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12th June, 2012 MATHEMATICS FOR ECONOMICS Sharmini Balachandran, BA Hons (Cantab) Mathematics For Economics Introduction: How important is Mathematics for Economics? Economics Interview Questions How is Mathematics tested in an Economics interview? Interview example questions Mathematics required for an Economics Degree Mathematics for Microeconomics Mathematics for Macroeconomics Econometrics What can students do to prepare? How Important is Mathematics for Economics? You cannot avoid Maths in an Economics degree! Core elements: Pure Mathematics and Statistics Mathematics is part of an economist’s toolkit Model economic relationships Demonstrate well-known empirical results Solve for economic equilibrium BUT an economist must also be able to interpret results and evaluate conclusions ECONOMICS INTERVIEW QUESTIONS How Are Maths Skills Tested At Interview? ‘On the spot’ mental arithmetic and brain teasers Problem solving – Calculus with an economic application Graphical interpretation Probability questions ‘On The Spot’ Maths Simple questions designed to test how quickly candidates think ‘Curve ball’ questions that can get candidates flustered Question 1: What is 49 squared? ‘On The Spot’ Maths Solution 1 Look for shortcuts 49 2 (50 1)2 (50 1)(50 1) 2500 (2 50) 1 2500 100 1 2401 Problem Solving – Economic Applications Use algebraic techniques and calculus to solve economics problems Linear and non-linear functions Linear programming Differentiation and integration Economics applications Demand and supply analysis Profit maximization Calculate consumer or producer surplus Calculating elasticities Demand And Supply Question 2: Firm BA operates in the airline industry, facing the following demand and supply curves. p denotes price. q denotes quantity. Demand: p 1000 2q Supply: p 200 2q Solve for the equilibrium price and quantity in the airline industry. Demand And Supply Solution 2: Solve using linear programming. DS 1000 2q 200 2q 800 4q q* 200 Substituting equilibrium quantity into the demand curve: p 1000 (2 200) p* 600 Profit Maximisation Question 3: A monopolist faces the following economic relationships: Total Cost: Total Revenue: Demand Curve: TC(q) 200q 15q2 TR(q) 1200q 10q2 p 1200 10q What output maximises its profit? b) What is the profit-maximising price? c) What is the firm’s maximal profit? a) Profit Maximisation Solution 3: This is an optimisation problem a) The monopolist sets MC=MR MC(q) TC '(q) 200 30q MR(q) TR'(q) 1200 20q 200 30q 1200 20q 50q 1000 q* 20 b) Substituting q* into the demand curve: p 1200 (10 20) 1000 Profit Maximisation c) Profit is total revenue minus total cost, therefore: TR TC 1200q 10q 2 (200q 15q 2 ) 1000q 25q 2 Substituting in q* (1000 20) 25(20 2 ) 20000 10000 10000 Graphical Interpretation Candidates are given graphs and asked to identify/describe the economic relationship Explain or show how an economic relationship can be modeled with mathematics Draw the graph of a mathematical function Total Revenue Curve q Graphical Interpretation Question 4: Describe the relationship between total revenue and sales (output)? What mathematical function would you use to model this relationship? Solution 4: This is a quadratic function Revenue is an increasing function up to output q and decreasing thereafter Graph is an ‘inverted parabola’ we know that the quadratic term must be negative The general form is: TR(q) a bq cq 2 Probability Questions Probability of multiple events occurring at once Mutually exclusive events Independent events Conditional probability Expected values Question 5: Firm ABC is a monopoly in the electricity industry. The economy can be in a ‘boom’ phase or ‘recession’. The probability of a boom is 0.4. ABC charges a price of £2 per watt in a boom and charges £1 in a recession. What is the expected price per watt of electricity? Probability Questions Solution 5: The expected price is the probability weighted mean of the price in the two scenarios ‘boom’ and recession. E( p) (0.4 £2) (0.6 £1) £1.40 MATHEMATICS REQUIRED FOR AN ECONOMICS DEGREE Mathematics for Microeconomics Consumer Theory Indifference curves and budgetary constrainsts – maximise utility with respect to a budget Intertemporal consumption decisions – allocating consumption between time periods Producer Theory Pricing and output decisions – profit maximisation Constant, increasing & decreasing returns to scale Mathematics For Macroeconomics Aggregate Supply & Demand IS-LM framework – solve for equilibrium Intertemporal aggregate consumption Fiscal Policy Intertemporal government spending decisions Monetary Policy Money demand and supply equilibrium Econometrics Descriptive statistics Linear Regression Time Series Hypothesis Testing Confidence Intervals What Can Students Do To Prepare? Choose the right A-Levels – too much Maths and Statistics is never a bad thing! Be comfortable with calculus Preliminary reading in Mathematics before the course starts is a must Resources Introductory Text ‘Mathematics for Economics and Finance – Methods and Modelling’ – Anthony & Biggs 1st Year: ‘Mathematics for Economists – An introductory textbook’ – Pemberton & Rau 2nd Year: ‘Mathematical Statistics with Applications’ – Miller & Miller ‘Mathematics for Economists’ – Simon & Blume (advanced)