Solow–Swan model - Revision history

Issa Rice: /* Mathematical formalism */

2017-09-16T19:50:27Z

Mathematical formalism

← Older revision		Revision as of 19:50, 16 September 2017
Line 55:		Line 55:
	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 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 table of comparative statics~~ in ~~a diff section~~		A closed form is possible,[https://web.stanford.edu/~chadj/closedform.pdf] but it is possible to play around with the model in non-closed forms to extract useful information.

	~~A closed form is possible,[https://web.stanford.edu/~chadj/closedform.pdf] but it is possible to play around with the model in non-closed forms to extract useful information.~~		==Table of comparative statics==

	==History==		==History==

Issa Rice: /* Mathematical formalism */ move

2017-09-16T19:49:20Z

Mathematical formalism: move

← Older revision		Revision as of 19:49, 16 September 2017
Line 55:		Line 55:
	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 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".~~		TODO table of comparative statics in a diff section

	A closed form is possible,[https://web.stanford.edu/~chadj/closedform.pdf] but it is possible to play around with the model in non-closed forms to extract useful information.		A closed form is possible,[https://web.stanford.edu/~chadj/closedform.pdf] but it is possible to play around with the model in non-closed forms to extract useful information.

Issa Rice: /* Variables in the model */

2017-09-16T06:30:53Z

Variables in the model

← Older revision		Revision as of 06:30, 16 September 2017
Line 30:		Line 30:
	\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\| \|\|		\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\| \|\|
	\|-		\|-
	\| Population growth \|\| <math>n = \dot L / L</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\| \|\|		\| Population growth \|\| <math>n = \dot L / L</math> \|\| <math>\text{Time}^{-1}</math> \|\| (−∞, ∞) \|\| \|\|
	\|-		\|-
	\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\| \|\|		\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\| \|\|

Issa Rice: /* Variables in the model */

2017-09-16T06:28:58Z

Variables in the model

← Older revision		Revision as of 06:28, 16 September 2017
Line 36:		Line 36:
	\| Capital per worker \|\| ''k'' = ''K''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|		\| Capital per worker \|\| ''k'' = ''K''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|
	\|-		\|-
	\| Fraction saved \|\| ''s'' \|\| Unitless \|\|		\| Fraction saved \|\| ''s'' \|\| Unitless \|\| [0, 1] \|\|
	\|-		\|-
	\| Output per worker \|\| ''y'' = ''Y''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|		\| Output per worker \|\| ''y'' = ''Y''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|

Issa Rice: /* Variables in the model */

2017-09-16T06:28:29Z

Variables in the model

← Older revision		Revision as of 06:28, 16 September 2017
Line 12:		Line 12:

	{\| class="sortable wikitable"		{\| class="sortable wikitable"
	! Name !! Variable !! Unit !! ~~Interval~~ !! Rival input? !! Variable type !! Notes		! Name !! Variable !! Unit !! Set of possible values !! Rival input? !! Variable type !! Notes
	\|-		\|-
	\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| [0, ∞) \|\| – \|\| Endogenous \|\|		\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| [0, ∞) \|\| – \|\| Endogenous \|\|

Issa Rice: /* Variables in the model */

2017-09-16T06:27:15Z

Variables in the model

← Older revision		Revision as of 06:27, 16 September 2017
Line 14:		Line 14:
	! Name !! Variable !! Unit !! Interval !! Rival input? !! Variable type !! Notes		! Name !! Variable !! Unit !! Interval !! Rival input? !! Variable type !! Notes
	\|-		\|-
	\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| \|\| – \|\| Endogenous \|\|		\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| [0, ∞) \|\| – \|\| Endogenous \|\|
	\|-		\|-
	\| Physical capital (capital stock) \|\| ''K'' \|\| \|\| \|\| Yes \|\| Endogenous \|\| Physical capital includes things like machines, computers, buildings, etc.		\| Physical capital (capital stock) \|\| ''K'' \|\| \|\| [0, ∞) \|\| Yes \|\| Endogenous \|\| Physical capital includes things like machines, computers, buildings, etc.
	\|-		\|-
	\| Labor \|\| ''L'' \|\| \|\| \|\| Yes \|\| Exogenous \|\|		\| Labor \|\| ''L'' \|\| \|\| [0, ∞) \|\| Yes \|\| Exogenous \|\|
	\|-		\|-
	\| Technology (knowledge) \|\| ''A'', ''T'' \|\| \|\| \|\| No \|\| Exogenous \|\|		\| Technology (knowledge) \|\| ''A'', ''T'' \|\| \|\| \|\| No \|\| Exogenous \|\|
Line 44:		Line 44:
	\| Production function \|\| ''F'' \|\| \|\| \|\| \|\|		\| Production function \|\| ''F'' \|\| \|\| \|\| \|\|
	\|-		\|-
	\| Elasticity of output with respect to capital \|\| ''α'' \|\| Unitless \|\| \|\|		\| Elasticity of output with respect to capital \|\| ''α'' \|\| Unitless \|\| (0, 1) \|\|
	\|}		\|}

Issa Rice: /* Mathematical formalism */

2017-09-16T06:24:56Z

Mathematical formalism

← Older revision		Revision as of 06:24, 16 September 2017
Line 56:		Line 56:

	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".		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,[https://web.stanford.edu/~chadj/closedform.pdf] but it is possible to play around with the model in non-closed forms to extract useful information.

