1. {doc}`Cattle cycles <cattle_cycles>`

1. {doc}`Shock Non Invertibility <hs_invertibility_example>`

>

### Overview of the Models

In saying that "complete markets are all alike", Robert E. Lucas, Jr. was noting that all of them have

* Given the price system and given $h_{-1},\, k_{-1}$, the allocation solves the representative household’s problem and

the problems of the two types of firms.

>

Versions of the two classical welfare theorems prevail under our assumptions.

We exploit that fact in our algorithm for computing a competitive equilibrium.

**Step 1:** Solve the planning problem by using dynamic programming.

> The allocation (i.e., **quantities**) that solve the planning problem **are** the

> competitive equilibrium quantities.

The allocation (i.e., **quantities**) that solve the planning problem **are** the

competitive equilibrium quantities.

**Step 2:** use the following formulas to compute the **equilibrium price system**

* $n_w + n_y$ versus $n_y$

>

Compare spaces spanned

* $H(y^t) \subset H(w^t,v^t)$

* $H(y^t) = H(a^t)$

>

**Kalman Filter:**.

Kalman gain:

- $\color{blue}{\Pi}$ is nonsingular, and

- the absolute values of the eigenvalues of $\color{blue}{(\Delta_h - \Theta_h \Pi^{-1}\Lambda)}$ are strictly less than $1/\sqrt\beta$.

>

**Key invertibility property:** A canonical household service

technology maps a service process $\{s_t\}$ in $L_0^2$

into a corresponding consumption process $\{c_t\}$ for which the

1. {doc}`Cattle cycles <cattle_cycles>`

1. {doc}`Shock Non Invertibility <hs_invertibility_example>`

>

We'll start with an example of a **partial equilibrium** in which we posit demand and supply curves

Suppose that we want to capture the dynamic demand curve:

* note how the demand equations are cast in terms of the matrices in our standard preference representation

>

Now let's turn to supply.

A representative firm takes as given and beyond its control the

- Many consumption goods and services

- A single capital good with $R \beta =1$

- The physical production technology

>

$$

\begin{aligned}

\phi_c \cdot c_t+i_t&=\gamma k_{t-1}+e_t \\