Mathematical Economics - Aktualności

Mathematical EconomicsDr Wioletta Nowak, room 205 /user/12141/students-resources

SyllabusMathematical Theory of DemandUtility Maximization ProblemExpenditure Minimization ProblemMathematical Theory of ProductionProfit Maximization ProblemCost Minimization ProblemGeneral Equilibrium TheoryNeoclassical Growth ModelsModels of Endogenous Growth TheoryDynamic Optimization

SyllabusMathematical Theory of Demand Budget ConstraintConsumer PreferencesUtility FunctionUtility Maximization ProblemOptimal ChoiceProperties of Demand FunctionIndirect Utility Function and its PropertiesRoy’s Identity

SyllabusMathematical Theory of Demand Expenditure Minimization ProblemExpenditure Function and its PropertiesShephard's LemmaProperties of Hicksian Demand FunctionThe Compensated Law of DemandRelationship between Utility Maximization andExpenditure Minimization Problem

SyllabusMathematical Theory of Production Production Functions and Their Properties Perfectly Competitive Firms Profit Function and Profit MaximizationProblem Properties of Input Demand and OutputSupply

SyllabusMathematical Theory of Production Cost Minimization Problem Definition and Properties of Conditional FactorDemand and Cost Function Profit Maximization with Cost Function Long and Short Run Equilibrium Total Costs, Average Costs, Marginal Costs,Long-run Costs, Short-run Costs, Cost Curves,Long-run and Short-run Cost Curves

SyllabusMathematical Theory of ProductionMonopolyOligopoly Cournot Equilibrium Quantity Leadership – Slackelberg Model

SyllabusGeneral Equilibrium Theory Exchange Market Equilibrium

SyllabusNeoclassical Growth Model The Solow Growth Model Introduction to Dynamic Optimization The Ramsey-Cass-Koopmans Growth ModelModels of Endogenous Growth TheoryConvergence to the Balance Growth Path

Recommended Reading Chiang A.C., Wainwright K., Fundamental Methods ofMathematical Economics, McGraw-Hill/Irwin, Boston,Mass., (4th edition) 2005. Chiang A.C., Elements of Dynamic Optimization, WavelandPress, 1992. Romer D., Advanced Macroeconomics, McGraw-Hill, 1996. Varian H.R., Intermediate Microeconomics, A ModernApproach, W.W. Norton & Company, New York, London,1996.

The Theory of Consumer Choice The Budget Constraint The Budget Line Changes (Increasing Income,Increasing Price) Consumer Preferences Assumptions about Preferences Indifference Curves: Normal Good, PerfectSubstitutes, Perfect Complements, Bads, Neutrals The Marginal Rate of Substitution

Consumers choose the best bundle ofgoods they can afford How to describe what a consumer can afford? What does mean the best bundle? The consumer theory uses the concepts of abudget constraint and a preference map toanalyse consumer choices.

The budget constraint – the two-good case It represents the combination of goods thatconsumer can purchase given current pricesand income. x1 , x 2 , x i 0, i 1, 2 - consumer’sconsumption bundle (the object of consumerchoice) p1 , p 2 , p i 0, i 1, 2- market pricesof the goods

The budget constraint – the two-good case The budget constraint of the consumer (the amount ofmoney spent on the two goods is no more than the totalamount the consumer has to spend)p1x1 p 2 x 2 I I 0 - consumer’s income (the amount of money theconsumer has to spend)p1x1 - the amount of money the consumer is spending on good 1p 2 x 2 - the amount of money the consumer is spending on good 2

Graphical representation of the budget set and the budget line The set of affordable consumption bundles atgiven prices and income is called the budget setof the consumer.

The Budget Line

The Budget Line Changes Increasing (decreasing) income – an increase (decrease) inincome causes a parallel shift outward (inward) of the budgetline (a lump-sum tax; a value tax)

The Budget Line Changes Increasing price – if good 1becomes more expensive,the budget line becomessteeper. Increasing the price of good1 makes the budget linesteeper; increasing the priceof good 2 makes the budgetline flatter. A quantity taxA value tax (ad valorem tax)A quantity subsidyAd valorem subsidy

Exercise 1

Consumer Preferences

Consumer PreferencesPs ( x, y) X X x y relation of strict preferenceI ( x, y) X X x y relation of indifference P ( x, y) X X x y relation of weak preference

Assumptions about Preferences

Assumptions about Preferences

Assumptions about Preferences

Assumptions about Preferences

The relations of strict preference, weak preference andindifference are not independent concepts!

Exercise 2

Exercise 3

Indifference Curves The set of all consumption bundles that areindifferent to each other is called anindifference curve. Points yielding different utility levels are eachassociated with distinct indifference curves.

Indifference curves are

Indifference curve for normal goods

Perfect substitutes Two goods are perfectsubstitutes if the consumeris willing to substitute onegood for the other at aconstant rate. The simplest case of perfectsubstitutes occurs when theconsumer is willing tosubstitute the goods on aone-to-one basis. The indifference curves hasa constant slope since theconsumer is willing to tradeat a fixed ratio.

Perfect complements Perfect complements aregoods that are alwaysconsumed together infixed proportions. L-shapedcurves.indifference

Bads: a bad is a commodity that consumer doesn’t like

Neutrals: a good is a neutral good if the consumerdoesn’t care about it one way or the other

The Marginal Rate of Substitution (MRS) The marginal rate of substitution measures the slope of theindifference curve.

The Marginal Rate of Substitution (MRS)

The Marginal Rate of Substitution (MRS) The MRS is different at each point along theindifference curve for normal goods. The marginal rate of substitution betweenperfect substitutes is constant.

Mathematical Economics Dr Wioletta Nowak, room 205 C . Mathematical Theory of Production . Fundamental Methods of Mathematical Economics, McGraw-Hill/Irwin, Boston, Mass., (4th edition) 2005. Chiang A.C., Elements of Dynamic Optimization, Waveland Press, 1992.