<p>The production function is <span class="math-container">$F(K_t,N_t)=AK_t^\alpha N_t^{1-\alpha}$</span> and depreciation <span class="math-container">$(\delta)$</span> is equal to 1. The given preferences are as follows: <span class="math-container">$$U(c_1,l_1,c_2,l_2)=\gamma log(c_1)+(1-\gamma)log(l_1)+\beta[\gamma log(c_2)+(1-\gamma)log(l_2)]$$</span></p>
<p>From which <span class="math-container">$\gamma\in(0,1)$</span>,<span class="math-container">$\beta\in(0,1)$</span>. We assume <span class="math-container">$\forall t:l_t + n_t ≤ 1$</span> and all non-negativities on all variables. How would we solve the Social Planner?</p>
<p>Since <span class="math-container">$\delta=1$</span> we would have the constraint for period 1: <span class="math-container">$$c_1+k_2=AK_1^\alpha N_1^{1-\alpha}$$</span>, from which <span class="math-container">$$c_1=AK_1^\alpha N_1^{1-\alpha} -k_{2}$$</span>.
For period 2, <span class="math-container">$$c_2=AK_2^\alpha N_2^{1-\alpha}$$</span> since we don't have anymore capital for next period.
Now we need to find the allocation <span class="math-container">$\{c_1,c_2,l_1,l_2\}$</span> by doing the Lagrangian.</p>
<p><span class="math-container">$$U(c_1,l_1,c_2,l_2)=\gamma log(c_1)+(1-\gamma)log(l_1)+\beta[\gamma log(c_2)+(1-\gamma)log(l_2)]-\lambda_1[c_1-AK_1^\alpha N_1^{1-\alpha} -k_{2}]-\lambda_2[c_2-AK_2^\alpha N_2^{1-\alpha}]$$</span></p>
<p>Finding F.O.Cs wrt <span class="math-container">$c_1,c_2,k_2:$</span></p>
<p>{<span class="math-container">$c_1$</span>}: <span class="math-container">$\frac{\gamma}{c_1}=\lambda_1$</span></p>
<p>{<span class="math-container">$c_2$</span>}: <span class="math-container">$\frac{\beta\gamma}{c_2}=\lambda_2$</span></p>
<p>{<span class="math-container">$k_2$</span>}: <span class="math-container">$\lambda_2A\alpha k_2^{\alpha-1}N_2^{1-\alpha}=\lambda_1$</span></p>
<p>Generating <span class="math-container">$\frac{\lambda_1}{\lambda_2}$</span> we would get <span class="math-container">$$c_2=A\alpha k_2^{\alpha-1}N_2^{1-\alpha}c_1\beta$$</span></p>
<p>That we substitute in the original constraint to get <span class="math-container">$$k_2=c_1\alpha\beta$$</span> to which we substitute in <span class="math-container">$$k_2+c_1=Ak_1(1-l_1)^{1-\alpha}$$</span>
Some algebra and we get <span class="math-container">$$c_1=\frac{Ak_1^\alpha(1-l_1)^{1-\alpha}}{\alpha\beta+1}$$</span>
<span class="math-container">$$c_2=k_2^{\alpha}N_2^{1-\alpha}$$</span></p>
<p>For <span class="math-container">$l_t$</span> we substitute <span class="math-container">$c_1,c_2$</span> in the FOCs that equal each <span class="math-container">$\lambda$</span> respectively along with the FOCs of <span class="math-container">$l_1,l_2$</span> and get the following, again after the algebra:
<span class="math-container">$$l_1=\frac{1-\gamma}{\gamma(\alpha\beta+1)(1-\alpha)+1-\gamma}$$</span>
<span class="math-container">$$l_2=\frac{1-\gamma}{\gamma(1-\alpha)(1-\alpha)+1-\gamma}$$</span></p>
<p>Phew! This was hella lot of algebra and I don't know if I made a mistake or not. I hope this is correct but feel free to point my errors.</p>
<ul>
<li>Define the Arrow-Debreu equilibrium for this economy. Find the equilibrium prices and quantities.</li>
</ul>
<p>Besides the market clearing and prices and allocations conditions how do we continue from here :)</p>

