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Chapter 2: Demand Theory Demand Theory Utility Maximization Problem • Consumer maximizes his utility level by selecting a bundle 푥 (where 푥 can be a vector) subject to his budget constraint: max 푢(푥) 푥≥0 s. t. 푝 ∙ 푥 ≤ 푤 • Weierstrass Theorem: for optimization problems defined on the reals, if the objective function is continuous and constraints define a closed and bounded set, then the solution to such optimization problem exists. 2 Utility Maximization Problem • Existence: if 푝 ≫ 0 and 푤 > 0 (i.e., if 퐵푝,푤 is closed and bounded), and if 푢(∙) is continuous, then there exists at least one solution to the UMP. – If, in addition, preferences are strictly convex, then the solution to the UMP is unique. • We denote the solution of the UMP as the argmax of the UMP (the argument, 푥, that solves the optimization problem), and we denote it as 푥(푝, 푤). – 푥(푝, 푤) is the Walrasian demand correspondence, which 퐿 specifies a demand of every good in ℝ+ for every possible price vector, 푝, and every possible wealth level, 푤. 3 Utility Maximization Problem • Walrasian demand 푥(푝, 푤) at bundle A is optimal, as the consumer reaches a utility level of 푢2 by exhausting all his wealth. • Bundles B and C are not optimal, despite exhausting the consumer’s wealth. They yield a lower utility level 푢1, where 푢1 < 푢2. • Bundle D is unaffordable and, hence, it cannot be the argmax of the UMP given a wealth level of 푤. 4 Properties of Walrasian Demand • If the utility function is continuous and preferences 퐿 satisfy LNS over the consumption set 푋 = ℝ+, then the Walrasian demand 푥(푝, 푤) satisfies: 1) Homogeneity of degree zero: 푥 푝, 푤 = 푥(훼푝, 훼푤) for all 푝, 푤, and for all 훼 > 0 That is, the budget set is unchanged! 퐿 퐿 푥 ∈ ℝ+: 푝 ∙ 푥 ≤ 푤 = 푥 ∈ ℝ+: 훼푝 ∙ 푥 ≤ 훼푤 Note that we don’t need any assumption on the preference relation to show this. We only rely on the budget set being affected. 5 Properties of Walrasian Demand x2 – Note that the preference relation can be linear, and α w w homog(0) would α p = p 2 2 still hold. x (p,w), unaffected x α w = w 1 α p1 p1 Advanced Microeconomic Theory 6 Properties of Walrasian Demand 2) Walras’ Law: 푝 ∙ 푥 ≤ 푤 for all 푥 = 푥(푝, 푤) It follows from LNS: if the consumer selects a Walrasian demand 푥 ∈ 푥(푝, 푤), where 푝 ∙ 푥 < 푤, then it means we can still find other bundle 푦, which is ε–close to 푥, where consumer can improve his utility level. If the bundle the consumer chooses lies on the budget line, i.e., 푝 ∙ 푥′ = 푤, we could then identify bundles that are strictly preferred to 푥′, but these bundles would be unaffordable to the consumer. 7 Properties of Walrasian Demand x2 – For 푥 ∈ 푥(푝, 푤), there is a bundle 푦, ε–close to 푥, such that 푦 ≻ 푥. Then, 푥 ∉ 푥(푝, 푤). x x1 8 Properties of Walrasian Demand 3) Convexity/Uniqueness: a) If the preferences are convex, then the Walrasian demand correspondence 푥(푝, 푤) defines a convex set, i.e., a continuum of bundles are utility maximizing. b) If the preferences are strictly convex, then the Walrasian demand correspondence 푥(푝, 푤) contains a single element. 9 Properties of Walrasian Demand Convex preferences Strictly convex preferences Unique x( p , w ) x x(,) p w 10 UMP: Necessary Condition max 푢 푥 s. t. 푝 ∙ 푥 ≤ 푤 푥≥0 • We solve it using Kuhn-Tucker conditions over the Lagrangian 퐿 = 푢 푥 + 휆(푤 − 푝 ∙ 푥), ∗ 휕퐿 휕푢(푥 ) ∗ = − 휆푝푘 ≤ 0 for all 푘, = 0 if 푥푘 > 0 휕푥푘 휕푥푘 휕퐿 = 푤 − 푝 ∙ 푥∗ = 0 휕휆 휕푢(푥∗) • That is in the interior optimum, = 휆푝푘 for every 휕푥푘 good 푘, which implies 휕푢(푥∗) 휕푢(푥∗) 휕푢(푥∗) 휕푥푙 푝푙 푝푙 휕푥푙 휕푥푘 휕푢(푥∗) = ⇔ 푀푅푆 푙,푘 = ⇔ = 푝푘 푝푘 푝푙 푝푘 휕푥푘 11 UMP: Sufficient Condition • When are Kuhn-Tucker (necessary) conditions, also sufficient? – That is, when can we guarantee that 푥(푝, 푤) is the max of the UMP and not the min? • Answer: if 푢(∙) is quasiconcave and monotone, and 퐿 has Hessian 훻푢(푥) ≠ 0 for 푥 ∈ ℝ+, then the Kuhn- Tucker FOCs are also sufficient. 12 UMP: Sufficient Condition • Kuhn-Tucker conditions are x2 sufficient for a max if: 1) 푢(푥) is quasiconcave, w p2 i.e., convex upper contour set (UCS). x * ∈ x (p,w) 2) 푢(푥) is monotone. Ind. Curve 3) 훻푢(푥) ≠ 0 푥 ∈ ℝ퐿 for +. p1 slope = - – If 훻푢 푥 = 0 for some 푥, p2 then we would be at the “top of the mountain” (i.e., w x1 blissing point), which p1 violates both LNS and monotonicity. 