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

Limits Theorem: if and only if

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

Continuity and

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

Differentiation Functions 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.

Differentiation Functions of Single Variable Example:

Differentiation Functions of Single Variable Some common examples Polynomials: Natural logarithm:

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

Differentiation Multivariate Functions The multivariate function z = f(x,y) is defined to be 1.Increasing in x if and only if

Differentiation Multivariate Functions The multivariate function z = f(x,y) is defined to be 2.Decreasing in x if and only if

Differentiation Multivariate Functions The multivariate function z = f(x,y) is defined to be 3.Constant in x if and only if

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

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.

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.

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.

Optimization Without Constraints Utility Maximization First-Order Conditions (FOC)

Optimization Without Constraints Second Order Conditions: You don’t have to worry about for this class! I will make sure that they are satisfied.

Optimization Without Constraints Example: Cobb-Douglas

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

Optimization With Constraints Constrained utility maximization: subject to

Optimization With 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:

Optimization With Constraints So, the problem becomes subject to

Optimization With 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.

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