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Finance and Economics Discussion Series: 2007-53 Screen Reader version ^♣

Documentation of the Research and Statistics Division's Estimated DSGE Model of the U.S. Economy: 2006 Version

Rochelle M. Edge, Michael T. Kiley, and Jean-Philippe Laforte*

Keywords: Two-sector growth model, sticky-prices, sticky-wages, habit-persistence, investment adjustment costs, variable utilization, Bayesian estimation

Abstract:

This paper provides documentation for the large-scale estimated DSGE model of the U.S. economy used in Edge, Kiley, and Laforte (2007). The model represents part of an ongoing research project (the Federal Reserve Board's Estimated, Dynamic, Optimization-based--FRB/EDO--model project) in the Macroeconomic and Quantitative Studies section of the Federal Reserve Board aimed at developing a DSGE model that can be used to address practical policy questions and the model documented here is the version that was current at the end of 2006. The paper discusses the model's specification, estimated parameters, and key properties.

This paper contains documentation for the large-scale estimated DSGE model of the U.S. economy that was employed in Edge, Kiley, and Laforte (2007) to discuss the use of DSGE models in a policy oriented environment. The model represents part of an ongoing research project in the Macroeconomic and Quantitative Studies section of the Federal Reserve Board aimed at developing a DSGE model that can be used to address practical policy questions. The outline of the paper is as follows. Section 1 provides a brief qualitative description of the model. Section 2 outlines the model's production, capital evolution, and preference technologies. Section 3 describes the economy's decentralization, and section 3 defines equilibrium in the model. Section 4 lists the data that is used in estimating the model. Section 5 reports the model's key estimation results, which include the estimated parameter values, variance decompositions, impulse response functions, and implied paths of model variables. The precise equations that characterize equilibrium in this model are contained in the appendix. Appendix A presents the equations of the symmetric model, appendix B reports the equations of the symmetric and stationary model, and appendix C gives the solution to the model's steady-state. Finally, since the model contains a large number of parameter and variable names a key is given in appendices D, E, and F.

Before moving to our presentation of the model, we note that we anticipate that the DSGE model developed in this paper (and subsequent versions of it) will serve as a complement to the analyses that are currently performed using existing large-scale econometric models, such as FRB/US model, as well as smaller, ad hoc models that we have found useful for more specific questions. Our model, while quite a bit more detailed and disaggregated than most existing DSGE models, is nonetheless incapable of addressing many of the questions addressed in a very large model like FRB/US and cannot therefore serve as the sole model for policy purposes. We suspect that model-based analyses are enhanced by consideration of multiple models (and, indeed, our experience suggests that often we learn as much when models disagree than when they agree). The use of multiple models will allow us to examine the robustness of policy strategies across models with quite different foundations, which we view as important given the significant divergences of opinion regarding the plausibility of various types of models.

1 Model Overview and Motivation

Figure 1 provides a graphical overview of the economy described by our model. The model possesses two final goods: slow-growing "CBI" goods--so called because most of these goods are used for consumption (C) and because they are produced by the business and institutions (BI) sector--and fast-growing "KB" goods--so called because these goods are used for capital (K) accumulation and are produced by the business (B) sector. The goods are produced in two stages by intermediate- and then final-goods producing firms (shown in the center of the figure). On the model's demand-side, there are four components of spending (each shown in a box surrounding the producers in the figure): consumer nondurable goods and services (sold to households), consumer durable goods, residential capital goods, and non-residential capital goods. Consumer nondurable goods and services and residential capital goods are purchased (by households and residential capital goods owners, respectively) from the first of economy's two final goods producing sectors (good CBI producers), while consumer durable goods and non-residential capital goods are purchased (by consumer durable and residential capital goods owners, respectively) from the second sector (good KB producers). We "decentralize" the economy by assuming that residential capital and consumer durables capital are rented to households while non-residential capital is rented to firms. In addition to consuming the nondurable goods and services that they purchase, households also supply labor to the intermediate goods-producing firms in both sectors of the economy.

Our assumption of a two-sector production structure is motivated by the trends in certain relative prices and categories of real expenditure apparent in the data. As reported in Table 1, expenditures on consumer non-durable goods and non-housing services and residential investment have grown at roughly similar real rates of around 3-1/2 percent per year over the last 20 years, while real spending on consumer durable goods and on nonresidential investment have grown at around 6-1/2 percent per year. The relative price of residential investment to consumer non-durable goods and non-housing services has been fairly stable over the last twenty years (increasing only 1/2 percent per year on average, with about half of this average increase accounted for by a large swing in relative prices over 2003 and 2004). In contrast, the prices of both consumer durable goods and non-residential investment relative to those of consumer non-durable goods and non-housing services have decreased, on average, about 3 percent per year. A one-sector model is unable to deliver long-term growth and relative price movements that are consistent with these stylized facts. As a result, we adopt a two-sector structure, with differential rates of technical progress across sectors. These different rates of technological progress induce secular relative price differentials, which in turn lead to different trend rates of growth across the economy's expenditure and production aggregates. We assume that the output of the slower growing sector (denoted $X^{cbi}_{t}$ ) is used for consumer nondurable goods and services and residential capital goods and the output of a faster growing sector (denoted $X^{kb}_{t}$ ) is used for consumer durable goods and non-residential capital goods, roughly capturing the long-run properties of the data summarized in Table 1.

The DSGE models of Christiano et al. [2005] and Smets and Wouters [2004] did not address differences in trend growth rates in spending aggregates and trending relative price measures, although an earlier literature--less closely tied to business cycle fluctuations in the data--did explore the multi-sector structure underlying U.S. growth and fluctuations.¹ Subsequent models have adopted a multi-sector growth structure, including Altig et al. [2004], Edge, Laubach, and Williams [2003], and DiCecio [2005]; our model shares features with the latter two of these models.

The disaggregation of production (aggregate supply) leads naturally to some disaggregation of expenditures (aggregate demand). We move beyond a model with just two categories of (private domestic) final spending and disaggregate along the four categories of private expenditure mentioned earlier: consumer non-durable goods and non-housing services (denoted $E^{cnn}_{t}$ ), consumer durable goods (denoted $E^{cd}_{t}$ ), residential investment (denoted $E^{r}_{t}$ ), and non-residential investment (denoted $E^{nr}_{t}$ ).

While differential trend growth rates are the primary motivation for our disaggregation of production, our specification of expenditure decisions is related to the well-known fact that the expenditure categories that we consider have different cyclical properties. As shown in Table 2, consumer durables and residential investment tend to lead GDP, while non-residential investment (and especially non-residential fixed investment, not shown) lags. These patterns suggest some differences in the short-run response of each series to structural shocks. One area where this is apparent is the response of each series to monetary-policy innovations. As documented by Bernanke and Gertler [1995], residential investment is the most responsive component of spending to monetary policy innovations, while outlays on consumer durable goods are also very responsive. According to Bernanke and Gertler [1995], non-residential investment is less sensitive to monetary policy shocks than other categories of capital goods spending, although it is more responsive than consumer nondurable goods and services spending.

Beyond the statistical motivation, our disaggregation of aggregate demand is motivated by the concerns of policymakers. A recent example relates to the divergent movements in household and business investment in the early stages of the U.S. expansion following the 2001 recession, a topic discussed in Kohn [2003]. We believe that providing a model that may explain the shifting pattern of spending through differential effects of monetary policy, technology, and preference shocks is a potentially important operational role for our disaggregated framework.

2 Production, Capital Evolution, and Preferences

In this section we present the production, capital evolution, and preference technologies for our model. The long-run evolution of the economy is determined by differential rates of stochastic growth in the production sectors of the economy; its short-run dynamics are influenced by various forms of adjustment costs. Adjustment costs to real aggregate variables are captured in the economy's preference, production, and capital evolution technologies presented in this section. Adjustment costs to real sectoral variables and nominal variables are captured in the decentralization of the model presented in the following section.

