# Solving first-order conditions to this social planner's problem

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.

• 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? Apr 8, 2023 at 12:59
• @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
Apr 9, 2023 at 7:04
• Sorry it should be $q_{t}^{c}$ instead actually but I guess you already figured it out.
– OGC
Apr 9, 2023 at 23:13

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.