3
$\begingroup$

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.

—————

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

$\endgroup$
8
  • $\begingroup$ Don't you need a constraint where $\sum a = K$ and $\sum l = L$ to indicate that within a period (K) and at all times (L), these factors are in fixed supply? $\endgroup$
    – BKay
    Aug 4, 2022 at 13:31
  • $\begingroup$ The land ownership market does not make any sense given that you assume a representative agent. $\endgroup$ Aug 4, 2022 at 14:25
  • $\begingroup$ What you mean dear @Alalalalaki ? I cant see what you mean. How can I correct my solution according to your idea? Can you please show the solution in detail? Many thanks if you will show its solution $\endgroup$
    – studentp
    Aug 4, 2022 at 15:24
  • $\begingroup$ @BKay I have no idea about your suggestion. I am very new learner of this topic. So can you please post your answer in order to correct my answer? I will be glad. Thank you. $\endgroup$
    – studentp
    Aug 4, 2022 at 15:26
  • $\begingroup$ Are agents aware that utility depends on aggregate assets? $\endgroup$
    – BrsG
    Aug 4, 2022 at 17:30

1 Answer 1

3
+200
$\begingroup$

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.

$\endgroup$
5
  • $\begingroup$ I am really impressed in your explanation. I see it clearly. Many thanks:) $\endgroup$
    – studentp
    Oct 27, 2022 at 17:18
  • $\begingroup$ I also have a similar question. And I cannot deal with that question as well. I am really confused in all three parts. And your explanation is very good. If I you mind, can you please look at that question as well? I will be really glad. Thank you so much. economics.stackexchange.com/questions/53265/… $\endgroup$
    – studentp
    Oct 27, 2022 at 17:21
  • $\begingroup$ I am preparing an important preliminary exam. If you help me, I will be really happy. Many thanks in advance. $\endgroup$
    – studentp
    Oct 27, 2022 at 19:53
  • $\begingroup$ @studentp Would you like to tell which exam it is? I find you posting some interesting questions from time to time. $\endgroup$
    – Rick_Morty
    Oct 28, 2022 at 9:10
  • $\begingroup$ Phd prelim exam @Citrus I am only asking the question parts which either I cant totally understand or solve. $\endgroup$
    – studentp
    Oct 28, 2022 at 12:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.