Greg taylor

Formal Modelling in the Social Sciences
Greg Taylor
— Oxford Internet Institute —
Summer Doctoral Programme 2014
G. Taylor (OII) Modelling 1 / 48

What is a model?
model, n. and adj. A simpliﬁed or idealized description or conception of a
particular system, situation, or process, often in mathematical terms, that is
put forward as a basis for theoretical or empirical understanding, or for
calculations, predictions, etc.; a conceptual or mental representation of
something.
Models are abstractions and simpliﬁcations of reality.

What is a model?
something.
World is very, very complex.
Objective: distil world to a core of interesting elements to admit a
tractable analysis.

What is a model?
something.
tractable analysis.
The trade-off:
1. retain the features that make model recognisable.
2. be simple enough to permit clear understanding.

What is a model?
something.
tractable analysis.
The trade-off:
1. retain the features that make model recognisable.
2. be simple enough to permit clear understanding.
For our purposes, a model is equipped with an apparatus that
allows analysis (is not purely descriptive).

Where are models used?
In the social sciences modelling is most heavily concentrated in
economics, where it has been the dominant means of theory building
for a century and a half.

Where are models used?
In the social sciences modelling is most heavily concentrated in
economics, where it has been the dominant means of theory building
for a century and a half.
These methods have spread to other ‘nearby’ disciplines, e.g. political
science, marketing, and computer science.

Outline
Models as tools for logic
Models as laboratories
Models for concrete communication
Models as uniﬁed paradigms for knowledge
Other advantages

Logical consistency
Consider the following statements:
1. If prices are high then manufacturers will produce more since it is
more proﬁtable for them to do so.
High prices correspond to high production.

Logical consistency
2. If prices are high then people will buy less; if people buy less then
manufacturers will produce less because they tend to produce
only what they can sell.
High prices correspond to low production.

Logical consistency
2. If prices are high then people will buy less; if people buy less then
manufacturers will produce less because they tend to produce
only what they can sell.
High prices correspond to low production.
It is often hard to assess the logical consistency of even moderately
sophisticated rhetorical arguments about empirical relationships.
But also hard to diagnose what goes wrong: premise vs. conclusion.

Smith’s ‘Wealth of Nations’ (1776) Book 1, Chap. VII
When the price of any commodity is neither more nor less than what is sufficient to pay the rent of the land, the wages of the
labour, and the profits of the stock employed in raising, preparing, and bringing it to market, according to their natural rates, the
commodity is then sold for what may be called its natural price.
The commodity is then sold precisely for what it is worth, or for what it really costs the person who brings it to market; for
though in common language what is called the prime cost of any commodity does not comprehend the profit of the person who
is to sell it again, yet if he sells it at a price which does not allow him the ordinary rate of profit in his neighbourhood, he is
evidently a loser by the trade; since by employing his stock in some other way he might have made that profit. His profit,
besides, is his revenue, the proper fund of his subsistence. As, while he is preparing and bringing the goods to market, he
advances to his workmen their wages, or their subsistence; so he advances to himself, in the same manner, his own subsistence,
which is generally suitable to the profit which he may reasonably expect from the sale of his goods. Unless they yield him this
profit, therefore, they do not repay him what they may very properly be said to have really cost him.
Though the price, therefore, which leaves him this profit, is not always the lowest at which a dealer may sometimes sell his
goods, it is the lowest at which he is likely to sell them for any considerable time; at least where there is perfect liberty, or where
he may change his trade as often as he pleases.
The actual price at which any commodity is commonly sold is called its market price. It may either be above, or below, or
exactly the same with its natural price.
The market price of every particular commodity is regulated by the proportion between the quantity which is actually
brought to market, and the demand of those who are willing to pay the natural price of the commodity, or the whole value of the
rent, labour, and profit, which must be paid in order to bring it thither. Such people may be called the effectual demanders, and
their demand the effectual demand; since it may be sufficient to effectuate the bringing of the commodity to market. It is different
from the absolute demand. A very poor man may be said in some sense to have a demand for a coach and six; he might like to
have it; but his demand is not an effectual demand, as the commodity can never be brought to market in order to satisfy it.
When the quantity of any commodity which is brought to market falls short of the effectual demand, all those who are
willing to pay the whole value of the rent, wages, and profit, which must be paid in order to bring it thither, cannot be supplied
with the quantity which they want. Rather than want it altogether, some of them will be willing to give more. A competition will
immediately begin among them, and the market price will rise more or less above the natural price, according as either the
greatness of the deficiency, or the wealth and wanton luxury of the competitors, happen to animate more or less the eagerness of
the competition. Among competitors of equal wealth and luxury the same deficiency will generally occasion a more or less eager
competition, according as the acquisition of the commodity happens to be of more or less importance to them. Hence the
exorbitant price of the necessaries of life during the blockade of a town or in a famine.
When the quantity brought to market exceeds the effectual demand, it cannot be all sold to those who are willing to pay the
whole value of the rent, wages and profit, which must be paid in order to bring it thither. Some part must be sold to...

