<p>Assume an infinite horizon representative agent economy with the following consumer preferences <span class="math-container">$u(c_t)$</span></p>
<p>The production technology of this economy uses capital and land, which is fixed amount in aggregate <span class="math-container">$\bar{L}$</span>.</p>
<p><span class="math-container">$$Y_t=F(K_t, L_t)= K_t^aL_t$$</span></p>
<p>where, <span class="math-container">$L_t$</span> 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 <span class="math-container">$r_t$</span>. 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 <span class="math-container">$m_t$</span>. The land is tradeable, that is there exist a competitive market for land among households, at market price <span class="math-container">$q_t$</span>. The market for land opens after production happens, such that an household decides the amount of land ownership for period <span class="math-container">$t + 1$</span>, <span class="math-container">$l_{t+1} $</span> at the end of period t.</p>
<p>Note that land does not depreciate and is not consumable, capital however depreciates at rate <span class="math-container">$\delta$</span></p>
<p>The question asks for defining Recursive competitive equilibrium.</p>
<p>—————</p>
<p>I start with defining variables</p>
<p><span class="math-container">$a$</span> is individual asset
<span class="math-container">$K$</span> is aggregate asset</p>
<p>The choice ( control) variables are (<span class="math-container">$a’,c$</span>).</p>
<p>The individual states are (<span class="math-container">$a,l$</span>)</p>
<p>The aggregate state is (<span class="math-container">$K$</span>).</p>
<p>Next, I want to write the Bellman’s equation for this economy</p>
<p><span class="math-container">$$V(a, l, K)= max \{ u(c) + \beta V(a’, l’, K’)$$</span></p>
<p>Subject to <span class="math-container">$$a’+m.l’=q.l +r.a +(1-\delta)a-c$$</span>
<span class="math-container">$$K’=G(K)$$</span></p>
<p>And prices are determined competitively as follows:</p>
<p><span class="math-container">$$q=F_L(K,L) $$</span> and <span class="math-container">$$r=F_K(K,L) $$</span></p>
<p>My question is that the budget constraint for this economy is true or it has some mistakes?</p>
<p>I'd appreciate any hints for setting up these problems.</p>
<blockquote>
<blockquote>
<p><strong>Last edit</strong></p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p>I think that the budget constraint which I constructed is</p>
</blockquote>
</blockquote>
<blockquote>
<blockquote>
<p><span class="math-container">$$c+a’+ql’=ml+ra+(1-\delta)a$$</span></p>
</blockquote>
</blockquote>
<p>Please only help me to write budget constraint</p>

CC BY-SA 4.0
2022-08-04T13:31:35.600Z
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
CC BY-SA 4.0
2022-08-04T14:25:38.690Z
The land ownership market does not make any sense given that you assume a representative agent.
Alalalalaki
CC BY-SA 4.0
2022-08-04T15:24:49.813Z
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
studentp
CC BY-SA 4.0
2022-08-04T15:26:20.387Z
@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.
studentp
CC BY-SA 4.0
2022-08-04T17:30:17.860Z
Are agents aware that utility depends on aggregate assets?
BrsG
CC BY-SA 4.0
2022-08-04T17:33:51.003Z
Yes, they are aware. @BrsG
studentp
CC BY-SA 4.0
2022-08-05T13:45:02.187Z
Dear @BKay I will be happy if you will help me to solve this question. Please:)
studentp
CC BY-SA 4.0
2022-08-08T15:04:31.430Z
@Alalalalaki can you help me for this question please:) I really need to solve this question. thank you.
studentp
<macroeconomics><self-study><bellman-equations><recursive-macroeconomics>
Write down budget constraint
studentp
# Write down budget constraint

(Asked by: username_0 on 2022-08-02)

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

**Tags:** macroeconomics, self-study, bellman-equations, recursive-macroeconomics

> **Comments:**
> **username_1**: 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?  
> **username_2**: The land ownership market does not make any sense given that you assume a representative agent.  
> **username_0**: What you mean dear @username_2 ? 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  
> **username_0**: @username_1 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.  
> **username_3**: Are agents aware that utility depends on aggregate assets?  
> **username_0**: Yes, they are aware. @username_3  
> **username_0**: Dear @username_1 I will be happy if you will help me to solve this question. Please:)  
> **username_0**: @username_2 can you help me for this question please:) I really need to solve this question. thank you.  

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## 0 answers

