Chapter 5: Utility Maximization Problems
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1 Econ 01 Price Theory Chapter : Utility Maximization Problems Instructor: Hiroki Watanabe Summer / 9 1 Introduction 2 Solving UMP Budget Line Meets Indifference Curves Tangency Find the Exact Solutions Extreme Cases Perfect Complements Summary 2 / 9
2 So far we have discussed: $(Chapter 2 (Chapter (Chapter Now combine the above to predict Greg s consumption behavior given his budget constraint and preferences. Situational background: Greg wants to have as many cheesecakes and tea as possible. Greg s budget constraint does not allow him to choose (x C, x T =(infinite, infinite. Greg has to make choices. / 9 We assume that Greg picks the most preferred combination among what he can afford. The framework to analyze Greg s choice behavior is called utility maximization problem (UMP for short. / 9
3 Greg s decision making process is summarized as follows: Utility Maximization Problem Greg chooses the bundle (x C, x T that gives him the highest utility level among the affordable bundles. In other words, he max u(x C, x T subject to p C x C + p T x T = m. / 9 1 Introduction 2 Solving UMP Budget Line Meets Indifference Curves Tangency Find the Exact Solutions Extreme Cases Perfect Complements Summary 6 / 9
4 Budget Line Meets Indifference Curves How exactly do we find Greg s optimal bundle x = (x C, x T? Consider the case when u(x C, x T = x C x T m = 60, (p C, p T = (,. 7 / 9 Budget Line Meets Indifference Curves 20 1 Tea (x T Cheesecakes (x C 1 8 / 9
5 Budget Line Meets Indifference Curves Tea (x T Cheesecakes (x C / 9 Budget Line Meets Indifference Curves Tea (x T Cheesecakes (x C / 9
6 Budget Line Meets Indifference Curves 11 / 9 Figure: Tangency Tangency Condition At the optimal bundle, the indifference curve is tangent to the budget constraint (for standard preferences, i.e., they both have the same slope at x. 12 / 9
7 Tangency What does the tangency condition imply? The slope of IC denotes MRS (MWTP: the cups of tea Greg is willing to give up for one slice of cheesecake. The slope of budgte line denotes relative price (op. cost: the cups of tea Greg has to give up for one slice of cheesecake. Greg idea of the tea s worth coincides with market s idea of the tea s worth when he chooses the right amount. 1 / 9 Tangency What if the tangency condition is not met? Suppose MRS is larger than the relative price (IC is steeper than the budget line. Greg is willing to give up 0 cups of tea for a slice of cheesecake, while he has to sell only cups of tea to buy a slice of cheesecake. It s wise to increase cheesecake consumption and reduce cups of tea. The current bundle is not optimal. 1 / 9
8 Tangency Suppose MRS is smaller than the relative price (IC is steeper than the budget line. Greg is willing to give up.1 cups of tea for a slice of cheesecake, while he has to sell cups of tea to buy a slice of cheesecake. It s wise to reduce cheesecake consumption and increase cups of tea. The current bundle is not optimal. 1 / 9 Find the Exact Solutions UMP Where exactly is the bundget line tangent to the indifferent curve? We need a little bit of calculus to find Greg s optimal bundle. max u(x C, x T = x C x T subject to x C + x T = 60. MRS at (x 1, x 2 is given by x 2 x 1. The relative price is. 16 / 9
9 Find the Exact Solutions Tangency condition: x T x C = x T = x C. (1 There are lots of bundles (x C, x T satisfying (1. The bundle also has to be affordable: x C + x T = 60 x C + x C = 60. x T = 7. and x C = 10. Conclude x = (10, / 9 Find the Exact Solutions Exercise Greg spends his income ($ on coins (x C and tea (x T. Price is given by (p C, p T = (2, 1. His preferences are represented by u(x C, x T = x C + x T. 1 Write down the budget line. 2 Write Greg s utility maximization problem. What is the tangency condition in this example? (MRS at (x C, x T is 2 x T. Which bundle will Greg buy? 18 / 9
10 2 Find the Exact Solutions 7 66 Tea (x T Coins (x C 19 / 9 1 Introduction 2 Solving UMP Budget Line Meets Indifference Curves Tangency Find the Exact Solutions Extreme Cases Perfect Complements Summary 20 / 9
11 There are some exceptions where tangency condition does not apply. 21 / 9 Consider six-packs and bottles of Corona (x 6, x 1. Greg s MRS is 6 everywhere. Consider three cases: 1 Market rate of exchange is higher than 6. 2 Market rate of exchange is lower than 6. Market rate of exchange is exactly / 9
12 Case ➊ Let s say the market rate of exchange is 2 (i.e., 0 bottles of Corona is traded for 1 six-pack. Greg is willing to give up 6 bottles for 1 six-pack. Greg will spend all his income on six-packs. This type of solution is called a corner solution (e.g., (8, 0, (, 0, (0, 68 etc. 2 / Bottles (x Six Packs (x 6 2 / 9
13 Case ➋ Let s say the market rate of exchange is 1 (i.e., 1 bottle of Corona is traded for 1 six-pack. Greg is willing to give up 6 bottles for 1 six-pack. Greg will spend all his income on bottles. 2 / Bottles (x Six Packs (x 6 26 / 9
14 Case ➌ The market rate of exchange is 6 (i.e., 6 bottles of Corona is traded for 1 six-pack. Greg is willing to give up 6 bottles for 1 six-pack. Greg can choose any bundle on his budget line. 27 / Bottles (x Six Packs (x 6 28 / 9
15 Discussion If a bottle of Corona is sold at $2, a six-pack is usually sold at less than $12. Does that mean nobody buys Corona by the bottle? Dharma s MRS might be less than 6. She is willing to give up 1 six-pack for bottles (MRS=. So, there are Corona sold by the bottle, and there are people who buy Corona by the bottle. 29 / 9 Exercise Greg has $12 and spend his income on Coke & Pepsi (x C, x P. Suppose Coke is sold at $ while Pepsi is sold at $2. 1 Find Greg s optimal bundle. 2 Confirm your answer using indifference curves and the budget line. 0 / 9
16 Pepsi (x P Coke (x C 1 / 9 Perfect Complements For perfect complements, MRS is not defined. That does not mean the solution to UMP does not exist. Consider left gloves (x 1 and right gloves (x 2. 2 / 9
17 Perfect Complements 6 2 Right Gloves (x Left Gloves (x 1 / 9 Perfect Complements Optimal bundle x for left gloves and right gloves is 1 on the budget line and 2 on the degree line. / 9
18 Perfect Complements Exercise Consider a bundle (cereal, milk= (x C, x M. Greg says he can t have cereals without milk and the only time he drink milk is when he eats his cereals. Greg s preferred cereal-milk ratio is 2 to. (p C, p M = (6,. m = 2. 1 Draw his budget line. 2 Draw indifference curves at (2, and (, 6. Mark the bundle Greg will choose on your graph. / 9 Perfect Complements Milk (x M Cereals (x C 6 / 9
19 Perfect Complements Greg s preferred bundles are lined up on the ray x M = 2 x C. The optimal bundle is 1 on the ray x M = 2 x C and 2 on the budget line x M = 2 x C / 9 1 Introduction 2 Solving UMP Budget Line Meets Indifference Curves Tangency Find the Exact Solutions Extreme Cases Perfect Complements Summary 8 / 9
20 Set up the problem. Tangency condition for standard cases. Extreme cases. 9 / 9
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