Modelling paradigm
Assumptions Calculation Results
correct
Who is involved?
What are their objectives?
How are they constrained?
New environment→new assumptions.

Modelling paradigm
correct
Who is involved?
Who is involved?

Modelling paradigm
correct
Who is involved?
Who is involved?
correct

Modelling paradigm
correct
Who is involved?
Who is involved?
correct
New environment/phenomenon→new assumptions.

First assertion
“If prices are high then people will buy less...”

Demand curve
0 1
0
1
D
Quantity
Price

Demand curve
0 1
0
1
D
Quantity
Price
0.25
0.75

Demand curve
0 1
0
1
D
Quantity
Price
0.25
0.75
0.75
0.25

Second assertion
“If prices are high then manufacturers will produce more...”

Supply curve
0 1
0
1
S
Quantity
Price

Third assertion
“[suppliers] tend to produce only what they can sell.”

High market price
0 1
0
1
D
S
Quantity
Price

High market price
0 1
0
1
D
S
Quantity
Price
qd
p
qs
0
0
0
Excess Supply

Low market price
0 1
0
1
D
S
Quantity
Price
qs
p
qd
0
0
0
Excess Demand

Equilibrium market price
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Paradox resolved: high price =⇒ high quantity
0 q 1
0
p
1
D
S
Quantity
Price
q
p
D

Paradox resolved: high price =⇒ low quantity
0 q 1
0
p
1
D
S
Quantity
Price

Paradox resolved: high price =⇒ low quantity
0 q 1
0
p
1
D
S
Quantity
Price
q
p
S

How general is it?
Are these results an artefact of my chosen demand and supply curves?
†See Appendix for rigorous proof of this. 18 / 48

How general is it?
D(p; d0) satisﬁes D1(p; d0) < 0, D2(p; d0) > 0, and D(0; d0) > 0.
S(p) satisﬁes S (p) > 0 and S(0) < D(0; d0).
D(·; ·) and S(·) are continuous.

How general is it?
D(p; d0) satisﬁes D1(p; d0) < 0, D2(p; d0) > 0, and D(0; d0) > 0.
S(p) satisﬁes S (p) > 0 and S(0) < D(0; d0).
D(·; ·) and S(·) are continuous.
Equilibrium condition: D(p; d0) = S(p) =⇒ p = D−1(S(p); d0).
S (p) > 0, D1(p; d0) < 0 =⇒
∂
∂p
D−1
(S(p); d0) < 0.
Suppose we increase d0 (shift demand curve outward):
D1(p; d0) < 0, D2(p; d0) > 0 =⇒
∂
∂d0
D−1
(S(p); d0) > 0†
=⇒
∂p
∂d0
> 0.
So price increases. What about quantity? Using the chain rule:
∂S(p∗)
∂d0
=
∂p∗
∂d0
S (p∗
) > 0.

Putting the model to work
Note: although model is mathematical, it is essentially qualitative.
The model resolves the apparent paradox, and makes the underlying
mechanism much clearer.
It is the insistence on understanding such mechanisms that
differentiates social science from ‘market research’.

Model instantly tells us about effects of things that shift supply or
demand.
Example: ﬂooding in Thailand

demand.
Example: ﬂooding in Thailand =⇒ reduced supply of hard disk
drives

demand.
Example: ﬂooding in Thailand =⇒ reduced supply of hard disk
drives =⇒ higher hdd prices.

Example: EVE developers nerf cruise missile launcher

Example: EVE developers nerf cruise missile launcher =⇒ reduced
demand for cruise missiles

Example: EVE developers nerf cruise missile launcher =⇒ reduced
demand for cruise missiles =⇒ lower cruise missile prices.

Example: Steve Balmer quits Microsoft

Example: Steve Balmer quits Microsoft =⇒ increased future
proﬁtability

Example: Steve Balmer quits Microsoft =⇒ increased future
proﬁtability =⇒ increased demand for Microsoft shares =⇒ increase
in share price.

Outline
Other advantages

As well as explaining price changes ex post, the model can be used for
ex ante experimentation.
Let’s look at a hypothetical policy scenario to see how understanding
the model can help us.

Suppose there is a policy objective to decrease some illegal or socially
undesirable activity that is highly addictive/compulsive (e.g. online
gambling).

Suppose there is a policy objective to decrease some illegal or socially
undesirable activity that is highly addictive/compulsive (e.g. online
gambling).
Could (a) target sellers/suppliers with more severe penalties, (b)
target buyers/users with penalties, education programmes, or
remedial measures.