13 UMP: Violations of Sufficient Condition 1) 풖(∙) is non-monotone: – The consumer chooses bundle A (at a corner) since it yields the highest utility level given his budget constraint. – At point A, however, the tangency condition 푝1 푀푅푆 1,2 = does not 푝2 hold. 14 UMP: Violations of Sufficient Condition 2) 풖(∙) is not quasiconcave: – The upper contour sets (UCS) are not convex. x2 푝1 – 푀푅푆 1,2 = is not a 푝2 sufficient condition for a A max. C – A point of tangency (C) B gives a lower utility level Budget line than a point of non- x1 tangency (B). – True maximum is at point A. 15 UMP: Corner Solution • Analyzing differential changes in 푥푙 and 푥푙, that keep individual’s utility unchanged, 푑푢 = 0, 푑푢(푥) 푑푢(푥) 푑푥푙 + 푑푥푘 = 0 (total diff.) 푑푥푙 푑푥푘 • Rearranging, 푑푢 푥 푑푥푘 푑푥푙 = − = −푀푅푆푙,푘 푑푥푙 푑푢 푥 푑푥푘 푑푢 푥∗ 푑푢 푥∗ 푝푙 푑푥푙 푑푥푘 • Corner Solution: 푀푅푆푙,푘 > , or alternatively, > , i.e., 푝푘 푝푙 푝푘 the consumer prefers to consume more of good 푙. 16 UMP: Corner Solution • In the FOCs, this implies: 휕푢(푥∗) a) ≤ 휆푝 for the goods whose consumption is 휕푥 푘 푘 ∗ zero, 푥푘= 0, and 휕푢(푥∗) b) = 휆푝 for the good whose consumption is 휕푥 푙 푙 ∗ positive, 푥푙 > 0. • Intuition: the marginal utility per dollar spent on good 푙 is still larger than that on good 푘. 휕푢(푥∗) 휕푢(푥∗) 휕푥푙 = 휆 ≥ 휕푥푘 푝푙 푝푘 17 UMP: Corner Solution • Consumer seeks to consume good 1 alone. x2 • At the corner solution, the indifference curve is steeper than the budget line, i.e., w p1 p2 slope of the B.L. = - 푝1 푀푈1 푀푈2 p2 푀푅푆1,2 > or > 푝2 푝1 푝2 • Intuitively, the consumer slope of the I.C. = MRS1,2 would like to consume more of good 1, even after spending his entire wealth on good 1 alone. w x1 p1 18 UMP: Lagrange Multiplier • 휆 is referred to as the “marginal values of relaxing the x2 constraint” in the UMP (a.k.a. “shadow price of wealth”). • If we provide more wealth to w 1 p2 the consumer, he is capable of w 0 reaching a higher indifference p2 2 1 curve and, as a consequence, {x +: u(x) = u } obtaining a higher utility level. {x 2: u(x) = u 0} – We want to measure the + 0 1 change in utility resulting from w w x1 p p a marginal increase in wealth. 1 1 Advanced Microeconomic Theory 19 UMP: Lagrange Multiplier • Analyze the change in utility from change in wealth using chain rule (since 푢 푥 does not contain wealth), 훻푢 푥(푝, 푤) ∙ 퐷푤푥(푝, 푤) • Substituting 훻푢 푥(푝, 푤) = 휆푝 (in interior solutions), 휆푝 ∙ 퐷푤푥(푝, 푤) 20 UMP: Lagrange Multiplier • From Walras’ Law, 푝 ∙ 푥 푝, 푤 = 푤, the change in expenditure from an increase in wealth is given by 퐷푤 푝 ∙ 푥 푝, 푤 = 푝 ∙ 퐷푤푥 푝, 푤 = 1 • Hence, 훻푢 푥(푝, 푤) ∙ 퐷푤푥 푝, 푤 = 휆 푝 ∙ 퐷푤푥 푝, 푤 = 휆 1 • Intuition: If 휆 = 5, then a $1 increase in wealth implies an increase in 5 units of utility. 21 Walrasian Demand • We found the Walrasian demand function, 푥 푝, 푤 , as the solution to the UMP. • Properties of 푥 푝, 푤 : 1) Homog(0), for any 푝, 푤, 훼 > 0, 푥 훼푝, 훼푤 = 푥 푝, 푤 (Budget set is unaffected by changes in 푝 and 푤 of the same size) 2) Walras’ Law: for every 푝 ≫ 0, 푤 > 0 we have 푝 ∙ 푥 = 푤 for all 푥 ∈ 푥(푝, 푤) (This is because of LNS) 22 Walrasian Demand: Wealth Effects • Normal vs. Inferior goods 휕푥 푝,푤 > 0 normal 휕푤 < inferior • Examples of inferior goods: – Two-buck chuck (a really cheap wine) – Walmart during the economic crisis 23 Walrasian Demand: Wealth Effects • An increase in the wealth level produces an outward shift in the budget line. 휕푥 푝,푤 • 푥 is normal as 2 > 0, while 2 휕푤 휕푥 푝,푤 푥 is inferior as < 0. 1 휕푤 • Wealth expansion path: – connects the optimal consumption bundle for different levels of wealth – indicates how the consumption of a good changes as a consequence of changes in the wealth level 24 Walrasian Demand: Wealth Effects • Engel curve depicts the consumption of a particular good in the horizontal axis and Wealth, w wealth on the vertical axis. Engel curve for a normal good (up to income ŵ), but inferior good for all w > ŵ • The slope of the Engel curve is: – positive if the good is normal Engel curve of a normal – negative if the good is inferior good, for all w. ŵ • Engel curve can be positively slopped for low wealth levels and become negatively slopped x afterwards.
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