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**UNIT I: Theory of the Consumer**

4/14/2017 UNIT I: Theory of the Consumer Introduction: What is Microeconomics? Theory of the Consumer Individual & Market Demand 6/25

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**Theory of the Consumer Preferences Indifference Curves**

4/14/2017 Theory of the Consumer Preferences Indifference Curves Utility Functions Optimization Income & Substitution Effects How do consumers make optimal choices? How do they respond to changes in prices and income?

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4/14/2017 Theory of the Consumer We said last time that microeconomics is built on the assumption that a rational consumer will attempt to maximize (expected) utility. But what is utility? Over time, economists have moved away from a notion of cardinal utility (an objective, measurable scale, e.g., height, weight) and toward ordinal utility, built up from a simple binary relation, preference.

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4/14/2017 Preferences We start by assuming that a rational individual can always compare any 2 alternatives (“consumption bundles” or “market baskets”). We call this basic relationship preference: For any pair of alternatives, A and B, either A > B A is preferred to B A < B B is preferred to A A = B Indifference e.g., 2 apples & 3 oranges.

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**Preferences “Well-behaved” preferences are **

4/14/2017 Preferences “Well-behaved” preferences are (i) Connected: For all A & B, either A>B; B>A; A=B (ii) Transitive: If A > B & B > C, then A > C (iii) Monotonic: More is always preferred to less (free-disposition) (iv) Convex: Combinations are preferred to extremes

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**Preferences X A B A = ( 2 , 3 ) B = ( 3 , 2 ) A ? B 3 2 2 3 4/14/2017**

Y X A = ( 2 , 3 ) B = ( 3 , 2 ) A ? B A B 3 2

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**Preferences X A B A = (Xa, Ya) B = (Xb, Yb) A ? B Ya Yb Xa Xb**

4/14/2017 Preferences Y X A = (Xa, Ya) B = (Xb, Yb) A ? B A B Ya Yb Xa Xb

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**Preferences X A C B A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A ? B**

4/14/2017 Preferences Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A ? B C > A C > B A C B Ya Yb Xa Xb

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**Preferences X A C B A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A = B**

4/14/2017 Preferences Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A = B C > A C > B A C B Ya Yb Xa Xb

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**Preferences X A C B Convexity A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya)**

4/14/2017 Preferences Y X Convexity A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A = B D > E D > B D > A A C B Ya Yb D E Xa Xb

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**Preferences X A C B A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A = B = E**

4/14/2017 Preferences Y X Indifference curves A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A = B = E D > E D > B D > A A C B Ya Yb D E Xa Xb

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**Indifference Curves X U = XY Indifference Curve:**

4/14/2017 Indifference Curves Y X U = XY Generally, ICs are: Downward sloping Convex to origin Inc utility, further from origin Cannot cross Indifference Curve: The locus of points at which a consumer is equally well-off, U. 3 2

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**Indifference Curves X U = XY Generally, ICs are: Downward sloping**

4/14/2017 Indifference Curves Y X U = XY Generally, ICs are: Downward sloping Convex to origin Inc utility, further from origin Cannot cross 3 2 U = 9 U = 6 U = 4

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**Indifference Curves X A B C U = XY Generally, ICs are:**

4/14/2017 Indifference Curves Y X U = XY Generally, ICs are: Downward sloping Convex to origin Inc utility, further from origin Cannot cross A = B; A = C; B > C !! A B C 3 2

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4/14/2017 Indifference Curves Perfect Substitutes Perfect Complements

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**Indifference Curves Recall: Remember our simple example: Oranges**

4/14/2017 Indifference Curves Remember our simple example: Recall: Oranges Freddie likes oranges twice as much as apples 25 100 Apples

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**Indifference Curves What does his utility function look like?**

4/14/2017 Indifference Curves What does his utility function look like? Y Utility = No. of Apples + 2(No. of Oranges) U = X Y MUx = 1 MUy = 2 X Generally, any set of preferences can be described by many utility functions -2 +1 5 4 Freddie is willing to trade 2 apples for 1 orange U = 12

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**Indifference Curves What does his utility function look like?**

4/14/2017 Indifference Curves What does his utility function look like? Y Utility = No. of Apples + 2(No. of Oranges) U = X Y MUx = 1 MUy = 2 X -2 +1 5 4 Freddie is willing to trade 2 apples for 1 orange U = 12

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4/14/2017 Indifference Curves Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade between 2 goods. The amount of Y he is willing to give up for 1 unit of X. Y Utility = No. of Apples + 2(No. of Oranges) U = X Y MUx = 1 MUy = 2 MRS = - MUx/MUy = - ½ X -2 +1 5 4 Freddie is willing to trade 2 apples for 1 orange

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4/14/2017 Indifference Curves Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade between 2 goods. The amount of Y he is willing to give up for 1 unit of X. Y Utility = No. of Apples + 2(No. of Oranges) U = X Y MUx = DU/DX MUy = DU/DY MRS = - MUx/MUy = - ½ X DX DY Generally, this rate will not be constant; it will depend upon the consumer’s endowment.

