interest rate.
One motivation for our work is to see how far the basic theory of intertemporal choice and dynamic optimization can go in explaining certain aspects of the macro economy. While recent research points out anomalies in consumption behavior (mostly at the individual level) that are difficult to explain in the context of rational intertemporal behavior (see for example Frederick et al. 2002, Thaler 1994, Akerlof 2002, Angeletos eta a. 2001), the basic framework may still be useful for explaining other behavior, especially at the aggregate level (Browning and Crossley 2001). In particular, it may be a reasonable framework for evaluating monetary and fiscal policies that can have aggregate effects. Our analysis should help guide policy evaluation in this regard.
The second section of the paper lays out the theory. Although our analysis is partial equilibrium in that we focus on markets for durable goods, we embed these markets in a simple general equilibrium framework. Section 3 describes the VAR model and reports the estimated dynamic response functions assuming long-run monetary neutrality. We use data on personal consumer expenditures on durable goods as well as aggregate housing in the model.
2 We find that money supply shocks have important effects on these durable goods markets, particularly expenditures. The final section maps the theory to the empirical analysis. The theoretical model does a reasonable, but not perfect, job of explaining the estimated dynamic responses.
2 Bernanke and Gertler (1995) and Christiano, Eichenbaum and Evans (1999) for similar recent evidence on durable good spending. Lastrapes (2002), Baffoe-Bonnie (1998), Harter-Dreiman
(2003) and Iacoviello (2000) use VAR models to examine the housing market. Although our focus is on aggregate markets, monetary transmission is likely to have different effects on disaggregated markets, especially housing; see for example, Evenson (2003) and Fratantoni and Schuh (2003).
– 2 –

2. The theoretical model
We consider an economy with a representative household and firm. Although our focus is on the markets for durable goods, we lay out the general equilibrium structure to define concepts and clarify how the overall economy fits together.
We assume that the household maximizes expected intertemporal utility by choosing optimal time paths for consumption of non-durable goods ( c t
), the stock of (non-housing) durable goods
( d t
), the stock of housing ( h t
) and real money balances ( µ t
). We follow convention (dating back to
Wicksell) in assuming that the service flow of durable goods is proportional to the stocks currently held. Real money provides utility directly to the household, presumably in the form of transactions services. The household’s objective at time 0 is
V
0
= E
0
∞
X
β t
σ
σ − 1
ν ( · )
σ − 1
σ t =0
ν ( · ) = [(1 − γ ) c
ρ t
+ γ
1 d
ρ t
+ γ
2 h
ρ t
+ γ
3
µ t
ρ
]
1
ρ
(1) where γ = γ
1
+ γ
2
+ γ
3
, σ is the intertemporal elasticity of substitution of total expenditures, and
=
1
1 − ρ is the intratemporal elasticity of substitution among the three consumption goods and real money balances. The objective function is intertemporally additively separable, but the CES form of ν ( · ) allows for intratemporal non-separability between non-durable and durable goods.
3 As approaches 1( ρ → 0), the period utility function takes on the Cobb-Douglas form, given that the sum of the linear weights in ν ( · ) is one. If σ approaches 1 as well, the utility function becomes log-linear.
The household’s intertemporal budget constraint is a t +1
= (1 + r t
)[ a t
+ y t
− c t
− p dt c dt
− p ht c ht
− τ t
− µ t
+ µ t − 1
(1 + π t
)
− 1
] , t = 1 , . . . , ∞ , (2) where a is real financial wealth, r is the real, after-tax yield on these assets, y is the household’s real income from its ownership of the factors of production (which we assume are supplied inelastically), c d and c h are flow real expenditures on durable goods and housing respectively, p dt and p ht are the
3 An objective function of similar form is used by Ogaki and Reinhart (1998), who stress the importance of such non-separability in estimating intertemporal elasticity of substitution.
– 3 –

real prices of durable goods and housing, τ measures lump sum taxes collected by the government sector, and π t +1
= p t +1 p t
− 1 is the per-period rate of inflation in terms of the nominal price of non-durable goods, p . All real magnitudes are measured in terms of non-durable goods. In terms of the stocks of durable goods and housing, real desired expenditures are c dt
= [ d t
− (1 − δ d
) d t − 1
] c ht
= [ h t
− (1 − δ h
) h t − 1
] .
(3)
(4)
The constant rates of depreciation of the stocks of durable goods and housing are represented by
δ d and δ h
, respectively. For our purposes, the important distinction between non-durable goods on the one hand and durables and housing on the other is that both δ d and δ h are presumably less than 1. In this case, durables and housing are not only a source of utility, but a source of wealth as well.
This model is a straightforward extension of the forward-looking theory of aggregate consumption that allows for variation in the durability of goods.
4
The simple budget constraint ignores many of the complexities that might affect the household’s accumulation of assets. For example, we ignore many of the institutional features of the housing market and housing finance, such as the role of land, maintenance costs, the tax treatment of mortgage interest, the long-term nature of mortgage debt, and the possibility of different financial instruments, that may be important.
5
However, our reliance on this basic structure is sufficient for our focus on the role of durable goods and housing as assets.
The household maximizes (1) subject to (2), non-negativity constraints on non-durable consumption and the stocks of durable goods, housing, and real money, and a limit on borrowing by the household. The Euler equations from this constrained optimization problem, along with
4
See for example Obstfeld and Rogoff (1996, pp. 96-99) for durable goods, and Miles (1994) for housing.
5
Miles (1994) and Poterba (1984) account for some of these features. By ignoring the term structure of debt in this model, the household in effect is forced to ‘roll-over’ assets each period so that the short-term interest rate is the only yield that matters. While it may be of interest to allow for long-term debt in this model, Poterba (1984, p. 736) argues for the relevance of the short-term rate in making rational decisions on the purchases of durable goods. The short-term rate is especially relevant in models of aggregate expenditures, since the appropriate margin is on deciding when to make housing expenditures – this period or next.
– 4 –

