SEMAINE D ETUDE SUR LE ROLE DE IL ANALYSE ECONOMETRIOUE ETC. 307

Section 6 is devoted to the case of a linear utility function,
a case that is often considered, for instance when one chooses
to maximize a discounted sum of the future consumption

stream.
The formulas found in section 3 are no longer applicable
as such, because the solution of the recurrence equations leads
to non-feasible programs. However, the general approach can
be maintained if one introduces upper and lower bounds on
consumption per head c, and labor input per head #,.
Depending on the values of the parameters, the shape of
the optimal program varies a great deal. I consider precisely
a few cases which may be of some interest. For instance, if
the initial endowment of capital is small and if leisure has some
value even when consumption is at its minimum, the optimal
program may imply that the labor input be not pushed at its
maximum in the first periods, but be increased progressively
as capital accumulates, consumption being nevertheless kept at
its minimum until a sufficiently high capital stock has been
reached (see figure 5 in the text).
Notwithstanding the fact that it permits interesting insights.
the assumption of a linear utility function is not quite satisfactory.
 In all cases, the optimal program exhibits some discontinuities
 in its time shape, discontinuities that go against common
 sense. For instance, consumption may switch in one
period from its minimum to its maximum value.
This unsatisfactory feature is partly due to the simplifications
 made in the model, notably to the assumption of a onecommodity,
 one-sector world, and to the assumption of independence
 among the utilities for different periods. However,
optimal programs in more elaborate models would present similar,
 even though less extreme, discountinuities as long as
linear utilities would be assumed. The unescapable conclusion
seems to be that. for the problems considered here, one must

Malinvaud - pag.

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