Lab 1 answer key
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ECON MISC
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Economics
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Mar 2, 2021
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Lab 1: World Population Growth 1.[Multiple choice] Which of the following descriptions is best for the history of human population growth? A.Constant exponential growth at a rate slightly larger than zero. B.Thousands of years of near zero growth, followed by centuries of accelerating growth, with a recent slowing of growth. C.Thousands of years of essentially zero growth, followed by a one-time increase in growth rates. D.Alternating lengthy periods of positive and negative growth. Correct answer is B. 2.[A numerical answer] How large would the world population be today if annual growth rates had always been .0005 larger for the last 10,000 years? Hint 1:The rules of exponents tell us e(R+d)∗t=eRt∗e dt Hint 2: To calculatee10000∗.0005 , you can use a calculator or any other tool. You don't have to use R. Hint 3: You should get a population roughly 150 times as large as today. Note: Imagine we increased the reproduction rate of each generation by a tiny bit -- so, for example, instead of each woman having on average 1 surviving daughter, imagine she had 1.015 surviving daughters. If generations were 30 years in length, then this would mean an increase in the growth rate by log(1.015/1.00)/30 = .0005. So, the calculation we just did tells us what would have happened to the human population if fertility rates had always been just slightly, only 1.5 percent, higher. Answer: Approximately 1.09 trillion people. Acceptable answers include using the time frame (-8000, 2000) or using actual current population rather thanP 2015 . The fastest way to do this is to understand that: Pt=P0 e rt We know the population at time t (the year 2015), time 0 (the year -8000), and the intervaltbetween them (10,015 years, or 10,000 years for simplicity - you'll get an answer in the acceptable range for this question either way). From this, we can solve forr,add .0005 to therwe get,and plug in P0 andtwith the new largerrto recalculate for a new P'2015 , the larger hypothetical population in 2015. OR we could recognize that, by the rules of exponents and the transitive property, the hypothetical larger 2015 population, P'2015, is equal to the given P2015times e(.0005*10,000) .
3.[A numerical answer] How large would the world population in 2115 be if current exponential growth rates continue? Use 7.3 billion as the population size in 2015 and assume R = 0.01. You can do this calculation however you choose (with R, by hand with a calculator, or any other way.) Answer: Approximately 20 billion people. We are given P2015 and r, so we can use the exponential growth modelPt=P0 ert wheret = 2115-2015 = 100 years. Pt=P0 ert =¿ 7.3 billion * e(.01*100)= 19.84 billion Another way is to start from the growth rate and solve forP t . P r=log(¿¿t/P 0 )∗(1/t) ¿ However, be careful of using the natural logarithm (ln) in your calculator or statistical program. 4.(Numbered as 5. in rStudio) [A short answer.] Do you think this estimate of the world population in 2115 is likely to be too high or too low? Explain your reasoning in a sentence. Answer: The estimate istoo high, because growth rates have been declining in recent years (a constant rate of r = .01 is too high). Another answer: The estimate is too high because of resource constraints, there aren't enough resources, or the earth cannot support 19 billion people. (This answer is not as good, as we will learn later in the semester, but it's still acceptable.) 5.(number as 6 in rStudio) [Optimum Population Exercise] Imagine there's an island that can only sustain a few people. The marginal product starts at 5 units for the first person and declines by 1 unit for each additional person until the MP of the 6th person is zero. Each person needs to consume 2 units per year to subsist. (Hint: if you are having trouble here with the definitions of the optimum populations, consult the lecture slides.) a.Average product:See above. To obtain this, find the total product at each population size, which is the sum of the marginal products up to each population size. Then, divide by the population. b.Economic optimum: 1 person, where the average product is at its maximum. c.Power optimum: 3 or 4 people (both give a ruler a maximum power of 6 units. Recall power is defined as total output-total subsistence.) d.Maximum sustainable: 7 people (we cannot have half people). 6.(Numbered as 7. in rStudio) [non-graded] About how much time (in hours and minutes) did it take you to complete this lab?
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