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Intro Solving UMP Extreme Cases Econ 3023 Microeconomic Analysis Chapter 5: Utility Maximization Problem Instructor: Hiroki Watanabe Fall 2012 Watanabe Econ 3023 Intro 5 UMP Solving UMP 1 Introduction 2 Solving UMP 3 Extreme Cases 4 Now We Know Watanabe Econ 3023 Intro Extreme Cases 5 UMP Solving UMP Extreme Cases 1 Introduction Consumer Theory Overview 2 Solving UMP 3 Extreme Cases 4 Now We Know Watanabe Econ 3023 5 UMP 1 / 43 2 / 43 3 / 43 Intro Solving UMP Extreme Cases Consumer Theory Overview 1 What do consumers face? 2 What do consumers want? 3 How do consumers resolve conflict above? Chapter 2 & 10 Chapter 3 & 4 Chapter 5 Watanabe Econ 3023 Intro 5 UMP Solving UMP Extreme Cases 4 / 43 Consumer Theory Overview Consumers choose the best combination of goods they can afford. The framework to analyze their choice behavior is called utility maximization problem (UMP). Watanabe Econ 3023 Intro 5 UMP Solving UMP Extreme Cases 5 / 43 Consumer Theory Overview Definition 1.1 (Utility Maximization Problem) Liz chooses the bundle (C , T ) that gives her the highest utility level among the affordable bundles. In other words, she max(C , T ) Watanabe Econ 3023 subject to pC C + pT T = m. 5 UMP 6 / 43 Intro Solving UMP Extreme Cases 1 Introduction 2 Solving UMP Budget Line Meets Indifference Curves Tangency To Find the Exact Solution 3 Extreme Cases 4 Now We Know Watanabe Econ 3023 Intro 5 UMP Solving UMP 7 / 43 Extreme Cases Budget Line Meets Indifference Curves How exactly do we find Liz’s optimal bundle ∗ , ∗ )? ∗ = (C T Consider a case when Utility function (C , T ) = C T Income m = 60, Price (pC , pT ) = (3, 2). Watanabe Econ 3023 Intro 5 UMP Solving UMP 8 / 43 Extreme Cases Budget Line Meets Indifference Curves 30 Budget Line Tea xT (cups) 25 20 15 10 5 0 0 Watanabe Econ 3023 5 10 Cheese xC (slices) 5 UMP 15 20 9 / 43 Intro Solving UMP Extreme Cases Budget Line Meets Indifference Curves 30 5 Indifference Curves50 50 0 25 45 0 Tea xT (cups) 40 0 35 0 20 30 0 250 15 200 10 150 100 5 50 0 0 Watanabe 5 Econ 3023 Intro 10 Cheese xC (slices) 15 20 5 UMP Solving UMP 10 / 43 Extreme Cases Budget Line Meets Indifference Curves 30 5 Indifference Curves50 50 Budget Line 0 25 45 0 Tea xT (cups) 40 0 35 0 20 30 0 250 15 200 10 150 100 5 50 0 0 Watanabe 5 Econ 3023 Intro 10 Cheese xC (slices) 15 20 5 UMP Solving UMP Extreme Cases 11 / 43 Tangency Fact 2.1 (Tangency Condition) For standard preferencesa , the indifference curve is tangent to the budget line at the optimal bundle, i.e., they share the same slope at ∗ . a This condition does not apply to certain types of preferences. See Exercise 3.2 for example. Watanabe Econ 3023 5 UMP 12 / 43 Intro Solving UMP Extreme Cases Tangency What does tangency condition imply? 