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 $u"<0<u'$ 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.

  • 2
    $\begingroup$ 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? $\endgroup$ Apr 8, 2023 at 12:59
  • $\begingroup$ @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. $\endgroup$
    – OGC
    Apr 9, 2023 at 7:04
  • $\begingroup$ Sorry it should be $q_{t}^{c}$ instead actually but I guess you already figured it out. $\endgroup$
    – OGC
    Apr 9, 2023 at 23:13

1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.