Monetary RBC

#economics #macro

Oh, Hyunzi. (email: wisdom302@naver.com)
Korea University, Graduate School of Economics.
2024 Spring, instructed by prof. Eo, Yunjong.

Classical Monetary Model

  • Monetary is payed for the consumption
  • is a monetary required for one unit of consumption.
  • Flexible price is assumed (no rigidity)
  • Perfectly competitive market on both (single) goods and labor
  • Consumer can transfer its wealth to future by bonds

Benchmark Model

Households

UMP

  • : consumption, : labor supply, : price for consumption
  • : price of bond, : demand for bond, where
  • : tax or subsidy

Lagrangian function: F.O.C.

Equilibrium

Euler equation: Consumption-Labor Substitution:

Log-linearization

  1. Consumption-labor substitution:
  2. Euler equation: where , , and .

Firms

Production Function

where Taking log-linearization on the production function, Real wage per labor is log-linearization is

PMP

F.O.C. By the log-linearization, we have

Equilibrium

  1. : consumption-labor substitution

  2. : Euler equation

  3. : production function

  4. : real wage

  5. : total production equals total consumption

  6. : real interest rate

  7. : labor Determination

  8. : product determination

  9. : real interest rate

  10. : real wage

  11. : technology development

Monetary Policy and Price Determination

Now we incorporate the monetary policy to the Benchmark model. We divide the case by how the central bank determines the its monetary policy by controlling the nominal interest rate or monetary supply . The basic foundation of the monetary reaction follows the equation 11 and 12.

  1. : monetary demand
  2. : fisher equation

Case 1) Exogenous Nominal Interest Rate

Assume is exogenously determined and follows stationary process. We Assume that nominal interest rate is determined by

  1. : price determination

This implies that the price level is not uniquely determined, since is affected by an arbitrary sunspot shock . This is called price lever indeterminacy.

Case 2) Taylor Principle

Assume is determined by the endogenous variables, i.e. Taylor principle: Then the deviation from the real interest rate is
14) : monetary policy shock 15) : price determination Therefore, must hold to handle the sunspot shock. If not (), then the price is still determined by the sunspot shock, resulting indeterminacy in price level.

Case 3) Exogenous Monetary Policy

Suppose that the central bank implements the monetary policy by controlling the monetary supply.

  1. : price determination Thus the price level is determined by the expected monetary growth rate ().

  2. : nominal interest rate determination Assume that the monetary supply follows AR(1) process as Then the solution is

  • .
  • .
    Thus if , then and both increases. However, this cannot explain the liquidity effect ( decreases when increases).
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Table Of Contents
Monetary RBC
Benchmark Model
Households
UMP
Equilibrium
Log-linearization
Firms
Production Function
PMP
Equilibrium
Monetary Policy and Price Determination
Case 1) Exogenous Nominal Interest Rate
Case 2) Taylor Principle
Case 3) Exogenous Monetary Policy