The Home Production Puzzle:

The Role of Human Capital

and Adjustment Cost

Jaihyun Nahm

＊Dr. Nahm, Jaihyun, Korea Institute of Finance, 6th FL., KFB Building, 4-1, 1-Ga, Myung-Dong, Chung-Gu, Seoul 100-021, Korea, (Tel)82-2-3705-6249, (Fax)82-2-3705-6345, (E-mail) jnahm@kif.re.kr.

Korea Institute of Finance

Seoul, Korea

January 2005

Table of Contents

Abstract

Ⅰ. Introduction	1

Ⅱ. Model	2

1. Household	2

2. Firm	4

3. Government	5

Ⅲ. Equilibrium	5

1. Equilibrium	5

2. Euler Equation	7

Ⅳ. Calibration	8

1. Steady State	8

2. Calibration	9

Ⅴ. Simulation	10

1. Simulation	10

2. Sensitivity Analysis	14

Ⅵ. Conclusion	16

The Home Production Puzzle: The Role of Human Capital and Adjustment Cost

by

Nahm, Jaihyun, Korea Institute of Finance

Abstract

The standard general equilibrium Beckerian home production model has a counterfactual implication for the co-movement of the market and the home investments. This paper shows that the puzzle can be resolved by introducing human capital accumulation and the cost of adjusting the stock of market capital in a standard home production model. By incorporating these two features, the model is able to account for the observed positive co-movements in the two investments, even when the assumption that the home production function is Cobb-Douglas. This assumption is important because it implies that the model is consistent with the long run behavior of durables and nondurables in US data.

Ⅰ. Introduction

Since Becker (1965), there have been several papers which incorporate home production in the one-sector RBC model. By introducing a non-market sector as an extra margin for allocating resources, a temporary shock to the market sector results in not only intertemporal substitution of resources but also a richer intratemporal substitution between non-market and market activities than a simple one-sector RBC model. In particular, it enables the model to better match some of the business cycle characteristics of labor market data.

On the other hand, this has a counterfactual implication for investment activities. The US annual data clearly show that the market and home investments move together with a correlation of 0.40. However, the model predicts a strong negative correlation. Greenwood & Hercowitz (1991) first tried to resolve this "home production puzzle" by assuming highly correlated technology shocks between the market and home sectors and a general Constant Elasticity of Substitution (CES) home production function instead of the usual Cobb-Douglas function. Nonetheless, the puzzle was not solved since it lead to a significant deterioration in the performance of the model. The variability of the time spent in the market sector became too low, even lower than that of the one-sector RBC model.

For example, Benhabib, Rogerson & Wright (1991) predicts that the correlation between market and home investment is -0.82.

There have been some efforts to fix this problem. Among them, many researchers assumed a CES home production function and set the CES parameter arbitrarily. Greenwood et al. (1993) and Einarsson & Marquis (1997) set the CES parameter value less than one, and McGrattan et al. (1997) and Chang (2000) set it larger than one. The value of the CES parameter is critical in each model. However, it is not possible to estimate it directly due to the lack of home labor and home consumption data. Hence, all the papers set the parameter so as to obtain an overall best fit for the second moments of their models.

Greenwood et al.(1993) picked the substitute elasticity between home labor and home capital as 0.67, and Einarsson & Marquis (1997)  as 0.60. McGrattan et al.(1997) set it as 1.37 and Chang (2000) as 2.59.

Greenwood & Hercowitz (1991) said " Little independent evidence is available to assess the reasonableness of the value chosen for the elasticity of substitution between capital and time in household production. And it would seem dificult on theoretical grounds alone to argue about the appropriate value for this parameter, ... Clearly, further empirical work is needed to gauge the value of this and other parameters governing household activity."

The US data, however, clearly show that the price of durables relative to nondurables declines while the expenditure share of durables is constant over time. Only the Cobb-Douglas function can reconcile this balanced growth fact. I will show in this paper that the positive correlation between the two investments can be attained using a Cobb-Douglas home production function with the addition of a second countercyclical non-market sector and some adjustment cost for market capital accumulation.

Kydland(1995) also emphasized this stylized fact and Christiano & Fisher(1995) showed this fact in their Fig. 1b and 2b.

Dellas & Sakellaris (1995) provided evidence for countercyclical activity in the formal training sector by using college enrollment data. I will introduce this schooling sector (human capital sector) as an extra time use margin.  Adding this new margin accounts for the data better by increasing the variability of time spent working in the market sector. Even after incorporating the home sector, this variability is usually much smaller in the model than in the data.