2.1 The Production Technology

As noted in the previous section our model economy produces two final goods and services: slow-growing "consumption" goods and services $X^{cbi}_{t}$ and fast-growing "capital" goods $X^{kb}_{t}$ . "Capital" goods are produced by businesses; "consumption" goods and services are produced by businesses and institutions. These final goods are produced by aggregating (according to a Dixit-Stiglitz technology) an infinite number of differentiated inputs, $X_{t}^{s}(j)$ for , distributed over the unit interval. Specifically, final goods production is governed by the function

$\displaystyle X^{s}_{t} =\left(\int_{0}^{1} X_{t}^{s}(j)^{\frac{\Theta^{x,s}_{t}-1} {\Theta^{x,s}_{t}}}dj\right)^{\frac{\Theta^{x,s}_{t}} {\Theta^{x,s}_{t}-1}}, s=cbi,kb.$

(1)

The term $\Theta^{x,s}_{t}$ is the stochastic elasticity of substitution between the differentiated intermediate goods inputs used in the production of the consumption or capital goods sectors. Letting $\theta^{x,s}_{t} \equiv \ln \Theta^{x,s}_{t}-\ln \Theta^{x,s}_{\ast}$ denote the log-deviation of $\Theta^{x,s}_{t}$ from its steady-state value of $\Theta^{x,s}_{\ast}$ , we assume that

$\displaystyle \theta_{t}^{x,s}=\epsilon_{t}^{\theta,x,s}$

(2)

where $\epsilon_{t}^{\theta,x,s}$ is an i.i.d. shock process.

The th differentiated intermediate good in sector (which is used as an input in equation 1) is produced by combining each variety of the economy's differentiated labor inputs $\{L^{s}_{t}(i,j)\}_{i=0}^{1}$ with the sector's specific utilized non-residential capital stock $K^{u,nr,s}_{t}(j)$ . (Utilized non-residential capital, $K^{u,nr,s}_{t}(j)$ , is equal to the product of physical non-residential capital, $K^{nr,s}_{t}(j)$ , and the utilization rate, $U^{nr,s}_{t}(j)$ ). A Dixit-Stiglitz aggregator characterizes the way in which differentiated labor inputs are combined to yield a composite bundle of labor, denoted $L^{s}_{t}(j)$ . A Cobb-Douglas production function then characterizes how this composite bundle of labor is used with capital to produce--given the current level of multifactor productivity $MFP^{s}_{t}$ in the sector --the intermediate good $X^{s}_{t}(j)$ . The production of intermediate good is governed by the function:

$\displaystyle X_{t}^{m,s}(j)\!=\! \left(K^{u,nr,s}_{t}(j)\right)^{\alpha} \!\left( Z^{m}_{t}Z^{s}_{t} L_{t}^{x,s}(j)\right)^{1-\alpha} \!\!{\textrm{where }}L_{t}^{x,s}(j)=\!\left(\int_{0}^{1} \!\! L_{t}^{x,s}(i,j)^{\frac{\Theta^{l}_{t}\!-\!1}{\Theta^{l}_{t}}} di \! \right) ^{\frac{\Theta^{l}_{t}}{\Theta^{l}_{t}\!-\!1}} \!\!\!\!\!\!\!\! s=cbi,kb$

(3)

and where we assume $Z^{cbi}_{t}\equiv 1$ . The parameter $\alpha$ in equation (3) is the elasticity of output with respect to capital while $\Theta^{l}_{t}$ denotes the stochastic elasticity of substitution between the differentiated labor inputs. Letting $\theta^{l}_{t} \equiv \ln \Theta^{l}_{t}-\ln \Theta^{l}_{\ast}$ denote the log-deviation of $\Theta^{l}_{t}$ from its steady-state value of $\Theta^{l}_{\ast}$ , we assume that

$\displaystyle \theta_{t}^{l}=\epsilon_{t}^{\theta,l}$

(4)

where $\epsilon_{t}^{\theta,l}$ is an i.i.d. shock process.

The level of productivity in the capital goods producing sector has two components. The $Z^{m}_{t}$ component represents an economy-wide productivity shock, while the $Z^{kb}_{t}$ term represents a productivity shock that is specific to the capital goods sector. The level of productivity in the consumption goods producing sector has only the one economy-wide component, $Z^{m}_{t}$ , since we assume $Z^{cbi}_{t}=1$ . The exogenous productivity terms contain a unit root, that is, they exhibit permanent movements in their levels. We assume that the stochastic process $Z^{s}_{t}$ evolves according to

$\displaystyle \ln Z^{s}_{t} - \ln Z^{s}_{t-1} =\ln \Gamma^{z,s}_{t} =\ln \left(\Gamma^{z,s}_{\ast} \cdot \exp[\gamma^{z,s}_{t}] \right) =\ln \Gamma^{z,s}_{\ast} + \gamma^{z,s}_{t}, s = kb,m,$

(5)

where $\Gamma^{z,s}_{\ast}$ and $\gamma^{z,s}_{t}$ are the steady-state and stochastic components of $\Gamma^{z,s}_{t}$ . The stochastic component $\gamma^{z,s}_{t}$ is assumed to evolve according to

$\displaystyle \gamma^{z,s}_{t}=\rho^{z,s}\gamma_{t-1}^{z,s}+\epsilon_{t}^{z,s}.$

(6)

where $\epsilon_{t}^{z,s}$ is an i.i.d shock process, and $\rho^{z,s}$ represents the persistence of $\gamma^{z,s}_{t}$ to a shock. In line with historical experience, we assume a more rapid rate of technological progress in capital goods production by calibrating $\Gamma^{z,k}_{\ast} \geq 1$ .

2.2 Capital Stock Evolution

As already noted, there are three types of physical capital stocks in our model economy: non-residential capital, $K^{nr}_{t}$ , residential capital, $K^{r}_{t}$ , and consumer durables capital, $K^{cd}_{t}$ .

Purchases of the economy's fast-growing "capital" good can be transformed into either non-residential capital, $K^{nr}_{t+1}$ , which can be used to produce output in either sector of the economy, or into the economy's consumer-durable capital stock, $K^{cd}_{t+1}$ , from which households derive utility. Purchases of the economy's slow-growing "consumption" good can be transformed into residential capital.

The evolution of the economy's three capital stocks are given below in equations (8), (9), and (10). We assume that there is some stochastic element affecting the efficiency of investment--reflected in the term $A^{s}_{t}$ , for and --in the capital accumulation process. These shocks are uncorrelated with each other and exhibit only transitory movements from their steady-state unit mean.² Letting $a_{t}^{s}\equiv\ln A_{t}^{s}$ denote the log-deviation of $A^{s}_{t}$ from its steady-state value of unity, we assume that:

$\displaystyle a_{t}^{s}=\rho^{a,s}a_{t-1}^{s}+\epsilon_{t}^{a,s}, s = nr,cd,r.$

(7)

We also asssume that there are adjustment costs (captured by the final term in equations 8, 9, and 10) that are implied by capital installation. The evolution of capital is given by:

$\displaystyle K^{nr}_{t+1}(k)$	$\displaystyle =$	$\displaystyle (1\!-\!\delta^{nr})K^{nr}_{t}(k)\!+\!A^{nr}_{t}E^{nr}_{t}(k) -\! \frac{100\cdot\chi^{nr}}{2} \left(\frac{E^{nr}_{t}(k)\!-\! E^{nr}_{t-1}(k)\Gamma^{x,kb}_{t}} {K^{nr}_{t}}\right)^{\!2}\!\!K^{nr}_{t},$	(8)
$\displaystyle K^{cd}_{t+1}(k)$	$\displaystyle =$	$\displaystyle (1\!-\!\delta^{cd})K^{cd}_{t}(k)\!+\!A^{cd}_{t}E^{cd}_{t}(k) -\! \frac{100\cdot\chi^{cd}}{2} \left(\frac{E^{cd}_{t}(k)\!-\! E^{cd}_{t-1}(k)\Gamma^{x,kb}_{t}} {K^{cd}_{t}}\right)^{\!2}\!\!K^{cd}_{t}, \mathrm{and}$	(9)
$\displaystyle K^{r}_{t+1}(k)$	$\displaystyle =$	$\displaystyle (1\!-\!\delta^{r})K^{r}_{t}(k)\!+\!A^{r}_{t}E^{r}_{t}(k) -\! \frac{100\cdot\chi^{r}}{2} \left(\frac{E^{r}_{t}(k)\!-\! E^{r}_{t-1}(k)\Gamma^{x,cbi}_{t}} {K^{r}_{t}}\right)^{\!2}\!\!K^{r}_{t}.$	(10)

