In this section, we review the concept of the transmission coefficients and provide a brief description of the BPS method used to estimate these coefficients from PSID data. We also discuss possible biases and concerns associated with this method and how our exercises can help answer these questions. All the key formulas relevant to this paper and the details of how we implement the BPS method are provided in Appendix1.D. For more information about this method, we refer the readers to the original BPS paper.
The focus of this paper is on how household consumption and labor income respond to unexpected wage shocks, the first order effects of which are captured by the transmission coefficients,κ’s, in Equation (1.3.1). ∆ct ∆y1,t ∆y2,t = κc,u1 κc,u2 κc,v1 κc,v2 κy1,u1 κy1,u2 κy1,v1 κy1,v2 κy2,u1 κy2,u2 κy2,v1 κy2,v2 ∆u1,t ∆u2,t v1,t v2,t (1.3.1)
The left hand side variables of this equation include the consumption response,∆ct, and the
labor supply response,∆yj,tto wage shocks of various persistence.12 On the right hand side
are the transitory wage shocks,∆uj,t, and permanent wage shocks,vj,t. The transmission
coefficients measure directly how consumption and labor income respond to wage shocks of different persistence. For example,κc,vj measures how consumption responds to earner
j’s permanent wage shocks. Aκc,vj value of 0.4 means that40% of earnerj’s permanent wage shocks pass through to household consumption, and hence60%of them are insured. If transitory and permanent wage shocks are separately observable, equation (1.3.1), 12Note that all the changes here are measured in percentage term, i.e., first differences of logs. Also note
thatctandyj,tare the residuals of log consumption and log labor income of earnerjat agetafter controlling
and thus the insurance coefficients, could be estimated directly. However, in practice, only the sum of transitory and permanent wage shocks are observed directly in the data:
∆wj,t = ∆uj,t+vj,t.
Because ∆wj,t is only observed once for each individual at each time, it is technically
impossible to recover the two types of shocks without additional information. One contri- bution of BPS is to provide an empirically applicable method to estimate these transmission coefficients. The authors show that if one log-linearizes the first order conditions (assuming interior solutions) and the intertemporal budget constraint of a two-earner household life cycle model very similar to the one described in Section1.2.1, the authors are able to derive formulas for the transmission coefficients as functions of Frisch elasticities and smoothing parameters which can be calculated directly from the data. For example, the transmission coefficientκc,v1 can be calculated using
κc,v1 =
(−ηc,p+ηc,w1 +ηc,w2)[ηc,w1 −(1−β)(1−πt)(s1,t+ηh,w1)]
ηc,p−ηc,w1 −ηc,w2 + (1−β)(1−πt)(ηh,p+ηh,w1 +ηh,w2)
+ηc,w1 (1.3.2)
where theη’s andη’s are Frisch elasticities and combinations of them.13 As the authors argue, the entityπtis well approximated by the share of asset in the total discounted wealth
for a household at aget, that is:
πt ≈
Assett
Assett+Human W ealtht
where human wealth is the expected value of the discounted future labor income stream of the household. Whereassj,t is approximately the share of earner j’s human wealth in the
13In particular,η h,p≡P 2 j=1sjηhj,p,ηh,w1≡ P2 j=1sjηhj,w1 andηh,w2 ≡ P2 j=1sjηhj,w2.
total human wealth of the household, i.e.,
sj,t ≈
Human W ealthj,t
P2
i=1Human W ealthi,t
BPS add the parameterβto capture sources of insurance for the household that are not explicitly modeled (neither by them nor by us), such as the insurance provided by networks of relatives and friends as well as social insurance such as unemployment benefits and food stamps. The higher the values ofπtandβ are, the less does consumption respond to wage
shocks, and therefore the better these shocks are insured. The baseline results of BPS are estimated assumingβ = 0.
To estimate the transmission coefficients, the BPS method includes four steps: (1) Esti- mate the variance-covariance matrices of the permanent and transitory shocks directly from wage data alone (with results documented in section 1.2.3). (2) Calculate the smoothing parameters πt andsj,t directly from the asset and labor income data. (3) Conditional on
the results obtained from the first two steps, and using the empirical second order moments for ∆ct and∆yj,t, a Generalized Method of Moment (GMM) is employed to jointly es-
timate all the Frisch elasticities η’s (and the external insurance coefficient β if assumed unknown). (4) Calculate the estimates of the transmission coefficients for each household using the formulas derived, and take the sample averages of them as the final results.
For the general case with non-separability between consumption and labor in the utility function no prior restrictions are imposed on the Frisch elasticities to be estimated. How- ever, the assumption of separability in the utility function translates into the restrictions for the cross Frisch elasticities to be zero in the GMM estimation.
Several assumptions are imposed along the way in the BPS approach which delivers a transparent and operational methodology without imposing a specific functional form of the utility function. Violations of these assumptions of course may result in biases in the
estimation results. For example, in order to log-linearize the Euler equations requires inte- rior solutions to the saving and labor supply decisions from the households’ problem. This assumption does not hold if borrowing constraints of households are frequently binding or one of the household member decides not to participate in the labor market for a good number of households.14
As part of the contribution of the paper we evaluate, in section 1.5, the quality of the BPS methodology for environments with endogenous labor supply using simulated data from our life cycle model for which we know the true parameters, Frisch elasticities and thus partial insurance coefficients. But first we study, in the next section 2.4, and after briefly documenting basic life cycle profiles from the model, the model-implied wage- shock insurance coefficients and compare them to their empirical counterpart.