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.

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

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

$$

At $t$ the household allocates its income to the following

purchases

purchases between the following two categories:

* consumption $c_t$

* net investment $k_{t+1} -(1-\delta)k_t$

$$

\left(c_t + (k_{t+1} -(1-\delta)k_t)\right)

$$

Here $\left(k_{t+1} -(1-\delta)k_t\right)$ is the household's

net investment in physical capital and $\delta \in (0,1)$ is

\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:

* $w_t$ is measured in units of the time $t$ good per unit of time $t$ labor hired

* $\eta_t$ 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

### Definitions

- A **price system** is a sequence

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

- An **allocation** is a sequence

$\{c_t,k_{t+1},n_t=1\}_{t=0}^T = \{\vec{c}, \vec{k}, \vec{n}\}$.

- A **competitive equilibrium** is a price system and an allocation

for which

with the following properties:

- Given the price system, the allocation solves the household's

problem.

- Given the price system, the allocation solves the firm's

problem.

The vision here is that an equilibrium price system and allocation are determined once and for all.

In effect, we imagine that all trades occur just before time $0$.

## Computing a Competitive Equilibrium

We compute a competitive equilibrium by using a **guess and

```{math}

:label: eq-price

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

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

```

```{math}

\eta_t = f'(K_t)

```

At these prices, let the capital chosen by the household be

At these prices, let capital chosen by the household be

```{math}

:label: eq-pr4

c_t^* + k_{t+1}^* - (1-\delta) k_t^* = F(\tilde k_t^*, \tilde n_t^*)

$$

We shall verify that for $t=0,\dots,T$ the allocations chosen

We shall verify that for $t=0,\dots,T$ allocations chosen

by the household and the firm both equal the allocation that solves

the planning problem:

### Verification Procedure

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

Our approach is 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