The parameter $\delta^{s}$ denotes the depreciation rate for either the non-residential (

), consumer durables (

), or residential (

) capital stocks. Investment adjustment costs are zero for non-residential capital when $E^{nr}_{t}=E^{nr}_{t-1}\Gamma^{x,kb}_{t}$ but rise to above zero (at an increasing rate that depends on $\chi^{nr}$ ) as these values diverge. The relative values of $E^{cd}_{t}$ and $E^{nr}_{t-1}\Gamma^{x,kb}_{t}$ have similar implications for the adjustment costs entailed in the accumulation of consumer durables capital, as do the relative values of $E^{r}_{t}$ and $E^{r}_{t-1}\Gamma^{x,cbi}_{t}$ for the accumulation of residential capital. The parameters $\chi^{cd}$ and $\chi^{r}$ have the same interpretation as $\chi^{nr}$ . Adjustment costs are quite important in models such as ours in ensuring gradual responses of investment to shocks.

2.3 Preferences

The th household derives utility from four sources: consumer non-durable goods and non-housing services, $E^{cnn}_{t}(i)$ , the flow of services from consumer-durable capital, $K^{cd}_{t}(i)$ , the flow of services from residential capital $K^{r}_{t}(i)$ , and its leisure time, which is equal to what remains of its time endowment after $L^{cbi}_{t}(i)+L^{kb}_{t}(i)$ hours are spent working. The preferences of household are separable over all of the arguments of its utility function. The utility that the household derives from the three components of its goods and services consumption is influenced by its habit stock for each of these consumption components, a feature that has been shown to be important for consumption dynamics in similar models. Household 's habit stock for its consumption of non-durable goods and non-housing services, is equal to a factor $h^{cnn}$ multiplied by its consumption last period $E^{cnn}_{t-1}(i)$ . The household's habit stock for its other components of consumption is defined similarly. In summary, the preferences of household are represented by the utility function:

		$\displaystyle \mathcal{E}_{0}\!\sum_{t=0}^{\infty}\!\beta^{t}\!\! \left\{\varsigma^{cnn}\Xi_{t}^{cnn}\ln(E^{cnn}_{t}(i)\!-\!h^{cnn} E^{cnn}_{t-1}(i)) +\!\varsigma^{cd}\Xi_{t}^{cd}\ln(K^{cd}_{t}(i)\!-\!h^{cd} K^{cd}_{t-1}(i)) \right.$
		$\displaystyle \left. +\varsigma^{r}\Xi_{t}^{r}\ln(K^{r}_{t}(i)\!-\!h^{r} K^{r}_{t-1}(i)) -\!\varsigma^{l}\Xi_{t}^{l} \frac{(L^{cbi}_{t}(i)\!+\!L^{kb}_{t}(i))^{1+\nu}}{1+\nu} \right\}\!\!.$	(11)

The parameter $\beta$ is the household's discount factor, $\nu$ denotes its inverse labor supply elasticity, while $\varsigma^{cnn}$ , $\varsigma^{cd}$ , $\varsigma^{r}$ , and $\varsigma^{l}$ are scale parameter that tie down the ratios between the household's consumption components. The stationary, unit-mean, stochastic variables $\Xi^{cnn}_{t}$ , $\Xi^{cd}_{t}$ , $\Xi^{r}_{t}$ , and $\Xi^{l}_{t}$ represent aggregate shocks to the household's utility of its consumption components and its disutility of labor.³ Letting $\xi^{x}_{t} \equiv \ln \Xi^{x}_{t}-\ln \Xi^{x}_{\ast}$ denote the log-deviation of $\Xi^{x}_{t}$ from its steady-state value of $\Xi^{x}_{\ast}$ , we assume that

$\displaystyle \xi_{t}^{x}=\rho^{\xi,x} \xi_{t-1}^{x}+\epsilon_{t}^{\xi,x}, x=cnn,cd,r,l.$

(12)

The variable $\epsilon_{t}^{\xi,x}$ is an i.i.d. shock process, and $\rho^{\xi,x}$ represents the persistence of $\Xi^{x}_{t}$ away from steady-state following a shock to equation (12).

3 The Decentralized Economy

The economy's decentralization, which is also depicted in Figure 1, is as follows:

A representative firm in each of the economy's two final-goods producing sectors purchases intermediate inputs from the continuum of intermediate goods producers and produces the sector's final goods output.

Each firm in the economy's two intermediate-goods producing sectors rents non-residential capital from the capital owners and hires each type of differentiated labor from households so as to to produce its differentiated output. Because each firm is a monopolistically-competitive supplier of its own output, it is able to set the price at which it sells this output.

Each household purchases output from final-goods producers in the slow-growing "consumption" goods sector, which it then uses as non-durable goods and non-housing services, and rents consumer durables capital and residential capital from the capital owners. Because each household is a monopolistically-competitive supplier of its own labor, it is able to set the wage at which it supplies its labor services.

Each non-residential and consumer durables capital owner purchases output from the fast-growing "capital" final-goods sector and transforms it into either non-residential or consumer durables capital. Each residential capital owner purchases output from the slow-growing "consumption" final-goods and transforms it into residential capital.

The monetary authority sets the nominal interest rate given an interest rate feedback rule with smoothing of the policy response to endogenous variables.

The government and foreign economic agents demand a share of the economy's output.

We describe in this section the behaviour of agents listed above.

3.1 Consumption and Capital Final Goods Producers

The representative, perfectly competitive firm in the final consumption good sector owns the production technology described in equation (1) for , while the representative, perfectly competitive firm in the final capital goods sector owns the same technology for . The final-good producer in sector solves the cost-minimization problem of:

$\displaystyle \min_{\left\{ X_{t}^{s}(j)\right\} _{j=0}^{1}} \int_{0}^{1}P_{t}^{s}(j)X_{t}^{s}(j)dj\mathrm{ subject to } \left( \int_{0}^{1}\left(X_{t}^{s}(j)\right)^{\frac{\Theta^{x,s}_{t}-1} {\Theta^{x,s}_{t}}}dj\right)^{\frac{\Theta^{x,s}_{t} } {\Theta^{x,s}_{t}-1}} \!\!\geq X^{s}_{t}, \mathrm{for} s=cbi,kb.$

(13)

3.2 Consumption and Capital Intermediate Goods Producers

Each intermediate-good producing firm $j \in [0,1]$ and owns the production technology described in equation (3). It is convenient to think of the intermediate good producing firm as solving three problems: two factor-input cost-minimization problems and one price-setting profit-maximization problem. The two cost-minimization problems faced by the representative firm in sector s are:

$\displaystyle \min_{\left\{ L^{s}_{t}(i,j)\right\} _{i=0}^{1}}\!\! \int_{0}^{1}\!\!W_{t}^{s}(i)L^{s}_{t}(i,j)di\mathrm{ subject to }\! \left(\int_{0}^{1}\left(L^{s}_{t}(i,j)\right)^{\frac{\Theta^{l}_{t}-1} {\Theta^{l}_{t}}}\!\!di\right)^{\frac{\Theta^{l}_{t}} {\Theta^{l}_{t}-1}} \!\!\!\!\!\!\!\!\geq L^{s}_{t}(j), \mathrm{for} s=cbi,kb,$

