3/1/2016. Calculus: A Reminder. Calculus: A Reminder. Calculus: A Reminder. Calculus: A Reminder. Calculus: A Reminder. Calculus: A Reminder

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1 1 Intermediate Microeconomics W3211 Lecture 5: Choice and Demand Introduction Columbia Universit, Spring 2016 Mark Dean: 2 The Stor So Far. 3 Toda s Aims 4 We have now have had a first attempt at solving the consumer problem Provides a recipe based on finding three tpes of solution Corner solutions Tangenc points Kinks 1. Provide some more mathematicall sophisticated tools to find tangenc points Varian Ch. 5 appendix, Feldman and Serrano Ch. 3 appendix 2. Introduce the concept of a demand function, which measures how the amount of good a consumer purchaces changes with Prices Varian Ch. 6, Feldman and Serrano Ch Discuss how demand changes with income Varian Ch. 6, Feldman and Serrano Ch. 4 Solving the Consumer s Problem 6 One of the reasons that calculus is so useful is that it allows us to find the optimal solutions for constrained and unconstrained optimization problems Calculus and the Consumer s Problem 1: Constrained to unconstrained problems This relies on derivatives Onl works for finding points of tangenc Not corner solutions Not kinks So these tools are useful, but don t forget ou still need to worr about other solutions 5 1

2 7 8 f(x) Imagine ou have an unconstrained optimization problem Choose x to maximize f(x) How can ou use calculus to solve this? f'(x) f''(x) f(x) A maximum of the function will occur where it is flat i.e. where the derivative is zero Is it alwas the case that the derivative being zero means that ou have found a maximum? f'(x) f''(x) Flat points of the function occur either at maxima or minima To differentiate between the two, check the second derivative If ou have found a point at which The first derivative is zero The second derivative is negative Then ou have found a local maximum However ou still have to worr about Other local maxima 2

3 13 14 Flat points of the function occur either at maxima or minima To differentiate between the two, check the second derivative If ou have found a point at which The first derivative is zero The second derivative is negative Then ou have found a local maximum However ou still have to worr about Other local maxima Corner solutions Flat points of the function occur either at maxima or minima To differentiate between the two, check the second derivative If ou have found a point at which The first derivative is zero The second derivative is negative Then ou have found a local maximum However ou still have to worr about Other local maxima Corner solutions Kinks From Constrained to Unconstrained Problems How does this help us? Tells us how to solve unconstrained optimization problems But we have a constrained optimization problem Choose, to Maximize, Subject to Answer: We can substitute in using the budget constraint to make it an unconstrained problem Note in order to do so we are assuming the budget constrain holds with equalit Monotonic preferences 17 From Constrained to Unconstrained Problems An example: Choose, to Maximize, = Subject to Using the budget constraint we get Substitute this into the utilit function = Can choose an (such that and are greater than or equal to zero) to maximize utilit Don t have to worr about the budget constraint adjusts automaticall to changes in 18 3

4 From Constrained to Unconstrained Problems 19 The Advantages of This Approach 20 Problem becomes Choose to Maximize = Taking derivatives gives 2 )= )=2 Second derivative is negative, so first order condition will give us a local maximum 2 0 Implies (using the budget constraint for ) and Unsurprisingl, gives the same answer as the recipe from last week What is the advantage of this approach? One thing is the second order conditions Helps us to identif when we have found a local minima Consider concave preferences, = + You should check ou agree, but these preferences give utilit that looks like The Advantages of This Approach 21 The Advantages of This Approach 22 So a tangenc point is actuall a minimum The second order conditions can help us spot this Choose, to Maximize, = + Subject to Becomes Choose to Maximize = Taking derivatives gives )=2 2 )=22 0 Second order condition fails, tells us we have a maximum Karush Kuhn Tucker 24 Mathematicians have developed even more powerful tools for solving constrained optimization problems We won t have time to go over them thoroughl in this class Calculus and the Consumer s Problem 2: Karush Kuhn Tucker The ultimate tool for constrained optimization problems But I want ou to be aware of them, for two reasons 1. The can make it easier to solve problems we will deal with 2. You will certainl need them if ou take economics further You will not need to use these techniques to solve questions in exams But ou should use them if ou want You ma need them to solve homework questions 23 4

5 Karush Kuhn Tucker 25 Karush Kuhn Tucker 26 Imagine ou have a constrained optimization problem Choose, to Maximize, Subject to f, 0 Choose,, to Maximize,,, f, How do we solve this problem? (in our case f, ) Treat it like an unconstrained optimization problem It turns out that ou can solve this problem b solving a related unconstrained maximization problem Take first derivatives and set them to zero! Choose,, to Maximize,,, f, is the Lagrange Multiplier You can treat it just like another thing to choose (like and ) Karush Kuhn Tucker 27 Karush Kuhn Tucker 28 A worked example:,, Choose, to Maximize, = Subject to First set up the Lagrange Function,, Then take derivatives Then take derivatives First two equations gives the (ver familiar looking!) Last equation gives the (ver familiar looking) Advantages of Karush Kuhn Tucker 29 Karush Kuhn Tucker 30 Wh have I bothered to tell ou this? What is the advantage of this approach? Imagine that we had three goods Choose,, to Maximize,, Subject to Substitution method won t work (we have too man unknowns) Can t draw indifference curves in two dimensions,,,,, Then take derivatives However the KKT approach will still work Four equations, four unknowns will generall be able to find a solution 5

