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w o r k i n g p a p e r 0 0 0 7 R Home Production Meets Time-to-build by Paul Gomme, Finn Kydland, and Peter Rupert FEDERAL RESERVE BANK OF CLEVELAND Working Paper 00-07 R Home Production Meets Time-to-Build by Paul Gomme, Finn Kydland and Peter Rupert Paul Gomme and Peter Rupert are at the Federal Reserve Bank of Cleveland. Finn Kydland is at Carnegie-Mellon University, Pittsburgh, Pennsylvania. Working papers of the Federal Reserve Bank of Cleveland are preliminary materials circulated to stimulate discussion and critical comment on research in progress. They may not have been subject to the formal editorial review accorded official Federal Reserve Bank of Cleveland publications. The views stated herein are those of the authors and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System Working papers are now available electronically through the Cleveland Fed's home page on the World Wide Web: http://www.clev.frb.org. June 2000 Home Production Meets Time-to-Build by Paul Gomme, Finn Kydland and Peter Rupert An innovation in this paper is to introduce a time-to-build technology for the production of market capital into a model with home production. Our main finding is that the two anomalies that have plagued all household production models—the positive correlation between business and household investment, and household investment leading business investment over the business cycle—are resolved when time-to-build is added. Home Production Meets Time-to-build Paul Gomme, Finn Kydland and Peter Rupert December 1999 Abstract: An innovation in this paper is to introduce a time-to-build technology for the production of market capital into a model with home production. Our main finding is that the two anomalies that have plagued all household production models—the positive correlation between business and household investment, and household investment leading business investment over the business cycle—are resolved when time-to-build is added. Keywords: home production, time-to-build, business cycles, investment Affiliations: Gomme, Federal Reserve Bank of Cleveland and CREFE/UQAM; Kydland, Carnegie-Mellon University and Federal Reserve Bank of Cleveland; Rupert, Federal Reserve Bank of Cleveland. The views stated herein are those of the author and are not necessarily those of the Federal Reserve Bank of Cleveland or of the Board of Governors of the Federal Reserve System. 1 Introduction The household sector is large. For example, Greenwood and Hercowitz (1991) report that household capital actually exceeds market capital. Further, Benhabib, Rogerson, and Wright (1991) estimate that the output of the household sector may be as much as half that of the market sector, and that labor hours in the home sector are almost as great as in the market sector. Additionally, home investment (purchases of consumer durables and residential housing) exceeds that of market investment (purchases of nonresidential structures, equipment and inventories). Consequently, as suggested by Benhabib et al. (1991) and Greenwood and Hercowitz (1991), it seems plausible that accounting for home production and its interaction with market production may be important for understanding many macroeconomic phenomena. Yet, it seems fair to say that household production has not really taken hold in the profession despite the many papers that have pursued that idea and refined the necessary measurements. A reason may be that, in light of household-production theory, there are too many anomalies in the data. This paper attempts to settle the issue of the importance of household production for understanding the business cycle. One key anomaly is that there is a positive correlation between market and home investment in the U.S. data, while the basic home production models of Benhabib et al. (1991) and Greenwood and Hercowitz (1991) predict a negative comovement. Greenwood and Hercowitz (1991) obtain a positive correlation by assuming: (a) that the household production function exhibits strong complementarity between home capital and home labor; (b) that preferences allow a high degree of substitutability between market and home consumption goods; and (c) that the shocks to market and home productivity are perfectly correlated.1 1 As shown in Greenwood, Rogerson, and Wright (1995), perfect correlation in the shocks is not necessary to 1 More importantly, the U.S. data reveal that household investment leads the cycle by about one quarter while market investment