(14)

and

		$\displaystyle \min _{\left\{ L^{s}_{t}(j),K^{u,nr,s}_{t}(j)\right\} } W^{s}_{t}L^{s}_{t}(j)+R^{nr,s}_{t}K^{u,nr,s}_{t}(j)$
		$\displaystyle \mathrm{ subject to } \left(Z^{m}_{t}Z^{s}_{t}L^{s}_{t}(j)\right)^{1-\alpha} \left(K^{u,nr,s}_{t}(j)\right)^{\alpha}\geq X^{s}_{t}(j), \mathrm{for} s=cbi,kb, \mathrm{where} Z^{cbi}_{t}\equiv 1.$	(15)

The profit-maximization problem faces by the firm is given by:

		$\displaystyle \max_{\{P^{s}_{t}(j),X^{s}_{t}(j)\}_{t=0}^{\infty}} \mathcal{E}_{0}\sum_{t=0}^{\infty}\beta^{t} \frac{\Lambda^{cnn}_{t}}{P^{cbi}_{t}} \left\{ P^{s}_{t}(j)X^{s}_{t}(j) -\!MC^{s}_{t}(j) X^{s}_{t}(j) \right.$
		$\displaystyle \left. -\frac{100\cdot\chi^{p}}{2} \left(\frac{P^{s}_{t}(j)}{P^{s}_{t-1}(j)}\!-\!\eta^{p}\Pi^{p,s}_{t-1} -\!(1\!-\!\eta^{p})\Pi^{p,s}_{\ast}\right)^{2}\!\!P^{s}_{t}X^{s}_{t} \right\}$
		$\displaystyle \mathrm{subject to} X^{s}_{\tau}(j) =\!\left(\!\frac{P^{s}_{\tau}(j)}{P^{s}_{\tau}}\!\right)^{-\Theta^{x,s}_{\tau}}\!X^{s}_{\tau}, \mathrm{for} \tau=0,1,...,\infty, \mathrm{and} s = cbi,kb.$	(16)

The variable $MC^{s}_{t}(j)$ represents the marginal cost of producing a unit of $X^{s}_{t}(j)$ . The profits function incorporates price setting adjustment costs (the size which depend on the parameter $\chi^{p}$ and the lagged and steady-state inflation rate). The constraint against which the firm maximizes its profits is the demand curve it faces for its differentiated good; this is derived from the final goods producing firm's cost-minimization problem.

3.3 Capital Owners

Capital owners possess the technologies described in equations (8) and (9) for transforming the economy's fast-growing "capital" good into either non-residential capital, $K^{nr}_{t+1}$ , or the consumer-durable capital stock, $K^{nr}_{t+1}$ , and in equation (10) for transforming the economy's slow-growing "consumption" good into the economy's residential capital stock $K^{nr}_{t+1}$ .

The non-residential capital owner solves:

		$\displaystyle \max_{\{E^{nr}_{t}(k),K^{nr}_{t+1}(k),K^{nr,cbi}_{t}(k),K^{nr,kb}_{t}(k), U^{cbi}_{t}(k),U^{kb}_{t}(k)\}_{t=0}^{\infty}}$
		$\displaystyle \mathcal{E}_{0}\sum_{t=0}^{\infty} \beta^{t} \frac{\Lambda^{cnn}_{t}}{P^{cbi}_{t}} \left\{R_{t}^{nr,cbi}U^{cbi}_{t}(k)K^{nr,cbi}_{t}(k) +\!R_{t}^{nr,kb}U^{kb}_{t}(k)K^{nr,kb}_{t}(k)\!-\!P^{kb}_{t}E^{nr}_{t}(k) \right.$
		$\displaystyle \left. -\kappa \left(\frac{(U^{cbi}_{t}(k))^{1+\psi}-1}{1+\psi} \right)P^{kb}_{t}K^{nr,cbi}_{t} -\kappa \left(\frac{(U^{kb}_{t}(k))^{1+\psi}-1}{1+\psi} \right)P^{kb}_{t}K^{nr,kb}_{t} \right\}$
		$\displaystyle \mathrm{subject to}$
		$\displaystyle K^{nr}_{\tau+1}(k) =\! (1-\delta^{nr})K^{nr}_{\tau}(k)\!+\!A^{nr}_{\tau}E^{nr}_{\tau}(k) -\! \frac{100\cdot\chi^{nr}}{2} \left(\frac{E^{nr}_{\tau}(k)\!-\! E^{nr}_{\tau-1}(k)\Gamma^{y,kb}_{t} }{K^{nr}_{\tau}}\right)^{\!2}\!\!K^{nr}_{\tau}$
		$\displaystyle K^{nr,cbi}_{\tau}(k)\!+\!K^{nr,kb}_{\tau}(k) =\! K^{nr}_{\tau}(k), \mathrm{for} \tau=0,1,...,\infty.$	(17)

We assume that the capital owner decides on both the amount of capital that it will rent to firms and the rate of utilization at which this capital is used by firms. (Recall that the firm's choice variable in 15 is utilized capital $K^{u,nr,s}_{t}=U^{s}_{t}K^{nr,s}_{t}$ .) Raising the rate of utilization will boost the capital owners rental income but will incur a cost reflected in the last two terms in the capital owners profit function. We assume that $\kappa=R^{nr}_{\ast}/P^{kb}_{\ast}$ , which implies that in the steady-state equilibrium utilization is unity. The consumer durables capital owner solves:

		$\displaystyle \max_{\{E^{cd}_{t}(k),K^{cd}_{t+1}(k)\}_{t=0}^{\infty}\}} \mathcal{E}_{0}\sum_{t=0}^{\infty} \beta^{t} \frac{\Lambda^{cnn}_{t}}{P^{cbi}_{t}} \left\{R^{cd}_{t}K^{cd}_{t}(k)-P^{kb}_{t}E^{cd}_{t}(k) \right\} \mathrm{subject to}$
		$\displaystyle K^{cd}_{\tau+1}(k) =(1-\delta^{cd})K^{cd}_{\tau}(k)\!+\!A^{cd}_{\tau}E^{cd}_{\tau}(k) -\! \frac{100\cdot\chi^{cd}}{2} \left(\frac{E^{cd}_{\tau}(k)\!-\! E^{cd}_{\tau-1}(k)\Gamma^{x,kb}_{\tau} } {K^{cd}_{\tau}}\right)^{\!2}\!\!K^{cd}_{\tau}$
		$\displaystyle \mathrm{for} \tau=0,...,\infty,$	(18)

while the residential capital owner solves:

		$\displaystyle \max_{\{E^{r}_{t}(k),K^{r}_{t+1}(k)\}_{t=0}^{\infty}\}} \mathcal{E}_{0}\sum_{t=0}^{\infty} \beta^{t} \frac{\Lambda^{cnn}_{t}}{P^{cbi}_{t}} \left\{R^{r}_{t}K^{r}_{t}(k)-P^{cbi}_{t}E^{r}_{t}(k) \right\} \mathrm{subject to}$
		$\displaystyle K^{r}_{\tau+1}(k) =(1-\delta^{r})K^{r}_{\tau}(k)\!+\!A^{r}_{\tau}E^{r}_{\tau}(k) -\! \frac{100\cdot\chi^{r}}{2} \left(\frac{E^{r}_{\tau}(k)\!-\! E^{r}_{\tau-1}(k)\Gamma^{x,cbi}_{\tau} } {K^{cd}_{\tau}}\right)^{\!2}\!\!K^{cd}_{\tau}$
		$\displaystyle \mathrm{for} \tau=0,1,...,\infty.$	(19)

3.4 Households

The household possesses the utility function--defined over three components of goods and services consumption and leisure--described by equation (11).The representative household solves the problem:

