Download Active Learning in Calculus: ConcepTests

Mathematics for
the Social and Behavioral Sciences:
Deborah Hughes Hallett
Department of Mathematics, University of Arizona
Harvard Kennedy School
Economics
Examples for students of policy-making and
international development
•
Algebra and precalculus
•
Logarithmic differentiation and elasticity
•
Multivariable optimization under constraints
•
Real Analysis
Precalculus
George has demand function DG(p) = 12 – 2p. Rachel’s demand function for
the same good is DR(p) = 9 – p. Find the equation for total demand for both
George and Rachel, then draw a single graph that includes George’s demand
function, Rachel’s demand function, and their combined demand function.
Harvard Kennedy School API-101 Diagnostic Problems
Precalculus
Suppose that the supply function for a good is given by S(p) = 2p. The
government introduces a subsidy to suppliers of $5 per unit.
(a) What is the equation for the new supply curve as a function of price paid by
consumers?
(b) Illustrate the old and new supply curves in a graph.
Since Pnew = Pold + 5 , we have with Pold on vertical axis
S(Pnew) = 2Pnew = 2(Pold + 5) = 2Pold + 10.
Precalculus
Suppose that the demand function for a good is given by D(p) = 220 – 10p.
The government imposes a tax on consumers of 10% per unit sold.
(a) What is the equation for the new demand curve as a function of price paid
to firms?
(b) Illustrate the old and new demand curves in a graph.
Consumer faces: ptotal = 1.1 * pfirm
New demand function is D(p) = 220 – 10*1.1*p = 220 – 11p.
Precalculus
The WIC (Women, Infants, Children) program provides food coupons for lowincome families with small children. Suppose a family’s income is $1600 a
month, and they get $200 worth of coupons every month regardless of how
much food they actually buy. WIC coupons can only be spent on food.
(a) Draw this family’s budget line with expenditure on food on the horizontal
axis and all other expenses on the vertical axis.
(b) Draw a set of indifference
curves that would illustrate a
family that would rather have
$200 in cash than in coupons.
(c) Draw a set of indifference
curves that would illustrate
another family that would be
indifferent between having
$200 in cash and in coupons.
Harvard Kennedy School API-101
Logarithmic Scales
y, ln(CPI)
How fast has the CPI (Consumer Price Index) grown over the last century?
y = 0.033t + 2.12
6
5
4
3
2
1
0
0
50
t, years since 1913
100
How Such Data Might Appear in a Math Course
CPI, with 1982-84 defined to be 100
250
y = 8.31e0.033x
CPI
200
150
100
50
0
0
20
40
60
Years since 1913
80
100
Logarithmic Differentiation
𝒚′
• Relative Rate of Change:
𝒚
=
𝒅
(𝐥𝐧 𝒚)
𝒅𝒕
Example: Production, 𝑄, is given in terms of capital, 𝐾, and labor,
𝐿, by
𝑄 = 𝐴𝐾 𝛼 𝐿𝛽 ,
where 𝐴 is a technology coefficient. All quantities depend on time 𝑡.
Differentiate logarithmically with respect to time.
We have ln 𝑄 = ln 𝐴 + 𝛼 ln 𝐾 + 𝛽 ln 𝐿, so
𝑄′ 𝐴′
𝐾′
𝐿′
= +𝛼 +𝛽
𝑄
𝐴
𝐾
𝐿
In other words, the relative rate of change of production is a linear
combination of the relative rates of 𝐴, 𝐾, and 𝐿.
Differentials, 𝒅𝒚
• The differential 𝑑𝑦 means very small change in 𝑦
• Units are units of 𝑦
• Differentials are to analyze the impact of small changes only. (Why
only small?) Economists use them to study the effect of an impact of
a shock to a system in equilibrium
• Example: Let 𝑀𝐶(10,000) = $27 m per unit. Then
𝑑𝐶
= 𝑀𝐶 = 27.
𝑑𝑞
So if 𝑑𝑞 = 10, the change in cost is
𝑑𝐶 = 27 ∙ 𝑑𝑞 = 270 m dollars.
In other words, after 10,000 units have been produced, an additional
10 units cost about an additional $270 m.
Logarithmic Differentiation and Elasticity
% change in quantity demanded
𝑑𝑞/𝑞
Price elasticity of demand 𝐸 =
=
% change in price
𝑑𝑝/𝑝
and
𝑑 ln 𝑞
𝑑𝑞
1
𝑞
= , so 𝑑 ln 𝑞 =
𝑑𝑞
.
𝑞
So we have
𝑬=
𝑝 𝑑𝑞
𝑞 ∙ 𝑑𝑝
=
𝒅 𝐥𝐧 𝒒
𝒅 𝐥𝐧 𝒑
Example: Find elasticity for the demand curve 𝑞 = 𝐴𝑝−𝑘 , where 𝐴 and 𝑘
are positive constants.
Since ln 𝑞 = ln 𝐴 − 𝑘 ln 𝑝, we have 𝐸 =
𝒅 𝐥𝐧 𝒒
𝒅 𝐥𝐧 𝒑
= −𝑘 = 𝑘.