lags by about a quarter. As shown below, both the Benhabib et al. (1991) and Greenwood and Hercowitz (1991) models predict exactly the opposite pattern: that home investment lags the cycle and that market investment leads. This phase shift pattern is such an interesting and striking feature of the data that home production models that cannot replicate this phase shift must be considered a failure. The principal innovation of this paper is to introduce a time-to-build technology for the production of market capital into an otherwise standard household production model. With time-to-build, initiating a market investment project in the current period yields useful capital several periods hence. Furthermore, starting a project today implies a commitment of resources to this project not only in the current period, but in all periods leading to project completion. By way of contrast, home production is subject to a standard one period time-to-build; that is, investment today yields home capital in the next period. In the basic home production model, an improvement in market productivity leads, on impact, to a sharp rise in market investment, and a fall in home investment. This pattern arises because market investment allows greater future market output. As a result, it is only in subsequent periods that home investment rises. With time-to-build, only a fraction of the total resources for market investment are needed in the impact period. That is, the impact response on market investment is spread out over the length of time it takes to complete a project. Loosely speaking, time-tobuild makes it prohibitively costly to quickly bring on line new units of market capital. At the same time, time-to-build reduces the cost (in terms of consumption and leisure) of increasing generate the positive correlation in market and home investment, although the shocks must be very highly correlated. 2 the market capital stock at longer horizons. These two effects induce a positive comovement between market and home investment. The same mechanisms also operate in regard to the leadlag patterns of the two investment series. So, it appears that time-to-build is an essential feature of reasonably calibrated household production models to match the cyclical properties of market and home investment. 2 The Economic Environment 2.1 Households The representative household has preferences over market consumption, c Mt , home consumption, cHt , market hours, hMt , and home hours, hHt , summarized by ∞ E0 ∑ βt U cMt cHt hMt hHt 0 β 1 (1) t 0 The momentary utility function has the following form: U c M cH h M h H ω lnC cM cH C cM cH ω 1 ω ln 1 hM hH 1 h M hH 1 γ 1 ω 1 γ if γ 1 (2) if 0 γ 1 or γ 1 1 where the consumption aggregator is ψ 1 ψ C cM cH if ξ 0 cM cH ξ ψcM 1 ξ 1 ξ ψ cH (3) if ξ 0 or 0 ξ 1 Households face a number of constraints. First, its budget constraint is cMt xMt xHt 1 τK rt kMt 1 τH wt hMt δM τK kMt τt (4) Here, kMt is the household’s stock of market capital, rt is the rental price of capital, wt is the real wage rate, and xMt and xHt are investment in market and home capital, respectively. Capital income is taxed at the rate τK while labor income is taxed at the rate τH . Notice that in (4) the tax rate τK 3 applies to gross capital income. The term δM τK kMt captures the depreciation allowance built into the U.S. tax code. Finally, τt is a lump-sum transfer from the government. Second, as in Kydland and Prescott (1982), capital projects are subject to a J-period timeto-build technology constraint. Specifically, starting a project at date t requires investment of resources at dates t, t 1, ... t J 1, with the capital finally being ready for use at date t J. A project j periods from completion requires a fraction φ j of the total resources required for that project. Let s jt be the number of projects which are j periods from completion at date t. Then, total market investment is J ∑ φ j s jt xMt (5) j 1 Further, the project commitments evolve according to s j 1 t j 2 J s jt 1 (6) That is, a project which is j periods from completion at date t will be j 1 periods from completion in the next period. Third, the household’s capital stocks evolve according to kMt 1 kHt 1 δM kMt 1 1 δH kHt s1t and (7) xHt (8) where δM and δH are the depreciation rates of market and home capital, respectively. Recall that s1t represents the number of projects which are one period from completion as of the beginning of period t. Finally, home production is described by cHt H kHt hHt ; zHt 4 (9) The home