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Econ/Demog c175
1
Optimum Population Size
Economic Demography (DEMOG/ECON C175)
Prof. Ryan EdwardsSpring 2020, Class 2
January 23, 20201/22/20 11:46 AM
• News sources are reporting that China is shutting down public transportation in Wuhan, China
• Wuhan is home to about 9 million people, which is roughly = New York City
• Yesterday evening, BBC reported 500 confirmed cases of coronavirus “2019-nCoV” and 17 deaths
Current Events
1/22/20 4:32 PM
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Paging Dr. John Snow
1/22/20 4:33 PM
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Per BBC:The virus originated in a seafood market in Wuhan that "conducted illegal transactions of wild animals", authorities said.
1. Population numbers and RStudio
2. Sauvy reading material
You should stick around for #2 because it’ll be on the exam!
Plan today
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• Because of staffing turbulence, parts of our RStudio labs are deprecated
• I’m sorry, and thanks for your patience
• You can probably figure out which parts are out of date. The lab titles, for example! In doubt? Ask Ryan
• We have plans TBA
RStudio notes
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• You can use “File: Knit Document” but it will give you everything plopped into an HTML document
• A better approach is to copy and paste into a new document and submit that
• Whatever you decide, please make sure when submitting on Gradescope that you identify which parts of the PDF have which answers
RStudio workflow
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• Do you like spreadsheets better than R?
• I like them both
• Let’s look at the data in a Google Sheets document in this folder on our public drive… after I mention some algebra
Hands-on with the data
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• It’s super easy if you can become comfortable with this math:
N(t) = N(0) eRt
log N(t) = log N(0) + R tR = [log N(t) – log N(0)]/t
• But you might be more accustomed to calculating a growth rate like this:
R’ = [N(t) – N(0)] / N(0)
• These are usually pretty similar, if R is small, because then eR ≈ 1 + R
Growth rate algebra
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• Are we talking about just the growth rate or the average annual growth rate?
• Lab 1 and other stuff is all about average annual growth rates, which look like this:
N(1900) = 1,650N(2000) = 6,217
R = [log(6,217) – log(1,650)] / 100 ≈ 0.013265122
Key thing to remember
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Our agenda• What do we mean by the optimum?• Two specific examples from Sauvy:
– economic optimum– power optimum
• Also:– minimum population– maximum population
• Ordering the optima by population size• Population “evolution”
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A happy medium?
“If there is such a thing as over-population and such a thing as under population, it follows that between the two there must be such a thing as just the right population”
-- H.P. Fairchild
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What size population is best?A slippery question
• Who is asking it?– A peasant, a worker, the average citizen?– A king, a capitalist, a “society”?
• What do they want to maximize?– average welfare– longevity– cultural power– military power– sum of all welfare
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Our approach• Multiple answers ® optima (not optimum)• Our goal: a broadly applicable framework
to help us think about optimum population• We focus on two specific examples
(1) Economic optimum (2) Power optimum
• Introduce concepts (MP, TP, ...) that we'll see again.
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“Economic” optimum (issues)
• Many possible economic objective functions • Sauvy’s “economic optimum” maximizes average
standard of living• Note:
– Assume we can define “standard of living” – Take an individual perspective: size benefits not
“society” but rather individual via standard of living– Assume inequality doesn’t exist or it’s not part of the
objective function
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Where is economic optimum?• We assume fixed technology and fixed resources• Under-population = can’t exploit the technology• Over-population = run into resource constraints• Example: how big should Econ/Demog c175 be?
OePopulation size (N)
standardof living(average)
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Critiques of the idea of optimum population size
(1) We assumed fixed technology; but maybe each density has its own appropriate technology (later we’ll read Boserup)
(2) Perhaps maximum is not a single point -- rather, a wide range
(2) a flat objective functionstandardof living(average)
(1) multiple maxima
pop size pop size1/22/20 11:27 AM
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Power optimum (issues)• Let’s not think about average individual but about
the sum of the society• Can think of a ruler, or of entire society
( "La France" )• Sauvy defines power as the sum of production
beyond sustenance(for pyramids, armies, culture, ...)
• Question: Do more people always mean more power?
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Marginal productivity• MP(i) = output of the i’th additional person (the additional
increment of production) (NOT the average)• power(N) =
sum(marginal output of first N people) -their subsistence requirements
Population size N
marginaloutput
subsistence power(i)
N(i)
-
=
total product
subsistence
power
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The Power Optimum
Population size
output
Op
marginal product
subsistence.
What population size maximizes power?
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Why does equality with subsistence mark the power maximum?
• Below Op, each additional person increases social surplus• Above Op, each person is a “net cost”
Population size
output
Op
marginal product
subsistence req.
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Ordering the optima
Oecon
Population size
output
Opow Omax
marginal product
average product
subsistence req.
Econ.optimum < Power optimum < Max. population
Omin
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Population “evolution”?
If technology and other factors remain constant -- a big “if” -- then a growing population will experience the following stages:1. The minimum (viable) population2. Maximum marginal productivity3. Maximum standard of living (econ opt)4. Maximum power (power opt)5. Maximum (viable) population
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Government and choice of optimum
“Before the 19th century this desire for power used to be so predominant that it hampered for a long time the understanding of the economic optimum. At about the time when this was understood, regal authority began to decline.”
(Sauvy p. 62)
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Conclusions• Theoretical optimum, not a requirement
(doesn’t mean we should ship off our population if we are over the optimum)
• Ignores feedback between pop size and technological level• We assume resources never get destroyed, just that there’s
congestion …• BUT, still a very useful framework:
– the relative positions of different optima in general– we can understand how optimum depends on objective
(whose perspective)
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Total Product: Understanding Sauvy's Figure 14
• Total prod = sum (MP)TP(N) = ∑1
N MP(i)• Tangent TP = MP
TP' = MP• Angle from origin to TP is
AP. Why?
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marginal product
Total Product(not to scale)
Average Product
Marginal Product
TP picture will be useful when working with Solow modelMP picture will be useful with Malthus, Migration, and elsewhere
(Note: MP and AP labels reversed in earlier
version)
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