Solow–Swan model

The Solow–Swan model is a long-run economic growth model.

Model assumptions

• single-sector economy
• closed economy (no trade)
• no taxation

etc.

Variables in the model

Name	Variable	Unit	Set of possible values	Rival input?	Variable type	Notes
Output	Y	Units of GDP (dollar?)	[0, ∞)	–	Endogenous
Physical capital (capital stock)	K		[0, ∞)	Yes	Endogenous	Physical capital includes things like machines, computers, buildings, etc.
Labor	L		[0, ∞)	Yes	Exogenous
Technology (knowledge)	A, T			No	Exogenous
Consumption	C
Investment	I
Amount saved	S
Growth of X	$g_{X}={\dot {X}}/X={\frac {\frac {\partial X}{\partial t}}{X}}$	${\text{Time}}^{-1}$
Population growth	$n={\dot {L}}/L$	${\text{Time}}^{-1}$	(−∞, ∞)
Depreciation (rate?)	δ, d, D	Unitless
Capital per worker	k = K/L				Endogenous
Fraction saved	s	Unitless	[0, 1]
Output per worker	y = Y/L				Endogenous
Time	t	Time, e.g. years
Production function	F
Elasticity of output with respect to capital	α	Unitless	(0, 1)

Mathematical formalism

$Y=K^{\alpha }(AL)^{1-\alpha }$ (sometimes also $Y=AK^{\alpha }L^{1-\alpha }$)

${\dot {K}}=sY-\delta K$

TODO show that the model satisfies (1) constant returns to scale; (2) diminishing returns to capital; (3) diminishing returns to labor; (4) the Inada conditions.

A closed form is possible,[1] but it is possible to play around with the model in non-closed forms to extract useful information.

External links

• Solow–Swan model (Wikipedia)