I considered an adjustment cost for market capital accumulation because it seems apparent that market capital has a higher adjustment cost than home capital. Kydland & Prescott (1982) pointed out that the average construction period of plants is nearly two years, while consumer durables are in production for a much shorter period of time, on average. The introduction of this market capital adjustment cost reduces the variability of market investment by transferring some part of the market investment response to home investment during the first period after a shock. This substitution increases home investment during the first period after a shock, which makes the correlation between the two investments positive. It also improves the model by changing the second moments of the two investments in the "right" direction, with the exception of the standard deviation of home investment.

The paper is organized as follows. Section Two presents the model.   Section Three formalizes the equilibrium. Section Four explains the calibration method with data description. Section Five gives simulation results, and Section Six ends with a conclusion.

Ⅱ. Model

1. Household

In an economy, there is a [0,1] continuum of identical infinitely lived individuals who maximize their expected lifetime utility as given by

[eqn: maxE _{0} sum _{t=0} ^{INF } beta ^{t} u LEFT ( c _{mt} ,c _{ht} RIGHT )]                             (1)

where [eqn: c _{mt}] represents consumption goods purchased in the market, and [eqn: c _{ht}] is consumption goods acquired at home. The momentary utility function is assumed to have logarithmic form.

[eqn: u(c _{mt} ,~c _{ht} )=(1- mu )ln`c _{mt} + mu ln`c _{ht}]                         (2)

Unlike the one-sector RBC model, the household in this model has three alternatives to allocate its total time. The total time is normalized to one, therefore,

[eqn: n _{mt} +n _{st} +n _{ht} =1]                                (3)

where market hour [eqn: n _{mt}] is hours spent for market production, school hour [eqn: n_{st}] is time allocated for formal training, and home hour [eqn: n _{ht}] is hours for home production.

As Greenwood & Hercowitz (1991) or Einarsson & Marquis (1997), we do not introduce the leisure in the model. This simplifies the quantitative analysis and we can think leisure is included in the home hour of the model.

In this model, long run growth is determined endogenously through the following human capital accumulation function:

Perli & Sakellaris (1998) calculated the average share of capital in educational output is only 0.08. So I exclude physical capital as a factor input.

[eqn: s _{t+1} =s _{t} (1+ rho n _{st} )]                                (4)

where [eqn: s _{t}] is human capital at time [eqn: t] , and [eqn: rho] is a technology parameter.

The household is constrained to a standard budget equation.

[eqn: c _{mt} +i _{mt} +i _{ht} =(1- tau _{k} )R _{t} k _{mt} +(1- tau _{n} )W _{t} s _{t} n _{mt} +J _{t}]                 (5)

where [eqn: i _{mt}] and [eqn: i _{ht}] are market and home investment, [eqn: k _{mt}] is the stock of market capital, [eqn: tau _{k}] and [eqn: tau _{n}] are taxes on gross capital and labor income, [eqn: J _{t}] is the government lump-sum transfer, [eqn: R _{t}] is the rental rate on the market capital, and [eqn: W _{t}] is the real wage rate on effective units of market labor, [eqn: s _{t} n _{mt}].

There is no tax on the home capital in this model. McGrattan et al.(1997) pointed out that the property tax on the residential capital as well as interest payments on the home mortgage is deductible against income taxes. I assume that the net effect is approximately zero as they did.

The individual is also subject to the home production constraint,

[eqn: c _{ht} =k _{ht} ^{alpha ₂} (n _{ht} s _{t} ) ^{1- alpha ₂}]                                (6)

where [eqn: k _{ht}] is the stock of home capital. As Greenwood et al (1993) emphasized, this constraint implies that there are no uses for home-produced output other than consumption - it cannot be sold or transformed into market or home capital, as the market-produced output can. This is a key asymmetry between the market and home sectors; only the former produces capital. Meanwhile, I assume the home production function is in Cobb-Douglas form. This is one of the main differences from other papers, which assume a general CES home production function.

The home capital stock has a typical law of motion, but the market capital stock evolves with the following adjustment cost.

[eqn: k _{mt+1} =(1- delta _{m} )k _{mt} +i _{mt} - {gamma } over {2} ( {i _{mt}} over {k _{mt}} - delta _{0} )²k _{mt}]                     (7)

[eqn: k _{ht+1} =(1- delta _{h} )k _{ht} +i _{ht}]                              (8)

I also assume that [eqn: delta _{h}] is equal to [eqn: delta _{m}], and [eqn: delta _{0}] is set such that there is no adjustment cost in the steady state.