production function has the form η 1 η if ζ 0 e zH k H h H H kh hH ; zH ζ ezH ηkH ζ 1 ζ η hH 1 (10) if ζ 0 or 0 ζ 1. The home productivity shock evolves as zH t ρH zHt 1 εHt εHt N 0 σ2H (11) 2.2 Firms Goods producing firms act competitively and seek to maximize profits, F KMt HMt ; zMt rt KMt wt HMt (12) The production function is Cobb-Douglas, α 1 F KM HM ; zM ezM KM HM α (13) and the market productivity shock evolves according to zMt 1 ρM zMt εMt εMt N 0 σ2M (14) 2.3 Government In this economy, the government raises revenue via labor and capital taxes, lump-sum rebating the proceeds to households: τt τK rt KMt τH wt HMt δM τK KMt (15) As discussed in Greenwood et al. (1995), the reason for including taxes is that they have important implications for the calibration procedure; this issue is discussed in more detail in Section 3. 5 3 Calibration The model is calibrated using the procedure set out by Kydland and Prescott (1982). In particular, as many parameters as possible are set in advance based on either a priori information concerning their magnitude, or so as to match certain long run averages observed in the postwar U.S. economy. The set of parameters which need to be assigned values are summarized in Table 2. 2 Except for the parameters governing time-to-build, the values are either the same as in Model 1 of Greenwood et al. (1995) or calibrated to match the same long run averages. To start, a model period corresponds 1 to one quarter. Setting the discount factor, β to 1 06 4 thus generates an annual real interest rate of 6 percent in steady state. The coefficient of relative risk aversion, γ, is set to one which implies logarithmic preferences. The home production function and consumption aggregator are assumed to be Cobb-Douglas; thus, ξ ζ 0. Evidence on U.S. Solow residuals motivates setting ρM 0 95 and σM 0 00763; see Prescott (1986). Absent hard evidence to guide the choice of the stochastic process describing the home technology shock, it is assumed that the home shock process is the same as that of the market shock, i.e., ρH 0 95 and σH 0 00763. The correlation between the innovations to the market and home shocks (i.e., between εMt and εHt ) is set to 2/3. In the home production literature to date, the value of this correlation has important implications for the cyclical behavior of home and market investment. In particular, Greenwood and Hercowitz (1991) require virtually a perfect correlation for their household production model to predict a positive correlation between the two investment series. The parameterization of our baseline model—in particular, the logarithmic preferences and 2 Table 2 summarizes parameter values for the 4-period time-to-build model. For the benchmark home production model, the calibration procedure implies slightly different values for the parameters ω, ψ, η and α. 6 Cobb-Douglas home production function and consumption aggregator—imply enough separability that the home production shock only affects home consumption and aggregated consumption. That is, none of the market variables respond to a home productivity shock. Consequently, apart for home and aggregated consumption, none of the baseline model results would change if the home productivity shock were simply dropped from the model. Following Kydland and Prescott (1982), when time-to-build is in play, it takes four quarters to complete a market investment project, and each period 1/4 of the total resources are used. Thus, J 4 and φ j 14 for j 1 2 3 4. Models without time-to-build correspond to a one period time- to-build. That is, J 1 and φ1 1. The longer gestation time for market investment projects visà-vis home investment projects is motivated by empirical plausibility: construction of residential structures, for example, often take roughly a quarter while nonresidential structures such as office towers take a year or so to build, while factories may take longer. The parameters ω, ψ, α, η, δH and δM are chosen such that in steady state: 1. Market hours, hM , is 13 and home hours, hH , is 1 4. These values are consistent with evi- dence from time use surveys. 2. Market capital, kM , is four times market output while home capital, kH , is five times market output. 