Theory-Based Models

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Theory-Based Models

Solow Growth Model

A long-run growth framework linking saving, depreciation, population growth, and productivity to the steady-state capital stock and output path.

Intermediatephase path

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Theory-Based Models

Proof

Route section

Read the derivation as a document, with the math typeset directly and the intermediate chains tucked behind expandable steps.

Sections

Setup and notation

The baseline branch uses a Cobb-Douglas production function and a standard capital-accumulation equation.

The same route also houses Golden Rule, AK, MRW, and Malthusian extensions.

y = A k^{\alpha}

\dot k = s f(k) - (n + g + \delta)k

Capital accumulation

Saving adds to capital while population growth, technology growth, and depreciation form the break-even drain.

break\text{-}even = (n + g + \delta)k

Steady-state conditions

Steady state is the point where net capital accumulation is zero.

sAk^{\alpha} = (n + g + \delta)k

k^* = \left(\frac{sA}{n + g + \delta}\right)^{\tfrac{1}{1-\alpha}}

Golden Rule branch

Golden Rule capital maximizes steady-state consumption rather than simply satisfying a chosen saving rule.

c^*(k) = f(k) - (n + g + \delta)k

f'(k_{GR}) = n + g + \delta

AK, MRW, and Malthusian extensions

AK removes diminishing returns, MRW adds human capital, and Malthusian dynamics replace capital deepening with a demographic feedback around subsistence.

\dot k = (sA - n - g - \delta)k

y = A k^{\alpha} h^{\beta}

\frac{\dot N}{N} = \phi\left[\frac{y}{N} - \bar y_{sub}\right]

Back to explore Open compare

Theory-Based Models

Loading Theory-Based Models

Theory-Based Models

Solow Growth Model

A long-run growth framework linking saving, depreciation, population growth, and productivity to the steady-state capital stock and output path.

Intermediatephase path

Catalog Overview Explore Proof Compare

Theory-Based Models

Proof

Route section

Read the derivation as a document, with the math typeset directly and the intermediate chains tucked behind expandable steps.

Sections

Setup and notation

The baseline branch uses a Cobb-Douglas production function and a standard capital-accumulation equation.

The same route also houses Golden Rule, AK, MRW, and Malthusian extensions.

y = A k^{\alpha}

\dot k = s f(k) - (n + g + \delta)k

Capital accumulation

Saving adds to capital while population growth, technology growth, and depreciation form the break-even drain.

break\text{-}even = (n + g + \delta)k

Steady-state conditions

Steady state is the point where net capital accumulation is zero.

sAk^{\alpha} = (n + g + \delta)k

k^* = \left(\frac{sA}{n + g + \delta}\right)^{\tfrac{1}{1-\alpha}}

Golden Rule branch

Golden Rule capital maximizes steady-state consumption rather than simply satisfying a chosen saving rule.

c^*(k) = f(k) - (n + g + \delta)k

f'(k_{GR}) = n + g + \delta

AK, MRW, and Malthusian extensions

AK removes diminishing returns, MRW adds human capital, and Malthusian dynamics replace capital deepening with a demographic feedback around subsistence.

\dot k = (sA - n - g - \delta)k

y = A k^{\alpha} h^{\beta}

\frac{\dot N}{N} = \phi\left[\frac{y}{N} - \bar y_{sub}\right]

Back to explore Open compare