Following Baxter (1996), many papers including Atkeson & Kehoe (1999) or Chang (2000) assumed the capital adjustment cost term is zero in the steady state.

2. Firm

All firms are assumed to have a constant return to scale production functions. Therefore, we can normalize the number of firms to unity. The production function is given by

[eqn: Y _{t} =Z _{t} K _{Mt} ^{alpha } (N _{Mt} S _{t} ) ^{1- alpha }]                             (9)

where [eqn: Y _{t}] is the aggregate per capita output, [eqn: K _{Mt}] is the aggregate per capita stock of market capital, [eqn: N _{Mt}] is the aggregate per capita hours for market production, and [eqn: S _{t}] is the aggregate per capita human capital stock.  [eqn: Z _{t}] is the productivity shock that is assumed to follow as

[eqn: lnZ _{t+1} = kappa lnZ _{t} +ɛ _{t+1} ~~~~~ɛ SIM N(0, ~sigma _{ɛ} ²)]                 (10)

The perfect competition and profit maximization condition determine the rental rate and wage rate.

[eqn: R _{t} =F₁(K _{Mt} ,N _{Mt} S _{t} ,Z _{t} )]                             (11)

[eqn: W _{t} =F₂(K _{Mt} ,N _{Mt} S _{t} ,Z _{t} )]                             (12)

3. Government

I also assume that the government redistributes tax revenue uniformly across households with lump-sum transfers.

[eqn: J _{t} = tau _{k} R _{t} K _{Mt} + tau _{n} W _{t} S _{t} N _{Mt}]                          (13)

Ⅲ. Equilibrium

1. Equilibrium

I distinguish between the economy-wide per capita aggregate state variables [eqn: Z~and~X~=~(K _{M} ,K _{H} ,S)], and the household's individual state variables [eqn: x=(k _{m} ,k _{h} ,s)], in standard fashion. I further distinguish variables over which the household has control, [eqn: d=(c _{m} ,c _{h} ,k _{m} prime ,k _{h} prime ,n _{m} ,n _{h} ,n _{s} ,s prime )], from their counterparts, [eqn: D=(C _{M} ,C _{H} ,K _{M} prime ,K _{H} prime ,N _{M} ,N _{H} ,N _{S} ,S prime )], by using lower-case and upper-case letters, respectively. In equilibrium, it is true that [eqn: X=x], but the problem the household solves makes a distinction between these two.  The state variables for the household are [eqn: (x,X,Z)]. Let [eqn: v(x,X,Z)] be the household's optimum value function, and let the primes denote the next period values. The household's problem is to choose a vector d that solves the problem:

[eqn: V(x,X,Z)=max _{d} LEFT{ { U(c _{m} ,c _{h} )+ beta EV(x prime ,X prime ,Z prime )} RIGHT]

[eqn: s.t.]

[eqn: n _{m} +n _{s} +n _{h} =1]

[eqn: c _{m} +i _{m} +i _{h} =(1- tau _{k} )R(X,Z)k _{mt} +(1- tau _{n} )W(X,Z)sn _{m} +J(X,Z)]

[eqn: c _{h} =k _{h} ^{alpha ₂} (n _{h} s) ^{1- alpha ₂}]

[eqn: s prime =s(1+ rho n _{s} )]

[eqn: k _{m} prime =i _{m} +(1- delta _{m} )k _{m} - {gamma } over {2} ( {i _{m}} over {k _{m}} - delta _{0} )²k _{m}]

[eqn: k _{h} prime =i _{h} +(1- delta _{h} )k _{h}]

[eqn: S prime =S(1+ rho N _{S} )]

[eqn: K _{M} prime =I _{M} +(1- delta _{m} )K _{M} - {gamma } over {2} ( {I _{M}} over {K _{M}} - delta _{0} )²K _{M}]

[eqn: K _{H} prime =I _{H} +(1- delta _{h} )K _{H}]

[eqn: lnZ prime = kappa lnZ+ɛ]

and the initial conditions

[eqn: (k _{m0} ,k _{h0} ,s _{0} ,K _{M0} ,K _{H0} ,S _{0} ,Z _{0} )]                          (14)

Let [eqn: d(x,X,Z)] be the policy function that gives the optimal decision for this problem. All households are identical and must be, in equilibrium, [eqn: d(X,X,Z)=D(X,Z).]