CC BY-SA 4.0
2021-02-24T14:29:28.957Z
Is there a relationship between $n_t$ and $N_t$? The planner's problem usually involves maximizing utility subject to *resource* constraint(s). So you should determine what those are.
Herr K.
CC BY-SA 4.0
2021-02-24T16:21:43.900Z
So I guess $n_t=1-l_t$ to which we substitute in  $c_t=Ak_t^\alpha (1-l_t)_t^{1-\alpha} -k_{t+1}$ and from this we substitute $c_1$ and $c_2$ in the utility? I'm a bit confused with all of this :\
Maybeline Lee
CC BY-SA 4.0
2021-02-26T02:46:11.083Z
The problem you provided seems under-specified. For example, where does $K_1$ come from? What's the aggregation rule that takes lower case variables ($c_t,n_t,k_t$) to the upper case variables ($C_t,N_t,K_t$)?
Herr K.
CC BY-SA 4.0
2021-02-26T02:55:17.857Z
For Arrow-Debreu equilibrium, you need to incorporate market-clearing conditions in each period: wages should clear the labor market, rents should clear the capital market, and prices should clear the goods market.
Herr K.
CC BY-SA 4.0
2021-02-26T11:14:07.923Z
For AD would we have the following: max $u(c_1,l_1)+\beta u(c_2,l_2)$ s.t $p_1(c_1+k_2)\leq p_1(r_1k_1+w_1n_1)$ and $p_2(c_2)\leq p_2(r_2k_2+w_2n_2)$? And then again with the Lagrangian by finding focs w.r.t $c_t,l_t,k_2$?
Maybeline Lee
CC BY-SA 4.0
2021-02-26T14:26:52.120Z
Market clearing means prices equalize supply and demand. For example, by solving an agent's maximization problem, you'd get optimal consumption as a function of good's price (maybe among other things) $c_t^*(p_t,\dots)$, and this is the demand for goods from one agent. You'd have to aggregate all agent's demand to obtain the *market* demand for goods $C_t(p_t,\dots)$. The supply of goods is given by the production function $F(K_t,N_t)$ whose arguments ($K_t,N_t$) are functions of the good's price.
Herr K.
CC BY-SA 4.0
2021-02-26T14:26:56.207Z
In equilibrium, the market clearing price in the good's market should be such that agents maximize utility by consuming an appropriate amount of goods and firms maximize profit by producing and selling exactly the amount consumed by the agents. And this is just the good's market. The same must also be true for the labor market, where aggregate labor supply by the agents is equal to labor demand by the firm, and the capital market, etc.
Herr K.
CC BY-SA 4.0
2021-02-26T14:34:12.783Z
I'm not sure if this level of nuance is what you intended to get to, or perhaps whoever gave you this problem simply wanted you to be able to apply the Lagrange method of optimization. If you give the full description of the problem (in particular, is it representative agent & representative firm, unit mass of agents, etc?), you'll have a better chance of getting a clearer answer.
Herr K.
CC BY-SA 4.0
2021-02-26T15:49:10.813Z
Yeah, I already found the allocations for the ADE as well. But you're right, from what I've discussed with others it's just pure derivation and not logic. It's a 2 period-economy with a representative consumer and one firm.
Maybeline Lee
<macroeconomics><utility><production-function>
Social Planner problem: two period
Maybeline Lee
# Social Planner problem: two period

(Asked by: username_0 on 2021-02-23)

The production function is $F(K\_t,N\_t)=AK\_t^\alpha N\_t^{1-\alpha}$ and depreciation $(\delta)$ is equal to 1. The given preferences are as follows: $$U(c\_1,l\_1,c\_2,l\_2)=\gamma log(c\_1)+(1-\gamma)log(l\_1)+\beta[\gamma log(c\_2)+(1-\gamma)log(l\_2)]$$

From which $\gamma\in(0,1)$,$\beta\in(0,1)$. We assume $\forall t:l\_t + n\_t ≤ 1$ and all non-negativities on all variables. How would we solve the Social Planner?

Since $\delta=1$ we would have the constraint for period 1: $$c\_1+k\_2=AK\_1^\alpha N\_1^{1-\alpha}$$, from which $$c\_1=AK\_1^\alpha N\_1^{1-\alpha} -k\_{2}$$.
For period 2, $$c\_2=AK\_2^\alpha N\_2^{1-\alpha}$$ since we don't have anymore capital for next period.
Now we need to find the allocation $\{c\_1,c\_2,l\_1,l\_2\}$ by doing the Lagrangian.

$$U(c\_1,l\_1,c\_2,l\_2)=\gamma log(c\_1)+(1-\gamma)log(l\_1)+\beta[\gamma log(c\_2)+(1-\gamma)log(l\_2)]-\lambda\_1[c\_1-AK\_1^\alpha N\_1^{1-\alpha} -k\_{2}]-\lambda\_2[c\_2-AK\_2^\alpha N\_2^{1-\alpha}]$$