		$\displaystyle \max_{\left\{E^{cnn}_{t}(i),K^{cd}_{t}(i),K^{r}_{t}(i),\{W^{s}_{t}(i),L^{s}_{t}(i)\}_{s=cbi,kb}, B_{t+1}(i)\right\}_{t=0}^{\infty}}\!\!$
		$\displaystyle \mathcal{E}_{0}\!\sum_{t=0}^{\infty}\!\beta^{t}\!\! \left\{\varsigma^{cnn}\Xi_{t}^{cnn}\ln(E^{cnn}_{t}(i)\!-\!h^{cnn} E^{cnn}_{t-1}(i)) +\!\varsigma^{cd}\Xi_{t}^{cd}\ln(K^{cd}_{t}(i)\!-\!h^{cd} K^{cd}_{t-1}(i)) \right.$
		$\displaystyle \left. +\varsigma^{r}\Xi_{t}^{r}\ln(K^{r}_{t}(i)\!-\!h^{r} K^{r}_{t-1}(i)) -\!\varsigma^{l}\Xi_{t}^{l} \frac{(L^{cbi}_{t}(i)\!+\!L^{kb}_{t}(i))^{1+\nu}}{1+\nu} \right\} \mathrm{subject to}$
		$\displaystyle R^{-1}_{\tau}B_{\tau+1}(i) =\! B_{\tau}(i)+\!\!\!\!\sum_{s=cbi,kb}\!\!\!\! W^{s}_{\tau}(i)L^{s}_{\tau}(i) +{\textit{Capital and Profits Income}}_{\tau}(i) -P^{cbi}_{\tau}E^{cnn}_{\tau}(i)$
		$\displaystyle -\!R^{cd}_{\tau}K^{cd}_{\tau}\!-\!R^{r}_{\tau}K^{r}_{\tau} -\!\sum_{s=cbi,kb}\frac{100\cdot\chi^{w}}{2} \left(\frac{W^{s}_{\tau}(j)}{W^{s}_{\tau-1}(j)}\!-\!\eta^{w}\Pi^{w,s}_{\tau-1} -\!(1\!-\!\eta^{w})\Pi^{w}_{\ast}\right)^{2} W^{s}_{\tau}L^{s}_{\tau}$
		$\displaystyle -\frac{100\!\cdot\!\chi^{l}}{2} \left(\frac{L^{cbi}_{\ast}}{L^{cbi}_{\ast}\!+\!L^{kb}_{\ast}}\!\cdot\! W^{cbi}_{\tau} +\frac{L^{kb}_{\ast}}{L^{cbi}_{\ast}\!+\!L^{kb}_{\ast}}\!\cdot\! W^{kb}_{\tau} \right)\! \left(\frac{L^{cbi}_{\tau}(i)}{L^{kb}_{\tau}(i)}\!-\!\eta^{l}\frac{L^{cbi}_{\tau-1}}{L^{kb}_{\tau-1}} -\!(1\!-\!\eta^{l}) \frac{L^{cbi}_{\ast}}{L^{kb}_{\ast}}\right)^{2} \frac{L^{kb}_{\tau}}{L^{cbi}_{\tau}},$
		$\displaystyle L^{cbi}_{\tau}(i) =\!\left(\!\frac{W^{cbi}_{\tau}(i)}{W^{cbi}_{\tau}}\!\right)^{\!-\Theta^{l}_{\tau}}\!\! L^{cbi}_{\tau}, {\textrm{and }} L^{kb}_{\tau}(i) =\!\left(\!\frac{W^{kb}_{\tau}(i)}{W^{kb}_{\tau}}\!\right)^{\!-\Theta^{l}_{\tau}}\!\! L^{kb}_{\tau}, \mathrm{for} \tau=0,1,...,\infty.$	(20)

The household's budget constraint incorporates wage setting adjustment costs (the size of which depend on the parameter $\chi^{w}$ and the lagged and steady-state wage inflation rate) and costs in altering the composition of labor supply. The constraint against which the household maximizes its utility is the demand curve it faces for its differentiated labor. This demand curve derives from the first of the intermediate goods producing firm's cost-minimization problems.

3.5 Market Clearing

There are a number of market clearing conditions that must be satisfied in our model. Market clearing in the slow-growing "consumption" goods and fast-growing "capital" goods sectors, given price- and wage-adjustment costs and variable utilization costs implies that

$\displaystyle X^{cbi}_{t}$	$\displaystyle =$	$\displaystyle \int_{0}^{1}\!\!\!E^{cnn}_{t}(i)di +\!\!\int_{0}^{1}\!\!\!E^{r}_{t}(k)dk +\frac{100\cdot\chi^{p}}{2} \left(\Pi^{p,kb}_{t}\!-\!\eta^{p}\Pi^{p,cbi}_{t-1} -\!(1\!-\!\eta^{p})\Pi^{p,cbi}_{\ast}\right)^{2}\!\!P^{cbi}_{t}X^{cbi}_{t}$
$\displaystyle$	$\displaystyle +$	$\displaystyle \frac{100\!\cdot\!\chi^{w}}{2} \left(\Pi^{w,cbi}_{t}\!\!-\!\eta^{w}\Pi^{w,cbi}_{t-1} -\!(1\!-\!\eta^{w})\Pi^{w,cbi}_{\ast}\right)^{2}\!\!W^{cbi}_{t}L^{cbi}_{t} -\!\kappa\left(\frac{U^{cbi}_{t}(k)^{1+\psi}\!-\!1}{1+\psi} \right)\!P^{cbi}_{t}K^{nr,cbi}_{t}$	(21)

and

$\displaystyle X^{kb}_{t}\!$	$\displaystyle =$	$\displaystyle \int_{0}^{1}\!\!\!E^{cd}_{t}(k)dk +\!\!\int_{0}^{1}\!\!\!E^{nr}_{t}(k)dk. +\frac{100\cdot\chi^{p}}{2} \left(\Pi^{p,kb}_{t}\!-\!\eta^{p}\Pi^{p,kb}_{t-1} -\!(1\!-\!\eta^{p})\Pi^{p,kb}_{\ast}\right)^{2}\!\!P^{kb}_{t}X^{kb}_{t}$
$\displaystyle$	$\displaystyle +$	$\displaystyle \frac{100\!\cdot\!\chi^{w}}{2} \left(\Pi^{w,kb}_{t}\!\!-\!\eta^{w}\Pi^{w,kb}_{t-1} -\!(1\!-\!\eta^{w})\Pi^{w,kb}_{\ast}\right)^{2}\!\!W^{kb}_{t}L^{kb}_{t} -\!\kappa\left(\frac{U^{kb}_{t}(k)^{1+\psi}\!-\!1}{1+\psi} \right)\!P^{kb}_{t}K^{nr,kb}_{t}\!\!\!\!\!\!\!\!.$	(22)

The market clearing conditions for the labor and non-residential capital supplied and demanded in sector

are given by

$\displaystyle L^{s}_{t}(i)=\!\!\int_{0}^{1}\!\!\!L^{s}_{t}(i,j)dj {\mathrm{and}} \!\!\int_{0}^{1}\!\!U(k)^{s}_{t}K^{nr,s}_{t}(k)dk=\!\!\int_{0}^{1}\!\!\!K^{u,nr,s}_{t}(j)dj \forall i \in [0,1] {\mathrm{and for}} s=cbi,kb.$

(23)

The market clearing conditions for consumer durables and residential capital are

$\displaystyle \int_{0}^{1}\!\!K^{cd}_{t}(k)dk=\!\!\int_{0}^{1}\!\!\!K^{cd}_{t}(i)di {\mathrm{and}} \!\! \int_{0}^{1}\!\!K^{r}_{t}(k)dk=\!\!\int_{0}^{1}\!\!\!K^{r}_{t}(i)di .$

(24)

3.6 Identities

The model also consists of the following identities:

