2.3 General Equilibrium
The life cycle model is used to simulate steady state equilibrium effects of the
restriction on time allocation and the supplementary tax. Since population
growth is not important to the analysis, each cohort size is kept constant and
normalized at unity. Prices and aggregate quantities are constant in steady state,
and the variables do not carry time indices unless necessary.
Consumption and investment decisions by the representative individuals re-
fleet the behavior of current generations in the steady state equilibrium. It is
therefore simple to derive the aggregate supply of labor services to the labor
market and the aggregate private demand for leisure, learning and consumption
in the steady state. The aggregate supply of labor services is equal to the weighted
sum of the supply of labor services over the life cycle for the two different types
of individuals:
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l = ε ¢ X lc,t + (1 - ɛ) ¢ X ln,t, (8)
t=0 t=0
where L is the aggregate supply of labor services in steady state, and ε is the
share of the population subject to conscription.
The production of final goods combines labor services and physical capital,
and the technology is represented by a Cobb-Douglas specification:
Y = Kφ ■ L(1"≠) (9)
where Y is aggregate output, K is the aggregate stock of physical capital, and φ
is the value share of physical capital. Each producer of goods maximizes profits
subject to the production technology (9), and the first order conditions imply
that the marginal product of a particular factor input is equal to the producer
price of that factor input.
Physical capital depreciates at rate δκ > 0. The capital stock in period t is
equal to the capital stock at the beginning of the previous period less depreciation
plus investment in the previous period. As the capital stock is constant in the
steady state, gross investment I in physical capital is given by:
I = δκ ■ K. (10)
Aggregate output is either invested or consumed by individuals or the public
sector, and the market clearing condition for output is:
Y = C + I + G, (11)
where C is aggregate private consumption and G is the public consumption. Fi-
nally, the government operates with a dynamic budget constraint that is balanced
in each period:
τl ∙ w ∙ L + ε ∙ (τ ∙ w(1 — τl) ∙ lc,o) = G. (12)