Constrained Optimization: Lagrangians
Manufacturing a good requires two inputs that cost $2 and $3, respectively,
per unit. A quantity 𝒙 of the first input and a quantity 𝒚 of the second input
produces 𝑸 = 𝒙𝒚 units of the good. What is the maximum production that can
be achieved with a budget of $12?
That is: The budget constraint is 𝐵 = 2𝑥 + 3𝑦 = 12.
Objective function is 𝑄 = 𝑥𝑦
We want to find the maximum value of 𝑄 = 𝑥𝑦 on the budget constraint.
The Lagrangian is 𝐿 = 𝑥𝑦 − 𝜆(12 − 2𝑥 − 3𝑦)
At the optimal point, the budget constraint is tangent to a curve of constant
production, 𝑄 = 𝑐.
For a constant, 𝜆, the Lagrange Multiplier, this leads to:
𝜕𝐿
𝜕𝑥
=
𝜕𝑄
𝜕𝑥
𝜕𝑄
𝜕𝑦
𝜕𝐵
− 𝜆 𝜕𝑥 = 0
𝜕𝐿
𝜕𝑦
=
𝜕𝐿
𝜕𝜆
= 12 − 2𝑥 − 3𝑦 = 0
−𝜆
𝜕𝐵
𝜕𝑦
=0
so 𝑦 − 2 𝜆 = 0
so 𝑥 − 3 𝜆 = 0
so 2𝑥 + 3𝑦 = 12
What argument can you give that that critical point is a global maximum?
Constrained Optimization: Meaning of Lagrange Multiplier 𝝀
Suppose we are optimizing 𝑄(𝑥, 𝑦) subject to the constraint
𝐵 𝑥, 𝑦 = 𝐶. At the optimum point
𝑄𝑥 = 𝜆𝐵𝑥 and 𝑄𝑦 = 𝜆𝐵𝑦
Thus,
𝑑𝑄 = 𝑄𝑥 𝑑𝑥 + 𝑄𝑦 𝑑𝑦 = 𝜆 𝐵𝑥 𝑑𝑥 + 𝐵𝑦 𝑑𝑦 = 𝜆𝑑𝐵
so
• An additional input of 𝒅𝑩 is multiplied by 𝝀 to give
the additional output of 𝒅𝑸.
• If the budget is increased by $1, the increase in
output is worth about $𝝀.
• Shadow price
Optimization on a Non-convex set
Consider the maximization problem with objective function
𝑢 𝑥, 𝑦 = 𝑥 ∝ 𝑦 1−∝ ,
with 0 < ∝ < 1
subject to a non-convex constraint region given by 𝑥 ≥ 0, 𝑦 ≥ 0 and with a two-part
boundary:
𝑥 + 𝑦 ≤ 10
or
𝑥 + 4𝑦 ≤ 20.
(a) For ∝ = 0.4 and ∝ = 0.8 find two critical points (𝑥 ∗ , 𝑦 ∗ ) on the constraint boundary.
Which gives the maximum value?
(b) For what values of ∝, if any, are there two maximizers, one on each part of the
constraint boundary?
Real Analysis
A function 𝑢 is concave (that is, concave down) if
𝑢(𝛼 𝑥1 + 1 − 𝛼 𝑥2 ) ≥ 𝛼𝑢 𝑥1 + 1 − 𝛼 𝑢(𝑥2 )
A function 𝑢 is quasiconcave if
𝑢 𝛼 𝑥1 + 1 − 𝛼 𝑥2 ≥ min(𝑢 𝑥1 , 𝑢 𝑥2 )
1.
2.
Prove that a quasiconcave function is concave.
For a strictly quasiconcave function, prove that a local max is a
global max. (Proof by contradiction.)
Politics and Current Affairs
For students in any field
•
Probability and Statistics
Important that the examples have importance
outside the mathematics
Descriptive Statistics
Who is the outlier?
Number of countries
Distribution of Physicians across Countries
70
60
50
40
30
20
10
0
66
14
15
14
10
16
8
16
3
1
0
0
Physicians per 10,000 people
“In the medical response to Ebola,
Cuba is punching far above its
weight” “…..165 health
professionals….the largest medical
team of any single foreign nation”
Washington Post Oct 4
http://www.cubaminrex.cu/en/cuba-health-professionals-arrived-sierra-leone-fight-ebola
0
1
Bayes’ Theorem and Prosecutor’s Fallacy
Sally Clark, UK
Life sentence 1999 for double
murder; released 2003
http://www.sallyclark.org.uk/
Duane Buck, Texas
Scheduled to be executed
Thursday, Sept 15, 2011.
http://www.guardian.co.uk/world/2011/sep/16/duanebuck-texas-executions
Sally Clarke and husband Steve pictured after
being cleared by the Court of Appeal in 2003
http://www.dailymail.co.uk/debate/columnists/article-492799/Honour-Our-leaders-dont-know-meaning-word.html
Hypothesis Test of Means
July–August 2014:
Grocery store workers picket;
Customers boycott
http://www.bbc.com/news/business-28580359
http://wearemarketbasket.com/
In Conclusion
•
Value to students is in the problems the
mathematics addresses
•
May use substantial mathematics
•
Need to know the basic mathematics well
•
Crucial to know what the mathematics means---as
important as knowing how to calculate
•
Use in economics is “very theoretical”