<p>For two periods, the objective function is</p>
<p><span class="math-container">$$...+\,\beta^{t} \left\{ \sigma[u(q_{t}) - q_{t}] + \theta U(C_{t}) + \bar{N} - N_{t} \right\} \\+ \beta^{t+1} \left\{ \sigma[u(q_{t+1}) - q_{t+1}] + \theta U(C_{t+1}) + \bar{N} - N_{t+1} \right\}+...$$</span></p>
<p>Differentiate with respect to <span class="math-container">$K_{t+1}$</span> while using direct substitution of the resource constraint, and apply the steady state &quot;star&quot; variables to get common factors and be left with</p>
<p><span class="math-container">$$\beta \left( \frac{\partial F}{\partial K} + 1-\delta\right) -1 = 0.$$</span></p>
<p>Differentiate with respect to <span class="math-container">$N_t$</span> and arrive at</p>
<p><span class="math-container">$$\theta U(C^*)\frac{\partial F}{\partial N} - 1 =0.$$</span></p>
<p>Now note that (verify)</p>
<p><span class="math-container">$$\frac{\partial F}{\partial K} = \frac{\partial N f(k)}{\partial K} = N\frac{f'(k)}{N}$$</span></p>
<p>and (verify)</p>
<p><span class="math-container">$$\frac{\partial F}{\partial N}=\frac{\partial N f(k)}{\partial N} = f(k) - Nf'(k)\frac{K}{N^2}.$$</span></p>
<p>I leave the last one totally for you.</p>

CC BY-SA 4.0
2023-04-09T17:41:10.530Z
2023-04-09T17:46:31.407Z
2023-04-09T17:46:31.407Z
Alecos Papadopoulos
<p>I am trying to derive the first-order condtions to this economic problem, where a unit mass of ex ante agents identical agents have preference given by
<span class="math-container">$$E_{0}\sum_{t=0}^{\infty} \beta^{t} \left\{ \sigma[u(q_{t}^{c}) - q_{t}] + \theta U(C_{t}) + \bar{N} - N_{t} \right\}.$$</span> The above equation is the one I am trying to maximize. This is the social planner's problem.</p>
<p>Here <span class="math-container">$\sigma$</span> is the probability of becoming a consumer or a producer. It's just a parameter not a choice variable. Assume <span class="math-container">$u"&lt;0&lt;u'$</span> and <span class="math-container">$u(0)=0$</span> and <span class="math-container">$u'(0)=\infty$</span>.</p>
<p>Let the production of the output be <span class="math-container">$Q_{t}=F(K_{t},N_{t})$</span> and let <span class="math-container">$f(k) \equiv F(K/N,1)$</span>, where <span class="math-container">$k\equiv K/N$</span>. The resource constraint is given by <span class="math-container">$$C_{t} = F(K_{t},N_{t})+(1-\delta)K_{t} - K_{t+1},$$</span> for all <span class="math-container">$t \geq 0$</span> with <span class="math-container">$K_{0}&gt;0$</span> given and where <span class="math-container">$0 \leq \delta \leq 1$</span>. So the maximization problem is subject to this resource constraint.</p>
<p>What I am struggling to figure out is how these first-order conditions were derived. So the authors have a sequence <span class="math-container">$\left\{ K_{t+1}, N_{t}, C_{t}, q_{t} \right\}_{t=0}^{\infty}$</span> that maximizes the above problem subject to the resource constraint with <span class="math-container">$K_{0}&gt;0$</span> given.</p>
<p>The authors compute the steady-state first-best allocation that constitutes a set of numbers <span class="math-container">$\left( K^{*},k^{*},C^{*},q^{*} \right)$</span> that satisfies the following:</p>
<p><span class="math-container">$$u'(q^{*}) = 1 ( \text{I understand this part},)$$</span></p>
<p><span class="math-container">$$\beta \left( f'(k^{*}) + 1-\delta) \right) = 1,$$</span></p>
<p><span class="math-container">$$\theta U'(C^{*}) \left( f(k^{*}) - f'(k^{*})k^{*} \right) = 1,$$</span></p>
<p><span class="math-container">$$\left( f(k^{*})k^{*} - \delta \right) K^{*} = C^{*},$$</span></p>
<p>where <span class="math-container">$N^{*} = K^{*}/k^{*}$</span>.</p>
<p>Any thoughts on how these first-order conditions were derived would be greatly appreciated.</p>

