Households have utility u(C,D,I) over consumption C education D and investment I. The house- hold has income Y which it sp ends on these three go o ds at cost 1. Therefore the household's maxi- mization problem is: maxC,D,I u(C,D,I) s.t. Y = C + D + I A. Suppose that the household's utility function is u(C,D,I) = C1−φ−θDθIφ, with 0 < φ + θ ≤1, φ > 0, and θ > 0. Using this functional form, combine the first order conditions for C,D,I into two equations, by eliminating the Lagrange multiplier. B:You now have three equations (the budget constraint and your answer from b) and three unknowns (C, D, and I). Use these equations to show that aggregate education and investment are always prop ortional to aggregate income Y .
Households have utility u(C,D,I) over consumption C education D and investment I. The house-
hold has income Y which it sp ends on these three go o ds at cost 1. Therefore the household's maxi-
mization problem is:
maxC,D,I u(C,D,I)
s.t. Y = C + D + I
A. Suppose that the household's utility function is u(C,D,I) = C1−φ−θDθIφ, with 0 <
φ + θ ≤1, φ > 0, and θ > 0. Using this functional form, combine the first order conditions
for C,D,I into two equations, by eliminating the Lagrange multiplier.
B:You now have three equations (the budget constraint and your answer from b) and
three unknowns (C, D, and I). Use these equations to show that aggregate education
and investment are always prop ortional to aggregate income Y .
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