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4/14/2017 Indifference Curves Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade between 2 goods. The amount of Y he is willing to give up for 1 unit of X. Y Utility = No. of Apples + 2(No. of Oranges) U = X Y MUx = DU/DX MUy = DU/DY MRS = - MUx/MUy = - ½ X Generally, this rate will not be constant; it will depend upon the consumer’s endowment. DX DY

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4/14/2017 Indifference Curves Marginal Rate of Substitution (MRS): The rate at which a consumer is willing to trade between 2 goods. The amount of Y he is willing to give up for 1 unit of X. Y Utility = No. of Apples + 2(No. of Oranges) U Along an indifference curve, DU = 0 Therefore, MUxDX + MUyDY = 0 MUxDX = MUyDY - (MUx/MUy)DX = DY DY/DX = - MUx/MUy = MRS = slope X Generally, this rate will not be constant; it will depend upon the consumer’s endowment. DX DY

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**Utility Functions Assume 1 Good:**

4/14/2017 Utility Functions Assume 1 Good: Utility: The total amount of satisfaction one enjoys from a given level of consumption (X,Y) U U = 2X X

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**Utility Functions Assume 1 Good:**

4/14/2017 Utility Functions Assume 1 Good: Marginal Utility: The amount by which utility increases when consumption (of good X) increase by one unit MUx = DU/DX U U = 2X MUx = DU/DX = 2 MUx DU DX X

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**Utility Functions Assume 1 Good:**

4/14/2017 Utility Functions Assume 1 Good: Diminishing Marginal Utility: Utility increases but at a decreasing rate U U = 2X MUx = DU/DX = 2 U U (X) DU DX DU DX X X

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**Utility Functions Now Assume 2 Goods: U = f(X,Y) U Y X U (Y) U (X)**

4/14/2017 Utility Functions Now Assume 2 Goods: U = f(X,Y) U U (Y) U (X) Y X

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**Utility Functions U U = f(X,Y) Y X Indifference curves U3 U2 U1 U0**

4/14/2017 Utility Functions U U = f(X,Y) U3 U2 U1 U0 Y U0 U U U3 Indifference curves X

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**Utility Functions U U = f(X,Y) Y X U3 U2 U1 U0 U0 U1 U2 U3 4/14/2017**

DX DY MRSyx X

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4/14/2017 Optimization We assume that a rational consumer will attempt to maximize her utility. But utility increases with consumption of all goods, so utility functions have no maximum -- more is always better! Y Utility = No. of Apples + 2(No. of Oranges) U X Indifference Curves depict consumer’s “willingness to trade” Increasing utility

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4/14/2017 Optimization We assume that a rational consumer will attempt to maximize her utility. But utility increases with consumption of all goods, so utility functions have no maximum -- more is always better! Y Utility = No. of Apples + 2(No. of Oranges) U X Indifference Curves depict consumer’s “willingness to trade” Indifference Curves depict consumer’s “willingness to trade” Slope = - MRS Budget Constraint depicts “opportunities to trade” Slope = - Px/Py

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4/14/2017 Optimization We assume that a rational consumer will attempt to maximize her utility. But utility increases with consumption of all goods, so utility functions have no maximum -- more is always better! Y Utility = No. of Apples + 2(No. of Oranges) U X Indifference Curves depict consumer’s “willingness to trade” Slope = - MRS Indifference Curves depict consumer’s “willingness to trade” Budget Constraint depicts “opportunities to trade” Slope = - Px/Py A At point A, MRS > Px/Py, so consumer should trade Y for X.

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4/14/2017 Optimization We assume that a rational consumer will attempt to maximize her utility. But utility increases with consumption of all goods, so utility functions have no maximum -- more is always better! Y Utility = No. of Apples + 2(No. of Oranges) U X Indifference Curves depict consumer’s “willingness to trade” Slope = - MRS Indifference Curves depict consumer’s “willingness to trade” Budget Constraint depicts “opportunities to trade” Slope = - Px/Py B At point B, MRS < Px/Py, so consumer should trade X for Y.

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4/14/2017 Optimization The optimal consumption bundle places the consumer on the highest feasible indifference curve, given her preferences and the opportunities to trade (her income & the prices she faces). Y Utility = No. of Apples + 2(No. of Oranges) U Y* X* X Indifference Curves depict consumer’s “willingness to trade” Slope = - MRS Budget Constraint depicts “opportunities to trade” Slope = - Px/Py C At point C, MRS = Px/Py, so consumer can’t improve thru trade.

