Basic Growth Theory

Notes on Growth Theory

1        Growth Accounting

• Notations: Y –output or GDP, K–capital, L–labor, A–technology or total factor productivity (TFP); subscript t denotes the time, e.g.  Yt  is the output at time  t.
• The change of variable X is defined by

X can be Y, K, L, or A.

• Growth rate γX of variable X is

∆Xt = Xt+1 − Xt,

γX =

Xt+1 − Xt

Xt

= ∆Xt . Xt

– For example, growth rate of GDP, γY  = ∆Yt/Yt, growth rate of capital γK = ∆Kt/Kt.

• The aggregate production function has the form

Yt = AtKαL1−α,

t        t

i.e. the Cobb-Douglas production function, where 0 < α < 1.

1. Basic Introduction of Natural Logarithm Function  ln

Deftnition 1.1. A real number y = ln x if ey = x.

Properties of natural logarithm   function

Proposition 1.1. For any two strictly positive real numbers a and b, ln ab = ln a + ln b. Proof.  Note that

therefore, ln a + ln b = ln ab.

eln a+ln b = .eln a. .eln b. = ab,

Proposition 1.2. For any strictly positive number a and any real number β, ln(aβ ) = β ln a. Proof.  Note that