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
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.

TODO talk about the point of the model. What do we want out of it? (1) We want to know what happens to all the endogenous variables given the equations and the exogenous variables; (2) we want to know what happens to the output as a whole when we adjust the parameters ("comparative statics"). The latter is what tells us things like "increase the savings rate to grow the economy".

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)