2.1 The household
The population is assumed to be constant and normalized to one. The representative
agent has logarithmic preferences over sequences of consumption:
∞
U (c0,c1, ..) = Et
βtln(ct)
(1)
t=0
where ct is the level of consumption in period t ∈ N0 and β ∈ (0, 1) is the subjective
discount factor. Expectations are formed over the sequence of shocks {εt}t∞=0 entering
goods production. The logarithmic utility function implies that the intertemporal elas-
ticity of substitution is equal to one. In each period, agents have a fixed endowment
of time, which is normalized to one unit. The variable ut denotes the fraction of time
allocated to goods production in period t. Furthermore, as agents do not benefit from
leisure, the whole time budget is allocated to the two production sectors. The fraction
1 - ut of time is spent in the schooling sector. Note that in any solution the condition
ut ∈ [0, 1]
(2)
has to be fulfilled. The variables ct and ut are the two control variables of the agent.
When maximizing her discounted stream of utility, the agent has to pay attention to the
following budget constraint:
τrrtkt + τwwtutht = ct + kt+1, ∀t ∈ N0, k0 > 0, (3)
where kt is the agent’s physical capital stock in period t. The terms τrrtkt and τwwtutht
are, respectively, the net returns on physical capital and work effort after taxation. We
assume that both parameters τr and τw are positive. If the parameter τr is smaller than
1, we have a tax on physical capital, if it is larger than 1, we have a subsidy. The same
is true for the parameter τw. If τw < 1, work effort is taxed, if τw > 1, work effort is
subsidized. Hence, the rates of taxation are given by τr - 1 and τw - 1, respectively. The
above constraint implies full depreciation of physical capital. The variables rt and wt are
market-clearing factor prices. Prices and tax rates are endogenous to the model. The
former via the market clearing mechanism and the latter via the government’s balanced
budget condition. Despite this fact, prices and tax rates are taken as given by the
representative agent. The left-hand side describes her income derived from physical
capital plus the income stemming from effective work, which is determined by the worker’s
level of human capital ht multiplied by the fraction of time spent in the goods sector in
period t, i.e. htut . We assume that the initial values k0 and h0 are strictly positive. On
the right-hand side, the spending of the agent’s earnings appears, which she can either
consume or invest. Another constraint the agent has to keep in mind is the evolution of
her stock of human capital when allocating 1 - ut to the schooling sector.
2.2 The schooling sector
The creation of human capital is determined by a linear technology in human capital
only:
ht+1 =B(1-ut)ht, ∀t ∈N0, h0 > 0, (4)
where we assume that B is positive1 . If we set ut in equation (4) equal to zero, we get
the potential stock of next period’s human capital. If we set ut equal to one, tomorrow’s
1 The case when B equals 0 corresponds to the neoclassical growth model.