Calvo Price

macro

The Calvo Price is a core mechanism from the NK-DSGE,

Rules

Firm change price with probability $1-\theta$. The ability to change price is completely random. This directly let the firms side not symmetric.

At time $t$, set price $P_{it}^{\ast }$ to max present value of dividends

$$P_{it}^{\ast }=\arg \underset{P_{i,t}}{\max }\sum _{k=0}^{\infty }{\theta }^k{\mathbb{E}}_t{Q}_{t,t+k}\left(P_{i,t}{Y}_{i,t+k}-W_{t+k}{L}_{it+k}\right)$$

Subject to

$$Y_{it+k}={\left(P_{it}/P_{t+k}\right)}^{-ϵ}{C}_{t+k}\text{and}{Y}_{i,t+k}=A_{t+k}{L}_{it+k}^{1-\alpha }$$

Here ${\theta }^k$ means the Probability of being stuck at the price level. Once once I am able to change the price, I will set a new $P^{\ast }$.

I Define ${\Psi }_{t+k}\left(Y_{t+k|t}\right)$ as costs at $t+k$ for firm that set prices at t

$$\max \sum _{k=0}^{\infty }{\theta }^k{\mathbb{E}}_t{Q}_{t,t+k}\left(P_t^{\ast }{Y}_{t+k|t}-{\Psi }_{t+k}\left(Y_{t+k|t}\right)\right)s.t.Y_{t+k|t}={\left(\frac{P_t^{\ast }}{P_{t+k}}\right)}^{-ϵ}{C}_{t+k}$$