advmacro Takeki Sunakawa toe

Timing of events

1 Capital inherited from the previous period is at hand, kt

2 Output is produced by the production function, yt= Atk^αt

3 The produced goods are divided into consumption and saving ct+ xt= yt

4 Capital depreciates at rate δ. Remaining capital and new investment determined new capital stock. kt+1= (1 − δ)k^t+ xt.

Here, ktis capital stock at thebeginningof the period t.

October 28, 2015 1 / 5

Timing of events, cont’d

1 Capital inherited from the previous period is at hand, ˜kt₋₁

2 Output is produced by the production function, yt= At^˜k^αt₋₁

3 The produced goods are divided into consumption and saving ct+ xt= yt

4 Capital depreciates at rate δ. Remaining capital and new investment determined new capital stock. ˜kt= (1 − δ)˜k^t−1^{+ xt}

Here, ˜ktis capital stock at theendof the period t. Note that ˜kt= kt+1.

Timing of events in stochastic setting

1 Capital inherited from the previous period is at hand, kt. Technology level At

is realized.

2 Output is produced by the production function, yt= Atk^αt.

3 The planner forms expectation in the next period’s At+1. The produced goods are divided into consumption and saving ct+ xt= yt.

4 Capital depreciates at rate δ. Remaining capital and new investment determined new capital stock. kt+1= (1 − δ)k^t+ xt.

Here, ktis capital stock and Atis realized technology level at thebeginningof the period t.

October 28, 2015 3 / 5

Timing of events in stochastic setting

In Dynare ....

In the literature of the business cycle theory, the endogenous state variable, i.e., capital, is expressed by using kt= ˜kt₋₁.

Dynare uses ˜kt₋₁ and At₋₁ as state variables.

For example, kt+1= (1 − δ)k^t+ xtor ˜kt= (1 − δ)˜k^t₋₁+ xt is written as k

= (1-delta)*k(-1) + x. Also, yt= Atk^αt = Atk^˜t^α₋₁ is written as y = A*k(-1)^alpha.

Note that Atin known at the beginning of the period t, whereas ˜kt= kt+1 is not.

October 28, 2015 5 / 5