Assume an infinite horizon representative agent economy with the following consumer preferences $u(c_t)$
The production technology of this economy uses capital and land, which is fixed amount in aggregate $\bar{L}$.
$$Y_t=F(K_t, L_t)= K_t^aL_t$$
where, $L_t$ is the land input and production function has the usual properties. The household owns the land and capital in this economy. Capital stock is rented to firms for production with a rate of return $r_t$. The land, at each period, can be lent out to firms at the competitive markets to be used in production with the rate of return $m_t$. The land is tradeable, that is there exist a competitive market for land among households, at market price $q_t$. The market for land opens after production happens, such that an household decides the amount of land ownership for period $t + 1$, $l_{t+1} $ at the end of period t.
Note that land does not depreciate and is not consumable, capital however depreciates at rate $\delta$
The question asks for defining Recursive competitive equilibrium.
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I start with defining variables
$a$ is individual asset $K$ is aggregate asset
The choice ( control) variables are ($a’,c$).
The individual states are ($a,l$)
The aggregate state is ($K$).
Next, I want to write the Bellman’s equation for this economy
$$V(a, l, K)= max \{ u(c) + \beta V(a’, l’, K’)$$
Subject to $$r.a’+m.l’=q.l +r.a +(1-\delta)a-c$$ $$K’=G(K)$$
And prices are determined competitively as follows:
$$q=F_L(K,L) $$ and $$r=F_K(K,L) $$
My question is that the budget constraint for this economy is true or it has some mistakes?
I'd appreciate any hints for setting up these problems.
PS: Cross-Post I have asked for it on both websites because there this question didn’t attract attention. I hope someone on this website will give hint for this question. Thank you.
Capital can be freely converted from consumption goods, land can be purchased and sold at price m’, so the budget constraint should be:
$a’+l’m’+c = (q+m’)l + (1-\delta +r) a $
On thing to then note is that you can pin down aggregate capital accumulation by noting that the rate of return on both assets, in every period will be equal in equilibrium, that is:
$(1-\delta +r) a = (m’+q)/m$
This should leave you with enough equations to solve the problem.