3. Market investment, xM , is 11.8% of market output while home investment, xH , is 13.5% of market output. The tax rates are set to τK 0 70 and τH 0 25. Along with the restrictions above, these tax rates imply the values for ω, ψ, α, η, δH and δM given in Table 2. At first blush, the tax rate 7 on capital may seem quite high. It is, however, well within the range of effective capital income tax rates reported by Feldstein, Dicks-Mireaux, and Poterba (1983). Further, Greenwood et al. (1995) argue that τK should also incorporate the effects of the cornucopia of regulations faced by business. They also point out that τK is an important parameter for generating a reasonable capital share parameter in the market sector (given the restrictions above, in particular the market capital to market output ratio). Models without home production do not seem to have such a problem (related to income taxation) for they calibrate to a much higher capital-output ratio since market and home capital are lumped together. 4 Findings Since the parameterization of the baseline model is chosen to be consistent with several other papers incorporating home production, it also shares their successes and failures. Although attention will be focused on the cyclical pattern of market and home investment, a fairly comprehensive set of business cycle moments can be found in Tables 3 (for the U.S. economy) and 4 (for the household production-only baseline model). For all tables of business cycle moments, the data has been detrended by taking logarithms and Hodrick-Prescott filtering. For a more complete assessment of the baseline model’s strengths and weaknesses, see Model 1 of Greenwood et al. (1995). One feature, emphasized by Greenwood and Hercowitz (1991) and Greenwood et al. (1995), is the contemporaneous correlation between market and home investment. Table 6 reports that for the U.S. economy, this correlation is 0 41 while the “standard” household production model predicts a value of 0 10. It was this failure of the standard model that led Greenwood and Hercowitz (1991) to make the following assumptions: (1) the market and home shocks are perfectly correlated; (2) 8 a high degree of substitutability in the consumption aggregator (ξ complementarity in the home production function (ζ 2 3); and a high degree of 1 2). These assumptions are problematic. For instance, while there is little direct evidence on the size of the correlation between the market and home shocks, indirect evidence suggests that it is less than perfect. Regulatory changes, by way of example, are unlikely to have the same effect on market and home production. Furthermore, as pointed out by Kydland (1995), it is hard to reconcile any deviation from Cobb-Douglas (for either the home production function or consumption aggregator) with the fact that the price of durable goods relative to nondurables has exhibited a secular decline while the expenditure share of durables has remained fairly constant.3 Benhabib et al. (1991) face similar challenges in regard to their parameter choices.4 Less attention has been placed on the lead-lag patterns of the investment series. In the U.S. data, home investment leads the cycle by one quarter while market investment lags by one quarter. By way of contrast, the baseline model predicts that home investment lags output by a quarter and that market investment is coincident-to-leading. That is, the baseline home production model predicts that investment is out of phase relative to the U.S. data. 4.1 Reintroducing Time-to-build As stated in Section 3, the time-to-build version of the baseline model has a time-to-build for market investment of four quarters, and a standard one quarter time-to-build for home investment. Business cycle moments for this model are contained in Tables 5 and 6. Regarding the business 3 It would be fairly straightforward to add to our model a relative price of durables, qt q0 eq1t . For q1 0, the relative price of durables falls over time. It is well-known that Cobb-Douglas preferences imply constant expenditure shares. Consequently, given the utility function, (2), we would require that both the consumption aggregator and home production function to be of the Cobb-Douglas variety. 