1 2 Trinity on the budget side Trinity on the preference side Liz’s idea of the tea’s worth coincides with market’s idea of the tea’s worth when she chooses the right amount. Watanabe Econ 3023 Intro 5 UMP Solving UMP 13 / 43 Extreme Cases Tangency What if tangency condition is not met? Watanabe Econ 3023 Intro 5 UMP Solving UMP 14 / 43 Extreme Cases Tangency 30 5 Indifference Curves50 50 Budget Line 0 Relative Price 450 MRS 40 25 Tea xT (cups) 0 35 0 20 30 0 250 15 200 10 150 100 5 50 0 0 Watanabe Econ 3023 5 10 Cheese xC (slices) 5 UMP 15 20 15 / 43 Intro Solving UMP Extreme Cases To Find the Exact Solution Where exactly is the budget line tangent to the indifferent curve? We need two conditions to find the point of tangency: Fact 2.2 (Finding the Solution) For standard utility functions, the solution can be found using 1 tangency condition and 2 budget constraint Watanabe Econ 3023 Intro 5 UMP Solving UMP 16 / 43 Extreme Cases To Find the Exact Solution Problem 2.3 (UMP) max(C , T ) = C T subject to MRS at (C , T ) is given by The relative price is 1 In Watanabe 3C + 2T = 60. −T 1 . C −3 . 2 what follows, I’ll get you MRS except for perfect substitutes. Econ 3023 Intro 5 UMP Solving UMP 17 / 43 Extreme Cases To Find the Exact Solution Tangency condition: −T C = −3 2 ⇒ T = 3 2 C . (1) There are lots of bundles (C , T ) satisfying (1). Watanabe Econ 3023 5 UMP 18 / 43 Intro Solving UMP Extreme Cases To Find the Exact Solution 30 5 Indifference Curves50 50 Budget Line 0 25 45 0 Tea xT (cups) 40 0 35 0 20 30 0 250 15 200 10 150 100 5 50 0 0 Watanabe 5 10 Cheese xC (slices) Econ 3023 Intro 15 20 5 UMP Solving UMP Extreme Cases 19 / 43 To Find the Exact Solution To pin down the solution, use budget constraint: 3 60 = 3C + 2 C . 2 Conclude ∗ = (10, 15). Watanabe Econ 3023 Intro 5 UMP Solving UMP Extreme Cases 20 / 43 To Find the Exact Solution Exercise 2.4 (UMP) Liz spends her income ($5) on coins (C ) and tea (T ). Price is given by (pC , pT ) = (2, 1). His preferences are p represented by (C , T ) = C + T . 1 Write Liz’s utility maximization problem. 2 What is the tangency condition in this example? p (MRS at (C , T ) is −2 T ). 3 Which bundle will Liz choose? Watanabe Econ 3023 5 UMP 21 / 43 Intro Solving UMP Extreme Cases To Find the Exact Solution 5 2.5 Indifference Curves Budget Line 4.5 2.0 3 T 0 4. Tea x (cups) 4 2 3. 5 5 1. 1 3. 0 1.0 0 0 Watanabe 0.5 2.5 0.5 Econ 3023 Intro 1 1.5 Coins xC 2 5 UMP Solving UMP Extreme Cases 1 Introduction 2 Solving UMP 3 Extreme Cases When Tangency Condition Does Not Hold Perfect Substitutes Perfect Complements 4 Now We Know Watanabe Econ 3023 Intro 5 UMP Solving UMP Extreme Cases 2.5 22 / 43 23 / 43 When Tangency Condition Does Not Hold There are some exceptions where tangency condition does not apply. Watanabe Econ 3023 5 UMP 24 / 43 Intro Solving UMP Extreme Cases Perfect Substitutes Consider six-packs and bottles of Corona (6 , 1 ). Lizs MRS is 6 everywhere (why?) Consider three cases: 1 2 3 Watanabe Market rate of exchange is higher than 6. Market rate of exchange is lower than 6. Market rate of exchange is exactly 6. Econ 3023 Intro 5 UMP Solving UMP 25 / 43 Extreme Cases Perfect Substitutes Market rate of exchange is higher than 6. 1 Let’s say the market rate of exchange is larger than 6, say 12. Liz will spend all her income on bottles. This type of solution is called a corner solution (e.g., (8, 0), (35, 0), (0, 68) etc.) Watanabe Econ 3023 Intro 5 UMP Solving UMP 26 / 43 Extreme Cases Perfect Substitutes 60 Indifference Curves 84 Budget Line 7 8 48 72 Bottles x1 66 60 36 54 30 24 48 24 42 18 12 36 12 6 0 0 Watanabe Econ 3023 1 2 3 Six−Packs x6 5 UMP 4 5 27 / 43 Intro Solving UMP Extreme Cases Perfect Substitutes If market rate of exchange is lower than 6... 2 Watanabe Econ 3023 Intro 5 UMP Solving UMP 28 / 43 Extreme Cases Perfect Substitutes 60 Indifference Curves 84 Budget Line 7 8 48 72 Bottles x1 66 60 36 54 30 24 48 24 42 18 12 36 12 6 0 0 Watanabe 1 Econ 3023 Intro 2 3 Six−Packs x6 4 5 5 UMP Solving UMP Extreme Cases 29 / 43 Perfect Substitutes This type of solution is called a corner solution (e.g., (8, 0), (35, 0), (0, 68) etc.) Watanabe Econ 3023 5 UMP 30 / 43 Intro Solving UMP Extreme Cases Perfect Substitutes Market rate of exchange is exactly 6... 3 Watanabe Econ 3023 Intro 5 UMP Solving UMP 31 / 43 Extreme Cases Perfect Substitutes 60 Indifference Curves 84 Budget Line 7 8 48 72 Bottles x1 66 60 36 54 30 24 48 24 42 18 12 36 12 6 0 0 Watanabe 1 Econ 3023 Intro 2 3 Six−Packs x6 4 5 5 UMP Solving UMP Extreme Cases 32 / 43 Perfect Substitutes Exercise 3.1 (Perfect Substitutes) Liz has $12 and spend his income on Coke & Pepsi (C , P ). Suppose Coke is sold for $4 while Pepsi is sold for $2. 1 Find Liz’s optimal bundle. 2 Confirm your answer using indifference curves and the budget line. Watanabe Econ 3023 5 UMP 33 / 43 Intro Solving UMP Extreme Cases Perfect Substitutes 6 Indifference Curves Budget Line 6 5 4 Bottles x 1 21 3 3 2 1 Econ 3023 Intro 1.5 Six−Packs x6 18 0.5 15 0 0 Watanabe 12 9 1 2 2.5 3 5 UMP Solving UMP 34 / 43 Extreme Cases Perfect Complements For perfect complements, MRS is not defined. What to do with them then? Consider left gloves (L ) and right gloves (R ). Watanabe Econ 3023 Intro 5 UMP Solving UMP 35 / 43 Extreme Cases Perfect Complements 12 4 Indifference Curves Budget Line 10 8 8 2 Right Gloves xR 10 6 6 4 4 2 2 0 0 Watanabe 2 Econ 3023 4 6 8 Left Gloves xL 5 UMP 10 12 36 / 43 Intro Solving UMP Extreme Cases Perfect Complements Optimal bundle ∗ for left gloves and right gloves is 1 2 Watanabe on the budget line and on the 45 degree line. Econ 3023 Intro 5 UMP Solving UMP 37 / 43 Extreme Cases Perfect Complements Exercise 3.2 (Perfect Complements) Consider a bundle (cereal, milk)= (C , M ). Liz says he can’t have cereals without milk and the only time she drinks milk is when she eats his cereals. Liz’s preferred cereal-milk ratio is 2 to 3. (pC , pM ) = (6, 4). m = 24. 1 Draw her budget line. 2 Draw indifference curves at (2, 3) and (4, 6). 3 Mark the bundle Liz will choose on your graph. Watanabe Econ 3023 Intro 5 UMP Solving UMP 38 / 43 Extreme Cases Perfect Complements 6 Indifference Curves Budget Line Right Gloves x R 5 10 4 8 3 6 2 4 1 2 0 0 Watanabe Econ 3023 1 2 Left Gloves xL 5 UMP 3 4 39 / 43 Intro Solving UMP Extreme Cases Perfect Complements Liz’s preferred bundles are lined up on the ray M = 23 C . The optimal bundle is Watanabe on the ray M = 2 on the budget line M = Econ 3023 Intro 1 Introduction 2 Solving UMP 3 Extreme Cases 4 Now We Know Econ 3023 Intro and −3 2 C + 6. 5 UMP Solving UMP Watanabe 3 2 C 1 Extreme Cases 5 UMP Solving UMP Extreme Cases 40 / 43 41 / 43 How to set up the utility maximization problem. Tangency condition for standard cases. Optimal bundles for extreme preferences. Watanabe Econ 3023 5 UMP 42 / 43 Intro Solving UMP Extreme Cases Index budget constraint, 16 corner solution, 26, 30 optimal bundle, 8 tangency condition, 12, 16 Watanabe Econ 3023 utility maximization problem, 5 ∗ , see optimal bundle 5 UMP 43 / 43