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library(quizify)
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# Overview

In this lab we will work with the neo-classical growth model developed by Solow and Swann. Our goals are:

1. To derive some analytical properties of the model, assuming a Cobb-Douglas production function.

2. To understand the dynamics of income and savings using a simulation app.

3. To understand how the capital intensification may or may not lead to increasing inequality, depending on the nature of the production function.

The order of this lab will be slightly different in that we will start with some mathematical derivations and then go on to do some computing afterwards.

(In order to display the equations, you may need to click on them with the mouse.)

# 1. Introduction

In the Solow model, the capital per worker and output per worker are in their steady state when investment exactly balances population growth and depreciation. This is when

$sy\left(k\right)=\left(n+d\right)k.$

# 2. Analytical questions

Note: In this lab, we are beginning with some analytic, pencil and paper problems. Please answer them on Gradescope. You can even open up the Gradescope page using an R-command.

browseURL("https://gradescope.com/courses/14565")
1. Let the production function be Cobb-Douglas, with $y\left(k\right)={k}^{0.4}$$y(k)=k^{0.4}$. Assume that the savings rate is 20 percent, population growth is 1 percent, and depreciation is 4 percent.

(Express your answer as a general formula in terms of the symbols “s”, “k”, “n”, “d”, and the exponent $\alpha$$\alpha$ of the Cobb-Douglas production function.

For example, for the steady-state level of capital per worker, we substitute the production function $y\left(k\right)={k}^{a}$$y(k) = k^a$ into the steady-state condition above to give

$sy\left(k\right)=s{k}^{\alpha }=\left(n+d\right)k$
Re-arranging, then gives us a formula for k in the steady state:
${\left(\frac{s}{n+d}\right)}^{\frac{1}{1-\alpha }}=k$

You can then substitute the values given in the problem and provide your numerical answer.)

1. What is the steady-state level of capital per worker?

2. What is the steady-state of output per worker?

3. What is the steady-state level of consumption per worker?

1. Now assume population growth is instead -0.5 % (approximately the growth rate when every couple has 1.7 children), but that all other parameters stay the same.
1. What is the new steady-state output per worker? Is it higher or lower than with faster population growth? [A numerical answer and 1 sentence response is fine.]

2. What is the new steady-state of consumption per worker? Is it higher or lower than with faster population growth? [A numerical answer and 1 sentence response is fine.]

# 3. Questions with the App

For the following question, you may find it useful to experiment with the “Solow_2017” app available at:

browseURL("http://shiny.demog.berkeley.edu/josh/solow_2017/")

(You can also use the app to get a rough check your answers to the analytical questions above.)

Assume your initial parameters are the default s, n, d, alpha, d, and k values on the app.

1. Assume a technological innovation, like the availability of electricity, increases output per person at all levels of the capital/labor ratio by 30% (You can implement this in the app by moving the slider on “Output level y’” from 1 to 1.3).
1. Describe in words what happens in the short run to output. (E.g., how large is the immediate increase in output?) [ 1 sentence ]

2. Describe in words what happens in the long-run. (If the long-run steady state is higher than the short-term level of output right after the technology change, what is causing this additional increase?) [2 sentences]

# 4. Growth and Inequality

In this part of the lab we will use R to calculate the shares of income from capital and from labor using two different production functions. In a competitive market, the returns to capital and labor are equal to their marginal product. In our case, we are working with per capita income, so the only variable is the amount of capital per worker, k. The derivative (change) of the production with respect to k is the marginal product and the rate of return of capital.

## i. Cobb-Douglas

Here we use a production function of the form

$y\left(k\right)={k}^{\alpha }$

k <- seq(1, 20, .1)
alpha <- 1/3
y.of.k <- k^alpha
## numerical slope = rises/runs
mp.k.numeric = diff(y.of.k)/diff(k)
plot(k[-1], mp.k.numeric)
## analytic derivative, taking derivatitive of k^alpha with respect to k.
mp.k.analytic = alpha * k^(alpha - 1)
lines(k, mp.k.analytic, col = "red", lwd = 2)
## we see they match quite well

Q1.1 Is the marginal product on capital A. Constant with increases in capital B. Rising with increases in capital C. Declining with increases in capital

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
quiz.check(answer1.1)

