Mathematics and Economics in Social Sciences
Explore the intersection of mathematics and economics in social and behavioral sciences through examples in algebra, precalculus, and real analysis. Solve problems related to demand functions, supply curves, subsidies, and taxes. Understand the impact of programs like WIC on family budgets. Discover the growth of CPI using logarithmic scales.
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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 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
Logarithmic Scales How fast has the CPI (Consumer Price Index) grown over the last century? y = 0.033t + 2.12 6 5 y, ln(CPI) 4 3 2 1 0 0 50 100 t, years since 1913
How Such Data Might Appear in a Math Course CPI, with 1982-84 defined to be 100 250 y = 8.31e0.033x 200 150 CPI 100 50 0 0 20 40 60 80 100 Years since 1913
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 ln? = ln? + ? ln? + ? ln?, so ? ?=? In other words, the relative rate of change of production is a linear combination of the relative rates of ?,?,and ?. ?+ ?? ?+ ?? ?
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 ??(10,000) = $27 m per unit. Then ?? ??= ?? = 27. So if ?? = 10, the change in cost is ?? = 27 ?? = 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 % change in price ??/? ??/? Price elasticity of demand ? = = and ? ln ? 1 ?, so ? ln? = ?? ?. = ?? So we have ? =? ? ?? ? ??? ? ??? ??= Example: Find elasticity for the demand curve ? = ?? ?, where ? and ? are positive constants. ? ?? ? ? ?? ? Since ln? = ln? ?ln?, 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 ? = 2? + 3? = 12. Objective function is ? = ?? We want to find the maximum value of ? = ?? on the budget constraint. The Lagrangian is? = ?? ?(12 2? 3?) At the optimal point, the budget constraint is tangent to a curve of constant production, ? = ?. For a constant, ?, the Lagrange Multiplier, this leads to: ?? ??= ?? ?? ? ?? ??= 0 so ? 2 ? = 0 ?? ??= ?? ?? ? ?? ??= 0 so ? 3 ? = 0 ?? ??= 12 2? 3? = 0 so 2? + 3? = 12 What argument can you give that that critical point is a global maximum?
Constrained Optimization: Meaning of Lagrange Multiplier ? 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
Optimization on a Non-convex set Consider the maximization problem with objective function ? ?,? = ? ?1 , subject to a non-convex constraint region given by ? 0,? 0 and with a two-part boundary: ? + ? 10 or (a) For = 0.4 and = 0.8 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? with 0 < < 1 ? + 4? 20.
Real Analysis A function ? is concave (that is, concave down) if ?(? ?1 + 1 ? ?2) ?? ?1 + 1 ? ?(?2) A function ? is quasiconcave if ? ? ?1+ 1 ? ?2 min(? ?1, ? ?2 ) 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? Distribution of Physicians across Countries 66 70 Number of countries 60 50 40 30 16 16 15 14 14 10 8 20 3 1 1 0 0 0 10 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
Bayes Theorem and Prosecutors 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. 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.guardian.co.uk/world/2011/sep/16/duane- buck-texas-executions
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
