Introduction of RBC model

#economics #macro

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

Real Business Cycle Model

Real: economic shock is caused by the real economic component
- real exogenous shocks can affect real endogenous variables.
- real exogenous: technology(), or government spending().
- real endogenous: output(), consumption(), savings(), or investment().
Business cycle: business cycle is the reaction of the shocks imposed on the efficient market, which itself is a equilibrium.

Assumptions

Baseline Model

large number of identical, price-taking firms
infinitely living single representative household
all variables are real variables

Production

Cobb-Douglas Function

: output, : capital, : labor, : technology.

Capital Stock

: capital depreciation rate.
Closed economy: .
: government purchase equals to the lump-sum taxes

Profit Maximization

and

: real wages, : real interest rate.

Households

Utility and Budget

: instantaneous utility function
- : consumption per member of households
- : working hours per member of households
- : leisure(non-working hours) per member
: discount rate
: population
- population growth rate: .
- , i.e. , where .
: number of households
: number of members of the household

We assume a single representative household with one member (, ) then the utility function becomes,

Utility Maximization

Lagrangian: F.O.C. thus and

Euler equation: Labor supply equation:

Exogenous Variables

Technology

: technological progress rate
: trend of technology
: departure from trend, follows AR(1) process
- : positive correlation between previous' uncertainty, while the effect of the shocks disappears gradually over time.
- .

Government Purchase

: population growth rate, : technological progress rate
: trend of .
: shocks on , follows AR(1) process

Simplified Model

Assumption: (no government), ( depreciation).

thus we have

Constant Savings and Labor

Saving Rate

From , we have since , i.e. , If is constant at , then we have thus

Labor

From , we have as , by the assumption of Cobb-Douglas function, , thus i.e.

Log-linearization

From the production function , by taking logs on both sides, Since , we have as we derived , we have Also, we have , and .

Thus, Now define the cycle of as where is the steady state, which is Therefore, the cycle of is this implies since we have , thus the departure of log output from its normal path is which follows AR(2) process.

Calibration and Simulation

Hansen and Wright (1992)

Calibration: Choosing parameter values on the basis of micro(macro)-economic evidence.
, per quarter and per quarter.
, , .

Pasted image 20240417180206.png

blue line: cycle of output ()
red dash: cycle of productivity ()
both lines are de-trended, i.e. .
Y-axis: percentage deviation from the trend (-4%, -2%, 0%, 2%, 4%, ...)
- since , by the Taylor approximation,
- similarly, from , we have where is .

General Case

State variables (at period ): exogenous variables and control variables in previous periods, not controllable.
Control variables (at period ): the variables which can be controlled.
the linear system between state and control variables: