# PROBLEM SET 3

# Problem Set 3

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

and h*Apply

before this

event occurs. Justify your answer by using a 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. At constant growth 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 At, 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 using a 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 one-time

jump in at

b. Graph the path of yt and ct against time (or better yet, ln(yt) and ln(ct), which will be linear) for the event.

analyzed in Part A.

c. Repeat parts a and 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 age 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, and

ln(y), c, ln(c), actual investment, break-even investment,?k, and gy 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 gy 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 for 200 years.

Note that in all cases, the savings rate switches to 0.3 at the 25th year. Make sure to incorporate this into 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 only need to specify each column’s formula once, then copy and paste it 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

of human capital.

2

a. Give the income and consumption levels in year 200 for each of the three models. In which model is the

increase in the 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 model graphs are similar. How do

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 and 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, 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 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 and slope.)

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