Write down budget constraint

Question

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 $$a’+m.l’=q.l +r.a +(1-\delta)a-c$$ $$K’=G(K)$$

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.

Last edit

I think that the budget constraint which I constructed is

$$c+a’+ql’=ml+ra+(1-\delta)a$$

Please only help me to write budget constraint

Answer 1

I think the confusion is all coming from the HH's side of the problem. So let's start by looking at the HH's problem. Let $\{r_t,q_t,m_t\}$ be given. Suppose that the HH starts period $t$ with $k_t$ amount of capital and $l_t$ amount of land.

First let's look at capital. The only thing the HH can do is rent it out as there is no investment decision, therefore the HH receives $k_tr_t$

The land decision is a little more involved. The HH can either sell the land in the open market and receive $q_t$, or rent the land and receive $m_t$. TLet $x_t$ be the amount of land that the HH sells (if $x_t<0$ then the HH is buying). Then the HH receives $x_tq_t+(l_t)m_t$. It's important that the HH sells the land after production. This means that the HH rents $l_t$ to the firm and then decides how much to sell in the open market.

Therefore, the budget constraint is

$$c_t=x_tq_t+l_tm_t+k_tr_t$$

Let's look at the resource constraint, If they sell $x_t$ units of land, then the HH has $l_t-x_t$ units of land tomorrow. Therefore, $x_t=l_t-l_{t+1}$. In addition capital depreciates, $k_{t+1}=k_t(1-\delta)$

The HH solves

$$\max_{c_t} \sum_t\beta^tu(c_t)$$

subject to

$$c_t=q_t(l_t-l_{t+1})+l_{t}m_t+k_tr_t$$ $$k_{t+1}=(1-\delta)k_t$$

If we wrote this equation as a Bellman it would be

$$V(a,l,K)=\max\{u(c)+\beta V(a',l',K')\}$$

$$c=q(l-l')+lm+ra$$ $$a'=(1-\delta)a$$

This should be your budget constraint.