Theory-Based Models

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Theory-Based Models

Advancedphase path

Overlapping Generations Model

A two-period overlapping-generations framework linking saving, population growth, and capital accumulation across cohorts.

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

Sections

Setup and notation

The route uses a two-period overlapping-generations structure: workers save when young and consume when old.

y = A k^{\alpha}

w = (1 - \alpha)A k^{\alpha}

r = \alpha A k^{\alpha - 1} - \delta

Household saving problem

The young save a share of wage income, which becomes the next cohort’s capital once population growth is accounted for.

s_y = s \cdot w

Law of motion for capital

With population growth, the next generation inherits diluted capital unless saving rises enough to offset it.

k' = \frac{s(1 - \alpha)A k^{\alpha}}{1 + n}

Steady state and demographics

Higher saving raises steady-state capital; faster population growth lowers capital per young worker.

k^* = \left(\frac{s(1-\alpha)A}{1+n}\right)^{\tfrac{1}{1-\alpha}}

Theory-Based Models

Theory-Based Models

Advancedphase path

A two-period overlapping-generations framework linking saving, population growth, and capital accumulation across cohorts.

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

Sections

Setup and notation

The route uses a two-period overlapping-generations structure: workers save when young and consume when old.

y = A k^{\alpha}

w = (1 - \alpha)A k^{\alpha}

r = \alpha A k^{\alpha - 1} - \delta

Household saving problem

The young save a share of wage income, which becomes the next cohort’s capital once population growth is accounted for.

s_y = s \cdot w

Law of motion for capital

With population growth, the next generation inherits diluted capital unless saving rises enough to offset it.

k' = \frac{s(1 - \alpha)A k^{\alpha}}{1 + n}

Steady state and demographics

Higher saving raises steady-state capital; faster population growth lowers capital per young worker.

k^* = \left(\frac{s(1-\alpha)A}{1+n}\right)^{\tfrac{1}{1-\alpha}}