the constraint in (2) and the transversality (complementary slackness) condition, form a system of non-linear difference equations that determine the household’s optimal decision rules for c t
, d t
, h t
,
µ t and a t +1
.
In this paper, we focus on the following Euler equations, which reflect the marginal trade-off between consumption of non-durable goods with durable goods and housing, respectively:
ρ − 1
γ
1
1 − γ d t c t
= p dt
1 −
1 − δ d
1 + r t
E t p dt +1 p dt
ρ − 1
γ
2
1 − γ h t c t
= p ht
1 −
1 − δ h
1 + r t
E t p ht +1 p ht
.
(5)
(6)
The left-hand-side of each equation is the marginal rate of substitution of non-durable consumption for the stock of each durable good. The right-hand-side is the user cost of durable goods: the marginal opportunity cost in non-durable goods of purchasing a unit of, say, d at real price p dt
, incurring the physical depreciation of this unit ( δ d
), and selling the remaining quantity at the anticipated price E t p dt +1 next period, with the proceeds discounted to the current period at the real interest rate r t
. The form of the user cost distinguishes durable goods – as assets – from non-durable consumption: if δ = 1, user cost reduces to the relative price of the durable in terms of the non-durable.
We suppose that the representative firm maximizes its value by producing all three goods – nondurable, durable and housing – using inputs supplied inelastically by the household. We ignore the firm’s investment in capital goods. Let y ct denote the flow supply of the non-durable good, and y dt and y ht be the flow supply, or gross investment, in durables and housing [i.e.
y dt
= d s t
− (1 − δ d
) d t − 1
, with a similar expression holding for housing]. Then the household’s real income is defined to be y t
= y ct
+ p dt y dt
+ p ht y ht
. Instead of deriving the firm’s flow supply of these goods explicitly from the firm’s objective, we simply posit the following supply curves for durable goods and housing (as in, for example, Poterba 1984 or Miles 1996): y dt
= p
φ d dt y ht
= p
φ h ht
.
(7)
(8)
We rely on this simplification given our focus on durable goods and housing as assets from the household’s perspective. These supply curves assume that the firm faces increasing costs of production in the long run; for example, if land is in fixed supply, increasing production of housing
– 5 –

would require an increase in the relative price of housing. Although explicit costs of adjusting durable good and housing stocks are not built into the household’s problem (as in, for example,
Eberly 1994, Bernanke 1985 and Lam 1991), the presence of these upward-sloping supply curves in effect imposes external adjustment costs on the household.
6
The government sector in this model is trivial. It exogenously issues fiat money to finance net transfers to households: τ t
= − (
M t
− M t − 1 p t
) , where M is the nominal stock of money.
The economy comprises markets in financial assets, money, non-durable goods, durable goods, and housing. In equilibrium, each of these markets clears, which determines the four prices: r , p , p d and p h
(because of Walras’ Law, one of market clearing conditions is redundant). Because neither firms nor the government issues financial assets, the household’s financial wealth must be zero in equilibrium. The general equilibrium conditions, are, then, a t +1
= 0
M t p t
= µ t y ct
= c t y dt
= c dt y ht
= c ht
(9)
(10)
(11)
(12)
(13)
Our analysis is partial equilibrium in that we focus only on equilibrium conditions (12) and
(13). In particular, we set the stock demand for durable goods in (5) equal to the implied stock supply in (7), do the same for housing in (6) and (8), and solve for the equilibrium paths of real price and output for durable goods and housing, conditional on the path for non-durable goods, c t
, and the real interest rate, r t
. By conditioning the dynamics of the durables and housing markets
6 See Romer 2001, p. 380 for a discussion of internal and external adjustment costs. Interesting extensions to this supply behavior would be to a) allow gross investment to respond directly to interest rates as a cost of financing working capital; b) impose internal adjustment costs on the firm’s production of durables and housing, as in Topel and Rosen (1988), which could distinguish between long-run and short-run supply elasticities; and c) relax the implicit assumption that durable goods and housing are separable in the firm’s cost function, so that these goods could be complements or substitutes in production. Most of these extensions will seriously complicate the equilibrium solution below, so are left for future research.
– 6 –

