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The formulas found in section 3 are no longer applicable 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 satisfac- 
tory. In all cases, the optimal program exhibits some discon- 
tinuities in its time shape, discontinuities that go against com- 
mon sense. For instance, consumption may switch in one 
period from its minimum to its maximum value. 
This unsatisfactory feature is partly due to the simplifica- 
tions made in the model, notably to the assumption of a one- 
commodity, one-sector world, and to the assumption of inde- 
pendence among the utilities for different periods. However, 
optimal programs in more elaborate models would present si- 
milar, 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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