Assignment #1 is worth 10% of the nal grade. Late assignments will not be accepted.
Notes: You are encouraged to work together with your classmates but: (1) You must try the
problems on your own rst; and (2) The answers handed in must be written by you and reect your
own understanding.
1. Negative Population Growth. (35 points) Many high income countries have fertility rates
below replacement. Declining fertility in developing countries makes it possible and likely that
we will see overall negative population growth rates worldwide in the coming decades. This
question would like you to consider the implications for economic growth (i.e., growth in output
per worker).
(a) Explore the possible implications of negative population growth in the Solow model.
(b) Are the semi-endogenous and endogenous growth models of Romer Chapter 3.1-3.3 appropriate for studying a world with negative population growth? Why or why not?
(c) If you were a Social Planner trying to maximize the welfare of current and future generations
(assuming each generation has an equal weight), what factors would you consider when you
design an optimal policy with respect to the fertility rate?
• Please use this question as a learning opportunity to think through the models and the
main driving forces of growth in each. You can also use this as an opportunity to try to
read some additional literature that expands on the models we have learned. Suggested
resources are: Jones 2020; Jones (video); Economist article; Book on Ageing and Growth.
2. Ramsey Model (35 points)
(a) Explain intuitively why g (the growth rate of A) enters the Euler equation negatively (see
Romer eqn 2.21). Explain how a change in g aects the trade o between consuming today
vs tomorrow.
(b) Compare the eect of a permanent rise in the discount rate ρ and a permanent rise in the
CRRA parameter θ in the Ramsey model. What are the dierences and similarities? How
do these changes impact the modied golden-rule capital stocks relative to their golden rule
counterparts?
(c) Consider the Ramsey model where at some time t0, the government starts subsidizing
investment income at rate τ (an unanticipated change), so the real rate of return on savings
is now higher and equal to r(t) = (1 + τ )f
0
(k(t)). In order to nance this subsidy, the
government uses lump-sum taxes. Use the phase diagram to show the eects on c
∗ and k

.
3. Diamond Model (30 points)
(a) Consider the Diamond model with the modication that now, individuals live to period 2
with probability ψ, where 0 < ψ < 1. With probability (1 − ψ), they die at the end of
period 1. Write their optimization problem and derive the new Euler equation. What is
the eect on optimal savings?
(b) Consider an increase in the CRRA preference parameter θ from θ1 to θ2 where we assume
that 1 < θ1<θ2. If there was dynamic ineciency before the change in preferences, does
the change increase or decrease the size of the ineciency? Use both algebra and intuition
1

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