LM Curve and Money Market Equilibrium

The lecture continues the development of the IS-LM model, focusing on building up to the LM curve.
Topics covered:
- The money market in the short run.
- The theory of liquidity preference.
- How monetary policy impacts interest rates.
- Derivation of the LM curve, which represents equilibrium in the money market.

Prices are flexible and adjust to maintain money market equilibrium in the long run.
Quantity equation: $MY = PV$
- Where:
  - M = Money supply
  - Y = Output
  - P = Price level
  - V = Velocity of money
An increase in the money supply (M ↑) leads to an increase in the price level (P ↑), given that V is constant and Y is determined by economic fundamentals.
The Fisher Equation relates changes in the money supply to the nominal interest rate:
- $↑ P, ↑ π =⇒ ↑ i$
- $i = r + π$
  - i = nominal interest rate
  - r = real interest rate
  - π = inflation

In the short run, prices are fixed, so the real interest rate adjusts to restore equilibrium.
Equilibrium in the money market is given by:
- $(M/P)^D := m^D = L(Y, i)$
  - Where L is the liquidity preference function
  - ∂L/∂Y > 0 (Money demand increases with income)
  - ∂L/∂i < 0 (Money demand decreases with interest rate)

Two Assets:
- Money: Does not pay interest but can be used for transactions.
- Bonds: Pay interest but cannot be used for transactions.
  - Assume maturity less than 1 year (e.g., Treasury bill).
- Bond Pricing:
  - $P_I$ - Initial price (auction in the Primary market)
  - $P_F$ - Face value (what the government promises to pay at maturity)
  - $r = (P<em>F − P</em>I) / P_I$ - Rate of return
  - When the initial price increases, the rate of return decreases.
Money Supply:
- Exogenous and directly controlled by the Central Bank.
Real Money Demand:
- $(M/P)^D = L(Y, r)$
  - ∂L/∂Y > 0
  - ∂L/∂r < 0

Money market equilibrium equates the supply and demand of real money balances:
- $M/P = L(Y, r)$

If the central bank increases the money supply:
- $(M/P)^S ↑ =⇒ Excess Supply$
- M/P > L(Y, r)
Individuals hold too much cash and buy bonds.
Demand for bonds ↑ =⇒ Price of bonds ( $P_I$ ) ↑ =⇒ r ↓ =⇒ L(Y, r) ↑
This continues until $M/P = L(Y, r)$ , and the money market is back in equilibrium.

The Central Bank controls the real interest rate by adjusting the money supply through monetary policy.
Theory of Liquidity Preference shows how, in the short run, the money supply leads to changes in the real interest rate.
- To increase the real interest rate, decrease the money supply.
- To decrease the real interest rate, increase the money supply.
Central banks often directly set a nominal interest rate, which, given price rigidities, effectively sets a real interest rate.

The short-run and long-run theories predict different relationships between M and i.
- Short Run:
  - $↓ M =⇒$ Excess demand for cash =⇒ People sell T-Bills =⇒ $P_I ↓ =⇒ r (and i) ↑$
- Long Run:
  - $↓ M =⇒ ↓ P$ (Quantity theory) =⇒ $i ↓$ (Fisher equation).

Each point on the LM curve represents a combination of Y and r in which the money market is in equilibrium.

The LM curve is implicitly given by the money market equilibrium because the demand for money is a function of both Y and r.
Given a specific functional form for the demand for money, one can solve for the LM curve.
Example:
- Suppose $L(Y, r) = kY − hr$
- In equilibrium: $M/P = kY − hr$
- $Y = (1/k)(M/P) + (h/k)r$
- Solving for r:
  - $r = (k/h)Y − (1/h)(M/P)$
  - This is the equation of the LM curve.

Depends on monetary policy ( $m^S$ ).
Changes in these components shift the LM curve:
- $↑ m^S =⇒$ shift to the right
- $↓ m^S =⇒$ shift to the left
- $∆Y = (1/k)∆m = β∆m$

The LM curve is an upward sloping line in (Y, r) space:
- $↑ Y =⇒ ↑ m^D$ , $↓ B^D$ , $↑ r$
- $r = (k/h)Y − (1/h)m$
- dr/dY = k/h > 0
The higher $k$ , the higher the money demand transaction motive, and the higher the LM slope.
The lower $h$ , the lower the money demand sensitivity to r, and the higher the LM slope.