IS-LM Model and Keynesian Cross Notes

Output in the Short Run: The Goods Market

Introduction to Short Run and IS-LM Model

In the short run, where prices are fixed, the focus shifts to understanding output determination using the IS-LM and AS-AD models. The initial focus is on constructing the IS-LM model, specifically addressing:

1. How output is determined in the short run using the Keynesian cross.
2. The impact of fiscal policy on output.
3. Derivation of the IS curve, representing equilibrium in the goods market.

IS-LM Model: A Short Run General Equilibrium Model

The IS-LM model is a short-run general equilibrium model.

• Partial Equilibrium: Supply equals demand in a single market, ensuring all buyers and sellers can transact at the prevailing price without excess or shortages.
• General Equilibrium: Supply equals demand across all markets in the economy, recognizing the interdependence of prices across markets.
• The IS-LM model formalizes Keynes’s "The General Theory of Employment, Interest and Money" (1936).
• It was formalized by John Hicks (Nobel laureate 1972) and Hansen (1949) to synthesize Keynesian and neoclassical theories.

Relevance of IS-LM Model

Despite the evolution towards Dynamic Stochastic General Equilibrium (DSGE) models in modern macro research, the IS-LM model remains influential in policy circles.

• During the 2007-09 financial crisis, policy responses in the White House were primarily based on the IS-LM model, augmented by a liquidity trap analysis, according to Larry Summers.

Core Components of the IS-LM Model

The IS-LM model assumes three major markets within the economy:

1. Goods Market (IS): Where goods and services are traded.
2. Money Market (LM): Where the nominal interest rate is determined.

The IS-LM model tracks two equilibrium conditions that must be satisfied simultaneously.

• It describes combinations of the real interest rate $(r)$ and output $(Y)$ that achieve equilibrium in both the goods and money markets.

Key Ingredients of the IS-LM Model

The main ingredients of the IS-LM model include:

1. Two Markets: Goods market and money market.
2. Fixed Aggregate Price Level: Short-run analysis assumes fixed prices.
3. Closed Economy: Net exports are zero $(NX = 0)$, implying savings equals investment $(S = I)$.

• The initial focus is on the goods market to develop the IS curve.
• The subsequent focus will be on the money market to build the LM curve.

IS Curve: Equilibrium in the Goods Market

The IS curve represents the combinations of $(r)$ and $(Y)$ that arise when the goods market is in equilibrium, i.e., when the supply of goods and services equals the demand.

• It is built using Keynesian analysis to determine how output $(Y)$ is determined in a closed economy.
• The Keynesian cross model determines the equilibrium level of $(Y)$ in the short run for a given $(r)$.
• Output is determined by spending plans; increased spending leads to increased production and hiring.

Keynesian Cross: Notation

• Actual Expenditure: $(Y)$ = Real GDP – the total spending on goods and services by households, firms, and the government.
• Planned Expenditure (Aggregate Demand): $(E = C + I + G)$ – the desired or planned level of spending by households, firms, and the government.
• $(I)$ represents planned investment.

Components of Planned Expenditure

Planned expenditure is given by: $E = C + I + G$

• Government spending $(G)$ and taxes $(T)$ are assumed to be exogenous.
• Consumption $(C)$ depends on $(Y)$, and investment $(I)$ depends on $(r)$.
• Keynesian consumption function: $C = C_0 + c(Y − T)$
• Planned investment function: $I = I_0 − br$
• Policy variables are $(G)$ and $(T)$.

Keynesian Consumption Function

The Keynesian consumption function is described as: $C = C_0 + c(Y − T)$

• Taxes $(T)$ are lump sum.
• Disposable income is defined as $Y_d := Y − T$.
• Autonomous consumption $(C<em>0)$ is the consumption level needed to survive or when $Y</em>d = 0$.
• Marginal propensity to consume $(c)$ measures how much consumption changes with income, given by $\frac{\partial C}{\partial Y}$. It is assumed to be constant with $0 < c < 1$.

Planned Investment Function

The planned investment function is given by: $I = I_0 − br$

• Autonomous investment $(I_0)$ is the planned investment when the real interest rate $(r)$ is 0. It is also the level of planned investment when the nominal interest rate $(i)$ is 0 (Fisher equation).
• There is a negative relationship between $(r)$ and $(I)$ represented by $(−b)$. As $(r)$ increases, investment becomes more expensive, reducing investment.
• The value of $(b)$ captures the sensitivity of planned investment to changes in the real interest rate.
• A larger value of $(b)$ indicates higher sensitivity of $(I)$ to changes in $(r)$.

Planned Expenditure Equation

$E = C<em>0 + c(Y - T) + I</em>0 - br + G$

Keynesian Cross: Equilibrium

In equilibrium, actual expenditure equals planned expenditure $(Y = E)$.

$Y = C<em>0 + c(Y − T) + I</em>0 − br + G$

Solving for $(Y)$ yields the equilibrium level of output:

$Y^* = \frac{C<em>0 + I</em>0 + G}{1 − c} − \frac{c}{1 − c}T − \frac{b}{1 − c}r$

Keynesian Cross: Equilibrium Restoration

When plans don’t align such that $Y \neq E$, equilibrium is restored through unplanned changes in inventories, which then induce changes in the level of production.