on the equilibrium response of c t and r t to exogenous money supply shocks, we focus attention on the role of user cost in determining optimal behavior.
7
In our empirical analysis, we estimate the dynamic effects of nominal money shocks on the variables in the model above. The theoretical model, however, does not specify the ways in which money may be non-neutral in this economy. For example, we might suppose that nominal wages were rigid in the short-run and that firms determine the level of employment. Then the household’s inelastically supplied endowment of labor becomes irrelevant for equilibrium, and the firm would have incentives to adjust its production of all goods in response to a money supply shock. Our analysis takes the possibility of short-run monetary neutrality as given, without taking a stand on the form of rigidity (whether it be wage rigidity, price rigidity, limited participation, etc.).
Indeed, we see this as an advantage that allows us to concentrate on the intertemporal behavior of households in the markets for durable goods and housing.
To obtain closed form solutions and to make the theory’s implications comparable to the empirical estimates, we rely on the common method of using log-linear approximations to the Euler equations and supply curves (Campbell 1994). To this end, we rely on the following approximation: z a t z a e a ˆ t
≈ ˜ a
(1 + a ˆ t
) ,
(14) where ˜ is the constant, non-stochastic steady-state value of z , and ˆ t
= ln ( z t
) − ln (˜ ) is the log deviation of z t from this steady-state. Using this result to approximate the Euler equation (5), we obtain p dt
= u d
( ρ − 1)( ˆ t
− ˆ ct
) + (1 − u d
)[ E t
(ˆ dt +1
) − ˆ t
] (15) where u d
= (
˜ + δ d p d and ˜ are the durables price and stock such that p dt +1
= p dt and d t
= d t − 1
,
7
A complete general equilibrium analysis would entail solving (9) through (13) for all equilibrium quantities and prices in terms of all exogenous shocks. As long as we can identify truly exogenous shocks – here we make this claim for monetary shocks – we can think of our approach as substituting the equilibrium (reduced form) responses of c and r to such shocks into the equilibrium conditions for the durable goods and housing markets. One implication of the conditional approach is that σ , the intertemporal elasticity of substitution, plays no role since it cancels out in the Euler equations
(5) and (6).
– 7 –

and ˜ is, likewise, the steady-state real interest rate.
8
It is clear from (5) that u d is the steady-state user cost of durable goods, per unit of the real price. If d were not durable, so that u d
= 1, then the forward-looking component in (15) (the second term on the right-hand-side) would play no role in the demand for d . The log-linear approximation of the supply of durable goods in (7), in terms of the stock d , is d t
= (1 − δ d
) ˆ t − 1
+ δ d
φ d p dt
.
(16)
Equations (15) and (16) comprise a linear expectational difference equation system (in log deviations from steady-state) that describes the equilibrium path of the market for durable goods.
d t and substitute into
(15) to reduce the system to a single (second order) difference equation in price.
After some straightforward manipulation, this yields
ˆ dt
= a
0 d
E t p dt +1
+ a
1 d p dt − 1
+ a
2 d
E t − 1 p dt
+ x dt
(17) where a
0 d
=
1 − u d
1 − u d
( ρ − 1) δ d
φ d
1 − δ d a
1 d
=
1 − u d
( ρ − 1) δ d
φ d a
2 d
(1 − u
= −
1 − u d d
)(1 − δ
( ρ − 1) δ d d
)
φ d
(18)
(19)
(20) and x dt
= u d
(1 − ρ )[ y ct
− (1 − δ d
) y ct − 1
] − (1 − u d
)[ r t
1 − u d
( ρ − 1) δ d
φ d
− (1 − δ d
) r t − 1
]
.
(21)
Equation (17) is identical to (A1’) in Blanchard and Fischer (1989, p. 262), who show that its solution is governed by the roots of a
0 d
λ
2 d
+ ( a
2 d
− 1) λ d
+ a
1 d
= 0 .
(22)
8 p d and ˜ , implies
˜
= [ δ d
− 1
( u d
γ
1
)
φ d (1 − γ )
− φ d
(1 − ρ ) φ d c
]
1
1+(1 − ρ ) φd
˜ d
= [ u d
γ
1
/ (1 − γ )]˜
1 − ρ c d
( ρ − 1)
.
Thus, the model implies a unique steady-state. Note that in the steady-state gross investment is not zero: y dt
= δ d
˜
.
– 8 –

Under the reasonable conditions that 0 < δ d
< 1, 0 < r < 1, φ d
> 0, and ρ < 1, a
0 d and a
1 d will lie between zero and one, and a
2 d will be negative, which implies that both roots are real and positive.
Furthermore, one root is stable and the other unstable, so the system exhibits saddlepath stability.
Solving the unstable root forward, the stable solution to (17) is p dt
= λ d 1 p dt − 1
+
1
1 − a
0 d
λ d 1
∞
X
( λ
− i
2 d
E t x t + i
) + i =0 a
2 d a
0 d
∞
X
( λ
− i
2 d
E t − 1 x t + i − 1
) , i =1
(23) where λ
1 is the stable root, and which shows how the equilibrium durable goods price depends on the expected future path of x t
. Using the law of iterated expectations, the projection of the equilibrium price at time t + k on information at time t is
E t p dt + k
= λ d 1
E t p dt + k − 1
+
1
1 − a
0 d
λ d 1
1 + a
2 d a d 0
λ
2 d
∞
X
( λ
− i
2 d
E t x t + k + i
) .
i =0
Our interest is in the dynamic effect of an exogenous shock on this projection:
∂E t
ˆ
∂e dt t
+ k
= λ d 1
∂E t
ˆ dt
∂e
+ t k − 1
+
1
1 − a
0 d
λ d 1
1 + a
2 d a d 0
λ
2 d
∞
X
λ
− i
2 d i =0
∂E t x t + k + i
∂e t
.
(24)
(25)
Finally, the dynamic responses of the production of durable goods are obtained directly from (7) after transforming to log deviations:
∂E t
ˆ dt + k
∂e t
= φ d
∂E t
ˆ dt + k
∂e t
.
(26)
The partial derivatives in (25) and (26) are the theoretical counterparts to the impulse response functions estimated in the next section.
9
Clearly, analogous expressions can be derived for the housing market as well. The model implies that money supply shocks affect the market for durables only through their effect on non-durable production and the real interest rate.
As an illustration of how the model works, Figure 1 shows the simulated theoretical dynamic responses over 24 time periods of p dt + k and y dt + k to a permanent reduction of 0 .
3% (relative to its initial steady-state value) in the real interest rate, given selected parameter values and holding the response of non-durable production constant. The responses trace out the downward sloping saddlepath implied by the model (for an illustration, see Poterba 1984, Figure I, p. 737): the
9 Hamilton (1994, pp. 319-20) interprets impulse response functions as revisions to conditional forecasts.
– 9 –

