Problem
Set 3

MSU
EC 410

Prof.
Ahlin due 11/10/15

1a. Use the H-augmented Solow model to
determine the a) instantaneous impact on GDP per capita, b) instantaneous
impact on consumption per capita, c) long-run impact on GDP per capita, d)
long-run impact on consumption per capita, e) impact on long-run GDP per capita
growth rate, and f) impact on long-run GDP growth rate of a permanent and instantaneous increase in the
fraction of national resources devoted to investment in human capital, q. Assume the country begins at its steady state
values of k^* and h^* before this event occurs. Justify your answer by use of graph and/or
equation.

1b. How does each answer compare to the answer
the original Solow model would give when s increases, both qualitatively (whether the amount goes up or down) and
quantitatively (the amount by which
it goes up or down)?

2. Consider the Solow model with total factor productivity A_t
constantly growing at rate g>0.

a. Determine
the a) instantaneous impact on GDP per capita, b) instantaneous impact on
consumption per capita, c) long-run impact on GDP per capita (i.e. compare the
level of GDP per capita with and without the parameter change, in the
long-run), d) long-run impact on consumption per capita (i.e. compare the level
of consumption per capita with and without the parameter change, in the
long-run), and e) impact on long-run GDP per capita growth rate of a one-time
and instantaneous increase (jump) in productivity A_t,
through a significant and non-repeatable invention. Assume the country begins at its “steady
state value” of k^* before this event occurs. Justify your answer by use of graph and/or
equation. [Hint: this should not be
considered a change in g, since productivity resumes growth at rate g after the
one-time jump; it should be modeled as a onetime jump in A_t.]

b. Graph
the path of y_t and c_t against time (or better yet, ln(y_t)
and ln(c_t), which will be linear) for the event analyzed in part a.

c.
Repeat parts a&b for a permanent, instantaneous increase
in the growth rate of productivity, g.

3.
Growth Simulations. See PS3GrowthSimulationQuestion.xlsx posted
on D2L. Fill in 200 years of data using
the H-D model, the Solow model, and the H-Solow model using the functions and
parameters given in the “GrowthCalculations” worksheet. The savings rate in all cases increases to
30% at year 25.

Specifically:

3.1.
For the H-D model, A=0.25, n=0.01, d=0.04, and
s=0.2. Capital per person starts at
$4000. Fill out k, y,
ln(y),
c,
ln(c),
actual investment, break-even investment, ?k,
and g_y
for 200 years.

3.2.
For the Solow model, f(k) = k^1/3,
A=50, n=0.005, d=0.02, and s=0.2.
Capital per person starts at $8000.
Fill out k, y, c, actual investment,
break-even investment, ?k,
and g_y
for 200 years.

3.3.
For the H-Solow model, f(k,h) = k^1/3h^1/3,
A=5, n=0.01, d=0.04, and s=0.2. Physical
capital per person starts at $4000, human capital per person starts at
2000. Fill out k, h,
y,
c,
?k,
?h,
and g_y
for 200 years.

Note that in all cases, the savings rate s switches at 0.3 at the
25th year. Make sure to incorporate this
in your answers. [Hint: it will only
affect the consumption formula and the actual investment column formula for the
H-D and Solow models, and it will only affect the consumption formula and the ?k column formula for
the HSolow model.] [Hint: Of course, you
need only specify each column’s formula once, then copy and paste down the
column for all the years. The formulas are
pretty straightforward, and can be found by looking back at the key equations
for each model. It is simplest for
actual investment not to recalculate income, but simply use the fact that
actual investment equals a fixed fraction of income, sy in the case of
physical capital and qy in the case of human capital.]

1

a.
Give the income and consumption levels in year
200 for each of the three models. In
which model is the increase in s most effective? In which model is it least effective? Justify your answer.

b.
Look at the graphs for the three models (which
are in the other worksheets and should be filled out automatically from the
data you generate in the GrowthCalculations worksheet). Look at both H-D graphs, but focus on the one
using logs. Discuss one significant way
in which all three models’ graphs are similar.
How do the Solow and H-Solow graphs differ?

4. Imagine that
a bank will only lend if it can earn a rate of return of 6% on a loan. Further, imagine it incurs administrative
costs of $40 per loan it makes, regardless of the size of the loan. Throughout the problem, assume for simplicity
that the loans are all repaid with certainty, i.e. there is no risk.

a.
If the bank makes five loans – of $100, $200,
$500, $1000, and $10,000 – what are the respective interest rates it must
charge to break even on each loan?

b.
Imagine the bank makes the same loans but must
charge all borrowers the same interest rate.
What interest rate will it charge to break even overall? Which borrowers pay less, which pay more in
this case than in part a.? This practice
of making losses on some loans and profits on others is called
“cross-subsidization”. c. How
might competition between banks eliminate any one bank’s ability to
cross-subsidize smaller borrowers?
Specifically, ci) could a rival lender lure away any of the customers of
a bank carrying out the policy of part b., and cii) how would this affect the
ability to cross-subsidize of a bank carrying out the policy of part b.?

d.
It may not be accurate to assume
that every loan incurs the same administrative cost, irrespective of size.

Larger loans may require more work. Redo part a. under the assumption that the
administrative cost of a loan is

$40 per loan plus 1% of the size of the loan. (Thus a loan of $5000 would cost the bank $40
+ 1%*$5000 = $90, while a loan of $500 would cost the bank $40 + 1%*$500 =
$45. The cost structure is still linear,
but with a positive intercept andslope.)

2