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UNIT I: Theory of the Consumer

UNIT I: Theory of the Consumer. Introduction: What is Microeconomics? Theory of the Consumer Individual & Market Demand. 9 /7. Theory of the Consumer. Preferences Indifference Curves Utility Functions Optimization Income & Substitution Effects. How do consumers make optimal choices?.

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UNIT I: Theory of the Consumer

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  1. UNIT I: Theory of the Consumer • Introduction: What is Microeconomics? • Theory of the Consumer • Individual & Market Demand 9/7

  2. 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

  3. 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.

  4. 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.

  5. 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

  6. 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?

  7. 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?

  8. Preferences Now consider points D and E Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) A ? B ? D ? E D > E D ? B D ? A A C B Ya Yb D E Xa Xb

  9. Preferences Now consider points D and E Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) D > E D ? B D ? A Convexity: D > A D > B A C B Ya Yb D E Xa Xb

  10. Preferences Now consider points D and E Y X A = (Xa, Ya) B = (Xb, Yb) C = (Xb, Ya) D > E D > B D > A A C B Ya Yb D E Xa Xb

  11. 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

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

  13. 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

  14. 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 !! U = XY A B C 3 2 2 3

  15. Indifference Curves Perfect Substitutes Perfect Complements

  16. Indifference Curves Remember our simple example: Recall: Oranges Freddie likes oranges twice as much as apples 25 100 50 Apples

  17. Indifference Curves What does his utility function look like? Y Utility = No. of Apples + 2(No. of Oranges) U = X + 2 Y MUx = 1 MUy = 2 X 5 4 Generally, any set of preferences can be described by many utility functions Freddie likes oranges twice as much as apples U = 12 2 4

  18. Indifference Curves What does his utility function look like? Y Utility = No. of Apples + 2(No. of Oranges) U = 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

  19. 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 another unit of X. Y Utility = No. of Apples + 2(No. of Oranges) U = X + 2 Y 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

  20. 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

  21. Utility Functions Assume 1 Good: Marginal Utility: The amount by which utility increases when consumption (of good X) increases. MUx = DU/DX U U = 2X MUx = DU/DX = 2 MUx DU DX X

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

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

  24. Utility Functions U U = f(X,Y) U3 U2 Marginal Rate of Substitution MRS = - the slope of an indifference curve (at a point) U1 U0 Y Indifference curves U0 U1 U2 U3 DY X MRSYX = DX

  25. Utility Functions U U = f(X,Y) U3 U2 MRS = the rate at which the individual is willing to trade Y for X U1 U0 Y Indifference curves U0 U1 U2 U3 MUX DY X MRSYX = = MUY DX

  26. Utility Functions U U = f(X,Y) U3 U2 MRS = the rate at which the individual is willing to trade Y for X U1 DUY DUx Y Indifference curves U0 U1 U2 U3 MUX DY DU/DX X = = DU/DY MUY DX

  27. 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 A At point A, can the consumer increase his utility? How?

  28. 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 A At point A, MRS > Px/Py, so consumer should trade Y for X.

  29. 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 B At point B, MRS < Px/Py, so consumer should trade X for Y.

  30. 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 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.

  31. 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.

  32. 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).

  33. 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.

  34. 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.

  35. 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*=300 600 X*=1200

  36. 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.

  37. 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**= 600 12001200

  38. 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**= 600 1200 1200 S

  39. 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**= 600 1200 1200

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

  41. Next Time 9/14Individual and Market Demand Pindyck & Rubenfeld, Chapters 2 (review) & 4 Besanko, Chapters 2 (review) & 5

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