Empirical life cycle models of labour supply and - Statistisk sentralbyrå

Social and Economic Studies 91 Empirical Life Cycle Models
marginal wage and interest rate requires observations on At_ i , and we need
panel data for (at least) two years. The calculation of the after-tax discount
rate Dt , cf. equation (19), requires panel data for even more periods, and the
after-tax wage and interest rate, and the user price of durables, are now
endogenous. In order to simplify estimation, income taxes are often omitted.
Blomquist (1985) shows that the lifetime wealth constraint (4) then becomes
separable in goods and prices across periods, and the only way a price change
can influence demand in other periods, is through the wealth effect (4). The
responses to price changes in other periods are then of a very special form, since
there is no effect via the discounted prices corresponding to Dtpt , Dtqut and
Dtmt , cf. Blomquist.
The possibility of constraints in the credit market also complicates estimation of
the Frisch demands considerably. Comparing equations (7) to (9) with (20) to
(22), we notice that the substitution of the Euler equation (18) into the first
order conditions introduces the borrowing constraint multipliers for all earlier
periods into these conditions. Since these multipliers are latent, this
substitution probably introduces more problems that it solves, and the
specifications (7) to (9) seem to be more attractive.
In order to use the Frisch demands to explain differences in demand across
households, the relation between Ao and all the exogenous variables must be
determined. Substituting the A-constant demand functions (23) to (25) into the
lifetime wealth constraint in the case of no binding constraints in any market in
any period, leads to
dTAT - Ao
Edt {wtHt [(I.+ p) . tÀ0Dont] - St - PtC t [(1- AoDtPt]
t=o
(26) - qt[Kt[(1 + Aoptqut] - (1 - (5f)Kt-1[(1+ P) t_ l Aopt-iqut-1]1}
where S At denotes that the A-constant demand functions are also substituted
into the tax function. This function implicitly defines )to as a function of the
household's initial value of assets and durables, terminal value of assets, all
prices for all periods, and the time preference rate and the other parameters
determining lifetime preferences. If there are binding constraints in some
markets, the shadow prices of these constraints are also included.
In most cases this equation cannot can be given a closed form solution with
respect to Ao. Even if we could, cf. section 2.3.3, the fact that existing panel
data do not contain complete lifetime price paths, including the interest rate
and the income tax rate paths, and seldom initial and terminal assets and
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