# Write down budget constraint

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.

—————

$$a$$ 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

• 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?
– BKay
Commented Aug 4, 2022 at 13:31
• The land ownership market does not make any sense given that you assume a representative agent. Commented Aug 4, 2022 at 14:25
• 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 Commented Aug 4, 2022 at 15:24
• @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. Commented Aug 4, 2022 at 15:26
• Are agents aware that utility depends on aggregate assets?
– BrsG
Commented Aug 4, 2022 at 17:30

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.

• I am really impressed in your explanation. I see it clearly. Many thanks:) Commented Oct 27, 2022 at 17:18
• 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/… Commented Oct 27, 2022 at 17:21
• I am preparing an important preliminary exam. If you help me, I will be really happy. Many thanks in advance. Commented Oct 27, 2022 at 19:53
• @studentp Would you like to tell which exam it is? I find you posting some interesting questions from time to time. Commented Oct 28, 2022 at 9:10
• Phd prelim exam @Citrus I am only asking the question parts which either I cant totally understand or solve. Commented Oct 28, 2022 at 12:20