Supply-side policy
0 q 1
0
p
1
D
S
Quantity
Price
q
p

Supply-side policy
0 q 1
0
p
1
D
S
Quantity
Price
q
p
S
↓ Supply

Supply-side policy
0 q 1
0
p
1
D
S
Quantity
Price
q
p
D
↓ Demand

Ineffective supply-side crackdowns
“Despite attempts to prohibit Internet gambling – most recently
through the Unlawful Internet Gambling Enforcement Act of 2006 –
American consumers nevertheless are wagering online more than $100
billion annually with operators based outside the U.S. [...] As a result,
rather than protecting consumers, the UIGEA has simply left the
millions of Americans who continue to ﬁnd a way to gamble online
unprotected and vulnerable to unscrupulous gambling operators.”
— Jared Polis, U.S. Congress

Outline
Other advantages

Slippery concepts
Many concepts in the social and related sciences are inherently hard to
pin down (information, knowledge, bias, etc.).

Slippery concepts
Often, the biggest danger is not having the wrong deﬁnition, but
rather having no deﬁnition at all!

Slippery concepts
Usually easier to deﬁne things that can be measured (e.g. intelligent
vs. knowledgeable).

Slippery concepts
Usually easier to deﬁne things that can be measured (e.g. intelligent
vs. knowledgeable). Models can help us by making more things
measurable in a standardised fashion.

Willingness to Pay
0 1
0
1
D
Quantity
Price

Willingness to Pay
0 1
0
1
D
Quantity
Price
↓ q
↑ p
curve measures WTP

Willingness to Pay
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Consumer surplus
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Producer surplus
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Total surplus
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Total surplus
0 q∗ 1
0
p∗
1
D
S
efﬁcient!
Quantity
Price

Welfare
We have a definition of welfare (there are others). Also definitions of
consumer surplus, producer surplus, and efficiency.

Welfare
We have a definition of welfare (there are others). Also definitions of
consumer surplus, producer surplus, and efficiency.
The definitions are made concrete by the model.
The newly defined concept immediately becomes a useful analytical
tool:
Empirically: Most estimates put the consumer surplus from home
Internet use at about 2–4% of GDP.
Theoretically...

Outline
Other advantages

Monopolised markets
We have been looking at a competitive market (recall that it was
competitive pressure that forced the price to equilibrium).

Monopolised markets
competitive pressure that forced the price to equilibrium). We say
ﬁrms are price takers.

Monopolised markets
Some industries have one dominant ﬁrm—a monopoly. Such a ﬁrm
gets to dictate the price.

Monopolised markets
Some industries have one dominant ﬁrm—a monopoly. Such a ﬁrm
gets to dictate the price.
For this we need a new model.

Monopolist 1: producer surplus
0 q∗ 1
0
p∗
1
D
S
Quantity
Price

Monopolist 2: a better price?
0 q q∗ 1
0
p
p∗
1
D
S
Quantity
Price

Monopolist 2
0 q q∗ 1
0
p
p∗
1
+
-
D
S
Quantity
Price

Monopolist 3: welfare
0 q q∗ 1
0
p
p∗
1
D
S
Quantity
Price

Monopolist 3: dead weight loss
0 q q∗ 1
0
p
p∗
1
D
S
Quantity
Price

Monopolised markets
New market structure demanded a new model.

Monopolised markets
New market structure demanded a new model. But by working within
the same analytical paradigm,
(i) we were able to reuse existing tools and results;

Monopolised markets
(ii) we were able to compare the results of the new model to those of
the old in a very explicit fashion.

Monopolised markets
Over time, this helps to build a body of knowledge that is uniﬁed in
the sense that it is mutually compatible and expressed in a common
analytical language.

Monopolised markets
Over time, this helps to build a body of knowledge that is uniﬁed in
the sense that it is mutually compatible and expressed in a common
analytical language.
Important in topical studies such as ours since it prevents reinvention
of the wheel.

Outline
Other advantages

Other advantages of formal models
We have seen that models are tools for logic, experimentation,
communication, and uniﬁcation.

Models can help us in a number of other ways:
Encourage scientiﬁc transparency by making underlying
assumptions explicit and concrete.

Facilitate ‘theoretical reproducibility’.

Abstraction often helps separate the superﬁcial from the
fundamental and identify the latter.

Generate theoretically-grounded and theoretically-meaningful
hypotheses for empirical work.

Generate theoretically-grounded and theoretically-meaningful
hypotheses for empirical work.
Provide structure for empirical work.

Appendix: Proof
It was asserted that, for differentiable D(·, ·),
D1(p; d0) < 0, D2(p; d0) > 0 =⇒
∂
∂d0
D−1
(S(p); d0) > 0.
Suppose, on the contrary, D−1(S(p); d) > D−1(S(p); d ) for d < d. We
want to show that this leads to a logical contradiction. This and
D2(p, d) > 0 imply
D[D−1
(S(p); d); d] > D[D−1
(S(p); d ); d] > D[D−1
(S(p); d ); d ],
where the ﬁrst inequality comes from applying D(·, d) to both sides
and the second follows from D2(p, d) > 0. Noting that
D(D−1(x, d), d) = x by deﬁnition, we have x > D[D−1(S(p); d ); d] > x,
which can never be true.

Greg taylor

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