Model Extensions

The full CU model is obtained by extending the basic CU model with home production, indirect labor, capital and investment adjustment costs, and exogenous expenditure.

Home Production

I assume that households use their time to produce goods or services at home. The total number of hours available is normalized to one. The amount of home produced goods or services is given by c_h,t = Z_h,t(1 − l_t), where Z_h,t is the productivity of time spent at home. The one-period utility function of the representative household is now given by

u (zc,t, ct, c_h,t) =







φe^zc,t(^ct/(^φezc,t))^1−γ−1

1−γ + ¯ωc_h,t, γ 6= 1 φe^zc,tln (^ct/(φe^zc,t)) + ¯ωc_h,t, γ = 1

(1.63)

which replaces the utility function (1.18) in the household’s problem. The inclusion of home production enriches the model dynamics: an increase in the productivity at home

increases the opportunity cost of supplying labor and thus reduces the labor supply.

Later in section 1.5.1, I will relate the productivity at home to the labor productivity of firms z_l,t: Z_h,t = e^φlzl,t. The parameter φ_lis usually set to zero in standard models. In this case, an increase in labor productivity does not increase the opportunity cost of supplying labor. Hence, output moves positively with labor productivity. However, if φ_l is positive, an increase in labor productivity also increases the productivity at home and reduces the labor supply. Particularly, if φ_l is equal to one, an increase in labor productivity will not cause an increase in output but will be fully absorbed by a decrease in working hours.

Hence, the magnitude of φ_l affects the importance of labor productivity shocks in driving business cycles. I will estimate φ_l later in section 1.5.2.

Indirect Labor

I also assume that the Leontief production function of firm j ∈ [0, 1] at time t is now given by

where l_v,j,t is the amount of direct labor, l_f,j,t is the amount of indirect labor, and the inverse of αf,t is the productivity of the indirect labor.

Direct labor produces goods or services directly. The hours of direct labor fluctuate with output and can be easily adjusted within a period. Examples of direct labor positions are machine operators, assembly line operators, and cleaners.

Indirect labor, by contrast, supports the production process of firms, but is not directly involved in the active conversion of materials into finished products. Like capital stock, in-direct labor is predetermined. Examples of inin-direct labor positions are production supervi-sor, managerial and various administrative labor positions, such as accounting, marketing, and human resource positions.

By definition, the production capacity of firm j at time t is determined by the short run fixed factors, i.e., indirect labor and capital stock:

¯

where αf,t can also be understood as the amount of indirect labor required to form a unit of capacity.

The introduction of indirect labor allows output to be more volatile than labor. Hence, labor productivity measured by output to labor ratio can be pro-cyclical under demand shocks.

Capital and Investment Adjustment Costs

To get a persistent and hump-shaped investment response, I introduce capital and invest-ment adjustinvest-ment costs. The capital stock of firm j is accumulated according to

kj,t+1 = kj,t(1 − δ) + ij,t(1 − S (ij,t, ij,t−1, kj,t)) , (1.66)

where S is the adjustment cost function.

Following Hayashi (1982) and Christiano et al. (2005), I assume that the adjustment cost function is given by

where φk ≥ 0 and φ_i ≥ 0 are parameters that capture the curvature of the capital and investment adjustment costs respectively.

This functional form implies that to change investment or to deviate from the investment level that maintains the current level of capital is costly. Hence, the adjustment costs are zero in steady state, but the dynamics around the steady state will be influenced by the curvature of these two adjustment cost components.

Exogenous Expenditure

Following Smets and Wouters (2007), government spending and net exports are treated as exogenous expenditure. I abstract away from the out and (or) crowding-in effects of government spendcrowding-ing on consumption and crowding-investment by assumcrowding-ing that the exogenous expenditure is produced by an independent sector that requires direct labor only. Let gt be the exogenous expenditure and lg,t be the corresponding labor hired. We have l_g,t= α_g,tg_t, where α_g,tis the direct labor required per unit of exogenous expenditure.

The final product of the economy is Yt= ct+ it+ gt and the total amount of hours worked is l_t= l_f,t+ l_v,t+ l_g,t.

Exogenous Shocks

I introduce four types of exogenous shocks in total.

First, the exogenous expenditure is given by g_t = ge^zg,t, where z_g,t follows an AR(1) process with an i.i.d. Normal disturbance: zg,t = ρgzg,t−1+ eg,t. eg,t is a shock to the exogenous expenditure.

Second, the labor productivity z_l,tdetermines the productivity of all types of labor. Specif-ically, αv,t = αve^−zl,t, αf,t = αfe^−zl,t, and αg,t = αge^−zl,t. In addition, the productivity of the time spent at home is also affected by the labor productivity: Z_h,t = e^φlzl,t. z_l,t is assumed to follow an AR(1) process with an i.i.d. Normal disturbance: zl,t= ρlzl,t−1+ el,t. e_l,t is a shock to the labor productivity.

Third, there is an investment demand z_i,t that affects the subjective discount factor, i.e., the importance of the present relative to the future. The lifetime preference of the representative household is now given by:

E0

∞

X

t=0

β^te^−zi,tu (zc,t, ct, c_h,t)

!

. (1.68)

I assume that z_i,t follows an AR(2) processes with an i.i.d. Normal disturbance: z_i,t = ρi,1zi,t−1+ ρi,2(zi,t−1− z_i,t−2) + ei,t. The assumption is designed to capture both the low-frequency movement and the high-low-frequency zigzag pattern in investment. e_i,t is a shock to the investment demand.

Finally, as in the basic CU model, there is a consumption demand zc,t that affects the marginal utility of consumption relative to the marginal dis-utility of labor. I assume that zc,t follows an AR(1) process with an i.i.d. Normal disturbance and is also affected by the disturbance to the investment demand as follows: z_c,t = ρ_cz_c,t−1+ e_c,t+ ρ_cie_i,t. The parameter ρciallows us to capture the divergence between consumption and capital in the U.S. from 1968 to 1993.²⁰ e_c,t is a shock to the consumption demand.

All shocks, e_c,t, e_i,t, e_g,t, and e_l,t, are mutually uncorrelated.

20Appendix 4.4 gives some further discussions and shows the estimated consumption series under in-vestment demand shocks ei,t.