relative price rises quickly, overshooting the new steady-state, then declines monotonically to a higher steady-state value. Flow production likewise rises and falls monotonically; it also exhibits a permanent increase since the stock of durable goods will rise, and y d is gross investment in durables
(see footnote 8).
3. The empirical model
In this section we estimate the dynamic effects of money supply shocks on the market for durable goods and housing using a standard VAR model, which includes the relevant variables in the theoretical model above. However, we emphasize at the outset that, in estimating these dynamic responses, we do not impose the full set of restrictions from that theory. Instead, we impose only a minimum set of restrictions based on the assumption of long-run monetary neutrality to justidentify the response functions. This set of restrictions is consistent with the model of consumer behavior above, as well as with most theories of the macro economy. We first briefly lay out the
VAR and the identifying restrictions, then discuss the data and empirical estimates.
Let q t be an 8 × 1 vector of endogenous variables, containing empirical counterparts to the real price of durable goods, production (expenditures) of durable goods, the real price of housing, production (expenditures) of housing, the nominal interest rate, production of non-durable goods, real money balances and the nominal stock of money (in that order). We assume that each variable in this system has one unit root (which we verify with the usual battery of tests on the interest rate and the log transformation of each of the other variables), so the following square-summable vector moving averages fully represent the stochastic data generating process:
∆ q t
= ( D
0
+ D
1
L + D
2
L
2
+ · · · ) u t
= D ( L ) u t
∆ q t
= ( I + C
1
L + C
2
L
2
+ · · · ) t
= C ( L ) t
,
(27)
(28) where Eu t u
0 t
= I and E t
0 t
= Σ. The first equation represents the economic structure, so the white noise shocks u t have economic content. One of these shocks is taken to represent money supply shocks – unpredictable and exogenous changes in money or interest rates that reflect behavior in the sectors determining the supply of money (e.g. the central bank). The second equation is the reduced form, so that t
= D
0 u t
, C i
= D i
D
− 1
0 and Σ = D
0
D
0
0
.
– 10 –

(28) can be estimated directly from the VAR representation of the data.
10
To identify the structural impulse response functions ( D i
), we assume that money supply shocks are neutral in the long-run. Long-run monetary neutrality is essentially a stylized fact (Lucas 1996), and has received much empirical support.
11 It is also consistent with theoretical model above. Given the variable ordering and that the variables in the VAR are all real except the nominal money stock, long-run monetary neutrality restricts the final column of D (1) = lim k →∞
∂q t + k
∂u t to be 0 except for the final element. That is, we identify a money supply shock as a shock that has a permanent effect only on nominal money, and not on the other variables in the system.
12 It is straightforward to show that although this set of restrictions is not sufficient to fully identify the structural model, it is sufficient to just-identify the responses to money supply shocks (the final columns of D i
) by computing the
Cholesky factor of the long-run covariance matrix, C (1)Σ C (1)
0
.
13
Infinite-horizon identifying restrictions have been criticized by Faust and Leeper (1997) and
Pagan and Robertson (1998), among others, as relying on weak instruments. However, the approach has the advantage of being consistent with a wide range of policy rules and informational assumptions, unlike most strategies that rely on short-run restrictions. In any case, our strategy of comparing the identified response functions to those predicted by theory can help us determine the plausibility of the identifying restrictions.
Our baseline VAR model includes the following empirical proxies: for durable goods price, the producer price index for total durable goods reported by the Bureau of Labor Statistics
(ftp://ftp.bls.gov/pub/time.series/wp/, series WPUDUR0110); for durable good expenditures, real
10
In the implementation of this empirical strategy, we generally assume, based on the standard trace and maximum eigenvalue tests, that there are no compelling cointegrating relationships; hence, we estimate a VAR in first differences. However, our results were robust to introducing a single cointegrating vector while maintaining our key identifying assumption discussed below.
11 e.g., Fisher and Seater 1993, King and Watson 1997, Boschen and Mills 1995, Bernanke and
Mihov 1998, and the recent survey by Bullard 1999. Coe and Nason (2004) point out that some of these tests may be uninformative.
12
Although the interest rate in q t is the nominal rate, its long-run behavior in response to a permanent change in the level of the money supply is assumed to be identical to real rate behavior, since such a change in the stock of money will not cause a permanent change in the inflation rate.
13
See Keating (1996) and Lastrapes (1998, appendix). For a more general discussion of partial identification in VARs, see Christiano, Eichenbaum and Evans (1999). The use of infinite horizon restrictions to identify VARs was pioneered by Blanchard and Quah (1989) and Shapiro and Watson
(1988).
– 11 –

