Growth model | Mathematics homework help

1. Consider a growth model with technology transfer. Human capital is accumulated according

to the following law of motion:

˙h

= μeyuAgh1−g

Assume g = 1 and interpret μ as the quality of the education system. Take a poor country

(h < A) that starts out on its balanced growth path. Suppose this country reforms its education

system so as to permanently increase its quality.

(a) Draw a diagram with Ah

on the horizontal axis and ˙

hh

on vertical axis and plot the relationship

between these variables using the above law of motion for h. Indicate the growth rate

of A with a horizontal line. (Hint: your diagram should be similar to the one on Figure

5.1 in the textbook except you are plotting different variables.)

Using this diagram, examine short-run and long-run effects of the increase in μ on the

growth rate of human capital, ˙

hh

.

(b) Plot the behavior of hA

over time. Why is this variable of interest?

(c) Plot the behavior of human capital, h, over time (use log scale). According to the model,

does the reform have a growth effect or a level effect on h?

2. Consider a Malthusian economy with law of motion for population

˙L

= q(y−c)L

where q reflects all factors other then income that influence the birth and death rates (e.g.

hygiene). Production function for this economy is

Y = BXbL1−b

where X is land (fixed), and B is a constant total factor productivity.

(a) Suppose the economy starts in a steady state. At time t = t⇤ a new land is discovered: X

permanently increases to the new level X0. Examine consequences of this discovery on

output per worker, population, and the growth rates of these variables over the transition

to a steady state.

(b) Examine consequences of a permanent improvement in hygiene, as reflected by a new

q0 > q, on output per worker, population, and the growth rates of these variables over the

transition to a steady state.

3. Consider the following aggregate production function:

Y = A(uKK)a (uHH)1−a

where a = 13

, A is total factor productivity (TFP), K is stock of physical capital, H = hL is

stock of human capital, and uj 2 (0,1] is utilization of capital stock j 2 {K,H}. The utilization

rate below 100% is intended to capture “waste” of productive resources for providing

private protection of property rights. Examples of capital diversion include reinforced fences,

monitoring equipment, guns; examples of human capital diversion include design of security

measures, protection services (guards), stress and injuries.

Consider two countries: a rich, developed country R and a poor, developing country P. Suppose

country R has output (income) per worker 20 times higher that of country P. Moreover,

human capital per worker differs by a factor of 2 (hR

hP = 2) while capital per worker is the same

in the two countries ( kR

kP = 1).

(a) Suppose that the utilization rate accurately captures all institutional differences across

countries. If the TFP is identical in both countries (AR = AP), how much the utilization

rate in capital has to differ between the two countries to explain the entire difference in

incomes per worker? That is, assume uR

H = uP

H, and find uˆK =

uR

K

uP

K

such that ˆ y = yR

yP = 20.

(b) Now suppose that, in each country, the utilization rate of human capital equals to the

utilization rate of physical capital: uiK

=uiH

=ui. If the TFP is identical in both countries

(AR =AP), how much the utilization rate has to differ between the two countries to explain

the entire difference in incomes per worker?

(c) Examine your answers (are they quantitatively plausible?). List 3 alternative institutional

explanations of differences in income across countries that would show up as differences

in A. (Hint: refer to Ch. 7).

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