# Problem Set 2 v 5 Solutions

```Problem Set 2
Demog/Econ 175
Professors Josh Goldstein and Ryan Edwards
February 9, 2015
DUE: Tuesday, February 17, 2015 by 2pm
**Late assignments will not be accepted**
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
https://piazza.com/class/i4kgcntt6bmrv?cid=18. If you see that someone has
asked a question that you have encountered and solved, feel free to post a
response. If an existing response doesn’t look completely correct to you, you are
encouraged to edit it to make it more complete or more correct. Please do not
give answers to the problem set on piazza.
Th
No late problem sets will be accepted!
1. Malthus a) Briefly define the “positive” and “preventative” checks on population growth according to Malthus. The positive check is the response of mortality to wages. The preventative check is the
response of fertility to wages.
Th
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
b) Draw a graph demonstrating the relationship between wages and the crude death rate as well as the crude birth rate. Find the wage at which population will neither grow nor shrink. Below this graph, draw a second graph plotting the demand for labor. Locate the equilibrium population size and the equilibrium wage. Label them P* and W* respectively. c) Malthus thought that in China there was close to universal marriage at a young age. He wrote, “It is said that early marriages very generally prevail through all the ranks of the Chinese. Yet Dr. Adam Smith supposes that population in China is stationary. These two circumstances appear to be irreconcilable.” Let's compare the Chinese demographic regime to the English one. Say that England went from the English marriage pattern to the Chinese marriage pattern and that we modeled this by raising the birth rate at all wages. What happens to population and https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
wages on your graph? Label these new points as P** and W**. Describe the sequence of events that occur when moving from (P*, W*) to (P**, W**). At the new equilibrium, is mortality higher or lower than before? Is fertility higher or lower than before? Could Adam Smith's "supposition" have been correct that the population is "stationary" (i.e., is neither growing nor shrinking)? Due to the change in fertility, births increase for every level of wages, shifting the birth schedule up to B’(w). Beginning at the original equilibrium, fertility is higher than mortality so the population increases. Increasing population size then leads to lower wages, which in turn decreases births and increases deaths. After the shift, mortality and fertility are both higher. So, Adam Smith’s “supposition” could have been correct as the population grows, but then becomes stationary again.
Th
d) According to Malthus, would the population of England return to the initial equilibrium of (P*, W*) in the long run after the changes from question (c)? Why or why not? No. Population does not return to the initial equilibrium because the increase in fertility is exogenous and permanent. In other words, for every level of wage, fertility or the birth rate is increased. This shifts the curve up and leads to a permanent increase in population size. e) It is estimated that 50,000 Huguenots fled persecution in France for England in https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
about 1670. On a new graph, show what effect Malthus would have predicted this would have. Show what happens to population size, wages, births, and deaths shortly after the influx of refugees. Label these new points P** and W**. Assume the Huguenots have the same fertility and mortality curves as the native English. f) According to Malthus, does population of England return to the initial equilibrium of (P*, W*) in the long run after the influx of Huguenots? Why or why not? Th
Yes. For temporary shocks to the system, population returns to the initial equilibrium in the long run because the large immigration \ causes a decrease in wages, and the growth rate becomes negative. Eventually, as more deaths occur than births, the population will return to the initial equilibrium point. Note that labor productivity has not changed, nor have the birth and death schedules in England. https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
2. Solow Th
Look at the graph above, which represents a production function (red) as well as a capital accumulation function (blue). In this graph, y is output per worker, k is capital per worker, n is the rate of population increase, and δ represents capital depreciation. In a steady state, investment, s*y(k) where s=savings rate, will equal the amount needed to balance population growth and depreciation, or (n+ δ)k. In this problem, we will find the level of savings that maximizes consumption, sometimes referred to as the "Golden Rule" level of savings. That is, we want the largest possible c. Remember c = y(k) − s ∗ y(k) and s ∗ y k = (n + δ)k at equilibrium so that we have consumption is maximized when y k − n + δ k is greatest. (a) Draw a curve on the graph representing s*y(k) such that consumption is maximized. Remember, consumption is the vertical distance from the blue line to the red line. This happens when their slopes are the same. https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Th
