Introduction and Math

Prof. Eric Sims

University of Notre Dame

Fall 2015

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Logistics

I Syllabus and related issues

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About Me

I Associate Professor, Dept. of EconomicsI B.A. Trinity University, 2003

I “Miracle in Mississippi” October 27, 2007

I Ph.D. University of Michigan, 2009I Don’t get the wrong pictureI Wife proud ND graduate – Lewis chickenI Signed Charlie Weis picture in office (oops?)I Michigan Sucks

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Out of Style

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In Style I

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In Style II

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In Style III

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My Effect on Notre Dame Football

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Notre Dame Winning % By Decade

ND WP

Sims arrival

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Introduction

I Macroeconomics:I Better called aggregate economicsI Focus on dynamic and intertemporal nature of economic

decision-makingI Economics is “micro”: “macro” just studies issues at

aggregated (country) level

I Key questions:I Why does the economy grow over time?I Why are some countries rich and others poor?I Why do economies experience recessions?I What is the role of government?I More recently:

I What the heck happened in 2007-2009?I Why is the recovery weak?

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Basic Approach of this Course

I Economists do it with modelsI But the models are used to help understand the real world

I Micro-founded macroeconomics. Agents Optimize

I Equilibrium approach – market-clearing as the relevantbenchmark

I Fundamentally dynamic

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Math

I “It is true that modern macroeconomics uses mathematicsand statistics to understand behavior in situations where thereis uncertainty about how the future will unfold from the past.But a rule of thumb is that the more dynamic, uncertain andambiguous is the economic environment that you seek tomodel, the more you are going to have to roll up your sleeves,and learn and use some math. That’s life.” – ThomasSargent, 2011 Nobel Prize Winner

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Math Topics

I Dynamic notation

I Exponents and logs

I Growth rates

I Calculus

I Optimization

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Notation

I Variable types: exogenous (determined outside model) andendogenous (determined in the model)

I Upper case letters

I Parameters: fixed values governing mathematical relationships

I Either low case letters or Greek letters

α “alpha”β “beta”δ “delta”γ “gamma”λ “lambda”θ “theta”σ “sigma”ω “omega”

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Time

I Time is discreet, with t = 0, 1, 2, . . .I xt is x observed at time t. xt+j is x observed j periods away

from t

I Summation notation:

S = xt + xt+1 + xt+2 + . . . xt+T

S =T

∑j=0

xt+j

I Alternatively:

S = x0 + x1 + x2 + . . . xT

S =T

∑t=0

xt

I Can go either forward or backward in time

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Exponents, Logs, and Growth Rates

I Some basic rules

I Growth rate: g xt = xt−xt−1

xt−1

I Fun facts: ln(1 + α) ≈ α and exp(α) ≈ 1 + α, for α small

I g xt ≈ ln xt − ln xt−1

I Growth rate of a product ≈ sum of the growth rates

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Calculus

I Function y = f (x)

I Derivative: how y changes as x changes. Derivative itself afunction

I Notation: dydx = f ′(x)

I Second derivative: derivative of a derivative: dy2

d2x= f ′′(x)

I Distinction between the derivative, f ′(x), which is a function,and the derivative evaluated at a point, f ′(x0), which is anumber

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Derivative Rules

I Powers, logs, exponentsI y = xa, dy

dx = axa−1

I y = ln x , dydx = 1

xI y = exp(x), dy

dx = exp(x)

I Derivative of a sum is sum of derivativesI y = f (x) + g(x), dy

dx = f ′(x) + g ′(x)

I Product rule, quotient ruleI y = f (x)g(x), dy

dx = f (x)g ′(x) + f ′(x)g(x)

I y = f (x)g (x)

, dydx = g (x)f ′(x)−f (x)g ′(x)

(g (x))2

I Chain rule:I Composite function: y = f (g(x))I Derivative is “derivative of outside times derivative of inside”:

dydx = f ′(g(x))g ′(x)

I y = ln(xa), dydx = 1

xa axa−1 = a

x

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Multivariate Derivatives

I Function of two variables: y = f (x , z)

I Partial derivative is change in y for a change in x (or z),holding z (or x) fixed

I Notation: ∂y∂x = fx (x , z)

I Example:y = xazb

∂y

∂x= axa−1zb,

∂y

∂z= bxazb−1

I Total derivative: total change in y is approximately sum ofpartials evaluated at a point times changes in each variableabout that point: dy ≈ fx (x0, z0)dx + fz (x0, z0)dz , wheredy = y − y0, dx = x − x0 and dz = z − z0

I Application: growth rate of a sum

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Optimization

I Pick x to either maximize or minimize f (x)

I First order condition: x∗ needs to satisfy f ′(x∗) = 0

I Second order condition: sign of f ′′(x∗) tells you whether youhave a maximum or a minimum

I Example: y = ln x − 2x

I Multivariate optimization works the same way: first orderconditions set partial derivatives with respect to each choicevariable equal to zero

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Constrained Optimization

I Want to optimize f (x , z), but there is some constraint that xand z must satisfy, e.g. x + z ≤ 1

I Our approach: assume constraint holds with equality,eliminate one of the choice variables (x or z), and then do anunconstrained optimization problem

I Application: simple consumer problem

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