Fall Semester ’05-’06
Akila Weerapana
Lecture 6: Assessing the Basic Solow Model
I. OVERVIEW
•In the last class we looked at a couple of comparative statics exercises. The first was the
impact of a change in the saving rate on capital and output in the economy. We showed that
the change in the saving rate did not have a long run impact on the growth rate of output.
•However, there was a short run increase in the growth rate, and as a result the economy
reached a higher level of steady state output. A change in the saving rate is therefore said
to have a level effect on output but NOT have a growth effect; i.e. it affects the steady
state level of output but not the long run growth rate of output.
•The second was the impact of a change in the population growth rate on capital and output
in the economy. We showed that the change in the population growth rate, while raising
both the level and the growth rate of total output in the economy, left the level of per-capita
output lower than it would have been in the absence of the increased population growth.
•Today’s class looks at some of the mathematical properties of the steady state to understand
how economic changes affect the endogenous variables of the model.
II. COMPARING STEADY STATES USING ALGEBRA
•In economics, we typically use calculus to do comparative statics exercises. Given an economic
model, we would solve for the endogenous variables as functions of the exogenous variables
and the parameters, then take derivatives (or partial derivatives) to show how the endogenous
variable will change when an exogenous variable or a parameter changes.
•However, thus far with the Solow model we have used diagrams instead of calculus to perform
the various comparative statics exercises. The reason for doing so is that the Solow model
is a dynamic model whose behavior is governed by a differential equation, the now familiar
˙
k=sy −(n+δ)kequation.
•Since differential equations is not a pre-requisite for this course, we won’t solve this equation
for the value of kat a given point in time. We can however, avoid the complications of
the differential equation and use calculus if we restrict our focus to the steady state. Why?
Because at the steady state ˙
k= 0, thus eliminating the troublesome left-hand side of the
above equation.
•The capital accumulation equation can now be written as (using * to denote steady-state
values) 0 = sy∗−(n+δ)k∗. This simplifies to
