ECO362H1 Lecture Notes - Lecture 7: Balanced-Growth Equilibrium, Equation, Capital Accumulation
Solow Model
1 Introduction
The Solow model is a simple model of income differences across countries and time. We will
use the Solow model as a foundation for discussing the process of development. The model
begins by making a number of simplifying assumptions that we will relax later on. Some of
the more important assumptions in the model are:
•Households consume and save according to a rule of thumb;
•Representative perfectly competitive firm;
•Single Sector;
•Exogenous productivity (TFP).
In later topics, we will relax each of the above points.
The theory of the Solow model explains cross-country differences in income as a con-
sequence of differences in capital-per-worker. These differences in capital per worker are
driven by differences in savings across countries. The main exploration of the Solow model
will then be to examine how underlying differences in savings rates can lead to differences
in income-per-capita.
We will also use the Solow model as a starting point to examine the role of productiv-
ity.1Similarly to differences in the savings rate, for now we will not seek to explain why
productivity may differ across countries or time. Instead, we will examine the consequences
of these differences on key outcomes.
Objectives:
1. Develop a framework to analyze cross-country income differences
1In this course, we will use the terms total factor productivity (TFP) and productivity interchangeably.
1
2. Introduce concepts of Stationary Equilibrium and Balanced Growth Path (BGP)
3. Agent optimization and equilibrium defintiion
Key Terms:
•Exogenous Variable - a variable that is external to the economy. E.g. TFP, labour
stock, preferences, production technology.
•Endogenous Variable - a variable that is determined internally in the economy. E.g.
capital stock, interest rate, wage rate, savings / investment.
Additional References:
•Weil Chapter 3
2 Solow Model
In this section, we outline the model, introduce the equilibrium concept and solve for the
equilibrium.
2.1 Description of the Economy
Production: Output Ytis produced using capital Ktand labour Nt:
Yt=Ft(Kt, Nt)
where Ftis a function that characterizes the process by which capital and labour are trans-
formed into output.
We write Ftwith a time tsubscript to allow for the production technology to change over
time - i.e. the same amount of (Kt, Nt)may be converted into more (or less) output Ytin
different periods. In addition, the function Fthas the following properties:
•Constant returns to scale in (Kt, Nt): for any λ > 0we have that Ft(λKt, λNt) =
λFt(Kt, Nt)
•Increasing in Ktand Nt:∂Ft
∂Kt≥0and ∂Ft
∂Nt≥0
•Decreasing returns in Ktand Nt:∂2Ft
∂K2
t≤0and ∂2Ft
∂N 2
t≤0
2
In general, we assume that differences across time are captured by a term At, which we
will refer to as total factor productivity (TFP).2This assumes that all variation across time
can be well expressed by a single variable that captures the relative efficiency with which
resources are converted in output. With this in mind, we can rewrite the production function
as:
Yt=F(Kt, AtNt)
where, for convenience, we write productivity as being labour augmenting.3We will refer to
AtNtas augmented labour.
In addition, we assume that Ftakes a Cobb-Douglas form such that we can further rewrite
output as:
Yt=Kα
t(AtNt)1−α(1)
where α∈(0,1) describes the returns to scale on capital while 1−α∈(0,1) describes the
returns to scale on labour. For the remainder of this note we will consider that production
takes the Cobb-Douglas Form.
Capital Accumulation: Capital accumulation (or Law of Motion for capital) can be writ-
ten as:
Kt+1 =Kt−δKt+Xt
where δ∈(0,1) is the depreciation rate of capital. The amount of capital lost from depreci-
ation is then equal to δKt. In the Solow model we assume that a constant fraction of output
s∈(0,1) is saved as capital for the next period, such that Xt=sYt. We can then rewrite
the capital accumulation equation as:
Kt+1 = (1 −δ)Kt+sYt(2)
Resource Constraint: Finally, we can write the resource constraint in the economy. The
2This abstracts from other types of temporal changes that we may think are important. For example,
two of these changes we may consider are: (1) changes in factor intensities across time (e.g. Automation /
robots); (2) changes in functional form across time (e.g. inclusion of other factors, such as land).
3This is without loss of generality. Note that we can define ˜
At=A1−α
tand rewrite the production
function as Yt=˜
AtKα
tN1−α
t.
3
Document Summary
The solow model is a simple model of income di erences across countries and time. We will use the solow model as a foundation for discussing the process of development. The model begins by making a number of simplifying assumptions that we will relax later on. Some of the more important assumptions in the model are: households consume and save according to a rule of thumb, representative perfectly competitive rm, single sector, exogenous productivity (tfp). In later topics, we will relax each of the above points. The theory of the solow model explains cross-country di erences in income as a con- sequence of di erences in capital-per-worker. These di erences in capital per worker are driven by di erences in savings across countries. The main exploration of the solow model will then be to examine how underlying di erences in savings rates can lead to di erences in income-per-capita.
