Technological progress, organizational change and the size of the Human Resources Department

3.2 Household and human capital accumulation

The household in the economy has a utility function given by:

c1-τ    1

u (ct) = ct------1 τ > 0

^{t j} 1 - τ

where c_t is consumption, and to simplify the analysis we then assume τ = 1, that is:

u (ct)=lnct                                 (7)

The household is then endowed with one unit of time supplied each period,
that is spent on working (the fraction T_t^s) or on human capital accumulation (the
fraction 1 - T_t^s), and the accumulation of human capital is described by the following
equation:

ht+ι = Et • h • (1 - /.             0 < δ < 1                  (8)

where Et is an efficiency parameter.

The household’s intertemporal optimization program in the decentralized econ-
omy is then given by¹¹:

∞

max             β^t ln c_t

{ct,T_t^s,at+1,ht+1}_t^∞_{=0       t=0}

a_t+1 =(1+r_t ) a_t + w_t h_tT_t^s - c_t

s.t.

ht+1 = Etht^δ (1 - Tt^s)^1-δ

where β is the discount factor (with 0 <β<1)andat represents the assets held at
time t.

The Lagrangian for this problem is:

L = X©et ln ct + β^tμ_t [(1 + rt) at + WthtTt^s - ct - at+i] +
t=0

+β^t λt ^hEth_t^δ (1 - T_t^s )^1-δ - ht+1 ^io

¹¹ plus the standard transversality conditions.

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