Review of RBC

#economics #macro

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

Simplified version of King, Plosser, and Rebelo, 1988.

Assumptions

Endogenous Variables

  • Non-negativity Constraints
  • : growth rate for .
    • At steady state, we have
    • : growth rate of work effort to be zero.
  • : variables per growth component.

Firms

Production Function

  • : capital stock, predetermined at period .
  • : labor input in period .
  • : technology.

Capital Accumulation

  • : gross investment.
  • : rate of depreciation of capital.

Technology Development

where .

Representative Household

Utility Function

where

  • : commodity consumption in period .
  • : leisure in period .

Budget Constraints

  • : total time allocation to work and leisure must not exceed the endowment, which is normalized to one.
  • : total uses of the commodity must not exceed output.

Assuming an interior solution,
using the capital accumulation and production function,

Equilibrium

Market Clearing Conditions are

  • Labor Market: .
  • Capital Market: .
  • Goods Market: .

Utility Maximization Problem

Lagrangian function: F.O.C. therefore we have Euler equation: Labor supply curve: By letting , we have and the Definition 1 (Frisch Elasticity) is

Profit Maximization Problem

note that where .

F.O.C. Capital demand curve is implying that the rental payment is equivalent to the capital share.
and the interest rate is And for the gross return ,

Labor demand curve is showing that the wage payment is equivalent to the labor share.
Since , the Marginal Product of labor () decreases, thus the wage goes down as increases.

Frisch-Elasticity

Definition (Frisch Elasticity).

The Frisch elasticity of labor supply measures the percentage change in hours worked due to the percentage change in wages, holding the marginal utility of wealth (i.e. the Lagrangian multiplier) as constant. where denotes real wage.

Theorem (calculate frisch elasticity).

The Frisch elasticity of labor supply can be calculated through

Proof.From the general household utility maximization problem of then the F.O.C. are Then, we have where the time subscript was ignored for simplicity.

Then, using the chain rule, we have Then, by combining the two, we have thus, then, This completes the proof. □

Log-linearization

From the previous problems and the constraints, we have the following equations where the endogenous variables are: and the exogenous variable is where

  1. Euler Equation
  2. Consumption-Labor Substitution
  3. Capital Accumulation
  4. Resources Constraint
  5. Production Formula
  6. Real Interest Rate
  7. Real Wage
  8. Technology Development where Using Taylor approximation,

Guess and Verify

Solution for Log Utility Function

The solutions are where the endogenous variables are: and the exogenous variable is where

  1. Euler Equation
  2. Consumption-Labor Substitution Rest of the solutions are identical to the case when the utility function is CRRA.

Linear Matrix System

The linearized solution is Now, denoting be a vector of endogenous variables (), and be a vector of exogenous variable , we can re-express the equation as where and since , we have .

Guess and Verify

Guess: Verify: Thus, the solution is:

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Table Of Contents
Review of RBC
Assumptions
Endogenous Variables
Firms
Production Function
Capital Accumulation
Technology Development
Representative Household
Utility Function
Budget Constraints
Equilibrium
Utility Maximization Problem
Profit Maximization Problem
Frisch-Elasticity
Log-linearization
Guess and Verify
Solution for Log Utility Function
Linear Matrix System
Guess and Verify