Economic growth and economic development 267

Introduction to Modern Economic Growth
(1) Rewrite the budget constraint of household h at time t, (5.17), including
these bonds.
(2) Prove an equivalent of Theorem 5.8 in this environment with the extended
set of bonds.
Exercise 5.11. Consider a two-period economy consisting of two types of households. NA households have the utility function
¡ ¢
¡ ¢
u ci1 + β A u ci2 ,

where ci1 and ci2 denotes the consumption of household i into two periods. The
remaining NB households have the utility function
¡ ¢
¡ ¢
u ci1 + β B u ci2 ,

with β B < β A . Each group, respectively, has income yA and yB at date 1, and can

save this to the second date at some exogenously given gross interest rate R. Show
that for general u (·), this economy does not admit a representative household.
Exercise 5.12. Consider an economy consisting of N households each with utility
function at time t = 0 given by
∞
X
t=0

i

¡
¢
β t u ci (t) ,

with β ∈ (0, 1), where c (t) denotes the consumption of household i at time t. The
economy starts with an endowment of Y units of the final good and has access to
no production technology. This endowment can be saved without depreciating or
gaining interest rate between periods.
(1) What are the Arrow-Debreu commodities in this economy?
(2) Characterize the set of Pareto optimal allocations of this economy.
(3) Does Theorem 5.7 apply to this economy?
N

(4) Now consider an allocation of Y units to the households, {y i }i=1 , such that
PN i
i=1 y = Y . Given this allocation, find the unique competitive equilibrium
price vector and the corresponding consumption allocations.

(5) Are all competitive equilibria Pareto optimal?
(6) Now derive a redistribution scheme for decentralizing the entire set of Pareto
optimal allocations?
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