Empirical life cycle models of labour supply and - Statistisk sentralbyrå

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Social and Economic Studies 9.1 Empirical Life Cycle Models as a profit maximization problem, the requirement that the Frisch demands are linear in pu 1/A implies that demand is linearly related to within-period full income. Full income at age t, Yt , is defined as the sum of the value of the household's initial time endowment 1E, and its interest incomes and asset decumulation, (39) Yt = wtr; rtAt-i - (At - At-i)• The profit function approach, however, allows for non-separable within period preferences. Blundell, Fry and Meghir (1985) show an analogous result within the same framework, that is, relaxing intraperiod additive separability leads to unitary within-period full income elasticities if the loglinear Frisch demands are to be loglinear in Ao. The use of Frisch demand functions that are derived either from a specification of direct utility or from the profit function, then introduces quite strong restrictions on intratemporal preferences. Intuitively we would consider that the use of highly aggregated Hicks composite goods reduces the restrictiveness of assuming intraperiod additive separability. Intraperiod additive separability can then turn out to be a reasonable approximation for life cycle models of highly aggregated Hicks composite goods. The assumption of intertemporal additive separability can also be questioned. According to Blundell (1987), the indirect within-period utility at age t corresponding to the profit function of Browning et al., is of the Gorman Polar form (40) G =I bt(124) where It is some concave monotonic transformation that determines the intertemporal allocation of the goods, and at (pt) and bt (pt) are particular concave linear homogeneous functions of the vector pt of prices discounted to period zero. Blundell uses this specification to discuss the restrictions on intertemporal substitution implied by the profit function of Browning, Deaton and Irish. He follows Browning (1985) and assumes that the intertemporal substitution possibilities best can be measured by the intertemporal elasticity of substitution, = GyIYGyy , where Gy OGIOY and Gyy °GSM'. The utility index (40 implies (41) (1)t l yt at (Et ) 4{ [Yt at(pdlibt(Pt )IMEt ) — at(pt )]lbt(p)lYt where denotes partial derivative. Since Yt at (pt) is dominated by Yt , this means that bt (pt) represents the substitution possibilities for rich people. This 39
Empirical Life Cycle Models Social and Economic Studies 91 result and the fact that the specification of the profit function implies that bt (pt ) is a linear function of all the prices of pt , means that the use of linear Frisch demands constraints intertemporal substitution for the high income groups. It can be argued that this matter is not very important since the substitution possibilities are also determined by the derivatives of the monotonic transformation It. Many works, however, let h be the identity transformation, and Blundell shows that the substitution elasticity for the loglinear and the exponential transformations approaches -1 and 0 as income increases. Linear or loglinear Frisch demands then also seem to restrict intertemporal substitution possibilities considerably. 2.3.1.4 Income taxation and MaCurdy's fixed effect approach As mentioned MaCurdy disregarded income taxation and assumed that the interest rate was constant across periods and equal for all persons. In this case the Frisch demands only include current variables. Without these simplifications the Frisch labour supply function corresponding to equation (30) becomes (42) ln Hit = Fics + am [Dit (1 P) tmit] 4- eie where we recall that Di t is the after-tax marginal discount rate defined in equation (19), and mit is the after-tax marginal wage rate. Since the discount rate Dit is endogenous, we can hardly assume it is constant during the life cycle, and in no way we can reasonably assume it is equal for all persons. Taking also into consideration that Dit depends on Rio , Ra, Rit , the introduction of income taxes immediately seems to complicate estimation considerably. However, first differencing the Frisch demands yields (43) A In Hit = ap -1- a[6. In mit - In (1 + Rit)} which illustrates that the method of differencing is attractive even in this case. MaCurdy's fixed effect approach can then very well be used even in the case of personal income taxation, but we must take account of that the introduction of income taxes makes mit and Rit endogenous to the consumer and dependent on At-1- 2.3.2 A Fixed Effect Tobit Model Heckman and MaCurdy (1980) extend MaCurdy's fixed effect approach by implementing a bivariate fixed effect Tobit model for married females' labour supply. This work uses the utility function (44) E(i + p)_ t [FaCr +Fit(T, - t=o 40
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