CC BY-SA 4.0
2023-04-08T12:59:24.057Z
Is there a difference between $q_c^t$ and $q_t$? Is there a difference between $u$ and $U$? If $N$ is population, is it increasing or constant?
Alecos Papadopoulos
CC BY-SA 4.0
2023-04-09T07:04:42.173Z
@AlecosPapadopoulos Thanks for your reply. Not really. That's how the authors wrote it in the paper. But the for the first result they are letting both of them equal.
OGC
CC BY-SA 4.0
2023-04-09T23:13:54.677Z
Sorry it should be $q_{t}^{c}$ instead actually but I guess you already figured it out.
OGC
<optimization>
Solving first-order conditions to this social planner's problem
OGC
# Solving first-order conditions to this social planner's problem

(Asked by: username_0 on 2023-04-08)

I am trying to derive the first-order condtions to this economic problem, where a unit mass of ex ante agents identical agents have preference given by
$$E\_{0}\sum\_{t=0}^{\infty} \beta^{t} \left\{ \sigma[u(q\_{t}^{c}) - q\_{t}] + \theta U(C\_{t}) + \bar{N} - N\_{t} \right\}.$$ The above equation is the one I am trying to maximize. This is the social planner's problem.

Here $\sigma$ is the probability of becoming a consumer or a producer. It's just a parameter not a choice variable. Assume $u"<0 and $u(0)=0$ and $u'(0)=\infty$.

Let the production of the output be $Q\_{t}=F(K\_{t},N\_{t})$ and let $f(k) \equiv F(K/N,1)$, where $k\equiv K/N$. The resource constraint is given by $$C\_{t} = F(K\_{t},N\_{t})+(1-\delta)K\_{t} - K\_{t+1},$$ for all $t \geq 0$ with $K\_{0}>0$ given and where $0 \leq \delta \leq 1$. So the maximization problem is subject to this resource constraint.

What I am struggling to figure out is how these first-order conditions were derived. So the authors have a sequence $\left\{ K\_{t+1}, N\_{t}, C\_{t}, q\_{t} \right\}\_{t=0}^{\infty}$ that maximizes the above problem subject to the resource constraint with $K\_{0}>0$ given.

The authors compute the steady-state first-best allocation that constitutes a set of numbers $\left( K^{\*},k^{\*},C^{\*},q^{\*} \right)$ that satisfies the following:

$$u'(q^{\*}) = 1 ( \text{I understand this part},)$$

$$\beta \left( f'(k^{\*}) + 1-\delta) \right) = 1,$$

$$\theta U'(C^{\*}) \left( f(k^{\*}) - f'(k^{\*})k^{\*} \right) = 1,$$

$$\left( f(k^{\*})k^{\*} - \delta \right) K^{\*} = C^{\*},$$

where $N^{\*} = K^{\*}/k^{\*}$.

Any thoughts on how these first-order conditions were derived would be greatly appreciated.

**Tags:** optimization
> **Comments:**
> **username_1**: Is there a difference between $q_c^t$ and $q_t$? Is there a difference between $u$ and $U$? If $N$ is population, is it increasing or constant?  
> **username_0**: @AlecosPapadopoulos Thanks for your reply. Not really. That's how the authors wrote it in the paper. But the for the first result they are letting both of them equal.  
> **username_0**: Sorry it should be $q_{t}^{c}$ instead actually but I guess you already figured it out.  

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## 1 answers

### Answer by username_1 on 2023-04-09. Score: 2 | Accepted answer

For two periods, the objective function is

$$...+\,\beta^{t} \left\{ \sigma[u(q\_{t}) - q\_{t}] + \theta U(C\_{t}) + \bar{N} - N\_{t} \right\} \\+ \beta^{t+1} \left\{ \sigma[u(q\_{t+1}) - q\_{t+1}] + \theta U(C\_{t+1}) + \bar{N} - N\_{t+1} \right\}+...$$

Differentiate with respect to $K\_{t+1}$ while using direct substitution of the resource constraint, and apply the steady state "star" variables to get common factors and be left with

$$\beta \left( \frac{\partial F}{\partial K} + 1-\delta\right) -1 = 0.$$

Differentiate with respect to $N\_t$ and arrive at

$$\theta U(C^\*)\frac{\partial F}{\partial N} - 1 =0.$$

Now note that (verify)

$$\frac{\partial F}{\partial K} = \frac{\partial N f(k)}{\partial K} = N\frac{f'(k)}{N}$$

and (verify)

$$\frac{\partial F}{\partial N}=\frac{\partial N f(k)}{\partial N} = f(k) - Nf'(k)\frac{K}{N^2}.$$

I leave the last one totally for you.


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