	==History==		==History==

Issa Rice: /* Variables in the model */

2017-09-16T06:23:39Z

Variables in the model

Issa Rice: /* Variables in the model */

2017-09-16T04:14:36Z

Variables in the model

← Older revision		Revision as of 04:14, 16 September 2017
Line 28:		Line 28:
	\| Amount saved \|\| ''S'' \|\| \|\|		\| Amount saved \|\| ''S'' \|\| \|\|
	\|-		\|-
	\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| 1 / ~~Time~~ \|\| \|\|		\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\|
	\|-		\|-
	\| Population growth \|\| <math>n = \dot L / L</math> \|\| 1 / ~~Time~~ \|\| \|\|		\| Population growth \|\| <math>n = \dot L / L</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\|
	\|-		\|-
	\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\|		\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\|

Issa Rice: /* Variables in the model */

2017-09-16T04:13:38Z

Variables in the model

← Older revision		Revision as of 04:13, 16 September 2017
Line 28:		Line 28:
	\| Amount saved \|\| ''S'' \|\| \|\|		\| Amount saved \|\| ''S'' \|\| \|\|
	\|-		\|-
	\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| 1/Time \|\| \|\|		\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| 1 / Time \|\| \|\|
	\|-		\|-
	\| Population growth \|\| <math>n = \dot L / L</math> \|\| \|\| \|\|		\| Population growth \|\| <math>n = \dot L / L</math> \|\| 1 / Time \|\| \|\|
	\|-		\|-
	\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\|		\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\|

← Older revision		Revision as of 06:23, 16 September 2017
Line 12:		Line 12:

	{\| class="sortable wikitable"		{\| class="sortable wikitable"
	! Name !! Variable !! Unit !! Rival input? !! Variable type !! Notes		! Name !! Variable !! Unit !! Interval !! Rival input? !! Variable type !! Notes
	\|-		\|-
	\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| – \|\| Endogenous \|\|		\| Output \|\| ''Y'' \|\| Units of GDP (dollar?) \|\| \|\| – \|\| Endogenous \|\|
	\|-		\|-
	\| Physical capital (capital stock) \|\| ''K'' \|\| \|\| Yes \|\| Endogenous \|\| Physical capital includes things like machines, computers, buildings, etc.		\| Physical capital (capital stock) \|\| ''K'' \|\| \|\| \|\| Yes \|\| Endogenous \|\| Physical capital includes things like machines, computers, buildings, etc.
	\|-		\|-
	\| Labor \|\| ''L'' \|\| \|\| Yes \|\| Exogenous \|\|		\| Labor \|\| ''L'' \|\| \|\| \|\| Yes \|\| Exogenous \|\|
	\|-		\|-
	\| Technology (knowledge) \|\| ''A'', ''T'' \|\| \|\| No \|\| Exogenous \|\|		\| Technology (knowledge) \|\| ''A'', ''T'' \|\| \|\| \|\| No \|\| Exogenous \|\|
	\|-		\|-
	\| Consumption \|\| ''C'' \|\| \|\|		\| Consumption \|\| ''C'' \|\| \|\| \|\|
	\|-		\|-
	\| Investment \|\| ''I'' \|\| \|\|		\| Investment \|\| ''I'' \|\| \|\| \|\|
	\|-		\|-
	\| Amount saved \|\| ''S'' \|\| \|\|		\| Amount saved \|\| ''S'' \|\| \|\| \|\|
	\|-		\|-
	\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\|		\| Growth of ''X'' \|\| <math>g_X = \dot{X}/X = \frac{\frac{\partial X}{\partial t}}{X}</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\| \|\|
	\|-		\|-
	\| Population growth \|\| <math>n = \dot L / L</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\|		\| Population growth \|\| <math>n = \dot L / L</math> \|\| <math>\text{Time}^{-1}</math> \|\| \|\| \|\|
	\|-		\|-
	\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\|		\| Depreciation (rate?) \|\| ''δ'', ''d'', ''D'' \|\| Unitless \|\| \|\|
	\|-		\|-
	\| Capital per worker \|\| ''k'' = ''K''/''L'' \|\| \|\| \|\| Endogenous \|\|		\| Capital per worker \|\| ''k'' = ''K''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|
	\|-		\|-
	\| Fraction saved \|\| ''s'' \|\| Unitless \|\|		\| Fraction saved \|\| ''s'' \|\| Unitless \|\|
	\|-		\|-
	\| Output per worker \|\| ''y'' = ''Y''/''L'' \|\| \|\| \|\| Endogenous \|\|		\| Output per worker \|\| ''y'' = ''Y''/''L'' \|\| \|\| \|\| \|\| Endogenous \|\|
	\|-		\|-
	\| Time \|\| ''t'' \|\| Time, e.g. years \|\|		\| Time \|\| ''t'' \|\| Time, e.g. years \|\| \|\|
	\|-		\|-
	\| Production function \|\| ''F'' \|\| \|\| \|\|		\| Production function \|\| ''F'' \|\| \|\| \|\| \|\|
	\|-		\|-
	\| Elasticity of output with respect to capital \|\| ''α'' \|\| Unitless \|\|		\| Elasticity of output with respect to capital \|\| ''α'' \|\| Unitless \|\| \|\|
	\|}		\|}