Theoretical Tools of Public Economics

Math Review

Theoretical versus Empirical Tools

• Theoretical tools: The set of tools designed to understand the mechanisms behind decision making. (Chapter-2)

• Empirical tools: The set of tools designed to analyze data and answer questions raised by theoretical analysis. (Chapter-3)

Math Review

1. Limits

2. Differentiation– Functions of single variable– Multivariate functions (Partial derivatives)

3. Applications in Economics

4. Optimization– Without constraints– Constrained optimization

Limits

• Informal definition: If f(x) is defined for all x near a, except possibly at a itself, and if we can ensure that f(x) is as close as we want to L by taking x close enough to a, we say that the function f approaches the limit L as x approaches a, and we write

Lxfax

lim

Limits

• Theorem:

if and only if

Lxfax

lim

Lxfxfaxax

limlim

Continuity

• Definition: Let f(x) be defined on the interval [a,b]. We say that f(x) is continuous on the interval [a,b] if and only if

for all and

cfxfcx

lim

bac ,

Continuity

and

afxfax

lim

bfxfbx

lim

Limits of Continuous Functions

• Two continuous functions on the interval [a,b]: f(x) and g(x)

– Limit of the sum– Limit of the difference– Limit of the product– Limit of the multiple– Limit of the quotient

DifferentiationFunctions of Single Variable

• The derivative of a function y = f(x) is another function f’ defined by

at all points for which the limit exists.

h

xfhxfdxdy

xfh

0lim'

DifferentiationFunctions of Single Variable

• Example:

2xxfy

DifferentiationFunctions of Single Variable

• Some common examples

• Polynomials:

• Natural logarithm:

1 kk kxxdxd

x

xdxd 1ln

DifferentiationMultivariate Functions

• The partial derivative of a multivariate function z = f(x,y) with respect to x is defined as

h

yxfyhxfxyxf

h

,,lim

,0

DifferentiationMultivariate Functions

• The multivariate function z = f(x,y) is defined to be

1. Increasing in x if and only if

0,

yxfx

DifferentiationMultivariate Functions

• The multivariate function z = f(x,y) is defined to be

2. Decreasing in x if and only if

0,

yxfx

DifferentiationMultivariate Functions

• The multivariate function z = f(x,y) is defined to be

3. Constant in x if and only if

0,

yxfx

Applications in Economics

• Example: Utility functions

• Definition: A utility function is a mathematical function representing an individual’s set of preferences, which translates her well-being from different consumption bundles into units that can be compared in order to determine choice.

Applications in Economics

• Example: Cobb-Douglas

where

Let and

1, YXYXU

1,0

5.0 1

Applications in Economics

• Marginal utility of x: The change in the utility function of the individual with a unit change in amount of x consumed.

YXUX

MU X ,

Applications in Economics

• Utility function increases in x (positive marginal utility) if x is a ‘good’. Example: apples

• Utility function decreases in x (negative marginal utility) if x is a ‘bad’. Example: pollution

Applications in Economics

• Diminishing marginal utility: the consumption of each additional unit of a good makes an individual less happy than the consumption of the previous unit.

0

XMUX

Applications in Economics

• Indifference curves: A graphical representation of all bundles of goods that make an individual equally well off. Because these bundles have equal utility, an individual is indifferent as to which bundle he consumes.

Applications in Economics

• Important properties of indifference curves:

– Consumers prefer higher indifference curves (more is better).

– Indifference curves are always downward sloping (diminishing marginal utility).

– Indifference curves do not intersect (transitivity).

Applications in Economics

• Marginal rate of substitution: The rate at which a consumer is willing to trade one good for another. The MRS is equal to the slope of the indifference curve, the rate at which the consumer will trade the good on the vertical axis for the good on the horizontal axis.

Y

XYX MU

MUMRS ,

OptimizationWithout Constraints

• Utility Maximization

• First-Order Conditions (FOC)

YXUYX

,max,

0, YXUX

MU X

0, YXUY

MUY

OptimizationWithout Constraints

• Second Order Conditions: You don’t have to worry about for this class! I will make sure that they are satisfied.

OptimizationWithout Constraints

• Example: Cobb-Douglas

OptimizationWith Constraints

• Budget Constraint: A mathematical representation of all the combinations of goods an individual can afford to buy if she spends her entire income.

IYpXp YX

OptimizationWith Constraints

• Constrained utility maximization:

subject to

IYpXp YX

YXUYX

,max,

OptimizationWith Constraints

• Since both X and Y are goods (marginal utility of both X and Y are positive), we know that the maximization can only take place along the line:

IYpXp YX

OptimizationWith Constraints

• So, the problem becomes

subject to

IYpXp YX

YXUYX

,max,

OptimizationWith Constraints

• How to solve?1. Given prices of the goods and income, solve for X (or Y) using

2. Substitute X (or Y) into the utility function

3. Find the first order condition with respect to X (or Y) and find the value of X (or Y) that maximizes utility.

4. Find the value of Y (or X) that maximizes utility from the budget constraint.

IYpXp YX

Application

• Impact of a change in the price of X – Find the initial values of X and Y that maximize utility using the initial

prices.– Find the values of X and Y that maximize utility after the price change

using the final prices.– The changes in X and Y will include

• Substitution effect• Income effect