personal consumption expenditures on durable goods (in chained 2000 dollars, and defined as tangible products, besides housing, that can be stored or inventoried with an average life of at least three years) from the National Income and Product accounts (www.bea.doc.gov/bea/dn/nipaweb/,
Table 2.11); for housing price, the median US sales price of new home sold; for housing production, new houses sold (the housing series are from the US Census Bureau, www.census.gov/const/www/); the nominal interest rate is the 3-month t-bill rate; non-durable output is the industrial production index; and the money stock is M1.
14 The price level used to deflate nominal magnitudes is the producer price index for all commodities. We obtain these macro data from the Federal Reserve
Bank of St. Louis Electronic Database (http://research.stlouisfed.org/fred/index.html). The data are quarterly, and the sample ranges from the first quarter of 1963 to the fourth quarter of 2004.
Each equation in the estimated VAR model includes a constant, seasonal dummies, and four common lags of each of the variables. Plots of the estimated residuals and consideration of the
Q-test and Breusch-Godfrey tests for serial correlation indicate that the four lags are sufficient to effectively whiten the residuals. Allowing for first-differencing and conditioning on lagged values implies an estimation range of 1964:2 to 2004:4 (163 observations and 127 degrees of freedom).
Figure 2 reports the estimated dynamic responses to money supply shocks, given our identification strategy, as a function of forecast horizon. These impulse response functions show how each variable in the system responds to a standard deviation money supply shock, on average over the sample, holding all other shocks constant. The dashed curves represent a one-standard error confidence interval around the estimated response functions, computed from a typical Monte Carlo integration exercise with 1000 replications. We plot the estimated response coefficients up to a forecast horizon of 24 quarters.
The estimates imply that a standard deviation shock increases the nominal stock of money by
0.27% immediately, and by about 2% in the long-run. The initial effect on real money is almost identical to the nominal money response, indicating that the overall price level does not respond to the shock. Evidently, the price level remains rigid over the first four quarters, then begins to adjust upwards; the response of real money peaks at quarter six, and declines thereafter to its
14 For more details on the measurement of durable goods, see the Guide to the NIPAs
(www.bea.doc.gov/bea/an/nipaguid.pdf) and Landefeld and Parker (1997).
– 12 –

original steady-state as average prices rise. Industrial production also shows a typical delayed response, rising only after the second quarter. Its response peaks at eight quarters after the shock, at 0.8%, before declining to its original value. The nominal liquidity effect – the response of the interest rate to money supply shocks – is roughly 46 basis points (annualized) at the second quarter horizon. The responses of these macro variables are consistent with previous findings of monetary non-neutrality in the short-run using VAR models (e.g. Christiano, Eichenbaum and Evans 1999).
Although not reported, our estimated variance decompositions are also generally consistent with the previous literature: money explains at most 8% of the variation of industrial production (at horizon eight). It contributes about twice this much to variation in real money balances, and over
55% to the short-run variation in the 3-month treasury bill rate. The contribution to the rate variation remains about 30% after one year.
The responses of the durable goods and housing market variables, in the first column of Figure
2, are our primary interest in this paper. Both markets exhibit a short-run rise in real price and production in response to a money supply shock; however, the responses are stronger in the housing market than the market for general durable goods. The real price of housing shows no initial response, but ultimately is 1% higher after six quarters compared to its no-shock value. House sales increase by 2% when the shock occurs, and 3.2% after 3 quarters. Real durable goods price has a maximum response of only 0.24% at three quarters, and a maximum production response of 1.25% at six quarters. Although the latter is smaller (and more delayed) than the housing sales response, it is still about 1.75 times larger than the response of overall industrial production; these large responses of durable expenditures relative to a measure of aggregate output support the notion that money supply shocks may be an important factor in the explaining the high degree of volatility of durable good expenditures over the business cycle. Although money supply shocks contribute little to the forecast error variance of real durable good price (no more than 2.5%) and real housing price (a maximum of over 8%), they have greater explanatory power for the variance of the respective production levels: a maximum of over 16% for each. It seems safe to infer that durable goods and housing play important roles in the transmission of money supply shocks to the real economy.
The dynamic multipliers for the nominal interest rate are reported in the figure; however, it is
– 13 –

the real interest rate response that is relevant for behavior in the durable goods markets. The real rate response can be inferred from the nominal interest rate response and the price level response
(the latter of which is simply the difference between the nominal money and real money responses), as in Gali (1992) and Lastrapes (1998). To see how, let k denote the forecast horizon of the dynamic response functions and π h,t + k the rate of inflation of the numeraire good (non-durables) at time t + k over the following h quarters; i.e.
π h,t + k
≡ (
1 h
)( lnp t + k + h
− lnp t + k
). Then,
∂π h,t + k
∂u mt
=
1 h
∂lnp t + k + h
∂u mt
−
∂lnp t + k
∂u mt
, (29) where u mt is the exogenous shock to the money supply. This equation gives the response of the quarterly inflation rate to the exogenous money impulse. But if agents use the VAR to form expectations, then (29) shows how the path of inflationary expectations will be revised in light of this shock. Hence, (29) can be interpreted as the response of expected inflation under this assumption of expectation formation. If R is the (continuously-compounded) nominal yield-tomaturity on h -period bonds and r the corresponding real yield, then
∂r t + k
=
∂u mt
∂R t + k
−
∂u mt
∂π h,t + k
∂u mt
.
(30)
That is, the real rate response is the difference between the nominal rate response (directly estimated from the identified VAR) and the response of expected inflation as computed in equation (29). We set h = 1 since our nominal interest rate measure is the 3-month t-bill.
Figure 3 shows the dynamic responses of inflation and the real interest rate, together with the nominal rate response from Figure 2. The real rate response closely tracks the nominal rate response over the first two quarters, but continues to fall as inflation begins to respond to the money shock. The peak real liquidity effect occurs at seven quarters, with an annualized response of around 80 basis points. The approach to the steady-state is also more gradual than for the nominal rate.
We emphasize that the estimated response functions presented here are not identified using the restrictions of the dynamic theory in the previous section. Yet, the responses for the durable goods and housing market variables are qualitatively consistent with that theory. We estimate that a positive money supply shock lowers nominal (and more importantly real) interest rates in
– 14 –