4 McGrattan, Rogerson, and Wright (1997) cannot address this issue since they ignore the price data. 9 cycle behavior of the two investment series, two results stand out. First, the correlation between home and market investment matches that observed in the U.S. data. Second, home investment is now coincident-to-leading (the U.S. data displays a lead of one quarter) while market investment is coincident with the cycle (the U.S. data displays a lag of one quarter). In both regards, the timeto-build version of the model more closely conforms with the U.S. data than the baseline model with household production only. Figure 1, which plots the response of output and the two investment series to a one standard deviation innovation to the market shock, clearly demonstrates why time-to-build makes such a difference with respect to the behavior of the investment series.5 For the baseline household production model, the immediate response is for the two investment series to move in opposite directions. The reason for this is the production asymmetry assumed in all home production models: market output can be used to augment the home capital stock, but home output can only be used as home consumption. Consequently, on impact the market capital stock is built up in order to produce more future output which is then used to build up the home capital stock. The initial increase in market investment occurs at the cost of lower home investment. Only in subsequent periods do the two investment series move in tandem. The effect of time-to-build is to mute the impact effect of the shock on market investment by drawing out the response over the four quarters it takes to build market capital. The smaller impact effect results from the fact that initiating a market investment project in the current period requires only 1/4 of the total resources used by the project. As a result, home investment need not take such a big hit in the initial period of the shock. In fact, as seen in Figure 1, the parameterization for 5 Responses to the home shock are not presented because, as mentioned in Section 3, the baseline parameterization implies that these variables do not respond to the home shock. 10 the baseline time-to-build model implies that home investment rises on impact, with diminishing effects in the three subsequent quarters as subsequent market investment projects are initiated. Intuitively, time-to-build reduces the cost, in terms of consumption and leisure, of market investment, thereby permitting greater home investment. 4.2 Other Home Production Models Meet Time-to-build This section anaylzes how incorporating time-to-build affects the results of the models discussed in the Introduction, namely, Benhabib et al. (1991), McGrattan et al. (1997) and Greenwood and Hercowitz (1991). The business cycle properties of market and home investment for each of these models, along with the U.S. data and the baseline model, are summarized in Table 6. The Benhabib et al. (1991) specification corresponds to ξ 23 (increased substitutability be- tween market and home consumption) with all other parameters calibrated as in Section 3. It should not be too surprising that without time-to-build, this model shares many of the same deficiencies as the baseline home production model. In particular, the investment series are out of phase relative to the data, and the two investment series are strongly negatively correlated. Furthermore, the volatility of the individual investment series is grossly counterfactual. Adding time-to-build to the Benhabib et al. (1991) specification has the following effects: (1) it lessens the negative correlation between home and market investment; (2) it reduces the volatility of both these series; and (3) home investment is now coincident with the cycle while market investment still leads. While not as successful as the baseline model, time-to-build nonetheless improves the coherence between the Benhabib et al. (1991) model and the U.S. data. McGrattan et al. (1997) is more-or-less an estimated version of Benhabib et al. (1991). In this 11 case, ξ 0 4 (still more substitutability between market and home consumption than the baseline model), and corr εMt εHt 0 (the innovations to the market and home shocks are uncorrelated). Qualitatively, this model’s performance is quite similar to the previous model—at least in regard to the cyclical behavior of the investment series. As above, adding time-to-build brings the investment volatilities closer to the data, and lessens the negative correlation between the investment series. As well, market and home investment are both coincident with the cycle. Once more, time-to-build brings the model closer to matching U.S. business cycle experience. Our parameterization of Greenwood and Hercowitz (1991) nearly matches that in Greenwood et al. (1995) (see Model 4). Specifically, ξ 2 3, ζ 1 2 (implying greater complementarity between home capital and labor than is present in the baseline model), and corr ε Mt εHt 0 995 which implies that the market and home shocks (zMt and zHt ) will be nearly perfectly correlated. Relative to the other straight