Finding F.O.Cs wrt $c\_1,c\_2,k\_2:$

{$c\_1$}: $\frac{\gamma}{c\_1}=\lambda\_1$

{$c\_2$}: $\frac{\beta\gamma}{c\_2}=\lambda\_2$

{$k\_2$}: $\lambda\_2A\alpha k\_2^{\alpha-1}N\_2^{1-\alpha}=\lambda\_1$

Generating $\frac{\lambda\_1}{\lambda\_2}$ we would get $$c\_2=A\alpha k\_2^{\alpha-1}N\_2^{1-\alpha}c\_1\beta$$

That we substitute in the original constraint to get $$k\_2=c\_1\alpha\beta$$ to which we substitute in $$k\_2+c\_1=Ak\_1(1-l\_1)^{1-\alpha}$$
Some algebra and we get $$c\_1=\frac{Ak\_1^\alpha(1-l\_1)^{1-\alpha}}{\alpha\beta+1}$$
$$c\_2=k\_2^{\alpha}N\_2^{1-\alpha}$$

For $l\_t$ we substitute $c\_1,c\_2$ in the FOCs that equal each $\lambda$ respectively along with the FOCs of $l\_1,l\_2$ and get the following, again after the algebra:
$$l\_1=\frac{1-\gamma}{\gamma(\alpha\beta+1)(1-\alpha)+1-\gamma}$$
$$l\_2=\frac{1-\gamma}{\gamma(1-\alpha)(1-\alpha)+1-\gamma}$$

Phew! This was hella lot of algebra and I don't know if I made a mistake or not. I hope this is correct but feel free to point my errors.

* Define the Arrow-Debreu equilibrium for this economy. Find the equilibrium prices and quantities.

Besides the market clearing and prices and allocations conditions how do we continue from here :)

**Tags:** macroeconomics, utility, production-function
> **Comments:**
> **username_1**: Is there a relationship between $n_t$ and $N_t$? The planner's problem usually involves maximizing utility subject to *resource* constraint(s). So you should determine what those are.  
> **username_0**: So I guess $n_t=1-l_t$ to which we substitute in  $c_t=Ak_t^\alpha (1-l_t)_t^{1-\alpha} -k_{t+1}$ and from this we substitute $c_1$ and $c_2$ in the utility? I'm a bit confused with all of this :\  
> **username_1**: The problem you provided seems under-specified. For example, where does $K_1$ come from? What's the aggregation rule that takes lower case variables ($c_t,n_t,k_t$) to the upper case variables ($C_t,N_t,K_t$)?  
> **username_1**: For Arrow-Debreu equilibrium, you need to incorporate market-clearing conditions in each period: wages should clear the labor market, rents should clear the capital market, and prices should clear the goods market.  
> **username_0**: For AD would we have the following: max $u(c_1,l_1)+\beta u(c_2,l_2)$ s.t $p_1(c_1+k_2)\leq p_1(r_1k_1+w_1n_1)$ and $p_2(c_2)\leq p_2(r_2k_2+w_2n_2)$? And then again with the Lagrangian by finding focs w.r.t $c_t,l_t,k_2$?  
> **username_1**: Market clearing means prices equalize supply and demand. For example, by solving an agent's maximization problem, you'd get optimal consumption as a function of good's price (maybe among other things) $c_t^*(p_t,\dots)$, and this is the demand for goods from one agent. You'd have to aggregate all agent's demand to obtain the *market* demand for goods $C_t(p_t,\dots)$. The supply of goods is given by the production function $F(K_t,N_t)$ whose arguments ($K_t,N_t$) are functions of the good's price.  
> **username_1**: In equilibrium, the market clearing price in the good's market should be such that agents maximize utility by consuming an appropriate amount of goods and firms maximize profit by producing and selling exactly the amount consumed by the agents. And this is just the good's market. The same must also be true for the labor market, where aggregate labor supply by the agents is equal to labor demand by the firm, and the capital market, etc.  
> **username_1**: I'm not sure if this level of nuance is what you intended to get to, or perhaps whoever gave you this problem simply wanted you to be able to apply the Lagrange method of optimization. If you give the full description of the problem (in particular, is it representative agent & representative firm, unit mass of agents, etc?), you'll have a better chance of getting a clearer answer.  
> **username_0**: Yeah, I already found the allocations for the ADE as well. But you're right, from what I've discussed with others it's just pure derivation and not logic. It's a 2 period-economy with a representative consumer and one firm.  

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## 0 answers