the short-run, which, according to the theory, lowers user cost and thus increases the demand for durable goods relative to nondurable goods. Indeed, we find an increase in relative price and expenditures on durable goods and housing in the short-run in response to this shock, albeit the price response for non-housing durable goods is small. But these findings beg the question – does the theory match the estimated responses quantitatively, given reasonable parameter values? We consider this question in the following section.
4. Matching the impulse responses to the theory
In this section, we interpret the estimated dynamic response functions in light of the theoretical model of section 2. In particular, we choose values for the theoretical parameters to match, as closely as possible, the model’s projected responses to a money supply shock of the durable goods and housing market variables to the estimated responses from the data. Specifically, collect the model’s free parameters in the vector ω
0
= ( ρ δ d
φ d
δ h
φ h
), where, recall, ρ determines the intratemporal elasticity of substitution among the goods, δ is the rate of depreciation and φ is the price elasticity of supply.
15 Let p d
( k, ω ) denote the theoretical multiplier for durables price
(equation 25) at horizon k , evaluated at a given set of values for the parameters in ω , and using the estimated responses for y ct and r t from the VAR. Let p d
( k ) be the corresponding impulse response coefficient from the weakly-identified VAR (reported in Figure 2). The error in using the model to predict the estimated responses, v pd
( k, ω ), depends on ω ; analogous forecast errors are constructed for durable goods expenditures, housing price and housing expenditures. If v is the
4 K × 1 vector of these errors stacked over horizons 1 , . . . , K , we choose ω to minimize the weighted sum of squared errors J = v
0 ˆ v Ω is a diagonal matrix containing the simulated standard given to estimates that are relatively uncertain according to the Monte Carlo simulation. This approach to estimating economic structure has been used recently by Rotemberg and Woodford
(1997), Christiano, Eichenbaum and Evans (1995), and Smets and Wouter (2002).
15 We set the other parameter in the model, ˜ (the steady-state real interest rate), to be 2% on an annual basis. When we treat ˜ as a free parameter, it is estimated to be over 6% at annual rates, which seems high. The estimate of ρ rises slightly, there is essentially no change in the estimates of both δ ’s and φ ’s, and there is very little gain in the improvement in fit. We therefore rely on the value of ˜ calibrated to a more reasonable value.
– 15 –

To implement this estimation procedure, we fit the response functions over a 24-quarter horizon
(i.e.
K = 24); the results were unaffected using 32 quarters. Equations (25) and (26) indicate that the response for any horizon k depends on an infinite sum of the expected future responses of x ; we truncate this summation at 10 years (the results were not sensitive to extending this summation out to 20 years since the sum converges relatively quickly). We assume the tax rate to be 5% on a quarterly basis in computing the after-tax real interest rate response. The BFGS hill-climbing method in RATS is used to minimize the objective function; convergence is relatively quick, and the estimates are not sensitive to starting values.
Figure 4 contains the VAR estimates (green curves, repeated from Figure 2) with standard errors (blue curves, also repeated) and the responses implied by the model (black curves) for the coefficient estimates reported in the first column of Table 1. For these values, the theoretical model closely matches the estimated responses for durable goods and housing expenditures, but is less able to track the price responses (although the order of magnitude of the price responses is correct).
The model hits the timing of the peak response exactly for both expenditure series, although the predicted house sales response does not exhibit as extreme a ‘hump-shaped’ response as the VAR estimates. For durables price, the initial predicted responses are close to the data, but the theory cannot account for the higher peak and quicker adjustment to the steady-state of the estimated responses. The theoretical housing price reaction to a money supply shock is larger than that implied by the VAR estimates; indeed, this is the only predicted response that falls outside the standard error bands. From the point of view of the theory, the findings of the VAR that the housing price in the data is slow to adjust to the money shock is a puzzle.
We emphasize that our approach, as in Smets and Wouters (2002), relies on the estimated responses of external ‘driving’ variables (output and the real interest rate in our case) to simulate the theory’s predictions. Because the responses of these driving variables to money supply shocks are transitory (by construction through our identification strategy), the theoretical responses in
Figure 4 converge to zero as the forecast horizon increases, in contrast to the hypothetical case in
Figure 3. But recall that Figure 3 considers as a thought experiment a permanent reduction in the interest rate. We can think of Figure 4 as plotting the model’s predictions when the demand for durable goods shifts to the right in the short-run as user cost falls and incomes rise in response to
– 16 –