(b) Now let’s do it mathematically. Say we are given a per capita production function y=k.5. Depreciation is .07. Population growth is .03. What is the savings rate that maximizes consumption? First you need to find the level of capital that maximizes consumption, and then the savings rate that brings you to this level. (HINT: These functions are graphed above. EXTRA DOUBLE BIG HINT: the slope of y(k) is .5k-­‐.5) .5k-­‐.5=.7+.3 .5k-­‐.5=.1 k-­‐.5=(1/5) (1/k.5)=(1/5) k.5=5 k*=25 s(25).5)=(.1)25 s=(2.5)/ ((25).5) s=2.5/5 s=50% https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
Th
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
(c) In the previous question the population growth rate was 0.03. Say a government was able to reduce its population growth rate to 1%, keeping savings at the same fraction of income. What would happen to the level of capital per worker in the new steady state? What would happen to consumption per worker? Is there an optimal population growth rate in the Solow model? Discuss briefly. (No need to do the actual calculations. Just think through the answer.) If the population growth rate decreased, it would lower the slope of the (n+d)k curve. This would cause consumption to worker to increase as well as capitol per worker. If we were using this model to optimize the growth rate, we would want growth to be at zero, which would allow us to maximize both consumption and capitol. It is important to remember that the Solow growth model isn’t to be used to optimize growth rates. Instead, it takes them as exogenous. https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
3. IPUMS For this question, use IPUMS ACS 2001-­‐2013. As with the first problem set’s question on the recession, we are hoping to see how the recession impacted different groups. (A) Look at employment before and after the recession began in 2008 (look at the variable empstat by year). By how much did employment fall between 2007 and 2010? By how much did unemployment rise? (Answer both questions in terms of percentage change). The number of employed shrank by 2%. The number of unemployed rose by 75% from 9.5 million to nearly 17 million. Recognizing that some people might have been trying to calculate changes in employment/unemployment rate, it is possible to look at total employed divided by those in the labor force. This does not include respondents who are N/A. By looking at just employed and unemployed you are able to see that the employment rate dropped by approximately 5% and the unemployment rate increased by approximately 70%. It is not acceptable to just use the percentages laid out by IPUMS. Further, percentage change is different than percentage POINT change. Growth from 2% to 3% is a one percentage point change, but a 50% increase. (B) Now we want to compare those who have a 4-­‐year college degree to those with only a high school diploma. Use the “educ” variable to compare the two groups before and after the recession. In particular, we will look at the years 2007 and 2010 (note, we are only looking at people who have completed 4 years of college or 4 years of high school. Be sure not to group them together with those who don’t have either). a. What was the absolute difference in unemployment in 2007 between the two groups? There were 3397348 more unemployed high school graduate than college graduates in 2007.
b. What was the absolute difference in unemployment in 2010? 5523635
Th
c. Who did the recession hit harder, the college educated, or those with only a high school education? The recession was likely harder on those with a high school education as the difference in unemployed between the 2 years rose by over 2 million people. If unemployment hit the college educate worse, we would expect to see a decrease in that number. If one looked at these numbers in relative terms, they would notice that the number of unemployed college grads more than doubled whereas the number of unemployed high school grads increased by less than 100%. This reminds us that the way you look at numbers (in relative or absolute terms) affects the story you tell. (C) Now let’s look at fertility. Look at change in fertility (fertyr) from 2007 to 2010. https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/
Th
sh is
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
a. Using the same design as (B), how does fertility compare across the groups? In 2007, high school grads had 737832 more babies than college grads. In 2010, high school grads had 636443 more babies. This is showing us that the reduction in fertility was greater in the high school group than the college group. This tells a similar story to the previous question b. What are all of the NAs representing in the fertyr variable? Not everybody can have a baby. The ACS asks “Has this person given birth to any children in the past 12 months?” If you are not a female of child-­‐bearing age or a man, this question would be not applicable to you. https://www.coursehero.com/file/11349428/Problem-Set-2-v-5-Solutions/