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## UNIT I: Theory of the Consumer

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**UNIT I: Theory of the Consumer**• Introduction: What is Microeconomics? • Theory of the Consumer • Individual & Market Demand 6/25**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? Next Time: Individual & Market Demand**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.**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.**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**Preferences**Y X A = (Xa, Ya) B = (Xb, Yb) A B Ya Yb Xa Xb**Preferences**Y X A = ( 2 , 3 ) B = ( 3 , 2 ) A B 3 2 2 3**Preferences**Y X A = (Xa, Ya) B = (Xb, Yb) A ? B A B Ya Yb Xa Xb What can we say about the relationship between A and B?**Preferences**Now consider point C Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A ? B C > A C > B A C B Ya Yb Xa Xb What can we say about the relationship between A and B?**Preferences**Now consider points D and E Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A ? B ? D ?E ? A C B Ya Yb D E Xa Xb**Preferences**Now consider points D and E Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) Convexity: D > A D > B A C B Ya Yb D E Xa Xb**Preferences**Now consider points D and E 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**Indifference Curves**Y X • 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 2 3**Indifference Curves**Y X • Generally, ICs are: • Downward sloping • Convex to origin • Inc utility, further from origin • Cannot cross U = XY 3 2 U = 9 U = 6 U = 4 2 3**Indifference Curves**Y X • 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 2 3**Indifference Curves**Perfect Substitutes Perfect Complements**Indifference Curves**Remember our simple example: Recall: Oranges Freddie likes oranges twice as much as apples 25 50 Apples**Indifference Curves**What does his utility function look like? Y Utility = No. of Apples + 2(No. of Oranges) U = 1 X + 2 Y X -2 +1 Generally, any set of preferences can be described by many utility functions 5 4 Freddie is willing to trade 2 apples for 1 orange U = 12 2 4**Indifference Curves**Marginal Utility (MU): The amount by which utility increases when consumption (of good X) increases = DU/DX Y Utility = No. of Apples + 2(No. of Oranges) U = 1 X + 2 Y MUx = 1 MUy = 2 X -2 +1 5 4 Freddie is willing to trade 2 apples for 1 orange U = 12 2 4**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 = -MUx/MUy Y U = X+2Y MUx = 1 MUy = 2 MRS = MUx/MUy = ½ X -2 +1 5 4 Freddie is willing to trade 2 apples for 1 orange 2 4**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 =MUx/MUy Y X DX DY Generally, this rate will not be constant**Utility Functions**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 another unit of X. Y X U = f(X,Y) Generally, this rate will not be constant; it will depend upon the consumer’s endowment. DX DY**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**Utility Functions**Assume 1 Good: Marginal Utility: The amount by which utility increases when consumption (of good X) increases by one unit MUx = DU/DX U U = 2X MUx = DU/DX = 2 MUx DU DX X**Utility Functions**Assume 1 Good: We generally assume diminishing marginal utility U U U = 2X MUx = DU/DX = 2 U (X) DU DX DU DX X X**Utility Functions**Now Assume 2 Goods: U = f(X,Y) U U (X) U (Y) Y X**Utility Functions**U U = f(X,Y) Y X**Utility Functions**U U = f(X,Y) U3 U2 U1 U0 Y Indifference curves U0 U1 U2 U3 X**Utility Functions**U U = f(X,Y) U3 U2 U1 U0 Y U0 U1 U2 U3 DX DY What does the slope of an indifference curve measure? X**Utility Functions**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 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**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 X Increasing utility**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 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 A At point A, can the consumer increase his utility? How?**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 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 A At point A, MRS > Px/Py, so consumer should trade Y for X.**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 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 B At point B, MRS < Px/Py, so consumer should trade X for Y.**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 Y* X* X Indifference Curvesdepict 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.**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.**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. We know: 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 corner solution).**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.**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.**Optimization: An Example**Graphically: Y X Maximize: U = X2Y Subject to: I = PxX + PyY I = 1800; Px = $1; Py = $2 Y* = 300, X* = 1200. 900 Y*=300 X*=1200**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.**Optimization: An Example**Graphically: Y X Now: U = X2Y I = 1800; Px’ = $2; Py = $2 Y* = 300, X* = 600. 900 Y**=300 X**= 600 1200**Optimization: An Example**Graphically: Y X Now: U = X2Y I = 1800; Px’ = $2; Py = $2 Y* = 300, X* = 600. 900 Y**=300 X**= 600 12001200**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. Y X 900 Y**=300 S X**= 600 1200 S**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 900 Y**=300 X**= 600 1200 1200**Individual & Market Demand**• Income & Substitution Effects (from last time) • Normal, Inferior, and Giffen Goods • Consumer Demand • Price Elasticity of Demand • Next Time: The Theory of the Firm**Individual & Market Demand**We have seen how consumers make optimal choices. A rational consumer will attempt to maximize utility subject to market conditions (relative prices) and income. That is, given I, Px, Py, she chooses X and Y to maximize U. Now, we want to ask, how do changes in prices effect these consumption decisions? X = f(Px). We will see that changes in prices affect quantities through two causal channels: Income and substitution effects.**Income & Substitution Effects**Bullwinkle Moose faces a choice between leisure (X) and income (Y). He can work up to 100 hours a week at a wage of $8/hr. Initially, he chooses to work 50 hrs/wk. Draw his budget constraint and current consumption bundle.**Income & Substitution Effects**Bullwinkle Moose faces a choice between leisure (X) and income (Y). He can work up to 100 hours a week at a wage of $8/hr. Initially, he chooses to work 50 hrs/wk. Y X Now his wage rises to$12/hr for the first 40 hrs/wk; it remains $8/hr above 40 hrs/wk. Draw his new budget constraint. 800 50 100**Income & Substitution Effects**Bullwinkle Moose faces a choice between leisure (X) and income (Y). He can work up to 100 hours a week at a wage of $8/hr. Initially, he chooses to work 50 hrs/wk. Y X Now his wage rises to$12/hr for the first 40 hrs/wk; it remains $8/hr above 40 hrs/wk. Will he work more than, less than, or equal to 50 hrs/wk? What is the income effect? His purchasing power is greater, so he will consume more leisure, work less. 1200 960 800 50 60 100