an increase in liquidity, but gradually shifts back to the left as these variables return to original levels.
The estimates in the first column of Table 1 appear to be reasonable. The estimate of ρ is
0.59, and is thus clearly larger than zero, which suggests that Cobb-Douglas preferences ( ρ = 0) are inappropriate. This value implies that the intratemporal elasticity of substitution between durables and non-durables ( =
1
1 − ρ
) is about 2.4.
16
As we would expect, the depreciation rate (and thus the user cost) for non-housing durable goods is larger than that for housing (by a factor of almost
6), although the values, being quarterly rates, are perhaps larger than expected. However, Startz
(1989, p. 361, 362) estimates that the annual rate of depreciation for aggregate durable goods is
0.98. Our estimate of δ d implies an annual depreciation rate of 0.97.
17
Finally, the estimates imply that the flow supplies of both durables and housing are sensitive to changes in relative prices: a
1% rise in the price of durables induces a 10% rise in quarterly expenditures on durables, while a similar rise in the price of housing leads to a 3.5% rise in new houses sold. The latter is consistent with other estimates of the supply of housing; for example, Topel and Rosen (1988, p. 735) and
Harter-Dreiman (2003, Table 4).
18
Table 1 also contains a rough measure of goodness of fit. As noted by Smets and Wouters
(2002, p. 970), if the estimated dynamic response coefficients from the VAR are assumed to be asymptotically normal, then T · J is distributed as χ
2 with, in our case, 19 degrees of freedom
(the number of horizons over which the model is fit (24) less the number of free parameters in ω ), where T is the number of time-series observations used in estimating the VAR. The value we obtain for this statistic, 21.19, implies a marginal significance level for this model of 0.33, indicating that there is no reason to reject the restrictions imposed by the theoretical model at typical test sizes.
The theoretical model implies that exogenous money supply shocks can influence the durable goods markets only through their effects on non-durable expenditures and the real interest rate.
The former reflects an income effect – a monetary stimulus increases aggregate income which leads to an increase in both durable and non-durable goods. The latter is a substitution effect – a
16
17
18
Ogaki and Reinhart (1998, p. 1091) estimate to be around 1.2
δ a
= 1 − (1 − δ q
)
4
.
Bruce and Holtz-Eakin (2001) point out the importance of estimating supply elasticities of housing to understand the effects of tax and other policies.
– 17 –

reduction in the real interest rate reduces user cost and increases the demand for durable goods relative to non-durable goods. Figures 5 and 6 consider the separate roles of these two channels in explaining the estimated response functions for durable goods and housing. Figure 5 shows the result of simulating the model’s dynamic responses assuming the real rate response is exactly zero, therefore isolating the effect of non-durable expenditures. Figure 6 isolates the role of the real interest rate analogously. In each case, the simulation uses the parameter estimates from the general model, reported in Table 1. The most noteworthy feature of the figures is the relative importance of the real interest rate in the housing market: the response of the real rate accounts for most of the variation in the estimated path for new house sales and prices. Indeed, the theoretical and empirical impulse responses are almost identical at the initial horizon for housing. The same is true for durable goods expenditure, but the gradual rise in this variable over the first 6 to 7 quarters after the shock is clearly due to the non-durables response.
Finally, we have considered alternative proxies for the variables in our system to gauge the robustness of our results. The final columns in Table 1 report parameter estimates for five alternative systems, which fairly indicate the range of our estimates. Column 2 reports the results for a VAR that substitutes the repeat-sales price index computed by the Office of Federal Housing Enterprize
Oversight (OFHEO), which controls for variation in the quality of homes sold, for the median sales price of new houses.
19 Column 3 uses the 10-year treasury note yield instead of the 3-month t-bill to consider the potential relevance of long-term yields; column 4 replaces the industrial production index with real personal consumption expenditures on non-durable goods; column 5 uses housing starts rather than new home sales; and the estimates in the final column use M2 instead of M1 to measure the nominal money stock. In general, the patterns of the estimated response functions and those generated from the theoretical model are the same as in the baseline case. There is noticeable variation in the estimates of ρ – in specifications 2, 5 and 6, it approaches 1.
φ d also shows substantial variation, ranging from 12.49 in model 2 to 2.89 in model 6. However, the housing market parameters are remarkably stable across specifications. In only one case – specification 5 – does the p-value fall below 0.19; we would ‘reject’ the theoretical model for a test size of 1%.
19 See Calhoun (1996) for a detailed description of this price index. The series is not available prior to the first quarter of 1975, so the VAR sample size is 115 observations for this specification.
– 18 –

Our overall assessment is that the forward-looking theory performs reasonably well in predicting the observed responses of the durable goods markets to money supply shocks, especially regarding expenditures in these markets. The largest discrepancies between theory and fact lie in the relative price responses. Yet, the theory has predictive power despite the many simplifications built into our model. Our assumptions about firm behavior allow money supply shocks to influence expenditures only through price, with little concern for dynamics or adjustment costs; generalizing this aspect of the model (as in Topel and Rosen 1988, for example), might improve the explanatory power of the theory. The model ignores many institutional factors, especially for the housing market, such as fixed mortgage payments, property taxes deductions, maintenance costs, and so on. And we ignore many complexities of the household’s intertemporal choice, such as borrowing constraints (Chah, Ramey and Starr 1995), illiquidity and imperfections of secondary markets
(Mishkin 1976) and precautionary saving (Wilson 1998). Extending the model along these lines and others may strengthen its ability to explain the facts, particularly the dynamics of price. But our results suggest that the model of rational intertemporal behavior can be a viable theoretical framework for many questions regarding the macroeconomy, and serve as an effective guide for policy, despite recent evidence of anomalies in consumer behavior (e.g., Frederick et al. 2002, Thaler
1994, Akerlof 2002, Angeletos et al.
2001). Browning and Crossley (2001) make a similar point.
5. Conclusion
This paper provides evidence on the effects of money on the markets for aggregate durable goods and housing in the US. We find that unexpected money supply shocks, assuming that these shocks are neutral in the long-run, have potentially important effects on these markets, and can account at least for some of the excess volatility of durable goods and housing expenditures relative to total output. We also find that, for plausible parameter values, the dynamic response functions implied by a basic model of forward looking consumer behavior are a reasonable match, especially for expenditures, to those estimated from a weakly-identified VAR. While the fit is not perfect, primarily for the price responses, and important extensions to the theoretical model need to be considered, we conclude that the basic rational, optimizing framework remains useful for aggregate analysis.
– 19 –