$\displaystyle W^{s}_{t}(i)= \Pi^{w,s}_{t}(i)W^{s}_{t-1}(i)$	$\displaystyle {\mathrm{and}}\!\!\!\!$	$\displaystyle W^{s}_{t}= \Pi^{w,s}_{t}W^{s}_{t-1} \forall i \in [0,1] {\mathrm{and for}} s=cbi,kb, {\mathrm{and}}$	(25)
$\displaystyle P^{s}_{t}(j)= \Pi^{p,s}_{t}(j)P^{s}_{t-1}(j)$	$\displaystyle {\mathrm{and}}\!\!\!\!$	$\displaystyle P^{s}_{t}= \Pi^{p,s}_{t}P^{s}_{t-1} \forall j \in [0,1] {\mathrm{and for}} s=cbi,kb.$	(26)

3.7 Real GDP Growth and GDP Price Inflation

Real GDP growth and the inflation rate of the GDP deflator are two variables of interest to policy-makers that do not automatically appear in our model. Real GDP growth is constructed as the divisia (share-weighted) aggregate of output in the economy, that is:

$\displaystyle H^{gdp}_{t}\!= \!\left(\left(\frac{X^{cbi}_{t}} {X^{cbi}_{t-1}}\right)^{P^{cbi}_{\ast}\widetilde{X}^{cbi}_{\ast}} \!\!\!\!\left(\frac{X^{kb}_{t}} {X^{kb}_{t-1}}\right)^{P^{kb}_{\ast}\widetilde{X}^{kb}_{\ast}} \!\!\!\!\left(\frac{\Gamma^{x,cbi}_{t}\!\!\cdot\widetilde{X}^{gf}_{t}} {\widetilde{X}^{gf}_{t-1}}\right)^{P^{cbi}_{\ast}\widetilde{X}^{gf}_{\ast}} \right)^{\frac{1} {P^{cbi}_{\ast}\widetilde{X}^{cbi}_{\ast} +P^{kb}_{\ast}\widetilde{X}^{kb}_{\ast} +P^{cbi}_{\ast}\widetilde{X}^{gf}_{\ast}}} \!\!\!\! \!\!\!\!.$

(27)

To a first-order approximation, this definition of GDP growth is equivalent to how it is defined in the U.S. NIPA. The variable $\widetilde{X}^{gf}_{t}$ in equation (28) represents stationary un-modelled output (that is, GDP other that $E^{cnn}_{t}$ , $E^{cd}_{t}$ , $E^{r}_{t}$ , and $E^{nr}_{t}$ ). Stationary un-modeled output is exogenous and is assumed to follow the process:

$\displaystyle \ln \widetilde{X}^{gf}_{t}-\ln \widetilde{X}^{gf}_{\ast} =\rho^{x,gf}\left(\ln \widetilde{X}^{gf}_{t}-\ln \widetilde{X}^{gf}_{\ast} \right) +\epsilon^{x,gf}.$

The inflation rate of the GDP deflator, represented by $\Pi^{p,gdp}_{t}$ , is defined implicitly by:

$\displaystyle \Pi^{p,gdp}_{t} H^{gdp}_{t} =\frac{P^{gdp}_{t}X^{gdp}_{t}}{P^{gdp}_{t-1}X^{gdp}_{t-1}} =\frac{P^{cbi}_{t}X^{cbi}_{t}+P^{kb}_{t}X^{kb}_{t}+P^{cbi}_{t}X^{gf}_{t}} {P^{cbi}_{t-1}X^{cbi}_{t-1}+P^{kb}_{t-1}X^{kb}_{t-1}+P^{cbi}_{t-1}X^{gf}_{t-1}}.$

(28)

3.8 Monetary Authority

The central bank sets monetary policy in accordance with a Taylor-type interest-rate feedback rule. Policymakers smoothly adjust the actual interest rate $R_{t}$ to its target level $\bar{R}_{t}$

$\displaystyle R_{t} = \left( R_{t-1}\right)^{\phi^{r}} \left( \bar{R}_{t} \right)^{1-\phi^{r}} \exp \left[ \epsilon^{r}_{t} \right],$

(29)

where the parameter $\phi^{r}$ reflects the degree of interest rate smoothing, while $\epsilon_{t}^{r}$ represents a monetary policy shock. The central bank's target nominal interest rate $\bar{R}_{t}$ is given by:

$\displaystyle \bar{R}_{t} = \left( \Pi^{p,gdp}_{t}/\Pi^{p,gdp}_{\ast} \right)^{\phi^{\pi,gdp}} \left( \Delta\Pi^{p,gdp}_{t} \right)^{\phi^{\Delta\pi,gdp}} \left( H^{gdp}_{t}/H^{gdp}_{\ast} \right)^{\phi^{h,gdp}} \left( \Delta H^{gdp}_{t}\right)^{\phi^{\Delta h,gdp}} R_{\ast}.$

(30)

where $R_{\ast}$ denotes the economy's steady-state nominal interest rate (which is equal to $(1/\beta)\Pi^{p,c}_{\ast}\Gamma^{z,{m}}_{\ast} (\Gamma^{z,kb}_{\ast})^{\alpha}(\Gamma^{z,cbi}_{\ast})^{1-\alpha}$ ) and $\phi^{\pi,gdp}$ , $\phi^{\Delta \pi,gdp}$ , $\phi^{h,gdp}$ , and $\phi^{\Delta h,gdp}$ denote the weights in the feedback rule.

4 Equilibrium

Before characterizing equilibrium in this model, we define three additional variables, the price of installed non-residential capital $Q^{nr}_{t}(k)$ , the price of installed consumer durables capital $Q^{cd}_{t}(k)$ , and the price of installed residential capital $Q^{r}_{t}(k)$ . These variables are the lagrange multiplier on the capital evolution equations that would be implied by the th capital owner's profit-maximization problems (described in equations 17, 18, and 19).

Equilibrium in our model is an allocation:

		$\displaystyle \left\{H^{gdp}_{t}\!, X^{cbi}_{t}\!,\{X^{cbi}_{t}(j)\}_{j=0}^{1}, X^{kb}_{t}\!,\{X^{kb}_{t}(j)\}_{j=0}^{1}, \{E^{nr}_{t}(k)\}_{k=0}^{1},\{E^{r}_{t}(k)\}_{k=0}^{1},\{E^{cd}_{t}(k)\}_{k=0}^{1}, \right.$
		$\displaystyle \{E^{cnn}_{t}(i)\}_{i=0}^{1}, \{\Lambda^{r}_{t}(i)\}_{i=0}^{1}, \{\Lambda^{cd}_{t}(i)\}_{i=0}^{1},\{\Lambda^{cnn}_{t}(i)\}_{i=0}^{1}, \{\Lambda^{l,cbi}_{t}(i)\}_{i=0}^{1},\{\Lambda^{l,kb}_{t}(i)\}_{i=0}^{1},$
		$\displaystyle \{L^{cbi}_{t}(i)\}_{i=0}^{1},\{\{L^{cbi}_{t}(i,j)\}_{i=0}^{1}\}_{j=0}^{1}, \{L^{kb}_{t}(i)\}_{i=0}^{1},\{\{L^{kb}_{t}(i,j)\}_{i=0}^{1}\}_{j=0}^{1}, \{U^{cbi}_{t}(j)\}_{j=0}^{1},\{U^{kb}_{t}(j)\}_{j=0}^{1},$
		$\displaystyle \{K^{u,nr,cbi}_{t}(j)\}_{j=0}^{1},\{K^{u,nr,kb}_{t}(j)\}_{j=0}^{1}, \{K^{nr,cbi}_{t}(k)\}_{k=0}^{1},\{K^{nr,kb}_{t}(k)\}_{k=0}^{1},\{K^{nr}_{t+1}(k)\}_{k=0}^{1},$
		$\displaystyle \left. \{K^{r}_{t+1}(k)\}_{k=0}^{1},\{K^{r}_{t+1}(i)\}_{i=0}^{1}, \{K^{cd}_{t+1}(k)\}_{k=0}^{1},\{K^{cd}_{t+1}(i)\}_{i=0}^{1} \right\}_{t=0}^{\infty}$

