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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

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Paging Dr. John Snow

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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• You can probably figure out which parts are out of date.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

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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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