<p>Writing the production function in this way was most popularized by Lucas (1988) in 'On the Mechanics of Economic Development', though he used an externality on human capital.</p>
<p>You can read through the paper (or find some lecture notes online) if you want to see how the model is solved, however, I will provide a loose sketch of why writing a production function in this way implies capital has positive externalities.</p>
<p>Essentially, a firm maximizes the following problem:
<span class="math-container">$$\max_{k_t,n_t} \; A K_t^{1-\alpha}k_t^\alpha n_t^{1-\alpha} - w_tn_t - r_t k_t$$</span>
That is, their production depends on average capital in the economy <span class="math-container">$K_t$</span>, and while in equilibrium (of a representative agent economy) we will have <span class="math-container">$k_t = K_t$</span>, they do not believe they have any influence over this.</p>
<p>However, an optimal social planner knows this and would maximize:
<span class="math-container">$$\max_{k_t,n_t} \; A k_t n_t^{1-\alpha} - w_tn_t - r_t k_t$$</span></p>
<p>After you solve the firms problem, the household's maximization problem, and use the market clearing conditions, you will find that <span class="math-container">$k_t &lt; k^*$</span> (where <span class="math-container">$k^*$</span> is the <span class="math-container">$k_t$</span> the planner would choose). That is, firms will choose inefficiently low capital (<span class="math-container">$k_t$</span>) because the positive effect of choosing more capital has on the average capital in the economy (<span class="math-container">$K_t$</span>) is not priced in.</p>

CC BY-SA 4.0
2022-07-24T19:19:25.693Z
Hi Fica thank you for great answer. I understand, thanks to you. I will put tick for this answer. Also if it is possible for you, can you please look at this question aswell? https://economics.stackexchange.com/questions/52159/question-on-keynesian-economics-from-sargent-s-book
studentp
CC BY-SA 4.0
2022-07-23T19:03:23.157Z
2022-07-23T19:03:23.157Z
Fića
<p>I have a following model of endogenous growth where each firm has the following technology;
<span class="math-container">$$y_t=AK_t^{1-\alpha} k_t^{\alpha} n_t^{1-\alpha}$$</span></p>
<p>The production function above defines an externality. I am asking you for Explaining what it is.</p>
<p>I also want you to write down both growth models, show that the equilibrium allocation of this model generates lower growth than optimal due to the positive externality of firm level capital accumulation by solving the centralized and competitive problems.</p>
<p>This model seems me very complicated. Therefore I am asking. If it has any special name or the same like that, please let me know. I will read something about this model from papers or lecture notes or problem sets.</p>
<p><strong>Note: I dont want to write whole model. I just want to learn its interpretations  and intuitively explanation.</strong></p>

CC BY-SA 4.0
2022-07-23T13:49:22.987Z
Your question is very unclear. Endogenous growth model is a very large genre, you cannot simply write down a production function and ask for others to write the whole model.
Alalalalaki
CC BY-SA 4.0
2022-07-23T14:57:51.010Z
No no I dont want to write whole model. I just want to learn its interpretation and intiutively explanation. I will add this note. Thank you. @Alalalalaki
studentp
<macroeconomics><self-study><endogeneity><endogenous-growth>
Endogenous growth model with externalities
studentp
# Endogenous growth model with externalities

(Asked by: username_0 on 2022-07-21)

I have a following model of endogenous growth where each firm has the following technology;
$$y\_t=AK\_t^{1-\alpha} k\_t^{\alpha} n\_t^{1-\alpha}$$

The production function above defines an externality. I am asking you for Explaining what it is.

I also want you to write down both growth models, show that the equilibrium allocation of this model generates lower growth than optimal due to the positive externality of firm level capital accumulation by solving the centralized and competitive problems.

This model seems me very complicated. Therefore I am asking. If it has any special name or the same like that, please let me know. I will read something about this model from papers or lecture notes or problem sets.

**Note: I dont want to write whole model. I just want to learn its interpretations and intuitively explanation.**

**Tags:** macroeconomics, self-study, endogeneity, endogenous-growth
> **Comments:**
> **username_1**: Your question is very unclear. Endogenous growth model is a very large genre, you cannot simply write down a production function and ask for others to write the whole model.  
> **username_0**: No no I dont want to write whole model. I just want to learn its interpretation and intiutively explanation. I will add this note. Thank you. @Alalalalaki  

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## 1 answers

### Answer by username_2 on 2022-07-23. Score: 4 | Accepted answer

Writing the production function in this way was most popularized by Lucas (1988) in 'On the Mechanics of Economic Development', though he used an externality on human capital.

You can read through the paper (or find some lecture notes online) if you want to see how the model is solved, however, I will provide a loose sketch of why writing a production function in this way implies capital has positive externalities.

Essentially, a firm maximizes the following problem:
$$\max\_{k\_t,n\_t} \; A K\_t^{1-\alpha}k\_t^\alpha n\_t^{1-\alpha} - w\_tn\_t - r\_t k\_t$$
That is, their production depends on average capital in the economy $K\_t$, and while in equilibrium (of a representative agent economy) we will have $k\_t = K\_t$, they do not believe they have any influence over this.

However, an optimal social planner knows this and would maximize:
$$\max\_{k\_t,n\_t} \; A k\_t n\_t^{1-\alpha} - w\_tn\_t - r\_t k\_t$$

After you solve the firms problem, the household's maximization problem, and use the market clearing conditions, you will find that $k\_t < k^\*$ (where $k^\*$ is the $k\_t$ the planner would choose). That is, firms will choose inefficiently low capital ($k\_t$) because the positive effect of choosing more capital has on the average capital in the economy ($K\_t$) is not priced in.

> **Comments:**
> **username_0**: Hi Fica thank you for great answer. I understand, thanks to you. I will put tick for this answer. Also if it is possible for you, can you please look at this question aswell? https://economics.stackexchange.com/questions/52159/question-on-keynesian-economics-from-sargent-s-book  

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