Furthermore the necessary and sufficient condition from the firm's problem can be used,

[eqn: R=F₁(K _{M} ,N _{M} S,Z)=R(X,Z)]

[eqn: W=F₂(K _{M} ,N _{M} S,Z)=W(X,Z)]                          (15)

to determine the wage and rental rate as functions of X and Z, which must be consistent with what the individual household takes as given.

Finally, I can use

[eqn: J= tau _{k} RK _{M} + tau _{n} WSN _{M} =J(X,Z)]                        (16)

to determine fiscal policy as a function of X and Z, which must be also consistent with what the individual household takes as given.

These considerations lead to formally defining an equilibrium for this economy as follows.

Definition

A recursive competitive equilibrium is a value function, [eqn: v(x,X,Z)], a policy function, [eqn: d(x,X,Z)], which gives decisions on [eqn: c _{m} (x,X,Z),~][eqn: c _{h} (x,X,Z),] [eqn: ~k _{m} prime (x,X,Z),] [eqn: k _{h} prime (x,X,Z),][eqn: n _{m} (x,X,Z),] [eqn: `n _{h} (x,X,Z),] [eqn: `n _{s} (x,X,Z),] [eqn: `s prime (x,X,Z)] for the representative household; an aggregate per capita policy function, [eqn: D(X,Z)], which gives aggregate decisions on [eqn: C _{M} (X,Z),~C _{H} (X,Z),~K _{M} prime (X,Z),~K_{H}′(X,Z),] [eqn: N _{M} (X,Z),~N _{H} (X,Z),~N _{S} (X,Z),~S prime (X,Z)]; factor price functions [eqn: W(X,Z),~R(X,Z)], and fiscal policy function, [eqn: J(X,Z)], such that these functions satisfy

i) the household problem ( 14 )

ii) the necessary and sufficient conditions for profit maximization, ( 15 )

iii) the fiscal policy equation ( 16 ), and

iv)	consistency of individual and aggregate decisions, i.e., the condition [eqn: d(X,X,Z)=D(X,Z).]

2. Euler Equation

The first order conditions with respect to [eqn: k _{mt+1} ,k _{ht+1} ,n _{mt} ,] and [eqn: s _{t+1}] give the following Euler equations.

[eqn: 1``=``E _{t} ` LEFT [ beta LEFT { 1``-`` gamma LEFT ( {i _{mt}} over {k _{mt}} ``-`` delta _{0} RIGHT ) RIGHT } {c _{mt}} over {c _{mt`+`1}} LEFT { (1``-`` tau _{K} `)R _{t+1} `+`` {1``-`` delta _{m} ``+`` {gamma } over {2} LEFT ( LEFT ( {i _{mt+1}} over {k _{mt+1}} RIGHT ) ^{2} ``-`` delta _{0} ^{2} ` RIGHT )} over {1``-`` gamma ` LEFT ( {i _{mt+1}} over {k _{mt+1}} ``-`` delta _{0} ` RIGHT )} RIGHT } RIGHT ]]  (17)

[eqn: 1= beta E _{t} LEFT ( {c _{mt}} over {c _{mt+1}} LEFT ( 1- delta _{h} RIGHT ) + {mu alpha _{2}} over {1- mu } {c _{mt}} over {k _{ht+1}} RIGHT )]                      (18)

[eqn: {1- mu } over {c _{mt}} (1- tau _{n} )W _{t} s _{t} = {mu (1- alpha ₂)} over {n _{ht}}]                         (19)

[eqn: 1= beta (1+ rho )E _{t} LEFT ( {c _{mt}} over {c _{mt+1}} {W _{t+1}} over {W _{t}} RIGHT )]                          (20)

For all the constraints and the Euler equations, I divided all the variables, except for three time variables [eqn: (n _{mt} ,~n _{st} ,~n _{ht} )] and two factor prices [eqn: (W _{t} ,~R _{t} )] by the human capital stock [eqn: S _{t}], to transform them into stationary variables.

The model was solved by the log-linearization of the Euler equations and other constraints around the steady states.

I revised Uhlig (1999)'s code for solving the model.

