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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. 6/24. Theory of the Consumer. Indifference Curves Utility Functions Optimization under Constraint 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 6/24

  2. Theory of the Consumer • Indifference Curves • Utility Functions • Optimization under Constraint • Income & Substitution Effects How do consumers make optimal choices? How do they respond to changes in prices and income?

  3. 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

  4. 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

  5. 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

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

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

  8. Utility Functions U U = f(X,Y) U3 U2 U1 U0 Y U0 U1 U2 U3 DX DY X

  9. 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 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

  10. 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 Increasing utility

  11. 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.

  12. 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.

  13. 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).

  14. 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.

  15. 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.

  16. 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

  17. 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.

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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.

  23. 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. 1200 960 800 50 60 100 1200

  24. 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 1200

  25. 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 substitution effect? At 50 hrs/wk., the new wage rate is the same as the old ($8/hr). => no substitution effect! 1200 960 800 Px/Py = 8 50 60 100 1200

  26. Income & Substitution Effects 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 Originally, the price of food is $1 and the price of all other goods is $2. Then the price of food rises to $2. 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. .

  27. Income & Substitution Effects 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 1200 S

  28. Income & Substitution Effects 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

  29. Income & Substitution Effects The move from A to B is the substitution effect; B to C is the income effect. B is a point on the original indifference curve, tangent to the new budget constraint, indicating the bundle the consumer would choose at the new prices. Y X 900 Y**=300 B A I S C X**= 600 1200 1200 S I

  30. Income & Substitution Effects U = X2Y We are looking for a point on the indifference curve that includes Y = 300, X = 1200, for which MRS = 1 (the new price ratio): At point B, MRS = 2Y/X = 1 => X = 2Y. Also, Ua = Ub = 432,000,000 U = X2Y 4Y3 = 432,000,000 Y3 = 108,000,000 Yb = 476; Xb = 952 Y X 900 Y**=300 B A I S C X**= 600 1200 1200 S I

  31. Income & Substitution Effects U = X2Y 4Y3 = 432,000,000 Y3 = 108,000,000 Yb = 476; Xb = 952 So the substitution effect is a decrease in X of 248 and an increase in Y of 176. The income effect is a decrease in X of 352 and a decrease in Y of 176. Y X 900 Y**=300 B A I S C X**= 600 1200 1200 S I

  32. Income & Substitution Effects How much would Pat be willing to pay to avoid this price increase? Y X 900 Y**=300 X**= 600 1200 1200

  33. Income & Substitution Effects To calculate this amount, start by finding the minimum income Pat needs to purchase a bundle on the new indifference curve. Y X 900 Y**=300 X**= 600 1200 1200

  34. Income & Substitution Effects The difference between the market price of this bundle and her income ( = 1800) is the amount she’d be willing to pay to avoid the price increase. We call this the equivalent variation measure of utility loss. Y X 900 Y**=300 X**= 600 1200 1200

  35. Normal & Inferior Goods Normal Good Income-Expansion Path For most goods, the quantity consumed will increase as income increases. We call these normal goods. Y Y = f(X) optimal ratio X

  36. Normal & Inferior Goods Normal Good Engels Curve Income-Expansion Path Y Income X = f(I) X X

  37. Normal & Inferior Goods Inferior Good Income-Expansion Path For some goods, consumption will decrease at higher levels of income (e.g., hamburger). We call these inferior goods. Y X

  38. Normal & Inferior Goods Inferior Good Engels Curve Income-Expansion Path Y Income X X

  39. Normal & Inferior Goods Normal Good Y Y Px increases from $1 to $2. The movement from A to B is the substitution effect. B B A A Px = 1 Px = 2 X X S S

  40. Normal & Inferior Goods For both normal and inferior goods, the substitution effect is negative: consumption will increase as price decreases. Inferior Good Normal Good Y Y B Px increases from $1 to $2. B A A Px = 1 Px = 2 X X S S

  41. Normal & Inferior Goods For normal goods the income effect is positive, and for inferior goods it is negative. Inferior Good Normal Good Y Y B B A A C C X X I I

  42. Normal & Inferior Goods For some inferior goods, the income effect is so large it outweighs the substitution effect (eg., ?). Giffen Good Px 2 1 … giving rise to a upward sloping demand curve. Y C B A A Px = 1 C Px = 2 X X S I

  43. Normal & Inferior Goods Do any of these cases violate the assumptions of well-behaved preferences that we look at last time? No. Well-behaved preferences can give rise to all sorts of demand curves (depending on income and prices).

  44. Consumer Demand U = X2Y I = 1800; Py = 2 Y X Px*** = $3 Y*** = 300, X*** = 400. 900 Y**=300 400 1200

  45. Consumer Demand : U = X2Y I = 1800; Py = 2 Demand Curve Y Px Find the equation for the demand curve. X = f(Px) 400 3 600 2 1200 1 3 2 1 X X Px = 3 2 1 400 600 1200

  46. Consumer Demand U = X2Y I = 1800; Py = 2 MUx = 2XY; MUy = X2 MRS = 2Y/X = Px/Py = Px/2 => Y = (1/4)PxX I = PxX + PyY 1800 = PxX + (2)(1/4)PxX = (3/2)PxX X = 1200/Px Demand Curve Y Px X = f(Px) 400 3 600 2 1200 1 Solve for Y & substitute 3 2 1 X X Px = 3 2 1 400 600 1200

  47. Consumer Demand : U = X2Y I = 1800; Py = 2 Demand Curve Price-Consumption Curve Y Px In this case, consumption of Y is unaffected by changes in Px. Cross-price elasticity is zero. 3 2 1 X X Px = 3 2 1 400 600 1200

  48. Consumer Demand Demand Curve Price-Consumption Curve Y Px … with a smaller response in demand. Or, cross-price elasticity can be positive ... DX 3 2 1 DPx X X

  49. Consumer Demand : U = X2Y I = 1800; Py = 2 Demand Curve Price-Consumption Curve Y Px DX 3 2 1 DPx X X Px = 3 2 1 400 600 1200

  50. Price Elasticity of Demand Price Elasticity of Demand (Ep) Measures how sensitive quantity demanded is to changes in price. Demand Equation: Qd = a – bP Ep = (%DQ)/(%DP) = (DQ/Q)/(DP/P) = DQ/DP(P/Q) = -b(P/Q) Ep < 1 Inelastic: Total expenditure increases as price increases. Ep > 1 Elastic: Total expenditure decreases as price increases. Ep = 1 Unit Elastic: Total expenditure doesn’t change

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