home production models, the Greenwood and Hercowitz (1991) specification performs quite well with regards to the cyclical behavior of home and market investment. For example, the correlation between these series nearly matches the U.S. data, the investment volatilities are close to that in the data, as is their phase pattern (market investment is coincident with the cycle rather than lagging while home investment is coincident-to-leading as opposed to a definite lead in the data). Adding time-to-build worsens the correlation between home and market investment slightly. However, this is the only model that matches the lag in market investment seen in the data. Furthermore, the other home production models tend to predict that market investment is more volatile than home investment whereas the opposite is true in the data. Adding time-to-build to the Greenwood and Hercowitz (1991) model helps on this dimension as well. 12 4.3 Sensitivity Analysis The results of a further set of experiments is summarized in Table 7. In particular, we explore the sensitivity of our results to a higher coefficient of relative risk aversion (γ 2), a higher discount factor (β 0 99) and a lower tax rate on capital income (τK 0 50). Qualitatively, the results for each experiment are quite similar to those seen for the baseline model. In particular, time-to-build has the following effects: (1) it improves the correlation between market and home investment; and (2) it brings the phase pattern of both investment series closer to that seen in the data. These results suggest that the improvements obtained by adding time-to-build to the baseline model are fairly general and are not artifacts of a judicious choice of parameter values. 5 Conclusion The standard home production model makes two counterfactual predictions: market and home investment are negatively correlated while the data exhibits a positive correlation; and that market and home investment are out of phase relative to the data. On this second point, in the data market investment lags the cycle by about one quarter while the basic home production model predicts that market investment is coincident-to-leading, and in the data home investment leads the cycle whereas the model predicts that a lagging pattern. These anomalies are largely resolved when time-to-build is added to the home production model. In particular, adding time-to-build produces a positive correlation between market and home investment (for our parameterization, it actually matches the U.S. data), and brings the phase pattern of the investment series more closely in line with the data. The slight leading pattern of market investment in the baseline home production 13 model is coincident under time-to-build while the lagging behavior of home investment becomes coincident-to-leading. The parameterization is otherwise standard. Specifically, the home production function is Cobb-Douglas as is the aggregator of market and home consumption. Kydland (1995) argued that any deviation from Cobb-Douglas is difficult to reconcile in the fact of key balanced growth facts, in particular the secular decline in the price of durables relative to nondurables and the constant expenditure share of durables. The successes of existing home production models has come at a high cost. For example, Benhabib et al. (1991) emphasis the role of household production in generating procyclical movement in the labor input in different sectors. This and other modest improvements in business cycle behavior is bought at the cost of the anomalies listed above. Further, as shown in Table 6, the volatility of market and home investment is grossly at variance with the data. Much the same can be said of McGrattan et al. (1997) As discussed in Subsection 4.2, adding time-to-build to these models moves each of them to greater conformance with the observed cyclical properties of market and home investment. Although the Greenwood and Hercowitz (1991) model has fewer problems—at least with regards to the cyclical pattern of the investment series—even here time-to-build makes positive contributions. 