Ⅳ. Calibration

1. Steady State

The Euler equations and other constraints around the steady states with the transformed variables after imposing equilibrium conditions are as follows:

[eqn: {1+ rho bar{N _{S}}} over {beta } -1+ delta _{m} =(1- tau _{K} ) bar{R} = rho + delta _{m}]                     (17')

[eqn: {1+ rho bar{N _{S}}} over {beta } -1+ delta _{h} = {mu alpha _{2}} over {1- mu } {bar{C _{M}}} over {bar{K _{H}}} = rho + delta _{h}]                     (18')

[eqn: {(1- mu )(1- tau _{N} )} over {mu (1- alpha _{2} )} = {bar{C _{M}}} over {bar{W}} {1} over {bar{N _{H}}}]                           (19')

[eqn: {1+ rho bar{N _{S}}} over {beta } =(1+ rho )]                                (20')

[eqn: bar{N _{M}} + bar{N _{H}} + bar{N _{S}}~=1]                                 (3')

[eqn: LEFT ( {bar{S _{t+1}}} over {S _{t}} RIGHT ) =1+ rho bar{N _{S}}]                                 (4')

[eqn: bar{Y}~= bar{C _{M}} + bar{I _{M}} + bar{I _{H}}]                                (5')

[eqn: bar{C _{H}} `= bar{K _{H} ^{alpha _{2}}} ` bar{N _{H} ^{1- alpha _{2}}}]                                 (6')

[eqn: delta _{m} + rho bar{N _{S}} = {bar{I _{M}}} over {bar{K _{M}}}]                                 (7')

[eqn: delta _{h} + rho bar{N _{S}} = {bar{I _{H}}} over {bar{K _{H}}}]                                 (8')

[eqn: bar{Y} = bar{K _{M} ^{alpha }} ` bar{N _{M} ^{1- alpha }}]                                 (9')

[eqn: bar{R}~= {alpha bar{Y}} over {bar{K}_{M}}]                                  (11')

[eqn: bar{W} `=(1- alpha ) {bar{Y}} over {bar{N _{M}}}]                               (12')

2. Calibration

I calibrated the structure parameters of the model by imposing restrictions on a few parameters either from the literature or from the steady-state value of the data. Specifically, from the literature, I pinned down parameters such as [eqn: beta =0.96], [eqn: kappa =0.90], [eqn: alpha =0.3], [eqn: tau _{n} =0.25], and [eqn: delta _{m} =0.078].

The discount factor corresponds to a 1% quarterly discount rate, and the AR(1) coefficient of the log of exogenous shock is equivalent to approximately a 97.5% quarterly value. I took the capital income share of GNP net housing as 30% and the labor income tax as 25%. The depreciation rate on the market capital was taken from Greenwood & Hercowitz (1991).

These numbers are consistent with many other papers including Einarsson & Marquis (1997), Hausman & Poterba (1986), and McGranttan et al.(1997).

I also used three steady state values around the balanced growth path from the annual data from 1959 to 2000, such as [eqn: bar{N _{M}} ~=0.24,~ {bar{I _{M}}} over {bar{Y}} `=0.15] and [eqn: {bar{K _{H}}} over {bar{K _{M}}}~=1.13].

I use annual data because yealy time horizon is preferable to quarterly when we include Endogenous Growth Model for explaining business cycle.

[eqn: K _{M}] is defined as the stock of the business non-residential capital. [eqn: K _{H}] is the combined stock of consumer durables and residential capital. Y is GNP minus service flow from the housing stock. [eqn: N _{M}] is the average ratio of total hours worked to total non-sleeping hours of the working age population form household survey. More detailed definitions are in Appendix II.

With a 1.5% annual growth rate, [eqn: beta] and the equation (20') gave [eqn: rho] as 0.0574. The gross capital income tax was determined to be 27% to satisfy the equation (17').

On the deterministic balanced growth path and the given value of [eqn: delta _{m} ,~beta] and growth rate described above, this corresponds to a tax rate on the net capital income of about 47%. This figure is between the effiecitve tax rates of 52 % on corporate capital income and 40 % on noncorporate capital income computed by Jorgenson and Yun (1986).

The value of [eqn: sigma _{epsilon }] was determined to match the standard deviation of output in the model with the US data which results in [eqn: sigma _{epsilon }]=0.0173 for Model 1 and 0.014 for Model 2. The value of [eqn: gamma] was set as 0.125 to match the standard deviation of market investment in the model with that of the data.

Since I am considering the Cobb-Douglas home production function, [eqn: alpha _{2}] and [eqn: mu] can be calculated from the equations (6'), (18'), and (19') as [eqn: alpha _{2}]= 0.1845 and [eqn: mu] = 0.6626.

