14.451 Lecture Notes Economic Growth

14.451 Lecture Notes
3.5 Comparative Statics and Impulse Responses
3.5.1 Additive Endowment (Figure 2)
• Suppose that each household receives an endowment e > 0 from God. Then, the
household budget is
c j
t + k j
t+1 = wt + rtk j
t +(1− δ)k j
t + e
Optimal consumption growth is thus given again by
which together with rt = f 0 (kt) implies
Uc(c j
t)
Uc(c j
t+1) = β[1 + rt+1 − δ]
ct+1
ct
= {β[1 + f 0 (kt+1) − δ]} θ
On the other hand, adding up the budget across households gives the resource con-
straint of the economy
kt+1 − kt = f(kt) − δkt − ct + e
• We conclude that the phase diagram becomes
˙ct
ct
= θ[f 0 (kt) − δ − ρ],
˙kt = f(kt) − δkt − ct + e.
• In the steady state, k ∗ is independent of e and c ∗ moves one to one with e.
• Consider a permanent increase in e by ∆e. This leads to a parallel shift in the ˙ k =0
locus, but no change in the ˙c =0locus. If the economy was initially at the steady
state, then k stays constant and c simply jumps by exactly e. On the other hand, if the
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