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**Optimization Two Conditions for Optimization under Constraint:**

1. PxX + PyY = I Spend entire budget 2. MRSyx = Px/Py Tangency MRSyx = MUx/MUy = Px/Py => MUx/Px = MUy/Py The marginal utility of the last dollar spent on each good should be the same.

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4/14/2017 Optimization Arlene has $100 income to spend on food (F) and Clothing (C). Food costs $2, and Clothing costs $10. She is currently consuming 30 units of F and 4 units of C. At this point, she is willing to trade 1 unit of C for 2 units of F. Is she choosing optimally? C Utility = No. of Apples + 2(No. of Oranges) I/Pc=10 U 4 I/Pf =50 F PfF + PcC = I 2(30) + 10(4) = 100

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4/14/2017 Optimization Arlene has $100 income to spend on food (F) and Clothing (C). Food costs $2, and Clothing costs $10. She is currently consuming 30 units of F and 4 units of C. At this point, she is willing to trade 1 unit of C for 2 units of F. Is she choosing optimally? C Utility = No. of Apples + 2(No. of Oranges) I/Pc=10 U 4 I/Pf =50 F Pf/Pc = 1/5 MRScf = 1/2

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**Optimization: An Example**

4/14/2017 Optimization: An Example Pat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function: U = X2Y If the price of food is $1 and the price of all other goods is $2, find Pat’s optimal consumption bundle. Pat should choose the combination of food and all other goods that places her on the highest feasible indifference curve, given her income and the prices she faces. This is the point where an indifference curve is tangent to the budget constraint (unless there is a comer solution).

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**Optimization: An Example**

4/14/2017 Optimization: An Example Pat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function: U = X2Y If the price of food is $1 and the price of all other goods is $2, find Pat’s optimal consumption bundle. Since Pat’s utility function is U = X2Y, MUx = 2XY and MUy = X2. MRS = (-)MUx/MUy = (-)2XY/X2 = (-)2Y/X. Setting this equal to the (-)price ratio (Px/Py), we find ½ = 2Y/X, X = 4Y. This is Pat’s optimal ratio of the goods, given prices.

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**Optimization: An Example**

4/14/2017 Optimization: An Example Pat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function: U = X2Y If the price of food is $1 and the price of all other goods is $2, find Pat’s optimal consumption bundle. To find Pat’s optimal bundle, we substitute the optimal ratio into the budget constraint: I = PxX + PyY, 1800 = (1)X + (2)Y, 1800 = (1)4Y + (2)Y = 6Y, so Y* = 300, X* = 1200.

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**Optimization: An Example**

4/14/2017 Optimization: An Example Graphically: Y X U = XY Maximize: U = X2Y Subject to: I = PxX + PyY I = 1800; Px = $1; Py = $2 Y* = 300, X* = 1200. 900 Y*=300 X*=1200

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**Optimization: An Example**

4/14/2017 Optimization: An Example Pat divides a monthly income of $1800 between consumption of food (X) and consumption of all other goods (Y). Pat’s preferences can be described by the following utility function: U = X2Y Now suppose the price of food rises to $2. MRS = (-)2Y/X. Setting this equal to the new (-)price ratio (Px/Py), we find 1 = 2Y/X, X = 2Y. Substituting in Pat’s new budget constraint: I = PxX + PyY, 1800 = (2)X + (2)Y, 1800 = (2)2Y + (2)Y = 6Y, so Y** = 300, X** = 600.

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**Optimization: An Example**

4/14/2017 Optimization: An Example Graphically: Y X U = XY U = X2Y I = 1800; Px = $1; Py = $2 Y* = 300, X* = 1200. 900 Y*=300 X*=1200

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**Optimization: An Example**

4/14/2017 Optimization: An Example Graphically: Y X U = XY Now: U = X2Y I = 1800; Px’ = $2; Py = $2 Y* = 300, X* = 600. 900 Y**=300 X**=

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**Income & Substitution Effects**

4/14/2017 Income & Substitution Effects Graphically: Because the relative price of food has increased, Pat will consume less food (and more of all other goods). This the substitution effect. But because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect. Y X U = XY 900 Y**=300 S X**= S

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**Income & Substitution Effects**

4/14/2017 Income & Substitution Effects Graphically: But because Pat is now relatively poorer (her purchasing power has decreased), she will consume less of both goods. This is the income effect. In this case, the 2 effects are equal and opposite for Y, additive for X. Y X U = XY 900 Y**=300 X**=

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**Next Time How to consumers respond to changes in income and prices?**

4/14/2017 Next Time How to consumers respond to changes in income and prices?

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**Next Time 7/2 Individual and Market Demand**

4/14/2017 Next Time 7/2 Individual and Market Demand Pindyck & Rubenfeld, Chapters 2 (review) & 4 Besanko, Chapters 2 (review) & 5

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Indifference Curves and Utility Maximization

Indifference Curves and Utility Maximization

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