Ⅴ. Simulation

1. Simulation

Model 1 is a standard two-sector RBC model with a Cobb-Douglas home production function. It is similar to Model I of Greenwood & Hercowitz (1991) except for the form of the home production function.  In this paper, the home production function has a unit elasticity of substitution between two inputs, and it does not have exogenous shock, which Greenwood & Hercowitz (1991) assumed to have high correlation with market production shock. It shows the usual home production puzzle; namely, the correlation between the two investments is -0.345, whereas the data is 0.4 in Table 1.  We also observe that market investment is excessively volatile in Table 2, and the first autocorrelation of market investment has the wrong sign in Table 3.

TABLE 1

Correlation between Market and Home Investment

Data (1959-2000)	Model 1	Model 2
0.40	-0.345 (0.153)	0.231 (0.125)

Note: Simulated figures show the average across 500 simulations, each covering 42 years.  The figures in the parentheses are the simulation standard errors.  The US data were divided by the working age population (16-64) logged and Hodrick-Prescott filtered with the smoothing parameter [eqn: lambda]=100.  The definitions of the variables and their sources are in Appendix II.

TABLE 2

Standard Deviation (%)

	Data	Model 1	Model 2
Output	2.21	2.21 (0.42)	2.21 (0.38)
Market consumption	1.77	0.97 (0.20)	0.95 (0.17)
Market investment	8.37	10.71 (1.47)	8.37 (1.05)
Home investment	8.15	6.64 (0.99)	4.62 (0.87)
Market hour	1.69	0.91 (0.17)	1.33 (0.23)

TABLE 3

First Order Autocorrelation

	Data	Model 1	Model 2
Output	0.55	0.467 (0.143)	0.499 (0.136)
Market consumption	0.71	0.514 (0.152)	0.537 (0.140)
Market investment	0.31	-0.051 (0.177)	0.194 (0.162)
Home investment	0.56	0.174 (0.182)	0.614 (0.104)
Market hour	0.58	0.475 (0.136)	0.518 (0.129)

TABLE 4

Correlation with Output

	Data	Model 1	Model 2
Market consumption	0.78	0.942 (0.019)	0.942 (0.017)
Market investment	0.79	0.727 (0.025)	0.855 (0.014)
Home investment	0.68	0.370 (0.162)	0.681 (0.104)
Market hour	0.92	0.969 (0.014)	0.975 (0.009)

FIGURE 1

We can identify them by looking at the impulse response function of the market and home investments to a one-standard deviation technology shock in Figure 1. Market investment reacts too fast in the first period, which makes market investment too volatile compared with the data. We, therefore, need to spread its response over longer periods. This will not only reduce the variance of market investment, but also increase the first autocorrelation of market investment. This is necessary for matching the data as we discussed in Table 2 and Table 3.

The initial response of market investment is negative, which results in the negative correlation between the two investments in Model 1. Therefore, some parts of market investment must be transferred from the market sector to the home sector in the first period of the shock to obtain desirable results.

FIGURE 2

Meanwhile, even with the home sector, the variability of the market hour is much smaller in the model than in the data. Table 2 shows that the standard deviation of the market hour in Model 1 is only 0.91%, while that of the data is 1.69%. The impulse response of the market and home hour in Figure 2 shows that the market hour in the first period after the shock is about 0.52% higher than steady state level. To increase the variability of the market hour, the initial reaction must be larger than this response. We can also see that all the increase of the market hour is driven by the decrease of the home hour.

To correct these problems, I added the human capital sector as an extra margin and added some capital adjustment cost to the market capital accumulation in Model 2. These changes improve the model on market hour and both investments series.

The decrease of the standard deviation of market investment due to the market capital adjustment cost leads to a correct sign of the first autocorrelation of market investment. Table 3 shows that the first autocorrelation is increased from -0.051 to 0.194. It also increases both the correlation with output and the first autocorrelation of home investment, as expected. The correlation is now exactly same as the data, and the first autocorrelation is also very close to the data, as seen in Table 3 and 4.

FIGURE 3

Figure 3 shows this result as well. The initial increase in market investment is reduced from more than 8% of the steady state value to less than 7%. It gives some persistence to the response of market investment.  The initial response of home investment is now positive. This transfer from market investment to home investment in the initial response results in the correlation between the two investments to the correct sign. Table 1 shows the correlation in Model 2 is 0.231 which is close to the data 0.40.

