Theory-Based Models

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Theory-Based Models

Advancedphase path

RBC Simplified

A stripped-down real business cycle setup linking productivity, capital accumulation, and intertemporal return conditions.

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

Sections

Setup and notation

The simplified RBC route keeps only the real side: production, capital accumulation, and an Euler-style return condition.

y = A k^{\alpha}

k' = (1 - \delta)k + i

Production and capital law

A productivity shock changes the scale of output today and shifts the desired capital path over time.

MPK = \alpha A k^{\alpha - 1}

Euler condition and steady return

In steady state, the marginal return on capital must align with the discount factor once depreciation is accounted for.

\frac{1}{\beta} - 1 + \delta = \alpha A k^{\alpha - 1}

Productivity and patience shifts

Higher productivity or greater patience both support a larger steady-state capital stock.

\frac{\partial k^*}{\partial A} > 0, \qquad \frac{\partial k^*}{\partial \beta} > 0

Theory-Based Models

Theory-Based Models

Advancedphase path

A stripped-down real business cycle setup linking productivity, capital accumulation, and intertemporal return conditions.

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

Sections

Setup and notation

The simplified RBC route keeps only the real side: production, capital accumulation, and an Euler-style return condition.

y = A k^{\alpha}

k' = (1 - \delta)k + i

Production and capital law

A productivity shock changes the scale of output today and shifts the desired capital path over time.

MPK = \alpha A k^{\alpha - 1}

Euler condition and steady return

In steady state, the marginal return on capital must align with the discount factor once depreciation is accounted for.

\frac{1}{\beta} - 1 + \delta = \alpha A k^{\alpha - 1}

Productivity and patience shifts

Higher productivity or greater patience both support a larger steady-state capital stock.

\frac{\partial k^*}{\partial A} > 0, \qquad \frac{\partial k^*}{\partial \beta} > 0