K .A. Konrad Journal of Public Economics 79 2001 503 –519
507
to i’s second-period expected utility. It is not specified exactly what constitutes educational investment. One possible interpretation is that this investment is
mental effort or forgone leisure in Period 1, which explains why these resources
9
may enter additively separably in the payoff function.
3. The laissez-faire outcome
Before considering time-consistent optimal income taxes we analyse the laissez- faire equilibrium outcome. If there is no government to impose restrictions on
earnings choices and to implement redistributive taxation, each individual maxi- mizes 1 subject to x 5 m
and x 5 m .
H H
L L
The first-order conditions which implicitly determine the optimal values e,m,m are:
H L
9
u9m 5 c m 2
H H
H
9
u9m 5 c m 3
L L
L
and p9e
h[um 2 c m] 2 [um 2 c m]j 5 1 4
H H
H L
L L
These conditions have the natural interpretations that earnings effort is optimally chosen so that marginal utility of income and marginal disutility of generating
income are equal for each type of productivity, and that the marginal benefit of investment in education equals the marginal investment cost. The marginal benefit
from an increase in e is the increase in probability to become the highly
i
productive type times the difference between utilities of persons with high and low productivities.
Note that the high effectiveness of the first marginal units of education investment implies that e . 0 in this equilibrium.
4. Taxation and time consistency
Consider now income taxation by a benevolent government. I do not solve the ex ante optimal time inconsistent tax problem here. The benevolent government
i
would choose a tax schedule T 5 Tm to maximize the expected utility U as in
i i
1 with: x 5 m 2 T
5
i i
i
9
The additive separability of payoff is made only for simplicity.
508 K
.A. Konrad Journal of Public Economics 79 2001 503 –519
subject to the government’s budget constraint and subject to the individually rational choices of income and investment in education, m and e . The solution of
i i
this problem is not time consistent, because the government would like to implement a different tax policy in Period 2 i.e. 20 years later when the
education decisions are made. Boadway et al. 1996 compare the outcome in this ex ante optimal tax policy and in the case with time-consistent taxation.
This paper analyses a different comparison: it considers only time-consistent optimal tax policy and compares the cases with complete and incomplete
information in the next two sections, showing that welfare with time-consistent optimal taxation is higher if the government is incompletely informed.
Summarizing, the time structure of games in these sections will be as follows. In Stage 1, individuals i [ [0,1] choose their educational efforts e . In Stage 2, nature
i
decides whether an individual is more or less productive, in the sense of having effort functions c or c , with pe individual i’s probability for becoming highly
H L
i
productive. Individuals learn their own productivity at this stage. This concludes Period 1. In Stage 3 government implements the Mirrlees 1971 optimal tax
policy for a given distribution of productivities. This tax policy is an income tax or subsidy as a function of observed gross income, and possibly, as a function of
productivity, if the government can observe individual productivity. In Stage 4 each individual chooses his or her actual gross income and pays taxes or receives
subsidies accordingly.
5. Complete information