and a sequence of values

		$\displaystyle \left\{\Pi^{p,gdp}_{t}\!, \Pi^{p,cbi}_{t}\!,\{\Pi^{p,cbi}_{t}(j)\}_{j=0}^{1}, \Pi^{p,kb}_{t}\!,\{\Pi^{p,kb}_{t}(j)\}_{j=0}^{1}, \Pi^{w,cbi}_{t}\!,\{\Pi^{w,cbi}_{t}(i)\}_{j=0}^{1}, \Pi^{w,kb}_{t}\!,\{\Pi^{w,kb}_{t}(i)\}_{j=0}^{1}, \right.$
		$\displaystyle \frac{P^{kb}_{t}}{P^{cbi}_{t}},\left\{\frac{P^{kb}_{t}(j)}{P^{cbi}_{t}}\right\}_{j=0}^{1}\!\!, \frac{W^{cbi}_{t}}{P^{cbi}_{t}},\left\{\frac{W^{cbi}_{t}(i)}{P^{cbi}_{t}}\right\}_{i=0}^{1}\!\!, \frac{W^{kb}_{t}}{P^{cbi}_{t}},\left\{\frac{W^{kb}_{t}(i)}{P^{cbi}_{t}}\right\}_{i=0}^{1}\!\!, \frac{R^{nr,cbi}_{t}}{P^{cbi}_{t}},\frac{R^{nr,kb}_{t}}{P^{cbi}_{t}}, \frac{R^{nr}_{t}}{P^{cbi}_{t}},$
		$\displaystyle \left. \frac{R^{r}_{t}}{P^{cbi}_{t}}, \frac{R^{cd}_{t}}{P^{cbi}_{t}}, \left\{\frac{MC^{cbi}_{t}(j)}{P^{cbi}_{t}}\right\}_{j=0}^{1}\!\!, \left\{\frac{MC^{kb}_{t}(j)}{P^{cbi}_{t}}\right\}_{j=0}^{1}\!\!, \left\{\frac{Q^{nr}_{t}(k)}{P^{cbi}_{t}}\right\}_{k=0}^{1}\!\!, \left\{\frac{Q^{r}_{t}(k)}{P^{cbi}_{t}}\right\}_{k=0}^{1}\!\!, \left\{\frac{Q^{cd}_{t}(k)}{P^{cbi}_{t}}\right\}_{k=0}^{1}\!\!,R_{t} \right\}_{t=0}^{\infty}$

that satisfy the following conditions:

The model's two representative final-good producing firms solve (13) for and ;
All intermediate-good producers $j \in [0,1]$ solve (14), (15), and (16) for and ;
All capital owners $k \in [0,1]$ solve (17), (18), and (19);
All households $i \in [0,1]$ solve (20);
The two final goods markets clear as in (22) and (23);
All intermediate goods markets clear (by construction);
The labor and non-residential capital markets clear as in (24);
The consumer durable and residential capital rental markets clear as in (25);
The identities given in (26) hold, but are modified slightly to
$\displaystyle \frac{W^{s}_{t}(i)}{P^{c}_{t}} =\frac{\Pi^{w,s}_{t}(i)}{\Pi^{p,c}_{t}}\cdot \frac{W^{s}_{t-1}(i)}{P^{c}_{t-1}} \mathrm{and} \frac{W^{s}_{t}}{P^{c}_{t}} =\frac{\Pi^{w,s}_{t}}{\Pi^{p,c}_{t}}\cdot \frac{W^{s}_{t-1}}{P^{c}_{t-1}} {\mathrm{for all}} i \in [0,1] {\mathrm{and for}} s=cbi,kb;$
The identities given in (27) hold, although are modified slightly to
$\displaystyle \frac{P^{s}_{t}(j)}{P^{c}_{t}} =\frac{\Pi^{p,s}_{t}(j)}{\Pi^{p,c}_{t}}\cdot\frac{P^{s}_{t-1}(j)}{P^{c}_{t-1}} \mathrm{and} \frac{P^{k}_{t}}{P^{c}_{t}} =\frac{\Pi^{p,k}_{t}}{\Pi^{p,c}_{t}}\cdot\frac{P^{k}_{t-1}}{P^{c}_{t-1}} {\mathrm{for all}} j \in [0,1] {\mathrm{and for}} s=cbi,kb;$
The monetary authority follows (30) and (31), where the growth rate of GDP and aggregate prices are defined by equations (28) and (29).

In solving these problems agents take as given the initial values of $K^{nr}_{0}$ , $K^{r}_{0}$ , $K^{cd}_{0}$ , and $R_{-1}$ , and the sequence of exogenous variables

$\displaystyle \left\{ \Gamma^{z,kb}_{t}\!,\Gamma^{z,m}_{t}\!, \Theta^{x,cbi}_{t}\!,\Theta^{x,kb}_{t}\!, \Theta^{l}_{t}\!, A^{nr}_{t}\!,A^{r}_{t},A^{cd}_{t}\!,\Xi^{cnn}_{t}\!,\Xi^{cd}_{t}\!,\Xi^{r}_{t},\Xi^{l}_{t}, E^{gf}_{t} \right\}_{t=0}^{\infty}$

implied by the sequence of shocks

$\displaystyle \left\{ \epsilon^{z,kb}_{t}\!,\epsilon^{z,m}_{t}\!, \epsilon^{\theta,x,cbi}_{t}\!,\epsilon^{\theta,x,kb}_{t}\!, \epsilon^{\theta,l}_{t}\!, \epsilon^{a,nr}_{t}\!,\epsilon^{a,r}_{t}\!,\epsilon^{a,cd}_{t}\!, \epsilon^{\xi,cnn}_{t}\!,\epsilon^{\xi,cd}_{t}\!,\epsilon^{\xi,r}_{t}\!,\epsilon^{\xi,l}_{t}\!, \epsilon^{gf}_{t}\!, \epsilon^{r}_{t} \right\}_{t=0}^{\infty}.$

We estimate the log-linearized, symmetric and stationary version of the model described above. Equilibrium in the symmetric and non-stationary version of the model is defined in appendix A of the paper, and equilibrium in the symmetric and stationary version of the model is defined in appendix B. The log-linearization of our model equations is performed symbolically by the software that we use to parse the model into its estimable form; we therefore do not included the log-linearized equations in this documentation. The steady-state solution to the symmetric and stationary version of the model is, however, an input into the model's estimation and so is presented in appendix C.

5 Data

The model is estimated using 11 data series listed below. Except where noted, the series are from the U.S. National Income and Product Accounts (U.S. NIPA) published by the Bureau of Economic Analysis.

Nominal gross domestic product.
Nominal consumption expenditures on non-durables and non-housing services.
Nominal consumption expenditure on durables.
Nominal residential investment expenditure.
Nominal business investment expenditure (which equals nominal gross private domestic investment minus nominal residential investment and thus includes inventory investment).
The rate of GDP price inflation.
The rate of inflation for prices of consumer non-durables and non-housing services (which represents inflation in the slow-growing "consumption" goods sector).
The rate of inflation for prices of consumer durables (which represents inflation in the fast-growing "capital" goods sector).
Hours, which equals hours of all persons in the non-farm business sector (from the Bureau of Labor Statistics) scaled up by the ratio of nominal production in our model to nominal non-farm business sector output.⁴
Wage inflation, which equals compensation per hour in the non-farm business sector (from the Bureau of Labor Statistics).
The federal funds rate (from the Federal Reserve Board).

Some of the series are not those used in previous research with the Federal Reserve's FRB/US model. However, price and nominal quantity indices for each of the model's expenditure and output variables can be retrieved easily from the U.S. NIPA. The construction of these series are as follows:

Nominal expenditures on consumer non-durable goods and non-housing services ( $P^{ECNN} ECNN$ ) is the sum of nominal personal consumption expenditures on non-durable goods and nominal personal consumption expenditures on services (Table 1.1 of the NIPA) with owner-occupied nonfarm dwellings and tenant-occupied nonfarm dwellings (Table 2.4) subtracted.

