$\{w_t,\eta_t\}_{t=0}^T= \{\vec{w}, \vec{\eta} \}$

where

- $w_t$ is a wage or rental rate for labor at time $t$

- $w_t$ is a wage, i.e., a rental rate, for labor at time $t$

- $\eta_t$ is a rental rate for capital at time $t$

In addition there is a vector $\{q_t^0\}$ of intertemporal prices where

- $q^0_t$ is the price of a good at date $t$ relative

to a good at date $0$.

- $q^0_t$ is the price at time $0$ of one unit of the good at date $t$.

We call $\{q^0_t\}_{t=0}^T$ a vector of **Hicks-Arrow prices**,

named after the 1972 economics Nobel prize winners.

But because $q^0_t$ is a **relative price**, the units in terms of

which prices are quoted are arbitrary, we are free to re-normalize them by multiplying all of them by a positive scalar, say $\lambda > 0$.

Because is a **relative price**. the unit of account in terms of which the prices $q^0_t$ are stated is; we are free to re-normalize them by multiplying all of them by a positive scalar, say $\lambda > 0$.

Units of $q_t^0$ could be set so that

Units of $q_t^0$ could be set so that they are

$$

\frac{\text{number of time 0 goods}}{\text{number of time t goods}}

or

$$

\sum_{t=0}^T q^0_t \left(c_t + (k_{t+1} -(1-\delta)k_t)-(w_t 1 + \eta_t k_t) \right) \leq 0

\sum_{t=0}^T q^0_t \left(c_t + (k_{t+1} -(1-\delta)k_t)\right) \leq \sum_{t=0}^T q^0_t(w_t 1 + \eta_t k_t) \

$$

The household faces price system $\{q^0_t, w_t, \eta_t\}$ as a price-taker and chooses an allocation to solve the constrained optimization problem:

$$

\begin{aligned}& \max_{\vec{c}, \vec{k} } \sum_{t=0}^T \beta^t u(c_t) \\ \text{subject to} \ \ & \sum_{t=0}^T q_t^0\left(c_t +\left(k_{t+1}-(1-\delta) k_t\right) -w_t -\eta_t k_t\right) \leq 0 \notag \end{aligned}

\begin{aligned}& \max_{\vec{c}, \vec{k} } \sum_{t=0}^T \beta^t u(c_t) \\ \text{subject to} \ \ & \sum_{t=0}^T q_t^0\left(c_t +\left(k_{t+1}-(1-\delta) k_t \right) - (w_t -\eta_t k_t) \right)\leq 0 \notag \end{aligned}

$$

Components of a **price system** have the following units:

* $\eta_t$ is measured in units of the time $t$ good per unit of time $t$ capital hired

* $q_t^0$ is measured in units of the time $t$ good per unit of a numeraire

* $q_t^0$ is measured in units of a numeraire per unit of the time $t$ good

### Definitions

```{note}

This allocation will constitute the **Big** $K$ to be in the present instance of the **Big** $K$ **, little** $k$ trick

that we'll apply to a competitive equilibrium in the spirit of [this lecture](https://python.quantecon.org/rational_expectations.html)

and [this lecture](https://python-advanced.quantecon.org/dyn_stack.html).```

and [this lecture](https://python-advanced.quantecon.org/dyn_stack.html).

```

In particular, we shall use the following procedure:

In particular, we guess that for $t=0,\dots,T$:

* obtain first-order conditions for the representative firm and the representative consumer.

* from these equations, obtain a new set of equations by replacing the firm's choice variables $\tilde k, \tilde n$ and the consumer's choice variables with the quantities $\vec C, \vec K$ that solve the planning problem.

* solve the resulting equations for $\{\vec{q}, \vec{\eta}, \vec{w}\}$ as functions of $\vec C, \vec K$.

* verify that at these prices, $c_t = C_t, k_t = \tilde k_t = K_t, \tilde n_t = 1$ for $t = 0, 1, \ldots, T$.

Thus, we guess that for $t=0,\dots,T$:

```{math}

:label: eq-price

q_t^0 = \beta^t u'(K_t)

q_t^0 = \beta^t u'(C_t)

```

```{math}

### Verification Procedure

Our approach is to stare at first-order necessary conditions for

Our approach is firsts to stare at first-order necessary conditions for

optimization problems of the household and the firm.

At the price system we have guessed, we'll then verify that both sets of first-order

```{note}

Because budget sets are affected only by relative prices,

$\{q_0^t\}$ is determined only up to multiplication by a

$\{q^0_t\}$ is determined only up to multiplication by a

positive constant.

```

**Normalization:** We are free to choose a $\{q_0^t\}$ that

makes $\lambda=1$ so that we are measuring $q_0^t$ in

**Normalization:** We are free to choose a $\{q_t^0\}$ that

makes $\lambda=1$ so that we are measuring $q_t^0$ in

units of the marginal utility of time $0$ goods.

We will plot $q, w, \eta$ below to show these equilibrium prices

c_ss = pp.f(k_ss) - pp.δ * k_ss

```

The above code from this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` lets us compute an optimal allocation for the planning problem that turns

out to be the allocation associated with a competitive equilibium.

The above code from this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` lets us compute an optimal allocation for the planning problem.

* from the preceding analysis, we know that it will also be an allocation associated with a competitive equilibium.

Now we're ready to bring in Python code that we require to compute additional objects that appear in a competitive equilibrium.