*Tuesday, September 21, 2021*

## Overview

Here we explore the famous growth models of an economy. This lesson will be the most technical of the semester, but remember there are no homeworks or exams that have you use the models to calculate equilibrium outcomes. It is more important that you understand the basic mechanisms and the major conclusions of the models. Modern economic development, macroeconomics, and public policy literatures and debates rely heavily on these models!

We look first at the famous Solow model, or “*exogenous* growth” model of Robert Solow. Physical capital plays a central role in the model, requiring an economy to accumulate an optimal stock of physical capital (relative to labor) in order to achieve a steady-state growth path. The more surprising element is that growth in output empirically is explained almost entirely (about 80%) by achievements in “technology” or “total factor productivity (TFP),” the unmeasurable “Solow residual.” In these models, the growth of TFP is given *exogenously.*

## Readings

### Optional/Referenced Readings:

## Slides

Below, you can find the slides in two formats. Clicking the image will bring you to the html version of the slides in a new tab. Note while in going through the slides, you can type `h` to see a special list of viewing options, and type `o` for an outline view of all the slides.

The lower button will allow you to download a PDF version of the slides. I suggest printing the slides beforehand and using them to take additional notes in class (*not everything* is in the slides)!

## Interactive Solow Model

This is an example that I wrote in R/Shiny in previously, to visualize what it is we are looking at in the Solow Model. As I find more time, I may update this and integrate it into our slides, but until then, I will just post a link.

You can adjust things about the (Cobb-Douglas) production function, savings rate, depreciation rate, population growth rate, and TFP growth rate, and see how it would affect the graph and the equilibrium.

Note I have not added calculations or the golden rule yet. Moving \(k\) around just shows where \(k\) would start, and then it would have to grow/shrink to get to \(k^*\).

## Math Appendix

### Growth Accounting

Take the production function of the general form: \[Y=f(A,L,K)\] where \(Y\) is output, \(K\) is total capital stock, \(L\) is total labor, and \(A\) is TFP.

Take standard assumptions about \(f\): it is increasing in \(A\), \(L\), & \(K\), and is homogenous of degree 1 (constant returns to scale), so we can assume perfect competition, whereby:

\[\begin{align*} \frac{dY}{dK}=MP_K&=r\\ \frac{dY}{dL}=MP_L&=w\\ \end{align*}\]

Totally differentiate the production function:

\[dY=F_A dA+F_K dK + F_L dL\]

where \(F_i\) denotes the partial derivative of \(F\) w.r.t. factor \(i\), aka the marginal product of \(i\). With perfect competition, this becomes:

\[\begin{align*} dY&=F_A dA + MP_K dK+ MP_L dL\\ dY&=F_A dA + r dK + w dL\\ \end{align*}\]

Divide by \(Y\) and convert each change into growth rates:

\[\frac{dY}{Y}=\left(\frac{F_A A}{Y}\right)\left(\frac{dA}{A}\right)+\left(\frac{rK}{Y}\right)\left(\frac{dK}{K}\right)+\left(\frac{wL}{L}\right)\left(\frac{dL}{L}\right)\]

Now, denote the growth rate (% change over time) of each factor \(i\) as \(g_i=\frac{di}{i}\):

\[g_Y=(\frac{F_A A}{Y})*g_A + (\frac{rK}{Y})*g_K+(\frac{wL}{Y})*g_L\]

Then \(\frac{rK}{Y}\) is the share of total income that goes to capital, denoted \(\alpha\), and \(\frac{wL}{Y}\) is the share that goes to labor, denoted as \(1-\alpha\). Re-express the equation as:

\[g_Y=\frac{F_A A}{Y}*g_A+ \alpha g_K+(1-\alpha)g_L\]

In principle, \(\alpha, g_Y, g_L\) are observable, but the Solow residual term \(\frac{F_A A}{Y}\) isn’t. We can measure it as the portion of increase in total income not accounted for by (weighted) growth of factor inputs:

\[\text{Solow Residual}=g_Y-\alpha g_K-(1-\alpha)g_L\]