14 References Benhabib, Jess, Rogerson, Richard, and Wright, Randall. “Homework in Macroeconomics: Household Production and Aggregate Fluctuations.” The Journal of Political Economy 99 (December 1991): 1166. Feldstein, Martin, Dicks-Mireaux, Louis, and Poterba, James. “The Effective Tax Rate and the Pretax Rate of Return.” Journal of Public Economics 21 (July 1983): 129–158. Greenwood, Jeremy, and Hercowitz, Zvi. “The Allocation of Capital and Time over the Business Cycle.” The Journal of Political Economy 99 (December 1991): 1188. Greenwood, Jeremy, Rogerson, Richard, and Wright, Randall. “Household Production in Real Business Cycle Theory.” In Thomas Cooley, editor, “Frontiers of Business Cycle Research,” Princeton, N.J.: Princeton University Press (1995): 157–174. Kydland, Finn. “Business Cycles and Aggregate Labor Market Fluctuations.” In Thomas Cooley, editor, “Frontiers of Business Cycle Research,” Princeton, N.J.: Princeton University Press (1995): 126–156. Kydland, Finn E., and Prescott, Edward C. “Time to Build and Aggregate Fluctuations.” Econometrica 50 (November 1982): 1345–1370. McGrattan, Ellen R., Rogerson, Richard, and Wright, Randall. “An Equilibrium Model of the Business Cycle with Household Production and Fiscal Policy.” International Economic Review 38 (May 1997): 267–90. 16 Prescott, Edward C. “Theory Ahead of Business Cycle Measurement.” Federal Reserve Bank of Minneapolis Quarterly Review 10 (Fall 1986): 9–22. 17 Appendix: Data Reference Table 1: Data sources Series Haver Analytics Mnenomic Gross Domestic Product GDPH Consumption Non-durables CNH Services CSH Housing Services CSRH Durables CDH Private Non-residential Fixed Investment FNH Private Residential Fixed Investment FRH Hours LHTPRIVA All data quarterly and seasonally adjusted at annual rates. Apart from hours, all data is real chained 1992 dollars. Hours corresponds to the Bureau of Labor Statistics aggregate hours, private nonagricultural wage and salary workers. Services Housing Services Consumption Non-durables Market Investment Private Non-resiential Fixed Investment Househole Investment Durables Private Residential Fixed Investment Productivity Gross Domestic Product 15 Hours Table 2: Baseline Parameters Preferences β 0 9855 ω 0 6755 γ 1 0 ψ 0 5583 ξ 0 0 Home Production η 0 3526 ζ 0 0 δH 0 027 Time-to-build J 4 φj 0 25 Market Production α 0 3267 δM 0 0295 Government τH 0 25 τK 0 70 Shocks ρM 0 95 ρH 0 95 σM 0 00763 σ 0 00763 H corr εMt εHt 0 6667 discount factor consumption-leisure weight coefficient of relative risk aversion market-home consumption weight CES parameter in consumption aggregator capital-labor weight CES parameter depreciation rate number of project periods fraction of resources used at stage j capital share depreciation rate tax rate on labor income tax rate on capital income market shock autocorrelation home shock autocorrelation standard deviation of market shock innovation standard deviation of home shock innovation correlation of the innovations 18 xt 4 0 13 0 07 0 46 0 46 0 25 0 00 0 41 0 59 xt 3 0 37 0 28 0 68 0 68 0 03 0 26 0 63 0 59 Table 3: US Economy: Selected Moments Cross Correlation of Real Output With xt 4 xt 3 xt 2 xt 1 xt xt 1 xt 2 0 13 0 37 0 62 0 85 1 00 0 85 0 62 0 34 0 53 0 68 0 79 0 80 0 67 0 48 0 14 0 04 0 29 0 56 0 80 0 87 0 82 0 14 0 04 0 29 0 56 0 80 0 87 0 82 0 47 0 61 0 74 0 80 0 76 0 52 0 24 0 29 0 47 0 68 0 84 0 90 0 76 0 53 0 11 0 11 0 38 0 66 0 88 0 91 0 80 0 49 0 48 0 41 0 26 0 10 0 26 0 48 19 Gross Domestic Product Market Consumption Market Investment Market Investment Household Investment Total Investment Aggregate Hours Productivity Standard Deviation 1 66 0 95 4 73 4 73 6 74 4 95 1 79 0 86 xt 4 0 10 0 26 0 11 0 20 0 11 0 21 0 03 0 01 0 01 0 01 0 57 0 67 0 68 0 17 xt 3 0 27 0 40 0 19 0 32 0 05 0 40 0 20 0 17 0 17 0 17 0 64 0 61 0 64 0 33 20 Output Market Consumption Home Consumption Aggregated Consumption Market Investment Home Investment Total Investment Market Hours Home Hours Total Hours Market Capital Home Capital Total Capital Productivity Table 4: Home Production Only: Selected Moments Standard Cross Correlation of Real Output With Deviation xt 4 xt 3 xt 2 xt 1 xt xt 1 xt 2 1 43 0 10 0 27 0 47 0 73 1 00 0 73 0 47 0 58 0 02 0 15 0 38 0 67 0 97 0 76 0 57 0 88 0 02 0 12 0 24 0 38 0 52 0 37 0 27 0 63 0 00 0 15 0 34 0 58 0 83 0 62 0 46 7 45 0 19 0 32 0 46 0 64 0 78 0 16 0 03 4 58 0 02 0 07 0 19 0 33 0 53 0 96 0 66 4 03 0 15 0 31 0 51 0 75 0 99 0 71 0 43 0 58 0 18 0 33 0 52 0 76 0 99 0 69 0 40 0 29 0 18 0 33 0 52 0 76 0 99 0 69 0 40 0 21 0 18 0 33 0 52 0 76 0 99 0 69 0 40 0 51 0 31 0 16 0 04 0 32 0 65 0 70 0 69 0 37 0 49 0 45 0 37 0 25 0 06 0 27 0 49 0 40 0 48 0 42 0 32 0 16 0 06 0 35 0 54 0 85 0 05 0 22 0 43 0 71 0 99 0 75 0 52 xt 4 0 16 0 24 0 05 0 15 0 26 0 50 0 12 0 10 0 10 0 10 0 63 0 66 0 74 0 20 xt 3 0 25 0 38 0 17 0 29 0 10 0 21 0 18 0 14 0 14 0 14 0 83 0 49 0 47 0 31 21 Table 5: Home Production with Time-to-build: Selected Moments