Endogenous growth model with externalities

Question

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.

Answer 1

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.