Nominal expenditures on consumer durable goods ( $P^{ECD} ECD$ ) is nominal personal consumption expenditures on durable goods (Table 1.1).

Nominal expenditures on residential investment ( $P^{ER} ER$ ) is nominal gross private domestic residential investment (Table 1.1).

Nominal expenditures on non-residential investment ( $P^{ENR} ENR$ ) is the sum of nominal gross private domestic non-residential investment and the change in nominal private inventories (Table 1.1).

Nominal production in the slow-growing part of the business and institutions sector ( $P^{XCBI} XCBI$ ) is the sum of nominal expenditures on consumer non-durable goods and non-housing services ( $P^{ECNN} ECNN$ ) and nominal expenditures on residential investment ( $P^{ER} ER$ ).

Nominal production in the fast-growing part of the business sector ( $P^{XKB} XKB$ ) is the sum of nominal expenditures on consumer durable goods ( $P^{ECD} ECD$ ) and nominal expenditures on non-residential investment ( $P^{ENR} ENR$ ).

In summary,

$\displaystyle P^{XCBI} XCBI$	$\displaystyle =$	$\displaystyle P^{ECNN} ECNN + P^{ER} ER, \mathrm{and} ,$	(31)
$\displaystyle P^{XKB} XKB$	$\displaystyle =$	$\displaystyle P^{ECD} ECD + P^{ENR} ENR.$	(32)

In the NIPA, there is a different price index for every expenditure and output variable. Our theoretical model has only one price per output good.⁵

To bring our data in line with our model the series must be modified slightly. Although the three price indices $P^{XCBI}$ , $P^{ECNN}$ , and $P^{ER}$ are not identical they do not display any dramatic relative price swings. Similarly, the indicies $P^{XKB}$ , $P^{ECD}$ , and $P^{ENR}$ , while not identical, do not exhibit any large swings. Consequently, we make the following modification to the data. We re-write equations (32) and (33) as

$\displaystyle P^{ECNN}\left[\frac{P^{XCBI}XCBI}{P^{ECNN}}\right]$	$\displaystyle =$	$\displaystyle P^{ECNN}ECNN + P^{ECNN}\left[\frac{P^{ER}ER}{P^{ECNN}}\right], \mathrm{and}$
$\displaystyle P^{ECD}\left[\frac{P^{XKB}XKB}{P^{ECD}}\right]$	$\displaystyle =$	$\displaystyle P^{ECD}ECD + P^{ECD}\left[\frac{P^{ENR}ENR}{P^{ECD}}\right] ,$

$\displaystyle \mathrm{and let } XCBI^{adj}\!\!\!\!$	$\displaystyle =$	$\displaystyle \frac{P^{XCBI}XCBI}{P^{ECNN}}, ER^{adj}=\frac{P^{ER}ER}{P^{ECNN}}, \mathrm{and}$
$\displaystyle XKB^{adj}\!\!\!\!$	$\displaystyle =$	$\displaystyle \frac{P^{XKB}XKB}{P^{ECD}}, ENR^{adj}=\frac{P^{ENR}ENR}{P^{ECD}}.$

Equations (32) and (33) can then be written as:

$\displaystyle P^{ECNN} XCBI^{adj} \!\!$	$\displaystyle =$	$\displaystyle P^{ECNN} ECNN + P^{ECNN} ER^{adj} , \mathrm{and}$	(33)
$\displaystyle P^{ECD} XKB^{adj} \!\!$	$\displaystyle =$	$\displaystyle P^{ECD} ECD + P^{ECD} ENR^{adj}.$	(34)

The above renormalization of the data implies that the series for real expenditures on residential investment ( $ER^{adj}$ ) and for expenditures on non-residential investment ( $ENR^{adj}$ ) are no longer the series published in the NIPA. The qualitative patterns in the data are unaltered by these normalizations, although the precise values of the series do change thereby resulting in a slight bias in the adjusted series. This divergence can be handled, if and when a focus on the behavior of these real series is desired, by the standard assumption that the data that we are using for residential investment and for nonresidential investment is measured with error.

Our decision for choosing the price index of consumer non-durables and non-housing services as our price index for the slow-growing sector and the price index of consumer durables as our price index for the fast-growing sector is that the PCE price deflator is ultimately the price index that we are most interested in from a policy perspective. Consequently, it is the components of this index that we wish to model.

6 The Empirical Model

6.1 Estimation

We take a log-linear approximation to our model, cast this resulting dynamical system in its state space representation for the set of (in our case 11) observable variables, use the Kalman filter to evaluate the likelihood of the observed variables, and form the posterior distribution of the parameters of interest by combining the likelihood function with a joint density characterizing some prior beliefs. Since we do not have a closed-form solution of the posterior, we rely on Markov-Chain Monte Carlo (MCMC) methods.

We add measurement errors processes, denoted $\eta_{t}$ , for all of the observed series used in estimation except the nominal interest rate and the aggregate hours series. The measurement errors explain less than 5 percent of the observed series.⁶

6.2 Model Parameters

The model' calibrated parameters are presented in Table 3, while the estimated parameters are presented in Tables 4 and 5. We based out decision on which parameters to calibrate and which to estimate on how informative the data were likely to be on the parameter, as well as identification and overparameterization issues. The first three columns of Table 4 and 5 outline our assumptions about the prior distributions of the estimated parameters, the remaining columns describe the parameters' posterior distributions, which we now proceed to discuss.

We consider first the parameters related to household-spending decisions. The parameters related to habit-persistence are uniformly large. For nondurables and services excluding housing, the habit parameter is about 0.8, close to the value in found by Fuhrer [2000]. For consumer durables capital the habit parameter is somewhat smaller, while for residential capital it is smaller still. Since most DSGE models do not consider utility functions with this level of disaggregation, there is little consensus on these values. In addition, simulations indicate that habit and adjustment cost parameters--both present in our model--are closely related, further complicating any comparison. Indeed, we estimate investment adjustment costs to be very significant for residential investment but of modest importance for consumer durables.⁷ Nonetheless, habit-persistence and investment adjustment costs are important in generating "hump-shaped" responses of these series to monetary policy shocks.⁸ The estimated value of the remaining preference parameter, the inverse of the labor supply elasticity, is, at a bit over one, a little higher than suggested by the balance of microeconomic evidence (see Abowd and Card [1989]).

With regard to adjustment cost parameters for non-residential investment, we estimate significant costs to the change in investment flows, which imply an elasticity of investment to marginal q of about 1/3. We also find an important role for the sectoral adjustment costs to labor: In our multisector setup, shocks to productivity or preferences in one sector of the economy result in strong shifts of labor towards that sector, which conflicts with the high degree of sectoral co-movement in the data. The adjustment costs to the sectoral mix of labor input ameliorate this potential problem, as in Boldrin et al. [2001].

Finally, adjustment costs to prices and wages are both estimated to be important, although prices appear "stickier" than wages. Our quadratic costs of price and wage adjustment can be translated into frequencies of adjustment consistent with the Calvo model; these are about six quarters for prices and about one quarter for wages. However, these estimates are very sensitive to the specifics of our model and would be altered by reasonable assumptions regarding "real rigidities" such as firm-specific factors or "kinked" demand curves. We find only a modest role for lagged inflation in our adjustment cost specification (around 1/3), equivalent to modest indexation to lagged inflation in other sticky-price specifications. This differs from some other estimates, perhaps because of the focus on a more recent post-1983 sample (similar to results in Kiley [2007] and Laforte [2007]).

6.3 Variance Decompositions

Tables 6 to 11 present forecast error variance decompositions at various (quarterly) horizons at the posterior mode of the parameter estimates for key variables and shocks. We run through the key results here.

Volatility in aggregate GDP growth is, in the near horizon, accounted for predominantly by economy-wide technology shocks, non-residential investment efficiency shocks, and exogenous spending shocks. In the far horizon, volatility is accounted for primarily by capital-specifi