Standard Cross Correlation of Real Output With Deviation xt 4 xt 3 xt 2 xt 1 xt xt 1 xt 2 Output 1 29 0 16 0 25 0 41 0 66 1 00 0 66 0 41 Market Consumption 0 55 0 05 0 16 0 34 0 61 0 97 0 75 0 55 Home Consumption 0 91 0 07 0 14 0 24 0 38 0 56 0 43 0 29 Aggregated Consumption 0 63 0 07 0 17 0 32 0 54 0 83 0 64 0 45 Market Investment 4 79 0 22 0 30 0 44 0 64 0 86 0 63 0 39 Home Investment 3 82 0 13 0 17 0 29 0 49 0 80 0 39 0 18 Total Investment 3 59 0 21 0 29 0 43 0 67 0 99 0 61 0 34 Market Hours 0 52 0 24 0 31 0 45 0 67 0 98 0 58 0 30 Home Hours 0 25 0 24 0 30 0 45 0 67 0 98 0 58 0 29 Total Hours 0 18 0 24 0 31 0 45 0 67 0 98 0 58 0 29 Market Capital 0 51 0 41 0 36 0 28 0 12 0 02 0 16 0 45 Home Capital 0 29 0 42 0 35 0 23 0 05 0 25 0 38 0 43 Total Capital 0 36 0 47 0 44 0 38 0 27 0 09 0 10 0 27 Productivity 0 79 0 11 0 21 0 38 0 64 0 99 0 71 0 48 xt 4 0 46 0 25 0 11 0 21 0 26 0 50 0 11 0 12 0 23 0 29 0 14 0 18 0 39 0 62 0 03 0 03 0 23 0 17 xt 3 0 68 0 03 0 05 0 40 0 10 0 21 0 11 0 18 0 04 0 15 0 14 0 28 0 09 0 31 0 18 0 09 0 55 0 23 0 23 GH TTB 0 37 GH 0 59 MRW TTB 0 82 23 MRW 0 76 BRW TTB 0 87 BRW 0 41 HP TTB 0 10 HP Only 0 41 US Data Table 6: Model Comparisons Cross Correlation of Real Output With xt 4 xt 3 xt 2 xt 1 xt xt 1 xt 2 0 14 0 04 0 29 0 56 0 80 0 87 0 82 0 47 0 61 0 74 0 80 0 76 0 52 0 24 0 19 0 32 0 46 0 64 0 78 0 16 0 03 0 02 0 07 0 19 0 33 0 53 0 96 0 66 0 22 0 30 0 44 0 64 0 86 0 63 0 39 0 13 0 17 0 29 0 49 0 80 0 39 0 18 0 15 0 22 0 30 0 41 0 24 0 09 0 10 0 09 0 10 0 10 0 13 0 13 0 35 0 25 0 33 0 24 0 25 0 30 0 26 0 21 0 12 0 18 0 07 0 00 0 11 0 37 0 17 0 09 0 18 0 26 0 37 0 50 0 46 0 10 0 12 0 10 0 10 0 09 0 10 0 08 0 55 0 40 0 33 0 32 0 39 0 51 0 59 0 36 0 15 0 10 0 07 0 03 0 05 0 28 0 16 0 16 0 13 0 26 0 43 0 64 0 86 0 62 0 38 0 14 0 25 0 39 0 54 0 77 0 44 0 24 0 02 0 12 0 25 0 45 0 71 0 81 0 74 0 15 0 22 0 36 0 57 0 83 0 26 0 10 Corr. Model Standard Deviation 4 73 6 74 7 45 4 58 4 79 3 82 23 02 20 51 10 30 11 56 16 25 12 74 8 40 7 19 5 25 3 97 3 71 5 21 Notes: A “TTB” suffix to a model denotes the time-to-build version of that model. “HP Only” refers to the baseline model. “BRW” refers to the Benhabib et al. (1991) home production model. “MRW” denotes the McGrattan et al. (1997) parameterization. “GH” denotes the Greenwood and Hercowitz (1991) parameterization. The “Corr.” column refers to the correlation between market and home investment. For each model, the first row of leads and lags with output is for market investment while the second is for home investment. xt 4 0 46 0 25 0 11 0 21 0 26 0 50 0 08 0 20 0 20 0 41 0 09 0 20 0 21 0 42 0 16 0 22 0 39 0 72 xt 3 0 68 0 03 0 05 0 40 0 10 0 21 0 01 0 40 0 16 0 17 0 01 0 40 0 16 0 15 0 13 0 37 0 03 0 38 0 07 0 50, TTB τK 0 60 0 50 24 τK 0 49 0 99, TTB β 0 10 0 99 β 0 46 2, TTB γ 0 13 2 γ 0 41 HP TTB 0 10 HP Only 0 41 US Data Table 7: Sensitivity Analysis Cross Correlation of Real Output With xt 4 xt 3 xt 2 xt 1 xt xt 1 xt 2 0 14 0 04 0 29 0 56 0 80 0 87 0 82 0 47 0 61 0 74 0 80 0 76 0 52 0 24 0 19 0 32 0 46 0 64 0 78 0 16 0 03 0 02 0 07 0 19 0 33 0 53 0 96 0 66 0 22 0 30 0 44 0 64 0 86 0 63 0 39 0 13 0 17 0 29 0 49 0 80 0 39 0 18 0 18 0 31 0 46 0 65 0 83 0 26 0 11 0 00 0 11 0 25 0 43 0 65 0 95 0 65 0 19 0 28 0 43 0 63 0 87 0 67 0 43 0 12 0 18 0 31 0 53 0 83 0 39 0 16 0 18 0 31 0 47 0 66 0 83 0 23 0 09 0 00 0 10 0 24 0 40 0 62 0 98 0 67 0 19 0 29 0 43 0 64 0 89 0 68 0 44 0 14 0 19 0 31 0 52 0 83 0 38 0 14 0 19 0 29 0 41 0 56 0 61 0 05 0 11 0 08 0 04 0 02 0 08 0 25 0 81 0 58 0 27 0 33 0 45 0 62 0 80 0 53 0 26 0 05 0 09 0 19 0 36 0 64 0 38 0 27 Corr. Model Standard Deviation 4 73 6 74 7 45 4 58 4 79 3 82 6 19 3 63 4 44 3 50 7 37 4 21 5 06 3 99 12 71 8 69 6 75 4 65 Notes: A “TTB” suffix to a model denotes the time-to-build version of that model. “HP Only” refers to the baseline model. γ refers to the coefficient of relative risk aversion (see equation (2)); β refers to the discount factor (see equation (1)) and τ K refers to the tax rate on capital income (see equation (4)) The “Corr.” column refers to the correlation between market and home investment. For each model, the first row of leads and lags with output is for market investment while the second is for home investment. Figure 1: Response of Investment to a Market Shock 8 7 6 5 4 3 2 1 0 -1 -2 -5 0 5 10 15 Output Market Investment 20 25 30 Home Investment (a) Home Production only 8 6 4 2 0 -2 -5 0 5 10 15 Output Market Investment 20 25 Home Investment (b) Home Production with Time-to-build 22 30
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