FIGURE 4

Due to the new margin, the standard deviation of the market hour is now increased from 0.91% to 1.33% in Table 2. The initial increase in the market hour is now more than 0.9% above the steady state level as seen in Figure 4. Most of this increased time is substituted from the school hour, while the home hour is almost constant. This is consistent with what Dallas & Sakellaris (1995) found from college enrollment data.

Meanwhile, both of the endogenous growth and the market capital adjustment cost are necessary for the desirable performance of the model.   The three-sector endogenous growth model without the capital adjustment cost performed poorly. The standard two-sector home production model with the market capital adjustment cost term performed better. However, Model 2 of the paper performed best, especially in the labor market.

The results are in Appendix I.

2. Sensitivity Analysis

These results are insensitive to specific parameterization.

If I take the usual 95% quarterly value of the first autocorrelation of the log of exogenous shock, the corresponding annual value is 0.81. Since some researchers, such as Einarsson & Marquis (1997), use this value for annual calibration, I check the sensitivity of my result to this change in Table 5.

I also explore the sensitivity to a different capital depreciation rate.  Baxter (1996) uses a 7.1% annual depreciation rate calculated from the July 1985 Survey of Current Business. Einarsson & Marquis (1997) use 5.5% and Cooley & Prescott (1995) use 4.8%. I report the case with a 6% annual depreciation rate in Table 5. I only report six second moments here, because in the other cases either there is no difference between the two Models or Model 2 is better than Model 1 by construction.

A 6% depreciation rate in Table 5 implies 22% gross capital income tax in the model.

The results of other second moments are available upon request.

For both cases, all the same results as in the previous subsection are found here, too. In Model 2, the standard deviation of the market hour increases and first autocorrelation of the market investment  has the correct sign. It also increases both the correlation with output and the first autocorrelation of home investment, even though the standard deviation of home investment becomes too low, as we observed in Table 2. The correlation of the two investments are very close to the data.

TABLE 5

Sensitivity to [eqn: kappa] and [eqn: delta _{m}]

		[eqn: kappa] = 0.81		[eqn: delta _{m}] = 0.06
	Data	Model 1	Model 2	Model 1	Model 2
[eqn: rho (I _{M} `,`I _{H} )]	0.40	-0.335 (0.154)	0.391 (0.091)	-0.561 (0.129)	0.393 (0.082)
[eqn: sigma _{I _{H}}]	8.15	6.98 (1.06)	4.68 (0.96)	8.67 (1.27)	4.22 (0.89)
[eqn: sigma _{N _{M}}]	1.69	1.03 (0.19)	1.50 (0.26)	0.98 (0.20)	1.39 (0.24)
[eqn: AR1(I _{M} )]	0.31	-0.07 (0.18)	0.18 (0.15)	-0.08 (0.17)	0.26 (0.14)
[eqn: AR1(I _{H} )]	0.56	0.17 (0.18)	0.65 (0.09)	0.05 (0.18)	0.72 (0.08)
[eqn: rho (Y,I _{H} )]	0.68	0.38 (0.16)	0.77 (0.07)	0.22 (0.16)	0.75 (0.07)

Table 6 shows the sensitivity analysis for the smoothing parameter used in Hp-filtering. Three different rules have been suggested for adjusting the HP-filter to annual data. Backus & Kehoe (1992) use a value of 100 for the smoothing parameter. I used this value since it is the literature standard.  Meanwhile, Correia, Neves & Rebelo (1992), and Cooley & Ohanian (1991) use a value of 400 for the annual data, and Ravn & Uhlig (1997) argue that the correct value should be 6.25.

Table 6 also shows that the results for each experiment are qualitatively the same as those seen before. The improvements obtained by adding the adjustment cost and the human capital sector to the basic two-sector RBC model are very general, and they are not artifacts of the specific parameterization.