Table 1. Estimated theoretical parameters
ρ
δ d
φ d
δ h
φ h
J
T
T · J p-value
1 2 3 4 5 6
0.59
0.99
0.40
0.56
10.05
12.49
0.12
3.58
0.13
0.15
3.93
0.18
0.49
0.36
9.96
0.11
3.40
0.15
0.84
0.38
6.33
0.13
3.47
0.12
0.95
0.51
7.05
0.16
3.61
0.22
163 115 163 163 163 163
21.19
20.70
24.28
19.07
36.51
15.16
0.33
0.35
0.19
0.45
0.01
0.71
0.93
0.30
2.89
0.15
4.91
0.93
Specification 1: baseline model
Specification 2: OFHEO repeat-sales price index
Specification 3: 10-year treasury note instead of 3-month t-bill
Specification 4: real PCE on non-durable goods instead of industrial production
Specification 5: Housing starts instead of new home sales
Specification 6: M2 instead of M1.
– 20 –

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– 24 –

Figure 1. Theoretical dynamic responses to hypothetical interest rate shock rho = 0.167 , r = 0.005 , delta = 0.083 , phi= 3.0
Real price of durable goods
0.050
0.040
0.030
0.020
0.010
0.000
0.018
0.015
0.013
0.010
0.007
0.005
0.003
0.000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Expenditures on durable goods
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.0200
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
-0.0050
0.0048
0.0040
0.0032
0.0024
0.0016
0.0008
0.0000
-0.0008
-0.0016
0.0160
0.0080
0.0000
-0.0080
0.0480
0.0400
0.0320
0.0240
Figure 2. Dynamic responses to money supply shocks
Sample: 1964:02 to 2004:04
Response of real price of durables Response of interest rate
0.0002
0.0000
-0.0002
-0.0004
-0.0006
-0.0008
-0.0010
-0.0012
-0.0014
-0.0016
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Response of durable good production Response of non-durable good production
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Response of real price of housing
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Response of real money
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Response of housing production
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Response of money
0.0250
0.0200
0.0150
0.0100
0.0050
0.0000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Figure 3. Inflation rate, real rate and nominal rate responses
0.0005
0.0000
-0.0005
-0.0010
-0.0015
-0.0020
-0.0025
0.0020
0.0015
0.0010
5 10 15 20 25 30 35 40 45 50 55 inflation real rate nominal rate
60 65

0.0200
0.0175
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
0.0048
0.0040
0.0000
-0.0008
-0.0016
0.0032
0.0024
0.0016
0.0008
Durable goods price
Figure 4. Estimated and simulated responses
0.0150
0.0125
0.0000
-0.0025
-0.0050
0.0100
0.0075
0.0050
0.0025
2 4 6 8 10 12 14 16 18 20 22 24
Housing price
2 4 6 8 10 12 14 16 18 20 22 24
Expenditures on durable goods New house sales
0.0480
0.0400
0.0320
0.0240
0.0160
0.0080
0.0000
-0.0080
2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24

0.0075
0.0050
0.0025
0.0000
-0.0025
0.0200
0.0175
0.0150
0.0125
0.0100
0.0048
0.0040
0.0032
0.0024
0.0016
0.0008
0.0000
-0.0008
-0.0016
Durable goods price
Figure 5. Estimated and simulated responses
Effects through non-durable output only
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
-0.0050
2 4 6 8 10 12 14 16 18 20 22 24
Expenditures on durable goods
Housing price
2 4 6 8 10 12 14 16 18 20 22 24
New house sales
0.0240
0.0160
0.0080
0.0480
0.0400
0.0320
0.0000
-0.0080
2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24

0.0075
0.0050
0.0025
0.0000
-0.0025
0.0200
0.0175
0.0150
0.0125
0.0100
0.0048
0.0040
0.0032
0.0024
0.0016
0.0008
0.0000
-0.0008
-0.0016
Durable goods price
Figure 6. Estimated and simulated responses
Effects through real interest rate only
0.0150
0.0125
0.0100
0.0075
0.0050
0.0025
0.0000
-0.0025
-0.0050
2 4 6 8 10 12 14 16 18 20 22 24
Expenditures on durable goods
Housing price
2 4 6 8 10 12 14 16 18 20 22 24
New house sales
0.0240
0.0160
0.0080
0.0480
0.0400
0.0320
0.0000
-0.0080
2 4 6 8 10 12 14 16 18 20 22 24 2 4 6 8 10 12 14 16 18 20 22 24