Now let’s see what happens to capital’s share of total output. We assume here that the rate of return is equal to its marginal product. So as we increase the capital per person, we will have two countervailing forces: the amount of capital will increase, but the rate of return will decrease. Let’s see what effect dominates or if the two effects cancel each other out.

mp.k <- mp.k.analytic
output.from.capital.per.worker <- mp.k * k
total.output.per.worker <- y.of.k
output.from.labor.per.worker <- y.of.k - mp.k * k
capital.share <- mp.k * k / y.of.k

Now let’s see what happens to capital share as we increase k

plot(k, capital.share)

Q1.2 Does the capital share of income A. Rise with increases in capital B. Stay constant with increases in capital D. Fall with increases in capital

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
quiz.check(answer1.2)

## ii. An alternative production function

Let’s modify the Cobb-Douglas production function slightly so that it is

$y\left(k\right)={k}^{\alpha }+k/10$

Modify the code below to work with this new production function. (Hint: the derivative with respect to k of ${k}^{\alpha }+k/10$$k^\alpha + k/10$ is $\alpha \ast {k}^{\alpha -1}+1/10.$$\alpha * k^{\alpha-1} + 1/10.$)

k <- seq(1, 20, .1)
alpha <- 1/3
y.of.k <- k^alpha ### <--- MODIFY THIS LINE.
## numerical slope = rises/runs
mp.k.numeric = diff(y.of.k)/diff(k)
plot(k[-1], mp.k.numeric)
## analytic derivative, taking derivatitive of k^alpha with respect to k.
mp.k.analytic = alpha * k^(alpha - 1) ### <--- MODIFY THIS LINE, TOO!
lines(k, mp.k.analytic, col = "red", lwd = 2)
## you should see that they match quite well

Q2.1 Is the marginal product on capital still declining A. Yes, but perhaps less quickly B. No, it is no longer declining

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
quiz.check(answer2.1)

Now let’s see what happens to capital’s share of total output.

## Note: here the variables carry over from the previous chunk. So, as
## long as you have executed all of the chunks up to here, they will
## be from the 2nd production function.
mp.k <- mp.k.analytic
output.from.capital.per.worker <- mp.k * k
total.output.per.worker <- y.of.k
output.from.labor.per.worker <- y.of.k - mp.k * k
capital.share <- mp.k * k / y.of.k
## graph our result to see what happens to capital share as we increase k
plot(k, capital.share)

Q2.2 Does the capital share of income A. Rise with increases in capital B. Stay constant with increases in capital D. Fall with increases in capital

##  "Replace the NA with your answer (e.g., 'A' in quotes)"
quiz.check(answer2.2)

# Part 5: Lab write up.

browseURL("https://gradescope.com/courses/14565")
1. (Question 1 in the analytical section at the beginning of lab)

2. (Question 2 in the analytical section at the beginning of lab)

3. (Question 3 in the questions with the app section above)

4. What level of population growth would maximize income per capita? Is this a plausible goal for a society? (Hint:you don’t need calculus for this problem. Try thinking about it with a diagram.) [1 or 2 sentences is enough here.]

## Robots and depreciation?

Many are worried about the effects of robots on inequality. But perhaps robots will also depreciate more quickly than earlier forms of productive capital. (The next two questions are on this topic.)

1. What would happen to capital per worker if we invested our savings in fast depreciating robots instead of slower-depreciating traditional productive capital? (Please make the (unrealistic?) assumption that per dollar robots have the same effect on production as traditional capital.) [Two or three sentences is fine.]

2. Assume, as per Piketty (and per our 2nd example of a production function above), that the capital share of output increases with capital intensification and decreases when capital per worker declines. Would our hypothetical case of faster depreciation increase inequality or reduce it? [Two or three sentences is fine.]

## Immigration

A country is considering two immigration policies. What would Solow’s model predict for each of these policies? (Note: both of these examples leave out consideration of the human capital of migrants, which in a more realistic model could be important.)

1. Allow a one-time wave of immigrants, but only those who bring exactly the amount of capital needed to leave the steady state capital/labor ratio unchanged. What would happen to per capita income in the short and long-run? (Hint: think about the time path of $k$$k$) [Answer in a sentence or two.]

2. Allow a one-time wave of penniless immigrants who bring no capital at all. What would happen to per capita income in the short and long run? (Hint: think about the time path of $k$$k$). [Answer in a sentence or two.]

Congratulations! You are finished with Lab 3.