Differentiating (12) and combining it with the Slutsky equation (3) for each type
of household yields
dp = αxχm :
(13)
(14)
dp ∆
dp -α(p — p)x1m
dx ∆
where
∆ = αS +(1— α)S — xm(xs — αx)) — (1 — α)x (xm — xm) — xsp. (15)
If x1 ≤ x2 and xs — αx) ≥ 0, then, given that S1 < 0 and S2 < 0, we have ∆ < 0.
mm
Since both types of household face the same market price p, the condition
x1m ≤ x2m holds for many common specifications of preferences. Thus, homothetic
and quasi-homothetic preferences with linear Engel curves have x1m = x2m. Fur-
thermore, much of DTP has been of necessities - for example, many foods and
housing. To the households employed in the non-state sector, without the direct
allocation from the plan-track, those goods may be considered more as luxuries
compared to the households with government subsidy, who pay less for those goods
in money terms. Therefore, we view x1m <x2m as a reasonable assumption, reflect-
ing a concave Engel curve. αx) is the total ration allocated by the central planning
system, which is part of the total supply of X . This gives our next proposition.
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