Mathematics for the Social and Behavioral Sciences:

Deborah Hughes HallettDepartment 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

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 axisS(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 low-income 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

Precalculus

0 10 20 30 40 50 60 70 80 90 1000123456

f(x) = 0.0324891629360638 x + 2.12139518432819

t, years since 1913

y, ln

(CP

I)

How fast has the CPI (Consumer Price Index) grown over the last century?

Logarithmic Scales

How Such Data Might Appear in a Math Course

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

f(x) = 8.31323670454018 exp( 0.0326061423178268 x )R² = 0.918537287245173

CPI, with 1982-84 defined to be 100

Years since 1913

CP

I

Logarithmic Differentiation

• Relative Rate of Change:

Example: Production, is given in terms of capital, , and labor, , by

,

where is a technology coefficient. All quantities depend on time . Differentiate logarithmically with respect to time.

We have , so

In other words, the relative rate of change of production is a linear combination of the relative rates of .

Differentials,

• The differential means very small change in • Units are units of • 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 m per unit. Then

So if , the change in cost is

.

In other words, after 10,000 units have been produced, an additional 10 units cost about an additional $270 m.

Logarithmic Differentiation and Elasticity

and , so

So we have

Example: Find elasticity for the demand curve , where and are positive constants.

Since , we have

Constrained Optimization: Lagrangians

Manufacturing a good requires two inputs that cost $2 and $3, respectively, per unit. A quantity of the first input and a quantity of the second input produces units of the good. What is the maximum production that can be achieved with a budget of $12?

That is: The budget constraint is . Objective function is We want to find the maximum value of on the budget constraint.

The Lagrangian is

At the optimal point, the budget constraint is tangent to a curve of constant production,

For a constant, the Lagrange Multiplier, this leads to:

so

so

so What argument can you give that that critical point is a global maximum?

Suppose we are optimizing subject to the constraint At the optimum point

and

Thus,

so • An additional input of is multiplied by to give the

additional output of • If the budget is increased by $1, the increase in

output is worth about $.• Shadow price

Constrained Optimization: Meaning of Lagrange Multiplier

Consider the maximization problem with objective function, with

subject to a non-convex constraint region given by and with a two-part boundary: or

(a) For and find two critical points 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?

Optimization on a Non-convex set

A function is concave (that is, concave down) if

A function is quasiconcave if

Real Analysis

1. Prove that a quasiconcave function is concave.

2. 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

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14 15 14 10 16 8 163 1 0 0 0 1

Distribution of Physicians across Countries

Physicians per 10,000 people

Num

ber o

f cou

ntrie

s

Descriptive Statistics

Who is the outlier?

http://www.cubaminrex.cu/en/cuba-health-professionals-arrived-sierra-leone-fight-ebola

“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

Duane Buck, TexasScheduled to be executed Thursday, Sept 15, 2011.

Bayes’ Theorem and Prosecutor’s Fallacy

http://www.guardian.co.uk/world/2011/sep/16/duane-buck-texas-executions

http://www.sallyclark.org.uk/

Sally Clark, UKLife sentence 1999 for double

murder; released 2003

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

http://www.bbc.com/news/business-28580359http://wearemarketbasket.com/

July–August 2014:Grocery store workers picket; Customers boycott

Hypothesis Test of Means

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”