Keynesian Cross: Comparative Statics

Comparative statics examine how equilibrium changes with changes in variables.

• Focus is on the effects of fiscal policy changes.
• Expansionary fiscal policy involves increasing government spending $(G)$ or decreasing taxes $(T)$.
• Contractionary fiscal policy (austerity) involves decreasing government spending $(G)$ or increasing taxes $(T)$.

Keynesian Cross: Fiscal Policy (G)

If the government undertakes expansionary fiscal policy by increasing $(G)$, the output $(Y)$ will increase.

Keynesian Cross: Fiscal Policy (G) - Government Expenditure Multiplier

Government expenditure multiplier:
$\frac{\partial Y}{\partial G} = \frac{1}{1−c} > 1$

For every $1 increase in $(G)$, $(Y)$ increases by more than $1.

The multiplier effect occurs because the initial increase in $(G)$ leads to a direct increase in $(Y)$, which in turn increases consumption $(C)$, further boosting $(Y)$.

$G ↑ ⇒ Y ↑ \text{ (direct effect) } ⇒ C ↑ \text{ (Keynesian consumption function) } ⇒ Y ↑ …$

Keynesian Cross: Fiscal Policy (G) - Illustration

Illustration: If $(G)$ increases by $100 million:

• $Y$ increases by $100 million $(Y = C + I + G)$.
• $C$ increases by c \cdot $100 \text{ million}. Output equals income, and consumers spend a fraction $(c)$ of their income.
• $Y$ increases by c \cdot $100 \text{ million } (Y = C + I + G).
• $C$ increases by c^2 \cdot $100 \text{ million}.

$\Rightarrow \Delta Y = (1 + c + c^2 + c^3 + … + c^n + …)\Delta G, \lim_{n \to \infty} \frac{\Delta Y}{\Delta G} = \frac{1}{1-c}$

Keynesian Cross: Fiscal Policy (T)

Suppose government undertakes contractionary fiscal policy by increasing $(T)$. This leads to a decrease in $(Y)$.

Tax multiplier:
$\frac{\partial Y}{\partial T} = \frac{−c}{1−c}$

For every $1 increase in $(T)$, $(Y)$ decreases by $\frac{c}{1−c}$.

$T ↑ ⇒ Y ↓ \text{ (direct effect) } ⇒ C ↓ \text{ (Keynesian consumption function) } ⇒ Y ↓ …$

Keynesian Cross: Comparing Multipliers

$\frac{1}{1 − c} > | \frac{c}{1 − c} |$

Given the same change in $(G)$ and $(T)$, the change in $(G)$ is more powerful in affecting the level of output.

This aligns with Keynesian economics, which suggests that during low output (e.g., recession), the government should intervene by increasing spending (financed by debt or raising taxes).

Keynesian Cross: IS Curve

The IS curve summarizes the relationship between the real interest rate and the level of output.

The Keynesian cross equilibrium shows the equilibrium level of output in the goods market, given that planned investment is a function of $(r)$.

$Y = \frac{C<em>0 + I</em>0 + G}{1 − c} − \frac{c}{1 − c}T − \frac{b}{1 − c}r$

Using the Keynesian cross, we can map all equilibrium levels of output $(E = Y)$ for each given $(r)$.

Keynesian Cross: IS Curve Derivation

The IS curve is derived by varying the real interest rate $(r)$ and observing the resulting changes in equilibrium output $(Y)$.

Keynesian Cross: IS Curve Characteristics

• Each point on the IS curve represents a combination of $(Y)$ and $(r)$ where the goods market is in equilibrium.
• It is a downward sloping line in the $(Y, r)$ space.

The equation for the IS curve is derived by rearranging the equilibrium output equation from the Keynesian cross model to solve for $(r)$.

$r = \frac{C<em>0 + I</em>0}{b} − \frac{c}{b}T + \frac{G}{b} − \frac{1 − c}{b}Y$

Position of the IS Curve

Each point on the IS curve represents a combination of $(Y)$ and $(r)$ where the goods market is in equilibrium.

The position of the IS curve in the $(Y, r)$ space depends on autonomous demand components:

$A = C<em>0 + I</em>0 + G − cT$

Changes in these components shift the IS curve:

• Increases in $I<em>0$, $C</em>0$, $G$, or decreases in $T$ result in an upward shift.
• Decreases in $I<em>0$, $C</em>0$, $G$, or increases in $T$ result in a downward shift.

$\Delta Y = \frac{1}{1 − c} \Delta A = \alpha \Delta A$

Slope of the IS Curve

The IS curve is a downward sloping line in the $(Y, r)$ space.

$↑ r ⇒ ↓ I, ↓ E, ↓ Y, …$

$r = \frac{C<em>0 + I</em>0}{b} − \frac{c}{b}T + \frac{G}{b} − \frac{1 − c}{b}Y$

$\frac{dr}{dY} = − \frac{1 − c}{b} < 0$

• A higher $(c)$ leads to a higher aggregate demand multiplier and a flatter IS curve, meaning the same variation in the interest rate will lead to larger output changes.
• A higher $(b)$ increases the impact of interest rates on investment and flattens the IS slope; the same change in the interest rate will have greater effects on output.