Econ 101 Macroeconomics11st Semester, AY 2014-2015

Solow Growth Model

1 Aggregate Variables

Y = C + IY C + S

=) S = I:

Y = F (K;L) ;

where F () is the standard neoclassical production function, which exhibits constant returns to scale (CRTS),FK ; FL > 0 and FKK ; FLL < 0:Gross investment is given by

I =K + K;

whereK dKdt and 2 (0; 1) is the constant depreciation rate of capital.

There is no unemployment in this economy, so that L denotes both population and total labor supply.Population grows at a constant rate; n > 0; i.e.,

L

L= n > 0:

2 Per-capita Variables

In per capita terms,

Y

L=C

L+I

L=) y = c+ i

Y

L=C

L+S

L=) y = c+ S

L:

Constant returns to scale implies that

zY = F (zK; zL) ; z =1

LY

L= F

K

L; 1

=) y = f (k) :

Per capita investment is given by

I

L=

K

L+

K

L=) i =

K

L+ k: (1)

To deriveK=L; we start with

1S. Daway

1

k KL=)

k

k=

K

K

L

Lk

k=

K

K n

k =

K

Kk nk

k =

K

L nk

K

L=

k + nk: (2)

Plugging in (2) into (1) and rearranging yields

k = fig (n+ ) k:

k T 0() i T (n+ ) k:

The term (n+ ) k is called the break-even level of investment. When i = (n+)k; per-capita investmentis just enough (or per capita capital is growing just enough) to equip the new entrants into the labor marketand to replace capital that has depreciated.

k = fy cg (n+ ) kk = ff (k) cg (n+ ) k (3)

k =

S

L

(n+ ) k: (4)

Assuming that savings are a constant fraction of output, denoted by s 2 (0; 1) ; i.e.,

S = sYS

L= s

Y

L=) S

L= sy

S

L= sf (k) : (5)

Plugging in (5) into (4) yields

k = sf (k) (n+ ) k; (6)

which is the fundamental neoclassical growth equation.

k = 0 =) sf (k) = (n+ ) k:

In graphical terms,

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What happens when s increases? What about if either n or increases?

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