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Consider an economy of a country whose growth process can be described by the following system of equations:
Production function
Q = Kα(AhL)1-α (1) Physical capital formation
K = (ZZ(-1))0.5 (2.1) Z = (1-δ)Z(-1) + sQ (2.2)
Human capital formation h = exp(φu) (3.1)
u = u(0)+μt (3.2)
Growth of labor L = L(0)(1+n)t (4)
The system has the following features:
(1) An aggregate production function in which physical capital input (K) and labor input (L), with labor-augmenting (i) technology (A) and (ii) human capital (h) determine aggregate real output (Q). In this closed economy, the population is equal to labor force, which is equal to labor input.
Q = Kα(AhL)1-α (1)
(2) Physical capital input is determined as follows:
(2.1) Physical capital input (K) is the geometric average of the beginning and the ending stocks (Z, Z(-1)) of physical capital.
K = (ZZ(-1))0.5 (2.1)
(2.2) Physical capital stock grows in accordance with Solow’s equation of motion.
Z = (1-δ)Z(-1) + sQ (2.2)
(3) Human capital is determined by education.
(3.1) Human capital per worker is a function of the average number of years of schooling of workers.
h = exp(φu) (3.1)
(3.2) The average number of years of schooling of workers grows linearly through time.
u = u(0)+μt (3.2)
(4) Labor input grows exponentially through time.
c. The variables are defined as follows:
Q = aggregate real output (flow) for year t in billions of $ per year
K = aggregate capital input (flow) for year t in billions of $ per year
Z = aggregate capital stock as of end of year t in billions of $
Z(-1) = Z as of the end of year t-1 (or the beginning of year t)
h = labor-augmenting average human capital of workers measured as a factor
that augments labor for year t
u = average number of years of schooling of workers for year t
L = labor input (flow) for year t
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A = index of technology that augments labor, but A is also determined by the
units chosen for L, u, and K.
The h variable may be understood as a factor by which schooling augments labor. For example, a value of h = 1 means that schooling does not augment labor. A value of h = 1.5 means that schooling makes an average worker as effective as 1.5 worker without any schooling.
The unit of A may be understood as a factor analogous to h. While their units have similar interpretation, A and h are entirely different goods and have different economic features, as we have discussed. Moreover, the unit of A also depends on the units chosen for L and K.
d. The assumed parameter values are as follows:
α = 0.33 physical capital share (equal to Solow’s original estimate)
δ = 0.05 annual depreciation rate of physical capital (assumption;
Jones uses this number)
φ = 0.1 productivity of schooling in the production of human capital
(based on labor market research; Jones uses this number)
μ = 0.15 annual average increase in u (assumption; this economy is
assumed to increase u from 6 to 10.5 in 30 years)
n = 0.02 annual rate of increase in L (assumption, similar to those in
LDCs, some have lower rate)
s = 0.3 saving rate (assumption, similar to historical average of
Korean saving rate, but is higher than international standards)
e. Initial values
A(1) = 0.530 (set to accommodate the assumed units of K, L, and u; and
also to make K/Q equal to about 1 for year 1)
Z(0) = 19 (set to make K/Q equal to about 1 for year 1)
u(0) = 6 (assumption, similar to Korean initial u, shortly after independence)
L(0) = 20 (assumption)
The initial values have been given to make this country’s per worker GDP equal to about $1,000 per year. Of course, we will see later, how this country’s per worker physical capital, physical capital-output ratio, and output per worker grow over time.
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问题;
a. Eventually, you will use the system and the initial conditions to generate data for Q and K for 30 years (t = 1, …, 30).
b. Following is the first of a series of assignments:
(1) Use the information above to generate L and h for t = 1 and calculate
L(1), u(1), and h(1).
(2) Write down a non-linear system of three equations
production function defining aggregate output (Q),
definition of physical capital input (K), and
equation of motion for physical capital stock (Z).
for period t = 1.
The system is non-linear—some equations (production function and the function that defines K) are log-linear; one of them (the equation of motion) is linear in Z(1-) and Q.
You can reduce this system of 3 equations (for Q, K, and Z) to a system of 2 equations (for Q and Z), by eliminating K.
This reduced system of 2 equations determines Q(1), and Z(1), with assumed parameter values, and initial conditions for A(1), Z(0), L(1), and h(1), which you derive from the education production function.
To this end, take the following steps:
(2.1) Write down a non-linear system of equation for Q and Z with given parameters and initial values for A(1), Z(0), L(0), and h(0).
(2.2) Solve the system of equations for Z(1) and Q(1). You need to think about how to solve this non-linear system, given the exogenous variables and the initial conditions.
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