GR5215
Ron Miller
Fall 2021
Assignment 2; due Thursday October 7 in class
1) Assume that the rate of exogenous technological progress is constant in the Solow
model, but assume that the production function is Y = F [B(t)K ,A(t)L], where
B(t)=ezt and A(T)=ext , with z ≥ 0 and x ≥ 0. Show that if z > 0 and a steady state exists,
the production function must take the Co
–Douglas form.
2) Transitional dynamics in the Solow model.
Simulate a discrete time version of the Solow model, where the unit of time is one year.
Assume a Co
-Douglas production function:
Population grows at 2% per year, annual depreciation is 3%, and the savings rate is 25%.
There is no technological change.
To start with assume α = 0.33.
a) Solve for the steady-state capital-labor ratio. (You can do this from the continuous
time equations we examined in class.)
) Simulate the model starting from L0=1 (that’s just a normalization) and K0 = 0.05
of the steady state K/L ratio. (You can use any software you want to do the simulation.
Excel will do.) Include the simulation results in your answer. About how many years
does it take to reach steady-state? (It won’t get there exactly – state how you have defined
convergence.)
c) Repeat the simulation with α = 0.7. Now how long does convergence take?
3) Consider the two-period consumption problem with exogenous income and given
interest rate r. Suppose that the government taxes interest income at the rate Ï„. The
government’s revenue will be zero in period 1 and τr(Y1 –C1) in period 2.
a) Write out the individual’s budget constraint.
Now suppose the government eliminates the taxation of interest income
and instead institutes lump-sum taxes of amounts T1 and T2 in the two periods.
) Write out the individual’s budget constraint under this alternate tax regime.
c) What condition must the new taxes satisfy so that the change does not affect the
present value of government revenue?
d) If the new taxes satisfy the condition in part (c) is the old consumption
undle, not affordable, just affordable, or affordable with room to
spare?
e) If the new taxes satisfy the condition in part (c) does first-period consumption
ise, fall, or stay the same?
4) a) Write down a version of the Solow growth model in which the labor force is a
fraction ! of the population. For simplicity, assume no population growth. Write
down the equation for steady state under the assumption that != !#, a constant.
Does this change have any substantive effect on the Solow model?
Y = AK αL1−α
GR5215
Ron Miller
Fall 2021
) Suppose ! grows at a constant rate, !$. How does this steady increase affect the
growth rate of per capita income? Solve for the steady state in this model.
c) Briefly explain how your answer to part (b) may help explain high growth rates in
some developing countries. Why is this source of growth necessarily temporary?
Graphically depict the transition from a positive !$ to zero labor force participation
growth.