Correlation between Skill Level and Time Preference
So far we have assumed a uniform distribution of all types implying that
the correlation between skill level and time preference is zero, whereas in
the first section the two-type model assumed perfect positive dependence
between skill and time preference. However, this may not be the case; the two
characteristics may be imperfectly correlated. At the same time, changing
the assumption of the correlation allows us to consider the robustness of our
results with respect to the distribution of types. Figures 1 and 2 present the
marginal tax rates on savings in the welfarist and paternalist cases with the
CES utility function.
From the figures we can see that the results for the marginal tax on savings
are fairly robust except for type 3. As the correlation between skill level and
time preference increases, or as the fraction of types 2 and 3 decreases, the
distortion on type 3’s savings decision increases significantly with both types
of government preferences. In the welfarist case, type 4 remains undistorted
regardless of the structure of the economy. Type 1 is also undistorted except
at very high levels of correlation, where his savings become taxed. In the
paternalistic case types 1 and 2 remain pooled and the tax on savings for type
4 is almost fixed at all levels of correlation. Note the two-type case (section 2)
which is equivalent to a correlation of one corresponds to the right boundaries
of Figures 1 and 2.