Consumer Theory. June 30, 2013

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1 Consumer Theory Ilhyun Cho, June 30, 2013 The main topic of consumer theory is how a consumer choose best consumption bundle of goods given her income and market prices for the goods, which is budget constraint. In order to analyze it, we impose assumptions on consumer s preference. Since preference is inconvenient for analyzing the consumer choice, we define utility function. Then the consumer s problem becomes This is the constrained optimization problem. maximize Utility subject to Budget constraint In section 1-3, we define consumer s problem. Section 1 is the assumptions on preferences. In section 2, from the preference we can define utility function, which is the consumer s objective function. Consumer wants to maximize the utility. However, consumer faces constraint by her income and the market prices, which is budget constraint described in section 3. From now on, the consumer s name is Hyoyeon. 1 Preferences 1.1 Assumptions Let s assume that Hyoyeon consumes only two goods x X and y Y. Choose three different bundles a = (x 1, y 1 ) b = (x 2, y 2 ) c = (x 3, y 3 ) Completeness Hyoyeon can rank between two bundles. a b, a b or a b Transitivity More is better If a b, b c, then a c If x 1 > x 2 and y 1 > y 2 then a b. 1.2 Indifference Curve Indifference Curve is the set of all bundles of goods that Hyoyeon feels indifferent. 1

2 2 Utility Function Hyoyeon s preference enables us to rank the bundle of goods. We can order the bundles from the most preferred bundle to the least preferred bundles including the bundles that Hyoyoen feels the same. However, it is not easy to analyze using the actual bundle. That is why we are introducing the utility. Utility is the real number that is assigned on the consumption bundle. The utility function is the function that takes consumption bundles and gives the numbers. The utility function assigns the bigger number to the preferred bundle and assigns the same number if Hyoyeon feels indifferent between bundles. 1 example 1 If X = {1, 2}, Y = {1, 2}, then Hyoyeon s possible consumption bundles are a = (1, 1), b = (1, 2), c = (2, 1), d = (2, 2) Suppose Hyoyeon ranks the bundles like a b c d. Then, Hoyeon s preference can be represented by the many 2 utility functions like below: U(x, y) = x + y U(x, y) = x + y + 2 U(x, y) = 2(x + y) U(x, y) = xy U(x, y) = xy + 2 U(x, y) = 2xy where x X, and y Y. The above utility function assigns the bigger number to the preferred bundled and the same number for the equally preferred bundles. example 2 If U(x, y) = xy then consider the below bundles The utility for each bundles are a = (1, 1), b = (1, 2), c = (2, 1), d = (2, 2) U(a) = U(1, 1) = 1 1 = 1 U(b) = U(1, 2) = 1 2 = 2 U(c) = U(2, 1) = 2 1 = 2 U(d) = U(2, 2) = 2 2 = 4 Thus, Hyoyeon ranks the bundles like a b c d. 1 It is known that if the Hyoyeon s preference satisfies the assumptions in section 1, we can define utility function. Furthermore, if the preference is continuous, we can define continuous utility function. 2 Since only the ordinal ranking is important for the utility, positive monotone transformation of the utility function represents the same utility function. 2

3 example 3 U (x, y) = x y This utility function represents the preferences for imperfect substitutes. Draw the indifference curve which gives utility level 4. example 4 U (x, y) = x + 2y This utility function represents the preferences for perfect substitutes. Draw the indifference curve which gives utility level 4. example 5 U (x, y) = min{x, 2y} This utility function represents the preferences for perfect complements. Draw the indifference curve which gives utility level 2. 3

4 3 Budget Constraint Suppose Hyoyeon s income is M and consumes only x and y goods. The price of good x is, and the price of good y is, which are determined in the market. Then, Hyoyeon s budget constraint is x + y = M Rearranging it, y = M x 4 Constrained Optimization From section 1-3, we can summarize the Hyoyeon s problem like below: maximize U(x, y) subject to x + y = M 4.1 Imperfect Substitutes *Note 1, the below functional form of utility function is called Cobb-Douglas function. *Note 2, in this problem, I used the MRS xy = MRT xy in the optimal bundle, which is worth to understand why this condition holds in the optimal bundle. I provided the proof. *Note 3, there are two ways to solve this problem. One is using the MRS xy = MRT xy in the optimal bundle, the other is using the Lagrangian method. maximize subject to x a y b x + y = M Method 1 : Using the fact MRS xy = MRT xy Using the MRS xy = MRT xy in the optimal bundle, we can get one equation. With the budget constraint, this is a system of two equations with two unknown. Thus, we can solve the problem. 4

5 example 1 maximize x y subject to x + 4y = 16 First using MRS xy = MRT xy, we can get one equation. MRS xy = MU x = y x MRT xy = = 1 4 (1) (2) From (1) and (2) we can get y x = 1 4 x = 4y (3) (3) and budget constraint constitutes the system of two equations and two unknown. Solving (4) and (5) we can get x = 8, y = 2. x = 4y (4) x + 4y = 16 (5) example 2 maximize x 1 2 y 1 2 subject to x + 4y = 16 First using MRS xy = MRT xy, we can get one equation. MRS xy = MU x = MRT xy = = x 1 2 y x 1 2 y 1 2 = y x (6) (7) From (6) and (7) we can get y x = 1 4 x = 4y (8) (8) and budget constraint constitutes the system of two equations and two unknown. Solving (9) and (10) we can get x = 8, y = 2. x = 4y (9) x + 4y = 16 (10) * Note 4, the example 1 and example 2 are the same optimization problem since only the ordinal ranking is important for the utility function and both utility function represents the same preference. 5

6 Method 2 : Using Lagrangian method Check the Math Review note. Poof of MRS xy = MRT xy MRS xy = MRT xy implies, Rearranging it, Proof) Suppose not, i.e., MUx MUy Without loss of generality, assume MUx MRS xy = MU x > MUy and MRT xy = MU x = MU x = Take ɛ of money which is spent on the good y and transfer it to spending on good x. Then it increases the utility without spending more. Thus, any bundle which is not satisfying MUx = MUy is not the optimal bundle. In other words, at the optimal bundle, MRS xy = MRT xy holds. 4.2 Perfect Substitutes maximize subject to ax + by x + y = M If px If px If px < a b, then spend all income on purchasing x good. > a b, then spend all income on purchasing y good. = a b, then any consumption bundle that satisfying the budget constraint are the optimal bundles. example 3 maximize x + 2y subject to x + 4y = 6 Since px p = 1 4 < 1 2 = a b, Hyoyeon spends all her income buying good x. Thus, her optimal bundle is x = 6 and y = 0. 6

7 4.3 Perfect Complements *Note 1, the below functional form of utility function is called Leontief function. *Note 2, this function is not differentiable since it has kink. maximize subject to min{ax, by} x + y = M Using ax = by in the optimal bundle, we get one equation addition to the budget constraint. Thus, we can solve the problem. example 4 maximize min{x, 2y} subject to x + 4y = 6 Using ax = by and budget constraint constitutes the system of two equation and two unknown. Solving (11) and (12) we can get x = 2, and y = 1. x = 2y (11) x + 4y = 6 (12) 7