6 A Word of Warning 31 Are KKT solutions alwas the solutions to the consumer s problem? No! KKT conditions find points of tangenc But the usual caveats appl Tangenc are neither necessar or sufficienc for optimalit Solutions ma still be at corners or kinks, or preferences ma be nonmonotonic Also, KKT conditions ma pick out minima instead of maxima Just as when we use differentiation to solve unconstrained optimization problems There are equivalent second order conditions, but ou don t need to worr about them for this course Demand Or, the effect of prices and income Remember this! I will tr to fool ou, and I will catch some of ou! 32 Defining Demand 33 Defining Demand 34 We now know how to solve the consumer s problem This solution depends on budget constraint The prices of the two goods, and the income We define the amount that the consumer will bu, given prices and incomes, as their demand,, Is the amount that the consumer will bu of good one if prices are and and income is * This is the demand function for Similarl,,,is the demand function for * More generall,, is the demand function for good i if the vector of prices p In the next half a lecture, we will think about how demand changes with prices and income How Does Demand Change with? 36 Question 1: how does,, change with? Keeping prices fixed Demand 1: The effect of income on demand 35 6

7 37 38 < < < < < < < < offer curve The curve that unites the optimal choices for various levels of income is called income offer curve or income expansion path Axes are and < < offer curve A plot of quantit demanded against income is called an curve. Axes are and Traditionall is on the vertical axis 7

8 43 44 < < < < offer curve offer curve * * 45 < < offer curve * 46 < < offer curve * < < offer curve * 47 and Cobb- Douglas Preferences * An example of computing the equations of curves; the Cobb-Douglas case. Ux ( 1, x2) x1 a x b 2. The ordinar demand equations are * a * b x1 ; x2. ( a b) p1 ( a b) p2 48 8

9 and Cobb- Douglas Preferences * a * b x1 ; x2. ( a b) p1 ( a b) p2 Rearranged to isolate, these are: ( a b) p 1 * a ( a b) p * 2 b curve for curve for 49 and Cobb- Douglas Preferences ( a b) p 1 * a curve * * for ( a b) p 2 * b curve for 50 and Perfectl- Complementar Preferences Another example of computing the equations of curves; the perfectl-complementar case. Ux ( 1, x2) min x1, x2. The ordinar demand equations are * * x1 x2. p1 p2 51 and Perfectl- Complementar Preferences * * x1 x2. p1 p2 Rearranged to isolate, these are: * ( p1p2) x1 * ( p1p2) x2 curve for curve for < < 9

10 55 56 < < < < < < 57 < < 58 * x * 1 < < 59 ( p1p2) x * 2 60 * ( p x 1 p2) * 1 * x * 1 * 10

Homotheticit 63 64 It turns out the linear curves come from preferences which are homothetic A consumer s preferences are homothetic if and onl if (, ) ( 1, 2 ) (k,k ) (k 1,k 2 ) for ever k > 0.

11 61 62 Some properties of the curves that we have seen so far: 1. The are straight lines 2. The are upward sloping Is this true in general? Some properties of the curves that we have seen so far: 1. The are straight lines 2. The are upward sloping Is this true in general? Homotheticit It turns out the linear curves come from preferences which are homothetic A consumer s preferences are homothetic if and onl if (, ) ( 1, 2 ) (k,k ) (k 1,k 2 ) for ever k > 0. That is, the consumer s MRS is the same anwhere on a straight line drawn from the origin. i.e. draw a straight line from the origin The slope of indifference curves is the same everwhere along that line MRS the same along a straight line As prices don t change, tangenc point the same along a straight line Offer curve and Engle Curves are straight lines -- A Nonhomothetic Example Are preferences alwas homothetic? No, in fact ou have come across an example of onhomothetic preferences: Quasilinear. For example: 65 Quasi-linear Indifference Curves Each curve is a verticall shifted cop of the others. Each curve intersects both axes. 66 Ux ( 1, x2) x1 x2. 11

12 ; Quasilinear Utilit 67 ; Quasilinear Utilit 68 curve for ~ ~ ~ * ; Quasilinear Utilit curve for 69 ; Quasilinear Utilit 70 curve for * * curve for ~ ~ ~ * Some properties of the curves that we have seen so far: 1. The are straight lines 2. The are upward sloping A good for which quantit demanded rises with income is called normal. Therefore a normal good s curve is positivel sloped. Is this true in general? 12

13 Normal goods offer curve 73 * It is possible to construct examples in which this is not the case 74 * : inferior goods

14 79 80 offer curve 81 curve for 82 curve for * curve for * * 83 Inferior Goods 84 A good for which quantit demanded falls as income increases is called income inferior. Therefore an income inferior good s curve is negativel sloped. So good can be inferior goods But wait: wh? Examples? 14

15 Inferior Goods 85 Elasticit of Demand 86 So good can be inferior goods How do we mathematicall describe the effect of income on demand? But wait: wh? Derivative of the demand function?,, Examples? Cheap objects: the wealthier ou are the less ou bu of them Ramen noodles Often we will instead use the income elasticit of demand,, We can thing of this as the effect of a percentage change in income on the percentage change in demand Elasticit of Demand 87 What is the advantage of using elasticities? One is that the do not depend on the units Imagine that we measure the demand for potatoes in kilos rather than pounds The demand at a given price would change So would the derivation of the demand function However, the elasticit would not change A percentage change is the same whether demand is measured in kilos or pounds Summar Equivalent concept of price elasticit of demand,, 88 Summar 89 Toda we have done the following 1. Provide some more mathematicall sophisticated tools to find tangenc points Derivative based approach Karush Kuhn Tucker Conditions 2. Defined the concept of demand, and shown how the demand for a particular good can change with income 15

Algebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES

Algebra I Notes Linear Functions & Inequalities Part I Unit 5 UNIT 5 LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES UNIT LINEAR FUNCTIONS AND LINEAR INEQUALITIES IN TWO VARIABLES PREREQUISITE SKILLS: students must know how to graph points on the coordinate plane students must understand ratios, rates and unit rate VOCABULARY: