Theory-Based Models

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Theory-Based Models

Introductorytwo curve intersection

IS-LM Model

Simultaneous equilibrium in the goods market and money market, tracing how spending and liquidity conditions jointly pin down output and the interest rate.

Catalog Overview Explore Proof Compare

Proof

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

Sections

Setup and notation

Start from a fixed-price short run with one goods-market condition and one money-market condition.

The route keeps the schedules linear so the geometry and algebra line up cleanly.

i = A - bY

Reduced-form IS schedule.

i = m_0 + m_1 Y

Reduced-form LM schedule.

Deriving the IS curve

Higher demand raises output, while higher rates lean against interest-sensitive spending.

Collecting the autonomous terms gives a downward-sloping relation in output-interest space.

Y = \bar A - \gamma i

i = \frac{\bar A}{\gamma} - \frac{1}{\gamma}Y

Deriving the LM curve

Money demand rises with activity, so higher output needs a higher rate to keep money demand equal to the fixed real money stock.

\frac{M}{P} = L(Y,i)

i = m_0 + m_1 Y

Equilibrium conditions

Set the IS and LM schedules equal to each other and solve the two-line system.

A - bY = m_0 + m_1 Y

Y^* = \frac{A - m_0}{b + m_1}

i^* = A - bY^*

Comparative statics

A higher demand intercept raises output and rates, while a tighter money intercept lowers output and raises rates.

\frac{\partial Y^*}{\partial A} = \frac{1}{b + m_1}

\frac{\partial i^*}{\partial A} = \frac{m_1}{b + m_1}

Liquidity-trap intuition

If the LM schedule becomes very flat, rates move little and demand shifts translate mostly into output.

Back to explore Open compare

Theory-Based Models

Loading Theory-Based Models

Theory-Based Models

Introductorytwo curve intersection

IS-LM Model

Simultaneous equilibrium in the goods market and money market, tracing how spending and liquidity conditions jointly pin down output and the interest rate.

Catalog Overview Explore Proof Compare

Proof

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

Sections

Setup and notation

Start from a fixed-price short run with one goods-market condition and one money-market condition.

The route keeps the schedules linear so the geometry and algebra line up cleanly.

i = A - bY

Reduced-form IS schedule.

i = m_0 + m_1 Y

Reduced-form LM schedule.

Deriving the IS curve

Higher demand raises output, while higher rates lean against interest-sensitive spending.

Collecting the autonomous terms gives a downward-sloping relation in output-interest space.

Y = \bar A - \gamma i

i = \frac{\bar A}{\gamma} - \frac{1}{\gamma}Y

Deriving the LM curve

Money demand rises with activity, so higher output needs a higher rate to keep money demand equal to the fixed real money stock.

\frac{M}{P} = L(Y,i)

i = m_0 + m_1 Y

Equilibrium conditions

Set the IS and LM schedules equal to each other and solve the two-line system.

A - bY = m_0 + m_1 Y

Y^* = \frac{A - m_0}{b + m_1}

i^* = A - bY^*

Comparative statics

A higher demand intercept raises output and rates, while a tighter money intercept lowers output and raises rates.

\frac{\partial Y^*}{\partial A} = \frac{1}{b + m_1}

\frac{\partial i^*}{\partial A} = \frac{m_1}{b + m_1}

Liquidity-trap intuition

If the LM schedule becomes very flat, rates move little and demand shifts translate mostly into output.

Back to explore Open compare

Loading Theory-Based Models

IS-LM Model

Proof

Collect demand terms

Solve for the rate

Loading Theory-Based Models

IS-LM Model

Proof

Collect demand terms

Solve for the rate