Arrow-Debreu CE

Macro

The Arrow-Debreu CE, which is also called ADCE, is a growth model which consists of a sequences of $\{c_t^h,h_t^h,k_t^h,k_t^f,h_t^f,p_t,w_t,R_t{\}}_{t=0}^{\infty }$ subject to three conditions:

1. Household Max
2. Firm Max

And there is a relatively interesting conditions, which is the:

3. Market Clear Condition.

Household Max.

Take the $\{p_t,w_t,R_t\}$ as given, the household chooses $\{c_t^h,h_t^h,k_t^h{\}}_{t=0}^{\infty }$ to maximize the intertemporal utility function:

${\max }_{\{c_t^h,h_t^h,k_t^h{\}}_{t=0}^{\infty }}U={\sum }_{t=0}^{\infty }{\beta }^{t}u(c_t)$

\begin{align} \sum_{\ell=1}^L (x'_\ell - \delta) p_\ell(n_k) &= \sum_{\ell=1}^L x'_\ell p_\ell(n_k) - \delta \sum_{\ell=1}^L p_\ell(n_k) \ &= \sum_{\ell=1}^L x'_\ell [p_\ell(n_k) - p_\ell] + \sum_{\ell=1}^L x'_\ell p_\ell - \delta \sum_{\ell=1}^L p_\ell(n_k) \end{align}

The BGP ADCE

The balanced growth path ADCE is a special case where everything grows at constant rate, but maybe not the same rate. For example, the technology grows at rate $g_A$, and the consumption grows at rate $g_c$, etc.

But finally, it turns out to be the same. Here’s a sketch proof:

Consider the household problem:

$$c_t=F(k_t,A_t)+(1-\delta )k_t-k_{t+1}$$

In the BGP ADCE, $c_t$ grows at constant rate, thus the RHS part also grows at constant rate. Note that $A_t$ grows at constant rate $g_A$, thus $F(k_t,A_t)$ must also grow at constant rate. And all the components grow at constant rate, thus $k_t$ must also grow at constant rate.