TABLE 6

Sensitivity to Smoothing Parameter

	[eqn: lambda] = 400			[eqn: lambda] = 6.25
	Data	Model 1	Model 2	Data	Model 1	Model 2
[eqn: rho (I _{M} `,`I _{H} )]	0.40	-0.228 (0.183)	0.191 (0.151)	0.44	-0.601 (0.102)	0.068 (0.092)
[eqn: sigma _{I _{H}}]	8.17	7.28 (1.29)	5.84 (1.10)	5.88	5.28 (0.79)	2.70 (0.42)
[eqn: sigma _{N _{M}}]	1.86	1.04 (0.23)	1.54 (0.29)	1.18	0.63 (0.11)	0.91 (0.12)
[eqn: AR1(I _{M} )]	0.45	0.04 (0.19)	0.25 (0.17)	0.05	-0.28 (0.16)	-0.08 (0.13)
[eqn: AR1(I _{H} )]	0.59	0.28 (0.19)	0.62 (0.12)	0.38	-0.14 (0.17)	0.39 (0.11)
[eqn: rho (Y,I _{H} )]	0.74	0.48 (0.16)	0.69 (0.11)	0.74	0.04 (0.16)	0.48 (0.10)

Ⅵ. Conclusion

Greenwood & Hercowitz (1991) had the following three suggestions for solving the home production puzzle. First, two highly correlated technology shocks for the market and the home sector are necessary. Second, capital and labor in the home production function should be more complementary than in the Cobb-Douglas technology. In particular, using the Cobb-Douglas home production function cannot explain the puzzle. Third, adjustment cost in market capital is not helpful.

Contrary to their recommendations, I assumed only one technology shock in the market production function, and this shock was enough to generate the necessary variability and co-movement in the aggregate economic variables. I did not assume a general CES form because it was not consistent with the balanced growth facts for durables and nondurables.  Also, even though the introduction of the adjustment cost in market capital accumulation reduced the variabilities of both investments, it helped the model's performance by changing the response to a technology shock of the market and home investments in the "right" direction. By introducing human capital accumulation and market capital adjustment cost in the standard home production model, I showed that the Cobb-Douglas home production function with a shock only to the market sector could resolve the home production puzzle.

I suggest two possibilities for future research.

First, the introduction of the capital adjustment cost decreased excessively the standard deviation of home investment. More research on the variability of home investment is necessary. Second, the model is not applicable for explaining a phase shift that exists between the home and market investments. The data show that home investment leads market investment, while the model predicts the opposite.

Because this lead-lag relationship is mainly due to the housing sector, and the standard deviation of residential investment is larger than other forms of fixed investment, incorporating an explicit housing sector to the model would be required for explaining these phenomena.

Appendix I

Model A: Two Sector with Market Capital Adjustment Cost

([eqn: gamma =0.1,~sigma _{epsilon } =0.0176])

Model B: Three Sector without Market Capital Adjustment Cost

([eqn: sigma _{epsilon } =0.013])

TABLE 1

Correlation between Market and Home Investment

Data (1959-2000)	Model A	Model B
0.40	0.187 (0.142)	-0.396 (0.132)

TABLE 2

Standard Deviation (%)

	Data	Model A	Model B
Output	2.21	2.21 (0.42)	2.21 (0.38)
Market consumption	1.77	0.99 (0.21)	0.91 (0.17)
Market investment	8.37	8.37 (1.19)	11.15 (1.32)
Home investment	8.15	4.28 (0.93)	7.89 (1.00)
Market hour	1.69	0.87 (0.16)	1.39 (0.24)

TABLE 3

First Order Autocorrelation

	Data	Model A	Model B
Output	0.55	0.442 (0.165)	0.537 (0.114)
Market consumption	0.71	0.492 (0.168)	0.568 (0.126)
Market investment	0.31	0.145 (0.181)	-0.030 (0.152)
Home investment	0.56	0.589 (0.125)	0.118 (0.160)
Market hour	0.58	0.445 (0.159)	0.555 (0.106)

TABLE 4

Correlation with output

	Data	Model A	Model B
Market consumption	0.78	0.952 (0.019)	0.923 (0.019)
Market investment	0.79	0.859 (0.021)	0.663 (0.020)
Home investment	0.68	0.641 (0.130)	0.405 (0.129)
Market hour	0.92	0.971 (0.014)	0.972 (0.010)

Appendix II

The following real variables are from DRI Basic Economics in chained 1992 dollars unit.

Output: Real GNP - Real gross housing product

Market consumption: Real personal consumption expenditures on non-durables and services excluding housing

Market investment: Real non-residential private investment

Home investment: Real residential private investment + Real personal consumption expenditure on durable goods

Market hour

= (total weekly hours of working)/(working population*7days*16 hours)

which is calculated from the following data:

average weekly hours of work from Household Survey (CITIBASE)

number of employed from Household Survey (CITIBASE)

population age 16-64 (DRI)

For calibration,

Market capital: from BEA Homepage Fixed Assets

Home capital: from BEA Homepage Fixed Assets

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