opportunity cost of fertility falls, on the one side, and discourages fertility
because of the reduction in the after-tax marginal productivity of capital, on
the other. Therefore, fertility moves unclearly, while consumption is unam-
biguously reduced.
2.3 Normative analysis
The problem of efficient taxation, known as the ”Ramsey problem”, is stud-
ied by using the so-called ’primal method’ in the version developed by Lucas
and Stokey (1983); such a method is based on the concept of implementabil-
ity constraint, which is obtained from the households’ intertemporal budget
constraint by expressing prices and taxes in terms of quantities through the
marginal conditions (4).12 Optimal taxation is analyzed under the assump-
tion that total government spending is fixed.
Plugging (3), (4a), (4b’) and λ = Uc[0] into (2), we get the implementabil-
ity constraint, given by
0∞[(c-q)Uc
(1-T)(Un - kUc)]e-ρtdt = koUc[O].
(9)
The efficient taxation of factor income is found by maximizing the utility
functional (1) subject to the implementability constraint (9) and the feasi-
bility constraint (6), once the time allocation constraint (3) is brought in.
Define the pseudo-welfare function as
[1-T(n)]
W(c,n,k,ф) = U(c,n) + φ{(c - q)Uc(c,n)--Tt, ʌ [U [Un(c,n) - kUc(c,n)U,
T (n)
where Φ is the Lagrange multiplier associated with (9). Φ is positive in the
case of distortionary taxation of labor income.
The second-best problem can be formulated in a formal way as follows:
12See Lucas (1990), and Chari and Kehoe (1999) for an application of such a method-
ology to the problem of optimal capital taxation.
11