Homework Assignment #1: Answer Key

2. (This question appears more difficult than it really is. Just read carefully, and follow along.
There are no mysteries here, but some interesting ideas.) Consider an economy with a production
function, = ( ) 1−− ,where is resources used in production, all other
variables are as usual, and are positive and + 1. Labor supply grows at the constant
rate, , and grows at the constant rate . We consume resources over time, so the growth
rate of ( ∆ )=−, where is a positive constant. From the production function it follows
that the growth rate of income is given by:
= + +(1− − )[ + ] (1)
where isthegrowthrateofincomeattime, etc. We are interested in a balanced growth
path where all the variables are growing at a constant rate and and grow at the same
rate. Using (1) solve for the balanced growth rate of income,
in terms of the parameters
of the model. Show that
is increasing in , for example.
brief answer By definition on a balanced growth path and aregrowingatthesamerate.
So = so using this in (1) we have:
=
+ +(1− − )[ + ]
=
− +(1− − )[ + ] (2)
where the second line follows from the definitions of the growth rates in the problem.
Solving (2) for
yields:
−
= (1− − )[ + ] −
(1 − − )[ + ] −
=
1 −
addition after class, Sept 22 I really should not have asked you do show that
is increasing
in First, you would need calculus which is not required for the course, and
second if you did differentiate (3) with respect to you would find it is negative not
positive. What I meant to ask is show that
is increasing in or and decreasing in
That would have been a better and easier question. I will grade accordingly.
(a) What happens to
if increases. Why How does
vary with
brief answer If increases resources are declining at a faster rate. From (3) it is
apparent that
falls. We can also see that if rises, so growth is more sensitive
to resources, then
also falls.
(b) The growth rate of output per worker is just − , so using your expression for
what is the expression for the growth rate of output per worker on the balanced growth
path,
Is this necessarily positive Explain.
brief answer Start with (3), and note that − = − , so
− ≡ − +(1− − )[ + ]
= −
1 −
− +(1− − )[ + ]
= − 1 −
1 −
1 −
2
(3)